t-Motives Hodge structures, transcendence and other motivic aspects 9783037191989, 3037191988

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Gebhard Böckle, David Goss, Urs Hartl and Matthew Papanikolas, Editors

This volume contains research and survey articles on Drinfeld modules, Anderson t-modules and t-motives. Much material that had not been easily accessible in the literature is presented here, for example the cohomology theories and Pink’s theory of Hodge structures attached to Drinfeld modules and t-motives. Also included are survey articles on the function field analogue of Fontaine’s theory of p-adic crystalline Galois representations and on transcendence methods over function fields, encompassing the theories of Frobenius difference equations, automata theory, and Mahler’s method. In addition, this volume contains a small number of research articles on function field Iwasawa theory, 1-t-motifs, and multizeta values. This book is a useful source for learning important techniques and an effective reference for all researchers working in or interested in the area of function field arithmetic, from graduate students to established experts.

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SCR Böckle et al. | Egyptienne F | Pantone 116, 287 | RB 31 mm

t-Motives: Hodge Structures, Transcendence and Other Motivic Aspects

t-Motives: Hodge Structures, Transcendence and Other Motivic Aspects

Gebhard Böckle, David Goss, Urs Hartl and Matthew Papanikolas, Editors

Series of Congress Reports

Series of Congress Reports

t-Motives: Hodge Structures, Transcendence and Other Motivic Aspects Gebhard Böckle David Goss Urs Hartl Matthew Papanikolas Editors

EMS Series of Congress Reports

EMS Congress Reports publishes volumes originating from conferences or seminars focusing on any field of pure or applied mathematics. The individual volumes include an introduction into their subject and review of the contributions in this context. Articles are required to undergo a refereeing process and are accepted only if they contain a survey or significant results not published elsewhere in the literature. Previously published: Trends in Representation Theory of Algebras and Related Topics, Andrzej Skowron´ski (ed.) K-Theory and Noncommutative Geometry, Guillermo Cortiñas et al. (eds.) Classification of Algebraic Varieties, Carel Faber, Gerard van der Geer and Eduard Looijenga (eds.) Surveys in Stochastic Processes, Jochen Blath, Peter Imkeller and Sylvie Rœlly (eds.) Representations of Algebras and Related Topics, Andrzej Skowron´ski and Kunio Yamagata (eds.) Contributions to Algebraic Geometry. Impanga Lecture Notes, Piotr Pragacz (ed.) Geometry and Arithmetic, Carel Faber, Gavril Farkas and Robin de Jong (eds.) Derived Categories in Algebraic Geometry. Toyko 2011, Yujiro Kawamata (ed.) Advances in Representation Theory of Algebras, David J. Benson, Henning Krause and Andrzej Skowron´ski (eds.) Valuation Theory in Interaction, Antonio Campillo, Franz-Viktor Kuhlmann and Bernard Teissier (eds.) Representation Theory – Current Trends and Perspectives, Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb and Christoph Schweigert (eds.) Functional Analysis and Operator Theory for Quantum Physics. The Pavel Exner Anniversary Volume, Jaroslav Dittrich, Hynek Kovarˇ ík and Ari Laptev (eds.) Schubert Varieties, Equivariant Cohomology and Characteristic Classes, Jarosław Buczyn´ski, Mateusz Michałek and Elisa Postinghel (eds.) Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis, Fritz Gesztesy, Harald Hanche-Olsen, Espen R. Jakobsen, Yurii Lyubarskii, Nils Henrik Risebro and Kristian Seip (eds.) Spectral Structures and Topological Methods in Mathematics, MIchael Baake, Friedrich Götze and Werner Hoffmann (eds.)

t-Motives: Hodge Structures, Transcendence and Other Motivic Aspects Gebhard Böckle David Goss Urs Hartl Matthew Papanikolas Editors

Editors: Gebhard Böckle Interdisciplinary Center for Scientific Computing Universität Heidelberg Im Neuenheimer Feld 368 69120 Heidelberg Germany

Urs Hartl Mathematisches Institut Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Germany

http://www.iwr.uni-heidelberg.de/~Gebhard.Boeckle/

https://www.uni-muenster.de/Arithm/hartl/

Matthew Papanikolas Department of Mathematics Texas A&M University College Station, TX 77843-3368 USA E-mail: [email protected]

2010 Mathematics Subject Classification (primary; secondary): 11G09; 11J93, 11R58, 13A35 Key words: Drinfeld modules, t -motives, Anderson t -modules, transcendence, Hodge–Pink-structures

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Preface In 1974 Drinfeld revolutionized the field of arithmetic over global function fields. He introduced a function field analogue of elliptic curves over number fields, which he called elliptic modules but are now eponymously named Drinfeld modules. For him and for many subsequent developments their main use was in the exploration of the global Langlands conjecture for automorphic forms over function fields. One of its predictions is a correspondence between automorphic forms and Galois representations. The deep insight of Drinfeld was that the moduli spaces of Drinfeld modules can be assembled in a certain tower such that the corresponding direct limit of the associated `-adic cohomologies would be an automorphic representation which at the same time carries a Galois action. This would allow him to realize the correspondence conjectured by Langlands in geometry. Building on this, Drinfeld himself proved the global Langlands’ correspondence for function fields for GL2 and later L. Lafforgue obtained the result for all GLn . In a second direction, the analogy of Drinfeld modules with elliptic curves over number fields made them interesting objects to be studied on their own right. One could study torsion points and Galois representations, one could define cohomology theories such as de Rham or Betti cohomology and thus investigate their periods as well as transcendence questions. A main advance in this direction is the introduction of t-motives by Anderson. Passing from Drinfeld modules to t-motives may be compared to the passage from elliptic curves to abelian varieties. But more is true. The category of t-motives is also a simple function field analogue of Grothendieck’s conjectured category of motives over number fields. It is this second direction which constitutes a main theme of the present volume, including advances on Galois representations, L-functions, transcendence results, Hodge structures and period domains. Many exciting developments in the arithmetic of functions fields in recent decades have centered around the notion of a t-motive. Some of the most important ones are: • New developments in the transcendence theory over function fields: For instance, it has been shown that the period matrix of a t-motive has transcendence degree equal to the dimension of its motivic Galois group, much in the same way that Grothendieck’s conjecture predicts the transcendence degree of the period matrix of an abelian variety to be equal to the dimension of its Mumford–Tate group. • Hodge structures for function fields: Defined by Pink in 1997, they allow him to define the analogue of the Mumford–Tate group of a t-motive and to formulate a Mumford–Tate conjecture for certain t-motives and to prove the conjecture for Drinfeld modules. Also Pink proved the Hodge-conjecture in this theory, stating, that the Mumford–Tate group is equal to the motivic Galois group. • Period domains: More recently Pink’s Hodge structures have been used extensively to lay foundations for period domains over function fields and state an analogue of Fontaine’s theory of crystalline Galois representations.

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• Galois representations: While the Tate-conjecture for a t-motive over a finitely generated field has been proved already in the early 90’s, only recently results on the openness of the image of Galois (l-adically and adelically) have been obtained. • Tannakian formalisms: Such have recently been described for t-motives in various contexts. They should ultimately link transcendence, Hodge structures and images of Galois representations. • L-series: There are now cohomological approaches to L-series attached to t-motives. Moreover, recently new results on the zeroes of these L-series have emerged. The above topics are tightly interwoven. The Tannakian formalism is used in transcendence theory as well as in a formulation of a Mumford–Tate conjecture based on function field Hodge structures (which is proved for Drinfeld modules). This in turn spurs the interest in Galois representations over function fields. All of the above topics have close relations to similar questions in number theory. The function field Hodge structures and analogues of Fontaine’s theory have influenced questions on period spaces for number fields. Other developments such as the transcendence theory have gone far beyond comparable results in number theory. The first part of this volume consists of survey articles on central topics in the arithmetic of function fields, the first three of which focus on properties of t-motives and Anderson t-modules. There is a brief introductory article on Drinfeld modules, t-modules, and t-motives by Brownawell and Papanikolas. The article by Hartl and Juschka on Pink’s theory of Hodge structures provides an extensive view of the interconnectedness between cohomology theories, Hodge–Pink structures, t-motives, and Anderson t-modules. The article by Hartl and Kim investigates local shtukas connected to Hodge–Pink structures and Galois representations and provides a function field analogue of Fontaine’s theory of p-adic crystalline Galois representations and Kisin’s theory of crystalline Galois deformation rings. There are three further survey articles on transcendence methods over function fields. Chang has provided an overview of techniques in transcendence theory arising from solutions of Frobenius difference equations and their t-motivic interpretations. Pellarin’s article gives an overview of Mahler’s method for deducing transcendence and algebraic independence in the context of function fields. Finally, Thakur has written a survey of how automata theory can be applied to transcendence problems. The remaining three articles of the volume are research articles. The article by Bandini, Bars, and Longhi focuses on Iwasawa theory over function fields, and in particular they formulate a Main Conjecture for abelian varieties in ZN p -extensions of function fields. Taelman constructs and proves fundamental results for 1-t-motifs, which can be viewed as function field analogues of 1-motives over number fields. The article by Thakur reviews the theory of multizeta values over function fields and presents recent results on linear relations among them and their period interpretations. The present volume grew out of the workshop, “t-motives: Hodge structures, transcendence and other motivic aspects,” held at the Banff International Research Station on September 27–October 2, 2009, which brought together researchers from

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across the globe to discuss progress in function field arithmetic and related topics. The workshop page https://www.birs.ca/workshops/2009/09w5094/files/ contains further material for download that is not covered in the present volume. Due to the long time it took for this collection to appear, it also seems appropriate to discuss some important further developments since the time of the workshop. • Having introduced good notions of class module and unit group shortly before the Banff meeting, the search of Taelman for a class number formula for special values of L-functions proved to be successful. We refer to [3] for a first decisive theorem by Taelman in this direction, where some crucial steps are inspired by a trace formula of V. Lafforgue. This spurred much research afterwards by Taelman and many others. Recently M. Mornev has given a cohomological reformulation of some of Taelman’s work. There is also recent work on special L-values by B. Anglès, C.-Y. Chang, C. Debry, F. Demeslay, A. El-Guindy, J. Fang, T. Ngo Dac, M. Papanikolas, F. Pellarin, and F. Tavares Ribeiro. • In [1], F. Pellarin introduced a new kind of Drinfeld modular form along with a new kind of L-function. The forms are now called vectorial Drinfeld modular forms over the Tate algebra, and the L-functions are named after Pellarin. Both constructions have not yet found analogues over number fields but proved extremely useful in function field arithmetic. Recent research is also due to B. Anglès, Q. Gazda, O. Gezmi¸s, D. Goss, N. Green, A. Maurischat, T. Ngo Dac, M. Papanikolas, R. Perkins, and F. Tavares Ribeiro. • The work started by Thakur [4] on multizeta values, on which his article in the present volume reports, proved to be extremely influential; perhaps also because of the motivic interpretation given jointly by him and Anderson. In particular many transcendence results on algebraic independence for the infinite or v-adic places have been proved in the meanwhile and many relations among such values are now understood. There has been much work in this area, particularly by B. Anglès, C.-Y. Chang, H.-J. Chen, N. Green, W.-C. Huang, Y.-L. Kuan, J. A. Lara Rodríguez, Y.-H. Lin, Y. Mishiba, T. Ngo Dac, M. Papanikolas, S. Shi, F. Tavares Ribeiro, G. Todd, and J. Yu. • Breuer and Pink started a program to develop foundations of higher rank Drinfeld modular forms. Work of Pink [2] and of some of his students, led to a good understanding of the Satake compactification of Drinfeld modular varieties of higher rank, algebraically and analytically. Very recent work on this and on higher rank Drinfeld modular forms is also due to D. Basson, F. Breuer, E.-U. Gekeler, S. Häberli, M. Papikian, S. Schieder, and F.-T. Wei. The above list is certainly not complete and we apologize for not mentioning the many other developments that took place in the last years in function field arithmetic and the future directions that have been started recently. G. Böckle, D. Goss, U. Hartl, M. Papanikolas 23 May 2019

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N.B. It is with great sorrow that we observe that David Goss passed away during the compilation of this volume. David was one of the early pioneers in modern approaches to function field arithmetic, and throughout recent decades he enthusiastically championed new developments in the subject and continually encouraged both junior and established researchers to reach for new discoveries. He was a wonderful colleague and friend.

References [1] F. Pellarin, Values of certain L-series in positive characteristic. Ann. Math. 176(3) (2012), 2055–2093. [2] R. Pink, Compactification of Drinfeld modular varieties and Drinfeld modular forms of arbitrary rank. Manuscripta Math. 140(3–4) (2013), 333–361. [3] L. Taelman, Special L-values of Drinfeld modules. Ann. Math. 175(1) (2012), 369–391. [4] D. S. Thakur, Function Field Arithmetic, World Scientific, River Edge, NJ, 2004.

Contents Part A. Survey articles 1 A rapid introduction to Drinfeld modules, t-modules, and t-motives . by W. Dale Brownawell, Matthew A. Papanikolas 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Exponential functions of algebraic groups . . . . . . . . . . . . . . . . 1.3 Drinfeld modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 t-Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 t-Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Pink’s theory of Hodge structures and the Hodge conjecture over function fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by Urs Hartl, Ann-Kristin Juschka 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hodge–Pink structures . . . . . . . . . . . . . . . . . . . . . . . 2.3 Mixed A-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Mixed dual A-motives . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Anderson A-modules . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7  -Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Local shtukas, Hodge–Pink structures and Galois representations by Urs Hartl, Wansu Kim 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Local shtukas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Divisible local Anderson modules . . . . . . . . . . . . . . . . . . . . 3.4 Tate modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Hodge–Pink structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Admissibility and weak admissibility . . . . . . . . . . . . . . . . . . 3.7 Torsion local shtukas and torsion Galois representations . . . . . 3.8 Deformation theory of Galois representations . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Frobenius difference equations and difference Galois groups by Chieh-Yu Chang 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 t-Motivic transcendence theory . . . . . . . . . . . . . . . . . . 4.3 Carlitz polylogarithms and special -values . . . . . . . . .

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4.4 Special values of geometric and arithmetic -functions 4.5 Periods and logarithms of Drinfeld modules . . . . . . . . 4.6 Transcendence problems with varying constant fields . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

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Automata methods in transcendence . . . . . . . . . . . . . . . . . . . . . . . . by Dinesh S. Thakur 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Automata: implications, equivalences: definitions and statements . 6.3 Sketches of proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Applications to function field arithmetic . . . . . . . . . . . . . . . . . . . 6.5 Comparison with other tools . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Refined transcendence classification based on strength of computers 6.7 Beyond function field real numbers . . . . . . . . . . . . . . . . . . . . . . 6.8 Strong characteristic dependence for algebraicity and real numbers References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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An introduction to Mahler’s method for transcendence and algebraic independence . . . . . . . . . . . . . . . . . . . . by Federico Pellarin 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Transcendence theory over the base field Q . . . . . 5.3 Transcendence theory in positive characteristic . . . 5.4 Algebraic independence . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part B. Research articles 7

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Iwasawa theory over function fields . . . . . . . . . . . . by Andrea Bandini, Francesc Bars, and Ignazio Longhi 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Control theorems for abelian varieties . . . . . . . 7.3 ƒ-Modules and Fitting ideals . . . . . . . . . . . . . 7.4 Modular abelian varieties of GL2 -type . . . . . . . 7.5 Class groups . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Cyclotomy by the Carlitz module . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-t-Motifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . by Lenny Taelman 8.1 Introduction & statement of the main results 8.2 Duality for torsion modules over kŒŒz . . . . 8.3 Effective t-motifs and abelian t-modules . .

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8.4 Algebraic theory of 1-t-motifs . . . . . 8.5 Uniformization and Hodge structures . 8.6 Transcendental theory of 1-t-motifs . . References . . . . . . . . . . . . . . . . . . . . . . .

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9 Multizeta in function field arithmetic . . . . . . . . . . . by Dinesh S. Thakur 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Multizeta values for function fields: Definitions 9.3 First kind of relations between multizeta . . . . . 9.4 Second kind of relations between multizeta . . . 9.5 Period interpretation and motivic aspects . . . . . 9.6 Updates added on 23 August 2011 . . . . . . . . . 9.7 Updates added on 5 February 2013 . . . . . . . . . 9.8 Updates added on 27 April 2015 . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part A

Survey articles

Chapter 1

A rapid introduction to Drinfeld modules, t-modules, and t-motives W. Dale Brownawell and Matthew A. Papanikolas 1.1 Introduction The theory of Drinfeld modules was initially developed to transport the classical ideas of lattices and exponential functions to the function field setting in positive characteristic. This impulse manifested itself in work of L. Carlitz [9], who investigated explicit class field theory for Fq .#/ and defined the later named Carlitz module, which serves as the analogue of the multiplicative group Gm . The theory was brought to rapid fruition by V. G. Drinfeld [20], who independently superseded Carlitz’s work and further extended the theory to higher rank lattices in his investigation of elliptic modules, now called Drinfeld modules. G. W. Anderson [1] saw correctly how to develop the theory of higher dimensional Drinfeld modules, called t-modules, and at the same time produced a robust motivic interpretation in his theory of t-motives. The present article aims to provide a brief account of the theories of Drinfeld modules and Anderson’s t-modules and t-motives. As such the article is not meant to be comprehensive, but we have endeavored to summarize aspects of the theory that are of current interest and to include a number of examples. For further information and more complete details, readers are encouraged to consult the excellent surveys [17, 32, 39, 50, 56].

1.2 Exponential functions of algebraic groups We begin with some preliminary remarks about commutative algebraic groups over C, starting with the multiplicative group Gm . The exact sequence exp

0 ! 2 i Z ! C ! C ! 1 exhibiting the uniformizability of Gm , is the starting point for a multitude of problems and their solutions in number theory. For example, the entire study of abelian extensions of Q is intertwined with exp.z/, whose division values are simply roots of unity. Moreover, this sequence is the starting point for transcendence questions involving exponentials and logarithms of algebraic numbers. The next natural step along these lines leads to the investigation of elliptic curves. We can associate to an elliptic curve E over C a rank 2 lattice ƒ  C, from which

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W. Dale Brownawell, Matthew A. Papanikolas

we can define the Weierstraß }-function }.z/ D

X 1 C z2

2ƒ ¤0



1 1  .z  /2 2

 :

This leads to the exact sequence expE

0 ! ƒ ! C ! E.C/ ! 0; where expE .z/ D Œ}.z/; } 0 .z/; 1, and ƒ is called the period lattice of E. When E is defined over a number field K, the division values of expE .z/, much as in the case of Gm , generate interesting extensions of K and are the focus of much study. Also, transcendence questions about periods and elliptic logarithms of algebraic points naturally arise from analogy with the Gm case. These investigations generalize in satisfying ways to general commutative algebraic groups over C, including algebraic tori and abelian varieties. For a commutative algebraic group G one has the exponential sequence expG

0 ! ƒ ! Lie.G/ ! G.C/ ! 0; where ƒ is a lattice in Lie.G/, and when G is defined over a number field, we can similarly study special values of expG and logarithms of algebraic points on G. Over the past few decades function field analogues have fostered many fruitful research programs: • Cyclotomic theory and explicit class field theory over function fields, • Drinfeld modular forms and modular varieties, • Drinfeld modules over finite fields, • Torsion modules and Galois representations, • Characteristic p valued L-series, • • • •

Heights and Drinfeld modules over global function fields, Effective bounds on isogenies of t-modules, Transcendence theory, Special functions,

• Shtukas and automorphic representations over function fields, • t-motives,  -sheaves, and Hodge structures. Any complete list of references on the above topics would necessarily be too long for the scope of this survey. However, we list here several useful sources that represent a broad picture and contain themselves references for further study: texts and monographs by Böckle and Pink [4], Gekeler [23], Goss [32], Laumon [42, 43], Rosen [50], and Thakur [56]; survey articles by Deligne and Husemoller [17], Goss [28, 30, 31],

A rapid introduction to Drinfeld modules, t-modules, and t-motives

5

Hartl [34], Hayes [39], Pellarin [45], and Thakur [55]; and research articles by Anderson [1], Anderson and Thakur [3], David and Denis [16], Denis [18], Drinfeld [21], Galovich and Rosen [22], Gekeler [24], Gekeler and Reversat [26], Goss [27, 29, 33], Hayes [38], Lafforgue [41], Pink [46, 47], Pink and Rütsche [48], Poonen [49], Taguchi and Wan [53], Thakur [54], and Yu [58, 60, 61]. Accounts of many of these topics are also included in the current volume. Acknowledgements. This survey was adapted from lecture notes originally written for the Arizona Winter School at the University of Arizona in 2008. We thank the AWS for permitting us to use them here. We further thank U. Hartl for making several suggestions that improved the exposition and for pointing out an error in an earlier version. Research of the second author was supported by NSF Grant DMS-0903838.

1.3 Drinfeld modules 1.3.1 Table of symbols. p Fq Fq Œt k k1 j  j1 k1 C1 k deg 

:D :D :D :D :D :D :D :D :D :D :D

a fixed prime finite field of q D pm elements polynomials in the variable t Fq .#/ D rational functions in the variable # Fq ..1=#// D 1-adic completion of k absolute value on k1 such that j#j1 D q algebraic closure of k1 completion of k1 with respect to j  j1 algebraic closure of k in C1 the function associating to each element of Fq Œ# its degree in # the q-power Frobenius map sending x 7! x q on a commutative Fq algebra R i .i / :D c q D the i th iterate of  appliedPto an element c 2 R, i 2 Z c Rf g :D the ring of twisted polynomials i ai  i , ai 2 R, where multiplication is given by a i  b j D ab .i /  i Cj P Rf g :D the ring of twisted polynomials i ai  i , ai 2 R, R perfect, where multiplication is given by a i  b j D ab .i /  i Cj 1.3.2 The Carlitz module. The Carlitz module C is the first example of a Drinfeld module. Defined by Carlitz [9] in 1935, it is given by the Fq -algebra homomorphism C W Fq Œt ! C1 f g

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defined so that C.t/ D # C : (Truth be told, Carlitz set C.t/ D #   , but the definition we are using is more prevalent today.) The natural point of view is that a twisted polynomial f D a0 C a1  C    C ad  d 2 C1 f g represents the Fq -linear endomorphism of C1 , d

x 7! f .x/ D a0 x C a1 x q C    C ad x q : In this way C makes C1 into an Fq Œt-module, where a  x D C.a/.x/, and in particular C.t/.x/ D #x C x q : Exponential functions enter the picture with the Carlitz exponential function expC .z/ :D

X zqi i 0

i

Di

i

; i

i 1

where D0 D 1 and Di D .# q  #/.# q  # q /    .# q  # q / for i  1. This i function converges for all z 2 C1 , and the recursion Di D .# q  #/Diq1 implies that expC .#z/ D # expC .z/ C expC .z/q D C.t/.expC .z//: More generally, one checks that the following diagram commutes for any a 2 Fq Œt: C1 z 7! a.#/z



C1

expC .z/

/ C1 

x 7! C.a/.x/

/ C1 :

expC .z/

Thus expC W C1 ! C1 is an Fq Œt-module homomorphism, where t acts on the domain by scalar multiplication by # and on the range by the endomorphism C.t/. When convenient we will use .C; C1 / to denote C1 with the Carlitz Fq Œt-module structure. The Carlitz exponential function uniformizes the Carlitz module as follows. As C1 is algebraically closed, it follows from the Weierstrass preparation theorem that the Carlitz exponential is surjective. Remarkably, Carlitz found its kernel to be all Fq Œ#-multiples of e  :D #

1   Y p p   q1 i 1 # 2 k1 # ; 1  # 1q

q1

i D1

(1.3.1)

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p q1 # is any fixed root of # (see [Gos96, §3.2], [Tha04, §2.5]). We then where have an exact sequence of Fq Œt-modules, expC

 ! C1 ! .C; C1 / ! 0: 0 ! Fq Œ#  e The quantity e  is called the Carlitz period. This uniformization sequence underlies the first transcendence results in positive characteristic: in 1941, L. I. Wade [57] showed that e  is transcendental over Fq .#/. One of Carlitz’s motivations for studying the Carlitz module was to explore explicit class field theory for the rational function field Fq .#/, as in [10]. For f 2 Fq Œt, we let C Œf  :D fx 2 C1 j C.f /.x/ D 0g denote the f -torsion on C , which is isomorphic to Fq Œt=.f / as an Fq Œt-module and which is a Galois module over the separable closure of Fq .#/. The Carlitz cyclotomic field is the field Fq .#; C Œf /, and there is an isomorphism 

W .Fq Œt=.f // ! Gal.Fq .#; C Œf /=Fq .#//; such that for an Fq Œt-module generator  2 C Œf  we have a ./ D C.a/./ for any a 2 .Fq Œt=.f // . Moreover, a coincides with the Artin automorphism for a, and in this way we obtain an explicit Galois action on a piece of the maximal abelian extension of Fq .#/ that agrees with class field theory. However, at 1 Carlitz’s cyclotomic extensions are at most tamely ramified. Only later did Hayes [36] complete the picture to obtain a full analogue of the Kronecker-Weber Theorem, by showing how Carlitz’s constructions could be used to describe the Galois action on the full maximal abelian extension of Fq .#/ and provide an explicit class field theory as well. Indeed, the similarities we observe between the Carlitz module and the multiplicative group identifies a theme that pervades the theory: in the dictionary between function fields and number fields we have C

! Gm ;

and this identification often occurs even when not completely anticipated. 1.3.3 Drinfeld modules. After Carlitz the situation became clearer through the work of V. G. Drinfeld [20, 21] and D. Hayes [36, 37] in the 1970’s. Drinfeld introduced what he called elliptic modules (now commonly called Drinfeld modules) because they have remarkable similarities with classical elliptic curves. Drinfeld simultaneously generalized Carlitz’s work in two directions: he extended the definitions to arbitrary rings of functions on curves over finite fields and to arbitrary rank lattices. While Drinfeld was unaware of Carlitz’s previous work, Hayes continued Carlitz’s work on explicit class field theory of arbitrary function fields of one variable over finite fields and developed a rank 1 theory that coincided with Drinfeld’s. (Consequently rank 1

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“sign normalized” Drinfeld modules are often referred to as “Drinfeld-Hayes” modules.) It is worth pointing out that Carlitz had conceived of a theory of exponential functions for lattices of rank higher than 1, as evidenced by his manuscript [11] not published until 1995. The essential construction of a Drinfeld module is the following. Let L be an arbitrary extension of Fq . Let A be the ring of functions on a smooth projective geometrically irreducible curve X=Fq that are regular away from a fixed Fq -rational point 1 2 X , and fix an Fq -algebra homomorphism W A ! L. In general 1 need not be Fq -rational, but we consider this case here for convenience. Readers new to the subject are encouraged to take simply A D Fq Œt, for which the theory is just as rich as in the general case but at times more straightforward. A Drinfeld A-module is then an Fq -algebra homomorphism 'W A ! Lf g for which '.a/ D .a/ 0 C higher order terms in  . As in the case of the Carlitz module, elements of the ring Lf g can be thought of as Fq -linear endomorphisms of the additive group of L, and thus we often identify ' with the A-module structure on L induced by ' and write .'; L/ for L with this new A-module structure. Geometrically, the Drinfeld module is simply the additive group Ga over L, but we think of ' as the pair .'; Ga /, where 'W A ! EndL .Ga /; and EndL .Ga / consists of all endomorphisms defined over L. If W A ! L is injective, then ' is said to have generic characteristic. If not, then ' is said to have characteristic p, where p D ker ¤ .0/. If '.A/  Kf g for some subfield K  L, we say that ' is defined over K. There is a non-negative integer r such that for every a 2 A, deg .'.a// D r deg.a/; where deg.a/ is normalized by jA=.a/j D q deg.a/ (see [Gos96, §4.5]). The integer r is called the rank of '. Although it is not immediate, we will see in the next section that there exist Drinfeld modules of any rank r for any ring A. In the case of the Carlitz module we have A D Fq Œt, the structure morphism W A ! C1 is defined by .a/ D a.#/, and the Carlitz module is a rank 1 Drinfeld A-module with generic characteristic. 1.3.4 Drinfeld modules and lattices. We continue with the notation of the previous section. In the case that we have an embedding W A ,! C1 , Drinfeld constructed Drinfeld A-modules in close analogy to the situation of elliptic curves and elliptic functions over C. We consider C1 to be an A-module via , and for this section we identify A with its image .A/ in C1 .

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Starting with an A-lattice ƒ  C1 , i.e. a discrete finitely generated projective A-submodule of C1 , of rank r > 0, we define the lattice function  Y 0 z ; z 2 C1 ; 1 expƒ .z/ WD z  2ƒ

where the product is taken over all non-zero lattice elements. The discreteness of ƒ ensures that expƒ .z/ converges for all z 2 C1 : only finitely many  2 ƒ lie within any given distance to the origin. Consideration of the partial products involving bounded  shows that the Drinfeld exponential function has an expansion of the form X i ai z q ; expƒ .z/ D z C i 1

and as such it is an Fq -linear power series. Thus, for c 2 Fq , expƒ .z1 C cz2 / D exp.z1 / C c expƒ .z2 /: Also for c 2 C1 non-zero, the product expansion for expƒ .z/ makes obvious that expcƒ .cz/ D c expƒ .z/: Moreover expƒ .z/ visibly parametrizes C1 and has kernel ƒ. So the sequence expƒ

0 ! ƒ ! C1 ! C1 ! 0 is exact. Now consider the case that ƒ1 , ƒ2 are A-lattices of the same A-rank, but ƒ1  ƒ2 . Then ƒ2 =ƒ1 is a finite dimensional Fq -vector space, say with coset representatives 0 .D 0/; : : : ; d 1 . Hence PŒƒ2 Wƒ1  .X / :D X

dY 1 i D1

1

X

!

expƒ1 .i /

is an Fq -linear polynomial in X with X as lowest term, and it provides the crucial functional relation expƒ2 .z/ D PŒƒ2 Wƒ1  .expƒ1 .z//; since both sides have the same simple zeros and the same leading terms. When ƒ1 , ƒ2 have the same rank and cƒ1  ƒ2 , then Œƒ2 W cƒ1  is finite. In that case, expƒ2 .cz/ D PŒƒ2 Wcƒ1  .expcƒ1 .cz// D PŒƒ2 Wcƒ1  .c expƒ1 .z//

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as both sides have the same zeros and the same leading terms. In particular, when ƒ1 D ƒ2 D ƒ and c D a 2 A, a ¤ 0, we can write expƒ .az/ D 'ƒ .a/ expƒ .z/

(1.3.2)

for 'ƒ .a/.z/ D PŒƒWaƒ .z/, where 'ƒ .a/ D a 0 C higher order terms in  lies in C1 f g and is non-zero. Thus we have defined a function 'ƒ W A ! C1 f g;

(1.3.3)

and we will see shortly that 'ƒ defines a Drinfeld A-module. Returning to the case cƒ1  ƒ2 , c 2 C1 , and writing PŒƒ2 Wcƒ1  .cz/ D for .c/ in C1 f g, we find 'ƒ2 .a/

.c/ expƒ1 .z/

.c/ .z/

D 'ƒ2 .a/ expƒ2 .cz/ D expƒ2 .acz/ D

.c/

expƒ1 .az/ D

.c/ 'ƒ1 .a/ expƒ1 .z/:

Since expƒ2 .z/ is a transcendental function (it has infinitely many zeros), we conclude that 'ƒ2 .a/

.c/

D

.c/ 'ƒ1 .a/;

for all a 2 A, and we say that .c/ 2 HomA .'ƒ1 ; 'ƒ2 /. Any non-zero element of C1 f g satisfying this property will be called an isogeny from 'ƒ1 to 'ƒ2 , and we also write W 'ƒ1 ! 'ƒ2 : Isogeneity is an equivalence relation. Note that, as c ¤ 0, .c/ D c 0 C higher order terms in  in the previous displayed line. It is not hard to see that if such an isogeny has the form D c 0 C higher order terms, then D .c/ . In particular, when ƒ1 D ƒ2 D ƒ and c 2 A, we have that .c/ D 'ƒ .c/ and 'ƒ .a/'ƒ .c/ expƒ .z/ D expƒ .acz/ D 'ƒ .ac/ expƒ .z/: Since a 7! 'ƒ .a/ is also additive, 'ƒ is thus a ring homomorphism, and we have a Drinfeld module structure induced by the effect on the range of expƒ , i.e. on C1 , of the A-action on ƒ. Furthermore, since the degree in z of 'ƒ .a/.z/ is Œƒ W aƒ D q r deg a , we see that the degree in  of 'ƒ is r. Thus 'ƒ is a Drinfeld A-module of rank r. One also checks, for ƒ1 , ƒ2 as above, that HomA .'ƒ1 ; 'ƒ2 / is an A-module, and so for any particular lattice ƒ the endomorphism ring of 'ƒ is the A-algebra EndA .'ƒ / :D HomA .'ƒ ; 'ƒ /. Since EndA .'ƒ / can be identified with those c 2 C1 such that cƒ  ƒ, it follows that EndA .'ƒ / is an integral domain and is a finitely generated A-module of projective rank at most r [Gos96, Ch. 4]. In generic characteristic Drinfeld demonstrated the striking fact that the considerations proceeding from the analytic to the algebraic are also reversible.

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Theorem 1.3.1 (Drinfeld’s Uniformization Theorem). Given a homomorphism ' W A ! C1 f g such that '.a/ D a C    C am  m , am ¤ 0, m D r deg.a/, there is a unique A-lattice ƒ such that ' D 'ƒ . Moreover rankA ƒ D r. A brief outline of how to see this is to use the condition e.#z/ D '.t/e.z/ to define a unique Fq -linear power series e' .z/ with leading term z. Then one shows that e' .z/ is Fq -linear and entire. Finally, from the functional equation, one sees that the zeros of e' .z/ form a discrete A-module, i.e. a lattice ƒ, and then it remains to show that e' .z/ D expƒ .z/ by the uniqueness of the solution to (1.3.2) having lowest term z. For a complete proof, see [32, 50, 56]. 1.3.5 The Weierstraß–Drinfeld correspondence. One is reminded of the situation of elliptic curves over the complex numbers, and the analogies are amazingly tight. Based on our various observations, we have the following dictionary.

Weierstraß

Drinfeld

Z 2-dim. lattice ƒ expƒ .z/ D .}.z/; } 0 .z//

A r-dim. A-lattice Qƒ expƒ .z/ :D z 02ƒ .1  z / analytic! Ga

elliptic curve E W y 2 D 4x 3  g2 x  g3 0 ! ƒ ! C ! Eƒ .C/ ! 0 Isogenies given as c s.t. cƒ1  ƒ2 Z  End.E /

0 ! ƒ ! C1 ! .'ƒ ; C1 / ! 0 Isogenies given as c s.t. cƒ1  ƒ2 when rankA ƒ1 D rankA ƒ2 'ƒ W A ! End.Ga / via 'ƒ .t/ D # 0 C higher terms 2 C1 f g

1.4 t-Modules By analogy with taking the step from elliptic curves to abelian varieties, one can ask questions about how to define higher dimensional Drinfeld modules properly. In 1986, Anderson [1] devised and solved this problem by defining t-modules. Moreover, his construction intrinsically includes many reasonable generalizations of Drinfeld modules to the higher dimensional setting, including direct products, tensor products, and extensions. Anderson also defined a category of companion objects called t-motives, which will be the subject of §1.5. 1.4.1 Definitions. Throughout this section we assume that A D Fq Œt. It is possible to define a theory of ‘A-modules’ and ‘A-motives’ for general A, but to simplify things we adhere to the Fq Œt case as in [1]. For a commutative Fq -algebra R, a matrix B 2 Matmn .R/, and i 2 N [ f0g, we set B .i / to be the matrix whose j k-entry

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is Bj.ik/ D Bjqk . In this way, when m D n, the map B 7! B .i / is an Fq -algebra homomorphism Matn .R/ ! Matn .R/ and we can define the ring of twisted polynomials Matn .R/f g so that B i  C j D BC .i /  i Cj : Thus we really have Matn .R/f g D Mat P n .Rf g/. Moreover, we can map Matn .R/f g to a subring of EndR .Gna /, where for Bi  i 2 Matn .R/f g and x 2 Matn1 .R/  X X Bi x .i / : Bi  i .x/ D (1.4.1) Now let L be an extension of Fq , and fix W A ! L. An Anderson t-module over L is then defined by an Fq -algebra homomorphism ˆW A ! Matd .L/f g; such that if we set ˆ.a/ WD @ˆ.a/ 0 C higher order terms in ; where @ denotes the differentiation map on Gda at the origin, then @ˆ.t/ D #Id C N , where N is a nilpotent matrix and Id is the d  d identity matrix. In this way A operates on Matd 1 .L/ D Gda .L/ via ˆ through (1.4.1), and we will often say that a t-module is given by the pair .ˆ; Gda / or .ˆ; Ld / to denote this action. We say that ˆ has dimension d , and thus a 1-dimensional t-module is simply a Drinfeld A-module. If L D C1 , then we can also define a unique exponential function Expˆ W Cd1 ! d C1 , via a power series in z1 ; : : : ; zd , ! z1 X .i / Expˆ .z/ D z C Bi z ; z D ::: ; Bi 2 Matd .C1 /; i 1

zd

satisfying for a D t and thus for all a 2 A, Expˆ .@ˆ.a/z/ D ˆ.a/ .Expˆ .z// : This functional equation uniquely determines the coefficients Bi , i  1. The function Expˆ converges on all of Cd1 , and if Expˆ is surjective, then E is said to be uniformizable. Surjectivity of the exponential map is somewhat subtle (see [And86, §2.2]) and is not guaranteed. However, all exponential maps occurring in this note are surjective; in other words all t-modules we will consider are uniformizable. The kernel of Expˆ is a @ˆ.A/-submodule ƒ of Cd1 , which is finitely generated and discrete. Just as in the case of abelian varieties, not every @ˆ.A/-lattice in Cd1 is the kernel of an exponential function for some t-module. We define the rank of ˆ

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to be the rank of ƒ as a @ˆ.A/-module. Thus if E is uniformizable, the exponential function induces a familiar exact sequence of A-modules Expˆ

0 ! ƒ ! Cd1 ! .ˆ; Cd1 / ! 0 For proofs of the above statements about Expˆ , see [1, 32]. We summarize the connections between Drinfeld modules and t-modules:

Drinfeld A-modules

Anderson t-modules

'.t/ 2 C1 f g '.t/ D # 0 C higher order terms @'.t/ D # — unique entire exp' W C1 ! C1 exp' .#z/ D '.t/.expƒ .z// P i exp' .z/ D z C i 1 bi z q ƒ D ker.exp' / finitely generated discrete A-submodule of C1 exp' always surjective on C1

ˆ.t/ 2 Matd .C1 /f g ˆ.t/ D @ˆ.t/ 0 C higher order terms @ˆ.t/ D #Id C N 2 Matd .C1 / Nd D 0 unique entire Expˆ W Cd1 ! Cd1 Expˆ .@ˆ.t/z/ D ˆ.t/ Expˆ .z/ P Expˆ .z/ D z C i 1 Bi z .i / ƒ D ker.Expˆ / finitely generated discrete @ˆ.A/-submodule of Cd1 surjectivity of Expˆ not guaranteed

Although the functional equation of the exponential function has a unique solution, as we have noted above, it is perhaps of passing interest that two different t-modules may have the same exponential function [6]. Let E D .ˆ; Gda / and F D .‰; Gm a / denote two t-modules over a field L. Then by a morphism f W F ! E over L, we mean a morphism of commutative algebraic d groups f W Gm a ! Ga over L commuting with the action of A: f ‰.t/ D ˆ.t/f: A sub-t-module of E is then defined to be the image of any closed immersion f W F ! E, which is itself isomorphic as an algebraic group to Gsa for some s, is invariant under the A-action, and is isomorphic to a t-module. When L D C1 , we can further describe sub-t-modules of a t-module E D .ˆ; Gda / as follows. By identifying both Lie.E/ (the tangent space at the origin of Gda ) and E with copies of Cd1 , we have Expˆ W Lie.E/ ! E: In this setting, a sub-t-module is a connected algebraic subgroup F of Gda such that • @ˆ.t/.Lie.F //  Lie.F /, • Expˆ .Lie.F // D F .C1 /. In other words, t-modules satisfy the usual Lie correspondence for algebraic groups.

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Now we turn to some interesting examples of t-modules: 1.4.2 Products of other t-modules. In particular, if '1 ; : : : ; 'n are Drinfeld modules, then taking ˆ.t/ D diag.'1 .t/; : : : ; 'n .t// sets N D 0 and gives rise to Expˆ .z/ D .exp'1 .z1 /; : : : ; exp'n .zn //tr . 1.4.3 Tensor powers of the Carlitz module. The tensor powers C ˝n , n  1, of the Carlitz module were defined and investigated extensively by Anderson and Thakur in [3]. That C ˝n is the n-fold tensor product of C relies on the tensor product construction in §1.5, but we can define these t-modules directly and extract many interesting properties without this information. We define C ˝n W A ! Matn .C1 /f g by C ˝n .t/ D #In C N C E; where 0

0 B :: B: N D B: @ :: 0

1 :: :

 :: : :: :





1 0 :: C :C C; 1A 0

0 1 0   0 :: C B :: :C B: E D B: : :: C @ :: :A 1   0

Of particular interest are the following two results obtained in [3]. First, if we let Expn W Cn1 ! Cn1 be the exponential function of C ˝n , then there is a vector 1  B :: C n : C n D B @  A 2 C1 ; 0

e n

such that ker.Expn / D dC ˝n .A/  n : Thus C ˝n has rank 1, and the n-th power of e  is the final coordinate of a generator of the period lattice for C ˝n .

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Anderson and Thakur also show that the Carlitz zeta value X 1 C .n/ WD 2 k1 ; an a2A monic

is closely involved with C ˝n . Moreover, they find explicit points 1 0  :: C B n ˝n n : C sn D B @  A 2 C1 ; Sn 2 .C ; Fq .#/ /; n C .n/ such that Expn .sn / D Sn : Here n 2 A is the Carlitz factorial [32, 56]. Thus, C .n/ is the coordinate of the logarithm of a point on C ˝n that is defined over Fq .#/. Furthermore they prove that Sn is a torsion point on C ˝n if and only if .q  1/ j n, which is intertwined with the result of Carlitz that .q  1/ j n )

C .n/ 2 Fq .#/ : e n

See [3] for more details. 1.4.4 t-modules arising from quasi-periodic functions. If one is led by Drinfeld to natural function field analogies with elliptic curves, one can also be inspired to pursue a further analogy with the elliptic situation—that of extensions of elliptic curves E by the additive group Ga , giving rise to quasi-elliptic functions: 0 ! Ga ! E ! E ! 0; where the exponential function of E is given by .z; u/ 7! .1; }.z/; } 0 .z/; u  .z// and .z/ is the quasi-periodic Weierstraß zeta function. Here z is the coordinate on Lie.E / and u is the coordinate on Lie.Ga /; the first three coordinates in the image of this formula are the projective coordinates on E and the fourth is the affine coordinate on Ga . The periods of this map are the pairs .!; /, where ! D n1 !1 C n2 !2 is a period of }.z/ and  D n1 1 C n2 2 is the corresponding quasi-period expressed in terms of a basis !1 ; !2 of periods for }.z/ and i D 2.!i =2/, i D 1; 2. Anderson, Deligne, Gekeler, and Yu developed a theory of quasi-periodic Drinfeld functions [25, 59], which we now describe. Quasi-periodic extensions of general tmodules were developed in [8]. Fix a Drinfeld A-module ' of rank r over C1 . A '-biderivation is an Fq -linear map ıW A ! C1 f g satisfying ı.ab/ D a.#/ı.b/ C ı.a/'.b/;

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for all a, b 2 A. The space D.'/ of '-biderivations splits as a direct sum D.'/ D Dsr .'/ ˚ C1 ı0 ˚ Dsi .'/; where • Dsr .'/ D fı 2 D.'/W deg ı.a/ < deg '.a/; 8a 2 Ag, • ı0 .a/ D '.a/  a 0, • Dsi .'/ D fıT 2 D.'/W ıT .t/ D T '.t/  #T; some T 2 C1 f g g. For each '-biderivation ı, there is a unique entire Fq -linear function Fı .z/, with no linear term, such that Fı .#z/ D #Fı .z/ C ı.t/ exp' .z/: The function Fı is said to be the quasi-periodic function related to ı. We note that Fı0 .z/ D exp' .z/  z and that when ı D ıT 2 Dsi .'/, FıT .z/ is simply FıT .z/ D T .exp' .z//. Corresponding to ı we define a t-module ˆı by   '.t/ 0 ˆı .t/ D ı.t/ # 0 with exponential function     exp' .z/ z : D Expˆı u u C Fı .z/ If ! is a period for exp' .z/, then .z; u/ D .!; Fı .!// is the corresponding period for Expı . Therefore for ' D 'ƒ , the period lattice of ˆı is   ! W! 2 ƒ : ker.Expˆı / D Fı .!/ In this way we can view .ˆı ; G2a / as an extension of .'; Ga / by Ga : 0 ! Ga ! .ˆı ; G2a / ! .'; Ga / ! 0: This sequence splits in the category of t-modules precisely when ı 2 Di n .'/ :D C1 ı0 ˚ Dsi .'/. One can take these developments further and consider extensions of ' by several copies of Ga . For example, we define the t-module 1 0 '.t/ 0 0 ::: 0 0 ::: 0 C Bı1 .t/ # 0 C B 0 .t/ 0 # : : : 0 C; ı B 2 ˆ.t/ D B :: :: C :: @ ::: : : : A 0 ::: 0 # 0 ıs .t/

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where ı1 ; : : : ; ıs 2 D.'/, which represents an extension of t-modules 0 ! Gsa ! .ˆ; GsC1 a / ! .'; Ga / ! 0: Its corresponding exponential function is 0 1 0 1 exp.z0 / z0 Bz1 C F1 .z0 /C Bz1 C C 7! B C: Expˆ W B : :: @ A @ :: A : zs C Fs .z0 /

zs

One finds that, in the category of t-modules, Ext1 .'; Ga / Š D.'/=Di n.'/ as C1 vector spaces. Gekeler [25] found the dimension of D.'/=Di n.'/ over C1 to be r  1, and so any extension ˆ that contains no non-trivial subextensions must have s  r  1. For the connections among quasi-periodic functions and transcendence see [5, 6, 8, 13, 14, 45, 59]. 1.4.5 t-modules from divided derivatives. Until now examples of t-modules have been presented which in some significant sense have analogues in the classical setting of abelian varieties or commutative algebraic groups. The present example is different in that it starts with a t-module or a Drinfeld module and creates an extension in which the coordinates of periods or logarithms of algebraic points are divided derivatives of the coordinates of periods or logarithms of algebraic points of the original t-module or Drinfeld module [6, 7, 19]. The divided derivatives are a family fDi g of Fq -linear operators defined first on Fq Œ# by the formula ! n # ni : Di # n D i sep

We extend Di to k1 by continuity and then to k1 . Then the divided derivatives satisfy the product formula X Di .a/Dj .b/: Dn .ab/ D i Cj Dn

For legibility we often write aŒn for Dn .a/.

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For simplicity of notation, we restrict ourselves to the case in which we begin with P a Drinfeld module '. If '.t/ D rhD0 ah  h , then the t-module ˆŒn , representing the divided derivatives of order up to n, is defined by 1 0 '.t/ 0 ::: ::: 0 B Œ1 :: C B' .t/ # :C C B B Œ2 :: C ˆŒn .t/ D B' .t/ ˆ2;1 .t/ : : : C; : C B C B :: :: :: @ : : : # 0A ' Œn .t/ ˆn;1 .t/ : : : ˆn;n1 .t/ # Pr Œi  h where (1) ' Œi  .t/ D hD0 ah  , (2) the matrix ˆŒn .t/ has zero superdiagonal terms, and (3) the i C 1; j C 1 terms below the main diagonal, with 1  j < i  n, h P are equal to ˆi;j .t/ D q h i=j ahŒi q j   h . (One checks that the subdiagonal terms in the second through n-th columns are always 1 unless q D 2.) The corresponding exponential function is 1 0 0 1 exp' .z0 / z0 Œ1 C B exp' .z0 / C z1 C Bz1 C B C: B C B ExpŒn @ :: A D B :: C : : A @ P h c Œnq h r .h/ P Œn bn=q zn zr exp' .z0 / C q h n rD1 ah sep

If exp' .u/ 2 k1 , then 1 0 1 exp' .u/ u BuŒ1 C Bexp' .u/Œ1 C B C B C ExpŒn B :: C D B C; :: @ : A @ A : 0

uŒn

exp' .u/Œn

using the fact that .x q /Œqi  D .x Œi  /q : So if  is a period of exp' , then 1  BŒ1 C C B B :: C @ : A 0

Œn is a period of ExpŒn .

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1.5 t-Motives Anderson gave definitions of two related, but different, kinds of motives. The first kind, called t-motives, were introduced in his seminal paper [1], and they have played a dominant role in function field arithmetic. The second, called dual t-motives have turned out to be more apt for recent transcendence considerations [2, 12, 44]. The reader should consult [32, 56] for additional information on t-motives. 1.5.1 t-Motives. Let L be an extension of Fq , let W Fq Œt ! L be an Fq -algebra homomorphism, and set # D .t/. Let LŒt;   :D Lf gŒt be the ring of polynomials in the commuting variable t over the non-commuting ring Lf g. Thus tc D ct;

t D  t;

 c D c q ;

c 2 L:

A t-motive M is a left LŒt;  -module which is free and finitely generated as an Lf gmodule for which there is an ` 2 N with .t  #/` .M=M / D f0g: Strictly speaking, if L is not perfect, we need to replace M by the LŒt-submodule it generates. Morphisms of t-motives are morphisms of left LŒt;  -modules. The rank d.M / of M as an Lf g-module is called the dimension of M . Every t-module .ˆ; Gda / gives rise to the unique t-motive M.ˆ/ :D HomqL .Gda ; Ga / Š Lf gd ; the module of Fq -linear morphisms of algebraic groups. The action of LŒt;   is given by .ct i ; m/ 7! c ı m ı ˆ.t i /; c 2 Lf g. Projections onto the d coordinates give an Lf g-basis for M , d D rankLf g M , and ` obviously need not be taken greater than d . A t-motive has an Lf g-basis m1 ; : : : ; md which we can use to express the taction via a matrix B.t/ 2 Matd .Lf g/. This is compatible with the above considerations because, if we represent an arbitrary element of M as 0 1 0 1 m1 m1 B :: C B :: C .k1 ; : : : ; kd / @ : A D k @ : A ; ki 2 Lf g; md md then according to the commutativity of t with elements of Lf g, for c 2 Lf g, 0 1 0 1 0 1 m1 m1 m1 B :: C i i B :: C i B :: C ct  k @ : A D ck  t @ : A D ckB.t/ @ : A : md md md

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A t-motive M is said to be abelian if it is free and finitely generated over LŒt. In this case, the rank of M is its rank r.M / as an LŒt-module. A t-module is called abelian if its associated t-motive is abelian. Theorem 1.5.1 (Equivalence of categories (Anderson [And86, Thm. 1])). The above correspondence between abelian t-motives and abelian t-modules over L gives an anti-equivalence of categories. An abelian t-motive M is called pure in the following situation. We set M..1=t// WD M ˝LŒt  L..1=t//; which possesses a naturally induced left LŒt;  -module structure. If M..1=t// contains a finitely generated LŒŒ1=t-submodule H , which generates M..1=t// over L..1=t// and which satisfies t u H D  v H; for some u, v 2 N, then M is pure. The weight of M is then w.M / WD

u d.M / D : r.M / v

When L D C1 , we can ask for criteria that govern when the t-module of an abelian t-motive is uniformizable. Anderson shows that the notion of rigid analytic triviality for t-motives characterizes uniformizability. For an integer n, we can define n-fold twisting of Laurent series in C1 ..t// by X X .n/ f 7! f .n/ W ai t i 7! ai t i ; and clearly each twisting map operates on both C1 Œt and C1 ŒŒt. Moreover, the Tate algebra C1 hti of all power series in t over C1 that converge on the closed unit disk in C1 is also stable under twisting. We can give C1 ..t// a “trivial” left C1 Œt;  -module structure by setting  .f / WD f .1/ ; and in this way C1 Œt, C1 hti, and C1 ŒŒt can also be given trivial left C1 Œt;  module structures. Now for an abelian t-motive M we can make M ˝C1 Œt  C1 hti into a left C1 Œt;  -module by having  act diagonally, using the above trivial  action on C1 hti. We say that M is rigid analytically trivial if   r.M /  ; M ˝C1 Œt  C1 hti ˝Fq Œt  C1 hti Š C1 hti as left C1 Œt;  -modules. Another way of putting this is as follows. Let m 2 Matr1 .C1 Œt/ be a basis for M as a C1 Œt-module, and let ‚ 2 Matr .C1 Œt/

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21

represent multiplication by  on M with respect to m. If ‡ 2 GLr .C1 hti/ is chosen so that ‡ m is a C1 hti-basis of M ˝C1 Œt  C1 hti fixed by the above action of  , i.e.  .‡ m/ D ‡ m, then we must have  .‡ m/ D ‡ .1/ ‚m D ‡ m; where the twisting ‡ .1/ is applied entry-wise. Thus M is rigid analytically trivial precisely when there exists ‡ 2 GLr .C1 hti/ that satisfies ‡ D ‡ .1/ ‚: We call such an ‡ a rigid analytic trivialization for M . We then have the following fundamental result of Anderson. Theorem 1.5.2 (Anderson [And86, Thm. 4]). The t-module associated to an abelian t-motive M over C1 is uniformizable if and only if M is rigid analytically trivial. Now when M1 and M2 are pure t-motives, Anderson constructs their tensor product as the t-motive with underlying module M1 ˝LŒt  M2 on which  acts diagonally. Then M1 ˝LŒt  M2 is also a pure t-motive with weight w.M1 ˝LŒt  M2 / D w.M1 / C w.M2 /: In this way the category of pure t-motives over L is a tensor category. It is not a Tannakian category (there is no trivial object nor are there dual objects); however, it is possible to enlarge the category of t-motives to a Tannakian category (see [44, 52]). The construction of these categories can be reviewed in the current volume in [35, Rmks. 3.15, 4.14]. 1.5.2 The t-motive of a Drinfeld module. Let ' be a rank r Drinfeld A-module defined over L by '.t/ D # 0 C a1  C    C ar  r : Let M.'/ :D Lf g, and as in the previous section we make M.'/ into the t-motive associated to ' by setting ct i  m WD cm'.t i /;

c 2 L; m 2 Lf g:

We observe that M.'/ is an abelian t-motive. Indeed we note that 1; ; : : : ;  r1 form an LŒt-basis for M.'/ (using the right division algorithm on Lf g). In fact M.'/ is pure of dimension 1, rank r, and weight 1=r [1, Prop. 4.1.1].

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1.5.3 The t-motive of C ˝n . By the previous section the t-motive for the Carlitz module is M.C / D Lf g, where t  m D m.# C  /;

m 2 Lf g:

Also M.C / is rank 1 over LŒt, generated by 1 2 Lf g. The function .t/ :D .#/q=.q1/

1  Y i D1

1

t



# .i /

2 C1 hti

converges on all of C1 and satisfies the difference equation .1/ D .t  #/ : Thus ‡ D .t  #/ is a rigid analytic trivialization for M.C /. The t-motive M.C ˝n / has rank 1, dimension n, and weight n. It is given by the n-fold tensor product M.C ˝n / D Lf g ˝LŒt     ˝LŒt  Lf g; on which  acts diagonally. It is rank 1 over LŒt, generated by 1 ˝    ˝ 1. The calculation .t  #/.1 ˝    ˝ 1/ D 1 ˝    ˝ .t  #/  1 ˝    ˝ 1 D 1 ˝    ˝  ˝    ˝ 1; where t  # is multiplied at any arbitrary entry, implies that all elements of the form 1 ˝  ˝  ˝ ˝ 1 with one entry  and the others 1 are the same in M.C ˝n /. Repeating this construction by multiplying by additional factors of t  # and applying an induction argument on the total degree of  ’s appearing, we find that m1 :D 1 ˝    ˝ 1 m2 :D  ˝ 1 ˝    ˝ 1 m3 :D  ˝  ˝ 1 ˝    ˝ 1 :: :: : : :D  ˝  ˝  ˝ 1 mn are a basis of M.C ˝n / as an Lf g-module. The action of t  # on this basis is .t  #/mi D mi C1 ; .t  #/mn D  m1 :

1i = ” with ‰ 2 GLr .C1 hti/ satisfying ˆ > : ; ‰ .1/ D ˆ‰ 9 8 = < 1 ‰.#/ provides periods : H) and quasi-periods of E ; : We remark that we say E is A-finite precisely when its dual t-motive is A-finite. This notion is dual to the property of E and its t-motive being abelian, but it is an open question whether E being abelian and E being A-finite are equivalent. See the article by Hartl and Juschka [35] in this volume for more details on these two properties. The final implication, proved for periods by Anderson in unpublished work, allows us to arrive back at the t-module E but now with precise information about its periods and quasi-periods. For examples in several contexts, see [2, 8, 13, 14, 15, 40, 44, 45, 51]. These examples come from transcendence theory, as difference equations of the type in (1.5.1) were studied extensively in [2, 44]. See the article by C.-Y. Chang [12] in this volume for additional information on these connections. 1.5.6 Dual t-motives for Drinfeld modules and t-modules. Suppose we have a Drinfeld A-module ' W A ! Lf g, A D Fq Œt, given by '.t/ D # C a1  C    C ar  r ; and associated dual t-motive H.'/. As in the previous section, we let H.'/ D Lf g and then the action of A on H.'/ is induced by X .i j / t  b j :D ai b i Cj ; b 2 L: i

By using this description we can define the dual t-motive H.C / that corresponds to the Carlitz module, and in this way for h 2 H.C / D Lf g, t  h D h.# C  /: Because L is perfect, Lf g has a right division algorithm, and using it we can show that 1 2 Lf g forms a basis for H.C / as an LŒt-module. From this point of view, given h 2 H.C /, we can write h D f  1;

f 2 LŒt;

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and then  h D  .f  1/ D f .1/   D f .1/ .t  #/  1; since on H.C / we have t  1 D # C  . Moreover, if we consider the image of x 2 L in H.C /=.  1/H.C /, we have the following calculation t  x D #x C .t  #/  x D #x C  .x q / D #x C x q C .  1/.x q / D C.t/.x/ C .  1/.x q /: Thus H.C / Š .C; L/ .  1/H.C / as A-modules. We note that .t/ from §1.5.3 satisfies the difference equation .1/ D .t  #/ ; which makes a rigid analytic trivialization of H.C / in the dual t-motive framework (see [2, 44]). We see further that by specializing at t D #, we obtain 1 D e  .#/ from (1.3.1), which exemplifies the link between rigid analytic trivializations of tmotives and periods of their corresponding t-modules. Now let ' be a rank 2 Drinfeld A-module given by .t/ D # C  C  2 ;

2 L:

Again using the functor from Drinfeld modules to dual t-motives, we arrive at the following construction. The dual t-motive H.'/ is identified with Lf g, on which t acts by the rule t  h D h.# C .1/  C  2 /;

h 2 Lf g:

The right division algorithm on Lf g implies that 1,  form an LŒt-basis of H.'/, and since .t  #/  1 D .1/  C  2 ; we have 

     0  1 D D 2   t #

   1 1  :   .1/

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For a, b 2 LŒt, we set Œa; b WD a  1 C b   2 Lf g: Now for x 2 L, h i .  1/Œx; 0 D x C x .1/  D x; x .1/ ; i h . 2  1/Œx; 0 D x C x .2/  2 D x .2/ .t  #/  x;  .1/ x .2/ : Thus tŒx; 0 D Œtx; 0 i h i h D tx C x q ;  .1/ x C  x q ; .1/ x h i D tx C x q ;  .1/ x C .  1/Œ x q ; 0 i h i h 2 2 D #x C x q C x q ; 0 C .t  #/x  x q ;  .1/ x C .  1/ Œ x q ; 0 i h 2 i h 2 D #x C x q C x q ; 0 C .  1/ Œ x q ; 0 C . 2  1/ x q ; 0 : From this we see that the action of t on H.'/=.  1/H.'/ is the same as the action of '.t/ on L, and we find that H.'/ Š .'; L/ .  1/H.'/  0  1 as A-modules. Setting ˆ D t # , the matrix that represents multiplication  .1/ by  on H.'/, it is possible to find a matrix ‰ 2 GL2 .C1 ŒŒt/, whose entries are entire functions and which satisfies the difference equation ‰ .1/ D ˆ‰: This makes ‰ into a rigid analytic trivialization for H.'/, and we find that   !1 1 ; ‰.#/1 D !2 2 where !1 , !2 , 1 , 2 are the fundamental periods and quasi-periods of ' as in §1.3.4 and §1.4.4. See [13, 45] for complete details.

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References [1] G. W. Anderson, t-motives. Duke Math. J. 53 (1986), 457–502. [2] G. W. Anderson, W. D. Brownawell, and M. A. Papanikolas, Determination of the algebraic relations among special -values in positive characteristic. Ann. Math. (2) 160 (2004), 237– 313. [3] G. W. Anderson and D. S. Thakur, Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132 (1990), 159–191. [4] G. Böckle and R. Pink, Cohomological theory of crystals over function fields, Euro. Math. Soc., Zürich, 2009. [5] W. D. Brownawell, Transcendence in positive characteristic. In Number theory (Tiruchirapalli, 1996), Contemp. Math., 210. Amer. Math. Soc., Providence, RI, 1998, 317–332. [6] W. D. Brownawell, Minimal extensions of algebraic groups and linear independence. J. Number Theory 90 (2001), 239–254. [7] W. D. Brownawell and L. Denis, Linear independence and divided derivatives of a Drinfeld module. II. Proc. Amer. Math. Soc. 128 (2000), 1581–1593. [8] W. D. Brownawell and M. A. Papanikolas, Linear independence of Gamma values in positive characteristic. J. Reine Angew. Math. 549 (2002), 91–148. [9] L. Carlitz, On certain functions connected with polyomials in a Galois field. Duke Math. J. 1 (1935), 137–168. [10] L. Carlitz, A class of polynomials. Trans. Amer. Math. Soc. 43 (1938), 167–182. [11] L. Carlitz, Chapter 19 of “The arithmetic of polynomials”. Finite Fields Appl. 1 (1995), 157– 164. [12] C.-Y. Chang, Frobenius difference equations and difference Galois groups. In t-motives: Hodge structures, transcendence and other motivic aspects (G. Böckle, D. Goss, U. Hartl, and M. Papanikolas, eds.), EMS Congr. Rep., Eur. Math. Soc., Berlin, 2020. [13] C.-Y. Chang and M. A. Papanikolas, Algebraic relations among periods and logarithms of rank 2 Drinfeld modules. Amer. J. Math. 133 (2011), 359–391. [14] C.-Y. Chang and M. A. Papanikolas, Algebraic independence of periods and logarithms of Drinfeld modules. With an appendix by B. Conrad. J. Amer. Math. Soc. 25 (2012), 123–150. [15] C.-Y. Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic. Adv. Math. 216 (2007), 321–345. [16] S. David and L. Denis, Isogénie minimale entre modules de Drinfeld. Math. Ann. 315 (1999), 97–140. [17] P. Deligne and D. Husemoller, Survey of Drinfeld modules. In Current trends in arithmetical algebraic geometry (Arcata, CA, 1985), Contemp. Math. 67. Amer. Math. Soc., Providence, RI, 1987, 25–91. [18] L. Denis, Hauteurs canoniques et modules de Drinfeld, Math. Ann. 294 (1992), 213–223. [19] L. Denis, Transcendance et dérivées de l’exponentielle de Carlitz. In Séminaire de Théorie des Nombres, Paris, 1991–92, Birkhäuser, Boston, MA, 1993, 1–21. [20] V. G. Drinfeld, Elliptic modules. Mat. Sbornik 94 (1974), 594–627, 656, Engl. transl. Math. USSR Sbornik 23(4) (1974), 561–592.

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[21] V. G. Drinfeld, Elliptic modules. II.. Mat. Sbornik 102 (1977), 182–194, 325, Engl. transl. Math. USSR Sbornik 31(2) (1977), 159–170. [22] S. Galovich and M. Rosen, Units and class groups in cyclotomic function fields. J. Number Theory 14 (1982), 156–184. [23] E.-U. Gekeler, Drinfeld Modular Curves, Lecture Notes in Math. 1231, Springer, Berlin, 1986. [24] E.-U. Gekeler, On the coefficients of Drinfeld modular forms. Invent. Math. 93 (1988), 667– 700. [25] E.-U. Gekeler, On the de Rham isomorphism for Drinfeld modules. J. Reine Angew. Math. 401 (1989), 188–208. [26] E.-U. Gekeler and M. Reversat, Jacobians of Drinfeld modular curves. J. Reine Angew. Math. 476 (1996), 27–93. [27] D. Goss, v-adic zeta functions, L-series and measures for function fields. Invent. Math. 55 (1979), 107–119. [28] D. Goss, The algebraist’s upper half-plane. Bull. Amer. Math. Soc. (N.S.) 2 (1980), 391–415. [29] D. Goss, Modular forms for Fr ŒT . J. Reine Angew. Math. 317 (1980), 16–39. [30] D. Goss, The arithmetic of function fields. II. The “cyclotomic” theory. J. Algebra 81 (1983), 107–149. [31] D. Goss, L-series of t-motives and Drinfeld modules. In The Arithmetic of Function Fields (Columbus, OH, 1991), de Gruyter, Berlin, 1992, 313–402. [32] D. Goss, Basic Structures of Function Field Arithmetic, Springer, Berlin, 1996. [33] D. Goss, A Riemann hypothesis for characteristic p L-functions. J. Number Theory 82 (2000), 299–322. [34] U. Hartl, A dictionary between Fontaine-theory and its analogue in equal characteristic. J. Number Theory 129 (2009), 1734–1757. [35] U. Hartl and A.-K. Juschka, Pink’s theory of Hodge structures and the Hodge conjecture over function fields. In t-motives: Hodge structures, transcendence and other motivic aspects (G. Böckle, D. Goss, U. Hartl, and M. Papanikolas, eds.), EMS Congr. Rep., Eur. Math. Soc., Berlin, 2020. [36] D. R. Hayes, Explicit class field theory for rational function fields. Trans. Amer. Math. Soc. 189 (1974), 77–91. [37] D. R. Hayes, Explicit class field theory in global function fields. In Studies in algebra and number theory, Adv. in Math. Suppl. Stud. 6, Academic Press, New York, 1979, 173–217. [38] D. R. Hayes, Stickelberger elements in function fields. Compositio Math. 55 (1985), 209–239. [39] D. R. Hayes, A brief introduction to Drinfeld modules. In The arithmetic of function fields (Columbus, OH, 1991), de Gruyter, Berlin, 1992, 1–32. [40] A.-K. Juschka, The Hodge conjecture for function fields, Diplomarbeit, Universität Münster, 2010. [41] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147 (2002), 1–241. [42] G. Laumon, Cohomology of Drinfeld Modular Varieties. Part I, Cambridge University Press, Cambridge, 1996.

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[43] G. Laumon, Cohomology of Drinfeld Modular Varieties. Part II, Cambridge University Press, Cambridge, 1997. [44] M. A. Papanikolas. Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171 (2008), 123–174. [45] F. Pellarin, Aspects de l’indépendance algébrique en caractéristique non nulle, Sém. Bourbaki, vol. 2006/2007. Astérisque 317(973) (2008), viii, 205–242. [46] R. Pink, Hodge structures over function fields, preprint, 1997. http://www.math.ethz.ch/~pink/. [47] R. Pink, The Mumford–Tate conjecture for Drinfeld-modules. Publ. Res. Inst. Math. Sci. 33 (1997), 393–425. [48] R. Pink and E. Rütsche, Adelic openness for Drinfeld modules in generic characteristic. J. Number Theory 129 (2009), 882–907. [49] B. Poonen, Local height functions and the Mordell-Weil theorem for Drinfeld modules. Compositio Math. 97 (1995), 349–368. [50] M. Rosen, Number theory in function fields, Springer, New York, 2002. [51] S. K. Sinha, Periods of t-motives and transcendence. Duke Math. J. 88 (1997), 465–535. [52] L. Taelman, On t-motifs, Ph.D. thesis, Rijksuniversiteit Groningen, 2007. [53] Y. Taguchi and D. Wan, L-functions of '-sheaves and Drinfeld modules. J. Amer. Math. Soc. 9 (1996), 755–781. [54] D. S. Thakur, Gamma functions for function fields and Drinfeld modules. Ann. Math. (2) 134 (1991), 25–64. [55] D. S. Thakur, On characteristic p zeta functions. Compositio Math. 99 (1995), 231–247. [56] D. S. Thakur, Function Field Arithmetic, World Scientific, River Edge, NJ, 2004. [57] L. Wade, Certain quantities transcendental over GF .p n; x/. Duke Math. J. 8 (1941), 701–720. [58] J. Yu, Transcendence and Drinfeld modules. Invent. Math. 83 (1986), 507–517. [59] J. Yu, On periods and quasi-periods of Drinfeld modules. Compositio Math. 74 (1990), 235– 245. [60] J. Yu, Transcendence and special zeta values in characteristic p. Ann. Math. (2) 134 (1991), 1–23. [61] J. Yu, Analytic homomorphisms into Drinfeld modules. Ann. Math. (2) 145 (1997), 215–233.

Chapter 2

Pink’s theory of Hodge structures and the Hodge conjecture over function fields Urs Hartl and Ann-Kristin Juschka In 1997 Richard Pink has clarified the concept of Hodge structures over function fields in positive characteristic, which today are called Hodge-Pink structures. They form a neutral Tannakian category over the underlying function field. He has defined Hodge realization functors from the uniformizable abelian t-modules and t-motives of Greg Anderson to Hodge-Pink structures. This allows one to associate with each uniformizable t-motive a Hodge-Pink group, analogous to the Mumford-Tate group of a smooth projective variety over the complex numbers. It further enabled Pink to prove the analog of the Mumford-Tate Conjecture for Drinfeld modules. Moreover, based on unpublished work of Pink and the first author, the second author proved in her Diploma thesis that the Hodge-Pink group equals the motivic Galois group of the t-motive as defined by Papanikolas and Taelman. This yields a precise analog of the famous Hodge Conjecture, which is an outstanding open problem for varieties over the complex numbers. In this report we explain Pink’s results on Hodge structures and the proof of the function field analog of the Hodge conjecture. The theory of t-motives has a variant in the theory of dual t-motives. We clarify the relation between t-motives, dual t-motives and t-modules. We also construct cohomology realizations of abelian tmodules and (dual) t-motives and comparison isomorphisms between them generalizing Gekeler’s de Rham isomorphism for Drinfeld modules.

2.1 Introduction According to Deligne [Del71, 2.3.8], a rational mixed Hodge structure H consists of a finite dimensional Q-vector space H , an increasing filtration W H of H by Qsubspaces, called the weight filtration, and a decreasing filtration F  HC of HC WD H ˝Q C by C-subspaces, called the Hodge filtration, such that GrpF GrqF GrW n HC D q q .0/ for p C q ¤ n where F HC is the complex conjugate subspace F HC HC . The rational mixed Hodge structures form a neutral Tannakian category [DM82, Definition II.2.19] over Q, whose fiber functor sends a rational mixed Hodge structure H to its underlying Q-vector space [Del94]. By Tannakian duality [DM82, Theorem II.2.11] there is a linear algebraic group H over Q, called the Hodge group of H , such that the Tannakian subcategory hhH ii generated by H is tensor equivalent to the category of Q-rational representations of H . We give more details and explanations on Tannakian theory in Sect. 2.1.2.

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If X is a smooth projective variety over the complex numbers C, its Betti cohomology group HnBetti .X; Q/ is a Q-vector space. Via the de Rham isomorphism HnBetti .X; Q/ ˝Q C Š HndR .X=C/ and the Hodge filtration on the latter, it becomes a rational (pure) Hodge structure. This provides a functor from smooth projective varieties over C to rational mixed Hodge structures. Deligne [Del74, § 8.2] extended this functor to separated schemes of finite type over C. If X is smooth projective and Z X is a closed subscheme of codimension p then Z defines a cohomology class p in H2p Betti .X; Q/\F . The Hodge conjecture [Hod52, Gro69b, Del06] states that every 2p cohomology class in HBetti .X; Q/ \ F p arises from a Q-rational linear combination of closed subschemes of codimension p in X . Besides the Betti and de Rham cohomology, there are various other cohomology theories for X . They are linked to each other via comparison isomorphisms. This inspired Grothendieck to propose a universal cohomology theory he called “motives” [Gro69a]. More precisely Grothendieck conjectured the existence of a Tannakian category of motives such that the cohomology functors like X 7! HnBetti .X; Q/ and X 7! HndR .X=C/ factor through this category of motives; see [Dem69, Kle72, Man68]. The motive associated with X is denoted h.X / and the various cohomology groups attached to X are called the realizations of the motive h.X /. In particular the Betti reL dim X n alization of h.X / is H .X / WD 2nD0 HBetti .X; Q/ equipped with its rational mixed Hodge structure. In terms of the conjectural category of motives, the Hodge conjecture is equivalent to the statement, that the Betti realization functor hhh.X /ii ! hhH .X /ii is a tensor equivalence, where hhh.X /ii is the Tannakian subcategory generated by h.X /. By Tannakian duality hhh.X /ii is tensor equivalent to the category of Q-rational representations of a linear algebraic group h.X / over Q which is called the motivic Galois group of X . The Betti realization functor corresponds to a homomorphism of algebraic groups H .X / ! h.X / over Q. By [DM82, Proposition 2.21] it is a closed immersion and the Hodge conjecture is equivalent to the statement that this homomorphism is an isomorphism. In this article we want to describe the function field analog of the above. There, a category of motives actually exists in the t-motives of Anderson [And86]. We slightly generalize them to A-motives in Sect. 2.3. An A-motive has various cohomology realizations. In this article we explain the Betti, de Rham and `-adic realization. The p-adic and crystalline realization is discussed in the survey [HK20] in this volume. In [Pin97b] Richard Pink invented mixed Hodge structures over function fields (which we call mixed Hodge–Pink structures) as an analog of classical rational mixed Hodge structures. He discovered the crucial fact that instead of a Hodge filtration one needs finer information to obtain a Tannakian category. This information is given in terms of a Hodge–Pink lattice. The definition is as follows. Let Fp D Z=.p/ for a prime p and let A D Fp Œt and Q D Fp .t/ be the polynomial ring and its fraction field. They are the analogs in the arithmetic of function fields of the integers Z and the rational numbers Q. (The theory is actually developed for slightly more general rings A and Q.) Let Q1 D Fp (( 1t )) be the completion of Q for the valuation 1 of Q which does not correspond to a maximal ideal of A. Let C Q1 be an algebraically closed, complete, rank one valued extension, for

33

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

example the completion of an algebraic closure of Q1 . The fields Q1 and C are the analogs of the usual fields R and C of real, respectively complex numbers. We denote the image of t in C by # and consider the ring C[[t  #]] of formal power series in the “variable” t  # and the embedding Q ! C[[t  #]], t 7! t D # C .t  #/. Definitions 2.2.3 and 2.2.7. A mixed Q-Hodge–Pink structure is a triple H D .H; W H; q/ with • H a finite dimensional Q-vector space, • W H H for  2 Q an exhaustive and separated increasing filtration by Q-subspaces, called the weight filtration, • a C[[t  #]]-lattice q H ˝Q C((t  #)) of full rank, called the Hodge–Pink lattice, which satisfies a certain semi-stability condition; see Definition 2.2.7. The Hodge– Pink lattice induces an exhaustive and separated decreasing Hodge–Pink filtration  ı F i HC HC WD H ˝Q; t 7!# C for i 2 Z by setting F i HC WD p \ .t  #/i q .t  #/p \ .t  #/i q , where p WD H ˝Q C[[t  #]]. The mixed Hodge–Pink structures with the fiber functor .H; W H; q/ 7! H form a neutral Tannakian category over Q; see Theorem 2.2.10. It was Pink’s insight that for this result the Hodge–Pink filtration does not suffice, but one needs the finer information present in the Hodge–Pink lattice. Any Hodge–Pink structure H generates a neutral Tannakian subcategory, and the algebraic group H obtained from Tannakian duality is called the Hodge–Pink group of H ; see Sect. 2.1.2. Hodge–Pink structures may arise from Drinfeld-modules or more generally from uniformizable abelian Anderson A-modules E D .E; '/ over C, where E Š Gda;C and 'W A ! EndC .E/ such that .'t  #/d annihilates the tangent space Lie E to E at 0 for some integer d ; see Definitions 2.5.2 and 2.5.5. Namely, E possesses an exponential function expE W Lie E ! E.C/ and if this function is surjective, E is uniformizable. In this case the finite (locally) free A-module ƒ.E/ WD ker.expE / sits in an exact sequence 0

/q

/ ƒ.E/ ˝A C[[t  #]] P i 

˝

i

bi .t  #/

 /

P i

/ Lie E

/0

bi  .Lie 't  #/i ./ I

see (2.5.34). If E is mixed (Definition 2.5.27) the Q-vector space H WD H1;Betti .E/ WD ƒ.E/ ˝A Q inherits an increasing weight filtration W H and we define the mixed Hodge–Pink structures of E as H1 .E/ WD .H; W H; q/ and H1 .E/ WD H1 .E/_ ; see Corollary 2.5.40. Similarly to the classical situation, one can also associate with E a pure (or mixed) A-motive M WD HomC .E; Ga;C /; see Definition 2.5.5. By an A-motive of rank r we mean a pair M D .M; M / where M is a (locally) free CŒt-module of rank 1 1 1  r and M W   M Œ t #  ! M Œ t #  is an isomorphism of CŒtŒ t # -modules; see

34

Urs Hartl, Ann-Kristin Juschka

Definition 2.3.1. Here   M WD Frobp;C M D M ˝CŒt ;  CŒt for the endomorphism   of CŒt sending t to t and b 2 C to b p . For an A-motive we define its  -invariants 1 ˚P

over Chti WD bi t i W bi 2 C; lim jbi j D 0 as i D0

i !1

  ˚

ƒ.M / WD M ˝CŒt  Chti WD m 2 M ˝CŒt  Chti W M .Frobp;C m/ D m : (2.1.1) An A-motive of rank r is uniformizable if its  -invariants form a (locally) free Amodule of rank r; see Definition 2.3.17 and Lemma 2.3.21. We explain the results of Papanikolas [Pap08] and Taelman [Tae09a] that the category A-UMotI of uniformizable A-motives up to isogeny together with the fiber functor M 7! ƒ.M / ˝A Q is a neutral Tannakian category over Q; see Theorems 2.3.27 and 2.4.23. Considering the Tannakian subcategory hhM ii generated by M , the algebraic group M associated by Tannakian duality, is called the motivic Galois group of M . In unpublished work, the following function field analog of the classical Hodge conjecture was formulated by Pink and proved by him for pure uniformizable Amotives and by Pink and the first author for uniformizable mixed A-motives. Pink’s proof was worked out for dual A-motives (see below) by the second author in her Diploma thesis [Jus10]. There is a realization functor H1 from uniformizable mixed A-motives M to mixed Hodge–Pink structures as follows. The Q-vector space H WD H1Betti .M ; Q/ WD ƒ.M / ˝A Q inherits an increasing weight filtration W H and ad . M / ˝CŒt  C[[t  #]]; see mits a canonical isomorphism hW H ˝Q C[[t  #]] ! 1 1 Proposition 2.3.30. We set q WD h ı M .M ˝CŒt  C[[t  #]]/ H ˝Q C((t  #)) and define the mixed Hodge–Pink structures of M as H1 .M / WD .H; W H; q/ and H1 .M / WD H1 .M /_ ; see Definition 2.3.32. The functor H1 restricts to an exact tensor functor from the Tannakian subcategory hhM ii of uniformizable mixed A-motives generated by M to the Tannakian subcategory hhH1 .M /ii of mixed Hodge–Pink structures generated by H1 .M /. This induces a morphism from the Hodge–Pink group H1 .M / of H1 .M / to the motivic Galois group M of M . Theorems 2.3.34 and 2.6.1 (The Hodge Conjecture over Function Fields). The morphism H1 .M / ! M is an isomorphism of algebraic groups. Equivalently, H1 W hhM ii ! hhH1 .M /ii is an exact tensor equivalence. The crucial part in the proof of this theorem is to show that each Hodge–Pink substructure H 0 H1 .M / is isomorphic to H1 .M 0 / for an A-sub-motive M 0 M . This is achieved by associating with H 0 a  -bundle over the punctured open unit disk. The theory of  -bundles was developed in [HP04] and is explained in detail in Sect. 2.7, where we also show how to associate a pair of  -bundles with a mixed Hodge–Pink structure, respectively with a uniformizable A-motive (or dual A-motive; see below). Using the classification [HP04, Theorem 11.1] of  -bundles and the rigid analytic GAGA-principle, one defines an A-motive M 0 such that H1 .M 0 / D H 0 . Large parts of this article are not original but a survey of the existing literature, which tries to be largely self-contained. In Sect. 2.2 we review Pink’s theory of

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

35

mixed Hodge–Pink structures. In Sect. 2.3 we define pure and mixed A-motives and slightly generalize Anderson’s [And86, § 2] theory of uniformization of t-motives to A-motives. Also we define and study the mixed Hodge–Pink structure H1 .M / of a uniformizable mixed A-motive M and its Betti, de Rham and `-adic cohomology realization, as well as the comparison isomorphisms between them. Actually the `adic realization is called “v-adic” by us where v A is a place taking on the role of the prime ` 2 Z and H1v is our analog of He1K t . : ; Z`/. For applications to transcendence questions like in [ABP04, Pap08, CY07, CPY10, CPTY10, CP11, CPY11, CP12], it turns out that dual A-motives are even more useful than A-motives; see the article of Chang [Cha20] in this volume for an introduction. A dual A-motive of rank r is a pair ML D .ML ; LML / where ML is a (locally) free CŒt1 1 1   ! ML Œ t #  is an isomorphism of CŒtŒ t # module of rank r and LML W .L ML /Œ t #   1 modules for L D . / . (Beware that a dual A-motive is something different then the dual M _ of an A-motive M ). A dual A-motive of rank r is uniformizable if its L  L L L invariants ƒ.M / WD M ˝CŒt  Chti , which are defined analogously to (2.1.1), form a (locally) free A-module of rank r; see Definition 2.4.14 and Lemma 2.4.16. Also the category of uniformizable dual A-motives with the fiber functor ML 7! ƒ.ML / ˝A Q is a neutral Tannakian category; see Theorem 2.4.23. Actually this is the category studied by Papanikolas [Pap08]. If ML is uniformizable and mixed, the Q-vector space H WD H1;Betti .ML ; Q/ WD ƒ.ML / ˝A Q inherits an increasing weight filtration W H  and admits a canonical isomorphism hML W H ˝Q C[[t #]] ! ML ˝CŒt  C[[t #]]; see 1  L Proposition 2.4.27. We set q WD h L ı L L .L M ˝CŒt  C[[t  #]]/ H ˝Q C((t  M

M

#)) and define the mixed Hodge–Pink structures of ML as H1 .ML / WD .H; W H; q/ and H1 .ML / WD H1 .ML /_ ; see Definition 2.4.30. This theory of pure and mixed dual A-motives, their theory of uniformization, their associated mixed Hodge–Pink structures, and their Betti, de Rham and v-adic cohomology realizations, as well as the comparison isomorphisms between them are explained in Sect. 2.4. In the longest Sect. 2.5 we recall the theory of abelian Anderson A-modules, which generalize Anderson’s [And86] abelian t-modules, and their associated Amotives including uniformizability, scattering matrices (Remark 2.5.34) and Anderson generating functions (Corollary 2.5.22, Example 2.5.35). Moreover, in Sections 2.5.2, 2.5.3 and 2.5.5 we reproduce from unpublished work of Anderson [ABP02] the theory of A-finite Anderson A-modules E including uniformization and the description of torsion points. These are the ones for which the CŒt-module ML .E/ WD HomC .Ga;C ; E/ is finitely generated, and hence a dual A-motive. As described above, we associate a mixed Hodge–Pink structure with a uniformizable mixed abelian, respectively A-finite, Anderson A-module and v-adic, Betti and de Rham cohomology realizations. The latter go back to Deligne, Anderson, Gekeler, Yu, Goss, Brownawell and Papanikolas. We generalize the approach of these authors in Sect. 2.5.7 and prove comparison isomorphisms between these cohomology realizations. We also explain in Theorem 2.5.47 how to recover Gekeler’s comparison isomorphism [Gek89, § 2] between Betti and de Rham cohomology from ours.

36

Urs Hartl, Ann-Kristin Juschka

Finally, in Sect. 2.6 we briefly report on applications to Galois representations and transcendence questions due to Anderson, Brownawell, Chang, Papanikolas, Pink, Thakur, Yu and others. Although this article is mainly a review of (un)published work, we nevertheless establish the following new results: the theory of mixed Anderson A-modules (Sect. 2.5.4) and the construction that associates with a uniformizable mixed (dual) A-motive a mixed Hodge–Pink structure (Sections 2.3.4, 2.4.4). Also we clarify the relation between a uniformizable M D .M;  mixed A-motive  M / and the associ 1 _ L ated dual A-motive M .M / WD HomCŒt  . M; CŒt =C / ; M in Propositions 2.4.3, 2.4.9, 2.4.17, 2.4.25 and Theorem 2.4.32 and most importantly in the following Theorem 2.5.13. Let E be an Anderson A-module over C which is both abelian and A-finite, and let M D .M; M / D M .E/ and ML D .ML ; LML / D ML .E/ be   _ its associated (dual) A-motive. Let ML .M / D HomCŒt  .  M; 1CŒt =C / ; M be the dual A-motive associated with M . Then there is a canonical isomorphism of dual  A-motives „W ML .M / ! ML .E/. We illustrate the general theory with various examples, most notably Examples 2.5.16 and 2.5.35 which for Drinfeld-modules explain Theorem 2.5.13 in concrete terms and relate it to scattering matrices. Moreover, we prove the compatibility of the cohomology realizations and comparison isomorphisms of A-motives, dual A-motives and abelian, respectively A-finite Anderson A-modules in Theorems 2.3.37, 2.4.36, 2.5.51 and Propositions 2.4.38, 2.5.45, 2.5.48, and we prove the compatibility with a change of the ring A in Remark 2.5.52, and with Gekeler’s comparison isomorphism [Gek89, § 2] in Theorem 2.5.47. In particular, we prove the following theorems. Theorem 2.5.38. Let E be a uniformizable mixed A-finite Anderson A-module over C and let ML D ML .E/ be its associated mixed dual A-motive. Then the mixed Hodge–Pink structures H1 .E/ and H1 .ML / are canonically isomorphic. Theorem 2.5.39. Let E be a uniformizable mixed abelian Anderson A-module over C and let M D M .E/ be its associated mixed A-motive. Consider the Hodge–Pink structure D .H; W H; q/ which is pure of weight 0 and given by H D 1Q=Fq D Q dt and q D C[[t  #]]dt. Then the mixed Hodge–Pink structures H1 .E/ and H1 .M / ˝ are canonically isomorphic. Theorem 2.5.41. Let E be a uniformizable mixed Anderson A-module over C which is both abelian and A-finite, and let M D M .E/ and ML .E/ be the associated (dual) A-motive. Then the isomorphisms above are also compatible with the isomorphisms  ML .E/ from Theorems 2.4.32, 2.5.38 and 2.5.39 and the isomorphism „W ML .M / !

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

37

from Theorem 2.5.13, in the sense that the following diagram commutes   H1 ML .M / O Theorem 2:4:32 Š H1 .M / ˝

H1 .„/ Š Theorem 2:5:39 Š



/ H ML .E/ 1



Š Theorem 2:5:38  / H .E/ 1

Finally, we give a criterion in Theorem 2.7.13 which characterizes those mixed Hodge–Pink structures that arise from uniformizable mixed A-motives. Various categories of motives over C play a part in this article. To give the reader an overview we list them in the following table. Note that the set of morphisms HomA-Mot .M ; N / between two A-motives M and N is a finitely generated A-module; see Remark 2.3.7(c). The same is true for dual A-motives; see Remark 2.4.4(d). Category A-Mot A-MMot

Description A-motives over C mixed A-motives

A-MotI

A-motives up to isogeny, that is with HomA-MotI .M ; N / WD HomA-Mot .M ; N / ˝A Q mixed A-motives up to isogeny

A-MMotI A-UMotI

uniformizable A-motives up to isogeny A-MUMotI uniformizable mixed A-motives up to isogeny A-dMot dual A-motives A-dMMot mixed dual A-motives dual A-motives up to isogeny, that is with HomA-dMotI .ML ; NL / WD HomA-dMot .ML ; NL / ˝A Q A-dMMotI mixed dual A-motives up to isogeny A-dMotI

A-dUMotI uniformizable dual A-motives up to isogeny A-dMUMotI uniformizable mixed dual A-motives up to isogeny

Def. Properties o 2.3.1 exact (Rem. 2.3.5(b) 2.3.8 and 2.3.12) 9 non-neutral 2.3.1 > > > = Tannakian (Prop. 2.3.4 and > > > 2.3.8 ; 2.3.11) 2.3.18 9 = neutral Tannakian 2.3.18 ; (Thm. 2.3.27) o 2.4.1 exact (Rem. 2.4.4(b) 2.4.6 9and 2.4.11) > 2.4.1 > > > > = non-neutral Tannakian > > (Prop. 2.4.3 and 2.4.6 > > > ; 2.4.10) 2.4.14 9 = neutral Tannakian 2.4.14 ; (Thm. 2.4.23)

38

Urs Hartl, Ann-Kristin Juschka

Acknowledgements. We are grateful to Richard Pink for teaching us his beautiful theories of mixed Hodge–Pink structures and  -bundles and to Greg Anderson, Dale Brownawell and Matthew Papanikolas for sharing their unpublished manuscript [ABP02] with us and allowing us in the present article to reproduce the theory of dual A-motives from it. We also profited much from discussions with Dale Brownawell, Matthew Papanikolas, Lenny Taelman. These notes grew out of two lecture series given by the first author in the fall of 2009 at the conference “t-motives: Hodge structures, transcendence and other motivic aspects” at the Banff International Research Station BIRS in Banff, Canada and at the Centre de Recerca Mathemàtica CRM in Barcelona, Spain in the spring of 2010. The first author is grateful to BIRS and CRM for their hospitality. He also acknowledges support of the DFG (German Research Foundation) in form of SFB 878 and Germany’s Excellence Strategy EXC 2044– 390685587 “Mathematics Münster: Dynamics–Geometry–Structure”. 2.1.1 Preliminaries. Throughout this article we will denote by Fq

a finite field with q elements and characteristic p,

C

a smooth projective geometrically irreducible curve over Fq ,

1 2 C.Fq /

a fixed closed point, (To simplify the exposition in this article 1 is supposed to be Fq -rational. The main results we present here hold, and are in fact proved in [Pin97b, HP18], without this assumption.)

. C D C X f1g . A D .C ; OC. /

the associated affine curve, . the ring of regular functions on C (the function field analog of Z),

Q D Fq .C /

the function field of C , viz. the field of fractions of A (the analog of Q),

z2Q

a uniformizing parameter at 1,

Q1 D Fq ((z))

the completion of Q at 1 (the analog of R),

A1 D Fq [[z]]

the ring of integers in Q1 ,

C Q1

an algebraically closed, complete, rank one valued extension, for example the completion of an algebraic closure of Q1 (the analog of the usual field of complex numbers),

cW Q ! C

the natural inclusion,



 D c .z/

the image of z in C, which satisfies 0 < jj < 1,

AC D A ˝Fq C

the base extension of A,

QC D Q ˝Fq C

the base extension of Q, distinguishing between z and  allows us to abbreviate the element z ˝ 1 of QC by z and the element 1 ˝ c  .z/ by ,

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

39

CC D C Spec Fq Spec C the resulting irreducible curve over C, J AC

the (maximal) ideal generated by a ˝ 1  1 ˝ c  .a/ for all a 2 A,

AC ŒJ 1

the ring of global sections on the open affine subscheme Spec AC X V.J / of CC ,

C[[z  ]]

the formal power series ring in the “variable” z  . It is canonically isomorphic to the completion of the local ring of CC at V.J /, see Lemma 2.1.3, and replaces the ring C[[t  #]] from the introduction,

C((z  ))

the fraction field of C[[z  ]],

Q1 ,! C[[z  ]]

the natural Fq -algebra homomorphism satisfying 7!z D 1 P  i  zi j P P C.z/ and given by ai z i 7!  a .z i j i j D0 i /j ,

 W CC ! CC

the product of the identity on C with the q-th power Frobenius on Spec C, which acts on points and on the coordinates of C as the identity, and on the elements b 2 C as b 7! b q ,

  W AC ! AC

the corresponding endomorphism a˝b 7! a˝b q for a 2 A and b 2 C,

 i  WD . /i

for a non-negative integer i 2 N0 ,

 i M

the pullback  i  M WD M ˝AC ; i  AC of an AC -module M under  ,

  .m/ WD m ˝ 1

the canonical image of m 2 M in   M WD M ˝AC ;  AC ,

L  WD .  /1

the endomorphism of AC inverse to   sending a ˝ b to p q a ˝ b for a 2 A and b 2 C which exists because C is perfect,

L i  WD .L /i

for a non-negative integer i 2 N0 ,

L i  M

the tensor product L i  M WD M ˝AC ;L i  AC for an AC module M ,

L  .m/ WD m ˝ 1 Cf g WD

the canonical image of m 2 M in L  M WD M ˝AC ;L  AC , n ˚P

bi  i W n 2 N0 ; bi 2 C the skew polynomial ring in

Cfg L WD

the variable  with the commutation rule  b D b q  for b 2 C, n ˚P

bi L i W n 2 N0 ; bi 2 C the skew polynomial ring in

i D0

i D0

the variable L with the commutation rule b L D b 1=q L for b 2 C.

40

Urs Hartl, Ann-Kristin Juschka

For any module M over an integral domain R and any non-zero element x 2 R we let RŒ x1  and M Œ x1  WD M ˝R RŒ x1  denote the localizations obtained by inverting x. Any homomorphism of R-modules M ! N induces a homomorphism of RŒ x1 -modules M Œ x1  ! N Œ x1  denoted again by the same letter. Remark 2.1.1. The ring homomorphisms   W AC ! AC and L  W AC ! AC are flat because they arise by base p change from the flat homomorphisms C ! C; b 7! b q , q respectively C ! C; b 7! b. For later reference we record the following two lemmas. Lemma 2.1.2. (a) If t 2 Q is a uniformizing parameter at a closed point P of C then Q is a finite separable field extension of Fq .t/. (b) There exists an element t 2 A such that Q is a finite separable field extension ideal v A one may even find such a t 2 A such of Fq .t/. For every maximal p that the radical ideal A  t of A  t is v. Proof. (a) The point P 2 C is unramified under the map C ! P1Fq corresponding to the inclusion Fq .t/ Q. Since all ramification indices are divisible by the inseparability degree, the latter has to be one. (b) Choose some a 2 A X Fq . Then Fq Œa ,! A is a finite flat ring extension and so Q=Fq .a/ is a finite field extension. If it is not separable, let pe be its inseparability e e degree. Then Fq .a/ is contained in Qp WD fx p W x 2 Qg by [Sil86, Proof of e Corollary II.2.12]. So there is a t 2 Q with a D t p . We even have t 2 A because A is integrally closed in Q. By considering the inseparability degree in the tower Fq .a/ Fq .t/ Q we see that Q=Fq .t/ is separable. n If a maximal ideal v A is given, there is a positive integer n such that vp D Aa A t D is a principal ideal. Continuing as above we obtain an element t 2 A with p A  a D v.  Lemma 2.1.3. Let K be a field and let c  W A ,! K be an injective ring homomorphism. Let z 2 Q X Fq be an element such that Q is a finite separable extension of Fq .z/, and let  D c  .z/. Then the power series ring K[[z  ]] over K in the “variable” z   is canonically isomorphic to the completion of the local ring of CK at the closed point V.J / defined by the ideal J WD .a ˝ 1  1 ˝ c  .a/W a 2 A/ AK . Proof. The completion of the local ring of CK at V.J / is lim AK =J n . Since this is  a complete discrete valuation ring with residue field K we only need to show that z   is a uniformizing parameter. Clearly, z   is contained in the maximal ideal. To prove the converse, let a 2 Q. Let f 2 Fq .z/ŒX  be the minimal polynomial of a over Fq .z/ and multiply it with the common denominator to obtain the polynomial

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

41

F .X; z/ 2 Fq ŒX; z. The two-variable Taylor expansion of F at .c  .a/; / 2 K 2 is F .X; z/  F .c  .a/; / C C

@F  .c .a/; /  .X  c  .a// @X

@F  .c .a/; /  .z  / mod J 2 @z

@F Plugging in a for X yields F .a; z/ D 0 and @X .a; z/ ¤ 0 by the separability of Q=Fq .z/. Under the injective homomorphism c  W Q ,! K we get F .c  .a/; / D 0 and @F .c  .a/; / ¤ 0. This shows that the element ac  .a/ 2 J =J 2 is a multiple of @X z  , and so z   generates the AK -module J =J 2 . By Nakayama’s Lemma [Eis95, Corollary 4.7] there is an element f 2 1 C J that annihilates the AK -module J =.z  /. Since f is invertible in lim AK =J n we have proved that z   generates the

maximal ideal of lim AK =J n .







2.1.2 Tannakian theory. As already alluded to in the introduction, a good framework to discuss Hodge structures is the theory of Tannakian categories. Also Pink’s results which we explain in this article use this language. Therefore, we briefly recall the definition and some facts about Tannakian categories from the articles of Deligne and Milne [DM82, Del90, Mil92]. Definition 2.1.4 ([Mil92, (A.7.1) and (A.7.2), page 222]). Let K be a field. A Klinear abelian tensor category C with unit object 1l is a Tannakian category over K if (a) for every object X of C there exists an object X _ of C , called the dual of X , and morphisms evW X ˝ X _ ! 1l and ıW 1l ! X _ ˝ X such that idX ˝ı

ev˝idX

.ev ˝ idX / ı .idX ˝ı/ D idX W X ! X ˝ X _ ˝ X ! X ı˝idX _

and

idX _ ˝ev

.idX _ ˝ev/ ı .ı ˝ idX _ / D idX _ W X _ ! X _ ˝ X ˝ X _ ! X _ ; (b) and for some non-zero K-algebra L there is an exact faithful K-linear tensor functor ! from C to the category of finitely generated L-modules. Any such functor ! is called an L-rational fiber functor for C . A K-rational fiber functor for C is called neutral. If C has a neutral fiber functor it is called a neutral Tannakian category over K. Remark 2.1.5. (a) According to [DM82, § 1] being a tensor category means that there is a “tensor product” functor C C ! C ; .X; Y / 7! X ˝Y which is associative and commutative, such that C has a unit object. The latter is an object 1l 2 C  together with an isomorphism 1l ! 1l ˝ 1l such that C ! C ; X 7! 1l ˝ X is an equivalence of categories. A unit object is unique up to unique isomorphism; see [DM82, Proposition 1.3]. One sets X ˝0 WD 1l and X ˝n WD X ˝X ˝n1 for n 2 N>0 .

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(b) Being K-linear means that HomC .X; Y / is a K-vector space for all X; Y 2 C . (c) Being abelian means that C is an abelian category. Then automatically ˝ is a bi-additive functor and is exact in each factor; see [DM82, Proposition 1.16]. (d) By [Del90, §§ 2.1–2.5] the conditions of Definition 2.1.4 imply that EndC .1l/ D K and that the tensor product is K-bilinear and exact in each variable. It further implies that Hom.X; Y / WD X _ ˝ Y is an internal hom in C , that is an object which represents the functor C ı ! VecK ; T 7! HomC .T ˝ X; Y /. This means that HomC .T ˝ X; Y / D HomC .T; Hom.X; Y //. Then C is a rigid abelian K-linear tensor category in the sense of [DM82, Definition 2.19]. ThisN further means that the n _ _ naturalN morphisms X ! .X / are isomorphisms and that i D1 Hom.Xi ; Yi / D N Hom. i Xi ; i Yi / for all Xi ; Yi 2 C . The definition of a neutral Tannakian category over K in [DM82, Definition 2.19] as a rigid abelian K-linear tensor category possessing a neutral fiber functor is equivalent to Definition 2.1.4. (e) A functor F W C ! C 0 between rigid abelian K-linear tensor categories is a tensor functor if F .1l/ is a unit object in C 0 and there are fixed isomorphisms F .X ˝ Y / Š F .X / ˝ F .Y / compatible with the associativity and commutativity laws.   _ _ A tensor functor automatically satisfies F .X / D F .X / and F H om.X; Y / D   Hom F .X /; F .Y / ; see [DM82, Proposition 1.9]. In particular, for an L-rational fiber functor ! this means !.1l/ Š L. If G is an affine group scheme over K, let RepK .G/ be the category of finitedimensional K-rational representations of G, that is K-homomorphisms of K-group schemes W G ! GLK .V / Š GLdimK V; K for varying finite dimensional K-vector spaces V . Together with the forgetful functor ! G W .V; / 7! V it is a neutral Tannakian category over K; see [DM82, Example 1.24]. Tannakian duality says that every neutral Tannakian category over K is of this form: Theorem 2.1.6 (Tannakian duality [DM82, Theorem 2.11]). Let C be a neutral Tannakian category over K with neutral fiber functor !; and let Aut˝ .!/ be the set of automorphisms of tensor functors of !; see [DM82, p. 116]. (a) There is an affine group scheme G over K that represents the functor Aut˝ .!/ on K-algebras given by Aut˝ .!/.R/ WD Aut˝ . where

R

R

ı !/ for all K-algebras R;

W VecK ! ModR ; V 7! V ˝K R; is the canonical tensor functor.

 (b) The fiber functor ! defines an equivalence of tensor categories C ! RepK .G/.

Definition 2.1.7. A subcategory C 0 of a category C is strictly full if it is full and contains with every X 2 C 0 also all objects of C isomorphic to X . A strictly full subcategory C 0 of a rigid tensor category C is a rigid tensor subcategory if 1l 2 C 0 and X ˝ Y; X _ 2 C 0 for all X; Y 2 C 0 .

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If C is a neutral Tannakian category over K and X 2 LC , the rigid tensor subcategory of C containing as objects all subquotients of all riD1 X ˝ni ˝ .X _ /˝mi for all r; ni ; mi 2 N0 is called the Tannakian subcategory generated by X and is denoted hhX ii. It is a neutral Tannakian category over K. Lemma 2.1.8 ([DM82, Proposition 2.20]). An affine K-group scheme G is (linear) algebraic, that is a closed subscheme of some GLn;K , if and only if there exists an ˝ G object X in Rep  K .G/ with RepK .G/ D hhX ii. In this case G D Aut .! / ,! G GLK ! .X / is a closed immersion, which factors through the centralizer of End.X /   inside GL ! G .X / . Proof. This was proved in [DM82, Proposition 2.20] except for the statement about the centralizer, which follows from the fact that G is the automorphism group of the  forgetful fiber functor ! G . A homomorphism f W G ! G 0 of affine K-group schemes induces a functor 0 ! W RepK .G 0 / ! RepK .G/; 7! ı f , such that ! G ı ! f D ! G . The same holds in the other direction: f

Lemma 2.1.9 ([DM82, Corollary 2.9]). Let G and G 0 be affine group schemes over 0 K and let F W RepK .G 0 / ! RepK .G/ be a tensor functor such that ! G ı F D ! G . 0 Then there is a unique homomorphism f W G ! G of affine K-group schemes such that F Š ! f . Under this correspondence various properties of group homomorphisms are reflected on the associated tensor functor. Proposition 2.1.10 ([DM82, Proposition 2.21]). Let f W G ! G 0 be a homomorphism of affine K-group schemes and let ! f W RepK .G 0 / ! RepK .G/ be defined as above. (a) f is faithfully flat if and only if ! f is fully faithful and for every object X 0 in RepK .G 0 / each subobject of ! f .X 0 / is isomorphic to the image of a subobject of X 0 . (b) f is a closed immersion if and only if for every object X of RepK .G/ there exists an object X 0 in RepK .G 0 / such that X is isomorphic to a subquotient of ! f .X 0 /.

2.2 Hodge–Pink structures In this section we present Pink’s definition [Pin97b] of the Tannakian category of mixed Q-Hodge structures. Pink first defines pre-Hodge structures which form an additive tensor category. This category is not abelian, so he introduces a semistability condition for pre-Hodge structures. The semistable ones form a neutral Tannakian

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category and will be called Hodge structures. Compared to the classical theories of the rational mixed Hodge-structures of Deligne [Del71] and the p-adic Hodge theory of Fontaine [Fon82] there is one important difference in Pink’s theory. In the classical theories, Hodge structures consist of a vector space over one field (with additional structures like weight filtration or Frobenius endomorphism) and a decreasing Hodge filtration defined over a separable extension of this field. In the function field setting C=Q is not separable and hence a semistability condition solely based on the Hodge filtration cannot be preserved under tensor products. This is Pink’s crucial observation and the reason why he replaces Hodge filtrations by finer structures and why we call all these structures Hodge–Pink structures. Definition 2.2.1. An exhaustive and separated increasing Q-filtration W H on a finite dimensional Q-vector space H is a collection of Q-subspaces W H H for  2 Q with W0 H W H whenever 0 < , such that the associated Q-graded vector space  M M S GrW W H= 0  Q and that W H D H for  0. Definition 2.2.3 (Pink [Pin97b, Definition 9.1]). A (mixed) Q-pre Hodge–Pink structure (at 1) is a triple H D .H; W H; q/ with • H a finite dimensional Q-vector space, • W H an exhaustive and separated increasing Q-filtration, • a C[[z  ]]-lattice q H ˝Q C((z  )) of full rank. The filtration W H is called the weight filtration, q is called the Hodge–Pink lattice, and rk H WD dimQ H is called the rank of H . The jumps of the weight filtration are called the weights of H . If GrW  H D H , then H is called pure of weight . A morphism f W .H; WH; q/ ! .H 0 ; W H 0 ; q0 / of Q-pre Hodge–Pink structures consists of a morphism f W H ! H 0 of Q-vector spaces satisfying f .W H / W H 0 for all  and .f ˝id/.q/ q0 . The morphism  f is called strict if f .W H / D f .H / \ W H 0 for all  and .f ˝ id/.q/ D q0 \ f .H / ˝Q C((z  )) .

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Remark 2.2.4. The Hodge–Pink lattice of a mixed Q-pre Hodge–Pink structure H D .H; W H; q/ induces an exhaustive and separated decreasing Z-filtration as follows. Define the tautological lattice p WD H ˝Q C[[z  ]] inside H ˝Q C((z  )) and consider the natural projection p  p=.z  /p D H ˝Q;c  C DW HC : The Hodge–Pink filtration F  HC D .F i HC /i 2Z of HC is defined by letting F i Hı C i i i be the image of p \ .z  / q in H for all i 2 Z; that is, F H D p \ .z  / q C C   .z  /p \ .z  /i q . One finds that any morphism is also compatible with the Hodge–Pink filtrations, but a strict morphism is not necessarily strictly compatible with the Hodge–Pink filtrations. The Hodge–Pink weights .!1 ; : : : ; !rk H / of H are the jumps of the Hodge–Pink filtration. They are integers. Equivalently they are the elementary divisors of q relLrk H eC!i and ative to p; that is, they satisfy q=.z  /e p Š i D1 C[[z  ]]=.z  / L rk H e e!i p=.z  / q Š i D1 C[[z  ]]=.z  / for all e 0. We usually assume that they are ordered !1  : : :  !rk H . A main source for Hodge–Pink structures are Drinfeld A-modules or more generally uniformizable mixed abelian Anderson A-modules (see Sect. 2.5). Example 2.2.5. (a) Let 'W A ! EndC .Ga;C / be a Drinfeld A-module [Dri76] of rank r over C where Ga;C is the additive group scheme. We set E D Ga;C and E D .E; '/, and we write Lie E for the tangent space to E at 0. Consider the exponential exact sequence of A-modules expE

0 ! ƒ.E/ ! Lie E ! E.C/ ! 0 ;

(2.2.1)

where ƒ WD ƒ.E/ WD ƒ.'/ WD ker.expE /; see Sect. 2.5.1 or the survey of Brownawell and Papanikolas [BP20, § 2.4] in this volume. ƒ.E/ Lie E D C is a discrete A-submodule of rank r. Clearly, ƒ.E/ generates the one dimensional C-vector space Lie E. Through the identification C[[z  ]]=.z  / D C we make Lie E into a C[[z  ]]-module. We obtain a C[[z  ]]-epimorphism on the right in the sequence 0

/q

/ ƒ ˝A C[[z  ]]

˝

P

i

bi .z  /

i



/ Lie E / b0  

/0

(2.2.2)

and we let q be its kernel. By sequence (2.2.2) the pair .ƒ; q/ determines the C-vector space Lie E with the A-action on it, and the A-lattice ƒ inside Lie E as the image of the A-homomorphism ƒ ,! ƒ ˝A C[[z  ]]  Lie E. Therefore the pair .ƒ; q/ also determines the Drinfeld A-module ' by sequence (2.2.1). We further set ( .0/ if  <  1r ; H WD H1 .E/ WD ƒ.E/ ˝A Q and W H D H if    1r :

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Then H1 .E/ WD .H; W H; q/ is a pure Q-pre Hodge–Pink structure of weight  1r . It satisfies that .z  /p q p and F 1 HC D HC F 0 HC F 1 HC D .0/. Since dimQ H D r and dimC .p=q/ D dimC Lie E D 1 we have dimC F 0 HC D r  1. As we will explain in Sect. 2.5.7 below, F 0 HC HC D H1;Betti .E; C/ is the Hodge filtration studied by Gekeler [Gek89, (2.13)] using the de Rham isomorphism H1Betti .E; C/ Š H1dR .E; C/. See Example 2.2.9 for a continuation of this example. Also in Sect. 2.5.6 we will generalize the present construction to Anderson’s abelian t-modules [And86]. Note that this parallels the case of complex abelian varieties X , whose Hodge structure H1;Betti .X; Q/ is pure of weight  12 . (b) More specifically, if C D P1Fq ; A D Fq Œt; # WD c  .t/ 2 C and E is the Carlitzmodule [BP20, § 2.2] with 't D # CFrobq;Ga , where Frobq;Ga W x 7! x q is the relative q-Frobenius of Ga;C D Spec CŒx over C, then r D 1 and 0 H WD H1 .E/ D Q ; GrW 1 H D H ; q D .z  /  p ; F HC D .0/ :

(c) In (a) and (b) the subspace F 0 HC determines q uniquely as its preimage under the surjection H ˝Q C[[z  ]]  HC because .z  /  p q. However, note that in general q is not determined by F  HC . For example let H D Q˚2 and q D .z  /2 p C C[[z  ]]  v0 C .z  /v1 for vi 2 HC with v0 ¤ 0. Then F 2 HC D HC F 1 HC D C  v0 D F 0 HC F 1 HC D .0/ : So the information about v1 is not preserved by the Hodge–Pink filtration. To continue with the general theory let H D .H; W H; q/ be a Q-pre Hodge– Pink structure. A subobject in the category of Q-pre Hodge–Pink structures is a morphism H 0 ! H whose underlying homomorphism of Q-vector spaces is the inclusion H 0 ,! H of a subspace. It is called a strict subobject if H 0 ! H is strict. Likewise a quotient object is a morphism H ! H 00 whose underlying homomorphism of Q-vector spaces is the projection H  H 00 onto a quotient space. It is called a strict quotient object if H ! H 00 is strict. For any Q-subspace H 0 H one can endow H 0 with a unique structure of strict subobject H 0 and H 00 WD H=H 0 with a unique structure of strict quotient object H 00 . The sequence 0 ! H 0 ! H ! H 00 ! 0 and any sequence isomorphic to it is called a strict exact sequence. With these definitions the category of Q-pre Hodge–Pink structures is a Q-linear additive category. Pink makes a suitable subcategory of it into a Tannakian category. In order to do this, he defines tensor products, internal hom and duals. Definition 2.2.6. Let H 1 D .H1 ; W H1 ; q1 / and H 2 D .H2 ; W H2 ; q2 / be two Q-pre Hodge–Pink structures. (a) The tensor product H 1 ˝ H 2 is the Q-pre Hodge–Pink structure consisting of the tensor product HP 1 ˝Q H2 of Q-vector spaces, the induced weight filtration W .H1 ˝Q H2 / WD 1 C2 D W1 H1 ˝Q W2 H2 and the lattice q1 ˝C[[z ]] q2 . One defines for n  1 the symmetric power Symn H and the alternating power ^n H as the induced strict quotient objects of H ˝n .

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e D Hom.H 1 ; H 2 / consists of the Q-vector space H e WD (b) The internal hom H e WD f h 2 H e W h.W1 H1 / HomQ .H1 ; H2 /, the induced weight filtration W H WC1 H2 8 1 g, and the lattice qQ WD HomC[[z ]] .q1 ; q2 /. The latter is e ˝Q C((z  )) via the inclusion a C[[z  ]]-lattice in H qQ ,! qQ ˝C[[z ]] C((z  ))    ! HomC((z )) q1 ˝C[[z ]] C((z  )); q2 ˝C[[z ]] C((z  ))    ! HomC((z )) H1 ˝Q C((z  )); H2 ˝Q C((z  ))  e ! H ˝Q C((z  ))

obtained by applying [Eis95, Proposition 2.10]. (c) The unit object 1l consists of the vector space Q itself together with the lattice q WD p and is pure of weight 0. The dual H _ of a Q-pre Hodge–Pink structure H is then Hom.H ; 1l/. The category of Q-pre Hodge–Pink structures is an additive tensor category but it is not abelian because not all subobjects and quotient objects are strict. Indeed, the category theoretical image (respectively coimage) of a subobject H 0 ,! H (respectively quotient object H  H 0 ) is the strict subobject (respectively strict quotient object) with same underlying Q-vector space as H 0 (respectively H 00 ). In order to remedy this, Pink defines semistability as follows. Definition 2.2.7. Let H D .H; W H; q/ be a Q-pre Hodge–Pink structure. 0 H1 WD H ˝Q Q1 consider the induced strict (a) for any Q1 -subspace H1 Q1 -subobject  0 0 0 H 01 WD H1 ; W H1 WD H1 \ .W H ˝Q Q1 / ;   0 ˝Q1 C((z  )) q0 WD q \ H1 and (using the induced Hodge–Pink filtration F  HC from Remark 2.2.4) set X i  dimC GriF HC0 degq H 01 WD degF HC0 WD i 2Z

D dimC degW H 01 WD

X

q0 p0  dim C p0 \ q0 p0 \ q0

0   dimQ1 GrW  H1

2Q

(b) H is called locally semistable or a (mixed) Q-Hodge–Pink structure (at 1) 0 if for any Q1 -subspace H1 H1 one has degq H 01  degW H 01 with 0 equality for H1 D .W H /1 for all . (c) We denote by Q-HP the full subcategory of all mixed Q-Hodge–Pink structures.

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Remark 2.2.8. (a) Alternatively degq H 01 can be computed as dimC q0 =rdimC p0 =r for any C[[z  ]]-lattice r which is contained in both q0 and p0 . In particular, if .z  /d p0 q0 for some d 2 Z with d  0, then degq H 01 D dimC q0 =.z  /d p0  0 0 , because dimC p0 =.z  /d p0 D d  dimQ1 H1 . d dimQ1 H1 (b) The piecewise linear function on Œ0; rk H  whose slope on Œi  1; i  is the i th smallest Hodge–Pink weight is called the Hodge polygon of H and is denoted HP .H /. Analogously one defines the weight polygon W P .H / of H using the weights of H . A Q-pre Hodge–Pink structure is locally semistable if and only if for every strict Q1 -subobject H 01 the weight polygon lies above the Hodge polygon, and both have the same endpoint whenever H 01 D .W H /1 ; see [Pin97b, Proposition 6.7]. Example 2.2.9. We continue with Example 2.2.5. (a) If E is the Carlitz-module over A D Fq Œt then H1 .E/ is a pure Q-Hodge–Pink structure of weight 1, because degW H1 .E/ D 1 D degq H1 .E/ and there are no non-trivial Q1 -subspaces of H1 .E/ ˝Q Q1 D Q1 . (b) The same is true for a Drinfeld A-module '. Indeed, assume that H1 .E/ is 0 not locally semistable. Then there is a non-trivial Q1 -subspace H1 H1 with W 1 0 0 degq H 1 > deg H 1 . Since H1 .E/ is pure of weight  r we find degW H 01 D 0  1r  dimQ1 H1 > 1 and degq H 01  0. Since .z  /p q p the same is true  0  0 0 ˝Q1 C[[z  ]]. So degq H 01 can for q D q \ H1 ˝Q1 C((z  )) and p0 D H1 0 0 0 0 only be non-negative if p D p \ q ; that is, p D q0 . This implies 0 HC0 D p0 =.z  /p0 D q0 =.z  /p0 q=.z  /p D ker.HC ! Lie E/ : H1

But ƒ.'/ Lie E is discrete, which by definition means that the natural morphism 0 ! Lie E must be H1 D ƒ.'/ ˝A Q1 ! Lie E is injective. Therefore, also H1 injective and we obtain a contradiction. One of the main results of Pink [Pin97b] is the following Theorem 2.2.10 ([Pin97b, Theorem 9.3]). The category Q-HP together with the Qrational fiber functor !0 W Q-HP ! VecQ , .H; W H; q/ 7! H , is a neutral Tannakian category over Q. See Sect. 2.1.2 for some explanations. Remark 2.2.11. (a) The assertion that Q-HP is abelian rests on the relatively easy fact that in Q-HP any subobject and quotient object is strict. (b) The difficult part of the proof is to show that the condition of local semistability is closed under tensor products. For this it is essential to work with Hodge–Pink lattices instead of Hodge–Pink filtrations. Indeed, if one works with triples .H; W H; F  HC /

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

49

consisting of Q-vector spaces H with weight filtration W H and decreasing Hodge– Pink filtrations F  HC and defines local semistability analogous to Definition 2.2.7, then this local semistability would not be closed under tensor products due to the inseparability of the field extension C=Q; see [Pin97b, Example 5.16]. This is Pink’s ingenious insight. This theorem allows to associate with each Q-Hodge–Pink structure H D .H; W H; q/ an algebraic group H over Q as follows. Consider the Tannakian subcategory hhH ii of Q-HP generated by H . By [DM82, Theorem 2.11 and Proposition 2.20] the category hhH ii is tensor equivalent to the category of Q-rational representations of a linear algebraic group scheme H over Q which is a closed subgroup of GLQ .H /. Definition 2.2.12. The linear algebraic Q-group scheme H associated with H is called the Hodge–Pink group of H . Pink proves that H is connected and reduced and that any connected semisimple group over Q can occur as H for a Q-Hodge–Pink structure [Pin97b, Propositions 9.4 and 9.12]. Note however, that in general H does not even need to be reductive. If the Hodge–Pink structure H comes from a pure (or mixed) uniformizable abelian t-module E, Pink (respectively Pink and the first author) also proved in unpublished work, that H equals the motivic Galois group of E as considered by Papanikolas [Pap08] and Taelman [Tae09a]; see Remark 2.4.26. If H comes from a pure dual A-motive, Pink’s proof was worked out by the second author in her Diploma thesis [Jus10]. We will explain these proofs in Theorems 2.3.34 and 2.4.33 below. In the special case when E is a Drinfeld module, there are further results of Pink on the structure of H ; see Sect. 2.6.

2.3 Mixed A-motives The functor E 7! H1 .E/ from Drinfeld A-modules to Q-Hodge–Pink structures from Examples 2.2.5 and 2.2.9 extends to the uniformizable abelian t-modules of Anderson [And86], the higher dimensional generalizations of Drinfeld-modules. We will define the functor in Sect. 2.5.6 below. In order to prove that H1 .E/ is a pure QHodge–Pink structure when E is a pure uniformizable abelian t-module, we need to review Anderson’s theory of t-motives [And86] or more generally A-motives. We do this first because it also allows to define mixed abelian t-modules and their associated mixed Q-Hodge–Pink structures. 2.3.1 A-Motives. Recall that we denote the natural inclusion Q ,! C by c  and consider the maximal ideal J WD .a ˝ 1  1 ˝ c  .a/ W a 2 A/ AC WD A ˝Fq C. The open subscheme Spec AC X V.J / of CC is affine. We denote its ring of global sections by AC ŒJ 1. For example if C D P1Fq and A D Fq Œt then J D .t  #/ for 1 # WD c  .t/. In this case AC ŒJ 1  D CŒtŒ t # .

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Definition 2.3.1. (a) An A-motive over C of characteristic c  is a pair M D .M; M / consisting of a finite projective AC -module M and an isomorphism of AC ŒJ 1 -modules  M W  M ŒJ 1  ! M ŒJ 1  :

where we set   M ŒJ 1  WD .  M / ˝AC AC ŒJ 1 and M ŒJ 1  WD M ˝AC AC ŒJ 1. A morphism of A-motives f W M ! N is a homomorphism of the underlying AC -modules f W M ! N that satisfies f ı M D N ı   f . The category of A-motives over C is denoted A-Mot. (b) The rank of the AC -module M is called the rank of M and is denoted by rk M . The virtual dimension dim M of M is defined as ı ı dim M WD dimC M .M \ M .  M //  dimC M .  M / .M \ M .  M // : (c) An A-motive .M; M / is called effective if M comes from an AC -homomorphism   M ! M . An effective A-motive has virtual dimension  0. (d) For two A-motives M and N over C we call QHom.M ; N / WD HomA-Mot .M ; N / ˝A Q the set of quasi-morphisms from M to N . (e) The category with all A-motives as objects and the QHom.M ; N / as Hom-sets is called the category of A-motives over C up to isogeny. It is denoted A-MotI. Remark 2.3.2. (a) If C D P1Fq , A D Fq Œt and AC D CŒt, we set # WD c  .t/ and then J D .t  #/. In this case, our effective A-motives are a slight generalization of Anderson’s t-motives [And86], which are called abelian t-motives in [BP20, §4.1]. Namely, Anderson required in addition, that M is finitely generated over the skewpolynomial ring Cf g, where  acts on M through m 7! M .  m/. (b) We will explain in Remark 2.3.7(c) below, that the set of morphisms HomA-Mot .M ; N / between A-motives M and N is a finite projective A-module of rank at most .rk M /  .rk N /. (c) By definition, for every quasi-morphism f 2 QHom.M ; N / there is an element a 2 A X f0g such that a  f 2 HomA-Mot .M ; N / is a morphism of A-Motives. Moreover, ˚ QHom.M ; N / D f W M ˝AC Quot.AC / ! N ˝AC Quot.AC /

such that f ı M D N ı   f ; where Quot.AC / denotes the fraction field of AC and f is a homomorphism of Quot.AC /-vector spaces. Indeed, the inclusion is obvious and the equality was proved in [BH11, Corollary 5.4] and also follows from [Pap08, Proposition 3.4.5] and [Tae09a, Proposition 3.1.2]. Note that this is not equivalent to the inclusion f .J n  M / N for n 0, as can be seen from f D idM ˝ a1 2 QEnd.M / for a 2 A X Fq .

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(d) The name for the category A-MotI stems from the fact that a morphism f W M ! N in A-Mot is an isogeny, that is injective with torsion cokernel, if and only if it becomes an isomorphism in A-MotI; see for example [Har17, Theorem 5.12] or [Tae09a, Proposition 3.1.2]. The tensor product of two A-motives M and N is the A-motive M ˝N consisting of the AC -module M ˝AC N and the isomorphism M ˝ N . The A-motive 1l.0/ with underlying AC -module AC and  D idAC is a unit object for the tensor product in A-Mot and A-MotI. Both categories possess finite direct sums in the obvious way. We also define the tensor powers of an A-motive M as M ˝0 D 1l.0/ and as M ˝n WD M ˝n1 ˝ M for n > 0. The dual of an A-motive M is the A-motive M _ D .M _ ; M _ / consisting of the AC -module M _ WD HomAC .M; AC / and the isomorphism _  M W   M _ ŒJ 1  D HomAC .  M; AC /ŒJ 1 ! M _ ŒJ 1 ;

1 h 7! h ı M :

If M D .M; M / and N D .N; N / are A-motives the internal hom Hom.M ; N / is the A-motive with underlying AC -module H WD HomAC .M; N / and H W  H ŒJ 1   1 ! H ŒJ 1; h 7! N ı h ı M . In particular, M _ D Hom.M ; 1l.0//. Moreover, there a canonical isomorphism ofP A-motives M _ ˝ N Š Hom.M ; N / sending P is _ _ _ i mi ˝ni 2 M ˝AC N to Œm 7! i mi .m/ni  2 HomAC .M; N /. Indeed, this is an isomorphism on the underlying finite locally free AC -modules, and it is obviously compatible with the isomorphisms  . This implies that there are morphisms in A-Mot X X evW M ˝ M _ ! 1l.0/ ; mi ˝ m_ m_ (2.3.1) i 7! i .mi / and i

i

ıW 1l.0/ ! M _ ˝ M D Hom.M ; M / ;

a 7! a  idM ;

(2.3.2)

which satisfy the conditions of Definition 2.1.4(a). We also note the following formulas for the rank and the virtual dimension rk 1l.0/ D 1 ; rk Hom.M ; N / D .rk M /  .rk N / ; rk M _ D rk M ;

dim 1l .0/ D 0 ; dim Hom.M ; N / D .rk M /  .dim N /  .rk N /  .dim M / ; dim M _ D  dim M ;

(2.3.3)

rk M ˝ N D .rk M /  .rk N / ;

dim M ˝ N D .rk N /  .dim M / C .rk M /  .dim N / ;

rk M ˚ N D .rk M / C .rk N / ;

dim M ˚ N D .dim M / C .dim N / ;

which follow easily from the elementary divisor theorem. Proposition 2.3.3. Let f W M ! M 0 be a morphism of A-motives. (a) Then     ker f WD ker f; M jker f ŒJ 1  and im f WD im f; M 0 jim f ŒJ 1  are A-motives, which are called the kernel, respectively image A-motive of f .

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(b) Let N D M 0 =f .M / and let Ntors N be the AC -torsion submodule. Then M 0  induces an isomorphism N=Ntors W   .N=Ntors/ŒJ 1 ! N=Ntors ŒJ 1 and coker f WD .N=Ntors; N=Ntors /

and

coim f WD ker.M 0 ! coker f /

are A-motives, which are called the cokernel, respectively coimage A-motive of f . The A-motive coim f equals the saturation fm0 2 M 0 W 9 h 2 AC ; h ¤ 0 with h  m0 2 f .M /g of im f and the natural inclusion im f ,! coim f is an isogeny, and hence an isomorphism in A-MotI. In particular, rk.im f / D rk.coim f /. Proof. Since AC is a Dedekind domain, the kernel and image of f and N=Ntors are again finite, locally free AC -modules and therefore A-motives with the inherited isomorphism  . That coim f is the saturation of im f follows from the definition of Ntors . Therefore, the inclusion im f ,! coim f is injective with torsion cokernel, hence an isogeny and an isomorphism in A-MotI by [Har17, Theorem 5.12] or [Tae09a, Proposition 3.1.2].  Proposition 2.3.4. The category A-MotI is a Q-linear (non-neutral) Tannakian category, and in particular, a rigid abelian tensor category.   Proof. Since   -invariants in AC equal A, we have EndA-Mot 1l.0/ D A and  the EndA-MotI 1l.0/ D Q. In particular, A-MotI is a Q-linear tensor category. If f W M ! N is a morphism in A-MotI we may multiply f with an element of Q and assume that f is a morphism in A-Mot. Therefore, it follows from Proposition 2.3.3 that A-MotI is abelian. To show that A-MotI is Tannakian we use the morphisms (2.3.1) and (2.3.2). In addition, we have to exhibit an exact faithful Q-linear fiber functor over some nonzero Q-algebra. For example, we can take the quotient field Quot.AC / of AC and the functor M D .M; M / 7! M ˝AC Quot.AC /. This functor is faithful, because M f

g

M ˝AC Quot.AC /. Moreover, it is exact, because a sequence 0 ! M 0 ! M ! M 00 ! 0 in A-MotI is exact if and only if f is injective, im f Š ker g, and im g Š M 00 in A-MotI. By the definition of morphisms in A-MotI as quasi-morphisms, these isomorphisms are in general not isomorphisms of the underlying AC -modules,  but they provide isomorphisms of the associated Quot.AC /-vector spaces. Remark 2.3.5. (a) Fiber functors over C, respectively Qv , are also provided by the de Rham cohomology realization H1dR .M ; C/, respectively the v-adic cohomology realization H1v .M ; Qv /; see Sect. 2.3.5. A neutral fiber functor only exists on the full subcategory of uniformizable A-motives; see Theorem 2.3.27 (b) The category A-Mot is an exact category in the sense of Quillen [Qui73, §2] if one f

defines the class E of short exact sequences to be those sequences 0 ! M 0 ! g

M ! M 00 ! 0 of A-motives whose underlying sequence of AC -modules is

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exact. Then f (respectively g/ is called an admissible monomorphism (respectively admissible epimorphism). Indeed, this means that f is the kernel of g and g is the cokernel of f in A-Mot, that every canonical split sequence 0 ! M 0 ! M 0 ˚ M 00 ! M 00 ! 0 lies in E, that E is closed under isomorphisms, pullbacks via morphisms N 00 ! M 00 and pushout via morphisms M 0 ! N 0 , and that the composition of admissible monomorphisms is an admissible monomorphism and the composition of admissible epimorphisms is an admissible epimorphism. All this is straight forward to prove. Moreover, with the analogous definition of E, also the subcategories of A-Mot consisting of A-motives which are effective, respectively effective and finitely generated over Cf g, are exact. Example 2.3.6. An effective A-motive of rank 1 with M .  M / D J  M is called a Carlitz–Hayes A-motive. It has virtual dimension 1. Carlitz–Hayes A-motives can be constructed as follows. Let P 2 CC be a (C-valued) point whose projection onto C is the point 1 2 C . (Under our assumption 1 2 C.Fq / there is a unique such point.) The divisor   .V.J //  .P / on CC has degree zero and induces a line bundle O .V.J //.P / . Since the endomorphism id  Frobq of the abelian variety Pic0C=Fq   is surjective, there is a line bundle L of degree zero on CC with O .V.J //  .P / D .id  Frobq /.L/ D L ˝  L_ in Pic0C=Fq .C/. The AC -module M WD .Spec AC ; L/   is locally free of rank one and the isomorphism   L Š L ˝ O .P /  .V.J // of  M ŒJ 1  with M .  M / D line bundles yields an isomorphism M W  M ŒJ 1  ! J  M . So M D .M; M / is a Carlitz–Hayes A-motive. If M is a Carlitz–Hayes A-motive and M 0 is any A-motive of rank 1, then M 0  . M 0 / D J d  M 0 for a uniquely determined integer d by the elementary divisor theorem. Under our assumption that 1 is Fq -rational, we claim that M 0 is isogenous to M ˝d . So in particular all Carlitz–Hayes A-motives are isomorphic in the category A-MotI. Namely, consider the A-motive N WD M 0 ˝ .M ˝d /_ of rank one. It  satisfies N W   N ! N and its  -invariants N0 WD ff 2 N W N .  f / D f g form a locally free A-module of rank one with N Š N0 ˝A 1l.0/. Indeed, one can extend N to a locally free sheaf N on CC of degree zero and, by reasons of  N at the one missing point degree, N will extend to an isomorphism N W  N ! 1C D CC X Spec AC . This means that the element N 2 Pic0C=Fq .C/ arises from an Fq -rational point N 0 of Pic0C=Fq . It follows that N Š N0 ˝A 1l.0/ for N0 WD .Spec A; N 0 / as claimed. Now the A-module N0 is isomorphic to an ideal of A which we again denote by N0 . Tensoring the inclusion N0 ,! A with M ˝d yields the desired isogeny M 0 Š N0 ˝A M ˝d ,! M ˝d . We may therefore denote any Carlitz–Hayes A-motive by 1l.1/. We also define 1l.n/ WD 1l.1/˝n for n  0 and 1l.n/ D 1l.n/_ for n  0. Then dim 1l.n/ D n. In the special case where C D P1Fq ; A D Fq Œt and # WD c  .t/ 2 C, all Carlitz–Hayes A-motives are already in A-Mot isomorphic to the Carlitz t-motive M D .M; M / with M D CŒt and M D t  #, because in this case the A-module N0 is free and isomorphic to A.

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Remark 2.3.7. (a) Every A-motive is isomorphic to the tensor product of an effective A-motive and a power of a Carlitz–Hayes A-motive. In fact, if M is an A-motive with M . M / J d  M Then M 0 WD M ˝ 1l.1/˝d satisfies M 0 .  M 0 / M 0 ; hence, M 0 is effective and M Š M 0 ˝ 1l.1/˝d . Note that rk M 0 D rk M and dim M 0 D dim M C d  rk M . (b) This implies that for A D Fq Œt the category A-Mot is equivalent to Taelman’s category t MC of t-motives [Tae09a, Def. 2.3.2] and A-MotI is equivalent to Taelman’s category t MoC of t-motives up to isogeny [Tae09a, §3]. Indeed, Taelman defines t MC as the category of effective A-motives with the formally adjoined inverse of a Carlitz–Hayes A-motive. (c) Let us explain why the set of morphisms HomA-Mot .M ; N / between A-motives M and N is a finite projective A-module of rank at most .rk M /  .rk N /. By (a) we may write M Š M 0 ˝ 1l.1/˝d and N Š N 0 ˝ 1l.1/˝d for effective A-motives M 0 and N 0 . Then HomA-Mot .M ; N / Š HomA-Mot .M 0 ; N 0 / and for the latter the statement was proved by [And86, Corollary 1.7.2]. 2.3.2 Purity and mixedness. We fix a uniformizing parameter z 2 Q D Fq .C / of C at 1. For simplicity of the exposition we assume that 1 2 C.Fq /. The main results we present here hold, and are in fact proved in [Pin97b, HP18], without this assumption. The assumption implies that there is a unique point on CC above 1 2 C , which we call 1C . The completion of the local ring of CC at 1C is canonically isomorphic to C[[z]]. Definition 2.3.8. (a) An A-motive M D .M; M / is called pure if M ˝AC C((z)) contains a C[[z]]-lattice M1 such that for some integers d; r with r > 0 the r map M WD M ı   .M / ı : : : ı  r1 .M /W  rM ! M induces an iso r W  r M1 ! M1 . In this case the weight of M is defined as morphism z d M d wt M D r . (b) An A-motive M is called mixed if it possesses an increasing weight filtration by saturated A-sub-motives W M for  2 Q (i.e. W M M is a saturated S AC -submodule) such that all graded pieces GrW  M WD W M = 0  Q for  0, and that W M D M for  0; compare Remark 2.2.2. (b) Every pure A-motive of weight  is also mixed with W0 M D .0/ for 0 < , and W0 M D M for 0  , and GrW  M D M. To explain this definition we use the notion of z-isocrystals over C; see [HK20, M ; M Definition 5.1]. These are defined to be pairs c M D .c b / consisting of a finite di c together with a C((z))-isomorphism  W   c mensional C((z))-vector space M M ! b M c M . They are also called Dieudonné-Fq ((z))-modules in [Lau96, § 2.4] and local isoshtukas in [BH11, § 8]. Some of the following results were proved by Taelman [Tae09a, Tae09b]. Proposition 2.3.11. Let M D .M; M / be an A-motive and consider the z-isocrystal   c is isomorphic to c WD M ˝AC C((z)) D M ˝AC C((z)); M ˝ id . Then M M L d c i M di ;ri where for d; r 2 Z; r > 0; .d; r/ D 1 and m WD d r e we set 0 cd;r WD M



˚r

C((z))

;  D d;r

z m

0

B B B WD B 1m Bz @

z 1m

1 C z m C C C C A 0

(2.3.4)

and where in the matrix the term z 1m occurs exactly mr  d times. In particular, (a) M is pure of weight  if and only if

di ri

D  for all i .

M WD (b) M is mixed if and only if the filtration W c

M di ri

cd ;r comes from M i i



e  M M with W c a filtration of M by saturated A-sub-motives W M D e e .W  M / ˝AC C((z)). In this case the filtration W  M equals the weight filtration W M of M and the dr i are the weights of M . In particular, the weight i filtration of a mixed A-motive M is uniquely determined by M . (c) Any A-sub-motive M 0 ,! M and A-quotient motive f W M  M 00 of a pure (mixed) A-motive M is itself pure (mixed) of the same weight(s), (by letting W M 0 WD M 0 \ W M , and letting W M 00 be the saturation of f .W M / inside M 00 , if M is mixed).

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(d) Any A-motive which is isomorphic in A-MotI to a pure (mixed) A-motive is itself pure (mixed). (e) The weight of a pure A-motive M is wt M D .dim M /=.rk M /. The tensor product of two pure A-motives M and N is again pure of weight .wt M / C .wt N /. (f) The category A-MMotI is a full Q-linear (non-neutral) Tannakian subcategory of A-MotI, and in particular, a rigid abelian tensor category. (g) Any morphism f W M 0 ! M between mixed A-motives satisfies f .W M 0 / W M . More precisely, the saturation of f .W M 0 / inside f .M 0 / equals f .M 0 / \ W M . Remark. We do not know whether in (g) the submodule f .W M 0 / f .M 0 / is always saturated, that is, whether the equality f .W M 0 / D f .M 0 / \ W M always holds. Proof. The fact that over the algebraically closed field C any z-isocrystal is isomorphic to a direct sum of standard ones is proved in [Lau96, Theorem 2.4.5]. It is analogous to the Dieudonné–Manin classification of F -isocrystals over an algebraically closed field of positive characteristic [Man63]. That the standard z-isocrystals in cd;r follows by an elementary compu[Lau96] are isomorphic to our standard ones M tation. c˚.rk M /=r with  D d and .d; r/ D 1, we can take for M1 the tautocŠM (a) If M r

d;r

˚.rk M /=r

c logical C[[z]]-lattice C[[z]]˚ rk M inside M d;r if there is an i with  D

d r

¤

di ri

phism for any C[[z]]-lattice M1 [Tae09a, Proposition 5.1.4].)

to see that M is pure. Conversely,

r r z ri d Mi

, then D z ri d z di r cannot be an isomorc. So M is not pure of weight . (Compare in M

(b) If M has a weight filtration W M M with respect to which it is mixed, then W c˚.rk Gr M /=r for  D d with .d; r/ D 1. (a) implies that .GrW  M / ˝AC C((z)) Š M d;r r Since the category of z-isocrystals is semi-simple by [Lau96, Theorem 2.4.5] the sequences 0

/.

S

0 0 and .di ; ri / D 1 for all i ; see Proposition 2.3.11. (a) If M is effective and M is a finitely generated module over the skew-polynomial ring Cf g, where  acts on M through m 7! M .  m/, then di > 0 for all i .

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(b) If di =ri  n for all i and M .  M / J d M for to  d 2 Z, then M extends  a locally freesheaf M on CC with  W  M ! M n  1C C d  V.J / , where the notation n  1C C d  V.J / means that we allow poles at 1C of order less than or equal to n and at V.J / of order less than or equal to d . (c) If M is not necessarily effective, then M is pure of weight  D dr with .d; r/ D r 1 if and only if M extends to a locally free sheaf M on CC such that z d M is  r an isomorphism  M 1 ! M 1 on the stalks at 1C . d

Proof. (a) (compare [Tae09b, Proposition 8]) We may assume that dr i  r i C1 for i i C1 all i . By the explicit description of c M di ;ri in (2.3.4) there is a C((z))-basis B of c and an integer s > 0 such that .M ˝ id/s is a diagonal matrix with entries M as z sdi =ri with respect to B . Assume that d1  0. Since M is finitely generated P a Cf g-module there are finitely many elements mi 2 M such that M D i;j 0 C  sj c D M ˝AC C((z)) the set M Œz D M ˝AC AC Œz D M . sj  mi /. By definition of M P k sj sj  c. We write mi with respect mi / is z-adically dense in M i;j;k0 C  z  . M

to the basis B as a vector .mi;1; : : : ; mi;r /T 2 C((z))˚r . Then the first coordiP sj nates of the elements of M Œz have the form i;j;k0 bi;j;k z k M . sj mi;1 / D P ksjd1 =r1 sj   mi;1 for bi;j;k 2 C. Since k  sjd1 =r1  0 for all i;j;k0 bi;j;k z j and k, all these terms lie in z N C[[z]] for a suitable N 2 Z. In particular, elements c with first coordinate outside z N C[[z]] can not belong to the z-adic closure of of M M Œz. This contradiction shows that our assumption was false and d1 > 0. (c) If the described extension M of M exists, then M ˝OCC C[[z]] is a C[[z]]-lattice r is an isomorphism and M is pure of weight dr by Definiinside c M on which z d M tion 2.3.8(a). We prove (b) and the remaining implication of (c). Since mi WD d dr i e  n for i

all i , we can define M by requiring that M ˝OCC C[[z]] is equal to the sum of the cd ;r in (2.3.4). Then M has the desired tautological C[[z]]-lattices C[[z]]˚ri inside M i i properties.  S n .M / is z-adically Proposition 2.3.15. If di > 0 for all i , then the set n2N>0 L n M c dense in M WD M ˝AC C((z)). Proof. We choose a finite flat inclusion Fq Œt ,! A and set zQ WD 1t . Then M is c D M ˝CŒt  C((z)). Q We a finite (locally) free CŒt-module, say of rank r and M choose a CŒt-basis B of M . By Proposition 2.3.11 there is a C((z))-isomorphism L b be the C((z))-basis c obtained from the standard basis c Q of M M Š ic M di ;ri . We let B of the c M di ;ri given by (2.3.4) and from the choice of a C[[z]]-basis Q of C[[z]]. The base   b and B is given by a matrix U 2 GLr C((z)) Q . There is an integer change between B rr 1 N N  0 such that U; U 2 zQ C[[Qz ]] . By our assumption di > 0 and the explicit

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cd ;r from (2.3.4) there is a positive integer s such that the matrix T form of the M i i s b lies in zQ 2N C1 C[[z]] with respect to the basis B Q rr . Therefore, the representing M s with respect to the basis B lies in zQ C[[Qz ]]rr . matrix  s .U /T U 1 representing M Now the proposition is a consequence of the following  s Lemma 2.3.16. If M 2 zQ C[[z]] Q rr . Then for all x 2 C((z)) Q r and for all n 2 N0 sn there exists a y 2 CŒtr such that x  L sn M .y/ 2 zQ nC1 C[[z]] Q r.

P P sn . sn .x// D i bi zQ i with bi 2 Cr and set y WD i 0 bi t i 2 Proof. Write M   P sn sn .y/ D L sn M Q i 2 zQ nC1 C[[z]] Q r because CŒtr . Then x  L sn M i >0 bi z P r i Q 2 zQ C[[z]] Q .  i >0 bi z 2.3.3 Uniformizability. In order to define the notion of uniformizability (also called rigid analytic triviality) for A-motives we have to introduce some notation of rigid analytic geometry as in [HP04, HP18]. See [Bos14] or [BGR84] for a general introduction to rigid analytic geometry. With the curve CC and its open affine part . C C one can associate by [BGR84, §9.3.4] rigid analytic spaces CC WD .CC /rig and . . CC WD .C C /rig D CC X f1C g where, using our convention that the point 1 2 C is Fq -rational, 1C 2 CC is the unique point above 1 2 C . By construction, the under. . lying sets of CC and CC are the sets of C-valued points of CC and C C , respectively. For any open rigid analytic subspace U CC we let O .U/ WD .U; OU / denote the ring of regular functions on U. The endomorphism  of CC induces endomorphisms . of CC and CC which we denote by the same symbol  . Let OC be the valuation ring of C and let C be its residue field. By the valuative criterion of properness every point of CC D CC .C/ D C.C/ extends uniquely to an OC -valued point of C and in the reduction gives rise to a C -valued point of C . This yields a reduction map red W CC ! C. C /. The curve CC is non-singular and, due to our convention 1 2 C.Fq /, the subscheme f1g Spec Fq Spec C CC consists of a single point which we call 1C . So [BL85, Proposition 2.2] implies that the preimage DC of 1C under red is an open rigid analytic unit disc in CC around 1C . Without the convention 1 2 C.Fq / the subscheme f1g Spec Fq Spec C CC decomposes into finitely many points and . there is a corresponding disc for each one of them; see [HP18, § 11]. Let further DC D DC Xf1C g be the punctured open unit disc around 1C in CC . By [BL85, Proposition 2.2] both discs have z as a coordinate. By Lemma 2.1.3 the power series ring C[[z  ]] is also canonically isomorphic to the completion of the local ring of CC at the closed point V.J /, respectively of DC . and DC at the point fz D g 2 DC . The complement CC X DC of DC in CC equals . the preimage of the open affine curve C C under the reduction map red and is hence affinoid. For example, if C D P1Fq and A D Fq Œt we can give the following explicit description

O.CC X DC / D Chti WD

1 ˚X i D0

bi t i W bi 2 C; lim jbi j D 0 i !1



(2.3.5)

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and CC X DC D Sp Chti is the closed unit disc inside C.C/ X f1C g D C on which the coordinate t has absolute value less or equal to 1. Also we can take z D 1t as the coordinate on the disc DC . For general C we may choose an element a 2 A X Fq and consider the finite flat morphisms Fq Œt ! A and CŒt ! AC which send t to a. Then O .CC X DC / D AC ˝CŒt  Chti and CC X DC D Sp.AC ˝CŒt  Chti/. . . The spaces CC , DC and DC are quasi-Stein spaces in the sense of Kiehl [Kie67, §2]. In particular, the global section functors are equivalences between the categories of locally free coherent sheaves on these spaces and the categories of finitely generated projective modules over their rings of global sections; see Gruson [Gru68, Chapter V, Theorem 1 and Remark on p. 85]. Definition 2.3.17. For an A-motive M , we define the  -invariants   ƒ.M / WD M ˝AC O .CC X DC / ˚

WD m 2 M ˝AC O .CC X DC / W M .  m/ D m : We also set H1 .M / WD ƒ.M / ˝A Q. Since the ring of   -invariants in O .CC X DC / is equal to A, the set ƒ.M / is an A-module. By [BH07, Lemma 4.2(b)], it is finite projective of rank at most equal to rk M . Therefore, also H1 .M / is a finite dimensional Q-vector space. Definition 2.3.18. An A-motive M is called uniformizable (or rigid analytically trivial) if the natural homomorphism hM W ƒ.M / ˝A O .CC X DC / ! M ˝AC O .CC X DC / ;

 ˝ f 7! f  ;

is an isomorphism. The full subcategory of A-MotI consisting of all uniformizable A-motives is denoted A-UMotI. The full subcategory of A-MMotI consisting of all uniformizable mixed A-motives is denoted A-MUMotI. Remark 2.3.19. If A D Fq Œt, then the category A-UMotI is canonically equivalent to the category t Mıa:t: of Taelman [Tae09a, Def. 3.2.8] in view of Remark 2.3.7.   Example 2.3.20. (a) 1l.0/ D .AC ;  D idAC / is uniformizable, because ƒ 1l.0/ D A and A ˝A O .CC X DC / D AC ˝AC O .CC X DC /. (b) Let C D P1Fq , A D Fq Œt, z D

# WD c  .t/ D

2 C. The Carlitz t-motive Q qi WD 1 M D .AC ; M D t  #/ is uniformizable. Namely, we set ` i D0 .1   t/ 2  . O.CC / and choose an element  2 C with q1 D . Then ` 2 ƒ.M / X f0g. has no zeroes outside DC it generates the O .CC X DC /-module M ˝AC Since `  O.CC X DC / D O.CC X DC / and so hM is an isomorphism and M is uniformizable. 1 , t

1 

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Urs Hartl, Ann-Kristin Juschka

The following criterion for uniformizability is well known. Lemma 2.3.21. Let M be an A-motive of rank r. (a) The homomorphism hM is injective and satisfies hM ı .idƒ.M / ˝ id/ D .M ˝ id/ ı   hM . (b) M is uniformizable if and only if rkA ƒ.M / D r. Proof. Assertion (a) follows for example from [BH07, Lemma 4.2(b)], and assertion (b) from [BH07, Lemma 4.2(c)].  Lemma 2.3.22. Let C D P1Fq , A D Fq Œt, AC D CŒt and # D c  .t/. Then O .CC X    1  represent M with DC D Chti; see (2.3.5). Let ˆ D .ˆij /ij 2 GLr CŒtŒ t # P respect to a CŒt-basis B D .m1 ; : : : ; mr / of M , that is M .  mj / D riD1 ˆij mi . Then M is uniformizable if and only if there is a matrix ‰ 2 GLr .Chti/ such that  ‰T D ‰T  ˆ ; In that case, ‰ is called a rigid analytic trivialization of ˆ. It is uniquely determined up to multiplication on the right with a matrix in GLr .Fq Œt/. The columns of .‰ T /1 are the coordinate vectors with respect to B of an Fq Œt-basis C of ƒ.M /. Moreover, with respect to the bases C and B the isomorphism hM is represented by .‰ T /1 . Remark 2.3.23. Here .: : :/T denotes the transpose matrix. The matrix ‰ will turn out to be Anderson’s scattering matrix and this is the reason why we work with ‰ T here; see Remark 2.5.34 below. Proof of Lemma 2.3.22. Assume that M is rigid analytically trivial and choose an Fq Œt-basis C of ƒ.M /. Let .‰ T /1 be the matrix representing the isomorphism  M ˝CŒt  Chti with respect to the bases C and B . Then hM W ƒ.M / ˝Fq Œt  Chti ! ˆ    .‰ T /1 D .‰ T /1 and ‰ 2 GLr .Chti/ is a rigid analytic trivialization. Conversely, if there is a rigid analytic trivialization ‰, then the columns of .‰ T /1 provide a Chti-basis of M ˝CŒt  Chti, with respect to which M is represented by the identity matrix ‰ T ˆ   .‰ T /1 D Idr . Therefore, the columns of .‰ T /1 form an Fq Œt-basis C of ƒ.M / and hM is represented with respect to the bases C and B by .‰ T /1 . Therefore, hM is an isomorphism and M is uniformizable.  Before we can conclude that A-UMotI and A-MUMotI are neutral Tannakian categories over Q with fiber functor M 7! H1 .M /, we need to state the following Proposition 2.3.24. (a) Every A-motive which is isomorphic to a uniformizable A-motive in A-MotI is itself uniformizable. (b) Every A-motive of rank 1 is uniformizable.

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(c) If M and N are uniformizable A-motives, then also M ˝ N and Hom.M ; N / and M _ are uniformizable with and ƒ.M ˝ N / Š ƒ.M / ˝A ƒ.N /     ƒ Hom.M ; N / Š HomA ƒ.M /; ƒ.N / and ƒ.M _ / Š HomA .ƒ.M /; A/ : (d) If M and N are uniformizable, the natural map QHom.M ; N / ! HomQ .H1 .M /; H1 .N //,   f ˝ a 7! H1 .f ˝ a/ WD a  hN 1 ı .f ˝ id/ ı hM jH1 .M / for f 2 HomA-Mot .M ; N / and a 2 Q, is injective.  N be an isomorphism of Proof. (a) Let M be uniformizable and let f W M ! A-motives in A-MotI. By multiplying f with an element of A we can assume that f W M ,! N is an A-sub-motive in A-Mot. Then f W ƒ.M / ,! ƒ.N / and rk M D rkA ƒ.M /  rkA ƒ.N /  rk N D rk M . So N is uniformizable by Lemma 2.3.21(b).

(b) is proved in [HP18, Propositions 12.3(b) and 12.5]. In the special case where C D P1Fq , A D Fq Œt and # D c  .t/ 2 C, assertion (b) follows from (c) and from Examples 2.3.6 and 2.3.20, because all t-motives of rank 1 are tensor powers of the Carlitz t-motive .CŒt; t  #/. (c) If M and N are uniformizable, then hM and hN induce an isomorphism hM ˝hN

ƒ.M / ˝A ƒ.N / ˝A O .CC X DC / ! M ˝AC N ˝AC O .CC X DC / satisfying .hM ˝ hN / ı .idƒ.M / ˝ idƒ.N / ˝ id/ D .M ˝ N ˝ id/ ı   .hM ˝ hN /. Therefore, the  -invariants are   ƒ.M ˝ N / D M ˝AC N ˝AC O .CC X DC / Š ƒ.M / ˝A ƒ.N / : Likewise, by applying [Eis95, Proposition 2.10], the uniformizability of M yields an isomorphism .hM _ /1

HomA .ƒ.M /; A/ ˝A O .CC X DC / ! M _ ˝AC O .CC X DC / satisfying .hM _ /1 ı .idHomA .ƒ.M /;A/ ˝ id/ D .M _ ˝ id/ ı   .hM _ /1 . Therefore, the  -invariants are   ƒ.M _ / D M _ ˝AC O .CC X DC / Š HomA .ƒ.M /; A/ : From this also the statement about Hom.M ; N / Š N ˝ M _ follows. (d) Since hM and hN are isomorphisms, f ˝a can be recovered from H1 .f ˝a/.



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Lemma 2.3.25. Let 0 ! M 0 ! M ! M 00 ! 0 be a short exact sequence of Amotives. Then M is uniformizable if and only if both M 0 and M 00 are. In this case the induced sequence of A-modules 0 ! ƒ.M 0 / ! ƒ.M / ! ƒ.M 00 / ! 0 is exact. Proof. The first assertion follows from Anderson [And86, Lemma 2.7.2 and 2.10.4].  0 / D ker id  W M 0 ˝AC O .CC X DC / ! For the second assertion observe that ƒ.M  0 M ˝AC O .CC X DC / . Since the map id  is surjective by [BH07, Proposition 6.1] the snake lemma proves the exactness of the sequence 0 ! ƒ.M 0 / ! ƒ.M / ! ƒ.M 00 / ! 0.  Remark 2.3.26. If a mixed A-motive M is uniformizable, then all filtration steps W M and factors GrW  M of the weight filtration of M are uniformizable by Lemma 2.3.25. Therefore M could equivalently be called a uniformizable mixed A-motive or a mixed uniformizable A-motive. Theorem 2.3.27. The category A-UMotI of uniformizable A-motives up to isogeny and its rigid tensor subcategory A-MUMotI of uniformizable mixed A-motives up to isogeny are neutral Tannakian categories over Q with fiber functor M 7! H1 .M /. Proof. By Propositions 2.3.24 and 2.3.11(f), A-UMotI and A-MUMotI are closed under taking tensor products, internal homs and duals, contain the unit object 1l.0/ for the tensor product, and M 7! H1 .M / is a faithful Q-linear tensor functor, which is exact by Lemma 2.3.25. Moreover, H1 .M / is finite-dimensional for any uniformizable A-motive M by Lemma 2.3.21(b). As strictly full subcategories of the Q-linear abelian category A-MotI also A-UMotI and A-MUMotI are Q-linear. Let f W M ! N be a morphism in A-UMotI. Then the kernel, cokernel, image and coimage of f in A-MotI are uniformizable by Lemma 2.3.25 and belong to A-UMotI.  Therefore, A-UMotI and A-MUMotI are abelian. This theorem allows to associate with each (mixed) uniformizable A-motive M an algebraic group M over Q as follows. Consider the Tannakian subcategory hhM ii of A-UMotI, respectively A-MUMotI generated by M . By Tannakian duality [DM82, Theorem 2.11 and Proposition 2.20], the category hhM ii is tensor equivalent to the category of Q-rational representations of a linear algebraic group scheme M over Q which is a closed subgroup of GLQ .H1 .M //. Definition 2.3.28. The linear algebraic Q-group scheme M associated with M is called the (motivic) Galois group of M . Example 2.3.29. The trivial A-motive 1l.0/ has trivial motivic Galois group 1l.0/ D .1/. For any A-motive 1l.n/ of rank 1 with n ¤ 0 (see Example 2.3.6) the motivic  1 . Indeed, since H Š Q, the group 1l.n/ 1 l.n/ Galois group equals 1l .n/ D Gm;Q  1 is a subgroup of GLQ H .1l.n// D Gm;Q . If it were a finite group, it would be annihilated by some positive integer d . This implies that it operates trivially on 1l.n/˝d Š 1l.d n/ 2 hh1l.n/ii. Therefore, 1l.d n/ must be a direct sum of the trivial object 1l.0/, that is 1l.d n/ Š 1l.0/, which is a contradiction.

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Pink’s theory of Hodge structures and the Hodge conjecture over function fields

2.3.4 The associated Hodge–Pink structure. We associate a mixed Q-Hodge–Pink structure with every uniformizable mixed A-motive. Note that (a variant of) this is used by Taelman [Tae20] in this volume to study 1-t-motives. i For i 2 N0 we consider the pullbacks  i  J D .a ˝11˝c .a/q W a 2 A/ AC . . i qi and . the points V. J / of C C and CC . They correspond. to theSpoints V.zi   / 2 DC and have 1C as accumulation point. Therefore, CC X i 2N0 V. J / is an . admissible open rigid analytic subspace of CC . Proposition 2.3.30. Let M be a uniformizable A-motive over C. ˚  . S (a) Then ƒ.M / equals m 2 M ˝AC O CC X i 2N0 V. i J / W M .  m/ D

m and the isomorphisms hM and   hM extend to isomorphisms of locally free sheaves hM W ƒ.M / ˝A OC. C XSi 2N

0

V. i  J /

 ! M ˝AC OC. C XSi 2N

0

V. i  J /

;

 ˝ f 7! f   ;   hM W ƒ.M / ˝A OC. C XSi 2N

>0

V. i  J /

 !   M ˝AC OC. C XSi 2N

>0

V. i  J /

;



 ˝ f 7! f    ; satisfying hM ı .idƒ.M / ˝ id/ D .M ˝ id/ ı   hM . ˚ . (b) If moreover M is effective, then ƒ.M / equals m 2 M ˝AC O .CC / W M .  m/

D m and the isomorphism hM extends to an injective homomorphism hM W ƒ.M / ˝A OC. C ! M ˝AC OC. C ;

 ˝ f 7! f  ;

with hM ı .idƒ.M / ˝ id/ D .M ˝ id/ ı   hM . At the point V.J / its cokernel satisfies coker hM ˝ C[[z  ]] D M=M . M /. Proof. (b) If M is effective, the claimed equality for ƒ.M / was proved in [BH07, Proposition 3.4]. This allows to extend hM to a homomorphism . . hM W ƒ.M / ˝A O .CC / ! M ˝AC O .CC / ;  ˝ f 7! f   : which satisfies hM ı .idƒ.M / ˝ id/ D .M ˝ id/ ı   hM and is injective because . O.CC / O.CC X DC /. Let D WD coker hM and consider the following diagram, in which the first row is exact because of the flatness of   ; see Remark 2.1.1. 0

/ ƒ.M / ˝A   O .C. C /

idƒ.M / ˝ id 0

  hM

Š

 / ƒ.M / ˝A O .C. C /

hM

/   M ˝A O .C. C / C _

/  D

M ˝ id

D

 / M ˝A O .C. C / C

 /D

/0

/0

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Urs Hartl, Ann-Kristin Juschka

By the snake lemma, D is injective and coker M Š coker D . The support of D is contained in DC . So we now look at the points in DC and use z as a coordinate on DC . Let ˛ ¤  and consider the point fz D ˛g in DC . Since fz D ˛g ¤ V.J /  1  and coker M is supported at V.J /, we find   D ˝ C[[z  ˛ q ]] D . D/ ˝ . C[[z  ˛]] Š D ˝ C[[z  ˛]]. Since the support of D is discrete on CC it cannot have a limit point on the affinoid space CC X fP 2 DC W jz.P /j < jjg. This S i implies D ˝ C[[z  ˛]] D .0/ for all ˛ … i 2N0 f q g and proves that hM is an S isomorphism outside i 2N0 V. i  J /. Moreover, .  D/ ˝ C[[z  ]] D .0/ and coker M D coker M ˝ C[[z  ]] Š D ˝ C[[z  ]], and so   hM is an isomorphism S outside i 2N>0 V. i  J /. (a) If M is not effective, then M is isomorphic to N ˝ 1l.n/ by Remark 2.3.7 for an effective A-motive N and some positive integer n. By Proposition 2.3.24 the Amotives 1l.n/ and N Š M ˝ 1l.n/ are uniformizable. Since N and 1l.n/ are effective, our proof of (b) yields isomorphisms hN W ƒ.N / ˝A OC. C XSi 2N

0

V. i  J /

h1l.n/ W ƒ.1l.n// ˝A OC. C XSi 2N

0

 ! N ˝AC OC. C XSi 2N

V. i  J /

0

and

V. i  J /

 ! 1l.n/ ˝AC OC. C XSi 2N

0

V. i  J /

:

Dualizing and inverting the second isomorphism and tensoring with the first yields the isomorphism 1 W ƒ.N / ˝A ƒ.1l.n//_ ˝A OC. C XSi 2N hN ˝ .h_ 1l.n/ /

0

V. i  J /

 ! N ˝AC 1l.n/_ ˝AC OC. C XSi 2N

0

V. i  J /

which satisfies 1 ı .idƒ.N / ˝ idƒ.1l.n// ˝ id/ hN ˝ .h_ 1l.n/ / 1 D .N ˝ .1_l.n/ /1 ˝ id/ ı   .hN ˝ .h_ /: 1l.n/ /

Combined with the isomorphisms N ˝AC 1l.n/_ Š M and ƒ.M / Š ƒ.N / ˝A ƒ.1l.n//_ , this yields the desired extension of hM ƒ.M / ˝A OC. C XSi 2N and proves ƒ.M / D

m .

˚

0

V. i  J /

 ! M ˝AC OC. C XSi 2N

0

V. i  J /

 . S m 2 M ˝AC O CC X i 2N0 V. i  J / W

M .  m/ D 

Corollary 2.3.31. In the situation of Lemma 2.3.22 let ‰ 2 GLr .Chti/ be a rigid analytic trivialization of ˆ. Then the entries of ‰ and ‰ 1 converge for all t 2 C with jtj < j#j. If M is effective, then the entries of ‰ 1 even converge for all t 2 C.

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

67

Proof. In view of J D .t  #/ this follows from the fact that hM is represented by the matrix .‰ T /1 .  Proposition 2.3.30 implies that   hM is an isomorphism locally at V .J / D . fz D g DC . This allows us to associate a Q-pre Hodge–Pink structure with any uniformizable mixed A-motive as follows. Namely, hM induces isomorphisms ƒ.M / ˝A C((z  ))

  hM ˝idC((z )) Š

idƒ.M / ˝ idC((z )) Š

 ƒ.M / ˝A C((z  ))

/   M ˝A C((z  )) C Š M ˝idC((z ))

hM ˝idC((z )) Š

 / M ˝A C((z  )) : C

(2.3.6) Here hM ˝ idC((z )) is an isomorphism because the three others are. Therefore, the preimage q WD .hM ˝ idC((z )) /1 M ˝AC C[[z  ]] is a C[[z  ]]-lattice in ƒ.M / ˝A C((z  )). is p WD ƒ.M / ˝A C[[z  ]] D   The tautological lattice . hM ˝ idC[[z ]] /1   M ˝AC C[[z  ]] . Definition 2.3.32. Let M be a uniformizable mixed A-motive with weight filtration W M . We set H1 .M / WD .H; W H; q/ with • H WD H1 .M / WD ƒ.M / ˝A Q, • W H WD H1 .W M / D ƒ.W M / ˝A Q H for each  2 Q,   • q WD .hM ˝ idC((z )) /1 M ˝AC C[[z  ]] . We call H1 .M / the Q-Hodge–Pink structure associated with M . (This name is justified by Theorem 2.3.34 below.) We also set H1 .M / WD H1 .M /_ in Q-HP. The functor H1 is covariant and H1 is contravariant in M . Remark 2.3.33. (a) If M D M .E/ is the A-motive associated with a Drinfeld Amodule E, then H1 .M / Š H1 .E/_ DW H1 .E/. We will prove this more generally for a uniformizable pure (or mixed) abelian Anderson A-module E in Theorem 2.5.39 below. (b) We draw some conclusions from the description of q and p WD ƒ.M /˝A C[[z ]] given before the definition: If J m  M .  M / M J n  M .  M / for some integers n  m, then .z  /m p q .z  /n p. For example, if M is effective, that is M .  M / M , then p q and there is an exact sequence of C[[z  ]]-modules hM ˝idC((z ))

0 ! p ! q ! M=M .  M / ! 0 : Note that M=M .  M / is a C[[z  ]]-module because it is annihilated by some power of z  . (c) In terms of Definition 2.2.7 the virtual dimension of M is dim M D degq H1 .M /.

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The following theorem is the main theorem of [HP18]. Theorem 2.3.34. Consider a uniformizable mixed A-motive M . (a) H1 .M / is locally semistable and hence indeed a Q-Hodge–Pink structure. (b) The functor H1 W M ! H1 .M / is a Q-linear exact fully faithful tensor functor from the category A-MUMotI to the category Q-HP. (c) The essential image of H1 is closed under the formation of subquotients; that is, if H 0 H1 .M / is a Q-Hodge–Pink sub-structure, then there exists a uniformizable mixed A-sub-motive M 0 M in A-MUMotI with H1 .M 0 / D H 0 . (d) The functor H1 defines an exact tensor equivalence between the Tannakian subcategory hhM ii A-MUMotI generated by M and the Tannakian subcategory hhH1 .M /ii Q-HP generated by its Q-Hodge–Pink structure H1 .M /. Assertions (c) and (d) are the function field analog of the Hodge Conjecture [Hod52, Gro69b, Del06]. We will prove Theorem 2.3.34 in Sect. 2.7 and discuss its consequences for the Hodge–Pink group H1 .M / in Sect. 2.6. Example 2.3.35. Let C D P1Fq , A D Fq Œt, z D 1t , # D c  .t/ D 1 2 C. Let   t # b ˚2 . Then M D .M; M / is mixed M D AC with M D ˆ WD 0 .t  #/3 W 3 with GrW 1 M D W1 M Š .AC ;  D .t  #// and Gr3 M Š .AC ;  D .t  #/ /. So M has weights 1 and 3. Moreover, M is uniformizable by Lemma 2.3.25 and Proposition 2.3.24(b). Q . qi WD 1 We set ` i D0 .1   t/ 2 O .CC / and choose an element  2 C with  q1 D . Then . ƒ.W1 M / D f 2 O .CC / W .t  #/  ./ D g D `   Fq Œt;  3 ƒ.GrW and 3 M / D .` /  Fq Œt;    f  ƒ.M / D `  Fq Œt ˚ . ` /3  Fq Œt

0



  .

` 3 for an f 2 O .CC / with .t  #/  .f / C b  3q   .` / D f . Putting  WD 1  0  f  and 2 WD . ` /3 , we get H.M / D 1  Q ˚ 2  Q and W1 H.M / D 1  Q.      With respect to the bases . 10 ; 01 / of M and .1 ; 2 / of ƒ.M / the isomorphism    f ` . Therefore, the Hodge– hM is given by the matrix .‰ T /1 WD 0 .` /3  Pink lattice is described by  q D

`  0

f .` /3 

1  t # p D 0

b .t  #/3

1  p:

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

69

Since ` has a simple zero at z D , one sees that q=p (which is also isomorphic to  coker M ) is isomorphic to C[[z  ]]=.z  / ˚ C[[z  ]]=.z  /3 if .t  #/jf (equivalently, if .t  #/jb) and isomorphic to C[[z  ]]=.z  /4 if .t  #/ − f (equivalently, if .t  #/ − b). So the Hodge–Pink weights of H1 .M / are .1; 3/ or .0; 4/, and the weight polygon lies above the Hodge polygon with the same endpoint W P .M /  HP .M / in accordance with Theorem 2.3.34(a) and Remark 2.2.8. In particular, if b D .t  #/  b 0 then the equation defining f shows that f . i Q Q vanishes at t D # q for all i 2 N0 , whence f D `  f for an f 2 O .CC / satisfying   .fQ/ C b 0  2q   .` /2 D fQ. 

2.3.5 Cohomology realizations. Let M D .M; M / be an A-motive of rank r over C. Anderson defined the Betti cohomology realization of M by setting H1Betti .M ; B/ WD ƒ.M / ˝A B

and

H1;Betti .M ; B/ WD HomA .ƒ.M /; B/

for any A-algebra B; see [Gos94, § 2.5]. This is most useful when M is uniformizable, in which case both are locally free B-modules of rank equal to rk M and H1 .M / D H1Betti .M ; Q/; see Lemma 2.3.21. By Theorem 2.3.27 this realization provides for B D Q an exact faithful neutral fiber functor on A-UMotI. Moreover, the de Rham cohomology realization of M is defined to be H1dR .M ; C/

WD   M=J    M

and

H1;dR.M ; C/ WD HomC .  M=J    M; C/:

We define a decreasing filtration of H1dR .M ; C/ by C-subspaces   1 F i H1dR .M ; C/ WD image of   M \ J i  M .M / in H1dR .M ; C/

for all i 2 Z;

which we call the Hodge–Pink filtration of M ; see [Gos94, § 2.6]. If M satisfies J  M M .  M / M then 1 .J  M /=J    M F 2 D .0/: F 0 D H1dR .M ; C/ F 1 D M

For example, this is the case if M is the A-motive associated with a Drinfeld Amodule. In this case the Hodge–Pink filtration coincides with the Hodge filtration studied by Gekeler, see Proposition 2.5.45(b) and Lemma 2.5.44. As noted in Remark 2.2.11 and Example 2.2.5(c), more useful than the Hodge– Pink filtration is actually the Hodge–Pink lattice q, and the latter cannot be recovered from the Hodge–Pink filtration in general. We therefore propose to lift the de Rham cohomology to C((z  )) and define the generalized de Rham cohomology realization of M by H1dR .M ; C[[z  ]]/ WD   M ˝AC C[[z  ]]   H1dR M ; C((z  )) WD   M ˝AC C((z  )) 

H1;dR .M ; C[[z  ]]/ WD HomAC . M; C[[z  ]]/     H1;dR M ; C((z  )) WD HomAC   M; C((z  )) :

and and and

70

Urs Hartl, Ann-Kristin Juschka

In particular by tensoring with themorphism C[[z  ]]  C; z   7! 0 we   get back 1 H1dR .M ; C/ D H ; C[[z  ]] ˝ C and H .M ; C/ D H M 1;dR 1;dR M ; C[[z  C[[z ]] dR  ]] ˝C[[z ]] C. We define the Hodge–Pink lattices of M as the C[[z ]]-submodules qM WD _ qM WD .M

  1 M .M / ˝AC C[[z  ]] H1dR M ; C((z  )) and     ˝ idC((z )) / HomAC .M; C[[z  ]]/ H1;dR M ; C((z  )) :

i H1;dR .M ; C/ of M are recovThen the Hodge–Pink filtrations F i H1dR .M  ; C/ and F  1 i M ered as the  ]] \ .z  / q in H1dR .M ; C/, respectively  dR M ; C[[z  images of H i of H1;dR M ; C[[z  ]] \ .z  / qM in H1;dR .M ; C/ like in Remark 2.2.4. All these 1 structures are compatible with the natural duality between HdR and H1;dR . The de Rham realization provides (covariant) exact faithful tensor functors

H1dR . : ; C/W

A-MotI ! VectC ;

M 7! H1dR .M ; C/

and

H1dR . : ; C[[z  ]]/W A-MotI ! ModC[[z ]] ; M 7! H1dR .M ; C[[z  ]]/ : (2.3.7) This is clear for H1dR . : ; C[[z]]/ and for H1dR . : ; C/ exactness follows from the snake lemma applied to multiplication with z on H1dR . : ; C[[z]]/. To prove faithfulness for H1dR . : ; C/ note that every morphism f W M 0 ! M can in A-MotI be factored into  coim.f / ,! M . If H1dR .f; C/ is the zero map the exactness M 0  im.f / ! 1 of HdR . : ; C/ shows that H1dR .im.f /; C/ D .0/. Since dimC H1dR .M ; C/ D rk M it follows that the A-motive im.f / has rank zero and therefore im.f / D .0/ and f D 0. . Finally, let v 2 C be a closed point. We say that v is a finite place of C . Let Av be the v-adic completion of A, and let Qv be the fraction field of Av . Consider the vadic completions AC;v WD lim AC =v n AC of AC and Mv WD lim M=v n M of M . Note   b idAv -linear map  W Mv ! Mv . that  W m 7! M .  m/ for m 2 M induces a   ˝ We let the  -invariants of Mv be the Av -module Mv WD fm 2 Mv j  .m/ D mg: ˚ rk M

It is isomorphic to Av and the inclusion Mv Mv induces a canonical    equivariant isomorphism Mv ˝Av AC;v ! Mv by [TW96, Proposition 6.1]. The v-adic cohomology realizations of M are given by H1v .M ; Av / WD Mv H1;v .M ; Av / WD HomAv .Mv ; Av /

and H1v .M ; Qv / WD Mv ˝Av Qv and

and

H1;v .M ; Qv / WD HomAv .Mv ; Qv / I

see [Gos94, § 2.3]. If M is defined over a subfield L of C (with L D C allowed) then they carry a continuous action of Gal.Lsep =L/ and the v-adic realization provides

71

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

(covariant) exact faithful tensor functors H1v . : ; Av /W A-Mot ! ModAv ŒGal.Lsep =L/ ; M 7! H1v .M ; Av /

and

H1v . : ; Qv /W A-MotI ! ModQv ŒGal.Lsep =L/ ; M 7! H1v .M ; Qv / :

(2.3.8)

 This follows from the isomorphism H1v .M ; Av / ˝Av AC;v ! Mv because Av AC;v is faithfully flat. Moreover, if L is a finitely generated field then Taguchi [Tag95b] and Tamagawa [Tam94, § 2] proved that    H1v . : ; Av /W Hom.M ; M 0 / ˝A Av ! HomAv ŒGal.Lsep =L/ H1v .M ; Av /; H1v .M 0 ; Av / (2.3.9) is an isomorphism for A-motives M and M 0 . This is the analog of the Tate conjecture for A-motives.

Proposition 2.3.36. Let M be a pure or mixed A-motive, which is defined over a finite field extension L of Q. Let P be a finite place of L, not lying above 1 or v, where M has good reduction, and let FP be its residue field. Then the geometric Frobenius FrobP of P has a well defined action on H1v .M ; Av / and each of its eigenvalues lies in the algebraic closure of Q in C and has absolute value .#FP / for a weight  of M . These eigenvalues are independent of v. Remark. The geometric Frobenius FrobP of P is the inverse of the arithmetic Frobe1 #FP nius Frob1 mod P for x 2 OL . P , which satisfies FrobP .x/  x Proof. Let v;M W Gal.Lsep =L/ ! AutAv H1v .M ; Av / be the associated Galois representation. By Gardeyn’s criterion [Gar02, Theorem 1.1] for good reduction, the inertia group of Gal.Lsep =L/ at P acts trivially on H1v .M ; Av / for every v ¤ 1 not lying below P , and therefore the Frobenius FrobP of P has a well defined action v;M .FrobP / on H1v .M ; Av /. Let M P be the reduction of M at P . Then there is  a canonical isomorphism H1v .M ; Av / ! H1v .M P ; Av / under which the action of FrobP corresponds to the action of the Frobenius endomorphism ŒF WFq 

MPP

WD MP ı   MP ı : : : ı  .ŒFP WFq 1/ MP W M P ŒJ 1   M P ŒJ 1 : D  ŒFP WFq  M P ŒJ 1  !

D  ŒFP WFq  on H1v .M P ; Av / is computed as Indeed, the action of Frob1 P 1 1  1 v;M .FrobP / WD h ı .FrobP / h via the vertical isomorphisms h in the following commutative diagram .M P /v ˝AF

O

h

P ;v

 ŒFP WFq  .M P /v

A

alg FP ;v

O

 .Frob1 P / h

Š

.M P /v ˝Av A

alg FP ;v

o

v;M .Frob1 P /˝id Š

˝AF

P ;v

Š

A

alg FP ;v

ŒF WFq   P MP Š

/

.M P /v ˝AF

O

h

.M P /v ˝Av A

alg FP ;v

A

alg FP ;v

P ;v Š

.M P /v ˝Av A

alg FP ;v

:

72

Urs Hartl, Ann-Kristin Juschka ŒF WFq 

In particular hı v;M .FrobP / D MPP

ıh on H1v .M P ;Av /. Since Q˝A EndFP .M P /

ŒF WF 

is a finite dimensional Q-algebra, MPP q satisfies a polynomial equation with coefficients in Q and its eigenvalues on H1v .M P ; Av / satisfy the same equation. In particular, these eigenvalues are independent of the place v ¤ 1 not lying below P . Now our formula for the absolute values of the eigenvalues was proved for pure M by Goss [Gos96, Theorem 5.6.10] and follows for mixed M , because the eigenvalues of FrobP coincide with the eigenvalues on the graded pieces GrW  M of M by considerations of triangular matrices. This motivates our convention that the weights of an effective A-motive are non-negative; see Proposition 2.3.11(e).  The morphism hM from Proposition 2.3.30 induces comparison isomorphisms between the Betti and the v-adic, respectively the de Rham realizations as follow. Theorem 2.3.37. If M is a uniformizable A-motive there are canonical comparison isomorphisms, sometimes also called period isomorphisms  H1v .M ; Av / ; hBetti; v W H1Betti .M ; Av / D ƒ.M / ˝A Av !

 ˝ f 7! .f   mod v n /n2N and      hBetti; dR WD   hM ˝ idC[[z ]] W H1Betti M ; C[[z  ]] ! H1dR M ; C[[z  ]] ; hBetti; dR WD   hM mod J

W

H1Betti .M ; C/

 ! H1dR .M ; C/ :

The latter are compatible with the Hodge–Pink lattices, respectively the Hodge–Pink filtration provided on the Betti realization H1Betti .M ; Q/ D H1 .M / via the associated Hodge–Pink structure H1 .M /. Proof. Since v ¤ 1 the points in the closed subscheme fvg Fq C CC do not specialize to 1C 2 CC and so this closed subscheme lies in the rigid analytic space  CC X DC . This yields isomorphisms O .CC X DC /=v n O .CC X DC / ! AC =v n AC for all n. The isomorphism hM induces a  -equivariant isomorphism  M ˝AC lim AC =v n AC D Mv : ƒ.M / ˝A lim O .CC X DC /=v n O .CC X DC / ! 



 Taking  -invariants on both sides and observing limO .CC X DC /=v n O  Did  Did .CC X DC / D lim AC =v n AC D Av provides hBetti; v .  The compatibility of the Betti–de Rham comparison isomorphism with the Hodge–Pink lattice and the Hodge–Pink filtration follows from diagram (2.3.6).  Remark 2.3.38. (a) If M D M .E/ is the A-motive associated with a Drinfeld Amodule E, the isomorphism hBetti; dR coincides with the period isomorphism studied

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

73

by Gekeler [Gek89, Theorem 5.14]; see Sect. 2.5.7, in particular Theorem 2.5.47 and Proposition 2.5.45. (b) Note that there are no A-homomorphisms between Av and C and therefore no  1 1 1 comparison isomorphism between H .M ; A / and H .M ; C/ or H ; C[[z  M v v dR dR  ]] . However, if one considers A-motives M over an algebraically closed, complete extension K of the v-adic completion Qvinstead of over  C, there is a comparison isomorphism between H1v .M ; Av / and H1dR M ; K((z)) ; see [HK20, Remark 4.16]. Example 2.3.39. Let C D P1Fq , A D Fq Œt, z D 1t , # D c  .t/ D 1 2 C, and let M D .CŒt; M D t  #/ be the Carlitz t-motive from Example 2.3.6. As in Example 2.3.20(b) we obtain . ƒ.M / D f 2 O .CC / W .t  #/  ./ D g D `   Fq Œt Q . qi q1 for ` WD 1 D . The comparison i D0 .1   t/ 2 O .CC / and  2 C with    isomorphism hBetti;dR D  hM ˝ idC[[z ]] sends the basis ` of H1Betti .M ; Fq Œt/ D  ƒ.M / to the element   .` / D   .` / 2 H1dR .M ; C[[z  ]]/ D C[[z  ]], re  Q q i 1 /j D  1 / 2 H1dR .M ; C/ D spectively to the element   .` i D1 .1    t D# C. The latter is the function field analog of the complex number .2i /1 , the inverse of the period of the multiplicative group Gm;Q . It is transcendental over Fq .#/ by a result of Wade [Wad41]. See Example 2.5.49 for more explanations.

2.4 Mixed dual A-motives For applications to transcendence questions like in [ABP04, Pap08, CY07, CPY10, CPTY10, CP11, CPY11, CP12], it turns out that dual A-motives are even more useful than A-motives; see the article of Chang [Cha20] in this volume for an introduction. Beware that a dual A-motive is something different then the dual M _ of an A-motive M . We clarify the relation between dual A-motives and A-motives, also in view of purity, mixedness and uniformizability in this section. 2.4.1 Dual A-motives. We continue with the conventions made in Sect. 2.3.1. In particular, we denote the natural inclusion Q ,! C by c  and consider the maximal ideal J WD .a ˝ 1  1 ˝ c  .a/ W a 2 A/ AC WD A ˝Fq C. The open subscheme Spec AC X V.J / of CC is affine. We denote its ring of global sections by AC ŒJ 1. Definition 2.4.1. (a) A dual A-motive over C of characteristic c  is a pair ML D .ML ; LML / consisting of a finite projective AC -module ML and an isomorphism of AC ŒJ 1-modules  ML ŒJ 1 LML W L  ML ŒJ 1  !

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where we set L  ML ŒJ 1 WD .L  ML / ˝AC AC ŒJ 1 and ML ŒJ 1 WD ML ˝AC AC ŒJ 1. A morphism of dual A-motives fLW ML ! NL is a homomorphism of the underlying AC -modules fLW ML ! NL that satisfies fL ı LML D LNL ı L  fL. The category of dual A-motives over C is denoted A-dMot. (b) The rank of the AC -module ML is called the rank of ML and is denoted by rk ML . The virtual dimension dim ML of ML is defined as ı ı dim ML WD dimC ML .ML \ LML .L  ML //dimC LML .L  ML / .ML \ LML .L  ML // : (c) A dual A-motive .ML ; LML / is called effective if LML comes from an AC -homomorphism L  ML ! ML . An effective A-motive has virtual dimension  0. (d) For two dual A-motives ML and NL over C we call QHom.ML ; NL / WD HomA-dMot .ML ; NL / ˝A Q the set of quasi-morphisms from ML to NL . (e) The category with all dual A-motives as objects and the QHom.M ; N / as Hom-sets is called the category of dual A-motives over C up to isogeny. It is denoted A-dMotI. Again, if C D P1Fq and A D Fq Œt, our effective dual A-motives are a slight generalization of the abelian dual t-motives in [BP20, §4.4], who in addition require L that ML is finitely generated over Cfg L where L acts on ML through m L 7! LML .L  m/. The tensor product of two dual A-motives ML and NL is the dual A-motive ML ˝ NL consisting of the AC -module ML ˝AC NL and the isomorphism LML ˝ LNL . The dual A-motive 1Ll.0/ with underlying AC -module AC and L D idAC is a unit object for the tensor product in A-dMot and A-dMotI. Both categories possess finite direct sums in the obvious way. We also define the tensor powers of a dual A-motive ML as ML ˝0 D 1Ll.0/ and as ML ˝n WD ML ˝n1 ˝ ML for n > 0. If ML D .ML ; LML / and NL D .NL ; LNL / are dual A-motives the internal hom Hom.ML ; NL / is the dual A motive with underlying AC -module HL WD HomAC .ML ; NL / and LHL W L HL ŒJ 1 ! . The dual of a dual A-motive ML is the dual A-motive HL ŒJ 1 ; hL 7! LNL ı hL ı L 1 L M _ ML WD Hom.ML ; 1Ll.0// consisting of the AC -module ML _ WD HomAC .ML ; AC / and the isomorphism .L _L /1 . M

Remark 2.4.2. The reader should be careful not to confuse dual A-motives ML with the duals M _ of A-motives M , which are again A-motives. In fact, the relation between A-motives and dual A-motives is the following. Let 1A=Fq be the A-module of Kähler differentials. Then 1AC =C D 1A=Fq ˝Fq C D   1AC =C D L  1AC =C  under the Fq -isomorphism Frobq;C W C ! C.

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Proposition 2.4.3. Every A-motive M D .M; M / induces a dual A-motive ML .M / WD .ML ; LML / where ML WD HomAC .  M; 1AC =C /; hence, L  ML D HomAC .M; 1AC =C /; LML WD .M /

_

WD HomAC .M ; 1AC =C /W .L ML /   L m L

7!

˝AC AC ŒJ

1

and

 ! ML ˝AC AC ŒJ 1 ; 

L m L ı M :

Every morphism f W M ! N of A-motives induces a morphism fL WD HomAC . f; 1AC =C /W ML .N / ! ML .M / of the associated dual A-motives. Conversely, every dual A-motive ML D .ML ; LML / induces an A-motive M .ML / WD .M; M / where M WD HomAC .L  ML ; 1AC =C /;

hence,   M D HomAC .ML ; 1AC =C /;

and

  M ˝AC AC ŒJ 1  ; M WD .LML /_ WD HomAC .LML ; 1AC =C /W . M / ˝AC AC ŒJ 1  !  m

7!

  m ı LML :

Every morphism fLW ML ! NL of dual A-motives induces a morphism f WD HomAC .L fL; 1AC =C /W M .NL / ! M .ML / of the associated A-motives. These mutually inverse functors induce exact tensor-anti-equivalences of categories A-Mot ! A-dMot and A-MotI ! A-dMotI. They map effective Amotives to effective dual A-motives and vice versa. In particular, the category A-dMotI is a Q-linear (non-neutral) Tannakian category, and hence a rigid abelian tensor category. The motivation to throw in the Kähler differentials is given by Theorem 2.5.13 below. Proof of Proposition 2.4.3. Since   and L  are flat by Remark 2.1.1 and M and ML are locally free, it follows from [Eis95, Proposition 2.10] that L  HomAC . M , 1AC =C / D HomAC .M; 1AC =C / and   HomAC .L  ML , 1AC =C / D HomAC .ML , 1AC =C /. With this observation the proposition is straight forward to prove, and the  final statements about the category A-dMotI follow from Proposition 2.3.4. Remark 2.4.4. (a) A neutral fiber functor only exists on the full subcategory of uniformizable dual A-motives; see Theorem 2.4.23 (b) The category A-dMot is an exact category in the sense of Quillen [Qui73, §2] if one calls a sequence of dual A-motives exact when its underlying sequence of AC modules is exact; compare Remark 2.3.5(b). The same is true for the subcategories of dual A-motives which are effective, respectively effective and finitely generated over Cfg. L (c) For the rank and (virtual) dimension of dual A-motives the formulas (2.3.3) hold correspondingly and rk ML .M / D rk M and dim ML .M / D dim M .

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(d) It can be proved directly, but also follows from Proposition 2.4.3 and Remark 2.3.7(c) that the set of morphisms HomA-dMot .ML ; NL / between two dual A-motives ML and NL is a finite projective A-module of rank at most .rk ML /  .rk NL /. Example 2.4.5. An effective dual A-motive of rank 1 with LML .L  ML / D J  ML is called a dual Carlitz–Hayes A-motive. Clearly, ML .1l.1// is a dual Carlitz–Hayes Amotive for any (non-dual) Carlitz–Hayes A-motive 1l.1/. Therefore, Example 2.3.6 proves the existence of dual Carlitz–Hayes A-motives and that they are all isomorphic in A-dMotI. So we may denote any one of them by 1Ll.1/. We also define 1Ll.n/ WD 1Ll.1/˝n for n > 0 and 1Ll.n/ WD 1Ll.n/_ for n < 0. If C D P1Fq , A D Fq Œt and # D c  .t/ 2 C, again all dual Carlitz–Hayes Amotives are already in A-dMot isomorphic to the dual Carlitz t-motive with ML D CŒt and LML D t  #. The latter is obtained via the functor ML . : / from the Carlitz t-motive M D .CŒt; M D t  #/ from Example 2.3.6. Every dual A-motive is isomorphic to the tensor product of an effective dual Amotive and a power of a dual Carlitz–Hayes A-motive. In fact, if ML is a dual Amotive with LML .L  ML / J d  ML , then NL WD ML ˝ 1Ll.1/˝d satisfies LNL .L  NL / NL ; hence, NL is effective and ML Š NL ˝ 1Ll.1/˝d . Note that rk NL D rk ML and dim NL D dim ML C d  rk ML . 2.4.2 Purity and mixedness. As in Sect. 2.3.2 we fix a uniformizing parameter z 2 Q D Fq .C / of C at 1 and assume that 1 2 C.Fq /. We denote the unique point on CC above 1 2 C by 1C . The completion of the local ring of CC at 1C is canonically isomorphic to C[[z]]. Definition 2.4.6. (a) A dual A-motive ML D .ML ; LML / is called pure if ML ˝AC C((z)) contains a C[[z]]-lattice ML 1 such that for some integers d; r with r > 0 the map r LM L  .LML / ı : : : ı L r1 .LML /W L rML ! ML L ı L WD LM  ML 1 . Then the weight of ML is induces an isomorphism z d L rL W L rML 1 ! M defined as wt ML D  d . r

(b) A dual A-motive ML is called mixed if it possesses an increasing weight filtration by saturated dual A-sub-motives W ML for  2 Q (i.e. W ML ML is ML WD W ; ML = a saturated AC -submodule) such that all graded pieces GrW P S W L L 0  Q

  Q for  0, and that W ML D ML for  0; compare Remark 2.2.2. (b) Every pure dual A-motive of weight  is also mixed with W0 ML D .0/ for L L 0 < , and W0 ML D ML for 0  , and GrW  M D M. Proposition 2.4.3 extends to mixed (dual) A-motives as follows. Proposition 2.4.9. A dual A-motive ML is mixed (pure) if and only if the corresponding A-motive M .ML / from Proposition 2.4.3 is mixed (pure). In that case the weights of M .ML / are the negatives of the weights of ML . More precisely, if ML is mixed with L weights 1< : : : 1< n then  the weight filtration on M D .M; M / D M .M / WD _ L HomAC .L M ; AC =C /; L L is given by M

˚

W M WD m 2 M D HomAC .L  ML ; 1AC =C /W m.L W0 ML / D 0 for all 0 <  ; (2.4.1)  that is, by W M D .0/ for all  < n , by W M D ker M  M .Wi ML / for all i C1   < i , and by W1 M D M . In particular the functors ML 7! M .ML / and M 7! ML .M / from Proposition 2.4.3 induce exact tensor-antiequivalences of categories A-dMMot ! A-MMot and A-dMMotI ! A-MMotI. Proof. First assume that ML is pure of weight  D  dr . This means that there is  a C[[z]]-lattice ML 1 ML ˝AC C((z)) such that z d L rL is an isomorphism L r ML 1 ! M ML 1 . Then M1 WD HomC[[z ]] .L  ML 1 ; C[[z]]dz/ is a C[[z]]-lattice in M.ML / ˝AC C((z)) D HomAC .L  ML ; 1 / ˝AC C((z)) D HomC((z )) .L  ML ˝AC C((z)); C((z))dz/ such that 

r1

.z d L rL /_ M

AC =C D zd  r L M.M /

 defines an isomorphism  r M1 ! M1 .

Therefore, M .ML / is pure of weight  D dr . Conversely, a C[[z]]-lattice M1 M.ML /˝AC C((z)) with z d  r

L/ M.M

 W  r M1 !

M1 induces the lattice ML 1 WD HomC[[z ]] .  M1 ; C[[z]]dz/ ML ˝AC C((z)) with  ML 1 . This proves that ML is pure of weight L r1.z d  r L /_ D z d L rL W L rML 1 ! M.M /

M

 if and only if M .ML / is pure of weight .

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Now we consider a mixed dual A-motive ML . Applying the exact contravariant functor ML 7! M .ML / gives for all  exact sequences  S  L L L 0M 0 ! M .GrW ! 0 / ! M .W / ! M W M M    0 1.  Gda;C of Fq -module schemes and Lemma 2.5.4. For every isomorphism W E ! d every norm k : k on C there exists a constant C > 0 such that expE maps f 2 Lie EW k Lie ./k < C g isometrically onto fx 2 E.C/W k .x/k < C g. The inverse of this isometry is a rigid analytic function  logE W fx 2 E.C/W k .x/k < C g ! f 2 Lie EW k Lie ./k  C g

satisfying logE .'a .x// D .Lie 'a /.logE .x// for all a 2 A and all x 2 E.C/ subject to the condition k .x/k; k .'a .x//k < C . It is called the logarithm of E. In particular ƒ.E/ D ker.expE / Lie E is a discrete A-submodule. Proof. Since all norms on Cd are equivalent by [Sch84, Theorem 13.3], we may assume that k : k is the maximum norm on Cd and on Cd d . If ı expE ı .Lie /1 D p P1 q i 1 i kEi kW i  1 g1 suffices and logE equals i D0 Ei  then C WD supf 1  P1  Q P P i 1 i n . 1 D 1 2 Cff ggd d where Cff gg WD C i i D0 Ei  / nD0  i D1 Ei  is the non-commutative power series ring with  b D b q  for b 2 C.

i D0



With every Anderson A-module E D .E; '/ is associated an AC -module as follows. This construction is due to Anderson [And86]; see also [BP20, § 4.1]. Let M WD M.E/ WD HomFq ;C .E; Ga;C / be the AC -module of Fq -linear homomorphisms of group schemes, where a 2 A and b 2 C act on m 2 M via aW m 7! m ı 'a

and

bW m 7!

b

ım:

The   -semi-linear endomorphism of M given by m 7! Frobq;Ga ım yields an AC linear homomorphism M W  M ! M . Note that after choosing an isomorphism E Š Gda;C of Fq -module schemes we obtain M.E/ Š Cf g1d from Lemma 2.5.1, P where i bi  i 2 Cf g1d with bi D .bi;1 ; : : : ; bi;d / 2 C1d corresponds to the i P morphism Gda;C ! Ga;C given by .x1 ; : : : ; xd /T 7! i;j bi;j xjq . In particular the endomorphism m 7! M .  m/ D Frobq;Ga ım of M corresponds to the endomorP P phism i bi  i 7!   . i bi  i / of Cf g1d which is injective. Since C is perfect,   is an automorphism of AC . So   W M !   M is an isomorphism and hence, M is injective. There is a natural isomorphism of AC -modules  HomC .Lie E; C/; M=M .  M / !

m mod M .  M / 7! Lie m : (2.5.2)

see [And86, Lemma 1.3.4], where a 2 A acts on Lie E via Lie 'a . Condition (2.5.1) in Definition 2.5.2(a) implies that J d D 0 on M=M .  M /, where J WD .a ˝ 1 

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

97

1 ˝ c  .a/W a 2 A/ AC . Therefore, M induces an isomorphism M W   M ˝AC  AC ŒJ 1 ! M ˝AC AC ŒJ 1. Let E be an Anderson A-module over C and define M .E/ WD  Definition 2.5.5. M.E/; M as above. If M.E/ is a finite locally free AC -module then E is called abelian and M .E/ is the (effective) A-motive associated with E. The rank of M .E/ is called the rank of E and is denoted rk E. For example, if C D P1Fq , A D Fq Œt, # D c  .t/ 2 C, and E D .Ga;C ; 't D # C  / is the Carlitz-module, then E is abelian of rank 1 and M .E/ D .CŒt; M D t #/ is the Carlitz t-motive from Example 2.3.6. Remark 2.5.6. (a) By [And86, Proposition 1.8.3] the rank of E is characterized by  ˚ rk E EŒa.C/ Š A=.a/ for every a 2 A. (b) If E is a Drinfeld A-module the rank of E from Definition 2.5.2(b) equals the rank from Definition 2.5.5 by [Gos96, § 4.5]. Anderson [And86, Theorem 1] proved the following Theorem 2.5.7. The contravariant functor E 7! M .E/ is an anti-equivalence from the category of abelian Anderson A-modules onto the full subcategory of A-Mot consisting of those effective A-motives .M; M / that are finitely generated over Cf g, where  acts on M through m 7! M .  m/. 2.5.2 The relation with dual A-motives. In unpublished work [ABP02] Greg Anderson has clarified the relation between Anderson A-modules and dual A-motives. For convenience of the reader we reproduce some of his results here (in our own words); see also [BP20, § 4.4]. Let E be a group scheme over C isomorphic to Gda;C , and let 'W A ! EndC .E/ be a ring homomorphism. The set ML WD ML .E/ WD HomFq ;C .Ga;C ; E/ of Fq -linear homomorphisms of group schemes is an AC -module, where a 2 A and b 2 C act on m L 2 ML via L aW m L 7! 'a ı m

and

bW m L 7! m L ı

b

:

L 7! mıFrob L There is a L  -semi-linear endomorphism of ML D ML .E/ given by m q;Ga ,  L L which induces an AC -linear homomorphism LML W L M ! M . Note that after choosing an isomorphism E Š Gda;C of Fq -module schemes we obtain ML .E/ Š Cf gd P from Lemma 2.5.1, where i bi  i 2 Cf gd with bi 2 Cd corresponds to the P qi morphism Ga;C ! Gda;C given by x 7! i bi x . In particular the endomorphism m L 7! LML .L  m/ L D m L ı Frobq;Ga of ML corresponds to the endomorphism

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Urs Hartl, Ann-Kristin Juschka

P

P bi  i 7! . i bi  i /   of Cf gd which is injective. Since C is perfect, L is an automorphism of AC . So L  W ML ! L  ML is an isomorphism and hence, LML is injective. There is the following alternative description of ML .E/. Let Cf L g be the nonp q commutative polynomial ring over C in the variable L with L b D b L for b 2 C. Consider in [BP20, § 4.4]) which P the -operation 0(called -operation P sends a matrix 0 B D i Bi  i 2 Cf grr with Bi 2 Crr to the matrix B  WD . i L i  .Bi /L i /T 2 0 CfL gr r . Here .: : :/T denotes the transpose. The -operation satisfies .BC / D 0 0 00 C  B  for matrices B 2 Cf grr and C 2 Cf gr r . It induces an isomorphism of AC -modules i

 W ML .E/ Š Cf gd ! Cfg L 1d ;

m L 7! m L ;

(2.5.3)

where a 2 A and b 2 C act on m L  2 CfL g1d via aW m L  7! m L   a

bW m L  7! b  m L:

and

Here a 2 Cf gd d D EndFq ;C .Gda;C / Š EndFq ;C .E/ is the matrix corresponding  Cfg L 1d the L  -semi-linear endomorto 'a . Under this isomorphism W ML .E/ ! L L  -semi-linear endomorphism phism m L 7! mıFrob L q;Ga of M .E/ corresponds to the  m L  7! L  m L  of Cfg L 1d . This gives ML .E/ the structure of a finite free left CfL gmodule which is independent of the isomorphism E Š Gda;C . Proposition 2.5.8. Let E be a group scheme over C isomorphic to Gda;C , and let 'W A ! EndC .E/ be a ring homomorphism. Set E D .E; '/ and let ML D ML .E/ and LML W L  ML ! ML be as above. Then there is a canonical exact sequence of Amodules 0

/ M L

m L 

/

ı1

ML m L 

/ L L .L  m/ L m L; M

/ E.C/

/ 0;

(2.5.4)

/ 0:

(2.5.5)

/ m.1/ L

and a canonical exact sequence of AC -modules 0

/ L  ML

LM L

/ ML

m L



ı0

/

Lie E / .Lie m/.1/ L

In particular, E D .E; '/ is an Anderson A-module if and only if LML induces an  isomorphism LML W L  ML ˝AC AC ŒJ 1  ! ML ˝AC AC ŒJ 1. In this case, ı0 . d factors through ML =J ML and extends to an AC -homomorphism ı0 W ML ˝AC O CC X S i i 2N>0 V. J /  Lie E.

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

99

 Under the above identifications E.C/ Š Cd and Lie E Š Cd and W ML .E/ ! 1d Cfg L these sequences take the form

0

L 1

/ Cfg L 1d 

m L

/

CfL g1d P / L m Lm L; ci L i 

ı1

/

Cd

P /

i

/ 0;

 i  .ci /T

i

and 0

L

/ CfL g1d

m L



/

Cfg L 1d P / L m L; ci L i 

ı0

/ Cd

/ 0:

/ cT 0

i

Proof. The map ı1 is A-linear because a  m L D 'a ı m L 7! .'a ı m/.1/ L D 'a .m.1//. L The map ı0 is a homomorphism of AC -modules because a  m L D 'a ı m L 7! Lie.'a ı L and m/.1/ L D Lie 'a .Lie m.1// bm L D m L ı

b

7! Lie.m L ı

b /.1/

D .Lie m L ı Lie

b /.1/

D b  .Lie m/.1/ L :

To prove that the composition of the two morphisms in (2.5.4) is zero, we compute L  m/.1/ L WD m L ı Frobq;Ga .1/  m.1/ L D m.1/ L  m.1/ L D 0 for all m L 2 ML . .LML .L m/  L To prove that ı0 ı LML D 0 in (2.5.5), note that since C is perfect, L W M ! L  ML is L D an isomorphism. Therefore, every element of LML .L  ML / is of the form LML .L  m/ L ı Frobq;Ga / D .Lie m/ L ı .Lie Frobq;Ga / D 0. m L ı Frobq;Ga and satisfies Lie.m Furthermore, ı1 is surjective because through every point x 2 E.C/ there is a morphism mW L Ga;C ! E with m.1/ L D x. For example if we identify the Fq -module  Gda;C D Spec CŒX1 ; : : : ; Xd  and Ga;C D Spec CŒY  we can take schemes W E ! L also satisfies .Lie m/.1/ L Dx mW L Xi 7! xi Y where .x/ D .x1 ; : : : ; xd /T . This m and this shows that ı0 is surjective. To show that (2.5.4) and (2.5.5) are exact, we keep this identification and the P i d L D induced P isomorphism ML .E/ Š Cf gd . If m i bi  2 Cf g satisfies 0 D m.1/ L D i bi , then L 0/  m L0 D m L 0  m L0 m L D LML .L  m P for m L 0 D i bi .1 C  C : : : C  i 1/. This proves that (2.5.4) is exact in the middle. Exactness on the left holds because multiplication with   1 is injective P on Cf gd . Clearly, (2.5.5) is exact on the left because LML is injective. If m L D i bi  i P P satisfies 0 D .Lie m/.1/ L D b0 then m L D . i bi  i 1/   D LML .L  i bi  i 1 / 2 LML .L  ML /, and this proves the exactness of (2.5.5). Moreover, under the -operation P P P i  i i T b  is sent to m L D c  L for c D  L .b / and so ı . m/ L D m L D i i i i 1 i i i bi D P i T T L D b0 D c0 . i  .ci / and ı0 .m/

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 Finally, LML induces an isomorphism LML W L  ML ˝AC AC ŒJ 1 ! ML ˝AC 1 AC ŒJ  if and only if the elements of J are nilpotent on Lie E. Since Lie E is a d -dimensional C-vector space, the latter is equivalent to condition (2.5.1) in DefML =J d ML , and exinition 2.5.2(a). If this holds, the morphism   . ı0Sfactors through tends to a homomorphism ı0 W ML ˝AC O CC X i 2N>0 V. i J /  Lie E because  . S O CC X i 2N>0 V. i J / =.J d / D AC =J d . 

Definition 2.5.9. Let E be an Anderson A-module over C and define ML .E/ WD   ML .E/; LML as above. If ML .E/ is a finite locally free AC -module then E is called A-finite and ML .E/ is the (effective) dual A-motive associated with E. The rank of ML .E/ is called the rank of E and is denoted rk E. Remark 2.5.10. By the analog of [And86, Proposition 1.8.3] (see Proposition 2.5.12  ˚ rk E for every a 2 A, below) the rank of E is characterized by EŒa.C/ Š A=.a/ where EŒa WD ker.'a W E ! E/. Together with Remark 2.5.6 this shows that for an Anderson A-module E which is both abelian and A-finite the Definitions 2.5.5 and 2.5.9 of the rank of E coincide.   The assignment E 7! ML .E/ D HomFq ;C .Ga;C ; E/; L m L 7! m L ı Frobq;Ga is a covariant functor because a morphism f W E D .E; '/ ! E 0 D .E 0 ; ' 0 / between abelian Anderson A-modules (which satisfies f ı 'a D 'a0 ı f ) is sent to ML .f /W ML .E/ ! ML .E 0 /;

m L 7! f ı m L;

which satisfies a  ML .f /.m/ L D 'a0 ı .f ı m/ L D f ı .'a ı m/ L D ML .f /.a  m/ L and  L  L L b  M .f /.m/ L D .f ı m/ L ı b D M .f /.b  m/ L and .LML .E 0 / ı L M .f //.L m/ L D  L L /.L m/ L for a 2 A; b 2 C and m L 2 M .E/. .f ı m/ L ı Frobq;Ga D .M .f / ı L L M .E /

The following result is due to Anderson; see [BP20, Theorem 4.4.1]. Theorem 2.5.11. (a) The functor ML . : /W E 7! ML .E/ from the category of Anderson A-modules to the category of pairs .ML ; LML / consisting of an AC -module  ML ŒJ 1  is ML and an isomorphism of AC ŒJ 1-modules LML W L ML ŒJ 1 ! fully faithful. (b) The functor ML . : / restricts to an equivalence from the category of A-finite Anderson A-modules onto the full subcategory of A-dMot consisting of those effective dual A-motives .ML ; LML / which are finitely generated as left CfL gmodules, where L acts on ML through m L 7! LML .L  m/. L Proof. (a) Let E and E 0 be Anderson A-modules and fix isomorphisms E Š Gda;C 0 and E 0 Š Gda;C of Fq -module schemes. Then under the identification HomFq ;C .E; E 0 / 0 Š Cf gd d from Lemma 2.5.1, a morphism f W E ! E 0 corresponds to a matrix

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0 F 2 Cf gd d and the induced morphism ML .f /W Cf gd Š ML .E/ ! ML .E 0 / Š 0 Cf gd corresponds to multiplication on the left with the matrix F . 0 Conversely, let gW Cf gd Š ML .E/ ! ML .E 0 / Š Cf gd be a morphism, that is LML .E 0 / ı L  g D g ı LML .E / . Since LML .E/ .L m/ L WD m L ı Frobq;Ga D m L   in 0 d d d ML .E/ Š Cf g , this means that the map gW Cf g ! Cf g is compatible with multiplication by Cf g on the right. Therefore, g corresponds to multiplication on 0 the left by a matrix G 2 Cf gd d . This means that g induces a morphism of Fq module schemes f W E ! E 0 with ML .f / D g. Since g commutes with the A-action on ML .E/ and ML .E 0 /, also f commutes with the A-action on E and E 0 , that is f is a morphism of Anderson A-modules. This proves the full faithfulness of ML . : /. (b) Let ML be a dual A-motive which is finitely generated over CfL g. Then ML is a finite free CfL g-module by the Cfg-analog L of [And86, Lemma 1.4.5], because it is a torsion free AC -module. Any CfL g-basis of ML provides an isomorphism ML Š HomFq ;C .Ga;C ; E/ DW ML .E/ compatible with LML and LML .E / , where E WD Gda;C with d WD rkCfL g ML . The action of a 2 A on ML commutes with LML . Therefore, it is P  given by multiplication on ML Š CfL g1d on the right by a matrix a D i Bi L i 2 P Cfg L d d . The map 'W A ! Cf gd d D EndFq ;C .E/, a 7! a WD . i  i  .Bi / i /T makes E into an A-module scheme. Sequence (2.5.5) shows that E D .E; '/ is an Anderson A-module which is A-finite, because ML Š ML .E/. 

Let E D .E; '/ be a (not necessarily A-finite) Anderson A-module and let L M D .ML ; LML / D ML .E/ be as in Definition 2.5.9. The following crucial description of the torsion points of E is Anderson’s “switcheroo”; see [ABP02] or [Jus10, Lemma 4.1.23]. Proposition 2.5.12. Let m L 2 ML and let x D ı1 .m/ L D m.1/ L 2 E.C/. Let a 2 AXFq . Then there is a canonical bijection

˚ 0 L 0/  m L0 D m L in ML =aML m L 2 ML =aML W LML .L  m ˚ 0

 ! x 2 E.C/W 'a .x 0 / D x   m L 0 7! ı1 a1 .m L Cm L 0  LML .L  m L 0 // ; (2.5.6)   where x 0 WD ı1 a1 .m L Cm L 0  LML .L  m L 0 // is defined by choosing any representative m L 0 2 ML of m L 0 2 ML =aML , taking m L 00 2 ML as the unique element with m L Cm L0   0 00 0 00 L / D am L , and setting x WD ı1 .m L /. LML .L m If m L D 0 both sides are A=.a/-modules and the bijection is an isomorphism of A=.a/-modules    .ML =aML /L ! EŒa.C/ ; m L 0 7! ı1 a1 .m L 0  LML .L  m L 0 // : Proof. First note that the map is well defined. Namely, any two representatives of m L 0 2 ML =aML differ by anL for an element nL 2 ML . Then the corresponding elements m L 00 differ by nL  LML .L n/ L which lies in the kernel of ı1 . Therefore, x 0

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is independent of the representative m L 0 2 ML . Moreover, x 0 WD ı1 .m L 00 / satisfies 'a .x 0 / D 'a .ı1 .m L 00 // D ı1 .am L 00 / D ı1 .m/ L D x. If m L D 0, then the map clearly is an A=.a/-homomorphism. L 00 2 ML with ı1 .m L 00 / D x 0 by If x 0 2 E.C/ with 'a .x 0 / D x is given, there is an m 00 00 0 L / D 'a .ı1 .m L // D 'a .x / D x D ı1 .m/ L (2.5.4) in Proposition 2.5.8 and then ı1 .am L 0/  m L 0 for an element m L 0 2 ML . This proves the implies that m L  am L 00 D LML .L  m surjectivity. L 02 2 ML be mapped to the same element x 0 2 E.C/ To prove injectivity let m L 01 ; m  and let m L 00i D a1 m L 0i / for i D 1; 2. Then ı1 .m L 001 / D x 0 D ı1 .m L 002 / L Cm L 0i  LML .L  m 00 00  implies by (2.5.4) in Proposition 2.5.8 that m L2 D m L 1 C LML .L n/ L  nL for an element   nL 2 ML . From this it follows that LM L  .m L 02 C anL  m L 01 /  .m L 02 C anL  m L 01 / D 0 0 0 L 2 C an. L  and the exactness of (2.5.4) on the left implies m L1 D m The relation between M .E/ and ML .E/ of an abelian and A-finite Anderson Amodule E is described by the following Theorem 2.5.13. Let E be an abelian Anderson A-module over C, let M D .M; M / D M .E/ and ML D .ML ; LML / D ML .E/ be as in Definitions 2.5.5 and 2.5.9. Let   _ be the dual A-motive from Proposition 2.4.3. ML .M / D HomAC .  M; 1AC =C / ; M Then there is a canonical injective AC -homomorphism „W HomAC .  M; 1AC =C / ,! ML ;

 7! m L

such that for every m 2 M mı m L D

1  X

 q i i 1 m/  i 2 EndFq ;C .Ga;C / D Cf g : (2.5.7) Res1  L i  .M

i D0

It is compatible with M and LML , that is, the following commutative diagram commutes: HomAC .  M; 1AC =C / O : ı M L  HomAC .  M; 1AC =C / D HomAC .M; 1AC =C /

„

/ ML O

LML 

L „

/ L  M L

(2.5.8) Moreover, „ is an isomorphism if and only if E is A-finite. In this case „ is an  isomorphism of dual A-motives „W ML .M / ! ML .E/.

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Proof. 1. To show that the sum in (2.5.7) belongs to Cf g we have to show that   i 1 m/ D 0 for all i 0 : Res1  L i  .M L cd ;r By Proposition 2.3.11 the z-isocrystal c M WD M ˝AC C((z)) is isomorphic to i M i i c with all di > 0 by Proposition 2.3.14(a). The explicit description of M di ;ri in (2.3.4) c WD M ˝AC C((z)) such that shows that there is a C[[z]]-lattice V of full rank in M j j s . s V /. V M . V / for all j  0, and a positive integer s with z 1 V M This implies nsj

L .nsCj 1/.M

V / zn V

for all integers n  0

and 0  j < s :

We extend  2 HomAC .  M; 1AC =C / to  2 HomC((z )) .  c M ; C((z))dz/. In particufor an integer N . For every m 2 M , there is an integer e lar, .  V / z N C[[z]]dz  .nsCj 1/ nsj  e neN with m 2 z V so that  L .M m/ 2 z C[[z]]dz. It follows that   nsj Res1  L nsCj 1 .M m/ D 0 for all n  N C e and all 0  j < s. 2. Fix an  2 HomAC .  M; 1AC =C /. To define m L 2 ML we choose an isomorphism

 Gda;C of Fq -module schemes, let prj W Gda;C ! Ga;C be the projection W E ! onto the j -th factor, and set mj WD prj ı 2 M.E/ D HomFq ;C .E; Ga;C / for j D 1; : : : ; d . We define m L 2 ML D HomFq ;C .Ga;C ; E/ via

ım L WD

1  X

 q i i i 1 mj /  Res1  L i  .M

i D0

!d 2 Cf g˚d : j D1

In particular, (2.5.7) holds when m D mj for j D 1; : : : ; d . To prove that (2.5.7) holds for all m 2 M we use that m1 ; : : : ; md form a Cf g-basis of M . Thus it suffices to show that (2.5.7) is compatible with (a) addition in M , (b) scalar multiplication by elements of C, and (c) multiplication with  . Since both sides of (2.5.7) are additive in m, (a) is clear. (b) Let m 2 M and b 2 C and assume that (2.5.7) holds for m. The left hand side equals .bm/ ı m L D b  .m ı m L /. On the right hand side we have Res1   q i  i  i 1   i  i 1  Res1  L .M m/ . Therefore, (2.5.7) also holds for L .M .bm// D b bm.

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(c) We assume that (2.5.7) holds for some m 2 M . The left hand side equals . m/ ı m L D   .m ı m L /. The right hand side for  m D M . m/ equals 1  X  q i i i 1 Res1  L i  .M ı M .  m//  i D0

D

1  X  q .i 1/ q i i Res1  L .i 1/.M m/  i D1

D

1  X  q i 1 i 1  i Res1  L .i 1/.M m/  ; i D1

i 1 1 1 1 because M D  i  M ı: : :ı  M ıM and in the first line the term Res1 .  m/ for i D 0 vanishes by [Vil06, Theorem 9.3.22] as .  m/ 2 1AC =C . Therefore, (2.5.7) also holds for  m. This establishes (2.5.7) for all m 2 M .

3. To prove that the assignment „W  7! m L defined in step 2 is C-linear, note that additivity is clear. Let b 2 C. Then b is sent to b  m L because ım L .b / WD

1  X

 q i i i 1 mj /  Res1 .b/ L i  .M

!d

i D0

D

j D1

 q i q i i i 1 mj / b  Res1  L i  .M

1  X i D0

!d

j D1

D ım L ı

b

DW ı .b m L / : 4. The map „ is also A-linear. Indeed, let a 2 A. Then a is sent to a  m L because prj ı ı m L .a / WD WD D

1  X  q i i i 1 mj /  Res1 .a/ L i  .M i D0 1  X

 q i i i 1 mj /  Res1  a  L i  .M

i D0 1  X

 q i i i 1 .mj ı 'a //  Res1  L i  .M

i D0

D m j ı 'a ı m L

DW prj ı ı .am L / :

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

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5. To prove that „ is compatible with M and LML we must show that „.L  ı i 1 1 1 1 M / D LML .L  m L /. This is true because M D  i  M ı : : : ı   M ı M implies i  i 1 i  i D L M , and hence, M ı L M ım L .L  ı M / WD

1  X

 q i i i 1 mj /  Res1 .L   ı M / L i  .M

i D0

D

1  X  q i i i Res1 .L / L i  .M mj / 

j D1

!d

i D0

D

j D1

1  X

Res1 L





.L

.i 1/

i .M mj //

q i

!d



i

i D0

D

!d

j D1

  i mj / Res1  L .i 1/.M

1  X i D1

q i 1

!d

  i 1

 j D1

D ım L ı Frobq;Ga DW ı LML .L  m L / ; where in the fourth line the term Res1 .  mj / for i D 0 vanishes again by [Vil06, Theorem 9.3.22] as .  mj / 2 1AC =C . „ is injective. If m L D 0, then (2.5.7) 6. We prove that the  AC -homomorphism  i 1 m/ D 0 for all i  0 and all m 2 M . We must show implies that Res1  L i  .M M ; C((z))dz/ is zthat  D 0. Since  2 HomAC .  M; 1AC =C / HomC((z )) .  c c  adically continuous with  M WD  M ˝AC C((z)), the preimage U WD 1 .C[[z]]dz/ c. By Proposition 2.3.15,   M c D is a z-adically open neighborhood of 0 in   M S i  i 1 U C i 2N0 L M .M /. Since the C-linear map Res1 ı  is zero on U and also on c. This implies that  D 0. the second summand, it is zero on all of   M 7. If „ is an isomorphism, then ML D ML .E/ is locally free over AC of rank equal to rk E, because M and hence HomAC .  M; 1AC =C / are, as E is abelian. So E is A-finite. 8. Conversely, assume that E is A-finite, that is, ML is locally free over AC of rank equal to rk E. Since also M and hence, HomAC .  M; 1AC =C / are locally free over AC of rank equal to rk E, as E is abelian, an argument analogous to [Tae09a, Proposition 3.1.2] shows that coker „ is annihilated by an element a 2 A (and not just by an element of AC ); see also [BH11, Corollary 5.4]. We use this to prove the surjectivity of „ in the next step. 9. To prove that „ is surjective, when E is A-finite, take for the moment an arbitrary element a 2 A X Fq and let  2 HomAC .  M; 1AC =C / be such that  .L  ı M / D a 0 for some 0 2 HomAC .  M; 1AC =C /, where L   2 HomAC .M; 1AC =C / and

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L   ı M 2 HomAC .  M; 1AC =C /, as M .E/ is effective. Then m L  LML .L  m L / D  am L 0 by parts 4 and 5 above. Moreover, let m 2 M be such that m  M . m/ D a m0 for some m0 2 M . Then we have a telescoping sum L

i

i 1 .M m/



 m D a

i X

j 1 0 L j  .M m/

for all i  0 :

j D0

  Since 0 .  m/; L  .  m0 / 2 1AC =C we have Res1 0 .  m/ D Res1 L    Theorem 9.3.22]. Finally, by part .  m0 / D 0 by [Vil06,  1 above there is an  n1 n1 m/ and 0 L n .M m/ lie in C[[z]]dz for all integer N such that  L n .M   N 1 n  N . Since a1 2 zFq [[z]], also a1  L N  .M m/ 2 C[[z]]dz. For all such n > N this implies n  q i  P i 1 .m ı m L 0 /.1/ D Res1 0 L i  .M m/ i D0

D

n P i P

i D0 j D0

D

n P i P i D0 j D0

D

n P i P

i D0 j D0



i P j D0

D

 q i  j 1 Res1   .L   ı M / L j  .M m0 /   j 1 Res1  L j  .M m0 /

q i  j 0  Res1 L  .L .j 1/ .M m //

n P i P i D0 j D0



q i  j 1 0  Res1 a 0 L j  .M m/

q i  j 1 0  Res1  L j  .M m/

n  iP 1 P i D0 j D0

D

P n j D0

D

P N j D0

D

N P j D0

 q i 1 j 1 Res1  L j  .M m0 /

 q n j 1 Res1  L j  .M m0 / q n  j 1 0  Res1  L j  .M m/

  j 1 Res1  L j  .M m0 /

  N 1 D Res1 a1  L N  .M m/  Res1 a1 .  m/ D  Res1 a1 .  m/ ;

(2.5.9)

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107

where the independence of n  N of the expression in the seventh line implies that this expression lies in Fq . Since .m ı m L 0 /.1/ D m ı1 .a1.m L  LML .L  m L // by definition of ı1 , it follows that the diagram (2.5.10) described in the next corollary is commutative. In this diagram the left horizontal arrow is injective, because if L   2 ML .M /=aML .M / satisfies .  m/ 2 a 1A=Fq for all m 2 .M =aM / , then .M =aM / ˝Fq C Š M =aM implies that .  m/ 2 a 1AC =C for all m 2 M , whence   2 aML .M /. This arrow is surjective because both HomA=.a/ .M =aM / ; 1A=Fq = L   a 1A=Fq and ML .M /=aML .M / are locally free A=.a/-modules of rank rk E, and hence, are finite dimensional Fq -vector spaces of the same dimension, because E is A-finite.  L   This implies that „ induces an isomorphism ML .M /=aML .M / ! ML .E/= L aML .E/ . Since .ML =aML /L ˝Fq C Š ML =aML for every dual A-motive ML over  ML .E/=aML .E/. In C, we conclude that „ is an isomorphism ML .M /=aML .M / ! particular if we take the element a 2 A from part 8 which annihilates the cokernel of „ this shows that coker.„/ D 0 and that „ is an isomorphism. Altogether we have proved the theorem  Along with the proof of the theorem we also showed the following Corollary 2.5.14. Let E be an abelian and A-finite Anderson A-module, and let (dual) A-motive. Let a 2 A and conML .E/ and M D M .E/ be its associated  _ sider the dual A-motive ML .M / WD HomAC .  M; 1AC =C / ; M from Proposition 2.4.3. Then the following diagram consisting of isomorphisms of A=.a/-modules is commutative HomA=.a/

  .M =aM / ; 1A=Fq =a 1A=Fq

o

 Š

 L .M /=aM L .M / L M

/ ML .E /=aML .E /L

„ Š

Š

Š





E Œa.C/



 h W m 7! .  m/

_ 

P

o

E Œa.C/ ;







/

„



ı1 a

m L

_ 

1

 .m L   LM L m L / ; L . 

(2.5.10)

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 L where the left horizontal arrow sends  2 ML .M /=aML .M / to h WD . ı   /j.M =aM / , where the right horizontal arrow is the isomorphism „ from Theorem 2.5.13, where the left vertical map is (up to a minus sign motivated by Theorem 2.5.47 below)  Anderson’s isomorphism [And86, Proposition 1.8.3] which sends h 2 HomA=.a/ .M =aM / , 1A=Fq =a 1A=Fq to the point P 2 EŒa.C/ satisfying m.P / D  Res1 a1 h.m/ for all m 2 .M =aM / , and where the right vertical  1 L  LML .L  m/ L from Proposition 2.5.12. map is the isomorphism m L 7! ı1 a .m Proof. The proof of the corollary was given in step 9 of the proof of Theorem 2.5.13.  The theorem naturally leads to the following Question 2.5.15. If E is an abelian and A-finite Anderson A-module, the inverse of the isomorphism „ from Theorem 2.5.13 defines a perfect pairing of AC -modules ML .E/ ˝AC   M.E/ ! 1AC =C ;

 m L ˝   m 7! „1 .m/. L m/ :

Is it possible to give a direct description of this pairing, that is an explicit formula of  L m/ in terms of m L and m ? the differential form „1 .m/. For Drinfeld Fq Œt-modules the question has an affirmative answer as follows. Example 2.5.16. Let C D P1Fq , A D Fq Œt, AC D CŒt, # WD c  .t/, and J D .t #/. Also we choose z D 1t as the uniformizing parameter at 1. Then 1AC =C D CŒt  dt and dt D  z12 dz. Let E D .E; '/ be a Drinfeld Fq Œt-module given by E D Ga;C and 't D

C

#

˛1

ı C:::C

˛r

ı r

with ˛i 2 C and ˛r ¤ 0. Then the powers m L k WD  k for k D 0; : : : ; r  1 form a CŒt-basis of ML D HomFq ;C .Ga ; E/ on which LML acts via LML . i / D  i C1 for 0  i < r  1 and LML . r1 / D  r D 't ı

q r

1=˛r

  r1 ı D .t  #/=˛rq



q r

#=˛r

q .r1/

q 1

˛1

q r

=˛r

:::

q r

˛r1

=˛r

r

1

 ˛1q

 ı

=˛rq

r

.r1/

q    : : :  ˛r1

=˛rq

r

  r1 :

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

109

Thus with respect to this basis of ML and the induced basis of L  ML the CŒt-linear map LML is given by the matrix 1 0 q r 0 .t  #/=˛r C B0 C B B q 1 q r C C ˛1 =˛r B1 C B C B C B L ˆ D B0 C C B C B C B 0 C B A @ q .r1/ q r 0 1 ˛r1 =˛r 0 In particular E is A-finite. On the other hand the powers mj WD  j for j D 0; : : : ; r1 also form a CŒt-basis of M D HomFq ;C .E; Ga / on which M acts via M . i / D  i C1 for 0  i < r  1 and M . r1/ D  r D

1=˛r

ı 't 

#=˛r



˛1 =˛r

ı :::

˛r1 =˛r

ı  r1

D .t  #/=˛r  ˛1 =˛r    : : :  ˛r1 =˛r   r1 : Thus with respect to this basis of M and the induced basis of   M the CŒt-linear map M is given by the matrix 0 1 0 0 .t  #/=˛r B C B1 ˛1 =˛r C B C B C B0 C B C ˆ D B C B C B C 0 B C @ A 0 0 1 ˛r1 =˛r In particular E is also abelian. Let ` 2 ML .M / D HomAC .  M; 1

AC =C

/ for ` D 0; : : : ; r  1 be the basis ı

dual to .  mj /j which is given by ` .  mj / D ıj ` dt D  zj2` dz, where ıj;` is the Kronecker delta. We want to compute the matrix representing the isomorphism „ from Theorem 2.5.13 with respect to the bases .` /` and .m L k /k . For this puri  i 1  pose we have to compute  L . m / 2  M ˝ C((z)) modulo z 2 , because j A C M L 2   ` j z C[[z]]   mj C[[z]]dz and the elements of the latter have residue 0

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1 z at 1. We set ˛i WD 0 for i > r and observe t # D 1#z 2 z C[[z]]. By in1 i  1 duction on i one easily verifies that the matrix ˆ  : : :  L ˆ , which represents i 1 1 1 L i  M WD M ı : : : ı L i  M with respect to the basis .mj /j , is congruent to

0

z  L B 1  #z B B  B i  ˛2Ci z  B L B 1  #z B B B B B B   B i  ˛r1 z B L B 1  #z B B i   ˛r z  B L B 1  #z B B B B B B i  ˛rCi z  @ L 1  #z i

˛

1Ci

 ˛ z  i L 1  #z  ˛ 1Ci z L .i 1/ 1  #z

L .i 1/ L .i 1/

L .i 1/

˛



r2 z

0

1

0C C C C C C C C C 0C C C C 1C C C C 0C C C C C C C C 0A

0

˛ri 1 z 1  #z ˛ri z 1  #z

1  #z ˛ z r1

1  #z z 1  #z

˛

1

˛1 z 1  #z ˛2 z 1  #z

.i 1/

r1Ci

˛r z 1  #z

0

modulo z 2 C[[z]]rr for i D 0; : : : ; r  1, and to 0 B0 B B B B B B B B B B B @ 0

0

L

.r1/

 ˛ z  r 1  #z

1 L 1  #z C C C  ˛ z  C C r C L .i rC1/ 1  #z C C C C 0 C C A 0 .i rC1/

˛

i rC2 z

for i D r  1; : : : ; 2r  2, and to the zero matrix for i  2r  1. It follows that   i 1 mj / D Res1 ` L i  .M L ` D  and hence, mj ı m

P2r2 i Dj



qj

L .i j / .˛`C1Ci j / 0

˛`C1Ci j  i D  j 

Pr1`

L ` i  j . These equations are equivalent to „.` / D m

for j  i ; for j > i ; q k

 k ˛kC`C1 for k D P q k D  r1` L k. kD0 ˛kC`C1 m kD0

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Therefore, „ is represented with respect to the bases .` /` and .m L k /k by the matrix 0 1 ˛1 ˛2 ˛r 1 B q 1 q 1 C B ˛2 ˛3 ˛rq 0 C B C B C B C X WD  B C 2 GLr .C/ GLr .CŒt/ : (2.5.11) 2r q B C ˛r B C B C @ q 1r A ˛r 0 0 Note that the compatibility of „ with M and LML from equation (2.5.8) corresponds to the equation 0 1 t # 0 0 1 1 C B B 0 ˛2q ˛rq C B C L  L  .X / ; (2.5.12) X  ˆT D  B C D ˆ 0 B C @ A q 1r 0 ˛r 0 0 which is easily verified. In particular, if 0 0 B B X 1 D B B 0 @ ˇr1;0

0

ˇ0;r1

1 C C C C A

ˇr1;r1

denotes the inverse of the matrix X from (2.5.11) then the pairing from Question 2.5.15 is explicitly given by r1 X kD0

Lk ˝ fLk m

r1 X

fj   mj 7!

j D0

r1 X

r1 X

fj ˇj;k fLk dt

j D0 kDr1j

with fLk ; fj 2 CŒt for 0  j; k  r  1. (b) More generally let E D .E; '/ be an Anderson Fq Œt-module given by E D Gda;C and 't D 0 C 1 ı  C : : : C s ı  s with i 2 Cd d , such that .0  #/d the Kronecker delta) the elements 0 ı1;  k B :: m L k; WD @ : ıd;  k

D 0. Assume that s 2 GLd .C/. Then (with 1

0

ı1; x q B C :: 7 ! @ AW x  :

k

ıd; x q

1 C A

k

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for  D 1; : : : ; d and k D 0; : : : ; s  1 form a CŒt-basis of ML D HomFq ;C .Ga ; E/. And the elements 0 1 x1 j B C mj; WD .ı1;  j ; : : : ; ıd;  j / W @ ::: A 7! x q xd for  D 1; : : : ; d and j D 0; : : : ; s  1 form a CŒt-basis of M D HomFq ;C .E; Ga /. A similar computation as in (a) shows that with respect to these bases of ML and M and the induced bases of L  ML and   M the CŒt-linear maps LML and M are given by the matrices 0 1 0 0 .t  0 /  L s .1 s / B C B C  s 1 B Idd C L .1 /  L .s / B C B C B C 0 L D B ˆ C B C B C B C 0 B C @ A .s1/ s 1 0 0 Idd L .s1 /  L .s / and 0

0

B B B Idd B B ˆ D B B 0 B B B @ 0

0

T .t  T0 /  .1 s / T T1  .1 s /

0 0

Idd

1 C C C C C C C C C C A

T Ts1  .1 s /

In particular E is A-finite and abelian of dimension d and rank r WD sd and pure of weight s. Let `; 2 ML .M / D HomAC .  M; 1AC =C / for  D 1; : : : ; d and ` D 0; : : : ; s  1 be the basis dual to .mj; /.j; / which is given by `; .  mj; / D ıj;` ı ; dt D ı ı  j;`z 2; dz. A similar computation as in (a) then shows that „ is represented with

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

113

respect to the bases .`; /.`;/ and .m L k; /.k; / by the matrix 0 B B B B X WD  B B B @

1

2

L  2

L  3

s

C 0 C C C C 2 GLr .C/ GLr .CŒt/ C C A 0

L  s

L .s2/ s L .s1/ s

1

0

which satisfies 0

T

X ˆ

t  0 0 B  L 2 B 0 D B B @ 0 L .s1/ s

0  L s 0 0

1 C C C D ˆ L  L  .X / : C A

0

Corollary 2.5.17. Let E D .E; '/ be a Drinfeld A-module. Then E is abelian and A-finite. Proof. Fix an element t 2 A X Fq and consider the finite flat ring homomorphism e e A WD Fq Œt ,! A. By restricting 'je A W A ! EndFq ;C .E/ we view E as a Drinfeld Fq Œt-module. Then M.E/ and ML .E/ are finite free modules over e AC D CŒt by Example 2.5.16. Therefore, they are finite and torsion free, hence locally free modules over the Dedekind domain AC .  2.5.3 Analytic theory of A-finite Anderson A-modules. We equip the C-vector 0 spaces of matrices Cd d and vectors Cd D Cd 1 with the maximum norm k : k given by k.xij /k WD maxf jxij jW all i; j g. Then kBC k  kBk  kC k for all matrices B; C . All norms on these spaces are equivalent by [Sch84, Theorem 13.3] and induce the same topology. 0

Lemma 2.5.18. Let f W Gda;C ! Gda;C be a homomorphism of Fq -module schemes over C. Then f induces a continuous Fq -linear map f W Gda;C .C/ D Cd ! 0 0 Gda;C .C/ D Cd . More precisely, there is a constant C 2 R0 such that kf .y/k  C  kyk for every y 2 Cd with kyk  1. 0

0

Proof. Under the isomorphism HomFq ;C .Gda;C ; Gda;C / Š Cf gd d from Lemma 2.5.1 P i d 0 d we write f D and Bi D 0 for i 0. Let C WD i 0 Bi  with Bi 2 C i

maxf kBi kW i  0 g. For y 2 Cd with kyk  1 we have k i  .y/k D kykq  kyk,

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and therefore

 X   Bi  i .y/ kf .y/k D  i 0

˚  max kBi  i  .y/kW i  0

˚  max kBi k  k i  .y/kW i  0  C  kyk : 0

Since f W Cd ! Cd is Fq -linear, this shows that f is continuous.



Definition 2.5.19. Fix an a 2 A X Fq and an x 2 E.C/. (a) A sequence x.0/ ; x.1/ ; x.2/ ; : : : 2 E.C/ is an a-division tower above x if 'a .x.n/ / D x.n1/

for all n > 0

and

'a .x.0/ / D x :

(b) An a-division tower .x.n/ /n0 is said to be convergent if for some (or, equiva lently, any) isomorphism W E ! Gda;C of Fq -module schemes, lim .x.n/ / n!1

D 0 in the C-vector space Gda;C .C/ D Cd . Proof. We must explain, why the definition in (b) is independent of . For this pur pose let W Q E ! Gda;C be another isomorphism. Then Q ı 1 2 AutFq ;C .Gda;C / induces a homeomorphism Q ı 1 W Cd ! Cd by Lemma 2.5.18. It follows that limn!1 k .x.n/ /k D 0 if and only if limn!1 k .x Q .n/ /k D 0 as claimed.  If E is A-finite (or abelian) then a-division towers exist above every x. This follows from Theorem 2.5.21 (or respectively Proposition 2.5.45(a) below). But there may or may not exist convergent ones. Theorem 2.5.20 ([ABP02]). Let E be an Anderson A-module over C, let x 2 E.C/, and let a 2 A X Fq . Then there is a canonical bijection  f convergent a-division towers above x g f  2 Lie EW expE ./ D x g !    (2.5.13)  7! expE Lie 'an1 ./ n2N0

  If W E ! Gda;C is an isomorphism of Fq -module schemes and Lie W Lie E ! Cd is the induced isomorphism of Lie algebras then   (2.5.14) lim Lie. ı 'anC1 ı 1 / .x.n/ / D .Lie /./ n!1

  holds in Cd for all  2 Lie E with x.n/ WD expE Lie 'an1 ./ for n  0.

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

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Remark. Equation (2.5.14) is the analog of the fact, that for a real or complex Lie group G the exponential function expG W Lie G ! G has derivative 1 near the identity element of G (with respect to any coordinate system). For example lim an n!1   exp.an /  1 D  for G D Gm , where  is any complex number and a 2 Z X f1; 0; 1g.   Proof. The element x.n/ WD expE Lie 'an1 ./ 2 E.C/ satisfies 'a .x.n/ / D   expE Lie 'an ./ . This equals x.n1/ when n > 0 and it equals x when n D 0, hence, .x.n/ /n is an a-division tower above x. By Lemmas 2.5.3 and 2.5.4, it is convergent and so the map is welldefined.    If ;  0 2 Lie E satisfy expE Lie 'an ./ D expE Lie 'an . 0 / for all n  0 then Lemma 2.5.3 implies that Lie 'an ./ and Lie 'an . 0 / converge to 0 in Lie E and therefore Lie 'an ./ D Lie 'an . 0 / for n 0 by Lemma 2.5.4. This implies  D  0 , and hence the map is injective. To prove surjectivity, let .x.n/ /n be a convergent a-division tower above x. Since .x.n/ / converges to 0 there is an n0 2 N0 such that logE .x.n/ / exists by Lemma 2.5.4     for all n  n0 . We set  WD Lie 'an0 C1 logE .x.n0 / / . Then Lie 'anC1 logE .x.n/ / D     Lie 'an0 C1 logE .'ann0 .x.n/ // D Lie 'an0 C1 logE .x.n0 / / D  for n  n0 by   Lemma 2.5.4. Therefore, x.n/ D expE Lie 'an1 ./ for all n  n0 , and for   n < n0 we compute x.n/ D 'an0 n .x.n0 / / D 'an0 n expE .Lie 'an0 1 .// D   expE Lie 'an1 ./ . It remains to prove (2.5.14). With respect to theP coordinate system and Lie i d d we write 'a as a matrix a WD ı 'a ı 1 D and i 0 a;i  2 Cf g P1 i 1 d d expE as a matrix i D0 Ei  WD ı expE ı .Lie / with a;i ; Ei 2 C and a;0 D Lie. ı'a ı 1/ and E0 D Idd . By replacing by Q WD Bı for a matrix B 2 GLd .C/ Cf gd d D EndFq ;C .Gda;C / we can write a;0 D c  .a/.Idd CN / with strictly upper triangular (nilpotent) N having only entries 0 and 1. This replacement is allowed because Q ı 1 DB is an automorphism C-vector space Cd . Then  of the nC1 P1 nC1 1 i n1 Lie. ı 'a ı / .x.n/ / D a;0 a;0 Lie ./ . We consider i D0 Ei   the maximum norm k : k on Cd and Cd d . For i > 0 the term nC1 Ei  i  n1 a;0 a;0  Lie ./ equals c  .a/nC1.Idd CN /nC1 Ei c  .a/q

i .nC1/

  .Idd CN /n1 i  Lie ./ ; i

i

(2.5.15)

and has norm less or equal to kEi k k Lie ./kq jc  .a/j.q 1/.nC1/ , because k : k is compatible with matrix multiplication and k Idd CN k D 1. Since jc  .a/j > 1 i and expE converges on all of Lie E, that is limi !1 kEi k k Lie ./kq D 0, the terms (2.5.15) go to zero uniformly in i when n ! 1. Therefore, limn!1 nC1 a;0   n1 P1 i n1 Lie ./ D Lie ./, proving a;0 Lie ./ D nC1 i D0 Ei  a;0 E0 a;0 (2.5.14). 

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From now on we assume that E is A-finite. The following theorem of Anderson [ABP02] is crucial for the theory of uniformizability. Let a 2 A X Fq and set ML a WD lim ML =an ML . If v1 ; : : : ; vs are the maximal ideals of A which con Q Q tain a then lim AC =.an / D siD1 AC;vi and ML a D siD1 ML ˝AC AC;vi . The lat ter equals the completion of ML at the closed subscheme V.a/ Q Spec AC . Since V.a/ CC X DC there are natural inclusions O .CC X DC / ,! siD1 AC;vi and ML ˝AC O .CC X DC / ,! ML a . Theorem 2.5.21 ([ABP02]). (a) Let E be an A-finite Anderson A-module and let .ML ; LML / D ML .E/ be its dual A-motive. Let m L 2 ML and x WD ı1 .m/ L D m.1/ L 2 E.C/. Then Proposition 2.5.12 defines a canonical bijection

a-division towers .x.n/ /n above x (2.5.16) as follows. Let m L 0 2 ML a satisfy LML .L  m L 0 / m L 0 D m. L For each n 2 N0 choose an m L 0n 2 ML with m L0  m L 0n mod anC1 ML a . There is a uniquely determined 00 nC1 00 L m L n 2 M with a m Ln D m L Cm L 0n  LML .L  m L 0n /. Then x.n/ WD ı1 .m L 00n /.

˚

L 0 /m L0 D m L m L 0 2 ML a W LML .L  m



 !

˚

(b) Let m L 0 correspond to the a-division tower .x.n/ /n under the bijection (2.5.16). Then the following are equivalent: (i) m L 0 2 ML ˝AC O .CC X DC / ML a ,  . S (ii) m L 0 2 ML ˝AC O CC X i 2N>0 V. i  J / ML a , (iii) .x.n/ /n is convergent,  Gda;C (iv) with respect to some (or, equivalently, any) isomorphism W E !  n of Fq -module schemes the sequence c .a/  .x.n/ / is bounded in the C-vector space Gda;C .C/ D Cd .

If these conditions hold and if  2 Lie E is the element from Theorem 2.5.20 that corresponds to the convergent a-division tower .x.n/ /n , that is x.n/ D   expE Lie 'an1 ./ for all n, then  D ı0 .m L 0 C m/ L for the map ı0 W ML ! Lie E from Proposition 2.5.8. Proof. 1. By Proposition 2.5.12 the definition of x.n/ is independent of the chosen L 0n1 D m L 0n and m L 00n1 D am L 00n to obtain 'a .x.n/ / D m L 0n . In particular we can take m 00 00 00 ı1 .am L n/ D ı1 .m L n1 / D x.n1/ and 'a .x.0/ / D ı1 .am L 0 / D ı1 .m/ L D x. This defines the bijection (2.5.16). Note that we explicitly describe its inverse in part 5 below. 2. To prove (b), note that trivially (b)(ii)H)(b)(i) and (b)(iv)H)(b)(iii), because jc  .a/j > 1.  3. To prove (b)(iii)H)(b)(iv) for any isomorphism W E ! Gda;C of Fq -module P 1 j d d D EndFq ;C .Gda;C / schemes, we write ı'a ı DW a D j 0 a;j  2 Cf g

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

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with a;j 2 Cd d and a;j D 0 for j 0. By replacing by Q WD B ı for a matrix B 2 GLd .C/ Cf gd d we can write a;0 D c  .a/.Idd CN / with strictly upper triangular (nilpotent) N having only entries 0 and 1. This replacement is allowed because Q ı 1 D B is an automorphism of the C-vector space Gda;C .C/. Consider the maximum norm kxk D maxfjxi jW i D 0 : : : d g for x D .x1 ; : : : ; xd /T 2 Cd and  1 the norm kyk WD k .y/k on y 2 E.C/ induced via . As 1 a;0 D c .a/ .Idd N C 1 2  1  N  : : :/ we find kxk D ka;0 a;0 xk  jc .a/j ka;0 xk  jc .a/j1jc  .a/j  kxk D kxk, whence ka;0 xk D jc  .a/jkxk. If n 0 then kx.n/ k 1 by assumpj tion (b)(iii), whence k j  .x.n/ /k D kx.n/ kq kx.n/ k for j > 0. So jc  .a/j > 1 implies ka;j  j  .x.n/ /k < jc  .a/j  kx.n/ k D ka;0 .x.n/ /k for n 0 and all P j > 0. Thus kx.n1/ k D k'a .x.n/ /k D k j 0 a;j  j  .x.n/ /k D jc  .a/j  kx.n/k for n 0, and this yields the boundedness of jc  .a/jn  kx.n/ k and c  .a/n  .x.n/ /.  4. To prove (b)(iv)H)(b)(ii) and (b)(i)H)(b)(iii) we choose an isomorphism W E !  d Ga;C of Fq -module schemes and consider the induced AC -isomorphism W ML ! L 7! m L  from (2.5.3). Moreover, under the finite flat ring homomorphism Cfg L 1d , m Q Fq Œt ! A; t 7! a we have AC =.an/ D AC ˝CŒt  CŒt=.t n/ and siD1 AC;vi D AC ˝CŒt  C[[t]], as well as O .CC X DC / D AC ˝CŒt  Chti; see (2.3.5). We also abbreviate # WD c  .a/ and for a real number s we use the notation

Ch #ts i WD

1 ˚P i D0

bi t i W bi 2 C; lim jbi j  j#jsi D 0 i !1



and

Chti WD Ch #t0 i :

(2.5.17) We consider ML as a finite (locally) free module over CŒt of rank r. We choose a CŒt-basis B of ML and use it to identify ML a Š C[[t]]˚r and ML ˝AC O .CC X DC / Š d and C1d , and consider the Chti˚r . Let k : k denote the maximum norms on Cr , CP norm kyk WD k .y/k on y 2 E.C/ and the norm k j cj L j kL WD supfkci kW i  0g on CfL g1d and ML where cj 2 C1d . For all s 2 R consider also the norm P k i bi t i ks WD supfkbi k jc  .a/jsi W i  0g on CŒt˚r and ML where bi 2 Cr . When s  s 0 these norms satisfy the inequalities k : ks  k : ks0 . Note that Ch #ts i˚r is the completion of CŒt˚r with respect to the norm k : ks , which therefore extends to Ch #ts i˚r . 5. We now assume that (b)(iv) holds for our fixed isomorphism . For each n 2 N0 we let .m L 00n / WD .x.n/ /T  L 0 2 C1d L 0 CfL g1d Š ML : L and x.1/ WD x. Then ı1 .m L 00n / D x.n/ for all n  1, and hence, We set m L 001 WD m 00 00 L n m L n1 / D 'a .x.n/ /x.n1/ D 0 implies that t m L 00n  m L 00n1 D yn  LML .L  yn / ı1 .t m for an element yn 2 ML for n  0. Moreover, the elements .t m L 00n  m L 00n1 / D .m L 00n /   00  T  T .a /  .m L n1 / D .x.n/ /  .a /  .x.n1/ / lie in the finite dimensional CL L of the entries of vector space W WD `j D0 C1d L j where ` is the maximal -degree P  d d  corresponding to 'a . If .yn / DW j cj L j 2 CfL g1d the matrix .a / 2 CfL g

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Urs Hartl, Ann-Kristin Juschka

 P P P  then .t m L 00n  m L 00n1 / D j cj L j  L  j cj L j D j cj  L  .cj 1 / L j . Writing P L 00n1 / DW `j D0 cQj L j we compute L  .cj 1 / D cj  cQj . Together with .t m L 00n  m Pj cj D 0 for j 0 this implies cj D 0 for all j  ` and cj D kD0 L .j k/ .cQk / for P ˚r n L j < `. So .yn / 2 W . In particular, the series 1 satisfies nD0 yn t in Ma Š C[[t]] 1 1 1   P P P LML L  . yn t n /  . yn t n / D .t n m L 00n1  t nC1 m L 00n / D m L 001 D m L; nD0

nD0

nD0

(2.5.18) P n whence m L0 D 1 nD0 yn t by Proposition 2.5.12. Moreover, our assumption (b)(iv) that jc  .a/jn  k .x.n/ /T kL is bounded together L 00n kL D jc  .a/jn  k .x.n/ /T  .a / kL  jc  .a/jn  k .x.n/ /T kL  with jc  .a/jn  kt m P k.a / kL implies that jc  .a/jn  k.t m L 00n  m L 00n1 / kL D k `j D0 c  .a/n  cQj L j kL D maxfkc  .a/n  cQj kW 0  j  `g is bounded, say by a constant C1  1. Therekj

kj

`

fore, kL .j k/ .cQk /k  jc  .a/jnq C1q  jc  .a/jn=q C1 for 0  k  j ` ` and thus kcj k  jc  .a/jn=q C1 , whence kyn kL  jc  .a/jn=q C1 . Fix an s with 0 < s < q ` . Since C is complete with respect to j : j the restrictions of the norms k : kL and k : ks to the finite dimensional C-vector space W are equivalent by [Sch84, Theorem 13.3]. Thus there is a constant C2 with k : ks  C2  k : kL on W . Since m L 00n 2 W we obtain in particular kt nC1 m L 00n ks D jc  .a/js.nC1/km L 00n ks  L 00n kL C2 D jc  .a/js.nC1/k .x.n/ /kC2 D jc  .a/jn.1s/Cs jc  .a/jn jc  .a/js.nC1/ km k .x.n/ /kC2 for all n, and hence, limn!1 kt nC1 m L 00n ks D 0. Moreover, kyn ks  ` kyn kL C2  jc  .a/jn=q C1 C2 for all n, whence limn!1 kyn t n ks D limn!1 kyn ks jc  .a/jsn D 0. This shows that even m L 0 2 Ch #ts i˚r and equation (2.5.18) holds in t ˚r Ch # s i . L 2 CŒtrr representing L L with respect to the basis B has deterThe matrix ˆ M  minant c  .t  #/d for a c 2 C due to the elementary divisor theorem and the L Š ML =L L .L  ML / Š Cd is annihilated by .t  #/d 2 J d . condition that coker ˆ M L ad 2 CŒtrr be the adjoint matrix which satisfies ˆ L ad ˆ L D c  .t  #/d Idr . Let ˆ  Q . 1  t Recall the element ` WD O . C / from Example 2.3.20(b) which 1  2 C i D0  qi #

satisfies ` D  #1 .t /d  m L0 2 y 0 WD   .`  





 #/   ` . Multiplying (2.5.18) with Ch #ts i˚r and applying   we obtain

y0 D  



d   .`  /

.#/d c

L ad , setting ˆ

 1 1   d L ad L ad y 0  .` / m L C ˆ ˆ  .#/d c .#/d c

Since   .y 0 / 2 Ch # tqs i˚r this shows that y 0 2 Ch # tqs i˚r and iteratively y 0 2 . Ch #ts0 i˚r for all s 0 D q k s, whence y 0 2 ML ˝AC O .CC / and m L 0 D   .` /d y 0 2  S . . d O.CC /. If P 2 CC X i 2N>0 V. i  J / is a point, that is P D V.I / ML ˝AC   .` / . for a maximal ideal I O .CC / with I ¤  i  J for all i 2 N>0 , such that P lies in the zero locus of   .` /, then we make the 

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

119

. Claim: m L 0 2 ML ˝AC OC. C ;P for the local ring OC. C ;P of CC at P . S When the claim holds for all those P , we derive m L 0 2 ML ˝AC O CC X i 2N>0  V. i J / , that is assertion (b)(ii). n To prove the claim let n 2 N>0 be the integer with t  # q 2 I , which exists nj because   .` / vanishes at P . Then t  # q 2 L j  I and L j  I ¤ J for all 0 <   ML ˝AC OC. C ;V.L j  I / is an isomorphism. j  n. Thus LML W L  ML ˝AC OC. C ;V.L j  I / !    For j D n it follows from  .` /jt D# ¤ 0 that   .` / 2 .OC. ;V.L n I / / and 



C

/d y 0 2 ML ˝AC OC. C ;V.L n I / . Therefore, m L 0 D   .`     1  1 . L L Cm L 0 / 2   LM m L 0 D   LM L .m L .M ˝AC OCC ;V.L n I / / D ML ˝AC OC. C ;V.L .n1/ I / and iteratively this yields m L 0 2 ML ˝AC OC. C ;V.L j  I / for j D n; : : : ; 0. So our claim and with it assertion (b)(ii) is proved. 6. Conversely, toPprove (b)(i)H)(b)(iii), we keep the notation from part 4 above i and write m L 0 as 1 2 C[[t]]˚r with bi 2 Cr and assume (b)(i), that is i D0 bi t n P limi !1 bi D 0 in Cr . For each n 2 N we set m L 0n WD bi t i 2 ML and m L 0>n WD 1 P i DnC1

i D0

i

bi t . Then

L  LML .L m L 0n / C m L 0n / D t n1 .LML .L  m L 0>n /  m L 0>n / 2 ML : m L 00n WD t n1 .m Note that the entries of m L 00n are polynomials in CŒt whose degree is bounded by a bound which is independent of n and only depends on the degrees of the entries of m L L 2 CŒtrr representing L L with respect to the basis B . It follows and of the matrix ˆ M that all m L 00n lie in a finite dimensional C-vector space V . By [Sch84, Theorem 13.3] the restrictions of k : k0 and k : kL to V are equivalent. From limi !1 bi D 0 it L km L 0>n k0 D 0. Thus kLML .L  m L 0>n /k0  kˆk L 0>n k1=q implies follows that limn!1 km 0 0 P 00 00 00  j L n k0 D 0, and hence, limn!1 km L n kL D 0. If .m L n/ D 2 limn!1 km j cj L 1d 00 then n

0 implies k m L k D maxfkc kW j  0g  1 and thus .x / D Cf g L j .n/ n L   P ı1 .m L 00n / D j  j  .cj /T satisfies L 00n kL : kx.n/ k  maxfk j  .cj /kW j  0g  maxfkcj kW j  0g D km Therefore, .x.n/ /n is convergent. Thus (b)(i) implies (b)(iii). 7. Finally, for the last statement of the theorem the notation from parts 4 and P1 we keep 0 i 6 above and assume moreover, that m L D b t satisfies (b)(ii). Let 1 < s < q. i D0 i S P . t ˚r i , Then Sp Ch #ts i CC X i 2N>0 V. i J / and this implies 1 i D0 bi t 2 Ch # s i  si that is limi !1 kbi k jc .a/j D 0; see (2.5.17). Fix a real number " > 0 with "  L q=.q1/ . Then there is an n0 2 N such that kbi k jc  .a/ji s=q  kbi k jc  .a/ji s < " kˆk s=q L L 0>n k  km L 0>n ks < "  kˆk "1=q and for all i  n0 . So n  n0 implies km s=q

kLML .L



m L 0>n /ks=q

L  kˆk  kL s=q



m L 0>n ks=q

s=q

L D kˆk  s=q

km L 0>n k1=q s

L < kˆk "1=q ; s=q

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P L and hence, km L 00n ks=q < jc  .a/j.nC1/s=q kˆk "1=q . We write .m L 00n / D i ci L i 2 s=q CfL g1d . This time we use that by [Sch84, Theorem 13.3] the restrictions of k : ks=q and k : kL to V are equivalent. So there is a constant C3 such that km L 00n kL WD L "1=q C3 for all n  n0 . By enlarging supfkci kW i  0g < jc  .a/j.nC1/s=q kˆk s=q  .nC1/s=q L kˆks=q "1=q C3  1. Therefore, kci k  1, n0 we may assume that jc .a/j i

whence k i  ci k D kci kq  kci kq for all i  1. So X     L q "C q : k ı1 .m L 00n /  .Lie / ı0 .m L 00n / k D k  i  .ci /T k < jc  .a/j.nC1/s kˆk 3 s=q i 1  By choosing the isomorphism W E ! Gda;C appropriately in the beginning we may 1  assume that Lie. ı 'a ı / D c .a/.Idd CN / for a nilpotent matrix N with only 0 and 1 as entries. This yields    L 00n //  .Lie /.ı0 .m L 00n //   (2.5.19) lim Lie. ı 'anC1 ı 1 / .ı1 .m n!1



L q "C q D 0 lim jc  .a/j.nC1/.1s/kˆk 3 s=q

n!1

  L 00n // D .Lie /./. By Theorem 2.5.20 we have limn!1 Lie. ı 'anC1 ı 1 / .ı1 .m So we must compute   L 00n // D .Lie / ı0 .t nC1 m L 00n / (2.5.20) Lie. ı 'anC1 ı 1 / ı .Lie /.ı0 .m   D .Lie / ı0 .m L Cm L 0n  LML .L  m L 0n //   L Cm L 0n / : D .Lie / ı0 .m  Since the projection ı0 W Ch #ts i˚d  ML =J d ML ! Lie E from Proposition 2.5.8 is 0 continuous with respect to k : ks and limn!1 km L m L 0n ks D limn!1 km L 0>n ks D 0, 0 0 we find limn!1 ı0 .m L Cm L n / D ı0 .m L Cm L /. In combination with (2.5.19) and L Cm L 0 / and establishes the theorem.  (2.5.20) this proves that  D ı0 .m

Corollary 2.5.22 ([ABP02]). Let C D P1Fq , A D Fq Œt, AC D CŒt and # D c  .t/.   Gda;C of Fq -module schemes Then O .CC XDC D Chti. Fix an isomorphism W E ! P 1 j d d D EndFq ;C .Gda;C / with and write ı 't ı DW t D j 0 t;j  2 Cf g  2 Cd d and t;j D 0 for j 0. For   0 consider the columns of the matrix Pt;j j d d as elements of Cf gd Š ML via . Note that this matrix j 0 t; Cj  2 Cf g is zero for  1. In the situation of Theorem 2.5.21 let .x.n/ /n be a t-division tower above x and let f WD

1 X nD0

.x.n/ /t n 2 C[[t]]d

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be the associated Anderson generating function. Then the bijection (2.5.16) from Theorem 2.5.21 sends .x.n/ /n to the element  X X t; Cj  j   .f / 2 ML t D ML ˝CŒt  C[[t]] : (2.5.21) m L0 D  1

j 0

Moreover, the t-division tower .x.n/ /n is convergent if and only if f 2 Chtid . Proof. In step 5 of the proof of Theorem 2.5.21 we obtain .x.n1/ / D  j  .x.n/ / and

P

j 0 t;j



L 00n1 / D .x.n/ /T  .t /  .x.n1//T .t m L 00n  m X X D .x.n/ /T  L j  .t;j /T L j   j  .x.n/ /T  Tt;j j 0

D

X

j 0

.L j  1/   j  .x.n/ /T  Tt;j

j 1

D

X

.L  1/ 

j 1 j D Ci

D

1 jX

 L i   j  .x.n/ /T  Tt;j

i D0

.L  1/ 

X X

  .x.n/ /T  L i  Tt; Ci



1 i 0

D .L  1/ 

X X

t; Ci   i    .x.n/ /

 :

1 i 0  Since also .t m L 00n  m L 00n1 L L  yn // D .1  L /  .yn / Proposition 2.5.8 P.yn  . P/ D implies that yn D  1 i 0 t; Ci   i    .x.n/ /. Multiplying with t n and summing over all n  0 yields

m L0 D

1 X nD0

yn t n D 

XX

1  X t; Ci   i    .x.n/ /t n

1 i 0

nD0

and establishes (2.5.21). Finally, if .x.n/ /n is convergent then by definition f 2 Chtid . Conversely, the latter together with (2.5.21) implies that m L 0 2 ML ˝CŒt  Chti.  By Theorem 2.5.21 this is equivalent to .x.n/ /n being convergent. The following corollary is the analog in terms of dual A-motives of Sinha’s diagram [Sin97, 4.2.3].

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Corollary 2.5.23. Let E be an A-finite Anderson A-module and let .ML ; LML / D  . S L 0 2 ML ˝AC O CC X i 2N>0 V. i  J / such ML .E/ be its dual A-motive. For every m L 0/  m L 0 2 ML we have that m L WD LML .L  m   L 0 C m/ L D ı1 .m/ L : expE ı0 .m Proof. This follows from the last statement of Theorem 2.5.21 and Theorem 2.5.20.  Corollary 2.5.24. The morphism ı0 W ML ! Lie E from Proposition 2.5.8 restricts to an A-isomorphism  L  . S ! ƒ.E/ D ker.expE / : ı0 W ML ˝AC O CC X i 2N>0 V. i J /  L . S Proof. Let m L 0 2 ML ˝AC O CC X i 2N>0 V. i  J / , that is m L WD LML .L  m L 0/  0 L D 0. By Theorems 2.5.21 and 2.5.20 both sides of the m L D 0. Then x WD ı1 .m/ claimed isomorphism are in bijection with the set of convergent a-division towers above 0. By the last statement of Theorem 2.5.21 the combined bijection equals ı0 , which is A-linear by Proposition 2.5.8.  2.5.4 Purity and mixedness. Before we define purity of Anderson A-modules which are abelian or A-finite in terms of the corresponding (dual) A-motives, we show that the functors E 7! M .E/ and E 7! ML .E/ are exact. Proposition 2.5.25. Let E 0 E be an Anderson A-submodule. Then the quotient E 00 WD E=E 0 exists as an Anderson A-module with dim E 00 D dim E  dim E 0 . (a) E is abelian if and only if both E 0 and E 00 are abelian. In this case rk E 00 D rk E  rk E 0 and the induced sequence of A-motives 0

/ M .E 00 /

/ M .E/

/ M .E 0 /

/0

is exact in the sense of Remark 2.3.5(b) (that is, the sequence of the underlying AC -modules is exact). (b) E is A-finite if and only if both E 0 and E 00 are A-finite. In this case rk E 00 D rk E  rk E 0 and the induced sequence of dual A-motives 0

/ ML .E 0 /

/ ML .E/

/M L .E 00 /

/0

is exact in the sense of Remark 2.4.4(b) (that is, the sequence of the underlying AC -modules is exact). Proof. Let E D .E; '/ and E 0 D .E 0 ; ' 0 /. Then the quotient E 00 WD E=E 0 is a smooth irreducible group scheme with dim E 00 D dim E  dim E 0 by [Bor69, Theorem II.6.8] and isomorphic to a power of Ga;C by [Ser88, Proposition VII.11]. It

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

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inherits an action ' 00 W A ! EndC .E 00 / of A satisfying (2.5.1) in Definition 2.5.2(a), because Lie E 00 D Lie E= Lie E 0 . Indeed, E ! E 00 is smooth because E 0 is smooth over C and so Lie E ! Lie E 00 is surjective by [BLR90, § 2.2, Proposition 8] with Lie E 0 contained in its kernel. By reasons of dimension Lie E 0 equals the kernel of Lie E  Lie E 00 . We obtain an exact sequence of Anderson A-modules / E0

0

f0

/E

f 00

/ E 00

/ 0:

(2.5.22)

(a) We apply the contravariant functor M . : / from Definition 2.5.5. This yields an exact sequence of AC -modules 0

/ M.E 00 /

/ M.E/

/ M.E 0 / :

(2.5.23)

It is exact on the left because E  E 00 is surjective. It is also exact in the middle by the universal mapping property of the quotient E 00 ; see [Bor69, II.6.1]. If E 0 and E 00 are abelian, that is M.E 0 / and M.E 00 / are finite locally free over the Dedekind domain AC , then also M.E/ is finite locally free and E is abelian. Conversely, if M.E/ is finite locally free, then also M.E 00 / is, and E 00 is abelian. If E is abelian it remains to prove that M.E/ ! M.E 0 / is surjective and E 0 is e WD M .E/=M .E 00 / which injects into M .E 0 /. abelian. We consider the quotient M e. Since M .E 0 / Since M .E/ is finitely generated both over AC and over Cf g, so is M e e has no Cf g-torsion the same holds for M , and so M is locally free over AC by e is an effective A-motive. If M e Š M .E 0 / [And86, Lemma 1.4.5]. Therefore, M 0 this will imply that E is abelian. By [And86, Theorem 1] there exists an abelian e with M e D M .E/ e and a morphism E e ! E induced from Anderson A-module E e e M .E/  M . Any Cf g-basis .m Q 1; : : : ; m Q dQ / of M provides an isomorphism m Q1   dQ e :::  m Q dQ W E ! Ga;C of Fq -module schemes, and if Ga;C D Spec CŒx then Q j .x/ for j D 1; : : : ; ıQ are free generators of the polynomial algebra the xQ j WD m e is a quotient of M .E/ the m e O / D CŒxQ 1 ; : : : ; xQ Q  over C. Since M Q j .x/ lie .E; e E d e ! E is a closed immersion. Let m0 be in the image of .E; OE /. Therefore, E j

the image of m Q j in M.E 0 /. Sending xQ j to .m0j / .x/ defines a C-homomorphism e ,! M .E 0 / induce e O / ! .E 0 ; OE 0 /. In this way the maps M .E/  M .E; e E morphisms E0

/E e

/E

/ E 00 :

e is the zero map, the closed immerSince the composite map M.E 00 / ! M.E/ ! M e e sion E ,! E factors through the kernel of E ! E 00 , which equals E 0 . So E 0 ! E 0 e e must be an isomorphism. This shows that M .E / D M .E/ D M onto which M .E/ surjects. Thus the sequence (2.5.23) is also exact on the right. From this also the formula for rk E 00 follows.

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(b) We apply the covariant functor ML . : / from Definition 2.5.9 to the sequence (2.5.22). This yields an exact sequence of AC -modules 0

/M L .E 0 /

/M L .E/

/ ML .E 00 / :

(2.5.24)

It is exact on the left because E 0 ,! E is a closed immersion. It is also exact in the middle because E 0 equals the fiber of E  E 00 above 0. If E 0 and E 00 are Afinite, that is ML .E 0 / and ML .E 00 / are finite locally free over the Dedekind domain AC , then also ML .E/ is finite locally free and E is A-finite. Conversely, if ML .E/ is finite locally free, then also ML .E 0 / is, and E 0 is A-finite. If E is A-finite it remains to prove that ML .E/ ! ML .E 00 / is surjective and E 00 is A-finite. We consider the quotient NL WD ML .E/=ML .E 0 / which injects into ML .E 00 /. Since ML .E/ is finitely generated both over AC and over CfL g, so is NL . Since ML .E 00 / has no CfL g-torsion the same holds for NL , and so NL is locally free over AC by the L -analog of [And86, Lemma 1.4.5]. Therefore, NL is an effective dual A-motive. If NL Š ML .E 00 / this will imply that E 00 is A-finite. By Theorem 2.5.11 there exists an e with NL D ML .E/ e and morphisms fQW E ! E e and A-finite Anderson A-module E 00 00 00 e L L L gW E ! E induced from M .E/  N ,! M .E / and satisfying f D g ı fQ. Since the composite map ML .E 0 / ! ML .E/ ! NL is the zero map, the morphism e is the zero morphism by Theorem 2.5.11. By the universal fQ ı f 0 W E 0 ,! E ! E mapping property [Bor69, II.6.1] of the quotient E=E 0 D E 00 the morphism fQW E ! e Again by the universal mapping e factors as fQ D hıf 00 for a morphism hW E 00 ! E. E 00 00 00 property, f D ghıf implies that gh D idE . Therefore, ML .g/ıML .h/ D idML .E 00 / and ML .g/ is surjective. As it is injective by construction we have NL Š ML .E 00 / and the proposition is proved.  Corollary 2.5.26. The category of abelian, respectively A-finite, Anderson A-modules is an exact category in the sense of Quillen [Qui73, §2] (see Remark 2.3.5(b) for explanations) if one calls the sequences E 0 ! E ! E 00 of Anderson A-modules exact where E 0 E is an Anderson A-submodule and E 00 WD E=E 0 is the quotient from Proposition 2.5.25. The functors E 7! M .E/ from Theorem 2.5.7, respectively E 7! ML .E/ from Theorem 2.5.11(b), are exact equivalences, that is, a sequence E 0 ! E ! E 00 is exact if and only if the induced sequence of A-motives, respectively dual A-motives, is exact. Proof. We start by proving the second assertion. By Proposition 2.5.25 the functors map exact sequences to exact sequences. Let E 0 ! E ! E 00 be a sequence of abelian Anderson A-modules whose associated sequence of A-motives 0 ! M .E 00 / ! M .E/ ! M .E 0 / ! 0 is exact in  the sense of Remark 2.3.5(b). Consider an isomorphism 0 D . 10 ; : : : ; d0 0 /W E 0 ! 0 Gda;C where i0 W E 0 ! Ga;C D Spec CŒx is the projection onto the i -th coordinate. Then i0 2 M .E 0 / and .E 0 ; OE 0 / is generated by i .x/. Since M .E/ surjects

Pink’s theory of Hodge structures and the Hodge conjecture over function fields

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onto M .E 0 /, we see that i0 lies in the image of .E; OE / ! .E 0 ; OE 0 /, and e WD E=E 0 be the quotient from hence E 0 ! E is a closed immersion. Let E 00 e both equal the kernel of Proposition 2.5.25. Then the A-motives M .E / and M .E/ 0 e M .E/  M .E / by Proposition 2.5.25. By Theorem 2.5.7 this shows that E 00 Š E, 0 00 and hence the sequence E ! E ! E is exact as desired. On the other hand, let E 0 ! E ! E 00 be a sequence of A-finite Anderson Amodules whose associated sequence of effective dual A-motives 0 ! ML .E 0 / ! ML .E/ ! ML .E 00 / ! 0 is exact in the sense of Remark 2.4.4(b), that is on the underlying AC -modules. Applying the snake lemma to 0

/ L  M L .E 0 / _ LM L .E 0 /

0



/ ML .E 0 /

/ L  M L .E/ _ LM L .E/



/M L .E/

/ L  M L .E 00 / _

/0

LM L .E 00 /

 /M L .E 00 /

/0

yields by (2.5.5) that the sequence on tangent spaces at the origin 0 ! Lie E 0 ! Lie E ! Lie E 00 ! 0 is exact. Analogously, (2.5.4) yields that the sequence 0 ! E 0 .C/ ! E.C/ ! E 00 .C/ ! 0 is exact. Both sequences together show that E 0 ,! e WD E=E 0 be the quotient from Proposition 2.5.25. E is a closed immersion. Let E e both equal the cokernel of ML .E 0 / ,! Then the dual A-motives ML .E 00 / and ML .E/ e and L M .E/ by Proposition 2.5.25. By Theorem 2.5.11(b) this shows that E 00 Š E, 0 00 hence the sequence E ! E ! E is exact as desired. The first statement now follows from Remark 2.3.5(b), respectively Remark 2.4.4(b).  Definition 2.5.27. (a) An abelian Anderson A-module E of dimension d and rank r is pure if M .E/ is pure. In this case, we set wt E D  wt M .E/ D  dr ; see [And86, Lemma 1.10.1]. (b) An abelian Anderson A-module E is mixed if it possesses an increasing weight filtration by abelian Anderson A-submodules W E for  2 Q such that  S 0E E WD W E= W is a pure abelian Anderson A-module of GrW 0     1 MO " .M / and call the local -shtuka the construction is slightly more complicated; compare the discussion in [BH11, after Proposition 8.4]. Namely, by continuity the map  extends to a ring homomorphism   W A" ,! R. We consider the canonical isomorphism F" [[z]] ! A" and the ideals Q i q ai D .a˝11˝.a/ W a 2 F" / A";R for i 2 Z=f Z, which satisfy i 2Z=f Z ai D Q i .0/, because i 2Z=f Z .X  aq / 2 Fq ŒX  is a multiple of the minimal polynomial of a over Fq and even equal to it when F" D Fq .a/. By the Chinese remainder theorem A";R decomposes Y A";R =ai : (3.2.1) A";R D i 2Z=f Z

Each factor is canonically isomorphic to R[[z]]. The factors are cyclically permuted by  because  .ai / D ai C1 . In particular  f stabilizes each factor. The ideal J decomposes as follows J  A";R =a0 D .z  / and J  A";R =ai D .1/ for i ¤ 0. We define the local -shtuka O at " associated with M as MO " .M / WD .MO ; MO / WD   f f M ˝AR A";R =a0 ; .M ˝ 1/ , where M WD M ı   M ı : : : ı  .f 1/ M . Of course if f D 1 we get back the definition of MO " .M / given above. Also note if M is effective, then M=M .  M / D MO =MO .O  MO /.

189

Local shtukas, Hodge–Pink structures and Galois representations

The local shtuka MO " .M / allows to recover M ˝AR A";R via the isomorphism 0 1 f 1 f 1 M M M .M ˝ 1/i mod ai W @  i  .M ˝AR A";R =a0 /; .M ˝ 1/f ˚ idA i D0

i D0

i ¤0 

! M ˝AR A";R ; because for i ¤ 0 the equality J A";R =ai D .1/ implies that M ˝1 is an isomorphism modulo ai ; see [BH11, Propositions 8.8 and 8.5] for more details. Note that M 7! MO " .M / is a functor. The philosophy is that MO " .M / encodes all the local information of M at " like the (dual) Tate module TLv M encodes all the local information of M at v ¤ ". The whole example also works over a base scheme S 2 NilpA" instead of S D Spec R; see [Har16, Example 7.2] and [HS15, § 6]. We will need the following lemma whose second part is proved more generally for S 2 NilpA" in [HS15, Lemma 2.3]. Lemma 3.2.3. Let MO D .MO ; MO / be a local shtuka of rank r over a valuation ring R as in Notation 3.1.1. (a) There is an integer d 2 Z such that det MO 2 .z  /d R[[z]] . (b) If MO is effective, the integer d from (a) satisfies d  0 and MO =MO .O  MO / is a free R-module of rank d which is annihilated by .z  /d . Proof. Compare [Har11, Proposition 2.1.3 and Lemma 2.1.2]. (a) Since R[[z]] is a local ring, we may choose a basis of MO and non-negative integers s; t such that the matrices of .z  /s  MO and .z  /t   1 with respect to this O M rs basis lie in Mr .R[[z]]/. Set f D .z  / det MO and g D .z  /rt det  1 . Then O M

f; g 2 R[[z]] satisfy fg D .z  /r.sCt / . Since R is an integral domain, .z  /R[[z]] is a prime ideal. So f 2 .z  /u  R[[z]] for a non-negative integer u, and so det MO 2 .z  /urs R[[z]] . (b) Let d be the integer from (a). Since M is effective we may take s D 0, and hence d  0. By Cramer’s (e.g. [Bou70, III.8.6, Formulas (21) and (22)]) the matrix   rule d lies in M R[[z]] . This implies that MO =MO .O  MO / is annihilated .z  / of  1 r O M

by .z  /d . By [Har11, Lemma 2.1.2] it is a finite free R-module. We can compute its rank after reducing modulo mR . Then we are over the  principal ideal  domain k[[z]] and the elementary divisor theorem tells us that dimk MO =MO .O  MO / ˝R k D   ordz .det MO mod mR / D ordz .z  /d mod mR D d .  More precisely [Har11, Proposition 2.1.3] (and the lemma) say that all (effective) local shtukas over Spec R are bounded (by .d; 0; : : : ; 0/ for d D rkR MO =MO .O  MO /) as in the following

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Definition 3.2.4. Let 1  : : :  r be a decreasing sequence of integers. A local shtuka .MO ; MO / of rank r over S is bounded by .1 ; : : : ; r / if   ^i MO ^i O  MO .z  /ri C1 C:::Cr  ^i MO for 1  i  r with equality for i D r : Although over a general base boundedness is not preserved under quasi-isogenies this is true over a valuation ring. Lemma 3.2.5. Let MO and MO 0 be two isogenous local shtukas over a valuation ring R as in Notation 3.1.1. If MO is bounded by .d; 0; : : : ; 0/, respectively satisfies .z  /a MO MO .O MO / .z  /b MO , then the same is true for MO 0 . Proof. Choosing bases of M and M 0 we write M and M 0 as matrices T; T 0 2 1  GLr .R[[z]]Œ z / and the quasi-isogeny f W M Œ 1z  ! M 0 Œ 1z  as a matrix F 2 GLr .R[[z]]Œ 1z / satisfying T 0 D F  T  O  .F /1 . Depending on the assumption on M , the matrix coefficients of T 0 , .z  /d det.T 0 / and .z  /d det.T 0 /1 , respectively 1 .z  /b T 0 and .z  /a .T 0 /1 lie in R[[z]]Œ 1z . Since they also lie in R[[z]]Œ z , we s t t s can write them as f =z D g=.z  / . The term .z  / f D z g lies in the prime ideal .z  / R[[z]], but z does not. Therefore g is divisible by .z  /t and all the  matrix coefficients g=.z  /t lie in R[[z]] as desired. Remark 3.2.6. If we are considering a local shtuka MO over Spec R for a valuation ring R as in Notation 3.1.1, we obtain for all i 2 N a local shtuka MO .i / WD MO ˝R R=. i / over Spec R=. i /. The MO .i / form a local shtuka over Spf R by which we mean a projective system .MO .i / /i 2N of local shtukas MO .i / over Spec R=. i / to gether with isomorphisms MO .i C1/ ˝R=. i C1 / R=. i / ! MO .i / . By [HV11, Proposition 3.16 and § 4] the functor MO 7! .MO .i / /i 2N is an equivalence between local shtukas bounded by .1 ; : : : ; r / over Spec R and over Spf R. In that sense the theory of local shtukas over Spec R is subsumed under the theory of bounded local shtukas over schemes S 2 NilpA" . Example 3.2.7. We discuss the case of the Carlitz module [Car35]. We keep the notation from Example 3.2.2 and set A D Fq Œt. Let Fq .#/ be the rational function field in the variable # and let  W A ! Fq .#/ be given by  .t/ D #. The Carlitz motive over Fq .#/ is the A-motive M D Fq .#/Œt; t  # . Now let " D .z/ A be a maximal ideal generated by a monic prime element z D z.t/ 2 Fq Œt. Then F" D A=.z/ and A" is canonically isomorphic to F" [[z]]. Let R F" [[]] be a valuation ring as in Notation 3.1.1 and let # D .t/ 2 R. The Carlitz motive has a model over R with good reduction given by the A-motive M D .RŒt; t  #/ over R. If degt z.t/ D 1, that is z.t/ D t  a for a 2 Fq , then F" D Fq ,  D #  a, and z   D t  #. So MO " .M / D .R[[z]]; z  /.

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 f 1  If degt z.t/ D f > 1, then MO " .M / D R[[z]]; .t  #/.t  # q /    .t  # q / . q q f 1 Here the product .t #/.t # /    .t # / D .z /u for a unit u 2 F" [[]][[z]] ,  because M . M / D .t  #/M implies that MO " .M / is effective and the d from Lemma 3.2.3 is 1. In order to get rid of u we denote the image of t in F" by . f 1 / Then F" D Fq ./ and z.t/ equals the minimal polynomial .t  /    .t  q of  over Fq . Moreover, t   mod zA" and #   mod F" [[]]. We compute in F" [[]][[z]]=./ z.t/ D .t  /    .t  q

f 1

/  .t  #/    .t  # q

f 1

/  .z  /u  zu mod  :

Since z is a non-zero-divisor in F" [[]][[z]]=./ it follows that u  1 mod  F" [[]][[z]]. We write u D 1 C u0 and observe that the product w WD

1 Y nD0

O n .u/ D

1 Y

O n .1 C u0 / D

nD0

1 Y   n 1 C  qO O n .u0 / nD0

converges in F" [[]][[z]] because F" [[]][[z]] is -adically complete. It satisfies w D u  O .w/ and so multiplication with w defines a canonical isomorphism .R[[z]]; z   / ! MO " .M /. We conclude that MO " .M / D .R[[z]]; z  /, regardless of degt z.t/.

3.3 Divisible local Anderson modules Let S 2 NilpA" and let MO D .MO ; MO / be an effective local shtuka over S . Set MO n WD .MO n ; MO n / WD .MO =z n MO ; MO mod z n /. For m 2 MO n set O O m WD m ˝ 1 2 M MO n ˝OS [[z ]]; O OS [[z]] DW O  MO n . Drinfeld [Dri87, § 2] associates with MO n a group scheme  ı   On (3.3.1) DrqO .MO n / WD Spec SymOS .MO n / m˝qO  MO .OM m/W m 2 M O D Spec OS Œm1 ; : : : ; mnr 

ı qO   mi  MO .OM O mi /W 1  i  nr ;

Lnr

(3.3.2)

OS  mi locally on S . It has the following properties • DrqO .MO n / Spec SymOS .MO n / is a finite locally free subgroup scheme over S of order nr, that is, the OS -algebra ODr .MO n / is a finite locally free OS qO L module of rank nr. Note that locally on S we have MO n D nr i D1 OS  mi and nr O Spec SymOS .Mn / Š Spec OS Œm1 ; : : : ; mnr  D Ga;S .

if MO n D

i D1

• DrqO .MO n / inherits from MO n an action of A" =.z n / D F" Œz=.z n /. • The Verschiebung map is zero on DrqO .MO n /.

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• There is a canonical  nisomorphism between  O coker MO n D M = z MO C MO .O MO / and the co-Lie module where eW S ! DrqO .MO n / is the zero section. See !Dr .MO n / WD e 1 O qO

DrqO .M n /=S

[HS15, Theorem 5.2(f)] for a proof. • DrqO .MO n / is a strict F" -scheme in the sense of Faltings [Fal02] and Abrashkin [Abr06]. See [Abr06, Theorem 2] for a proof, or [HS15, § 5]. n Conversely we recover MO n as the  A" =.z /-module  of F" -linear morphisms of S group schemes HomS-groups;F" -lin DrqO .MO n / ; Ga;S by [Abr06, Theorem 2] or [HS15, Theorem 5.2]. Moreover the structure as OS -module is given via the action of OS on the additive group scheme Ga;S and MO n corresponds to the map O W m 7! Fq;G O a;S =S ı O of Ga;S over S . More precisely, since m, where Fq;G O a;S =S is the relative q-Frobenius O .bm/ D b qO .m/ O for b 2 OS , the map O is -semilinear O and satisfies MO n .O O m/ D Mn O .m/ D Fq;G O a;S =S ı m. The canonical epimorphisms MO nC1 ! ! MO n induce closed immersions in W DrqO O O .M n / ,! DrqO .M nC1 /. The inductive limit DrqO .MO / WD lim DrqO .MO n / in the cate! gory of sheaves on the big fppf-site of S is a sheaf of F" [[z]]-modules that satisfies the following Definition 3.3.1. A z-divisible local Anderson module over S is a sheaf of F" [[z]]modules G on the big fppf-site of S such that (a) G is z-torsion, that is G D lim GŒz n , where GŒz n  WD ker.z n W G ! G/, !

(b) G is z-divisible, that is zW G ! G is an epimorphism, (c) For every n the F" -module GŒz n  is representable by a finite, locally free, strict F" -module scheme over S in the sense of Faltings [Fal02] and Abrashkin [Abr06], and (d) locally on S there exists an integer d 2 Z0 , such that .z  /d D 0 on !G where !G WD lim !GŒz n  and !GŒz n  D e 1GŒz n =S is the pullback under the 

zero section eW S ! GŒz n . A morphism of z-divisible local Anderson modules over S is a morphism of fppfsheaves of F" [[z]]-modules. The category of divisible local Anderson modules is F" [[z]]-linear. It is shown in [HS15, Lemma 8.2 and Theorem 10.8] that !G is a finite locally free OS -module and we define the dimension of G as rk !G . Note that in [Har05, Definition 6.2] and W. Kim [Kim09, Definition 7.3.1] different definitions of z-divisible local Anderson modules were given. Unfortunately, the latter definitions are both wrong, because the strictness assumption in (c) is missing. Remark 3.3.2. By [HS15, Theorem 8.3] the functor MO 7! DrqO .MO / is an antiequivalence between the category of effective local -shtukas O over S and the category of z-divisible local Anderson modules over S . Moreover, it is A" -linear and exact.

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Various properties are preserved under this anti-equivalence. More precisely, let MO be an effective local -shtuka O over S and let G D DrqO .MO /. Then the OS [[z]]-modules MO =MO .O  MO / and !G are canonically isomorphic. In particular, MO is étale, if and only if !G D .0/, if and only if all GŒz n  are étale. The map MO is topologically nilpotent, in the sense that locally on S there is an integer n such that im. nO W O n MO ! MO / z MO , if and only if G is a formal Lie M group. If MO is bounded by .d; 0; : : : ; 0/ we say that G is bounded by d . In this case .z  /d  !G D .0/ globally on S in axiom (d) and dim G D d as can be seen from the elementary divisor theorem applied to the pullback s  MO to a closed point sW Spec .s/ ! S , where .s/ is the residue field at s. Remark 3.3.3. If MO is a local shtuka over a valuation ring R as in Notation 3.1.1 and we view it as a projective system of local shtukas MO .i / WD .MO .i / ; MO .i / / WD MO ˝R R=. i / over Spec R=. i / as in Remark 3.2.6, then the following are equivalent. (a) The map  O is topologically nilpotent in the sense that  n . nMO /  MO C M

z MO for n 0.

O M

(b) For all i 2 N the map MO .i / is topologically nilpotent, that is  nO .i / .O nMO .i / / M z MO .i / for n 0. (c) There exists an i 2 N>0 for which the map MO .i / is topologically nilpotent. (d) DrqO .MO / WD lim DrqO .MO .i / / is a formal Lie group over Spf R. ! i

n .j 1/n Indeed, (a) yields  1CqO C:::CqO MO C z MO for all j 2 N. This implies (b), from which (c) follows trivially. Conversely if (c) holds for some i then  nO . n MO /  i MO Cz MO for n 0, whence (a). Finally, by Remark 3.3.2 assertion M (b) implies (d), and (d) implies that every DrqO .MO .i / / is a formal Lie group, whence (b).

 jOn . j nMO / M

Example 3.3.4. Let S D Spec B 2 NilpA" be affine and let d and r be positive integers. An abelian Anderson A-module of rank r and dimension d over S is a pair E D .E; '/ consisting of a smooth affine group scheme E over S of relative dimension d , and a ring homomorphism 'W A ! EndS-groups .E/; a 7! 'a such that (a) there is a faithfully flat morphism S 0 ! S for which E S S 0 Š Gda;S 0 as Fq -module schemes,  d (b) 'a  .a/ D 0 on !E WD e 1E=S for all a 2 A, where eW S ! E is the zero section, (c) the set M WD M.E/ WD HomS-groups;Fq -lin .E; Ga;S / of Fq -equivariant homomorphisms of S -group schemes is a locally free module over AB WD A ˝Fq B

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of rank r under the action given on m 2 M by A 3 aW M ! M; m 7! m ı 'a B 3 bW M ! M; m 7! b ı m In addition we consider the map  W m 7! Fq;Ga;S =S ı m on m 2 M , where Fq;Ga;S =S is the relative q-Frobenius of Ga;S over S . Since  .bm/ D b q  .m/ the map  is  semilinear and induces an AB -linear map M W  M ! M , which makes M .E/ WD  M.E/; M / into an effective A-motive over S in the sense of Example 3.2.2. The functor E 7! M .E/ is fully faithful and its essential image is described in [Har16, Theorem 3.5] generalizing Anderson’s description [And86, Theorem 1]. Now let E D .E; '/ be an abelian Anderson A-module over S and let MO WD O O at "; see Example 3.2.2. Let M " .M .E// be its associated effective local -shtuka n 2 N and let "n D .a1 ; : : : ; as / A. Then   EŒ"n  WD ker 'a1 ;:::;as WD .'a1 ; : : : ; 'as /W E ! E s is called the "n -torsion submodule of E. It is an A="n -module via A="n ! EndS .EŒ"n /; aN 7! 'a and independent of the set of generators of "n ; see [Har16, Lemma 6.2]. Moreover, by [Har16, Theorem 7.6] it is a finite S -group scheme of finite presentation and a strict F" -module scheme and there are canonical A="n -equivariant isomorphisms of finite locally free S -group schemes  DrqO .MO ="n MO / ! EŒ"n 

and

   HomS-groups;F" -lin EŒ"n  ; Ga;S MO ="n MO ! of torsion local shtukas in the sense of Definition 3.7.1 below. In particular, EŒ"1  WD lim EŒ"n  D DrqO .MO / is a z-divisible local Anderson module over S . !

Example 3.3.5. We continue with Example 3.2.7. Let A D Fq Œt and let " D .z/ A be a maximal ideal generated by a monic prime element z D z.t/ 2 Fq Œt. Let R D F" [[]] and let C WD .Ga;R ; '/ with 't D # CFq;Ga;R be the Carlitz module over R. That is, if Ga;R D Spec RŒx then 't .x/ D # x Cx q . Then M .C / D .RŒt; t #/ is the Carlitz Fq Œt-motive over R from Example 3.2.7 with associated local shtuka MO WD MO " .M .C // Š .R[[z]]; z  / at ". The "n -torsion submodule of C is C Œ"n  WD Ln ker.'z n / D Spec RŒx=.'zn .x//. We can compute MO =z n MO D i D1 R  mi with mi D z i 1 and MO .O O mi / D .z  /mi D mi C1  mi for 1  i < n and M MO .O O mn / D mn . By Example 3.3.4 this implies M

C Œ"n  Š DrqO .MO ="n MO / Š Spec RŒm1 ; : : : ; mn =.mqnO C mn ; mqiO  mi C1 C mi W 1  i < n/ : qO

On it z acts via 'z .mi / D mi C1 D mi C mi for 1  i < n and 'z .mn / D 0.

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Since MO .  MO / D .z  /MO  MO C z MO , Remark 3.3.3 implies that G WD C Œ"  Š DrqO .MO / is a formal Lie group over Spf R. Its dimension is 1 because !G Š MO =MO .O  MO / D R. Setting x WD m1 it follows that G Š Spf R[[x]] is the formal additive group scheme with the action of A" D F" [[z]] given by 'z .x/ D  x C x qO and 'a .x/ D a x for a 2 F" . So we can alternatively describe C Œ"n  as Spec RŒx=.'zn .x// with the latter expression for 'z . 1

3.4 Tate modules With a local shtuka one can associate a Galois representation. More precisely, let MO D .MO ; MO / be a local shtuka over Spec R for a valuation ring R as in Nota tion 3.1.1. Then MO induces an isomorphism MO W O MO ˝R[[z ]] K[[z]] ! MO ˝R[[z ]]  O K[[z]], because z   2 K[[z]] . So one could say that M ˝R[[z ]] K[[z]] is an “étale local shtuka over K”. Definition 3.4.1. With MO as above one associates the (dual) Tate module ˚

 TL" MO WD .MO ˝R[[z ]] K sep [[z]]/O WD m 2 MO ˝R[[z ]] K sep [[z]]W MO .OM O m/ D m and the rational (dual) Tate module ˚

 L O VL" MO WD m 2 MO ˝R[[z ]] K sep ((z))W MO .OM O m/ D m D T" M ˝A" Q" : One also sometimes writes H1" .MO ; A" / D TL" MO and H1" .MO ; Q" / D VL" MO and calls this the "-adic realization of MO . By Proposition 3.4.2 below, this defines a covariant functor TL" W MO 7! TL" MO from the category of local shtukas over R to the category RepA" Gal.K sep =K/ of continuous representations of Gal.K sep =K/ on finite free A" modules and a covariant functor VL" W MO 7! VL" MO from the category of local shtukas over R with quasi-morphisms to the category RepQ" Gal.K sep =K/ of continuous representations of Gal.K sep=K/ on finite dimensional Q" -vector spaces. For n 2 N we also define ˚

 .MO =z n MO /O .K sep / WD m 2 MO ˝R[[z ]] K sep [[z]]=.z n /W MO .OM O m/ D m D TL" MO =z n TL" MO ; which is a free A" =.z n /-module of rank equal to the rank of MO with TL" MO D lim .MO =z n MO /O .K sep /.



Proposition 3.4.2. TL" MO is a free A" -module of rank equal to rk MO and VL" MO is a Q" -vector space of dimension equal to rk MO . Both carry a continuous action of

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Gal.K sep =K/. Moreover the inclusion TL" MO MO ˝R[[z ]] K sep [[z]] defines a canonical isomorphism of K sep [[z]]-modules  MO ˝R[[z ]] K sep [[z]] TL" MO ˝A" K sep [[z]] !

(3.4.1)

which is functorial in MO and Gal.K sep =K/- and -equivariant, O where on the left O and on the right module module Gal.K sep =K/-acts on both factors and O is id ˝, Gal.K sep =K/ acts only on K sep [[z]] and O is .MO ı O O / ˝ . O In particular one can M sep =K/   Gal.K recover MO ˝R[[z ]] K[[z]] D TL" MO ˝A" K sep [[z]] as the Galois invariants. Remark 3.4.3. We actually prove the stronger statement that for every s > 1=qO the isomorphism (3.4.1) extends to a Gal.K sep =K/- and -equivariant O isomorphism  hW TL" MO ˝A" K sep h zs i ! MO ˝R[[z ]] K sep h zs i;

(3.4.2)

which is functorial in MO . Here for a field extension K of F" (()) and an s 2 R>0 we use the notation 1 ˚X

Kh zs i WD bi z i W bi 2 K; jbi j jjsi ! 0 .i ! C1/ : i D0

P qO i P i These are subrings of K[[z]] and the endomorphism W O i bi z 7! i bi z of K[[z]] restricts to a homomorphism W O Kh zs i ! Kh  zsqO i. Note that the O -equivariance of h means h ˝ idKh szqO i D MO ı O  h. 

Proof of Proposition 3.4.2 and Remark 3.4.3. That TL" MO is an A" -module and VL" MO is a Q" -vector space comes from the fact that the subring of -invariants O in K sep [[z]] is F" [[z]] D A" . We set r D rk MO , choose an R[[z]]-basis of MO , and write MO with respect to this   1 basis as a matrix T 2 GLr R[[z]]Πz  . Since z   is a unit in Kh  zsqO i for every P 1 i 1 i z , there is an inclusion R[[z]]Πz  ,! 1  s > 1=qO with .z  /1 D  1 i D0    z z Kh  sqO i, and we consider T as a matrix in GLr Kh  sqO i . We claim that there is   P i a matrix U 2 GLr K sep [[z]] with .U O / D T 1 U . We write T 1 D 1 i D0 Ti z and P1 n U D nD0 Un z . We must solve the equations .U O 0 / D T0  U0

and

.U O n/ D

n X

Tni  Ui :

(3.4.3)

i D0

By Lang’s theorem [Lan56, Corollary on p. 557] there exists a matrix U0 2 GLr .K alg / satisfying (3.4.3). Since the morphism GLr .K alg / ! GLr .K alg /; U0 7! U0 O .U0 /1 is étale, we actually have U0 2 GLr .K sep /. Then the second equation takes the form .U O 01 Un /  U01 Un D

n1 X i D0

O .U0 /1 Tni U0  .U01 Ui / :

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This is a system of Artin-Schreier equations for the coefficients of U01 Un which can be solved in K sep and this establishes our claim.  sep We show obtained r  that the matrix U 2 GLr K [[z]]  sep  in zthis way lies in GL z 1 O Since T 2 GLr Kh  sqO i GLr Kh zqO i K h  s i for every 1  s > 1=q.

O O j  c for all i , where jTi  qi j denotes the maximal there is a constant c  1 with jTi  qi qi O absolute value of the entries of the matrix Ti  . We write (3.4.3) as n X     O  n O / : D Tni  q.ni Ui  i  i.q1/ O Un  i D0

In view of jj < 1 this implies the estimate jUn  n jqO  c  maxf jUi  i j W 0  i  n g O from which induction yields jUn  n j  c 1=.q1/ for all n  0. In particular, if s > sep z 1=q, O then s qO > 1 and U 2 Mr K h  sqO i . But now the equation .U O / D T 1 U  sep z   sep z  shows that .U O / 2 Mr K h  sqO i , hence U 2 Mr K h  s i . A similar reasoning   with the equation .U O 1 / D U 1 T shows that also U 1 2 Mr K sep h zs i and   U 2 GLr K sep h zs i as desired.  MO Multiplication with U provides an isomorphism .K sep h zs i˚r ; O D id/ ! 

˝R[[z ]] K sep h zs i. Since .K sep h zs i/O D A" , it follows that TL" MO D U A˚r " is free of sep z ˚r L O rank r, and the inclusion T" M UK h s i induces the equivariant isomorphism 

(3.4.2). Since K sep h zs i K sep [[z]] this induces the isomorphism (3.4.1). The continuity of the Galois representation MO W Gal.K sep =K/ ! AutA" .TL" MO / means that for all n 2 N the subgroup f g 2 Gal.K sep =K/W MO .g/  id mod z n g is open. This is true because MO is given by the Galois action on the coefficients of  sep the matrix U , and so this subgroup contains the open subgroup Gal K =K.U0 ; : : : ;   Un1 / . Finally, the functoriality of h is clear. Remark 3.4.4. There is a statement similar to Proposition 3.4.2 for an étale local shtuka MO over a connected scheme S 2 NilpA" . For a geometric base point sN 2 S the (dual) Tate module and the rational (dual) Tate module TL" MO WD .MO ˝OS [[z ]] .Ns /[[z]]/O

and

VL" MO WD TL" MO ˝A" Q"

are free of rank rk MO and carry a continuous action of the étale fundamental group 1eK t .S; sN /. Moreover, the functor MO 7! TL" MO is an equivalence between the category of étale local shtukas over S and the category of representations of 1eK t .S; sN / on finite free A" -modules. Similarly, the functor MO 7! VL" MO is an equivalence between the category of étale local shtukas over S with quasi-morphisms and the category of representations of 1eK t .S; sN / on finite Q" -vector spaces; see [AH14, Proposition 3.4].

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Back in the situation over a valuation ring R as in Notation 3.1.1 there is the following Corollary 3.4.5. For local shtukas MO and MO 0 over R the A" -module HomR .MO ; MO 0 / is finite free of rank at most rk MO  rk MO 0 . Proof. First of all HomR .MO ; MO 0 / is A" -torsion free, because MO 0 is. The functor MO 7! TL" MO is faithful by (3.4.1), because MO MO ˝R[[z ]] K sep [[z]]. Since TL" MO is a free A" -module of rank rk MO , it follows that HomR .MO ; MO 0 / is a finitely generated A" -module, and hence free of rank at most rk MO  rk MO 0 .  Proposition 3.4.6. If MO D MO " .M / for an A-motive M D .M; M / over R as in Example 3.2.2, the "-adic (dual) Tate module of M defined by TL" M WD fm 2 M ˝AR  A";K sep W M .M m/ D mg is canonically isomorphic to TL" MO as representations of sep Gal.K =K/. This isomorphism is functorial in M . Q M ˝AR K sep [[z]] disProof. Consider the decomposition M ˝AR A";K sep D i 2Z=f Z Q M ˝AR K sep [[z]] satisfies cussed in Example 3.2.2. An element m D .mi /i 2

i 2Z=f Z f f   .M m0 / D M .M m/ D m if and only if mi C1 D M .M mi / for all i and m0 D M   L L O MO .O O m0 /. So the isomorphism T" M ! T" M is given by .mi /i 7! m0 . It clearly M



is functorial.

Remark 3.4.7. The arguments proving Remark 3.4.3 and Proposition 3.4.6 can also be applied to an A-motive M over K as in Example 3.2.2 which not necessarily has good reduction. Namely, for every s > 1=qO the decomposition (3.2.1) yields Y Y A";R =ai ˝R[[z ]] K sep h zs i D K sep h zs i : A";R ˝R[[z ]] K sep h zs i D i 2Z=f Z

M ˝AK K sep [[z]] extends i 2Z=f Z Q K sep h zs i/ D M ˝AK K sep h zs i.

Therefore the decomposition M ˝AK A";K sep D to a decomposition M ˝AK .A";R ˝R[[z ]] ^

i 2Z=f Z

Q

i 2Z=f Z

We denote the factor for i D 0 by M0 and equip it with the Frobenius 0^ WD ^ ^ f 1 1  mod a0 W O M0 Πz  ! M0 Πz . Since K sep h zs i is a principal ideal domain M ^

by [Laz62, Proposition 2], we may choose a K sep h zs i-basis of M0 and represent 0^   by a matrix T 2 GLr K sep h  zsqO i . Now the argument proving Proposition 3.4.6 and

^ f Remark 3.4.3 shows that TL" M D f m 2 M0 W 0^ .M m/ D m g and that there is sep a Gal.K =K/- and -equivariant O isomorphism

^  .M0 ; 0^ / D M ˝AK .A";R =a0 ˝R[[z ]] K sep h zs i/ hW TL" M ˝A" K sep h zs i ! (3.4.4) which is functorial in M .

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Definition 3.4.8. Let R be a valuation ring as in Notation 3.1.1. For a z-divisible local Anderson module G over R the Tate module T" G and the rational Tate module V" G are defined as   and T" G WD HomA" Q" =A" ; G.K sep /   V" G WD HomA" Q" ; G.K sep / Š T" G ˝A" Q" ; where the last isomorphism sends f ˝ z n 2 T" G ˝A" Q" to the homomorphism fQW Q" ! G.K sep / with fQ.a/ WD f .az n / for a 2 Q" . To see that it is indeed an isomorphism, note that it is clearly injective, because f can be recovered from fQ and n. Conversely, since every fQ 2 V" G satisfies fQ.1/ 2 G.K sep / D lim GŒz n .K sep/ by ! [HV11, Lemma 5.4], there is an n with fQ.1/ 2 GŒz n .K sep/, and so fQ.z n A" / D 0. This shows that fQ is the image of f ˝ z n for f 2 T" G with f .a/ WD fQ.az n /. T" G is an A" -module and V" G is a Q" -vector space. Both carry a continuous Gal.K sep=K/-action and G 7! T" G and G 7! V" G are covariant functors. For an abelian Anderson A-module E over R as in Example 3.3.4 the "-adic Tate module T" E and the rational "-adic Tate module V" E are defined as   and V" E WD T" E ˝A" Q" : T" E WD HomA Q" =A" ; E.K sep / Since every element of Q" =A" is annihilated by a power of ", we have T" E D T" EŒ"1 . After choosing a uniformizing parameter z of A" there are isomorphisms Y ˚ 

  T" G ! .Pn /n 2 GŒz n .K sep /W z.PnC1/ D Pn DW lim GŒz n .K sep/; z 

and

n

n2N0

Y ˚

 .Pn /n 2 G.K sep /W z.PnC1 / D Pn ; V" G ! n2Z

which send f W Q" ! G.K sep / to the tuple Pn WD f .z n /. These are indeed isomorphisms, because from .Pn /n we can reconstruct f as follows. Every a 2 Q" is of the form a D uz n for an integer n and a unit u 2 A" . Then f .a/ D u.Pn /. To describe the relation between T" G and TL" MO consider the A" -module HomF" .Q" =A" ; F" / which carries the trivial Galois action and is canonically isomorphic to b 1A =F under the map the module of continuous differential forms " "  b 1A =F ! HomF" .Q" =A" ; F" / ; " "

  ! 7! a 7! Res" .a!/ : 1

(3.4.5)

b A =F Š F" [[z]]dz After choosing a uniformizing parameter z of A" we can identify " " and the inverse map is given by HomF" .Q" =A" ; F" / ! F" [[z]]dz,  7!

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P1

1i i /z dz. In particular HomF" .Q" =A" ; F" / is non-canonically isomori D0 .z phic to A" . The following proposition generalizes Anderson’s result [And93, § 4.2] who treated the case where G is a formal Lie group.

Proposition 3.4.9. Let MO be an effective local shtuka over R, let G D DrqO .MO / be the associated z-divisible local Anderson module from Section 3.3 and view MO as lim HomR-groups;F" -lin .GŒz n  ; Ga;R /. Then there is a Gal.K sep =K/-equivariant per fect pairing of A" -modules 1

b A =F ; h : ; : iW T" G  TL" MO ! HomF" .Q" =A" ; F" / Š " "

hf; mi WD m ı f : (3.4.6) which identifies T" G with the contragredient Gal.K sep =K/-representation HomA" b 1A =F / of TL" MO . In particular T" G is a free A" -module of rank equal to .TL" MO ; " " rk MO that carries a continuous action of Gal.K sep =K/. Remark 3.4.10. (a) Note that indeed m ı f lies in HomF" .Q" =A"; F" /, because  m D MO . O m/ D Fq;G O a;S =S ı m implies that m ı f .a/ D Fq;G O a;S =S m ı f .a/ D M  qO m ı f .a/ in Ga .K sep / D K sep , and hence m ı f .a/ 2 F" for all a 2 Q" . (b) This pairing is functorial in MO in the following sense. If ˛W MO ! MO 0 is a morphism of local shtukas over R and DrqO .˛/W G 0 WD DrqO .MO 0 / ! DrqO .MO / D G is the induced morphism of the associated z-divisible local Anderson modules then TL" ˛W TL" MO ! TL" MO 0 and T" DrqO .˛/W T"G 0 ! T" G satisfy hf 0 ; TL" ˛.m/i D m ı DrqO .˛/ ı f 0 D hT" DrqO .˛/.f 0 /; mi for f 0 2 T" G 0 and m 2 TL" MO . Proof of Proposition 3.4.9. From [BH07, Lemma 2.4 and Theorem 8.6] we have a perfect pairing of A" =.z n /-modules GŒz n .K sep /  .MO =z n MO /O .K sep / ! HomA" .A" =.z n /; F" / ;   .Pn ; m/ 7! hn W a 7! m.a Pn / :    GŒz n .K sep/, f 7! Under the A" -isomorphism HomA" z n A" =A" ; G.K sep / ! n n Pn WD f .z / it transforms into a perfect pairing of A" =.z /-modules   HomA" z n A" =A" ; G.K sep /  .MO =z n MO /O .K sep / ! HomF" .z n A" =A" ; F" / ;   .f; m/ 7! hn W a 7! m ı f .a/ : Again the identification HomF" .z n A" =A" ; F" / D A" =.z n /dz induced from (3.4.5), shows that this is a free A" =.z n /-module of rank one. It follows that     HomA" z n A" =A" ; G.K sep / Š HomA" =.z n / .MO =z n MO /O .K sep /; A" =.z n /  dz

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is a free A" =.z n /-module of rank rk MO , because this holds for .MO =z n MO /O .K sep /. This implies that for varying n the inclusions z n A" z .nC1/ A" induce vertical maps in the following diagram   HomA" z n1 A" =A" ; G.K sep /  .MO =z nC1 MO /O .K sep /

/ A" =.z nC1 /  dz

    HomA" z n A" =A" ; G.K sep /  .MO =z n MO /O .K sep /

 / A" =.z n /  dz :

(3.4.7) By evaluating these pairings on a fixed choosen A" -basis of TL" MO D lim.MO =z n MO /O .K sep /, we obtain isomorphisms    rk MO  HomA" z n A" =A" ; G.K sep / ! A" =.z n /  dz ;



which are compatible for all n. Since the middle and the right vertical maps in diagram (3.4.7) are surjective, this shows that also the left vertical map is surjective. So the projective limit of this diagram is the pairing (3.4.6), and this yields an isomorphism    rk MO    b 1A =F : T" G D HomA" Q" =A" ; G.K sep / ! A" dz Š HomA" TL" MO ; " " This shows that (3.4.6) is a perfect pairing of free A" -modules. If g 2 Gal.K sep=K/ then hg.f /; g.m/i D g.m/ıg.f / D g.mıf / D mıf D hf; mi, because g acts trivially on m ı f . Therefore, the pairing (3.4.6) is Gal.K sep =K/-equivariant.  Example 3.4.11. We describe the "-adic (dual) Tate module TL" M D TL" MO " .M / of the Carlitz motive M D .RŒt; t  #/ from Example 3.2.7 by using the local shtuka MO WD MO " .M / D .R[[z]]; z  / computed there. For all i 2 N0 let `i 2 K sep O be solutions of the equations `q1 D  and `qiO C `i D `i 1 . This implies 0 P1 i O i < 1. Define the power series `C D j`i j D jjqO =.q1/ i D0 `i z 2 OK sep [[z]]. It satisfies .` O C / D .z  /  `C , but depends on the choice of the `i . A different choice yields a different power series `QC which satisfies `QC D u`C for a unit 

O .`Q /

`Q

O id u 2 .K sep [[z]] /D D A" , because .u/ O D O .`C / D `C D u. The field exC C tension F" (()).`i W i 2 N0 / of F" (()) is the function field analog of the cyclotomic p i tower Qp . p 1W i 2 N0 /; see [Har09, § 1.3 and § 3.4]. There is an isomorphism of topological groups called the "-adic cyclotomic character  ı    " W Gal F" (()).`i W i 2 N0 / F" (()) ! A" ;

P i sep [[z]] for g in the Galois which satisfies g.`C / WD 1 i D0 g.`i /z D " .g/  `C in K group. It is independent of the choice of the `i . The "-adic (dual) Tate module TL" MO of MO and M is generated by `1 C on which the Galois group acts by the inverse of

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the cyclotomic character. According to Proposition 3.4.9, Gal.K sep=K/ acts on the "-adic Tate module T" C of the Carlitz module C from Example 3.3.5 by " . So using the notation of Tate twists we may write TL" M D A" .1/ and VL" M D Q" .1/, as well as T" C D A" .1/ and V" C D Q" .1/. The isomorphisms (3.4.1) and (3.4.2) for s > 1=qO are given by sending the generator `1 of TL" MO to `1 2 K sep h zs i . C C We compute T" C explicitly. By Example 3.3.5 the group scheme C Œ"n  equals the kernel of the endomorphism 'zn of Ga;R D Spec RŒx where 'z .x/ D x C x qO . By definition 'z .`0 / D 0 and 'z .`i / D `i 1 for all i . This means that C Œ"n .K sep / D given by A="n  `n1 and T" C D A"  .`n1/n with the action of g 2 Gal.K sep=K/ P1 n g.`n1 /n D " .g/  .`n1 /n . In this respect the power series `C D nD0 `n z is the Anderson generating function from [And93, § 4.2] of the z-division tower to 1  dz. Indeed, if we write .`n1 /n . The pairing (3.4.6) sends .`n1 /n  `1 C P1 0 k Pn 0 0 D ` z , then ` ` D 1 and ` ` ele`1 C kD0 k kD0 k nk D 0 for n  1. The 0 0  sep ment .`n1/n 2 T" C corresponds to the element f 2 HomA" Q" =A" ; C .K / with P Pn 0 k 0 ı f .z n1 / D 1 f .z n1 / D `n . We compute `1 C kD0 `k 'z .`n / D kD0 `k `nk D ın;0 D Res" .z n1 dz/ for all n. This proves the claim. Definition 3.4.12. Let MO be a local shtuka over a valuation ring R as in Notation 3.1.1. We denote by K[[z  ]] the power series ring over K in the “variable” z   and by K((z  )) its fraction field. We consider the ring homomorphism R[[z]] ,! K[[z  ]]; z 7! z D  C .z  / and define the de Rham realization of MO as   H1dR MO ; K[[z  ]] WD O  MO ˝R[[z ]] K[[z  ]] ;   and H1dR MO ; K((z  )) WD O  MO ˝R[[z ]] K((z  )) H1dR .MO ; K/ WD O  MO ˝R[[z ]]; z7! K   D H1dR MO ; K[[z  ]] ˝K [[z ]] K[[z  ]]=.z  / :   The de Rham realization H1dR MO ; K((z  )) contains a full K[[z  ]]-lattice q WD  1 .MO ˝R[[z ]] K[[z  ]]/, which is called the Hodge–Pink lattice of MO . The de O M Rham realization H1dR .MO ; K/ carries a descending separated and exhausting filtration F  by K-subspaces called the Hodge–Pink filtration of MO . It is defined via p WD H1dR .MO ; K[[z  ]]/ and (for i 2 Z)  ı  F i H1dR .MO ; K/ WD p \ .z  /i q .z  /p \ .z  /i q H1dR .MO ; K/ :    If we equip H1dR MO ; K((z  )) with the descending filtration F i H1dR MO ; K((z   )) WD .z  /i q by K[[z  ]]-submodules, then F i H1dR .MO ; K/ is the image of     H1dR MO ; K[[z ]] \ F i H1dR MO ; K((z )) in H1dR .MO ; K/. Since z D  C.z / is invertible in K[[z  ]] the de Rham realization with Hodge–Pink lattice and filtration is a functor on the category of local shtukas over R with quasi-morphisms.

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Definition 3.4.13. If M D .M; M / is an A-motive over R as in Example 3.2.2 we use AK WD A ˝Fq K and AK =J D K, as well as the identification lim AK =J n D 

K[[z  ]] from [HJ20, Lemma 1.3]. Then the de Rham realization of M is defined as   H1dR M ; K[[z  ]] WD   M ˝AR lim AK =J n ; 

H1dR

    M ; K((z  )) WD H1dR M ; K[[z  ]] ˝K [[z ]] K((z  ))

and

H1dR .M ; K/ WD   M ˝AR AK =J : (See [HJ20, §§ 3.5 and 5.7] for a justification of this definition and the relation with the de Rham cohomology of a Drinfeld module, resp. abelian t-module, studied by Deligne, Anderson, Gekeler and Jing Yu [Gek89, Yu90], resp. Brownawell and Pa1 panikolas [BP02].) The  lattice of M is defined as q WD M .M ˝AR  Hodge–Pink n 1 lim AK =J / HdR M ; K((z  )) , and the Hodge–Pink filtration of M is defined 

via p WD H1dR .M ; K[[z  ]]/ and  ı  F i H1dR .M ; K/ WD p \ .z  /i q .z  /p \ .z  /i q   1 . J i M / ˝R K H1dR .M ; K/ I D image of   M \ M see [Gos94, § 2.6]. Note that the de Rham realization with Hodge–Pink lattice and filtration only depends on the generic fiber M ˝R K of M . It is a functor on the category of A-motives over K with quasi-morphisms. Remark 3.4.14. If MO WD MO " .M / is the associated local shtuka of M and f D ŒF" W Fq  as in Example 3.2.2, the map f 1

  M

D   M ı  2 M ı    ı  .f 1/ M W . f  M / ˝AR A";R =a0  ! .  M / ˝AR A";R =a0

is an isomorphism, because M is an isomorphism over A";R =ai for all i ¤ 0. There   f 1 fore it defines canonical functorial isomorphisms   M W H1dR MO ; K[[z  ]] !   f 1  H1dR M ; K[[z ]] and   M W H1dR .MO ; K/ ! H1dR .M ; K/, which are compatible f 1 1 with Hodge–Pink lattice and filtration because   M ı  1 D M . O M

For the next theorem note that there is a ring homomorphism Q" D F" ((z)) ,! K[[z  ]] sending z to z D  C .z  /, because  C .z  / is invertible in K[[z  ]]. Theorem 3.4.15. Let K be the completion of an algebraic closure K alg of K. There is a canonical functorial comparison isomorphism    h";dR W H1" .MO ; Q" / ˝Q" K((z  )) ! H1dR MO ; K((z  )) ˝K ((z )) K((z  )) ;

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  which satisfies h";dR H1" .MO ; Q" / ˝Q" K[[z  ]] D q ˝K [[z ]] K[[z  ]] and which is equivariant for the action of Gal.K sep=K/, where on the source of h";dR this group acts on both factors of the tensor product and on the target of h";dR it acts only on   K. However, if K is not perfect, h";dR does not allow to recover H1dR MO ; K((z  )) sep or H1dR .MO ; K/ from H1" .MO ; Q" / because the field of Galois invariants K Gal.K =K/

b

equals the completion K perf of the perfect closure of K by the Ax-Sen-Tate Theosep rem [Ax70, p. 417] and K((z  ))Gal.K =K/ D K perf ((z  )).

b

Remark. Regardless of the field isomorphism does not allow to   K the comparison 1 O 1 O recover H" .M ; Q" / from HdR M ; K[[z  ]] and q. Proof of Theorem 3.4.15. Note that the map z 7!  C .z  / induces ring homo1 P 1  ,! K((z  )) and K sep h z i ,! K[[z  ]], bn z n 7! morphisms R[[z]]Πz nD0

n   1 1   1   P  P  P P n ni n n i i n i bn .z  /   . The series  D b .z  / b n n i i i n nD0 i D0 i D0 nDi nDi   converges in K because j ni bn  n j  jbn  n j ! 0 for n ! 1. Thus we can take the functorial isomorphism h from (3.4.2) in Remark 3.4.3 and define 1 P

1  L O O  MO ˝R[[z ]] K((z  )) : h";dR WD .M O ı h/ ˝ idK ((z )) W T" M ˝A" K((z  )) ! 

   ı h TL" MO ˝A" MO ˝R[[z ]] K[[z  ]] D  1 Clearly q ˝K [[z ]] K[[z  ]] D  1 O O M M  K[[z  ]] .  Remark 3.4.16. If M D .M; M / is an A-motive over K as in Example 3.2.2 which not necessarily has good reduction, the functorial isomorphism h from (3.4.4) in Remark 3.4.7 defines a canonical functorial comparison isomorphism 1  ı h/ ˝ idK ((z )) W TL" M ˝A" K((z  )) !   M ˝AK K((z  )) ; h";dR WD .M

that is, a canonical functorial comparison isomorphism    H1dR M ; K((z  )) ˝K ((z )) K((z  )) ; h";dRW H1" .M ; Q" / ˝Q" K((z  )) !   which satisfies h";dR H1" .M ; Q" / ˝Q" K[[z  ]] D q ˝K [[z ]] K[[z  ]] and which is equivariant for the action of Gal.K sep =K/, where on the source of h";dR this group acts on both factors of the tensor product and on the target of h";dR it acts only on K. When M has good reduction, this comparison isomorphism is compatible with the comparison isomorphism from Theorem 3.4.15 and the isomorphisms      f 1 W H1dR MO .M /; K[[z  ]] ! H1dR M ; K[[z  ]] from Remark 3.4.14 and   M  H1" .M ; Q" / ! H1" .MO .M /; Q" / from Proposition 3.4.6.

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Remark 3.4.17. The comparison isomorphisms h";dR from Theorem 3.4.15 and Remark 3.4.16 are the function field analog for the comparison isomorphism HieK t .X L  Lalg ; Qp / ˝Qp BdR ! HidR .X=L/ ˝L BdR for a smooth proper scheme X over a complete discretely valued field L of characteristic 0 with perfect residue field of characteristic p. The existence of the latter comparison isomorphism was conjectured by Fontaine [Fon82, A.6] and proved by Faltings [Fal89]. It is equivariant for the action of Gal.Lsep =L/ and allows to compute HidR .X=L/ D HieK t .X L Lalg ; Qp / ˝Qp Gal.Lsep =L/ Gal.Lsep =L/ BdR , because BdR D L. Note that it does not allow to reconstruct i i alg HeK t .X L L ; Qp / from HdR .X=L/. The existence of this comparison isomorphism is also phrased by saying that the p-adic Galois representation HieK t .X L Lalg ; Qp / is de Rham. In our comparison isomorphism, K((z)) is the analog of BdR ; see [Har09, § 2.9], and we may thus say that for every A-motive M over K, which not necessarily has good reduction, the Q" -representation H1" .M ; Q" / of Gal.K sep =K/ is de Rham. Remark 3.4.18. The entries of a matrix representing the comparison isomorphism with respect to some bases are called the periods of MO , respectively of X . The transcendence degree of the periods of MO was related by Mishiba [Mis12] to the dimension of the Tannakian Galois group of MO following the approach of Papanikolas [Pap08]. Example 3.4.19. For the Carlitz motive from Example 3.4.11 we have H1" .M ; Q" / D 1 Q"  `1 C Š Q" and HdR .M ; K[[z  ]]/ D K[[z  ]] DW p. The Hodge–Pink lattice is q D .z  /1 p and the Hodge filtration satisfies F 1 D H1dR .M ; K/ F 2 D 1 1 .0/. With respect to the bases `1 C of H" .M ; Q" / and 1 of HdR .M ; K[[z  ]]/ the comparison isomorphism h";dR from Theorem 3.4.15 is given by the "-adic Carlitz period .z  /1 `1 O .`C /1 . It has a pole of order one at z D  because `C 2 C D     sep z  K h  i K[[z  ]] . So h";dR H1" .M ; Q" / ˝Q" K[[z  ]] D .z  /1 K[[z  ]] D q ˝K [[z ]] K[[z  ]]. Mishiba [Mis12, Example 6.6] shows that .` O C /1 is transcendental over Fq ./((z)). We next study properties of the (dual) Tate module functors. Theorem 3.4.20. Assume that R is discretely valued. Then the functor TL" W MO 7! TL" MO from the category of local shtukas over R to the category RepA" Gal.K sep =K/ of representations of Gal.K sep =K/ on finite free A" -modules and the functor VL" W MO 7! VL" MO from the category of local shtukas over R with quasi-morphisms to the category RepQ" Gal.K sep =K/ of representations of Gal.K sep =K/ on finite dimensional Q" vector spaces are fully faithful. We will give a proof at the end of the section.

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If we additionally require that MO is topologically nilpotent in the sense of Remark 3.3.3, then the above theorem was previously obtained by Anderson [And93, §4.5, Theorem 1] by a different method from ours. Note that if MO is “topologically nilpotent”, then the action of the (not necessarily commutative) polynomial ring R[[z]]fO g on MO extends to the action of the formal power series ring R[[z]]ffO gg, which is also a (not necessarily commutative) local ring. The theorem allows to make the following Definition 3.4.21. Let R be discretely valued. The full subcategory of RepQ" Gal .K sep =K/ which is the essential image of the functor VL" is called the category of equal characteristic crystalline representations. We explain the motivation for this definition in Remarks 3.5.13 and 3.6.17 below. Proposition 3.4.22. Let R be discretely valued, let MO be a local shtuka over R, and set VL WD VL" MO . Then the map   TL" MO 0 MO Œ 1z W local shtukas TL 0 VL W Gal.K sep =K/-stable ! over R of full rank full A" -lattices is a bijection. Let us now prove Theorem 3.4.20 and Proposition 3.4.22. We begin with a few lemmas. Lemma 3.4.23. Assume that R is discretely valued. Let MO be a finitely generated torsion-free R[[z]]-module (not necessarily free). We set MO 0 WD MO Œ 1z  \ .MO ˝R[[z ]] K[[z]]/, where the intersection is taken inside MO ˝R[[z ]] K((z)). Then MO 0 is free over 1 1 R[[z]] and we have MO Œ z  D MO 0 Œ z . In particular, if MO is equipped with an  MO Œ 1 , then MO 0 WD .MO 0 ;  O 0 / is a local shtuka, isomorphism  O W O  MO Œ 1  ! M

z

z

M

where MO 0 D MO . Furthermore, we have MO 0 D MO if MO is already free. Proof. Note that R[[z]]Œ 1z  is a principal ideal domain, being a 1-dimensional factorial ring, so the torsion-free module MO 0 Œ 1z  is free over R[[z]]Œ 1z . Likewise, MO ˝R[[z ]] K[[z]] is free over K[[z]].  Clearly MO MO 0 and MO Œ 1z  D MO 0 Œ 1z . Choose isomorphisms ˛W MO Œ 1z  !  K[[z]]˚r . After multiplying ˛ with a high R[[z]]Œ 1z ˚r and ˇW MO ˝R[[z ]] K[[z]] ! enough of the matrix A WD ˛ˇ 1 2  power  of z we may assume that 0all entries 0 O GLr K((z)) lie in K[[z]]. Then every m 2 M satisfies ˛.m0 / D A  ˇ.m0 / 2 .R[[z]]Œ 1z  \ K[[z]]/˚r D R[[z]]˚r . Let MO 00 WD ˛ 1 .R[[z]]˚r / so that MO 0 MO 00 . Since R[[z]] is noetherian, MO 0 is finitely generated over R[[z]]. Together with MO Œ 1z  D

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207

MO 0 Œ 1z  this implies that there is a power of z which annihilates MO 0 =MO . From MO 0 ˝R[[z ]] K[[z]] MO 00 ˝R[[z ]] K[[z]] Š K[[z]]˚r it follows that MO 0 ˝R[[z ]] K[[z]] has no z-torsion, whence MO 0 ˝R[[z ]] K[[z]] MO 0 ˝R[[z ]] K((z)) D MO ˝R[[z ]] K((z)). This implies MO 0 ˝R[[z ]] K[[z]] D MO ˝R[[z ]] K[[z]] and .MO 0 =MO / ˝R K D .MO 0 =MO / ˝R[[z ]] K[[z]] D .0/. So MO 0 =MO is annihilated by a power of  and thus by some power of 1 1  D MO 0 Œ z . z  , which shows that MO Œ z 0 O So it remains to show that M is free over R[[z]]. Note that by construction MO 0 is a reflexive R[[z]]-module; i.e. MO 0 is naturally isomorphic to the R-linear double dual .MO 0 /__ ; indeed, this follows from the fact that MO 0 Œ 1z  and MO 0 ˝R[[z ]] K[[z]] are free over R[[z]]Œ 1z  and K[[z]], respectively, and the equality R[[z]] D R[[z]]Œ 1z  \ K[[z]] with the intersection taking place in K((z)). Now, it is well known that a reflexive module over a regular 2-dimensional local ring is necessarily free; cf. [Ser58, §6, Lemme 6]. The last assertion follows from the equation R[[z]] D R[[z]]Œ 1z  \ K[[z]] in K((z)).  T n Lemma 3.4.24. Assume that n O .mR / D .0/. Then the base change functor MO 7! MO ˝R k from the category of local shtukas over R to the categories of local shtukas over k is faithful. T Remark. The assumption n O n .mR / D .0/ is satisfied if R is discretely valued. Proof. For local shtukas MO and MO 0 over R, let f W MO ! MO 0 be a morphism which becomes zero over k; i.e. f .MO / D .0/ in MO 0 ˝R R=mR . Since we have MO 0 ı O  f D   1 f ı MO , it follows inductively that f MO Œ 1z  D .0/ in MO 0 Œ z  ˝R R=O n .mR / D T MO 0 Œ 1z  ˝R R=O n .mR / for any n > 1. Now the assumption n O n .mR / D .0/ forces f D 0.  Lemma 3.4.25. Assume that R is discretely valued, and let MO and MO 0 be local shtukas over R. Let f W MO ! MO 0 be a morphism such that TL" f W TL" MO ! TL" MO 0 is an isomorphism. Then f is an isomorphism. Proof. To show that f is an isomorphism, it suffices to show that the determinant of f is an isomorphism. Therefore we may assume that MO and MO 0 are of rank 1 by replacing MO and MO 0 with their respective top exterior powers. Let us first show that f is an isogeny; i.e. f Œ 1z  is an isomorphism. By Nakayama’s lemma, it suffices to show the surjectivity of the following map induced by f Œ 1z : MO ˝R[[z ]] k((z)) ! MO 0 ˝R[[z ]] k((z)): Since both the source and the target are 1-dimensional vector spaces over k((z)), the above map is surjective as long as it is non-zero. The latter follows from Lemma 3.4.24 and the assumption that TL" f is an isomorphism, which implies that f ¤ 0. Let us now show that f is an isomorphism. By Proposition 3.4.2 it follows that f induces an isomorphism  f ˝ 1W MO ˝R[[z ]] K[[z]] ! MO 0 ˝R[[z ]] K[[z]]:

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Since f Œ 1z  is also an isomorphism, it follows from Lemma 3.4.23 that f is an isomorphism.  Proof of Theorem 3.4.20. Assume that R is discretely valued, and let MO and MO 0 be local shtukas over R. To prove Theorem 3.4.20, it suffices to show that for any Gal.K sep =K/-equivariant morphism gW TL" MO ! TL" MO 0 there exists a unique morphism f W MO ! MO 0 with TL" f D g. sep Since we have MO ˝R[[z ]] K[[z]] D .TL" MO ˝A" K sep [[z]]/Gal.K =K/ which matches MO ˝1 and 1˝ O by Proposition 3.4.2, it follows that g induces a uniquely determined morphism fK W MO ˝R[[z ]] K[[z]] ! MO 0 ˝R[[z ]] K[[z]] satisfying .MO 0 ˝ 1/ ı O fK D fK ı .MO ˝ 1/. Furthermore, any morphism f W MO ! MO 0 with TL" f D g has to satisfy f D fK jMO . Therefore, it suffices to show that fK .MO / MO 0 for any g. We define MO 100 ; MO 200 MO 0 ˝R[[z ]] K[[z]] as follows:   MO 100 WD fK .MO /Œ 1z  \ fK .MO / ˝R[[z ]] K[[z]]   MO 200 WD .fK .MO / \ MO 0 /Œ 1z  \ .fK .MO / \ MO 0 / ˝R[[z ]] K[[z]] : By Lemma 3.4.23 applied to the torsion free modules fK .MO / and .fK .MO / \ MO 0 /, we obtain local shtukas MO 001 and MO 002 with underlying R[[z]]-modules MO 100 and MO 200 , respectively, and the natural maps MO ! MO 001 , MO 002 ,! MO 001 , and MO 002 ,! MO 0 are morphisms of local shtukas. Clearly, we have MO 100 ˝R[[z ]] K[[z]] D MO 200 ˝R[[z ]] K[[z]] since both are equal to the image of fK , so the natural inclusion MO 002 ,! MO 001 induces  TL" MO 001 . We can now apply Lemma 3.4.25 to show that an isomorphism TL" MO 002 ! 00 00 MO 1 D MO 2 . Therefore, we obtain a map f W MO ! MO 0 by the following composition MO ! MO 001 D MO 002 ,! MO 0 ; which clearly extends to fK .



Proof of Proposition 3.4.22. Let MO be a local shtuka over R. We want to show that TL" induces a bijection from the set of local shtukas MO 0 MO Œ 1z  to the set of Galoisstable A" -lattices TL 0 VL" MO . The injectivity of TL" is clear from Theorem 3.4.20, so it suffices to show the surjectivity. Let TL 0 VL" MO be a Galois-stable A" -lattice. We want to show that there exists sep a local shtuka MO 0 MO Œ 1z  with TL" MO 0 D TL 0 . We set N WD .TL 0 ˝A" K sep [[z]]/Gal.K =K/ , which can be viewed as a K[[z]]-lattice in MO Œ 1z  ˝R[[z ]] K[[z]] by Proposition 3.4.2, and set MO 0 WD MO Œ 1z  \ N . By construction, MO 0 is finitely generated over R[[z]], and we have MO 0 D MO 0 Œ 1z  \ .MO 0 ˝R[[z ]] K[[z]]/. So by Lemma 3.4.23, MO 0 WD .MO 0 ; MO 0 / is a local shtuka where MO 0 is the restriction of MO .

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It remains to show that TL" MO 0 D TL 0 . By Proposition 3.4.2, it suffices to show that MO 0 ˝R[[z ]] K[[z]] D N , which follows from the left exactness of the following sequence: 0

/ MO 0 ˝R[[z ]] K[[z]]

/ N Œ1 z

/ N Π1 =N: z

(3.4.8)

Indeed, this sequence is the flat scalar extension of the left exact sequence 0 ! MO 0 ! MO Œ 1z  ! N Œ 1z =N ; note that we have N Œ 1z  D MO Œ 1z  ˝R[[z ]] K[[z]] and   .N Œ 1z =N / ˝R[[z ]] K[[z]] Š lim z r N=N ˝R[[z ]]=.z r / K[[z]]=.z r / ! r Š lim.z r N=N /Œ 1  Š N Œ 1z =N: ! r 

This concludes the proof.

3.5 Hodge–Pink structures From now on we work over the base scheme S D Spec R where R is a valuation ring as in Notation 3.1.1 with fraction field K and residue field k. In particular  D 0 in k. We assume that there is a section k ,! R and we fix one. As mentioned in the introduction we want to describe the analog of Fontaine’s classification of crystalline p-adic Galois representations by filtered isocrystals. In this function field analog we make the following Definition 3.5.1. A z-isocrystal over k is a pair .D; D / consisting of a finite di mensional k((z))-vector space together with a k((z))-isomorphism D W O D ! D. 0 0 A morphism .D; D / ! .D ; D0 / is a k((z))-homomorphism f W D ! D satisfying D0 ı O  f D f ı D . Here again O  D WD D ˝k ((z )); O k((z)) and we set OD x WD x ˝1 2 O  D for x 2 D. By the assumed existence of the fixed section k ,! R there is a ring homomorphism k((z)) ,! K[[z ]] ;

z 7!  C.z / ;

P i

bi z i 7!

1 P j D0

.z /j 

P i  i

j

bi  i j :

(3.5.1) We always consider K[[z  ]] and its fraction field K((z  )) as k((z))-vector spaces via (3.5.1). Definition 3.5.2. A z-isocrystal with Hodge–Pink structure over R is a triple D D .D; D ; qD / consisting of a z-isocrystal .D; D / over k and a K[[z  ]]-lattice qD in D ˝k ((z )) K((z  )) of full rank, which is called the Hodge–Pink lattice of D. The dimension of D is called the rank of D and is denoted rk D.

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A morphism .D; D ; qD / ! .D 0 ; D0 ; qD0 / is a k((z))-homomorphism f W D ! D 0 satisfying D0 ı O  f D f ı D and .f ˝ id/.qD / qD0 . 0 A strict  0 subobject D D0 is a z-isocrystal with  Hodge–Pink structure of the form 0 D D D ; D jO D0 ; qD \ D ˝k ((z )) K((z  )) where D 0 D is a k((z))-subspace with D .O  D 0 / D D 0 . Remark 3.5.3. This definition slightly deviates from [GL11, § 3 and § 7] and [Har11, § 2.3] where the Hodge–Pink lattice qD was defined to be a K[[z  ]]-lattice in O  D˝k ((z )) K((z)). The reason for our definition here is explained in Example 3.5.7 below. Definition 3.5.4. On a z-isocrystal with Hodge–Pink structure D there always is the tautological lattice pD WD D ˝k ((z )) K[[z  ]]. Since K[[z  ]] is a principal ideal domainL the elementary divisor theoremLprovides basis vectors vi 2 pD such that pD D riD1 K[[z  ]]  vi and qD D riD1 K[[z  ]]  .z  /i  vi for integers 1  : : :  r . We call .1 ; : : : ; r / the Hodge–Pink weights of D. Alternatively if e is large enough such that qD .z  /e pD or .z  /e pD qD then the Hodge–Pink weights are characterized by .z  /e pD =qD Š or qD =.z  /e pD Š

n M i D1 n M

K[[z  ]]=.z  /eCi ; K[[z  ]]=.z  /ei

i D1

The category of z-isocrystals with Hodge–Pink structure possesses tensor products and duals   D ˝ D 0 D D ˝k ((z )) D 0 ; D ˝ D0 ; qD ˝K [[z ]] qD0 ;   1 _ / ; HomK [[z ]] .qD ; K[[z  ]]/ ; D _ D D _ WD Homk ((z )) .D; k((z))/; .D   internal Hom’s, and the unit object k((z)); D D 1; qD D pD . The endomorphism ring of the unit object is F" ((z)). For every n 2 Z we consider the Tate object   1l.n/ WD D D k((z)) ; D D z n ; qD D .z  /n pD : The Hodge–Pink lattice qD induces a descending filtration of DK WD D ˝k ((z )); z7! K by K-subspaces as follows. Consider the natural projection ! pD =.z  /pD D DK : pD ! The Hodge–Pink filtration F  DK D .F i DK /i 2Z is defined by letting F i DK be the imageı in DK of pD \ .z  /i qD for all i 2 Z. This means, F i DK D pD \ .z  /i qD .z  /pD \ .z  /i qD .

Local shtukas, Hodge–Pink structures and Galois representations

211

Example 3.5.5. The Hodge–Pink lattice contains finer information than the Hodge– Pink filtration. For example let D D k((z))˚r and qD D .z/2 pD CK[[z]].v0 C .z  /v1 / pD for vectors v0 ; v1 2 K ˚r . Then F 2 DK D DK F 1 DK D K  v0 D F 0 DK F 1 DK D .0/. So the information about v1 is not contained in the Hodge–Pink filtration. Remark 3.5.6. In comparison with the Hodge–Pink structures at 1 from [HJ20, § 2] our Frobenius D replaces the weight filtration W H from [HJ20, Definition 2.3] and the Hodge–Pink weights from [HJ20, Remark 2.4] are the negatives of the Hodge– Pink weights we defined in Definition 3.5.4. Observe that the Frobenius D induces on D an increasing filtration given by Newton slopes similarly to [Zin84]. At the heart of the theory is the following e  / WD Example 3.5.7. Let MO D .MO ; MO / be a local shtuka over Spec R. We set .D; e D e  / as the underlying z-isocrystal. .MO ; MO / ˝R[[z ]] k((z)) and take .D; D / WD O  .D; e D e  / was This deviates from [GL11, § 3 and § 7] and [Har11, § 2.3] where instead .D; e D taken as the underlying z-isocrystal. Note however, that there is a canonical isomor e phism e D W .D; D / ! .D; e D ). The reason for our definition here is that we also define an integral structure H1cris .MO ; k[[z]]/ in Definition 3.5.14 below. In order to construct qD we will need a “comparison isomorphism”. This does not exist in general. However, if R is discretely valued it is constructed as follows. Consider the R-algebra ( 1 ) X R[[z; z 1 g WD bi z i W bi 2 R ; jbi j jjri ! 0 .i ! 1/ for all r > 0 : i D1

(3.5.2) It is a subring of K[[z  ]] via the expansion

1 P

bi z i D

i D1

1 P

 j

j D0

 P  1   i i  b .z  i j i D1

/j . The homomorphism (3.5.1) factors through R[[z; z 1 g. We view the elements of R[[z; z 1 g as functions that converge on the punctured open unit disc f0 < jzj < 1g. An example of such a function is ` WD

Y

.1 

 qO z

i

/ 2 F" [[]][[z; z 1 g R[[z; z 1 g ;

(3.5.3)

i 2N0

because the coefficient of z n is 1CqC:::C O qO n1

P

0i1 0 .1  i

O  / 2 F" (())[[z  ]] and qD D .` O 1 /.z  /1   qO 1 / 2 F" [[]] , we have .`  1 K[[z  ]] D .z  / pD . So the Hodge–Pink weight is 1 and   H.MO / D k((z)); z; .z  /1 pD D 1l.1/ : In order to describecomparison isomorphisms which relate the crystalline realization H1cris MO ; k((z)) of a rigidified local shtuka with the other realizations we let K be the completion of an algebraic closure of K, and we recall the definition of OK [[z; z 1 g from (3.5.2) and the elements ` 2 F" [[]][[z; z 1 g from (3.5.3) and `C 2 OK [[z]] from Example 3.4.11, which also satisfies `C 2 Kh z i and .` O C / D .z  /`C . We set ` WD `C ` 2 OK [[z; z 1 g :

(3.5.8)

Then .`/ O D z  ` because ` D .1  z /  O .` /, and g.`/ D " .g/  ` for g 2 sep Gal.K =K/ where " is the cyclotomic character from Example 3.4.11. With these properties ` is the function field analog of Fontaine’s period logŒe 2 e Brig of the multiplicative group, where e D .ei W i 2 N0 / is a compatible system of primitive pi -th roots of unity ei and Œe is its Teichmüller lift; see [Har09, § 2.7]. The following lemma is the function field analog of the fundamental exact sequence [CF00, Proposition 1.3(v)] of Qp -vector spaces / Qp

0

'D id /e B ˚ BC rig

dR

/ BdR

/0

(3.5.9)

'D id 'D id in which e Brig D Bcris ; see for example [Ber04b, § II.3.4].

Lemma 3.5.16. The following sequence of Q" -vector spaces is exact 0

O id / O [[z; z 1 gŒ`1 D ˚ K[[z  ]] K

/ Q"

a

 /

.a; a/

;

.f; g/



/ K((z  )) /

/0

f g:

Remark. Note that in [Har09, § 2.9, last line on p. 1745] the exactness of a similar sequence was stated. That sequence contains an error, as in the middle term “Œ`1 ” is missing. We prove the lemma simultaneously with the following P Corollary 3.5.17. For every c 2 K there is an f D i 2Z bi z i 2 OK [[z; z 1 g with P O / and c D f ./ WD i 2Z bi  i . f D z 1 .f

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Proof of Lemma 3.5.16 and Corollary 3.5.17. In Lemma 3.5.16 the exactness on the left is clear because the maps into each of the summands are ring homomorphisms. D O id To prove exactness in the middle let g D `bn 2 OK [[z; z 1 gŒ`1  \ K[[z  ]] 1 n n n n O / D z  ` we must have b D z .b/. O It follows with b 2 OK [[z; z g. Since .` from [Har11, Proposition 1.4.4] that b 2 Q" `n . This can also be seen directly as follows. For s  r > 0 consider the ring  1 P i ri si z s Kh  r ; z i WD bi z W jbi j jj ! 0.i ! 1/; jbi j jj ! 0 .i ! 1/ ; i D1

which is a principal ideal domain by [Laz62, Proposition 4]. Fix a real number r with 1=qO < r < 1. Then 1 < r qO < q. O Therefore the elements `C and .` O  / are  n z  r qO z  r qO n n units in Kh  r ; z i and the element .1  z / g D `C .` O  / b 2 Kh  r ; z i has a zero of order  n at z D  because g 2 K[[z  ]]. By [Laz62, Lemme 2] this 1 P rq O rq O shows that it is divisible by .1  z /n in Kh zr ;  z i. So g D bi z i 2 Kh zr ;  z i satisfies g D .g/ O D

1 P i D1

i D1

biqO z i

z

in Kh  r qO ;

 r qO i. z

It follows that bi D biqO , whence

O bi 2 F" . Now the convergence condition jbi j jjr qi ! 0 for i ! 1 implies that g 2 F" ((z)) D Q" as desired. Thus the sequence is exact in the middle. P We next prove Corollary 3.5.17. The condition f D z 1 .f O / D i 2Z biqO z i 1 P qO 1 qO i qO i implies bi D bi 1 D b0 , whence f D i 2Z b0 z i . The convergence condition on f for i ! 1 further implies that jb0 j < 1. Conversely this guarantees that f 2 OK [[z; z 1 g. Thus we must find an element b0 2 K with jb0 j < 1 and 1 X

i

b0qO  i D c :

(3.5.10)

i D1

We consider the valuation v on K with v.x/ WD log jxj= log jj, which satisfies jxj D O / and f ./ D c jjv.x/ and v./ D 1. If f 2 OK [[z; z 1 g satisfies f D z 1 .f then z n f 2 OK [[z; z 1 g satisfies z n f D z 1 O .z n f / and .z n f /./ D  n c for all n 2 Z. Therefore we may multiply c with an integral power of  and assume that 1 1 O < v.c/  1 C q1 , that is jj > jcjq1  jjqO . To solve equation (3.5.10) in this q1 O O case we iteratively try to find un 2 K such that n X

i

uqnO  i D c

(3.5.11)

i Dn

P for each n 2 N with n  2. Writing un D niD2 xn and i D n  j , multiplying (3.5.11) with  n and raising it to the qO n -th power we have to solve the equations 2n X j D0

n

 .2nj /qO .xn /qO

j

2n

n

n

D .xn /qO C: : :C .2n1/qO .xn /qO C 2nqO xn D cn (3.5.12)

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Local shtukas, Hodge–Pink structures and Galois representations 2

2

2n

n

for xn 2 K where c2 D  2qO c qO and cn D .un1 /qO   2nqO un1 for n  3. n We claim that there are solutions xn 2 K with jx2 j D jcj and jxn jqO D jcj jjqO < jcj jjqO  jcjqO for n  3. From this claim it follows that jun j D jcj and that the limit b0 WD limn!1 un exists in K with jb0 j D jcj < 1. In the expression X

qO i b0  i

C

n X

.b0  un /

qO i i

 D

i Dn

ji j>n

1 X

qO i b0  i

i D1

D

1 X



n X

i

uqnO  i

i Dn i

b0qO  i  c

i D1

the first sum goes to zero for n ! 1 because f 2 OK [[z; z 1 g. Since v.b0  un / D v.xnC1 / D qO 1 v.c/ C qO n the i -th summand in the second sum has valuation   i v .b0  un /qO  i D qO i 1 v.c/ C qO ni C i > .qO  1/.n  i / C 1 C i  n C 1 by Bernoulli’s inequality. So the second sum likewise goes to zero and b0 solves equation (3.5.10). To prove the claim we use the Newton polygon of (3.5.12), which is defined as the lower convex hull in the plane R2 of the points   O .2n  1/qO n / ; : : : ; .qO j ; .2n  j /qO n / ; : : : ; .qO 2n ; 0/ : 0; v.cn / ; .1; 2nqO n / ; .q; nj

on the The piecewise linear function passing through these points has slope  qOq1 O j j C1  for all j D 0; : : : ; 2n  1. In particular, these slopes are strictly interval ŒqO ; qO increasing. If n D 2 and v.c2 / D qO 2 v.c/ C 2qO 2 the Newton polygon starts with 1 the line segment of slope v.c/ on the interval Œ0; qO 2  because v.c/  1 C q1 2 O implies v.c2 /  1  v.c/ D .qO 2  1/v.c/ C 2qO 2 < 4qO 2 and v.c2 /  qO  v.c/ D .qO 2  q/v.c/ O C 2qO 2  3qO 2 , and because on the next interval ŒqO 2 ; qO 3  the slope is 1 > v.c/. So by the theory of the Newton polygon, (3.5.12) has a solution  q1 O x2 2 K with v.x2 / D v.c/. P On  If n  3 then n < 3.n  2/ C 1  .qO 2  1/.n  2/ C .qO  1/ implies 2n j DnC1 .q P n 2nj n nC2 n nC1 n 2nj n /  nqO < .qO  qO /.n  2/ C .qO  qO /  j D1 .qO  qO /, whence qO P 2n 1 2n q O O 2nj D q1 . Therefore the induction hypothesis v.un1 / D 2nqO n < j D1 q O 1 v.c/ > q1 yields O 2n

O v.uqn1 / D qO 2n v.c/ > v.c/ C

qO 2n 1 q1 O

n

> v.c/ C 2nqO n D v. 2nqO un1 /

and hence v.cn / D v.c/ C 2nqO n . It follows that the Newton polygon starts with the line segment of slope  v.c/  qO n1 on the interval Œ0; q O because v.c/ C 2nqO n C qO .q1/v.c/ O  qO n1 C 2nqO n < qO n1 slope is  qOq1  qO n1 >  v.c/ O qO

 qO n1 / D 1  . v.c/ qO

2nqO n , and because on the next

interval Œq; O qO 2  the

 qO n1 . Again by the theory of

the Newton polygon, (3.5.12) has a solution xn 2 K with v.xn / D n  3. This proves our claim and hence also the corollary.

v.c/ qO

C qO n1 for

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Finally, to prove exactness p on the right in Lemma 3.5.16 let c .z/n 2 K((z)) for c 2 K and fix an n-th root n c 2 K. By the there is an f 2 OK [[z; z 1 g ˇ corollary  ˇ p p ` ˇ 1 n n O / and f ./ D c  . z / zD D c  z 1 `C .` O  / ˇzD . Therewith f D z .f p D O id fore f n =`n 2 OK [[z; z 1 gŒ`1  and f =`  n c .z/1 mod K[[z]] implies c .z  /n  f n =`n 2 .z  /1n K[[z  ]]. In this way we can successively get rid  of all denominators in K((z  )) and this proves exactness on the right. Theorem 3.5.18. Let MO be a rigidified local shtuka over R. There are canonical functorial comparison isomorphisms between the de Rham and crystalline realizations    hdR;cris W H1dR .MO ; K[[z  ]]/ ! H1cris MO ; k((z)) ˝k ((z )) K[[z  ]] and    hdR;cris W H1dR .MO ; K/ ! H1cris MO ; k((z)) ˝k ((z )); z7! K ; which are compatible with the Hodge–Pink lattices and Hodge–Pink filtrations. They usually are not compatible with the integral structures H1dR .MO ; R/ WD O  MO =.z  /O  MO H1dR .MO ; K/ and H1cris .MO ; k[[z]]/. Proof. Taking the functorial isomorphism O  ıMO from Lemma 3.5.8, which is an 1 gŒ.` O  /1 , and using the inclusion isomorphism over O  R[[z; z 1 gŒ`1   R[[z; z 1 1 O  /  K[[z  ]], see Example 3.5.15, we define R[[z; z gŒ.`  D ˝k ((z )) K[[z  ]] : hdR;cris WD O  ıMO ˝ idK [[z ]] W O  MO ˝R[[z ]] K[[z  ]] !

We will see in Example 3.5.19 below that hdR;cris does not need to be compatible with the integral structures.  Example 3.5.19. To give an example in which hdR;cris is not compatible with the integral structures, we let A" D Fq [[z]] and assume that qO D q  3. As our base ring we take R D k[[]] with  D  qC1 . So the ramification index of  W A" ,! Ris   q C 1 > q. Over R we consider the local shtuka MO D R[[z]]˚2 ; MO D 0 z . 1 1 It is the local shtuka at " D .t/ Fq Œt associated with the Drinfeld Fq Œt-module E D .Ga;R ; '/ with 'W Fq Œt ! Rf g; 't D  C .1  / C  2 which has good 1 ordinary reduction. We compute its rigidification ıM , or rather ıM by following   the  z and proof of Lemma 3.5.8. We have T D 0 z and therefore S D 01 1 1 1       1=z 0 S 1 D z 1 11 z0 . We obtain C01 D T S 1 D =z 1 and T  S D 0  . 0  The Cm from (3.5.4) satisfy the recursion formula 1 Cm1  Cm1 D T    O m1.T /  O m .T  S /  O m .S 1 /    S 1 2 M2



m

q R[[z]] z mC1



:

1  1 D limm!1 Cm1 and we want evaluate We have ıM  toq1  O ıM at z D . We q 1 = 0 1 0 D … M2 .R/ and .C O m1  observe .C O 01 /jzD D 1  q = 1 

1

219

Local shtukas, Hodge–Pink structures and Galois representations 1 Cm1 /jzD 2 M2

  qmC1 z mC1

 R[[z]] jzD M2 .R/ for m  1, because

mC1

q  mC1

D

q mC1 .qC1/.mC1/

 2 R as q mC1  .q C 1/.m C 1/  0 for q  3 and m  1. 1 This shows that .ı O M /jzD … GL2 .R/, that is hdR;cris is not compatible with the integral structures. Note that this example is the function field analog of the example [BO83, Remark 2.10] which shows that also for a proper smooth scheme X over OL the com parison isomorphism HidR .X=OL / ˝OL L ! Hicris .X0 =W / ˝W L does not need to be compatible with the integral structures; see Remark 3.5.21. Theorem 3.5.20. Let MO be a rigidified local shtuka over R. There is a canonical functorial comparison isomorphism between the "-adic and crystalline realizations h";cris W H1" .MO ; Q" / ˝Q" OK [[z; z 1 gŒ`1     ! H1cris MO ; k((z)) ˝k ((z )) O [[z; z 1 gŒ`1  : K

The isomorphism h";cris is Gal.K sep =K/- and O -equivariant, where on the left module Gal.K sep =K/ acts on both factors and O is id ˝, O and on the right module Gal.K sep=K/ acts only on OK [[z; z 1 gŒ`1  and O is .D ı OD / ˝ O . In other words h";cris D D ı O  h";cris . Moreover, h";cris satisfies h";dR D .h1 dR;cris ˝ idK ((z )) / ı   .h";cris ˝ idK ((z )) /. It allows to recover H1" .MO ; Q" / from H1cris MO ; k((z)) as the   intersection inside H1cris MO ; k((z)) ˝k ((z )) K((z  ))    O D id   h";cris H1" .MO ; Q" / D H1cris MO ; k((z)) ˝k ((z )) OK [[z; z 1 gŒ`1  \ qD ˝K [[z ]] K[[z  ]] ;   where qD H1cris MO ; k((z)) ˝k ((z )) K((z  )) is the Hodge–Pink lattice of MO . Since  Gal.K sep =K/ k((z)) ¨ R[[z; z 1 g OK [[z; z 1 gŒ`1  , it does not allow to recover   H1cris MO ; k((z)) from H1" .MO ; Q" /. Remark 3.5.21. The comparison isomorphisms hdR;cris are the function field analogs  for the comparison isomorphism HidR .X=OL / ˝OL L ! Hicris .X0 =W / ˝W L between de Rham and crystalline cohomology of a proper smooth scheme X over a complete discrete valuation ring OL with perfect residue field of characteristic p and fraction field L of characteristic 0, where W is the ring of p-typical Witt vectors with coefficients in and X0 D X ˝OL ; see [BO83, Corollary 2.5]. If L0 is the fraction field of W then Hicris .X0 =L0 / D Hicris .X0 =W / ˝W L0 is an F -isocrystal, that is a finite dimensional L0 -vector space equipped with a Frobenius linear automorphism, and the Hodge filtration on HidR .X=OL / ˝OL L makes it into a filtered isocrystal via the comparison isomorphism. Less straightforward is the comparison isomorphism h";cris which is the function field analog of Fontaine’s comparison isomorphism HeiK t .X L Lalg ; Qp / ˝Qp  e Brig ! Brig . Namely, with a Gal.Lalg =L/-representation V Hicris .X0 =L0 / ˝L0 e

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such as HieK t .X L Lalg ; Qp / Fontaine associates a filtered isocrystal Dcris .V / WD alg .V ˝Qp Bcris /Gal.L =L/ and he calls V “crystalline” if dimL0 Dcris .V / D dimQp V .  Dcris .V / ˝L0 Bcris , In this case there is a comparison isomorphism V ˝Qp Bcris ! which is already defined over the subring e Brig Bcris ; see [Ber04b, II.3.5] and [Ber02, p. 228]. It was then conjectured by Fontaine [Fon82, A.11] and proved by Faltings [Fal89] that HieK t .X L Lalg ; Qp / is indeed crystalline and Dcris HieK t .X L  Lalg ; Qp / D Hicris .X0 =L0 /. The fundamental exact sequence (3.5.9) allows to also recover  Frob D id Brig HieK t .X L Lalg ; Qp / Š F 0 Hicris .X0 =L0 / ˝L0 e from Hicris .X0 =L0 /. In our comparison isomorphism, ` is the analog of logŒe 2 e Brig where e D .ei W i 2 N0 / is a compatible system of primitive pi -th roots of unity ei and Œe is its Teichmüller lift; see Example 3.5.15. Our ring OK [[z; z 1 gŒ`1  is the analog of e B ;  rig  0 1 O see [Har09, § 2.5 and § 2.7]. In that sense we could write F HdR M ; K((z )) WD q and   F 0 H1cris MO ; k((z)) ˝k ((z )) OK [[z; z 1 gŒ`1    WD H1cris MO ; k((z)) ˝k ((z )) OK [[z; z 1 gŒ`1  \ qD ˝K [[z ]] K[[z  ]] : Proof of Theorem 3.5.20. To construct h";cris we use for s 2 R>0 the notation  1 P i ri z 1 Kh  s ; z g WD bi z W bi 2 K ; jbi j jj ! 0 .i ! ˙1/ for all r  s i D1

(3.5.13) for the ring of rigid analytic functions with coefficients in K on the punctured closed disc f0 < jzj  jjs g of radius jjs . The ring Kh z ; z 1 g contains both the rings Kh z i  and R[[z; z 1 g, and so `C 2 Kh z ; z 1 g . In addition to ıMO we take the isomorphism ı h/ ˝ idKh z ;z 1 gŒ`1  h from Remark 3.4.3 and define h";cris WD .O ıMO ı  1 O M



   h";cris W TL" MO ˝A" Kh z ; z 1 gŒ`1  ! H1cris MO ; k((z)) ˝k ((z )) Kh z ; z 1 gŒ`1  : It satisfies h";cris D D ı O  h";cris and is functorial in MO . If we choose an A" -basis of   TL" MO and a k((z))-basis of H1cris MO ; k((z)) D D WD O  MO ˝R[[z ]] k((z)), we may write   h";cris and its inverse as matrices H; H 1 2 GLr Kh z ; z 1 gŒ`1  . Let also T 2     GLr k((z)) GLr OK [[z; z 1 g be the matrix of D with respect to the second basis. Then the equivariance of h";cris with respect to O implies that T .H O / D H . After 1 n each with a high enough power of ` we have ` H; `m H 1 2 multiplying H and H  z 1  n n 1 n m 1 m m 1 Mr Kh  ; z g with .` O H / D z T ` H and .` O H / D z ` H T . From

Local shtukas, Hodge–Pink structures and Galois representations

221

  n m 1 1 Lemma 3.5.22 below it follows that ` H; ` H 2 M O [[z; z g and H 2 r K   1 1 GLr OK [[z; z gŒ`  . So h";cris comes from an isomorphism h";cris W H1" .MO ; Q" / ˝Q" OK [[z; z 1 gŒ`1     ! H1cris MO ; k((z)) ˝k ((z )) OK [[z; z 1 gŒ`1  : By definition of h";dR WD . 1 ı h/ ˝ idK ((z )) in Theorem 3.4.15 and hdR;cris WD O M O  ıMO ˝ idK [[z ]] in Theorem 3.5.18 we obtain h";dR D .h1 dR;cris ˝ idK ((z )) / ı .h";cris ˝ idK ((z )) /. To prove the remaining statement we use the exact sequence of Q" -vector spaces from Lemma 3.5.16 and tensor it with H1" .MO ; Q" / to obtain the left column in the diagram 0

0



Š

O ; Q" / H1" .M

h";cris

 O ; Q" / ˝Q" H1" .M L

OK [[z; z 1 gŒ`1 

O ; Q" / ˝Q" K[[z   ]] H1" .M



O ; Q" / ˝Q" K((z   )) H1" .M

D O id

Š h";cris

Š h";cris Š h";cris

 / h";cris H1" .MO ; Q" /      / H1cris MO ; k((z)) ˝k((z)) OK [[z; z 1 gŒ`1  O D id L

/ qD ˝K [[z ]] K[[z   ]]  / H1cris MO ; k((z)) ˝k((z)) K((z   )) :

The second horizontal map is an isomorphism because h";cris is -equivariant. O The   1 O third horizontal map is an isomorphism because h";dR H" .M ; Q" / ˝Q" K[[z  ]] D q ˝K [[z ]] K[[z  ]] by Theorem 3.4.15 and because hdR;cris is compatible with the Hodge–Pink lattices by Theorem 3.5.18. From the right column it follows that    O D id   h";cris H1" .MO ; Q" / D H1cris MO ; k((z)) ˝k ((z )) OK [[z; z 1 gŒ`1  \ qD ˝K [[z ]] K[[z  ]] :



The following lemma is the function field analog of [Ber04a, Proposition I.4.1 and Corollary I.4.2]. Lemma 3.5.22. Let U 2 Mr .Kh zs ; z 1 g/ for some s > 0, and let V; W 2 Mr .R[[z; z 1 g/ such that .U O / D V U W . Then U 2 Mr .R[[z; z 1 g/.

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Remark 3.5.23. The lemma and its proof are valid more generally if R is replaced by an admissible formal R-algebra B ı in the sense of Raynaud, if K is replaced by B WD B ı ˝R K and j : j is a K-Banach norm on B with B ı D fb 2 BW jbj  1g; see [BL93]. 1 P bi z i 2 Kh zs ; z 1 g we consider the Proof of Lemma 3.5.22. For the elements i D1  P norm  i bi z i s WD maxi 2Z jbi j jjsi and we extend this to a norm on the matrix U by taking the maximum of the norms of the entries of U . It satisfies kV U ks  kV ks  kU ks . Since kzks D jjs < 1 there is an integer c such that kz c U ks  1. Likewise there is an integerPk such that kz k V ks ; kz k W ks  1. The entries of V and W are of the form i bi z i with jbi j  1. For those we have jbi jqO  n P P P jbi j and this implies kO n . i bi z i /ks D k i biqO z i ks  k i bi z i ks , and hence kO n .z k V /ks ; kO n .z k W /ks  1 for all n 2 N0 . We obtain    nC1 cC2k.nC1/  O .z U /s D O n .z k V /O n .z cC2k n U /O n .z k W /s    O n .z cC2k n U /  1 s

P

n P for all n 2 N0 by induction. Let i bi z i be an entry of U . Then i biqO z i CcC2k n is P  n the corresponding entry of O n .z cC2k n U /. The estimate 1   i biqO z i CcC2k n s D n n maxi jbi jqO jjs.i CcC2k n/ implies jbi j  jjs.i CcC2k n/qO for all i . For fixed i and n ! C1 the exponent Pgoes to 0 and so we find jbi j  1, that is bi 2 R for all i 2 Z. In order that i bi z i 2 R[[z; z 1 g we have toPverify that for all r > 0 the condition jbi j jjri ! 0 for i ! 1 holds. From i bi z i 2 Kh zs ; z 1 g we already know that this condition holds for all r  s. If r < s and i < 0 then jbi j jjri < jbi j jjsi and so the condition also holds for the remaining r < s. This  proves that U 2 Mr .R[[z; z 1 g/ as desired.

Example 3.5.24. For the Carlitz motive from Examples 3.4.19 and 3.5.15 the comparison isomorphisms from Theorems 3.5.18 and 3.5.20 between H1dR .M ; K[[z  with basis `1 on the ]]/ D K[[z  ]] with basis1 and H1" .M ; Q" / D Q"  `1 C C 1 one hand and Hcris M ; k((z)) D .k((z)); z; .z  /1 pD / with basis 1 on the other hand are given explicitly as follows. With respect to these bases hdR;cris is given by  O .`1 and h";cris is given by .` O 1 / D .` O  /1 .` O C /1 D z 1 `1 .  / 2 K[[z  ]] Q qO i 1 1  In particular, hdR;cris modulo z   is given by i 2N>0 .1   / 2 OK . So in this example hdR;cris is compatible with the integral structures.

3.6 Admissibility and weak admissibility It turns out that not all z-isocrystals with Hodge–Pink structure can arise via the functor H from (3.5.6). Namely a necessary condition is weak admissibility as in the following

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223

Definition 3.6.1. Let R be an arbitrary valuation ring as in Notation 3.1.1, let D D .D; D ; qD / be a z-isocrystal with Hodge–Pink structure over R and set r D dimk ((z )) D. (a) Choose a k((z))-basis of D and let det D be the determinant of the matrix representing D with respect to this basis. The number tN .D/ WD ordz .det D / is independent of this basis and is called the Newton slope of D. (b) The integer tH .D/ WD 1  : : :  r , where 1 ; : : : ; r are the Hodge–Pink weights of D from Definition 3.5.4, satisfies ^r qD D .z  /tH .D/ ^r pD and is called the Hodge slope of D. (c) D is called weakly admissible (or semi-stable of slope 0) if tH .D/ D tN .D/ and tH .D 0 /  tN .D 0 / for every strict subobject D 0 D. (d) D is called admissible if there exists a rigidified local -shtuka O MO over R with D D H.MO /. Remark 3.6.2. This definition parallels Fontaine’s definition [Fon79] of (weak) admissibility of filtered isocrystals. Namely, Fontaine calls a filtered isocrystal admissible if it comes from a crystalline Galois representation; compare Remark 3.5.13. In comparison with the Hodge–Pink structures at 1 from [HJ20, § 2] the Hodge slope tH .D/ corresponds to degq , the Newton slope tN .D/ corresponds to degW , and our notion of weak admissibility corresponds to Pink’s notion of “local semi-stability”; see [HJ20, Definition 2.7]. Example 3.6.3. For the Carlitz motive from Example 3.5.15 and its z-isocrystal with Hodge–Pink structure D D .k((z)); z; .z  /1 pD / one has tN .D/ D 1 D tH .D/. It is (weakly) admissible. To obtain a criterion for (weak) admissibility we need to recall a little bit of the theory of  -bundles from [HP04], see also [HJ20, § 7]. Here we consider a variant which also works if K is not algebraically closed. Namely, we do not consider the punctured open unit disc f0 < jzj < 1g but instead the punctured closed disc f0 < jzj  jjg of radius jj, and the ring Kh z ; z 1 g of rigid analytic functions with coefficients in K on it from (3.5.13). There are two morphisms from Kh z ; z 1 g to Kh zqO ; z 1 g, namely

W Kh z ; z 1 g ! Kh zqO ; z 1 g;

P

i i bi z 7!

X

bi z i

and

i

X qO P W O Kh z ; z 1 g ! Kh zqO ; z 1 g; i bi z i 7! bi z i : i

For a Kh z ; z 1 g-module F we define the two Kh zqO ; z 1 g-modules O  F WD F ˝Kh z ;z 1 g; O Kh zqO ; z 1 g and  F WD F ˝Kh z ;z 1 g;  Kh zqO ; z 1 g. 



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Definition 3.6.4. A -bundle O (on f0 < jzj  jjg) of rank r over K is a pair F D .F ; F / consisting of a free Kh z ; z 1 g-module F of rank r together with an 

 F of Kh zqO ; z 1 g-modules. isomorphism F W O F ! O is a homomorphism A homomorphism f W .F ; F / ! .G ; G / between -bundles f W F ! G of Kh z ; z 1 g-modules which satisfies F ı O  f D  f ı G .

The -invariants O of .F ; F / are defined as F O WD f f 2 F W F .O  f / D f g. It is a vector space over ff 2 Kh z ; z 1 g W .f O / D f g D F" ((z)) D Q" .  Example 3.6.5. 1. The trivial -bundle O over K is F 0;1 WD Kh z ; z 1 g; idKh

z  qO

;z 1 g

 .

2. More generally, for relatively prime integers d; r with r > 0 we let F d;r be the ˚r O -bundle over K consisting of Fd;r D Kh z ; z 1 g with 0

Fd;r

0

B B B WD B B B 0 @ z d

1 0

0

0

1

C C C 0 C: C 1C A 0

 O 3. We already saw in equation (3.5.8) that ` 2 F 1;1 . Lemma 3.6.6. If K is algebraically closed the evaluation at z D  induces an exact sequence of F" ((z))-vector spaces 0

/ F" ((z))  `



/ F 1;1

O

f .z/ 

/K

/ 0;

/ f ./

where K is an F" ((z))-vector space via the inclusion F" ((z)) ,! K, z 7! . Proof. The sequence is obviously exact on the left. Furthermore, ` maps to zero O  under the right morphism. To prove exactness in the middle let f 2 F 1;1 vanish i

at z D . Then it vanishes at z D  qO for all i 2 N0 and g WD f =` 2 Kh z ; z 1 g satisfies g D O  .g/ 2 F" ((z)) as desired. The exactness on the right was established in Corollary 3.5.17.  The structure theory of O -bundles over an algebraically closed complete field (for example over the completion K of an algebraic closure of K) was developed in [HP04] and [Har11, § 1.4].

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Local shtukas, Hodge–Pink structures and Galois representations

Theorem 3.6.7. We assume that K is algebraically closed (or we work over K). L F over K is isomorphic to i F di ;ri for pairs of relatively (a) Any -bundle O prime integers di ; ri with ri > 0, which P are uniquely determined by F up to F D permutation. They satisfy rk i ri and we define the degree of F as P deg F WD i di . 0

(b) There is a non-zero morphism F d 0 ;r 0 ! F d;r if and only if dr 0   ˚n satisfies deg F 0  dr  rk F 0 . (c) Any O -sub-bundle F 0 F d;r

d . r

Proof. Statements (a) and (b) are [Har11, Theorem 1.4.2 and L Proposition 1.4.5]. Statement (c) follows easily from (a) and (b). Namely, F 0 Š i F di ;ri by (a) with di  dr by (b) yields (c).  r i

Now let .MO ; ıMO / D .MO ; MO ; ıMO / be a rigidified local shtuka over R as in Definition 3.5.9, and let D D .D; D ; qD / be the associated z-isocrystal with Hodge–Pink structure over R. We define two -bundles O on f0 < jzj  jjg

E .MO / WD .E .MO /; E / WD O  MO ˝R[[z ]] k((z)) ˝k ((z )) Kh z ; z 1 g D .D; D / ˝k ((z )) Kh z ; z 1 g and

F .MO / WD .F .MO /; F / WD MO ˝R[[z ]] Kh z ; z 1 g :  W F .MO /Œ`1  ! E .MO /Œ`1 . We use There is a canonical isomorphism O  ıMO ı  1   O M z 1 1 O O this isomorphism to view F .M / as a Kh ; z g-submodule of E .M /Œ` . If MO is 

.F .MO //. Assume effective, that is if MO .O  MO / MO , we have E .MO / O  ıMO ı  1 O M O by the following diagram, that  O .O MO / ¨ MO . Then we visualize these -bundles M

in which the thick lines represent Kh z ; z 1 g-modules:

F.MO / MO ˝ Kh z ; z 1 g

O  MO ˝ Kh z ; z 1 g

R[[z]]

R[[z]]

E .MO / fjj  jzj > 0g zD

z D  qO

:::

Modules drawn higher contain the ones drawn below. All Kh z ; z 1 g-modules coinS i cide outside the discrete set i 2N0 fz D  qO g fjj  jzj > 0g. At those points S i in i 2N0 fz D  qO g where two modules are drawn at almost the same height, they also coincide. Indeed, F .MO / equals MO . Moreover, it contains MO .O  MO / and differs

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from it only at z D  by our assumption. Also E .MO / coincides with .O  MO / ˝R[[z ]] Kh z ; z 1 g at z D  because O  ıMO is an isomorphism of Kh z ; z 1 gŒO  `1  -modules and O  ` is a unit in K[[z  ]]. Finally, the strict inclusion MO .O MO / ¨ MO at z D  S i is transported by F and E to strict inclusions E .MO / F .MO / at i 2N0 fz D  qO g. The stalks at z D , or more precisely the tensor products with K[[z  ]] satisfy

E .MO / ˝Kh z ;z 1 g K[[z  ]] D pD

(3.6.1)



1 O O O  ıMO ı M O .F .M // ˝Kh z ;z 1 g K[[z  ]] D qD E .M / ˝Kh z ;z 1 g K((z  )) 



D

1 : pD Πz

We thus can view F .MO / as the modification of E .MO / at 1 , that is the inclusion qD pD Œ z 1 O O 1 O  ıMO ı M O .F .M // D E .M /Œ`  \

\

S

i 2N0 fz

i

D  qO g defined by

 j  Ej qD ˝K [[z ]]; O j K[[z   qO ]]

j 2N0

˚ i O i .f // 2 qD for all i  0 : D f 2 E .MO /Œ`1  W E . From this description it becomes clear that E .MO / and F .MO / only depend on the zisocrystal with Hodge–Pink structure D D .D; D ; qD / D H.MO / associated with MO . Namely, we can define

E .D/ WD .E .D/; E / WD .D; D / ˝k ((z )) Kh z ; z 1 g

and

F .D/ D .F .D/; F / WD E .D/Œ`1   \ j j  E qD ˝K [[z ]]; O j K[[z   qO ]] : \

(3.6.2)

(3.6.3)

j 2N0

  Definition 3.6.8. We call E .D/; F .D/ from (3.6.2) and (3.6.3) the pair of O bundles associated with D. The following proposition is analogous to [HJ20, Proposition 7.5] and was proved in [Har11, Lemma 2.4.5], where we used the notation P D E .D/ and Q D F .D/. Proposition 3.6.9. The degree (defined in Theorem 3.6.7(a)) satisfies tN .D/ D  deg E .D/

and

tH .D/ D deg F .D/  deg E .D/ :

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227

The criterion for (weak) admissibility is now the following Theorem 3.6.10.  Let D be a z-isocrystal with Hodge–Pink structure over R and let  E .D/; F .D/ be the associated pair of -bundles. O Then (a) D is admissible if and only if over K there is an isomorphism F .D/ Š rk D F˚ . 0;1 (b) D is weakly admissible if and only if deg F .D/ D 0  deg F .D 0 / for every strict subobject D 0 D. Proof. (b) directly follows from Proposition 3.6.9. (a) was proved in [Har11, Theorem 2.4.7]. We explain some parts of the proof. The implication “H)” is easy to see. Choose a rigidified local shtuka MO WD O .M ; MO ; ıMO / over R with D D H.MO /. We saw in Remark 3.4.3 that there is an  isomorphism hW TL" MO ˝A" K sep h z i ! MO ˝R[[z ]] K sep h z i, which is Gal.K sep=K/O and -equivariant. O Since K sep h z i Kh z ; z 1 g we obtain an isomorphism of  ˚ rk M  L O O ! T" M ˝A" F ! F .M / D F .D/ over K. bundles F 0;1

0;1

The converse implication “(H” is more complicated, as one has to construct rk D ; see [Har11, Theorem 2.4.7 and Proposithe local shtuka out of F .D/ Š F ˚ 0;1 tion 2.4.9].  Corollary 3.6.11. A z-isocrystal with Hodge–Pink structure over R which is admissible is also weakly admissible. Proof. This follows from the characterization of (weak) admissibility in Theorem 3.6.10 together with Theorem 3.6.7(c) applied to the -sub-bundle O F .D 0 / F .D/.  Whether the converse of Corollary 3.6.11 holds, depends on the field K. Before we state cases in which it holds, we discuss the following O D .R[[z]]; z / from Example 3.6.12. For the Carlitz motive and its local shtuka  M  z Example 3.6.3 we have E .MO / D F 1;1 and F .MO / D Kh  ; z 1 g; z   which is isomorphic over Kh z ; z 1 g to F 0;1 via an isomorphism given by multiplication with `C . Example 3.6.13. If the z-isocrystal .D; D / satisfies E .D; D / D F 1;r , for example if .D; D / is the crystalline realization of a Drinfeld module over k, then every weakly admissible Hodge–Pink lattice qD on .D; D / is already admissible. This can be shown in the same way as [Har13, Theorem 9.3(a)]. Example 3.6.14. We show that “weakly admissible does not imply admissible” if the field K is algebraically closed (actually perfect suffices) and complete. Let D D

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 0 1 k((z))˚2 ; D D z 3 . We search a Hodge–Pink lattice qD pD with dimK pD = 0 qD D 3 for which D D .D; D ; qD / is not admissible. This means the Hodge–Pink weights are .2; 1/ or .3; 0/. Note that any such D is weakly admissible, because tN .D/ D 3 D tH .D/ and there are no proper subobjects D 0 D, as the zisocrystal .D; D / is simple. We find qD by imposing the condition that there is a -sub-bundle O F 1;1 ˚2 F .D/ E .D/ D F 3;2 . Then F .D/ 6Š F 0;1 because there is no non-zero homomorphism F 1;1 ! F 0;1 by Theorem 3.6.7(b). Therefore Theorem 3.6.10 will imply that D is not admissible. One easily computes ) (   X qO 2i   u Hom F 1;1 ; E .D/ D f D W u 2 K; juj < 1 ; (3.6.4) zi  qO 2i 3 u i 2Z   O / D f  z 1 . This implies because f D ff12 must satisfy the equation D  .f

O 1 / D z 1 f2 , whence f1 D z 1 O 2 .f1 /. Writing f1 D O .f2 / D z 1 f1 and z 3 .f P 2 bi z i , we must have bi D .bi C1 /qO . So for b0 D u 2 K we obtain f1 D Pi 2Z qO 2i i z . Now the convergence condition for f 2 Kh z ; z 1 g implies juj < 1. i 2Z u Thus, if we take any u 2 K with 0 < juj < 1 and the corresponding f ¤ 0 from (3.6.4) and qD WD .z  /3 pD C K[[z  ]]  f the homomorphism f W F 1;1 ! E .D/ factors through F .D/ E .D/, as can be seen from (3.6.3). This implies that D is not admissible. On the positive side there is the following Theorem 3.6.15. If the field K satisfies the following condition (a) [GL11, Théorème 7.3]: K is discretely valued, or (b) [Har11, Theorem 2.5.3]: the value group of K is finitely generated, or

(c) [Har11, Theorem 2.5.3]: the value group of K does not contain an element n  x ¤ 1 such that x qO 2 jK j for all n 2 N0 , or

1

(d) [Har11, Theorem 2.5.3]: the completion K k alg of the compositum K  k alg n inside K alg does not contain an element a with 0 < jaj < 1 such that aqO 2 K k alg for all n 2 N0 ,

1

then every weakly admissible z-isocrystal with Hodge–Pink structure over R is admissible.

1

Proof. Clearly the conditions satisfy (a) H) (b) H) (c) H) (d), because the value groups of K and K k alg coincide. Under condition (d) the theorem was proved in [Har11, Theorem 2.5.3]. In terms of Example 3.6.14 the idea is that in this case one cannot have f 2 qD . 

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229

Remark 3.6.16. This theorem is the analog of the Theorem of Colmez and Fontaine [CF00, Théorème A], which states that in the theory of p-adic Galois representations every weakly admissible filtered isocrystal D D .D; D ; F  DL / over a discretely valued extension L of Qp with perfect residue field, comes from a representation of Gal.Lalg =L/ on a finite dimensional Qp -vector space; compare Remark 3.5.21. In the special case where the Hodge filtration satisfies F 0 DL D DL F 1 DL

F 2 DL D .0/ and all Newton slopes of the isocrystal .D; D / lie in the interval Œ0; 1 it was proved by Kisin [Kis06, Theorem 0.3] and Breuil [Bre00, Theorem 1.4] that D is (weakly) admissible if and only if D comes from a p-divisible group. Also in this situation there are weakly admissible filtered isocrystals D over Cp which do not come from a p-divisible group; see [Har13, Example 6.7]. Remark 3.6.17. Note that the Theorem of Colmez and Fontaine [CF00, Théorème A] from the previous remark actually says that a continuous representation of Gal.Lsep =L/ in a finite dimensional Qp -vector space is crystalline if and only if it is isomorphic to F 0 .D ˝L0 e Brig /Frob D id for a weakly admissible filtered isocrystal  D D .D; D ; F DL / over L. In the function field case, when K is discretely valued, we could therefore define the category of equal characteristic crystalline representations of Gal.K sep=K/ as the essential image of the functor  O D id D D .D; D ; qD / 7! D ˝k ((z )) OK [[z; z 1 gŒ`1  \ qD ˝K [[z ]] K[[z  ]] (3.6.5) from weakly admissible z-isocrystals with Hodge–Pink structure D to continuous representations of Gal.K sep =K/ in finite dimensional Q" -vector spaces. By Theorems 3.6.15, 3.5.20 and 3.4.20 and Proposition 3.5.12 this functor is fully faithful and this definition coincides with our Definition 3.4.21 above. Remark 3.6.18. The functor (3.6.5) can also be described via the associated O bundles. Namely if D is admissible, that is arises from a (rigidified) local shtuka MO over R with D D H.MO /, then F .D/ D F .MO / D MO ˝R[[z ]] Kh z ; z 1 g, and hence O  O  F .D/ ˝Kh z ;z 1 g Kh z ; z 1 g D MO ˝R[[z ]] Kh z ; z 1 g   O D TL" MO ˝A" F 0;1 D VL" MO I see the proof of Theorem 3.6.10(a). In the analogous situation of p-adic Galois representations, mentioned in the previous remarks, there is a similar description due to Fargues and Fontaine [FF13]. Namely, consider the punctured open unit disc f0 < jzj < 1g over K. Then every -bundle O on f0 < jzj  jjg extends canonically to a O -bundle over f0 < jzj < 1g; see [Har11, Proposition 1.4.1(b)]. The quotient f0 < jzj < 1g=O Z of f0 < jzj < 1g modulo the action of O exists in the category of Huber’s [Hub96] adic spaces and the category of vector bundles on f0 < jzj < 1g=O Z is equivalent to the category of -bundles O over f0 < jzj < 1g. Under this equivalence the Q" -vector space of global sections of the vector bundle on f0 < jzj < 1g=O Z equals the -invariants O of the associated -bundle. O

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In the theory of p-adic Galois representations the analog of the quotient f0 < jzj < 1g=O Z over Qp is the adic curve XFad;E constructed by Fargues [Far14]. Every weakly admissible filtered isocrystal D D .D; D ; F  DL / over L gives rise to two vector bundles E 1 and E1 on XFad;E which agree outside the point corresponding to O Z corresponding BC dR . These are analogous to the vector bundles on f0 < jzj < 1g= to our -bundles O E .D/ and F .D/, which agree outside the image of the point z D  on f0 < jzj < 1g=O Z corresponding to our analog K[[z  ]] of BC dR . By [Far14, Théorèmes 4.43 and 3.5] the p-adic Galois representation associated with D equals the global sections of E1 over XFad;E .

3.7 Torsion local shtukas and torsion Galois representations As a preparation for the equi-characteristic deformation theory, which will be discussed in Section 3.8, we need a “torsion version” of equi-characteristic Fontaine’s theory – or rather, a suitable function field analog of finite flat group schemes of p-power order. Let R be an arbitrary valuation ring as in Notation 3.1.1. From Lemma 3.7.4 onwards we will assume that R is discretely valued. The analogy between p-divisible groups and local shtukas suggests the following Definition 3.7.1. A torsion local shtuka (over R) is a pair MO D .MO ; MO / consisting of a finitely presented R[[z]]-module MO which is z-power torsion and free (necessar1 1   ! MO Œ z . ily of finite rank) as an R-module, and an isomorphism MO W O MO Œ z  O d O  O O O O If MO .O M / M then M is called effective, and if .z  / M MO .O M / M then we say that MO is of height 6 d . If MO .O MO / D MO then MO is called étale. A morphism of torsion local shtukas f W .MO ; MO / ! .MO 0 ; MO 0 / over R is a morphism of R[[z]]-modules f W MO ! MO 0 which satisfies  O 0 ı O  f D f ı  O . M

M

Remark 3.7.2. (a) Since  is MO -regular and MO is z-power torsion, MO injects into 1 . MO Πz (b) Since MO is finite free over R the pair .MO ; MO / with its R[[z]]-module structure can be defined in terms of finitely many parameters over R. Therefore it is defined over R0 [[z]] for some finitely generated and hence discretely valued A" -subalgebra R0 R. So we may assume that R is discretely valued. (c) The definition of torsion local shtukas over S 2 NilpA" is not completely straight1 forward, because we have MO Πz  D 0 for any finitely generated OS [[z]]-module MO killed by some power of z. Although it is possible to define torsion local shtukas over S 2 NilpA" (cf. Remark 3.7.6), we will only consider torsion local shtukas over R. The following example (together with Lemma 3.7.5) explains why we can regard torsion local shtukas over R as the function field analog of finite flat group schemes of p-power order.

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Example 3.7.3. Let f W MO 1 ! MO 0 be an isogeny of local shtukas over R; i.e. a morphism which is also a quasi-isogeny. Let MO WD coker f . Since f is a mor1 1   ! MO Πz . We claim that phism of local shtukas, MO 0 induces MO W O  MO Πz MO WD .MO ; MO / is a torsion local shtuka over R, which is effective (respectively, of height 6 d ) if MO 0 is effective (respectively, if .z  /d MO 0  O .O  MO 0 / MO 0 ). M0

Since R[[z]] R[[z]]Œ 1z  the map f is injective and there is an n such that z n MO 0 f .MO 1 /. In particular MO is killed by z n . Tensoring with the residue field k over R yields an exact sequence O O 0 ! TorR 1 .k; M / ! M1 ˝R[[z ]] k[[z]] f ˝ id

k ! MO 0 ˝R[[z ]] k[[z]] ! MO ˝R k ! 0 :

O As MO 1 and MO 0 are free we have MO i ˝R[[z ]] k[[z]] Š k[[z]]˚ rk M i for i D 0; 1 with rk MO 0 D rk MO 1 . Since MO ˝R k is killed by z n the map f ˝ idk is injecO O tive by the elementary divisor theorem. So TorR 1 .k; M / D .0/ and since M D   n O n O O O coker f W M1 =z M1 ! M0 =z M0 is finitely presented over R it is free of finite rank by Nakayama’s Lemma; e.g. [Eis95, Exercise 6.2].

From now on we assume that R is discretely valued. We will need the following Lemma 3.7.4. Assume that R is discretely valued, and let MO be a finitely generated module over R[[z]]=.z n / for some n. Then the following are equivalent (a) MO is flat over R; (b) The kernel of any surjective map MO 0 ! ! MO , where MO 0 is a free R[[z]]-module of finite rank, is free over R[[z]]. Proof. Since R[[z]] is regular, whence Cohen-Macaulay, the theorem of Auslander and Buchsbaum [Eis95, Theorem 19.9] tells us that proj:dimR[[z ]] MO C depthR[[z ]] MO D depth.R[[z]]/ D dim.R[[z]]/ D 2 for any finitely generated R[[z]]-module MO . If MO is flat over R, then  is MO -regular so the depth of MO is at least 1. Therefore, the projective dimension of MO is at most 1; in other words, MO admits a minimal projective resolution consisting of two terms, ! MO is an epimorphism for a free R[[z]]-module MO 0 then say FO1 ! FO0 . If MO 0 ! O O O ! MO / Š FO1 ˚ NO 0 by M0 Š F0 ˚ N0 for a free R[[z]]-module NO 0 and ker.MO 0 ! [Eis95, Theorem 20.2 and Lemma 20.1]. Conversely, assume that MO is not flat over R and killed by z n . Then the mR torsion of MO is of finite positive length since it is non-zero and killed by some powers of z and , so the depth of MO as a R[[z]]-module is zero. So the projective dimension of MO is 2; in other words, any projective resolution of MO consists of at least three terms. 

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The following lemma states that any torsion local shtuka over a discretely valued R arises from the construction in Example 3.7.3. This is analogous to Raynaud’s theorem [BBM82, 3.1.1], which states that any finite flat group scheme can be embedded, Zariski-locally on the base, in some abelian scheme. Lemma 3.7.5. Let MO be a torsion local shtuka over R. Then there exist local shtukas MO 0 and MO 1 , and an isogeny f W MO 1 ! MO 0 such that we have a short exact sequence f

0 ! MO 1 ! MO 0 ! MO ! 0 ; where each arrow is -equivariant. O If MO is effective of height 6 d (i.e. it satisfies d O  O .z  / M MO .O M / MO ), then we may take MO 0 and MO 1 to satisfy .z  /d MO i MO i .O  MO i / MO i . Proof. The proof is analogous to the proof of [Kis06, 2.3.4]. By replacing MO with .z  /n MO for some n, we may assume that MO satisfies .z  /d MO MO .O  MO / MO . We now choose a finite free R[[z]]=.z  /d -module L which surjects onto MO =MO .O  MO /, and a finite free R[[z]]-module MO 0 which fits into the following diagram 0

/ ker.M O0 ! ! L/

0

 / O  M O

M O

/M O0

/L

/0:

 /M O

 / MO = O .O  M O/ M

/0

Furthermore, by enlarging MO 0 we may arrange so that the left vertical map is also surjective; this can be done by replacing MO 0 with MO 0 ˚ MO 0 , where MO 0 is a finite free R[[z]]-module which surjects onto the kernel of L ! ! MO =MO .O  MO / and maps to zero in MO . Applying Lemma 3.7.4 to MO 0 ! ! L, ker.MO 0 ! ! L/ is free over R[[z]]. We de0 O O O ! L/ D N ˚ N into two free modules so that NO ˝R[[z ]] k D compose ker.M0 !  O !   MO is surjective by Nakayama. Then we may . M / ˝R[[z ]] k, and hence NO !  O  O  O !  M to a map  M0 ! NO which is also surjective by Nakayama lift  M0 ! and therefore a direct summand. We can now lift the latter map to an isomorphism    MO 0  ! NO ˚ NO 0 D ker.MO 0 ! ! L/, as both modules are free of same finite rank over R[[z]]. Therefore, we obtain a map  MO 0 W O MO 0 ! ker.MO 0 ! ! L/ ,! MO 0  lifting MO W O  MO ! MO .O MO /. Clearly, MO 0 WD .MO 0 ; MO 0 / is an effective local shtuka over R with .z  /d MO 0  O .O  MO 0 / MO 0 . Applying Lemma 3.7.4 M0

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233

to MO 0 ! ! MO , MO 1 WD ker.MO 0 ! ! MO / is free over R[[z]] so MO 1 WD .MO 1 ; MO 1 WD  O j O 1 / is an effective local shtuka with .z  /d MO 1  O .O MO 1 / MO 1 . M0 M1 Πz 

M1

Let f W MO 1 ! MO 0 be the morphism induced by the natural inclusion, then MO is the cokernel of f .  Remark 3.7.6. We have seen that the definition of torsion local shtukas over R does not straightforwardly apply to base schemes S 2 NilpA" . On the other hand, Lemma 3.7.5 and the anti-equivalence between effective local shtukas and z-divisible local Anderson modules (Remark 3.3.2) suggest the following notion as the function filed analog of finite flat group schemes of p-power order. We define a torsion local Anderson module over S 2 NilpA" to be a finite locally free A" -module scheme H over S such that for some Zariski covering fS˛ g of S there exists a z-divisible local Anderson module G˛ over S˛ and a monomorphism HS˛ ,! G˛ of fppf sheaves of A" -modules for each ˛. Note that the anti-equivalence of categories DrqO in Remark 3.3.2 can be extended to bounded effective local shtukas over R by limit as in Remark 3.2.6. If MO is a torsion local shtuka over R obtained as the cokernel of an isogeny f W MO 1 ! MO 0 of effective local shtukas, then we set DrqO .MO / WD ker.DrqO f W DrqO .MO 0 / ! DrqO .MO 1 //: Then DrqO .MO / is independent of the choice of f up to isomorphism, and this induces an anti-equivalence of categories between effective torsion local shtukas and torsion local Anderson modules over R. Indeed, it is possible to define DrqO directly without choosing f W MO 1 ! MO 0 as  ı  O DrqO .MO / WD Spec .SymR MO / m˝qO  MO .O M O m/W m 2 M I like in (3.3.1) in Section 3.3. And if H D DrqO .MO / for some effective torsion local shtuka MO over R, then we can recover MO as HomR-groups;F" -lin .H ; Ga;R / by [Abr06, Theorem 2] or [HS15, Theorem 5.2]. To a torsion local shtuka over R, we associate a torsion Galois representation as follows. Definition 3.7.7. Let MO be a torsion local shtuka over R. We define the (dual) Tate module of MO to be ˚

 TL" MO WD .MO ˝R[[z ]] K sep [[z]]/O WD m 2 MO ˝R[[z ]] K sep [[z]]W MO .OM O m/ D m : Proposition 3.7.8. For a torsion local shtuka MO over R, TL" MO is a torsion A" -module of length equal to rkR .MO /, which carries a discrete action of Gal.K sep =K/. Moreover the inclusion TL" MO MO ˝R[[z ]] K sep [[z]] defines a canonical isomorphism of

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K sep [[z]]-modules  MO ˝R[[z ]] K sep [[z]] TL" MO ˝A" K sep [[z]] !

(3.7.1)

which is Gal.K sep=K/- and O -equivariant, where on the left module Gal.K sep =K/ acts on both factors and O is id ˝, O and on the right module Gal.K sep =K/ acts only sep  O In particular one can recover MO ˝R[[z ]] K[[z]] D on K [[z]] and O is .MO ı O O / ˝ . M sep =K/   Gal.K as the Galois invariants. TL" MO ˝A" K sep [[z]] If we have a -equivariant O short exact sequence 0 ! MO 0 ! MO ! MO 00 ! 0 where each term is either a local shtuka or a torsion local shtuka over R, then we have a Gal.K sep =K/-equivariant short exact sequence 0 ! TL" MO 0 ! TL" MO ! TL" MO 00 ! 0: Proof. Let MO be a torsion local shtuka over R. To see that TL" MO carries a natural discrete action of Gal.K sep =K/, note that MO is killed by some power of z, say by z n , so the natural Gal.K sep =K/-action on MO ˝R[[z ]] K sep [[z]] D MO ˝R[[z ]] K sep [[z]]=.z n / is discrete and commutes with MO . f

! MO 0 ! MO ! Let us consider a O -equivariant short exact sequence 0 ! MO 1  O O O 0, where M is a torsion local shtuka over R and M 0 and M 1 are local shtukas over R. We obtain a Gal.K sep =K/-equivariant sequence TL f

" 0 ! TL" MO 1 ! TL" MO 0 ! TL" MO ! 0 ;

(3.7.2)

which by definition is exact on the left and in the middle. To see that it is also exact on the right, we obtain from Proposition 3.4.2  MO ˝R[[z ]] K sep [[z]] coker.TL" f / ˝A" K sep [[z]] !

- TL" MO ˝A" K sep [[z]]:

Combining these, it follows that the natural inclusion induces a Gal.K sep =K/- and O -equivariant isomorphism  MO ˝R[[z ]] K sep [[z]]: TL" MO ˝A" K sep [[z]] !

Taking O -invariants yields TL" MO D coker.TL" f / and so the sequence (3.7.2) is also exact on the right. The claim on the A" -length of TL" MO follows from this isomorphism. It now remains to show that TL" on the category of torsion local shtukas over R is exact. Indeed, it is clearly left exact by definition, and the right exactness follows from the length consideration. 

235

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Recall from Remark 3.7.6 that to an effective torsion local shtuka MO over R we associated a torsion local Anderson module H WD DrqO .MO / over R. Hence we have a finite-length A" -module H.K sep / equipped with a discrete Gal.K sep =K/-action. We will now discuss the comparison between H.K sep / and TL" MO in a way that is analogous to and compatible with Proposition 3.4.9. For this, we need some preparations. Lemma 3.7.9. Let H be the kernel of an isogeny G0 ! G1 of z-divisible local Anderson modules over R. Then there exists a natural Galois-equivariant short exact sequence 0 ! T" G0 ! T" G1 ! H.K sep / ! 0 : The surjective map on the right can be defined as follows. Given W Q" =A" ! G1 .K sep / (which is an element of T" G1 ), choose a lift Q W Q" ! G0 .K sep /. Then Q The Q we have .1/ 2 H.K sep /, and it only depends on  (not on the choice of ). Q surjective map on the right is given by  7! .1/. Proof. For any n, consider the exact sequence 0 ! A" ! "n A" ! "n A" =A" ! 0, where the first two over A" . By applying to it the long exact se  terms are projective quence for HomA"  ; H.K sep / , we obtain the following natural Galois-equivariant isomorphism     Ext1A" "n A" =A" ; H.K sep / Š coker HomA" "n A" ; H.K sep /

/ HomA A" ; H.K sep / "

7! .z n / Š

Š coker





H.K sep /

Š 7! .1/ Œz n 

/



H.K sep /

;

(3.7.3) where we let Gal.K sep =K/ act trivially on "n A" =A" . (Indeed, this isomorphism is independent of the choice of z.) In particular, if we choose n so that z n kills    H , then we have a natural isomorphism Ext1A" "n A" =A" ; H.K sep /  ! H.K sep /, n n1 and the natural inclusion " A" =A" ,! " A" =A" induces an isomorphism Ext1A"    n1   A" =A" ; H.K sep /  ! Ext1A" "n A" =A" ; H.K sep / , which induces the iden" tity map on H.K sep/. Recall that we have the following Galois-equivariant short exact sequence of A" modules 0 ! H.K sep / ! G0 .K sep / ! G1 .K sep / ! 0 : If z n kills H , then we have the following natural Galois-equivariant exact sequence via the Hom-Ext long exact sequence (independent of the choice of z):     0 ! HomA" "n A" =A" ; H.K sep / ! HomA" "n A" =A" ; G0 .K sep /     g ! HomA" "n A" =A" ; G1 .K sep / ! Ext1A" "n A" =A" ; H.K sep / ! 0 : (3.7.4)

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  The right exactness follows from the fact that Ext1A" "n A" =A" ; Gi .K sep / D 0 by the analog of (3.7.3) for Gi using that Œz n W Gi .K sep / ! Gi .K sep / is surjective as Gi is z-divisible. Now by taking the projective limit of the sequence (3.7.4) as we increase n and observing that both the kernel and the cokernel of g satisfy the MittagLeffler condition, we obtain     0 ! HomA" Q" =A" ; H.K sep / ! HomA" Q" =A" ; G0 .K sep /   g ! HomA" Q" =A" ; G1 .K sep / ! H.K sep/ ! 0 :   Since HomA" Q" =A" ; H.K sep / D 0, we obtain the short exact sequence as in the statement. It remains to explicitly describe the surjective map T" G1 ! H.K sep /. Choosing n so that z n kills H , the map, by construction, factors as follows:     T" G1 D HomA" Q" =A" ; G1 .K sep/ ! HomA" "n A" =A" ; G1 .K sep /    ! H.K sep /; ! Ext1A" "n A" =A" ; H.K sep /  where the first arrow is the restriction map and the second arrow is the connecting homomorphism in (3.7.4). Keeping in mind the description of Ext1A" "n A" =A" ;  H.K sep / in (3.7.3), it is straightforward that the description given in the statement matches with the connecting homomorphism of (3.7.4) given by the snake’s lemma.  Remark 3.7.10. One can prove Lemma 3.7.9 by directly checking that the map T" G1 ! H.K sep / is well defined and gives the desired short exact sequence. We    instead appealed to the isomorphism Ext1A" "n A" =A" ; H.K sep /  ! H.K sep/ for the sake of conceptual clarity. In order to relate (3.7.2) with the exact sequence in Lemma 3.7.9, we need a little digression on (some variant of) Pontryagin duality. Consider a short exact sequence of A" -modules 0 ! T0 ! T1 ! T ! 0 ; where T0 and T1 are finitely generated free A" -modules,and T is of  finite length. b1 , we get Then by applying to it the long exact sequence for HomA"  ; A" =F"     b1 b1 0 ! HomA" T1 ; A" =F" ! HomA" T0 ; A" =F"   b1 ! Ext1A" T; A" =F" ! 0 ;

(3.7.5)

    b1 b1 since HomA" T; D 0 and Ext1A" Ti ; D 0 for i D 0; 1 (as Ti are A" =F" A" =F" projective).

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Now, consider the following exact sequence b1 b1 b1 0 ! A" =F" ! A" =F" ˝A" Q" ! A" =F" ˝A" Q" =A" ! 0 : b1 Since is a rank-1 free module over A" , the last two terms are injective over A" =F" b1 A" . Using that T is torsion and ˝A" Q" is torsion free, we get the following A" =F" natural isomorphism      1 b1 b1 HomA" T; A" =F" ˝A" Q" =A" ! ExtA" T; A" =F" : Combining this isomorphism with the exact sequence (3.7.5), we obtain the following exact sequence:     b1 b1 0 ! HomA" T1 ; A" =F" ! HomA" T0 ; A" =F"   b1 ! HomA" T; A" =F" ˝A" Q" =A" ! 0 : (3.7.6) We can make explicit the surjective map in (3.7.6) in a similar way as Lemma 3.7.9. b1 b1 as a map T1 ! ˝ Q" (using the isomorNamely, we view W T0 ! A" =F" A" =F" A"  N T D T1 =T0 ! b1 ! T1 ˝A" Q" ), and take W ˝A" Q" =A" phism T0 ˝A" Q"  A" =F" N to be its reduction. Then the surjective map in (3.7.6) is defined by  7! . Applying (3.7.6) to (3.7.2) and the exact sequence in Lemma 3.7.9, we obtain the following commutative diagrams with exact columns: 0

0

0

0



Š

  / HomA" TL" M b 0; b 1A" =F"

b1 TL" M



Š

  / HomA" TL" M b 1; b 1A" =F"

b0 TL" M

T" G0

T" G1



H.K sep /



0





   1 / HomA" TL" M L" M b b b I T ; ˝ Q =A A " " " A" =F" 0

Š

. /



0



0

Š

  / HomA" T" G1 ; b 1A" =F"

Š

  / HomA" T" G0 ; b 1A" =F"

  / HomA" H.K sep /; b 1A" =F"˝A"Q" =A" I ./ Š



0

(3.7.7) where the horizontal isomorphisms on the first two rows are defined in Proposition 3.4.9. We now give an intrinsic description of the isomorphism ./ and .0 / in the above diagram, not depending on the choice of the isogeny MO 1 ! MO 0 . Assume that H is

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killed by z n . Then we can identify the Galois-equivariant A="n -isomorphism  H.K sep //; HomF" .A="n ; H.K sep// !

N defined by N 7! .1/. This isomorphism is compatible with increasing n with respect 0 ! A="n . to the natural projections A="nCn ! Let us consider the following A="n -linear isomorphism n  b1 A" =A" ! HomF" .A="n ; F" /; A" =F" ˝A" "

! 7! .a 7! Res" .aQ !//; Q

b1 where !Q 2 "n and aQ 2 A" are some lifts of ! and a, respectively. Note A" =F" Q modulo "n only depends on a and !. If we choose a unithat the residue Res" .aQ !/ formizing parameter z 2 A" , then this map can be identified with the reduction of the  b1 isomorphism A" =F" ! HomF" .Q" =A" ; F" / (introduced above Proposition 3.4.9)  "n A =A sending 1 to z1n , and we can write down via the isomorphism A="n ! Pn " i"1 i the inverse map as  7! dz ˝ . i D1 .z /z /. Let us now state the following comparison result analogous to Proposition 3.4.9:

Proposition 3.7.11. We use the notation as above, and view MO as HomR-groups;F" -lin .H ; Ga;R /; cf. Remark 3.7.6. Then the following Galois-equivariant pairing of A" -modules b1 H.K sep/  TL" MO ! A" =F" ˝A" Q" =A" ; .f; m/ 7! m ı f 2 HomF" .A="n ; F" / n b1 b1 Š A" =F" ˝A" " A" =A" A" =F" ˝A" Q" =A" I is perfect; in other words, it induces the following Galois-equivariant A" -linear isomorphisms:    b1 HomA" TL" MO ; and H.K sep / ! A" =F" ˝A" Q" =A"   sep 1  b TL" MO ! HomA" H.K /; A" =F" ˝A" Q" =A" : Furthermore, if we choose an isogeny MO 1 ! MO 0 whose cokernel is MO , then the above isomorphism coincides with the isomorphisms ./ and .0 / in the diagram (3.7.7). b1 Š A" dz, If we choose a uniformizing parameter z 2 A" and identify A" =F" sep then the above pairing non-canonically identifies H.K / with the Pontryagin dual of TL" MO as a torsion Gal.K sep =K/-representation. Proof. By Lemma 3.7.5 we can find an isogeny of effective local shtukas MO 1 ! MO 0 whose cokernel is the given effective torsion local shtuka MO . Therefore, it suffices to show that the pairing in the statement induces the isomorphisms ./ and .0 / in

Local shtukas, Hodge–Pink structures and Galois representations

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the diagram (3.7.7). Now, since the surjective vertical arrows in the diagram (3.7.7) are made completely explicit (cf. Lemma 3.7.9 and the discussion below (3.7.6)), we can directly check the commutativity of (3.7.7) with horizontal maps given by the pairings in Propositions 3.4.9 and 3.7.11.  Remark 3.7.12. As suggested in the proof of Proposition 3.7.11, one can easily show that Propositions 3.4.9 and 3.7.11 are equivalent. Indeed, it is possible to deduce Proposition 3.4.9 by directly proving Proposition 3.7.11 and apply it to H D GŒz n  for a z-divisible local Anderson module G over R. sep Definition 3.7.13. Let Reptors A" Gal.K =K/ denote the category of finite-length torsion A" -modules equipped with a discrete action of Gal.K sep=K/. Then for T 2 sep O Reptors A" Gal.K =K/, a torsion local shtuka model of T is a torsion local shtuka M over R equipped with a Gal.K sep =K/-equivariant isomorphism T Š TL" MO .

Contrary to the case of local shtukas over R, the functor TL" from the category of sep torsion local shtukas over R to Reptors A" Gal.K =K/ is not fully faithful. In particular, it is possible to have more than one torsion local shtuka model for T 2 sep O Reptors A" Gal.K =K/. For example, for any torsion local shtuka M over R, the Galois action on TL" MO trivializes after replacing R with some finite extension, so as R gets more ramified we obtain more torsion local shtuka models of the trivial torsion Galois module. Example 3.7.14. In this example, we find more torsion local shtuka models of A" =.z n / with trivial Gal.K sep =K/-action. We consider the following torsion local shtuka for any d 2 Z: MO dn WD .R[[z]]=.z n /; .z  /d /: Using the notation from Example 3.4.11, it follows that TL" MO dn K sep [[z]]=.z n / is an A" =.z n /-lattice generated by .`C /d mod z n with the Galois action given by the character ." /d mod z n . Moreover, if e 2 N is such that qO e  n and d D e 0 qO e .qO  1/d 0 then we have .`C /d mod z n D `d D ./qO d 2 K[[z]]=.z n /; in 0 other words, ." /d  1 mod z n . Therefore, in this case MO dn defines a torsion local shtuka model of the trivial Galois module A" =.z n / for any d 2 Z. Note that all these are pairwise non-isomorphic because .`C /d mod z n is not a unit in R[[z]]=.z n /. sep Proposition 3.7.15. The full subcategory of Reptors A" Gal.K =K/, consisting of torsion Galois representations which admit an effective torsion local shtuka model with height 6 d , is stable under subquotients, direct products, and twisted duality T 7! T _ .d / WD HomA" .T; Q" =A" / ˝ ." /d , where " is defined in Example 3.4.11.

The proposition, especially the assertion about subquotients, is not obvious since L T" may not be fully faithful. Recall that the analogous assertion for torsion representations of the Galois group of a p-adic field which admit finite flat group scheme

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models over the valuation ring can be obtained by working with the scheme-theoretic closure of the generic fiber. The following lemma provides an analog of the schemetheoretic closure, which is needed for the proof of Proposition 3.7.15. Lemma 3.7.16. Let MO be a torsion local shtuka over R, and consider a Gal.K sep =K/stable quotient TL" MO ! ! T 0 . We define MO 0 to be the image of the following map: ! T 0 ˝A" K sep [[z]]: MO ,! MO ˝R[[z ]] K sep [[z]] Š TL" MO ˝A" K sep [[z]] ! 1 1   ! MO 0 Œ z , and MO 0 WD Then, MO induces an isomorphism MO 0 W O  MO 0 Œ z .MO 0 ; MO 0 / is a torsion local shtuka over R. If MO is effective (respectively, effective with height 6 d ), then so is MO 0 . Furthermore, MO 0 is the unique quotient of MO such that T 0 D TL" MO 0 as a quotient of TL" MO .

Proof. Note that MO 0 is flat over the discrete valuation ring R and finitely presented over the noetherian ring R[[z]] because it is a submodule of T 0 ˝A" K sep [[z]] and a quotient of MO . The isomorphism MO 0 is obtained from the diagram 1  O  MO Œ z

MO Š  1  MO Œ z

/ / O  MO 0 Œ 1     z MO 0  Š    // O 0 1 

M Πz 

/ T 0 ˝A K sep [[z]] "

/ T 0 ˝A K sep [[z]] "

using that W O R ! R is flat. The diagram also shows that MO 0 is effective (respectively, effective with height 6 d ) if MO is. Finally, the uniqueness of MO 0 follows from Proposition 3.7.8.  Remark 3.7.17. In the setting of Lemma 3.7.16, assume furthermore that MO is effective. Then for a Galois-stable quotient T 0 of TL" MO , the associated MO 0 is effective. So 0 there is a unique A" -submodule scheme HK of the generic fiber of H WD DrqO MO 0 0 such that HK .K sep / is the Pontryagin dual of T 0 . Then DrqO MO 0 is the scheme0 in H , which can be seen from the uniqueness of the schemetheoretic closure of HK theoretic closure. Proof of Proposition 3.7.15. The claim on direct products is clear as TL" commutes with direct products. Let MO be an effective torsion local shtuka over R with height 6 d , and set T WD L T" MO . Lemma 3.7.16 shows that any Galois-stable quotient of T admits an effective torsion local shtuka model with height 6 d . Let T 0 T be a Galois-stable A" submodule. Then by Lemma 3.7.16, there exists a quotient MO 00 of MO corresponding to T =T 0 . Then Proposition 3.7.8 implies that MO 0 WD ker.MO ! ! MO 00 / can naturally 0 be viewed as a torsion local shtuka model of T , which is effective and of height 6 d .

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We construct a torsion local shtuka model of T _ .d / which is effective and of height 6 d . We first define MO _ WD .MO _ ; MO _ / as follows: MO _ WD HomR[[z ]] .MO ; R[[z]]Π1z =R[[z]]/ ;

(3.7.8)

1 _ 1 1 _  1 MO _ WD .M O  MO _ Œ z  Š .O MO /Œ z  ! MO _ Œ z : O / W 

Note that if z n MO D .0/ then HomR .M; R/ Š HomR .M; R/ ˝R[[z ]]=.z n /

 1 z n R[[z]]=R[[z]]

h˝a



/ MO _





/ m 7! a  h.m/ :

So as an R-module it is flat and finitely generated over R. By [Eis95, Proposition 2.10] we have   MO _ ˝R[[z ]] K sep [[z]] Š HomK sep [[z ]] MO ˝R[[z ]] K sep ; K sep [[z]]Œ 1z =K sep[[z]] Š HomA" .TL" MO ; Q" =A" / ˝A" K sep [[z]] ; and hence TL" MO _ Š HomA" .TL" MO ; Q" =A" /. Note that MO _ is not necessarily an effective torsion local shtuka, but we can slightly modify to get MO _;d WD .MO _ ; .z  /d MO _ /;

(3.7.9)

which is effective and of height 6 d if the same holds for MO . To see T _ .d / Š TL" MO _;d , note that .`C /d TL" MO _ D TL" MO _;d inside MO _ ˝R[[z ]] K sep [[z]], using the notation from Example 3.4.11.  Just as for finite flat group scheme models of torsion Galois representations of a p-adic field, there exists a natural notion of partial ordering on (equivalence classes of) torsion local shtuka models. sep Definition 3.7.18. We fix T 2 Reptors A" Gal.K =K/, and consider torsion local shtuka 0 models MO and MO of T . We write MO - MO 0 if there exists a (necessarily unique) map MO ! MO 0 which respects the identification TL" MO Š T Š TL" MO 0 . We say that MO and MO 0 are equivalent if MO - MO 0 and MO % MO 0 ; or equivalently, if there exists a necessarily unique isomorphism MO Š MO 0 which respects the identification TL" MO Š T Š TL" MO 0 . sep Lemma 3.7.19. For T 2 Reptors A" Gal.K =K/, we have the following:

(a) - defines a partial ordering on the set of equivalence classes of torsion local shtuka models of T .

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(b) In the set of equivalence classes of effective torsion local shtuka models of T with height 6 d , there exist unique maximal and minimal elements with respect to the partial ordering -. Furthermore, the formation of maximal and minimal effective torsion local shtuka models with height 6 d is functorial in the sense that any Galois-equivariant morphism of torsion Galois representations comes from a unique map of their maximal (respectively, minimal) effective torsion local shtuka models with height 6 d . (c) The set of equivalence classes of effective torsion local shtuka models of T with height 6 d is finite. Proof. For any torsion local shtuka models MO and MO 0 of T , the isomorphism TL" MO Š T Š TL" MO 0 enables us to identify the equivalence classes of MO and MO 0 as R[[z]]submodules of T ˝R[[z ]] K sep [[z]] which are stable under 1 ˝ O . One can easily check that MO C MO 0 and MO \ MO 0 can naturally be viewed as torsion local shtuka models of T , where O is obtained by the restriction of 1 ˝ O . Namely, MO C MO 0 is the torsion local shtuka from Lemma 3.7.16 associated with the quotient T" MO D T" MO 0 of T" .MO ˚ MO 0 /, and MO \ MO 0 is the kernel of MO ˚ MO 0 ! ! MO C MO 0 . This shows that - is a partial ordering, i.e. part (a). We next show that the set of equivalence classes of effective torsion local shtuka models of T admits a unique maximal element (if the set is non-empty). For an effective torsion local shtuka model MO of T we consider  ı  O DrqO .MO / WD Spec .SymR MO / m˝qO  MO .O M O m/W m 2 M I as in Remark 3.7.6. If MO 0 is another effective torsion local shtuka model of T with MO - MO 0 , then there exists a finite morphism DrqO .MO 0 / ! DrqO .MO / which is an e of DrqO .MO / in its isomorphism over Spec K. On the other hand, the normalization X O generic fiber is finite over DrqO .M / as the generic fiber is étale over K by the Jacobi criterion or because MO is étale over K. Indeed, the trace pairing .ODr

O

qO .M /

˝R K/  .ODr

O

q O .M /

˝R K/ ! K

is perfect by étaleness [EGA, IV4 , Proposition 18.2.3(c)], so the dual of ODr .MO / is q O finite over ODr .MO / and contains the normalisation of ODr .MO / . Thus the normalizaqO qO tion is finite. This shows that the set of equivalence class of effective torsion local shtuka models of T is bounded above with respect to -, because every such is cone . On the other hand, if tained in the effective torsion local shtuka corresponding to X 0 O O M and M are effective torsion local shtuka models of T , then so is the torsion local shtuka model MO C MO 0 . This shows the uniqueness of the maximal element. Let us show there exists a unique minimal effective torsion local shtuka model with height 6 d up to equivalence. Let MO 0 be a torsion local shtuka model of T _ .d / maximal among those effective and of height 6 d . Since MO 0 7! .MO 0 /_;d reverses

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the partial ordering by (3.7.8), it follows that .MO 0 /_;d is minimal among effective torsion local shtuka models of T with height 6 d . For the functoriality claim on maximal and minimal objects, we may assume that O M and MO 0 are maximal effective torsion local shtuka models of T D TL" MO and T 0 D TL" MO 0 , respectively. (The case of minimal objects is reduced to this since the functor (3.7.9) switches the maximal and the minimal objects.) Let f W T ! T 0 be a Gal.K sep =K/-equivariant map. If f is surjective, then the image of MO under f ˝1 MO ,! T ˝A" K sep [[z]] ! T 0 ˝A" K sep [[z]] is an effective local torsion shtuka model of T 0 with height 6 d , so it is contained in MO 0 by the maximality of MO 0 . If f C id

f is not surjective, then consider a surjective map T ˚ T 0 ! T 0 . Let MO 00 denote a torsion local shtuka model of T ˚ T 0 maximal among those effective and of height 6 d , then f C id induces a map MO 00 ! MO 0 . By maximality of MO 00 , there is a map MO ˚ MO 0 ! MO 00 , so we obtain . id;0/

MO ! MO ˚ MO 0 ! MO 00 ! MO 0 ; which induces f W T ! T 0 . This proves part (b). Let MO C and MO  respectively denote the maximal and minimal effective torsion local shtuka models of T with height 6 d . Since R is discretely valued, the quotient MO C =MO  is of finite length over R[[z]]. Since any equivalence class of effective torsion local shtuka models of T with height 6 d gives rise to a unique R[[z]]-submodule  of MO C =MO  , we obtain the desired finiteness claim (i.e. part (c)). Lemma 3.7.20. Let T be a finitely generated free A" -module equipped with a continuous action of Gal.K sep =K/. Then the following are equivalent: (a) There exists an effective local shtuka MO with .z  /d MO MO .O  MO / MO such that T Š TL" MO . (b) For each positive integer n, there exists an effective torsion local shtuka MO n with height 6 d (i.e. satisfying .z  /d MO n MO n .O  MO n / MO n ) such that T =.z n / Š TL .MO n /. Proof. It is clear that (a) implies (b), so let us assume (b). By choosing each MO n to be maximal among effective torsion local shtuka models with height 6 d , we may assume that for each n there exists a (not necessarily surjective) morphism MO nC1 ! ! T =.z n / by Lemma 3.7.19(b). MO n that induces the natural projection T =.z nC1 / ! 1 1  Set MO 0 WD lim MO n , equipped with MO 0 W   MO 0 Œ z  ! MO 0 Œ z  obtained as n O the limit of f O g. Note that the projective system .Mn / satisfies the Mittag-Leffler Mn

condition, because by Lemma 3.7.19(c) for each n the set of images MO n0 ! MO n is finite. Therefore MO 0 =z MO 0 is a quotient of MO n for n 1, and since MO 0 is z-adically separated and complete, it is finitely generated over R[[z]].

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We next consider the following injective map MO 0 D lim MO n ! lim.MO n ˝R[[z ]] K sep [[z]]/ Š lim.T =z n T ˝R[[z ]] K sep [[z]]/    n n n Š T ˝A" K sep [[z]]; where the non-trivial isomorphism is from Proposition 3.7.8. (The injectivity follows from the left exactness of the projective limit.) In particular, MO 0 is torsion-free. By K sep [[z]]-linearly extending it, we obtain a map MO 0 ˝R[[z ]] K sep [[z]] ! T ˝A" K sep [[z]];

(3.7.10)

which is injective by torsion-freeness of MO 0 . We claim that this map is an isomorphism. Indeed, the surjectivity follows since the image of MO 0 in MO n ˝R[[z ]] K sep [[z]] Š T =z n T ˝A" K sep [[z]] coincides with the image of MO n0 for n0 n by Lemma 3.7.19(c). One can now apply Lemma 3.4.23 to .MO 0 ; MO 0 / to obtain a local shtuka MO 00 WD 1 1 .MO 00 ; MO 00 / with MO 0 Œ z  D MO 00 Œ z . So we have .z  /d MO 00 MO 00 .O MO 00 / MO 00 . Furthermore, the isomorphism (3.7.10) implies that VL" MO 00 Š T Œ 1z . Then by Proposition 3.4.22, there exists a local shtuka MO isogenous to MO 00 with T D TL" MO . Finally, the property .z  /d MO MO .O  MO / MO could be checked up to isogeny, which concludes the proof.  Remark 3.7.21. Using the same notation in the proof of Lemma 3.7.20, one can also show that MO D MO 00 by using the isomorphism (3.7.10) and suitably adapting the exact sequence (3.4.8). We finish the section by discussing torsion local shtukas with coefficients. Let B be an A" -algebra with #B < 1, and set R[[z]]B WD R[[z]] ˝A" B. We define O W R[[z]]B ! R[[z]]B by B-linearly extending O W R[[z]] ! R[[z]]. Definition 3.7.22. A torsion local B-shtuka over R is a pair MO B D .MO B ; MO B / consisting of a finitely generated free R[[z]]B -module MO B , and an R[[z]]B -linear isomor MO B Œ 1 . We take the obvious notion of morphisms. phism  O W O  MO B Œ 1  ! MB

z

z

If B 0 is a B-algebra with #B 0 < 1 and MO B is a torsion local B-shtuka, then we define a torsion local B 0 -shtuka MO B ˝B B 0 WD .MO B ˝B B 0 ; MO B ˝ idB 0 /. We call MO B ˝B B 0 the scalar extension of MO B . We may view a torsion local B-shtuka MO B as a torsion local shtuka by forgetting the B-action. We say that MO B is effective (respectively, effective with height 6 d ; respectively, étale) if it is so as a torsion local shtuka. These properties are stable under scalar extensions.

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Lemma 3.7.23. (a) For any torsion local B-shtuka MO B , TL" MO B is a finitely generated free B-module of rank equal to rkR[[z ]]B MO B , where the B-action on TL" MO B is induced from the B-action on MO B . (b) For any B-algebra B 0 with #B 0 < 1, we have a natural Gal.K sep=K/equivariant isomorphism TL" .MO B ˝B B 0 / Š .TL" MO B / ˝B B 0 . Proof. Both claims are straightforward from the isomorphism (3.7.1).



Let MO D .MO ; MO / be a torsion local shtuka equipped with a B-action that commutes with MO , and assume that TL" MO is free over B. Note that it does not necessarily follow that MO is a torsion local B-shtuka; indeed, if B is non-reduced then one can create an example where MO is not free over R[[z]] ˝A" B by the same approach as in [Kim11, Remark 1.6.3]. On the other hand, we have the following lemma: Lemma 3.7.24. Let F be a finite extension of F" . Then any local shtuka MO equipped with an F-action is a torsion local F-shtuka; i.e. the underlying module MO is free over R[[z]]F . Proof. Since the assertion can be checked after increasing the residue field of R, we may assume that R contains F. Since z D 0 in F we have R[[z]]F Š R ˝F" F Š QŒFWF" 1 .i / R where R.i / is R viewed as an F-algebra via O i W F ! R. Likewise, we i D0 Q obtain the isotypic decomposition MO D i MO .i / for any local shtuka MO D .MO ; MO / with F-action. Since MO is -torsion free, each factor MO .i / is free over R.i / . It remains to show that the R.i / -rank of MO .i / is constant, which follows since MO restricts to an  ! MO .i / Œ 1  for any 1 6 i 6 ŒF W F"   1.  isomorphism O  MO .i 1/ Œ 1   z

z

3.8 Deformation theory of Galois representations Mazur [Maz89] has introduced the notion of Galois deformation rings (both for number fields and p-adic local fields), as an attempt to see the counterpart on the Galois representation side of p-adic deformations of modular forms; cf. Hida theory. Since Wiles’s proof of Fermat’s last theorem, which opened up its application to modularity of Galois representations, Galois deformation theory has become an indispensable technical tool in number theory. Among the important actors in modularity lifting theorems are p-adic local deformation rings with a certain condition in terms of Fontaine’s theory, such as flat deformation rings, crystalline deformation rings and potentially semi-stable deformation rings. In the second author’s thesis (cf. [Kim11, §4]), it was shown that there exists an equi-characteristic analog of flat deformation rings (or crystalline deformation rings) by working with torsion local shtukas and Hodge–Pink theory instead of finite flat group schemes and Fontaine’s theory. The existence of such equi-characteristic

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deformation rings is quite surprising because without imposing any deformation condition, the deformation functor would have infinite-dimensional tangent space; cf. Remark 3.8.2. Indeed, the main result we obtain is the finiteness of the tangent space when we impose a suitable deformation condition in terms of torsion local shtuka models. In this section, we indicate the main idea for the existence of the equi-characteristic deformation rings, and list some of their properties, such as smoothness and dimension, which are analogous to Kisin’s study of flat deformation rings; cf. [Kis09a]. Throughout this section, we assume that K is a finite extension of Q" . We fix a finite extension F of F" and a continuous homomorphism W N Gal.K sep=K/ ! GLr .F/: Let ArtF[[z ]] denote the category of Artinian local F[[z]]-algebras with residue field F. Definition 3.8.1. For B 2 ArtF[[z ]] , a framed deformation of N over B is a continuous homomorphism B W Gal.K sep =K/ ! GLr .B/ which reduces to N modulo the maximal ideal mB of B. For any morphism B ! B 0 in ArtF[[z ]] , we define the scalar extension B 0 of a framed deformation B to be the composite B

B 0 W Gal.K sep =K/ ! GLr .B/ ! GLr .B 0 /:  Let D N denote the set-valued functor on ArtF[[z ]] , where DN .B/ is the set of framed deformations of N over B, and for any morphism B ! B 0 in ArtF[[z ]] the map  0 0 D N .B/ ! DN .B / is defined by the scalar extension B 7! B . A deformation of N over B is an equivalence class of framed deformations of N 0 over B, where two framed deformations B and B are equivalent if and only if there 0 ! GLr .F// which conjugates B to B . Let DN denote exists g 2 ker.GLr .B/ ! a set-valued functor on ArtF[[z ]] , where DN .B/ is the set of deformations of N over B.

Remark 3.8.2. If K is replaced by a finite extension of Qp , then it is a well-known result of Mazur that D N is pro-representable by a complete local noetherian ring (and DN is pro-representable under some restriction on ). N For equi-characteristic K, on the other hand, these functors cannot be pro-represented by a complete local noethe2 rian ring because the reduced tangent spaces DN .FŒu=.u2// and D N .FŒu=.u // are infinite-dimensional. Indeed, we have DN .FŒu=.u2// Š H1 .K; N ˝ N_ / by [Maz89, p. 391], and the Galois cohomology group is infinite-dimensional over F, because if 1W Gal.K sep =K/ ! GL1 .F/ is the trivial representation H1 .K; 1/ Š Hom.Gal.K sep= K/; F/ is infinite-dimensional (by Artin-Schreier theory; see [NSW08, § VI.1]) and it maps to H1 .K; N ˝ N_ / with finite dimensional kernel, by the following long exact sequence H0 .K; N ˝ N_ =1/ ! H1 .K; 1/ ! H1 .K; N ˝ N_ / : of Definition 3.8.3. Let d be a non-negative integer. We define a subfunctor D;6d N

;6d  .B/ if and only if B admits an D N so that for B 2 DN .B/, we have B 2 DN

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effective torsion local shtuka model with height 6 d as a torsion A" ŒGal.K sep =K/module; in other words, there exists a torsion local shtuka MO satisfying .z  /d MO MO .O  MO / MO , equipped with an A" -linear Galois-equivariant isomorphism B Š D TL" MO . Since the condition defining the subfunctor D;6d N only depends on N the equivalence classes of framed deformations (i.e. deformations), we may similarly define a subfunctor D6d N of DN . Theorem 3.8.4 (cf. [Kim11, Theorem 4.2.1]). Let K be a finite extension of Q" . is pro-representable by a complete local Then the framed deformation functor D;6d N , and there exists a complete local noetherian F[[z]]noetherian F[[z]]-algebra R;6d N

6d algebra R6d N which is a pro-representable hull of the deformation functor DN in the sense of Schlessinger [Sch68, Definition 2.7]. If we have EndGal.K sep =K/ . / N D F, then 6d 6d DN is pro-representable by RN .

Proof. The statement is an equi-characteristic analog of [Kim11, Theorem 1.3], and the proof can be easily adapted by working with torsion local shtukas instead of torsion '-modules over S. Let us sketch the main ideas. Let F be a set-valued functor on ArtF[[z ]] such that F .F/ is a singleton. By Schlessinger’s theorem [Sch68, Theorem 2.11], the following conditions are equivalent to pro-representability of F by a complete local noetherian F[[z]]-algebra: (H1) For any surjective map B 0 ! ! B with length-1 kernel and a map B 00 ! B in ArtF[[z ]] , the natural map h W F .B 0 B B 00 / ! F .B 0 /F .B/ F .B 00 / is surjective. (H2) In the setting of (H1), h is bijective if B D F and B 0 D FŒu=.u2 /. (This implies that F .FŒu=.u2// is an F-vector space; cf. [Sch68, Lemma 2.10].) (H3) Under (H2), the F-vector space F .FŒu=.u2// is finite-dimensional. ! B with length-1 kernel, the natural map h W (H4) For any surjective map B 0 ! F .B 0 B B 0 / ! F .B 0 / F .B/ F .B 0 / is bijective. Furthermore, the existence of a pro-representable hull of F is equivalent to the conditions (H1)–(H3). The proof of the pro-representability result in [Maz89, §1.2] actually shows that  D N and DN satisfy (H1) and (H2), and (H4) is satisfied by DN for any N and by DN if EndGal.K sep =K/ . / N D F. Since the condition defining the sub-functors D;6d and N

D6d is stable under finite direct products and subquotients (by Proposition 3.7.15), N one can check without difficulty that D;6d (respectively, D6d N ) satisfies (H1), (H2), N  or (H4) if and only if the same holds for DN (respectively, for DN ).

It remains to verify (H3) for D;6d and D6d N N , which is the key step. Indeed, it suf-

6d 2 2 fices to show that D6d N .FŒu=.u // is a finite set, since DN .FŒu=.u // is the quotient ;6d c r .FŒu=.u2// WD ker GLr .FŒu= .FŒu=.u2// by the natural action of GL of D N

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.u2 // ! GLr .F/ ; cf. the definition of deformations as equivalence classes of framed 2 deformations in Definition 3.8.1. The finiteness of D6d N .FŒu=.u // can be obtained by repeating the argument in [Kim11, §1.6]. The main ideas behind the argument can be described roughly as follows: (a) Any 2 D6d .FŒu=.u2// admits an effective torsion local shtuka model MO N

with height 6 d with an action of FŒu=.u2 /, which induces the correct scalar action on ; indeed, viewing as a torsion A" ŒGal.K sep =K/-module, the maximal torsion local shtuka model MO C of with height 6 d (cf. Lemma 3.7.19) inherits the action of FŒu=.u2 / by Lemma 3.7.19(b). (b) Considering only the F-action on MO , it follows that MO is a torsion local Fshtuka by Lemma 3.7.24. Then by the FŒu=.u2 / action, MO fits in the following short exact sequence of torsion local F-shtukas: 0 ! MO 1 ! MO ! MO 0 ! 0; where MO 0 and MO 1 are effective torsion local F-shtuka models of N with height 6 d , and there exists an F-linear map u W MO 0 ! MO 1 of torsion local shtuka models of N (induced by the action of u). (c) By Lemma 3.7.19, there are only finitely many choices of MO 0 and MO 1 . (Recall that K is discretely valued.) Therefore, it suffices to show that for any fixed MO 0 and MO 1 as above there exist only finitely many extensions of MO 0 by MO 1 as effective torsion local F-shtukas with height 6 d . This finiteness assertion can be read off from [Kim11, §1.6.9ff ]. Note that the argument uses the finiteness of the residue field k of K as the extensions naturally form a k-vector space. This concludes the proof.  Remark 3.8.5. One can define a notion of “"-torsion A-motives” with good reduction at " (over certain open subschemes U of a finite cover of the curve C from 3.1.1) in an analogous way to torsion local shtukas over R. It is possible to formulate a global deformation problem (with the height condition at " analogous to D6d N ) and prove a suitable global analog of Theorem 3.8.4. For what follows, let us assume that EndGal.K sep =K/ . / N D F so that R6d proN

6d represents D6d N . Since RN is a complete local noetherian F[[z]]-algebra, it is possi-

rig over F((z)), where . : /rig deble to associate to it a rigid analytic variety .Spf R6d N / notes the rigid analytic generic fiber. Keeping the analogy between p-adic crystalline rig can representations and equi-characteristic crystalline representations, .Spf R6d N / be viewed as an equi-characteristic analog of a p-adic crystalline deformation space. Now we are going to prove some properties of equi-characteristic crystalline deformation spaces (such as smoothness and the dimension formula), which will strengthen the analogy with p-adic crystalline deformation spaces. If D6d is not pro-repreN

instead of R6d sentable, then we can work with the framed deformation ring R;6d N N .

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Let 6d W Gal.K sep =K/ ! GLr .R6d N / denote (a representative of) the universal deformation (obtained by taking the limit of the deformations over Artinian quotients of R6d N ). Then given a finite-dimensional F((z))-algebra B and a continuous F[[z]]homomorphism fB W R6d ! B, we obtain N

6d

fB

fB . 6d / W Gal.K sep=K/ ! GLr .R6d ! GLr .B/: N /  Let E be a finite extension of F((z)) with valuation ring OE , and fix a continuous F[[z]]-morphism fE W R6d ! E. We set E WD fE . 6d /. Since fE factors N through OE by compactness, E comes equipped with a distinguished Galois-stable OE -lattice. Lemma 3.8.6. Let B be an Artinian local F((z))-algebra with residue field E, and let ! B be a continuous F[[z]]-homomorphism. Then there exists an effective fB W R6d N local shtuka MO with .z  /d MO MO .O  MO / MO , with a Q" ŒGal.K sep =K/isomorphism fB . 6d / Š VL" MO . Conversely, let B W Gal.K sep =K/ ! GLr .B/ be a lift of E such that for some effective local shtuka MO with .z  /d MO MO .O MO / MO , there exists a Q" ŒGal.K sep =K/-isomorphism B Š VL" MO . Then there exists a unique continuous ! B lifting fE , such that there exists an isomorF[[z]]-homomorphism fB W R6d N  6d phism B Š fB . / which reduces to the identity map on E . Proof. For the first claim, observe that by the usual compactness argument one can find an F[[z]]-subalgebra B0 B, finitely generated as an F[[z]]-module, such that B0 Œ 1z  D B and fB factors through B0 . By construction, fB . 6d / has a distinguished Galois-stable B0 -lattice, whose Artinian quotients admit effective torsion local shtuka models with height 6 d . We now obtain the first claim by applying Lemma 3.7.20. For the converse, let B ı B denote the preimage of OE . Since E factors through GLr .OE /, it follows that B factors through GLr .B ı /. Since B ı is a directed union of F[[z]]-sub-algebras of B which are finitely generated as F[[z]]-modules, the usual compactness argument shows that there exists an F[[z]]-subalgebra B0 B, finitely generated as a F[[z]]-module, such that B0 Œ 1z  D B and B is factored by B0 W Gal.K sep=K/ ! GLr .B0 /. Now viewing B as a Galois representation over Q" isomorphic to VL" MO , its B0 lattice B0 yields an A" -lattice T 0 VL" MO . By Proposition 3.4.22 and Lemma 3.2.5 there exists an effective local shtuka MO 0 isogenous to MO with .z  /d MO 0 MO 0 .O MO 0 / MO 0 with an isomorphism B0 Š TL" MO 0 . By applying the universal propto the mod z n reduction of B0 for each n, we obtain a continuous erty of R6d N F[[z]]-morphism fB0 W R6d ! B0 giving rise to B0 . By composing fB0 with the N  natural inclusion B0 ,! B, we obtain the desired fB .

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Let us make a digression to the z-isocrystal with Hodge–Pink structure associated to an equi-characteristic crystalline representation “with coefficients in B”. Let B be a finite-dimensional Q" -algebra, and let B 2 RepB Gal.K sep =K/ be a representation which is equi-characteristic crystalline if viewed as a Galois representation over Q" , that is B Š VL" MO as Q" ŒGal.K sep =K/-modules for a local shtuka MO over R. By full faithfulness of VL" (Theorem 3.4.20), there exists a local shtuka MO WD .MO ; MO /, unique up to quasi-isogeny, such that B acts on MO by quasi-morphism and we have a BŒGal.K sep=K/-isomorphism B Š VL" MO . Lemma 3.8.7. In the above setting, MO Œ 1z  is free over R[[z]]B WD R[[z]] ˝A" B with rank equal to rkB B . Proof. One can repeat the proof of [Kis08, Proposition 1.6.1] via Fitting ideals, with S and E.u/ replaced with R[[z]] and z  .  Corollary 3.8.8. Let B and MO be as above. We define H. B / WD H.MO / WD .D; D ; qD /, where we give the unique rigidification to MO ; cf. Lemma 3.5.8. Then the admissible z-isocrystal with Hodge–Pink structure H. B / enjoys the following additional properties: (a) D is free over k((z))B WD k((z)) ˝Q" B and D is B-linear. (b) qD D ˝k ((z )) K((z  )) is a finitely generated free submodule over K[[z  ]]B WD K[[z  ]] ˝Q" B. Proof. This corollary follows directly from Lemma 3.8.7 and the construction of D and qD ; cf. (3.5.6).  Assume that EndGal.K sep =K/ . / N D F, and fix fE W R6d ! E for some finite N  6d extension E of F((z)). Set E WD fE . / and H. E / WD .DE ; DE ; qDE /, and write   H. E / ˝ H. E /_ D Ad.DE /; Ad.DE / ; qAd.DE / ; with pAd.DE / denoting the tautological lattice;  cf. Definition 3.5.2. Identifying  Ad.DE / ˝K [[z ]]E K((z  ))E with EndK ((z ))E DE ˝K ((z ))E K((z  ))E we can describe pAd.DE / and qAd.DE / as follows:    pAd.DE / D EndK [[z ]]E pDE / and qAd.DE / D EndK [[z ]]E qDE I   in other words, as a K[[z  ]]E -lattice of EndK ((z ))E DE ˝K ((z ))E K((z  ))E , pAd.DE / (respectively, qAd.DE / ) consists of endomorphisms preserving pDE (respectively, qDE ).

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Theorem 3.8.9. In the above setting, the completion of R6d N ˝F((z )) E with respect to the kernel of fE ˝ 1 is isomorphic to a formal power series ring over E of dimension 1 C dimE

pAd.DE / pAd.DE / \ qAd.DE /

rig (This is a finite number since K is a finite extension of Q" .) In particular, .Spf R6d N / is a smooth rigid analytic variety over F((z)).

Remark 3.8.10. If we do not assume that EndGal.K sep =K/ . / N D F, one can still show rig / is a smooth rigid analytic variety over F((z)), and its dimension at that .Spf R;6d N ! E is an E-point fE W R;6d N

r 2 C dimE

pAd.DE / ; pAd.DE / \ qAd.DE /

with the obvious notation, where r D dimF N D dimE E . This formula is compatible with Theorem 3.8.9 if EndGal.K sep =K/ . / N D F, since the natural morphism ;6d 6d ! Spf RN is formally smooth of relative dimension r 2  1. (In fact, Spf RN formal smoothness is clear, and to obtain the relative dimension it suffices to look at the tangent spaces. Since we have EndGal.K sep =K/ . / N D F, the stabilizer of the natc r .FŒu=.u2//-action on D;6d .FŒu=.u2// is the subgroup of scalar matrices ural GL N 2 b Gm .FŒu=.u //, which shows that D;6d .FŒu=.u2// is a PGLr .FŒu=.u2//-torsor

b

N

b

2 2 over D6d N .FŒu=.u //. Since PGLr .FŒu=.u // Š pglr .F/, where the latter is an 2 .r  1/-dimensional F-vector space, we obtain the desired numerology.) This result should be thought of as an equi-characteristic analog of the formal smoothness and the dimension formula for the (rigid analytic) generic fibers of crystalline deformation rings; cf. the case with N D 0 of [Kis08, §3]. Note also that in the dimension formula, we have terms involving the Hodge–Pink structure associated to E instead of the Hodge–Pink filtration, which supports that the Hodge–Pink structure is the right equi-characteristic analog of the Hodge filtration associated to a p-adic crystalline representation. (Compare with [Kis08, Theorems 3.3.4, 3.3.8].)

Example 3.8.11. Assume that F D F" so that R[[z]]F D R. Consider a local shtuka  1 1 2 O O M where M D R[[z]] and MO is given by the matrix 0 .z/d . Then the isomor  1 g phism ıMO from Lemma 3.5.8 equals 0 `d for some g 2 R[[z; z 1 gŒ`1   and so    .g/1 O K[[z  ]]. One can check that N WD TL" MO ˝A" F" qD D pD C .z  /d .` /d O 

is a non-split extension of ." /d mod z by the trivial character. Therefore, we have EndGal.K sep =K/ . / N D F if and only if qO  1 does not divide d , which we assume; cf. Example 3.7.14.

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Clearly, TL" MO defines an A" -point f W R6d ! A" . By Theorem 3.8.9, this N

rig at f is given by implies that the dimension of .Spf R6d N /

  1 C dimQ" K[[z  ]]=.z  /d D 1 C d  ŒK W Q" ; because EndK [[z ]] .pD / D K[[z  ]]22 and EndK [[z ]] .pD / \ EndK [[z ]] .qD / D ˚a b  2 K[[z  ]]22 W b c d

O  /2d C .d  a/..g/ O  1/.` O  /d mod .z  /d :  c..g/ O  1/2 .` Let us begin the proof of Theorem 3.8.9. Let B be an Artinian local E-algebra with residue field E, and let I B be an ideal annihilated by the maximal ideal of  B. We fix a lift fB=I W R6d ! B=I of fE , and set B=I WD fB=I . 6d /. N In order to study a lift of fB=I to a B-point, we introduce the following complex concentrated in degrees 0 and 1:   .DE 1;i ncl/ pAd.DE /  : (3.8.1) C . E / WD Ad.DE / ! Ad.DE / ˚ pAd.DE / \ qAd.DE / Let H0 . E / and H1 . E / respectively denote the 0th and the 1st cohomology of C  . E /. Clearly, H0 . E / is naturally isomorphic to the E-vector space of E-linear endomorphisms of H. E /, which is isomorphic to EndGal.K sep =K/ . E / by full faithfulness results (Theorem 3.4.20, Proposition 3.5.12). Lemma 3.8.12. The set of maps fB W R6d ! B lifting fB=I is a principal homogeN 1 neous space under H . E / ˝E I ; in particular, there exists a lift fB of fB=I . Proof. By filtering I if necessary, it suffices to show the lemma when I is 1-dimensional as an E-vector space, which we assume from now on. Let u 2 I denote a principal generator. Let us first show the existence of a lift fB . Write H. B=I / D .D; D ; q/. Since D is free over k((z))B=I by Corollary 3.8.8, we pick a free k((z))B -module D that lifts D and lift D arbitrarily to D . Note that .D; D / is a z-isocrystal. Since q is free over K[[z  ]]B=I by Corollary 3.8.8, we choose a free K[[z  ]]B -module q lifting q and an injective map q ,! .z  /d p lifting the natural inclusion q ,! .z  /d p. (Here, p and p respectively denote the tautological lattices for D and D.) We set D WD .D; D ; q/. Let us first show that D is admissible. Indeed, choosing an E-basis u 2 I , we  have an isomorphism H. E / ! D ˝B I (with the obvious notation), so we have the following short exact sequence of z-isocrystals with Hodge–Pink structure: 0 ! H. E / ! D ! H. B=I / ! 0:

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Now, one can show that any extension of admissible z-isocrystals with Hodge–Pink structure is again admissible, using the characterization of admissibility in terms of the associated -bundle O F .D/ (Theorem 3.6.10(a)) and the classification of O bundles (Theorem 3.6.7(a)). Let MO denote a local shtuka with H.MO / Š D, which exists by admissibility. By construction, all the Hodge–Pink weights of MO are between 0 and d , so MO is effective with .z  /d MO MO .O  MO / MO . By full faithfulness (cf. Lemma 3.5.8 and Proposition 3.5.12), B acts on MO by quasi-morphism, so VL" .MO / has a natural B-action commuting with the Galois action. On the other hand, VL" .MO / is a free B-module lifting B=I by comparing the E-dimension and the minimal number of generators obtained by the Nakayama lemma. By Lemma 3.8.6 it follows that VL" .MO / defines a B-point of R6d N lifting fB=I . Let us now show that the set of fB lifting fB=I is a principal homogeneous space under H1 . E / Š H1 . E / ˝E I . Let D be as above. Given  2 Ad.DE / Š Endk ((z ))E .DE / and ı 2 pAd.DE / Š EndK [[z ]]E .pDE /, we obtain another z-isocrystal with Hodge–Pink structure as follows: D . ;ı/ WD .D; .1 C u /D ; .1 C uı/q/: By repeating the previous argument for D . ;ı/ instead of D, it follows that D . ;ı/ defines a B-lift of fB=I , thus we obtain an E-linear transitive action of Ad.DE / ˚ pAd.DE / on the set of B-lifts of fB=I . (Transitivity follows from Corollary 3.8.8.) One can easily check that D . ;ı/ and D . 0 ;ı 0 / define the same B-lift of fB=I if and only if .; ı/ and . 0 ; ı0 / define the same class in H1 . E /. This shows the desired claim.  Proof of Theorem 3.8.9. Lemma 3.8.12 shows that the completion of R6d N ˝F[[z ]] E at the maximal ideal corresponding to fE is a formal power series ring of dimension equal to dimE H1 . E /. It remains to compute dimE H1 . E /. Note that dimE H0 . E /  dimE H1 . E /  D dimE Ad.DE /  dimE Ad.DE / C dimE

pAd.DE / pAd.DE / \ qAd.DE /

 ;

and H0 . E / Š EndGal.K sep =K/ . E / is 1-dimensional since EndGal.K sep =K/ . F / Š F. This shows that dimE H1 . E / has the expected dimension. rig By construction of rigid analytic generic fibers (cf. [dJ95, §7.1]), .Spf R6d N / is the direct union of affinoid rigid analytic spaces Sp Bn , where Bn is the complemn ˝F[[z ]] F((z)) with respect to the norm such that R6d tion of R6d N N Œ z  is the set of norm 6 1 elements in R6d N ˝F[[z ]] F((z)). Since fSp Bn g is an admissible covering of

rig .Spf R6d N / , it remains to show that each Bn is a topologically smooth affinoid algebra over F((z)). Since Bn is Jacobson [BGR84, §5.2.6, Theorem 3] and all ideals of Bn are closed [BGR84, §5.2.7, Corollary 2], it suffices to show that the completion

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of Bn at any closed maximal ideal (i.e. classical points of Sp Bn ) is geometrically regular. (Note that the invertibility of Jacobian can be captured by the completions at a dense set of prime ideals.) By [dJ95, Lemma 7.1.9], any (closed) maximal ideal of Bn restricts to a maximal ideal of R6d N ˝F[[z ]] F((z)), and the completion of Bn at a maximal ideal is isomorphic ˝F[[z ]] F((z)) at the corresponding maximal ideal. Since to the completion of R6d N

Lemma 3.8.12 shows that the local ring of R6d N ˝F[[z ]] F((z)) at any maximal ideal is geometrically regular, we obtain the desired smoothness claim. 

We end the section with discussing the equi-characteristic analog of moduli of finite flat group schemes [Kis09a]. We first need the following definition: Definition 3.8.13. Let B be an A" -algebra with #B < 1 (not a finite Q" -algebra), and let TB be a finitely generated free B-module equipped with a discrete action of Gal.K sep =K/. Then a torsion local B-shtuka model of TB is a torsion local B-shtuka MO B equipped with a BŒGal.K sep =K/-isomorphism TB Š TL" MO B . We take the obvious notion of equivalence for torsion local B-shtuka models of TB . We can understand an equivalence class of torsion local B-shtuka models of TB as a certain R[[z]]B -lattice of TB ˝A" K sep [[z]] stable under 1 ˝ O and invariant under the Galois action, using the isomorphism (3.4.1). Theorem 3.8.14 ([Kim11, Proposition 4.3.1]). We fix N W Gal.K sep =K/ ! GLd .F/, and for simplicity we assume that EndGal.K sep =K/ . / N Š F. Then there exists a pro6d over Spec R with the following property. For any local jective scheme GR6d N N 6d R6d N -algebra B with #B < 1, HomR6d .Spec B; GRN / is functorially bijective N

with the set of equivalence classes of torsion local B-shtuka models of 6d ˝R6d B which are effective and of height 6 d .

N

can be represented by a closed subscheme of a certain affine Proof. Indeed, GR6d N Grassmannian for GLr , following the same idea as Kisin’s construction of moduli of finite flat group schemes; cf. [Kis09a, Proposition 2.1.10].  Remark 3.8.15. In the p-adic setting, Kisin’s construction of moduli of finite flat group schemes over flat deformation rings was “globalized” by Pappas and Rapoport; cf. [PR09]. Namely, the mod p local Galois representation N is allowed to vary in the moduli space constructed by Pappas and Rapoport, and we recover moduli of finite flat group schemes by “fixing ”. N Such “coefficient spaces” (which can be thought of as fixing a base of p-divisible groups and varying coefficients) bear somewhat striking similarities with moduli spaces of p-divisible groups (with fixed coefficients Zp and varying its base), commonly known as Rapoport-Zink spaces. For example, there exists a natural “period-morphisms” [PR09, §5] on the rigid analytic generic fiber of a Pappas-Rapoport coefficient space whose image was computed by Hellmann and the first author; see [Hel13, Theorem 7.8] and [HH13, Corollary 6.12].

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By repeating the construction in [PR09] or [HH13] by working with A" instead of Zp , one can obtain an equi-characteristic analog of Pappas-Rapoport coefficient spaces, which can roughly be thought of as the moduli space of local shtukas with fixed base R and varying coefficients. Such equi-characteristic coefficient spaces could be interesting objects to study; for example, one can ask about a description of the image of the rigid analytic period morphism, analogous to [Hel13, HH13].

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M. Kisin, Crystalline representations and F -crystals, in Algebraic geometry and number theory, pp. 459–496, Progress in Math. 253, Birkhäuser, Boston, MA, 2006; also available at www.math.harvard.edu/kisin.

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S. Lang, Algebraic groups over finite fields. Amer. J. Math. 78 (1956), 555–563.

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H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, Cambridge, 1986.

[Maz89] B. Mazur, Deforming Galois representations. In Galois groups over Q, pp. 385–437, Math. Sci. Res. Inst. Publ., New York, 1989; also available at http://www.math.harvard.edu/mazur/. [Mis12]

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Chapter 4

Frobenius difference equations and difference Galois groups Chieh-Yu Chang

4.1 Introduction This is a survey article on recent progress concerning transcendence problems over function fields in positive characteristic. We are interested in some special values that occur in the following two ways. One is the special values of certain special transcendental functions, e.g. Carlitz -values at positive integers, which are specialization of Goss’ two-variable -function, arithmetic (resp. geometric) -functions at proper fractions (resp. proper rational functions), which are specialization of Goss’ two-variable -function, and Drinfeld logarithms at algebraic points, etc. The other is from algebro-geometric objects that are defined over algebraic function fields. The suitable geometric objects here are Drinfeld modules and the special values are the entries of the period matrix of a Drinfeld module that is related to the comparison between the de Rham and Betti cohomologies of the given Drinfeld module. A natural question concerns the transcendence of these special values. In the 1980s and 1990s, Yu successfully developed methods of Gelfond–Schneider–Lang type, which can be applied to prove many important results on transcendence of the special values mentioned above. The breakthrough from transcendence of single values to linear independence of several special values is Yu’s sub-t-module theorem [38], which is an analogue of the subgroup theorem of Wüstholz [33]. Here t-modules are higher-dimension analogues of Drinfeld modules introduced by Anderson [1] and they play the analogous role of commutative algebraic groups in classical transcendence theory. The key ingredient when applying Yu’s sub-t-module theorem is to relate the special values in question to periods of certain t-modules. For more details we refer the readers to [38]. In 2004, Anderson–Brownawell–Papanikolas [3] developed a linear independence criterion over function fields, the so-called ABP criterion. It results from a system of Frobenius difference equations, which can be thought of as analogues of classical first-order linear differential equations. Passing from rigid analytically trivial abelian t-modules to rigid analytically trivial dual t-motives in the terminology of [3], we naturally have a system of Frobenius difference equations which parameterize the dual t-motives in question. As remarked in [3, §1.3.4], following this direction the ABP criterion may be regarded as a t-motivic translation of Yu’s sub-t-module theorem. In [22], Papanikolas developed a Tannakian formulation for certain kinds of Frobenius difference modules, which are called rigid analytically trivial pre-t-motives. Note that there is a fully faithful functor F from the category of rigid analytically

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trivial dual t-motives up to isogeny to the Tannakian category R of rigid analytically trivial pre-t-motives (see [22, Thm. 3.4.9]). The category of t-motives in the terminology of [22] is the strictly full Tannakian subcategory of R generated by the essential image of F . Given a system of Frobenius difference equations occurring from a rigid analytically trivial pre-t-motive M , Papanikolas further developed a Picard-Vessiot theory for this system and constructed its difference Galois group explicitly. He further proved that the difference Galois group in question is isomorphic to the Galois group of M from Tannakian duality. Using ABP criterion and Picard-Vessiot theory, Papanikolas achieved an analogue of Grothendieck’s periods conjecture for abelian varieties: the dimension of the Galois group of a rigid analytically trivial pre-t-motive that is an image of F is equal to the transcendence degree of the period matrix of the pre-t-motive. From such an equality, we shall say that this pre-t-motive has the GP (Grothendieck periods) property. This property is the central spirit of our t-motivic transcendence program. We will review this t-motivic transcendence theory in § 4.2. From § 4.3 to § 4.5, we will review the recent algebraic independence results on special -values, -values, and periods and logarithms of Drinfeld modules by using these t-motivic techniques. In § 4.6, we will review a refined version of the ABP criterion investigated by the author of the present article. We will see that not only rigid analytically trivial pre-t-motives that are images of F have the GP property, but that there is a bigger class of pre-t-motives that have the property. We will also review its application to transcendence problems concerning Carlitz -values with varying finite constant fields. Finally, we mention that in order to avoid some unnecessary confusions with the terminology of t-motives in [1] and [3, 22], in this article we will use the terminology of rigid analytically trivial pre-t-motives that have the GP property instead of using the terminology of t-motives in [22] or dual t-motives in [3]. Acknowledgements. The author thanks National Central University for financial support to attend the workshop “t-motives: Hodge Structures, Transcendence and Other Motivic Aspects” in Banff. He further thanks the organizers for such a wellorganized workshop and BIRS for its hospitality. Finally, the author is grateful to the referee for many helpful comments and suggestions.

4.2 t-Motivic transcendence theory 4.2.1 Notation and Frobenius twisting. Throughout this article, we adopt the following notation. Fq t; # A AC

D the finite field of q elements with characteristic p. D independent variables. D Fq Œ# D the polynomial ring in the variable # over Fq . D the set of all monic polynomials in A.

Frobenius difference equations and difference Galois groups

k k1 k1 kN C1 j  j1 T L Ga GLr=F Gm

263

D Fq .#/ D the fraction field of A. D Fq ..1=#//, the completion of k with respect to the place at infinity. D a fixed algebraic closure of k1 . D the algebraic closure of k in k1 . D the completion of k1 with respect to the canonical extension of 1. D a fixed absolute value for the completed field C1 with j#j1 D q. D ff 2 C1 ŒŒt W f converges on jtj1  1g (the Tate algebra over C1 ). D the fraction field of T. D the additive group. D for a field F , the F -group scheme of invertible r  r matrices. D GL1 D the multiplicative group.

P For n 2 Z, given a Laurent series f D i ai t i 2 C1 ..t// we define the n-fold P qn i Frobenius twist over f by the rule f .n/ WD i ai t . For each n, the Frobenius twisting operation is an automorphism of the Laurent series field C1 ..t// stabilizing N N and T. More generally, for any matrix B with entries several subrings, e.g. kŒŒt, kŒt .n/ in C1 ..t// we define B by the rule B .n/ ij WD Bij.n/ . P i A power series f D 1 i D0 ai t 2 C1 ŒŒt that satisfies p lim i jai j1 D 0 and Œk1 .a0 ; a1 ; a2 ;    / W k1  < 1 i !1

is called an entire power series. As a function of t, such a power series f converges on whole C1 and, when restricted to k1 , f takes values in k1 . The ring of entire power series is denoted by E. Let Ai 2 Matmi .L/ for i D 1; : : : ; n, and m WD m1 C    C mn . We define ˚niD1 Ai 2 Matm .L/ to be the block diagonal matrix, i.e., the matrix with A1 ; : : : ; An down the diagonal and zeros elsewhere. 4.2.2 Tannakian formulation. In this section we follow [22] for relative backN ground and terminology. Let k.t/Œ;  1 be the noncommutative ring of Laurent N polynomials in  with coefficients in k.t/, subject to the relation f D f .1/ ;

N 8 f 2 k.t/:

N The Laurent series field C1 ..t// carries the natural structure of a left k.t/Œ;  1.1/ N N module by setting  .f / D f . As such, the subfields L and k.t/ are k.t/Œ;  1submodules. For any subfield F of C1 ..t// that is invariant under  , we denote by F  the subfield consisting of all elements in F fixed by  . Note that we have N  D Fq .t/: L D k.t/ See [22, Lem. 3.3.2] for more details.

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N N A left k.t/Œ;  1-module that is finite dimensional over k.t/ is called a pre-tmotive M . We let P be the category of pre-t-motives by defining morphisms in P as N left k.t/Œ;  1-module homomorphisms of pre-t-motives. Conversely, every matrix N N in the way above. defines a pre-t-motive of dimension r over k.t/ in GLr .k.t// N Given a pre-t-motive M of dimension r over k.t/, let m 2 Matr1 .M / comprise N a k.t/-basis of M . Then multiplication by  on M is given by  m D ˆm N for some matrix ˆ 2 GLr .k.t//. There are several important objects in the category P : (i) Tensor products of pre-t-motives. Given two pre-t-motives M1 and M2 , we N define M1 ˝ M2 to be the pre-t-motive whose underlying k.t/-vector space is M1 ˝k.t N / M2 , on which  acts diagonally. (ii) The Carlitz motive. We define the Carlitz motive to be the pre-t-motive C N N itself, on which  acts by whose underlying k.t/-space is k.t/ f D .t  #/f .1/

N for f 2 k.t/:

(iii) Internal Hom. Given two pre-t-motives M1 and M2 , we set Hom.M1 ; M2 / WD Homk.t N / .M1 ; M2 /: N N Then Hom.M1 ; M2 / is a k.t/-vector space and we define a left k.t/Œ;  1module structure on Hom.M1 ; M2 / by setting   WD  ı ı  1

for 2 Hom.M1 ; M2 /:

N and give a  -action on it by (iv) Identity object. We let 1 WD k.t/ f D f .1/

N for f 2 k.t/:

It has the properties:  For any M 2 P , the natural isomorphisms , M ˝k.t N / 1 Š 1 ˝k.t N / M Š M; are isomorphisms of pre-t-motives ;  EndP .1/ D Fq .t/. (v) Duals. Given any M 2 P , we define M _ WD Hom.M; 1/: It has the property that .M _ /_ Š M .

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265

Finally, we define the notion of rigid analytic trivialization. Let M be a pre-tN motive and let ˆ 2 GLr .k.t// be the matrix representing the multiplication by  on N M with respect to a k.t/-basis m of M . We say that M is rigid analytically trivial if there exists ‰ 2 GLr .L/ such that ‰ .1/ D ˆ‰: The matrix ‰ is called a rigid analytic trivialization for ˆ. It is unique up to right multiplication by a matrix in GLr .Fq .t// (see [22, §4.1.6]). An example of a rigid analytically trivial pre-t-motive is the Carlitz motive C . Throughout this article we fix a choice of .q  1/-th root of # and define  1  q Y t .t/ WD .#/ q1 1  i 2 E: (4.2.1) #q i D1 Then we have .1/ D .t  #/ and hence is a rigid analytic trivialization for 1 .t  #/. Note that the value Q WD .#/ is a fundamental period of the Carlitz Fq Œtp module (see [3, Cor. 5.4.1]). Such as the transcendence of 2 1, Q is known to be transcendental over k by the work of Wade [31]. For more details and relative background, see [7]. Given a pre-t-motive .M; ˆ; m/ as above, we consider M  WD L ˝k.t N / M , where N we give M  a left k.t/Œ;  1-module structure by letting  act diagonally:  .f ˝ m/ WD f .1/ ˝  m;

8 f 2 L; m 2 M:

Define M B WD .M  / WD f 2 M  W  D g: Then M B is a vector space over Fq .t/ since L D Fq .t/. Note that the natural map L ˝Fq .t / M B ! M  is an isomorphism if and only if M is rigid analytically trivial (see [22, §3.3]). In this situation, the entries of ‰ 1 m comprise an Fq .t/-basis of M B , where ‰ is a rigid analytic trivialization for ˆ. Theorem 4.2.1. (Papanikolas, [22, Thm. 3.3.15]) The category of rigid analytically trivial pre-t-motives R forms a neutral Tannakian category over Fq .t/ with fiber functor M 7! M B . Given any M 2 R, let RM be the strictly full Tannakian subcategory of R generated by M . That is, RM consists of all objects of R isomorphic to subquotients of finite direct sums of M ˝u ˝ .M _ /˝v for various u; v. By Tannakian duality there is an affine algebraic group scheme M over Fq .t/ such that RM is equivalent to the category of finite dimensional representations of M over Fq .t/. The algebraic group M is called the (t-motivic) Galois group of M . In the next section, we will see that the Galois group M and the faithful representation M ,! GL.M B / coming from Tannakian duality can be described explicitly.

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4.2.3 Difference Galois groups. From now on, we denote by .M; ˆ; ‰; m/ the object M 2 R endowed with the system of difference equations ‰ .1/ D ˆ‰ for N N a given k.t/-basis m of M described above. Let r be the dimension of M over k.t/ N and let X be an r  r matrix with r 2 independent variables Xij . Define the k.t/algebra homomorphism N 1=det.X / ! L ‰ W k.t/ŒX; Xij 7! ‰ij : N Put Z‰ WD Spec Im‰ and note that Z‰ is a closed k.t/-subscheme of GLr=k.t N /. We define two matrices ‰1 ; ‰2 2 GLr .L ˝k.t N / L/ by .‰1 /ij WD ‰ij ˝ 1; .‰2 /ij WD 1 ˝ ‰ij : e WD ‰ 1 ‰2 2 GLr .L ˝ N L/ and define the Fq .t/-algebra homomorphism Put ‰ 1 k.t / ‰ W Fq .t/ŒX; 1=det.X / ! L ˝k.t N /L e 7! ‰ ij : Xij Define ‰ WD Spec Im ‰ :

(4.2.2)

N Then ‰ is a closed subscheme of GLr=Fq .t / . Finally, we denote by k.t/.‰/ the field N generated by all the entries of ‰ over k.t/. Theorem 4.2.2. (Papanikolas, [22]) Given .M; ˆ; ‰; m/ 2 R, let Z‰ ; ‰ be defined as above. Then we have: (a) ‰ is an affine algebraic group scheme over Fq .t/. (b) ‰ is smooth over Fq .t/ and is geometrically connected. N over k.t/. N (c) Z‰ is a torsor for ‰ F .t / k.t/ q

(d) dim ‰ D tr: degk.t N /

N k.t/.‰/.

(e) ‰ is isomorphic to the Galois group M of M . Moreover, the faithful representation M ,! GL.M B / is described as follows: for any Fq .t/-algebra R, M .R/ ,! GL.R ˝Fq .t / M B /  7! .1 ˝ ‰ 1 m 7! . 1 ˝ 1/.1 ˝ ‰ 1 m//: 4.2.4 ABP criterion and connection with difference Galois groups. We recall the linear independence criterion developed by Anderson–Brownawell–Papanikolas [3], the so-called ABP criterion.

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Theorem 4.2.3. (Anderson–Brownawell–Papanikolas, [3, Thm. 3.1.1]) N Let ˆ 2 Matr .kŒt/ be given such that det ˆ D c.t  #/s for some c 2 kN  . Fix N a column vector 2 Matr1 .T/ satisfying .1/ D ˆ . For every 2 Mat1r .k/ N such that such that .#/ D 0, there exists a vector P 2 Mat1r .kŒt/ P

D0

and P .#/ D :

In the situation of Theorem 4.2.3, we first note that by [3, Prop. 3.1.3] the condition of ˆ implies 2 Matr1 .E/. We further mention that the spirit of the ABP N criterion is that every k-linear relation among the entries of .#/ can be lifted to N a kŒt-linear relation among the entries of . Although the theorem above is a kind of linear independence criterion, by taking tensor products one is able to pass linear independence to algebraic independence. The ideas due to Papanikolas are presented as the following. Let ˆ and be given in Theorem 4.2.3. For any n  1, we consider the Kronecker tensor product ˝n . Then the entries of ˝n comprise all monomials of total degree n in the entries of . Fix any d  1 and take e 2 MatN 1 .E/ to be the column vector whose entries are the concatenation of 1 and each of the columns of ˝n N N e 2 MatN .kŒt/ for n  d . (Here N D .r d C1  1/=.r  1/). We define ˆ \ GLN .k.t// to be the block diagonal matrix e WD Œ1 ˚ ˆ ˚ ˆ˝2 ˚ : : : ˚ ˆ˝d ; ˆ then it follows that e e: e.1/ D ˆ e and e satisfy the conditions of the ABP criterion. Note that ˆ polynomial relation among the entries of .#/ can be lifted to a relation among the entries of . By calculating the Hilbert series panikolas showed that

N Thus, every kN kŒt-polynomial in question, Pa-

N N tr: degk.t N / k.t/. / D tr: degkN k. .#//; N .#// is the field generated by all entries of .#/ over k. N Combining this where k. identity with Theorem 4.2.2, one has the following important equality. Theorem 4.2.4. (Papanikolas, [22]) Suppose we are given .M; ˆ; ‰; m/ 2 R and N suppose that ˆ 2 Matr .kŒt/, det ˆ D c.t  #/s , c 2 kN  , and that ‰ 2 GLr .T/. Then we have N dim M D tr: degkN k.‰.#//: For the ‰ in the theorem above, we shall call ‰ 1 .#/ the period matrix of M and note that it is unique up to left multiplication by a matrix in GLr .k/. In §4.5.1, we will see an explicit connection between ‰ 1 .#/ and period matrices of Drinfeld modules. Therefore, Theorem 4.2.4 can be regarded as an analogue of Grothendieck’s

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periods conjecture for abelian varieties. For those .M; ˆ; ‰; m/ 2 R having the two properties:  each entry of ‰ is regular at #, N N D tr: deg N k.‰.#//,  tr: deg N k.t/.‰/ k.t /

k

we say that M has the GP property (Grothendieck’s periods property), since it follows that N dim M D tr: degkN k.‰.#//: (4.2.3) So using t-motivic transcendence theory displays in the following way. Suppose we are given a set S of certain special values in C1 . If S has t-motivic interpretation in the sense that there is an object .M; ˆ; ‰; m/ 2 R which has the GP property N N and k.‰.#//  S , then we may have hope to figure out all the k-algebraic relations among the entries of S , since we have the equality (4.2.3). However, computing the dimension of M in terms of the known relations among the special values in question might be difficult in general.

4.3 Carlitz polylogarithms and special -values 4.3.1 Carlitz polylogarithms. The first application of Theorem 4.2.4 is the breakthrough on algebraic independence of Carlitz logarithms due to Papanikolas. (For the background of Carlitz module, we refer the reader to [7]). Theorem 4.3.1. (Papanikolas, [11, Thm. 1.2.6]) Let C be the Carlitz Fq Œt-module and expC .z/ be its exponential function. Let 1 ; : : : ; m 2 C1 satisfy expC .i / 2 kN for all 1  i  m. If 1 ; : : : ; m are linearly independent over k, then they are N algebraically independent over k. The theorem above is an analogue of the classical Gelfond’s conjecture on algebraic independence of logarithms of algebraic numbers. Conjecture 4.3.2. Let 1 ; : : : ; m 2 C satisfy ei 2 Q for all 1  i  m. If 1 ; : : : ; m are linearly independent over Q, then they are algebraically independent over Q. Under the assumptions in the conjecture above, one only knows the Q-linear independence of 1; 1 ; : : : ; m by the celebrated work of Baker in the 1960s. The analogue of Baker’s work for Drinfeld modules of arbitrary rank was established by Yu [38]. We will discuss the algebraic independence results for Drinfeld modules of higher rank in §4.5. Back to Theorem 4.3.1, in order to apply Theorem 4.2.4 suitably we have to give a t-motivic interpretation of the Carlitz logarithms as well as the Carlitz polylogarithms. Classically if one considers the motive which is an extension of the trivial

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one by the n-th Tate twist for n 2 N, then its periods can be given in terms of the classical n-th polylogarithms, which is  log.1  z/ when n D 1. In the positive characteristic case, we have the same phenomena as the following. Given a positive integer n, the n-th Carlitz polylogarithm is defined as logŒn C .z/ WD z C

1 X

zq

i D1

.#  # q /n .#  # q 2 /n    .#  # q i /n

i

:

N Note that logŒ1 C .z/ is the Carlitz logarithm (see [4, 7]). Let ˛ 2 k satisfy j˛j1 < nq

q1 for which the series logŒn j#j1 C .˛/ converges. Let M be the pre-t-motive which is N of dimension 2 over k.t/, and on which multiplication by  is represented by   .t  #/n 0 ˆ WD : ˛ .1/ .t  #/n 1

Then M fits into the short exact sequence of pre-t-motives 0 ! C ˝n ! M  1 ! 0; where C ˝n is the n-th tensor power of the Carlitz motive C . To solve the system of difference equations ‰ .1/ D ˆ‰, we define the following power series L˛;n .t/ WD ˛ C

1 X

˛q

i D1

.t  # q /n .t  # q 2 /n    .t  # q i /n

i

:

(4.3.1)

Specializing L˛;n at t D #, one sees that L˛;n .#/ is exactly the n-th Carlitz polylogarithm at ˛. Let be given in (4.2.1). Then one has the following identity . n L˛;n /.1/ D ˛ .1/ .t  #/n n C n L˛;n Defining

 ‰ WD

n n L˛;n

0 1

(4.3.2)

 2 GL2 .T/;

then it is a solution matrix satisfying the desired system of difference equations ‰ .1/ D ˆ‰, and so M is rigid analytically trivial and has the GP property because of Theorem 4.2.4.  More generally, given m nonzero elements ˛1 ;    ; ˛m 2 k with j˛i j1 < nq

q1 , we let L˛i ;n .t/ be the series as in (4.3.1) for i D 1;    ; m. We define j#j1 0 1 .t  #/n 0  0 B ˛ .1/ .t  #/n 1    0 C B 1 C ˆn D ˆ.˛1 ;    ; ˛m / WD B :: :: : : :: C 2 MatmC1 .kŒt/; @ : : A : :

.1/ .t  #/n ˛m

0



1 (4.3.3)

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0 1 :: :

  :: :

1 0 0 C 2 GLmC1 .T/; :: C : A

n L˛m ;n 0



1

0

n B L˛1 ;n ‰n D ‰.˛1 ;    ; ˛m / WD B :: @ : n

(4.3.4)

then by (4.3.2) we have ‰n.1/ D ˆn ‰n :

(4.3.5)

Therefore ˆn defines a rigid analytically trivial pre-t-motive Mn that has the GP property. Note that Mn fits into the short exact sequence of pre-t-motives 0 ! C ˝n ! Mn  1˚m ! 0: From the definition of ‰n , which is identified with Mn by Theorem 4.2.2, we see that 80 9 1  0  0 ˆ > ˆ >

: : ˆ > : : : ;  0  1 Note that by Theorem 4.2.2 the Galois group of C ˝n is isomorphic to Gm since N n is transcendental over k.t/. Since C ˝n is a sub-pre-t-motive of Mn , we have a surjective map  W Mn  C ˝n Š Gm : More precisely, for any Fq .t/-algebra R the restriction of the action of any element  2 Mn .R/ to R ˝Fq .t / .C ˝n /B is the same as the action of the upper left corner of  . That is,  is the projection on the upper left corner of any element of Mn . N n .#// D k. N Q n ; logŒn .˛1 /; : : : ; logŒn .˛m // and that Mn has As we have that k.‰ C C the GP property, Theorem 4.3.1 is a consequence of the following (for n D 1). Theorem 4.3.3. ([22, Thm. n6.3.2], [16, Thm. 3.1]) Let notation and assumptions be o Œn Œn n as above. Set Nn D k-Span Q ; logC .˛1 /; : : : ; logC .˛m / . Then we have   Œn dim ‰n D tr: degkN kN Q n ; logŒn .˛ /; : : : ; log .˛ / D dimk Nn : 1 m C C The first equality of the theorem above is from Theorem 4.2.4. To prove the second equality, it suffices to consider the case when dim ‰n < m C 1. We sketch a proof due to Papanikolas as the following steps. (I) The defining equations of ‰n are given by degree one polynomials over Fq .t/. (II) By (I) and Theorem 4.2.2, the defining equations of Z‰n (cf. § 4.2.3) are given N by degree one polynomials over k.t/.

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271

N (III) From the definition of Z‰n , we have k.t/-linear relations among the functions 1; n ; n L˛1 ;n ; : : : ; n L˛m ;n . By specializing at t D # of these functions, we Œn Œn are able to obtain k-linear relations among Q n ; logC .˛1 /; : : : ; logC .˛m /. Finally, using (III) we can show that dimk Nn  dim ‰n . As we have Œn

Œn

N Q n ; log .˛1 /; : : : ; log .˛m //  dimk Nn ; tr: degkN k. C C the result of Theorem 4.3.3 follows. 4.3.2 Special -values. To motivate the contents of this section, let us consider the classical Riemann -function .s/. The values .n/ for integers n  2 are called special -values. A natural question p is to ask the nature of such special values. The Euler relations, i.e., .2m/=.2 1/2m 2 Q for m 2 N, imply the transcendence of the -function at even positive integers and hence answer half part of the question above. However, one believes that .2m C 1/ for m 2 N should be transcendental numbers, but it is still an open question. Conjecturally one further expects that all the Q-algebraic relations among the special -values are generated by Euler relations. Conjecture 4.3.4. Given an integer s > 2, we have tr: degQ Q ..2/; .3/; : : : ; .s// D s  bs=2c: We turn to the positive characteristic world. The primary goal of this section is to explain all the algebraic relations among the following characteristic p -values: X 1 C .n/ WD 2 Fq ..1=#//; n D 1; 2; 3;    : (4.3.6) an a2AC

These -values were introduced in 1935 by L. Carlitz [9], where he obtained the Euler–Carlitz relations: if n is divisible by q  1, then C .n/=Q n 2 k. We call the positive integer n even provided it is a multiple of q  1, since q  1 is the cardinality of the units A . Since Q is transcendental over k, C .n/ is transcendental over k for n even. Therefore, the situation of these positive characteristic -values at even positive integers is completely analogous to the situation of the Riemann -function at even positive integers. In the classical case, the special zeta value .n/ is the specialization of the n-th polylogarithm at 1. However, this simple connection between zeta values and polylogarithms becomes more subtle in the function fields setting. In [4], Anderson and Thakur proved that C .n/ is a k-linear combination of logŒn C at integral points (see Theorem 4.3.6). They further gave a logarithmic interpretation of the Carlitz -values. More precisely, let C ˝n be the n-th tensor power of the Carlitz module. Then they showed that for each positive integer n, C .n/ occurs (up to a multiple in A) as the last coordinate of the logarithm of C ˝n at an explicitly constructed integral point of C ˝n .

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In [37], Yu developed his theory of the so-called Eq -functions to show the transcendence of the last coordinate of the logarithm of C ˝n at any nonzero algebraic point, whence deriving the transcendence of C .n/ for all positive integers n, in particular for odd n (i.e., n is not divisible by q  1). Later in [38], Yu developed the so-called sub-t-module theorem, which can be N applied to determine all k-linear relations among these Carlitz -values and powers of the fundamental period . Q As expected the Euler–Carlitz relations for n divisible N by q  1 are the only k-linear relations among these transcendental values. Since we are in the characteristic p fields, the Frobenius p-th power relations naturally occur: for positive integers m and n, m

C .pm n/ D C .n/p : (4.3.7) Using Papanikolas’ theory described in 2, Chang and Yu [16] demonstrated that these, i.e., Euler–Carlitz relations and p-th power relations, account for all the algebraic relations among the -values C .n/; n D 1; 2; 3;    . Theorem 4.3.5. (Chang-Yu, [16, Cor. 4.6]) For any positive integer s, we have N ; Q C .1/;    ; C .s// D s  bs=pc  bs=.q  1/c C bs=p.q  1/c C 1: tr: degkN k. 4.3.2.1 Anderson–Thukar formula. As is mentioned above, the classical special -value at an integer n  2 is the n-th polylogarithm at 1, but this is not valid in general in the function field setting. In their seminal work, Anderson and Thakur established a formula that C .n/ is a k-linear combination of the n-th Carlitz polynq logarithms of 1; #;    ; # ln with ln < q1 . We will see later that this formula enables us to give a t-motivic interpretation for C .n/. Theorem 4.3.6. (Anderson–Thukar, [4, §3.9]) Given any positive integer n, one can nq , such that the following find a finite sequence hn;0 ;    ; hn;ln 2 k with ln < q1 identity holds ln X hn;i L# i ;n .#/: (4.3.8) C .n/ D i D0

4.3.2.2 The Galois group of -motive. In this section, we assume q > 2 since all positive integers are even for q D 2, and in this case all Carlitz -values are rational multiples of powers of Q by the formula of Carlitz. We first briefly sketch the strategy of proving Theorem 4.3.5 as follows. First, we recall that we have established:  Theorem 4.3.3;  Theorem 4.3.6. We then use these two theorems above to list the following major steps to prove Theorem 4.3.5.  We make change of basis for Nn in Theorem 4.3.3 so that C .n/ occurs as an element of a basis for Nn .  We take direct sum of the pre-t-motives Mn with varying n and compute its Galois group exactly. More details in the two steps are given as the following.

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Given a positive integer n not divisible by q  1 we set Nn WD k-spanfQ n ; L1;n .#/; L#;n .#/;    ; L# ln ;n .#/g:

(4.3.9)

Note that according to (4.3.8) we have C .n/ 2 Nn and so mn C 2 WD dimk Nn  2 since C .n/ and Q n are linearly independent over k. (Note that Q n … k1 for .q 1/ − n). It follows that we can pick a k-basis of Nn which involves Q n and C .n/. More precisely, for each odd n we fix once for all a finite subset f˛n0 ;    ; ˛nmn g  f1; #;    ; # ln g such that both fQ n ; Ln0 .#/;    ; Lnmn .#/g and fQ n ; C .n/; Ln1.#/;    ; Lnmn .#/g are bases of Nn over k; where Lnj .t/ WD L˛nj ;n .t/ for j D 0;    ; mn : To each such odd integer n, we define Mn to be the pre-t-motive which is of N dimension mn C 2 over k.t/, and on which multiplication by  is represented by the matrix ˆn D ˆ.˛n0 ;    ; ˛nmn / as in (4.3.3): Theorem 4.3.3 implies that the Galois group of Mn has dimension mn C 2. Since Mn has the GP property, mn C 2 also equals the transcendence degree over kN of N Q n ; Ln0 .#/;    ; Lnm .#// D k. N Q n ; C .n/; Ln1.#/;    ; Lnm .#//: k. n n In particular, the elements Q n ; C .n/; Ln1.#/;    ; Lnmn .#/ N are algebraically independent over k. Given any positive integer s, we define U.s/ WD f1  n  s j p − n; q  1 − ng: We further define the block diagonal matrices ˆ.s/ WD ˚n2U.s/ ˆn ; ‰.s/ WD ˚n2U.s/ ‰n : The matrix ˆ.s/ defines a pre-t-motive M.s/ WD Mˆ.s/ , which is the direct sum of the pre-t-motives Mn with n 2 U.s/. Note that ‰.s/ is a rigid analytic trivialization for ˆ.s/ and M.s/ has the GP property by Theorem 4.2.4. We shall call M.s/ a -motive   as we have seen that [snD1 fC .n/g  kN ‰.s/ .#/ . In [16], the authors computed the Galois group .s/ of this -motive explicitly.

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Theorem 4.3.7. (Chang-Yu, [16, Thm. 4.5]) Fix any s 2 N. Then we have an exact sequence of algebraic groups over Fq .t/: 1 ! V.s/ ! .s/  Gm ! 1; Q where V.s/ is isomorphic to the vector group n2U.s/ Gamn C1 . In particular, we have X dim .s/ D 1 C .mn C 1/: n2U.s/

P By the theorem above we find that 1 C n2U.s/ .mn C 1/ is exactly the transcendence degree over kN of the following field: S N ; k. Q n2U.s/ fLn0 .#/;    ; Lnmn .#/g/ S N ; D k. Q n2U.s/ fC .n/; Ln1 .#/;    ; Lnmn .#/g/: It follows that the set fg Q

[

fC .n/; Ln1.#/;    ; Lnmn .#/g

n2U.s/

N hence also fC .n/ j n 2 U.s/g is algebraically is algebraically independent over k, N independent over k. Counting the cardinality of U.s/ shows Theorem 4.3.5.

4.4 Special values of geometric and arithmetic  -functions 4.4.1 Geometric  -function. We first mention the classical case as our motivation for this section. We consider the classical Euler -function at proper fractions (note that  has poles at non-positive integers and gives rational values at positive integers) and call them the special -values. The celebrated Chowla–Selberg formula expresses nonzero periods of CM elliptic curves defined over Q as products of special -values. For a CM elliptic curve over Q, Chudnovsky showed the algebraic independence of a nonzero period and a nonzero quasi-period of such a curve, and hence deriving the transcendence of special -values at those proper fractions having denominators 2; 4; 6. However, every special -value is expected to be a transcendental number and this problem is still wild open. For more details, see [32]. We further discuss the question on the Q-algebraic relations among these special -values. As the -function satisfies the translation formula .z C 1/ D z.z/, reflection formula .z/.1  z/ D = sin.z/; and Gauss multiplication identities     n1 1 n1 1  z C D .2/ 2 n 2 nz .nz/ for an integer n  2: .z/ z C n n

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275

Specializations at proper fractions of the identities above give rise to natural families of algebraic relations among these special -values. The Rohrlich–Lang conjecture p asserts that all Q-algebraic relations among the special -values and 2 1 are explained by the identities satisfied by the -function (see [32]). However, Rohrlich– Lang conjecture can be also formulated as the assertion that all Q-linear relations p among the monomials of special -values and 2 1 follow linearly from the twoterm relations provided by the Deligne-Koblitz-Ogus criterion. Its transcendence degree formulation is conjectured as the following. Conjecture 4.4.1. For any integer n > 2 the transcendence degree of the field generated by the set  p 1 f2 1g [ .r/I r 2 Z X Z0 n over Q is 1 C

1 2

 #.Z=n/.

Now we turn to the function field setting. In his Harvard PhD thesis, Thakur studied the geometric -function over A (see [27]), which is a specialization of the two-variable -function of Goss [20],   1 Y z 1 .z/ WD 1C ; z 2 C1 : z n n2AC

Note that we still use the notation  for the Thakur -function in characteristic p setting when there are no confusions. The function is a function field analogue of the classical Euler -function. It is meromorphic on C1 with poles at zero and n 2 AC and satisfies several functional equations, which are analogous to the translation, reflection, and Gauss multiplication identities satisfied by the classical -function. In analogy with classical special -values, which are values of the Euler function at proper fractions, we consider the special geometric -values .r/ for N r 2 k X A. For x, y 2 C 1 we denote by x  y when x=y 2 k . Then we N have the following three types of k-algebraic relations among the special geometric -values obtained from specializations at the identities satisfied by the -function: for all r 2 k X A, a 2 A, g 2 AC with deg g D d , we have:  .r C a/  .r/; Q  2Fq .r/  ; Q 

Q a2A=.g/

. rCa /  Q g

q d 1 q1

.r/.

In the characteristic p world, the transcendence of special geometric -values was first observed by Thakur [27] in the case q D 2. Thakur showed that when q D 2, all N values of .r/, r 2 k X A, are k-multiples of Q and hence are transcendental over k. Thakur also related some special geometric -values to periods of Drinfeld modules, and hence deduced their transcendence using the work of Yu [34].

276

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Compared with the classical transcendence question on special -values, Sinha [26] gave an answer for a large class of special geometric -values. Precisely, he proved the transcendence of . fa C b/ whenever a, f 2 AC , deg a < deg f , and b 2 A. The basic strategy of Sinha is to relate the special -values in question to periods of certain t-modules with complex multiplication by Carlitz cyclotomic fields (such t-modules are typical examples of the Hilbert–Blumental–Drinfeld modules). The construction of Sinha’s t-modules is to use Anderson’s soliton functions [2], which play the analogous roles of Coleman’s functions for Fermat curves. Having such t-modules at hand, Sinha applied Yu’s theorem of Schneider-Lang type [35] to deduce the transcendence of the special -values mentioned above. Concerning the algebraic relations among special geometric -values, a natural question is to determine which algebraic relations arise from the functional equations of the -function. To answer this question, Thakur [27] adopted the approach of Deligne-Koblitz-Ogus to devise a diamond bracket criterion. Such criterion can deterN Here a geometric -monomial mine whether a given geometric -monomial is in k. is a monomial in Q and special geometric -values with positive or negative exponents. In [6], Brownawell and Papanikolas not only proved the transcendence of any special geometric -value, but also proved the linear independence in the sense that N all the k-linear relations among 1; Q and special geometric -values are those generated by diamond-bracket relations. The main ingredient is to extend Sinha’s approach to relate the -values in question to the coordinates of periods and quasi-periods of certain t-modules with complex multiplication. Then the next step is to analyze the structure of the t-module and appeal to Yu’s sub-t-module theorem [38] to show the desired result. Later on in [3], Anderson, Brownawell and Papanikolas extended the approaches in [26, 6] to create rigid analytically trivial pre-t-motives whose period matrices contain the geometric -monomials in question. They further applied the ABP-criterion to analyze the interplay between the relations among the geometric -monomials and the isogeny relations among the simple quotients of the pre-t-motives in question. The detailed study allows them to show that all algebraic relations over kN among special geometric -values arise from diamond bracket relations among geometric -monomials, and thus showed that all algebraic relations among special geometric -values can be explained by the standard functional equations. As a consequence, the transcendence degree of the field generated by special geometric -values in question can be obtained explicitly and this is the precise function field analogue of the Rohrlich–Lang conjecture. Theorem 4.4.2 (Anderson–Brownawell–Papanikolas, [3, Cor. 1.2.2]). For any f 2 AC of positive degree, the transcendence degree of the field    1 N k fg Q [ .r/I r 2 A X .f0g [ AC / f over kN is 1 C

q2 q1

 #.A=f / .

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4.4.1.1 t-Motivic interpretation of special geometric  -values. Fix an f 2 AC with positive degree. Let Af be the free abelian group on symbols of the form Œx, where x 2 f1 A=A. Every a 2 Af is expressed as the form aD

X

ma 2 Z;

ma Œa=f ;

a2A;deg a 1;

R 2 L.X; Y /;

(5.1.1)

with d integer, see [27]. In this text, we will also be interested in analogues of these functions over complete, algebraically closed fields other than C and for this purpose it will be advantageous to choose right away an appropriate terminology. Indeed, in the typical situation we will analyse, there will be a base field K, together with a distinguished absolute value that will be denoted by j  j. Over K there will be other absolute values as well, and a product formula will hold. We will consider the completion of K with respect to j  j, its algebraic closure that will be embedded in its completion K with respect to an extension of j  j. The algebraic closure of Kalg. , embedded in K, will also be endowed with a absolute logarithmic height that will be used to prove transcendence results. Here, an element of K is transcendental if it does not belong to Kalg. . 1 We borrowed this presentation from Masser’s article [31, p. 5], whose point of view influenced our point of view.

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If L is a finite extension of K in K and f 2 LŒŒx is a formal series solution of (5.1.1), we will say that f is a Mahler function over K. If f converges at ˛ 2 Kalg. X f0g (for the distinguished absolute value), we will say that f .˛/ 2 K is a Mahler value and ˛ is a base point for this value. In spite of the generality of this terminology, in this text we will restrict our attention to the base fields Q; K D Fq .#/ and C.t/ where C is the completion of an algebraic closure of the completion of K for the unique extension of the absolute value defined by jaj D q deg# a , with a 2 K. The interest of the method introduced by Mahler in [27] is that it can be generalised, as it was remarked by Mahler himself in [28], to explicitly produce finitely generated subfields of C of arbitrarily large transcendence degree. This partly explains, after that the theory was long-neglected for about forty years, a regain of interest in it, starting from the late seventies, especially due to the intensive work of Loxton and van der Poorten, Masser, Nishioka as well as other authors we do not mention here but that are quoted, for example, in Nishioka’s book [37]. In some sense, Mahler’s functions and Siegel’s E-functions share similar properties; large transcendence degree subfields of C can also be explicitly constructed by the so-called Siegel-Shidlowski theorem on values of Siegel E-functions at algebraic numbers (see Lang’s account on the theory in [26]). However, this method makes fundamental use of the fact that E-functions are entire, with finite analytic growth order. This strong assumption is not at all required when it is possible to apply Mahler’s method, where the functions involved have natural boundaries for analytic continuation; this is certainly an advantage that this theory has. Unfortunately, no complex “classical constant” (period, special value of exponential function at algebraic numbers. . . ) seems to occur as a complex Mahler’s value, as far as we can see. More recently, a variety of results by Becker, Denis [9, 17, 18, 19, 20] and other authors changed the aspect of the theory, especially that of Mahler’s functions over fields of positive characteristic. It was a fundamental discovery of Denis, that every period of Carlitz’s exponential function is a Mahler’s value, hence providing a new proof of its transcendency. This motivates our choice of terminology; we hope the reader will not find it too heavy. At least, it will be useful to compare the theory over Q and that over K. The aim of this paper is to provide an overview of the theory from its beginning (transcendence) to its recent development in algebraic independence and its important excursions in positive characteristic, where it is in “competition” with more recent, and completely different techniques inspired by the theory of Anderson’s t-motives (see, for example, the work of Anderson, Brownawell, Papanikolas, Chieh-Yu Chang, Jing Yu, other authors [6, 14, 39] and the related bibliographies). The presentation of the paper essentially follows, in an expanded form, the instructional talk the author gave in the conference “t-motives: Hodge structures, transcendence and other motivic aspects”, held in Banff, Alberta, (September 27–October 2, 2009). The author is thankful to the organisers of this excellent conference for giving the opportunity to present these topics, and thankful to the Banff Centre for the exceptional environment of working it provided. The author also wishes to express his gratitude to B. Adamczewski and P. Philippon for discussions and hints that helped to improve the presentation of this text, and to P. Bundschuh, H. Kaneko and

An introduction to Mahler’s method for transcendence and algebraic independence

299

T. Tanaka for a description of the algebraic relations involving the functions Lr of Section 5.4.1.1 they provided. Here is what the paper contains. In Section 5.2, we give an account of transcendence theory of Mahler’s values with Q as a base field; this is part of the classical theory, essentially contained in one of the first results by Mahler. In Section 5.3, we will outline the transcendence theory with, as a base field, a function field of positive characteristic (topic which is closer to the themes of the conference). Here, the main two features are some applications to the arithmetic of periods of Anderson’s t-motives and some generalisations of results of the literature (cf. Theorem 5.3.3). In Section 5.4, we first make an overview of known results of algebraic independence over Q of Mahler’s values, then we describe more recent results in positive characteristic (with the base field K D Fq .#/) and finally, we mention some quantitative aspects. The main features of this section are elementary proofs of two results: one by Papanikolas [39], describing algebraic dependence relations between certain special values of Carlitz’s logarithms, and another one, by Chieh-Yu Chang and Jing Yu, describing all the algebraic dependence relations of values of Carlitz–Goss zeta function at positive integers. This paper does not contain a complete survey on Mahler’s method. For example, Mahler’s method was also successful in handling several variable functions. To keep the size of this survey reasonable, we made the arguable decision of not describing this part of the theory, concentrating on the theory in one variable, which seemed closer to the other themes of the conference.

5.2 Transcendence theory over the base field Q 5.2.1 An example to begin with. The example that follows gives an idea of the method. We consider the formal series: fTM .x/ D

1 Y

n

.1  x 2 / D

nD0

1 X

.1/an x n 2 ZŒŒx

nD0

.an /n0 being the Thue-Morse sequence (an is the reduction modulo 2 of the sum of the digits of the binary expansion of n and, needless to say, the subscript TM in fTM stands for “Thue-Morse”). The formal series fTM converges in the open unit ball B.0; 1/ to an analytic function and satisfies the functional equation fTM .x 2 / D

fTM .x/ ; 1x

(5.2.1)

Y so that fTM is a Mahler’s function. In all the (in (5.1.1), d D 2 and R D 1X following, we fix an embedding in C of the algebraic closure Qalg. of Q. We want to prove:

Theorem 5.2.1. For all ˛ 2 Qalg. with 0 < j˛j < 1, fTM .˛/ is transcendental.

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Federico Pellarin

This is a very particular case of a result of Mahler [27] reproduced as Theorem 5.2.2 in the present paper. The proof, contained in 5.2.1.3 uses properties of Weil’s logarithmic absolute height reviewed in 5.2.1.2. It will also use the property that fTM is transcendental over C.x/, proved below in 5.2.1.1. 5.2.1.1 Transcendence of fTM . The transcendence of fTM over C.x/ can be checked in several ways. A first way to proceed appeals to Pólya–Carlson Theorem (1921), (statement and proof can be found on p. 265 of [48]). It says that a given formal series ' 2 ZŒŒx converging with radius of convergence 1, either has fz 2 C; jzj D 1g as natural boundary for holomorphy, or can be extended to a rational function of P .x/ the form .1x m /n , with P 2 ZŒx. To show that fTM is transcendental, it suffices to prove that fTM is not of the form above, which is evident from the functional equation (5.2.1), which implies that fTM has bounded integral coefficients. Indeed, if rational, fTM should have ultimately periodic sequence of the coefficients. However, it is well known (and easy to prove, see [47, Chapter 5, Proposition 5.1.2]) that this is not the case for the Thue-Morse sequence. Another way to check the transcendence of fTM is that suggested in Nishioka’s paper [35]. Assuming that fTM is algebraic, the field F D C.x; fTM .x// is an algebraic extension of C.x/ of degree, say n, and we want to prove that this degree is 1. It is possible to contradict this property observing that the extension F of C.x d / ramifies at the places 0 and 1 only and applying Riemann-Hurwitz formula. Hence, fTM is rational and we know already from the lines above how to exclude this case. 5.2.1.2 A short account on heights. Here we closely follow Lang [26, Chapter 3] and Waldschmidt [54, Chapter 3]. Let L be a number field. The absolute logarithmic height h.˛0 W    W ˛n / of a projective point .˛0 W    W ˛n / 2 Pn .L/ is the following weighted average of logarithms of absolute values: h.˛0 W    W ˛n / D

X 1 dv log maxfj˛0 jv ; : : : ; j˛n jv g; ŒL W Q v2ML

where v runs over a complete set ML of non-equivalent places of L, where dv D ŒLv W Qp  with vjQ D p the local degree at the place v (one then writes that vjp) (Lv ; Qp are completions of L; Q at the respective places so that if vj1, Lv D R or Lv D C according to whether the place v is real or complex), and where j jv denotes, for all v, an element of v chosen in such a way that the following product formula holds: Y j˛jdv v D 1; ˛ 2 L ; (5.2.2) v2ML

where we notice that only finitely many factors of this product are distinct from 1. A common way to normalise the j  jv ’s is to set jxjv D x if x 2 Q, x > 0, and vj1, and jpjv D 1=p if vjp.

An introduction to Mahler’s method for transcendence and algebraic independence

301

This formula implies that h does not depend on the choice of the number field L, so that we have a well defined function h W Pn .Qalg. / ! R0 : If n D 1 we also write h.˛/ D h.1 W ˛/. For example, we have h.p=q/ D h..1 W p=q// D log maxfjpj; jqjg if p; q are relatively prime and q ¤ 0. With the convention h.0/ WD 0, this defines a function h W Qalg. ! R0

(5.2.3)

satisfying, for ˛; ˇ 2 Qalg. : h.˛ C ˇ/  h.˛/ C h.ˇ/ C log 2; h.˛ˇ/  h.˛/ C h.ˇ/; h.˛ n / D jnjh.˛/;

n 2 Z:

More generally, if P 2 ZŒX1 ; : : : ; Xn  and if ˛1 ; : : : ; ˛n are in Qalg. , n X .degXi P /h.˛i /; h.P .˛1 ; : : : ; ˛n //  log L.P / C

(5.2.4)

i D1

where L.P / denotes the length of P , that is, the sum of the absolute values of the coefficients of P . Proofs of these properties are easy collecting metric information at every place. More details can be found in [54, Chapter 3]. Liouville’s inequality, a sort of “fundamental theorem of transcendence”, reads as follows. Let L be a number field, v an archimedean place of L, n an integer. For i D 1; : : : ; n, let ˛i be an element of L. Further, let P be a polynomial in n variables X1 ; : : : ; Xn , with coefficients in Z, which does not vanish at the point .˛1 ; : : : ; ˛n /. Assume that P is of degree at most Ni with respect to the variable Xi . Then, log jP .˛1 ; : : : ; ˛n /jv  .ŒL W Q  1/ log L.P /  ŒL W Q

n X

Ni h.˛i /:

i D1

The proof of this inequality is again a simple application of product formula (5.2.2): [54, Section 3.5]. It implies that for ˇ 2 Qalg. , ˇ ¤ 0, log jˇj  ŒL W Qh.ˇ/:

(5.2.5)

This inequality suffices for most of the arithmetic purposes of this paper (again, see [54] for the details of these basic tools).

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5.2.1.3 Proof of Theorem 5.2.1. Step (AP). For all N  0, we choose a polynomial PN 2 QŒX; Y Xf0g of degree  N in both X and Y , such that the order of vanishing .N / at x D 0 of the formal series FN .x/ WD PN .x; fTM .x// D c .N / x .N / C   

.c .N / ¤ 0/:

(not identically zero because fTM is transcendental by 5.2.1.1), is  N 2 . The existence of PN follows from the existence of a non-trivial solution of a homogeneous linear system with N 2 linear equations defined over Q in .N C 1/2 indeterminates. We will not need to control the size of the coefficients of PN and this is quite unusual in transcendence theory. Step (NV). Let ˛ be an algebraic number such that 0 < j˛j < 1 and let us suppose by contradiction that fTM .˛/ is also algebraic, so that there exists a number field L containing at once ˛ and fTM .˛/. Then, by the functional equation (5.2.1), for all n  0,   fTM .˛/ 2nC1 2nC1 2nC1 2nC1 FN .˛ / D PN .˛ ; fTM .˛ // D PN ˛ ; 2 L: .1  ˛/    .1  ˛ 2n / nC1

We know that FN .˛ 2 / ¤ 0 for all n big enough depending on N and ˛; indeed, fN is not identically zero and analytic at 0. Step (UB). Writing the expansion of fN at 0 fN .x/ D

X

0

cm x m D x .N / @c .N / C

X

1 c .N /Ci x i A

i 1

m .N /

with the leading coefficient c .N / which is a non-zero rational integer (whose size we do not control), we see that for all " > 0, if n is big enough depending on N; ˛ and ": log jFN .˛ 2

nC1

/j  log jc .N / j C 2nC1 .N / log j˛j C ":

Step (LB). At once, by (5.2.4) and (5.2.5), log jFN .˛ 2

nC1

/j

 ŒL W Q.L.PN / C N h.˛ 2

nC1

n

/ C N h.f .˛/=.1  ˛/    .1  ˛ 2 /// n X n  ŒL W Q.L.PN / C N 2nC1 h.˛/ C N h.fTM .˛// C h.1  ˛ 2 // i D0

 ŒL W Q.L.PN / C N 2nC2 h.˛/ C N h.fTM .˛// C .n C 1/ log 2/: The four steps allow to conclude: for all n big enough, 2

n1

log jc .N / j C .N / log j˛j C 2n1 "

 ŒL W Q.L.PN /2n1 C 2N h.˛/ C N 2n1h.fTM .˛// C .n C 1/2n1 log 2/:

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Letting n tend to infinity and using that .N /  N 2 (recall that log j˛j is negative), we find the inequality N log j˛j  2ŒL W Qh.˛/: But the choice of the “auxiliary” polynomial PN can be done for every N > 0. With N >

2ŒL W Qh.˛/ ; j log j˛jj

(5.2.6)

we encounter a contradiction. 5.2.1.4 A more general result. For R D N=D 2 C.X; Y / with N; D relatively prime polynomials in C.X /ŒY , we write hY .R/ WD maxfdegY N; degY Dg. With the arguments above, the reader can be easily prove the following theorem originally due to Mahler [27]. Theorem 5.2.2 (Mahler). Let L C be a number field, R be an element of L.X; Y /, d > 1 an integer such that hY .R/ < d . Let f 2 LŒŒx be a transcendental formal series such that, in L..x//, f .x d / D R.x; f .x//: Let us suppose that f converges for x 2 C with jxj < 1. Let ˛ be an element of L such that 0 < j˛j < 1. n Then, for all n big enough, f .˛ d / is transcendental over Q. n

nC1

Obviously, for all n big enough, L.f .˛ d //alg. D L.f .˛ d //alg. . It can happen, under the hypotheses of Theorem 5.2.2, that f .˛/ is well defined and algebraic for certain ˛ 2 Qalg. X f0g. For example, the formal series f .x/ D

1 Y

i

.1  2x 2 / 2 ZŒŒx;

i D0

converging for x 2 C such that jxj < 1 and satisfying the functional equation f .x 2 / D

f .x/ ; 1  2x

(5.2.7)

i

vanishes at every ˛ such that ˛ 2 D 1=2, i  0. In particular, f being non-constant and having infinitely many zeroes, it is transcendental. By Theorem 5.2.2, f .1=4/ D f .x/ limx!1=2 12x is transcendental. Remark 5.2.3. Nishioka strengthened Theorem 5.2.2 allowing the rational function R satisfying only the relaxed condition hY .R/ < d 2 (see [37, Theorem 1.5.1] for

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an even stronger result). The proof, more involved than the proof of Theorem 5.2.2, follows most of the principles of it, with the following notable difference. To achieve the proof, a more careful choice of the polynomials PN is needed. In step (AP) it is again needed to choose a sequence of polynomials .PN /N with PN 2 QŒX; Y  of degree  N in X and Y , such that the function FN .x/ D PN .x; f .x// vanishes at x D 0 with order of vanishing  c1 N 2 for a constant c1 depending on ˛ and f . Since for hY .R/  d the size of the coefficients of PN influences the conclusion, the use of Siegel’s Lemma is now needed to accomplish this choice [53, Section 1.3]. To make good use of these refinements we need an improvement of the step (NV), since s an explicit upper bound like c2 N log N for the integer k such that F .x d / D 0 for 2 s D 0; : : : ; k is required ( ). 5.2.2 Some further discussions. In this subsection we discuss about some variants of Mahler’s method and applications to modular functions (in 5.2.2.1). We end with 5.2.2.2, where we quote a criterion of transcendence by Corvaja and Zannier quite different from Mahler’s method, since it can be obtained as a corollary of Schmidt’s subspace theorem. 5.2.2.1 The “stephanese” theorem. We refer to [50] for a precise description of the tools concerning elliptic curves and modular forms and functions, involved in this subsection. Let J.q/ D

X 1 C 744 C ci q i 2 .1=q/ZŒŒq q i 1

be the q-expansion of the classical hauptmodul for SL2 .Z/, converging for q 2 C such that 0 < jqj < 1. The following theorem was proved in 1996; see [8]: Theorem 5.2.4 (Barré-Sirieix, Diaz, Gramain and Philibert). For q complex such that 0 < jqj < 1, one at least of the two complex numbers q; J.q/ is transcendental. This stephanese theorem (3 ) furnished a positive answer to Mahler’s conjecture on values of the modular j -invariant (see [30]). Although we will not say much more about, we mention that a similar conjecture was independently formulated by Manin, for p-adic values of J at algebraic ˛’s, as the series J also converges in all punctured p-adic unit disks, for every prime p. Manin’s conjecture is proved in [8] as well. Manin’s conjecture is relevant for its connections with the values of p-adic L-functions and its consequences on p-adic variants of Birch and Swinnerton-Dyer conjecture.

2 This

is not difficult to obtain; see 5.2.2.1 below for a similar, but more difficult estimate. this result is called stephanese theorem from the name of the city of Saint-Etienne, where the authors of this result currently live. 3 Sometimes,

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Mahler’s conjecture was motivated by the fact that the function J satisfies the autonomous non-linear modular equation ˆ2 .J.q/; J.q 2// D 0, where ˆ2 .X; Y / D X 3 C Y 3  X 2 Y 2 C 1488X Y .X C Y /  162000.X 2 C Y 2 / C 40773375X Y C 8748000000.X C Y /  157464000000000: Mahler hoped to apply some suitable generalisation of Theorem 5.2.2. It is still unclear, at the time being, if this intuition is correct; we remark that Theorem 5.2.2 does not apply here. The proof of Theorem 5.2.4 relies on a variant of Mahler’s method that we discuss now. We first recall from [50] that there exists a collection of modular equations ˆn .J.q/; J.q n// D 0;

n > 0;

with explicitly calculable polynomials ˆn 2 ZŒX; Y  for all n. The stephanese team make use of the full collection of polynomials .ˆn /n>0 so let us briefly explain how these functional equations occur. For q complex such that 0 < jqj < 1, J.q/ is the modular invariant of an elliptic curve analytically isomorphic to the complex torus C =q Z ; if z 2 C is such that =.z/ > 0 and e2iz D q, then there also is a torus analytic isomorphism C =q Z  C=.Z C zZ/. Since the lattice Z C nzZ can be embedded in the lattice Z C zZ, the natural map C =q Z ! C =q nZ amounts to a cyclic isogeny of the corresponding elliptic curves which, being projective smooth curves, can be endowed with Weierstrass models y 2 D 4x 3  g2 x  g3 connected by algebraic relations independent on the choice of z. At the level of the modular invariants, these algebraic relations for n varying are precisely the modular equations, necessarily autonomous, defined over Z as a simple Galois argument shows. Assuming that for a given q with 0 < jqj < 1, J.q/ is algebraic, means that there exists an elliptic curve E analytically isomorphic to the torus C =q Z , which is definable over a number field (it has Weierstrass model y 2 D 4x 3  g2 x  g3 with g2 ; g3 2 Qalg. ). The discussion above, with the fact that the modular polynomials ˆn are defined over Z, implies that J.q n / is algebraic as well. Arithmetic estimates involved in the (LB) step of the proof of Theorem 5.2.4 require a precise control, for J.q/ algebraic, of the height of J.q n / and the degree dn of Q.J.q n/; J.q// over Q.J.q//. the degree dn can be easily computed counting Q lines in F2p for p prime dividing n; it thus divides the number .n/ D pjn .1C1=p/ and is bounded from above by c3 n1C" , for all " > 0. As for the height hn D h.J.q n //, we said that the modular polynomial ˆn is related to a family of cyclic isogenies of degree n connecting two families of elliptic curves. We then have, associated with the algebraic modular invariants J.q/; J.q n/, two isogenous elliptic curves defined over a number field, and the isogeny has degree n. Faltings theorem asserting that the modular heights of two isogenous elliptic curves may differ of at most the half of the logarithm of a minimal degree of isogeny gives the bound c3 .h.J.q// C .1=2/ log n/ for the logarithmic height h.J.q n // (this implies the delicate estimates the authors do in [8]).

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With these information in mind, the proof of Theorem 5.2.4 proceeds as follows. As in Remark 5.2.3, we use standard estimates of the growth of the absolute values of the (integral) coefficients of the q-expansions of the normalised Eisenstein series E4 ; E6 of weights 4; 6 to apply Siegel’s Lemma and construct a sequence of auxiliary polynomials (AP) .PN /N 1 in ZŒX; Y  X f0g with degX PN ; degY PN  N , such that FN .x/ WD PN .x; xJ.x// vanishes with order  N 2 =2 at x D 0. The (UB) estimate is then exactly as in the Proof of Theorem 5.2.2. All the authors of [8] need to achieve their proof is the (NV) step; and it is here that a new idea occurs. They use that the coefficients of J are rational integers to deduce a sharp estimate of the biggest integer n such that FN .x/ vanishes at q m for all m D 1; : : : ; n  1. This idea, very simple and appealing to Schwarz lemma, does not seem to occur elsewhere in Mahler’s theory; it was later generalised by Nesterenko in the proof of his famous theorem in [33, 34], implying the algebraic independence of the three numbers ; e ; .1=4/ and the stephanese theorem (4 ). We will come back to the latter result in Section 5.4. 5.2.2.2 Effects of Schmidt’s subspace theorem. We mention the following result in [16] whose authors Corvaja and Zannier deduce from Schmidt’s subspace theorem. Theorem 5.2.5 (Corvaja and Zannier). Let us consider a formal series f 2 Qalg. ..x//X Qalg. Œx; x 1  and assume that f converges for x such that 0 < jxj < 1. Let L C be a number field and S a finite set of places of L containing the archimedean ones. Let A N be an infinite subset. Assume that: (a) ˛ 2 L, 0 < j˛j < 1 (b) f .˛ n / 2 L is an S -integer for all n 2 A. Then, lim inf n2A

h.f .˛ n // D1 n

This theorem has as an immediate application with A D fd; d 2 ; d 3 ; : : :g, d > 1 being an integer. If f 2 Qalg. ŒŒx is not a polynomial, converges for jxj < 1 and is n such that f .x d / D R.x; f .x// with R 2 Qalg. .X; Y / with hY .R/ < d , then, f .˛ d / is transcendental for ˛ algebraic with 0 < j˛j < 1 and for all n big enough. This implies a result (at least apparently) stronger than Theorem 5.2.2; indeed, the hypothesis that the coefficients of the series f all lie in a given number field is dropped.

5.3 Transcendence theory in positive characteristic The reduction modulo 2 in F2 ŒŒx of the formal series fTM .x/ 2 ZŒŒx is an algebraic formal series. In this section we will see that several interesting transcendental series 4 We take the opportunity to notice that a proof of an analog of the stephanese theorem for the so-called “Drinfeld modular invariant” by Ably, Recher and Denis is contained in [1].

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in positive characteristic are analogues of the series satisfying the functional equation (5.2.7). Let q D pe be an integer power of a prime number p with e > 0, let Fq be the field with q elements. Let us write A D Fq Œ# and K D Fq .#/, with # an indeterminate over Fq , and define an absolute value j  j on K by jaj D q deg# a , a being in K, so that j#j D q. Let K1 WD Fq ..1=#// be the completion of K for this alg. be an algebraic closure of K1 , let C be the completion of absolute value, let K1 alg. alg. K1 for the unique extension of j  j to K1 , and let K alg. be the algebraic closure of K embedded in C . There is a unique degree map deg# W C  ! Q which extends the map deg# W K  ! Z. Let us consider the power series fDe .x/ D

1 Y

n

.1  #x q /;

nD1

which converges for all x 2 C such that jxj < 1 and satisfies, just as in (5.2.7), the functional equation: fDe .x/ fDe .x q / D (5.3.1) 1  #x q (the subscript De stands for Denis, who first used this series for transcendence purposes). P bn 2n For q D 2, we notice that fDe .x/ D 1 nD0 # x ; where .bn /n0 D 0; 1; 1; 2; 1; 2; 2; 3; 1; 2; 2; 3; : : : is the sequence with bn equal to the sum of the digits of the binary expansion of n (and whose reduction modulo 2 precisely is Thue-Morse sequence of Section 5.2.1). It is very easy to show that fDe is transcendental, because it is plain that fDe has 2 infinitely many zeros # 1=q ; # 1=q ; : : : (just as the function occurring at the end of 5.2.1.4). We shall prove: Theorem 5.3.1. For all ˛ 2 K alg. with 0 < j˛j < 1, fDe .˛/ is transcendental. 5.3.1 Proof of Theorem 5.3.1. The proof of Theorem 5.3.1 follows the essential lines of Section 5.2.1, once the necessary tools are introduced. 5.3.1.1 Transcendence of functions. Not all the arguments of 5.2.1.1 work well to show the transcendence of formal series such as fDe ; in particular, the so-called Riemann–Hurwitz–Hasse formula does not give much information for functional equations such as fDe .x d / D afDe .x/ C b with the characteristic that divides d . Since in general it is hard to detect zeros of Mahler’s functions, we report another way to check the transcendency of fDe , somewhat making use of “automatic methods”, which can also be generalised as it does not depend on the location of the zeroes. To simplify the presentation, we assume, in the following discussion, that q D 2 but

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at the same time, we relax certain conditions so that, in all this subsection, we denote by # an element of C and by f# the formal series f# .x/ D

1 Y

n

.1  #x q / D

nD0

1 X

# bn x n 2 F ŒŒx C ŒŒx

nD0 n

with F the perfect field [n0 F2 .# 1=2 /, converging for x 2 C with jxj < 1. In particular, we have f# .x/ D fDe .x 1=2 / 2 F2 Œ#ŒŒx: We shall prove: Theorem 5.3.2. The formal series f# is algebraic over F .x/ if and only if # belongs to Fq , embedded in C . Proof. If # 2 Fq , it is easy to show that f# is algebraic, so let us assume by contradiction that f# is algebraic, with # that belongs to C X Fq . We have the functional equation: .1  #x/f# .x 2 / D f# .x/:

(5.3.2)

We introduce the operators X X k f D ci x i 2 F ..x// 7! f .k/ D ci2 x i 2 F ..x//; i

i

well defined for all k 2 Z. Since f .x 2 / D f .1/ .x/2 for any f 2 F ŒŒx, we deduce from (5.3.2) the collection of functional equations k

f#.1k/ .x/2 .1  # 1=2 x/ D f#.k/ .x/;

k  0:

(5.3.3)

For any f 2 F ŒŒx there exist two series f0 ; f1 2 F ŒŒx, uniquely determined, with the property that f D f02 C xf12 : We define Ei .f / WD fi (i D 0; 1). It is plain that, for f; g 2 F ŒŒx, Ei .f C g/ D Ei .f / C Ei .g/; .i D 0; 1/; E0 .fg/ D E0 .f /E0 .g/ C xE1 .f /E1 .g/; E1 .fg/ D E0 .f /E1 .g/ C E1 .f /E0 .g/; E0 .f 2 / D f; E1 .f 2 / D 0:

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Therefore, Ei .f 2 g/ D f Ei .g/;

i D 0; 1:

By (5.3.3) we get .k/

.1k/

k

.1k/

E0 .f# / D f# E0 .1  # 1=2 x/ D f# ; .k/ .1k/ 1=2k 1=2k .1k/ E1 .f# / D E1 .1  # x/f# D # f# ; and we see that if V is a F -subvector space of F ŒŒx containing f# and stable under the action of the operators E0 ; E1 , then V contains the F -subvector space generated .1/ .2/ by f# ; f# ; f# ; : : :. By a criterion for algebraicity of Sharif and Woodcock [49, Theorem 5.3] there is a subvector space V as above, with finite dimension, containing f# . The formal series f#.k/ are F -linearly dependent and there exists s > 0 such that f# ; f#.1/ ; : : : ; f#.s1/ are F -linearly dependent. Going back to the explicit x-expansion of f# , the latter condition is equivalent to the existence of c0 ; : : : ; cs1 2 F , not all zero, such that for all n  0: s1 X

ci # 2

ib

n

D 0:

i D0

The sequence b W N [ f0g ! N [ f0g is known to be surjective, so that the Moore determinant i

det..# 2 j //0i;j s1 vanishes. But this means that 1; #; # 2 ; : : : ; # s1 are F2 -linearly dependent (Goss, [23, Lemma 1.3.3]), or in other words, that # is algebraic over F2 ; a contradiction which completes the proof that f# and in particular fDe are transcendental over F .x/ (and the fact that the image of b has infinitely many elements suffices to achieve the proof). 5.3.1.2 Heights under a more general point of view. A good framework to generalise logarithmic heights to other base fields is that described by Lang in [26, Chapter 3] and by Artin and Whaples [7, Axioms 1, 2]. Let K be any field together with a proper set of non-equivalent places MK . Let us choose, for every place v 2 MK an absolute value j  jv 2 v and assume that for all x 2 K , the following product formula holds (cf. [26] p. 23): Y jxjv D 1; x 2 K ; (5.3.4) v2MK

with the additional property that if ˛ is in K , then j˛jv D 1 for all but finitely many v 2 MK . Let us suppose that MK contains at least one absolute value associated with either a discrete, or an archimedean valuation of K. It is well known that under these

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circumstances [7], K is either a number field, or a function field of one variable over a field of constants. Given a finite extension L of K, there is a proper set ML of absolute values on L, extending those of MK , again satisfying the product formula Y j˛jdv v D 1; (5.3.5) v2ML

where, if v is the place of K such P that wjK D v (one then writes wjv), we have defined dw D ŒLw ; Kv , so that wjv dv D ŒL W K. An analogue of the absolute logarithmic height h is available, by the following definition (see [26, Chapter 3]). Let .˛0 W    W ˛n / be a projective point defined over L. Then we define: X 1 dw log maxfj˛0 jw ; : : : ; j˛n jw g: h.˛0 W    W ˛n / D ŒL W K  w2ML

Again, we have a certain collection of properties making this function useful in almost every proof of transcendence over function fields. First of all, product formula (5.3.5) implies that h.˛0 W    W ˛n / does not depend on the choice of the field L and defines a map h W Pn .Kalg. / ! R0 : We write h.˛/ WD h.1 W ˛/. If the absolute values of MK are all ultrametric, it is easy to prove, with the same indications as in 5.2.1.2, that for ˛; ˇ 2 Kalg. : h.˛ C ˇ/; h.˛ˇ/  h.˛/ C h.ˇ/; h.˛ n / D jnjh.˛/; n 2 Z: More generally, if P is a polynomial in LŒX1 ; : : : ; Xn  and if .˛1 ; : : : ; ˛n / is a point of Ln , we write h.P / for the height of the projective point whose coordinates are 1 and its coefficients. We have: h.P .˛1 ; : : : ; ˛n //  h.P / C

n X .degXi P /h.˛i /:

(5.3.6)

i D1

Product formula (5.3.5) also provides a Liouville’s type inequality. Let ŒL W Ksep. be the separable degree of L over K. Let us choose a distinguished absolute value j  j of L and ˇ 2 L . We have: log jˇj  ŒL W Ksep. h.ˇ/:

(5.3.7)

The reason of the presence of the separable degree in (5.3.7) is the following. If ˛ 2 L is separable over K then log j˛j  ŒK.˛/ W Kh.˛/ D ŒK.˛/ W Ksep. h.˛/. s Let ˇ be any element of L . There exists s  0 minimal with ˛ D ˇ p separable and we get ps log jˇj  ŒK.˛/ W Kh.˛/ D ps ŒK.ˇ/ W Ksep. h.ˇ/.

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5.3.1.3 Transcendence of the values of fDe . We now follow Denis and we take K D K, MK the set of all the places of K and we choose in each of these places an absolute value normalised so that product formula (5.3.4) holds, with the distinguished absolute value j  j chosen so that j˛j D q deg# ˛ for ˛ 2 K  . As we already did in 5.2.1.3, we choose for all N  0, a polynomial PN 2 KŒX; Y , non-zero, of degree  N in both X; Y , such that the order of vanishing .N / < 1 of the function FN .x/ WD PN .x; fDe .x// at x D 0 satisfies .N /  N 2 . We know that this is possible by simple linear algebra arguments as we did before. Let ˛ 2 K alg. be such that 0 < j˛j < 1; as in 5.2.1.3, the sequence .PN /N need not to depend on it but the choice of N we will do does. By the identity principle of analytic functions on C , if " is a positive real number, nC1 for n big enough depending on N and ˛; l; ", we have FN .˛ q / ¤ 0 and log jFN .˛ q

nC1

/j  .N /q nC1 log j˛j C log jc .N / j C ";

where c .N / is a non-zero element of K depending on N (it is the leading coefficient of the formal series FN ). Let us assume by contradiction that fDe .˛/ 2 K alg. ; let L be a finite extension of K containing ˛ and fDe .˛/. By the variant of Liouville’s inequality (5.3.7) and from the basic facts on the height h explained above log jFN .˛ q

nC1

/j

  q nC1 / C Nh  ŒL W Ksep. deg# PN C N h.˛

fDe .˛/



.1  #˛ q /    .1  #˛ q nC1 /

:

Dividing by N q nC1 and using that .N /  N 2 we get, for all n big enough, N log j˛j C log jc .N / j C "  ŒL W Ksep. .deg# PN N 1 q n1 C .1 C .q  q n1 /=.q  1//h.˛/ C q n1 h.fDe .˛// C .n C 1/N 1q n1 h.#//: Letting n tend to infinity, we obtain the inequality:  N log j˛j  ŒL W Ksep. 1 C

 q h.˛/ q1

for all N > 0. Just as in the proof of Theorem 5.2.1, if N is big enough, this is contradictory with the assumptions showing that fDe .˛/ is transcendental.

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5.3.1.4 A first application to periods. The transcendence of values of fDe at algebraic series has interesting applications, especially when one looks at what happens with the base point ˛ D # 1 . Indeed, let e  D #.#/1=.q1/

1 Y

i

.1  # 1q /1

(5.3.8)

i D1

be a fundamental period of Carlitz’s module (it is defined up to multiplication by an element of F q ). Then, e  D #.#/1=.q1/ fDe .# 1 /1 ; so that it is transcendental over K.  is not in K, it suffices to choose If ˛ D # 1 , h.˛/ D log q so that to show that e N  4 if q D 2 and N  3 if q ¤ 2 in the proof above. Let us look, for q ¤ 2 given, at a polynomial (depending on q) P 2 AŒX; Y  X f0g with relatively prime coefficients in X of degree  3 in X and in Y , such that P .u; fDe .u// vanishes at u D 0 with the biggest possible order  > 9 (which also depends on q). It is possible to prove that for all q  4, P D X 3 .Y  1/3 2 Fq ŒX; Y : This means that to show that fDe .# 1 / 62 K, it suffices to work with the polynomial Q D Y  1 (5 ). Indeed, fDe .u/  1 D # uq C    . Therefore, for n big enough, if by nC1 nC1 contradiction fDe .# 1 / 2 K, then log jfDe .˛ q /  1j  .1 C q=.q  1//h.˛ q / which is contradictory even taking q D 3, but not for q D 2, case that we skip. If q D 3, we get a completely different kind of polynomial P of degree  3 in each indeterminate: P D 2 C # 2 C #X 3 C 2# 3 X 3 C 2# 2 Y C Y 3 C 2#X 3 Y 3 : It turns out that P .u; fDe .u// D .# 6  # 4 /u C    with  D 36 so that the order of vanishing is three times as big as the quantity expected from the computations with q  4: 12 D 33C3. How big can  be? It turns out that this question is important, notably in the search for quantitative measures of transcendence and algebraic independence; we will discuss about this problem in Section 5.4.1.

5 Other reasons, related to the theory of Carlitz module, allow to show directly that e  62 Fq ..# 1 // for q > 2.

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5.3.2 A second transcendence result. With essentially the same arguments of Section 5.3, it is possible to deal with a more general situation and prove the Theorem below. We first explain the data we will work with. Let F be a field and t an indeterminate. We denote by F hhtii the field of Hahn generalised series. This is the set of formal series X ci t i ; ci 2 F; i 2S

with S a well ordered subset of Q (6 ), endowed with the standard addition and Cauchy’s multiplication from which it is plain that every non-zero formal series is invertible. We have a field Fq hhtii-automorphism  W C hhtii ! C hhtii defined by ˛D

X

ci t i 7!  ˛ D

i 2S

X

q

ci t i :

i 2S

Assume that, with the notations of 5.3.1.2, K D C.t/, with t an independent b indeterminate. Let j  j be an absolute value associated with the t-adic valuation and K the completion of K for this absolute value. Let K be the completion of an algebraic b for the extension of j  j, so that we have an embedding of Kalg. in K. closure of K We have an embedding W K ! C hhtii (see Kedlaya, [24, PTheorem 1]); there exists a rational number c > 1 such that if ˛ is in K and .˛/ D i 2S ci t i , then j˛j D c i0 , where i0 D min.S /. We identify K with its image by . It can be proved that  K K,  Kalg. Kalg. and  K K. The definition of  implies immediately that, for all ˛ 2 K, j ˛j D j˛j:

(5.3.9)

We choose MK a complete set of non-equivalent absolute values of K such that the product formula (5.3.4) holds. On Pn .Kalg. /, we have the absolute logarithmic height whose main properties have been described in 5.3.1.2. There is a useful expression for the height h.˛/ of a non-zero element ˛ in Kalg. of degree D. If P D a0 X d C a1 X d 1 C    C ad 1 X C ad is a polynomial in C ŒtŒX  with relatively prime coefficients such that P .˛/ D 0, we have: 1 0 X 1 @ h.˛/ D log maxf1; j .˛/jgA ; (5.3.10) log ja0 j C D alg. WK

!K

where the sum runs over all the K-embeddings of Kalg. in K. The proof of this formula follows the same ideas as that of [54, Lemma 3.10]. 6

By definition, every nonempty subset of S has a least element for the order .

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Let ˛ be in Kalg. . From (5.3.9) and (5.3.10), it follows that: h. ˛/ D h.˛/:

(5.3.11)

Let us also consider, over the ring of formal series KŒŒx, the Fq hhtii-extension of  defined in the following way: X X f WD ci x i 7! f WD . ci /x qi : i

i

We can now state the main result of this section. Theorem 5.3.3. Let f 2 KŒŒx be converging for x 2 K, jxj < 1, let ˛ 2 K be such that 0 < j˛j < 1. Assume that: 1. f is transcendental over K.x/, 2. f D af C b, where a; b are elements of K.x/. Then, for all n big enough, . nf /.˛/ is transcendental over K. Proof. We begin with a preliminary discussion about heights. Let r D r0 x n C    C rn j be a polynomial in KŒx. We have, for all j  0,  j r D . j r0 /x q n C    C . j rn /. alg. Therefore, if ˛ is an element of K , we deduce from (5.3.6), (5.3.11) and from elementary height estimates: h.. j r/.˛//  h.1 W  j r0 W    W  j rn / C q j nh.˛/; n X  h. j ri / C q j nh.˛/ i D0



n X

h.ri / C q j nh.˛/;

i D0

where we have applied (5.3.11). Therefore, if a is a rational function in K.x/ such that . j a/.˛/ is well defined, we have h.. j a/.˛//  c1 C q j c2 ;

(5.3.12)

where c1 ; c2 are two constants depending on a; ˛ only. The condition on f implies that, for all k  0,  kf D f

k1 Y

k1 X

k1 Y

i D0

i D0

j Di C1

. i a/ C

. i b/

. j a/

(5.3.13)

(where empty sums are equal to zero and empty products are equal to one). Hence, the field

L D K.˛; f .˛/; .f /.˛/; . 2f /.˛/; : : :/ is equal to K.˛; . n f /.˛// for all n big enough.

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315

The transcendence of f implies that a ¤ 0. If ˛ is a zero or a pole of  k a and a pole of  k b for all k, then it is a simple exercise left to the reader to prove that j˛j D 1, case that we have excluded. Let us suppose by contradiction that the conclusion of the theorem is false. Then, ˛ is not a pole or a zero of  n a;  nb, . n f /.˛/ is algebraic over K for all n big enough, and L is a finite extension of K. An estimate for the height of this series can be obtained as follows. A joint application of (5.3.11), (5.3.12) and (5.3.13) yields: h.. k f /.˛//  h.f .˛// C

k1 X

h.. i a/.˛// C

i D0

k1 X

h.. i b/.˛//

i D0

 c3 C c4 k C c5 q k : Therefore, if P is a polynomial in KŒX; Y  of degree  N in X and Y , writing Fk k k k for the formal series  k P .x; f .x// D P  .x q ; . k f /.x// (P  is the polynomial obtained from P , replacing the coefficients by their images under  k ), we get: k

h.Fk .˛//  h.P  / C q k .degX P /h.˛/ C .degY P /h.. k f /.˛//  c7 .P / C c6 N q k ;

(5.3.14)

where c7 is a constant depending on P . Let N be a positive integer. There exists PN 2 KŒX; Y  with partial degrees in X; Y not bigger than N , with the additional property that FN .x/ WD PN .x; f .x// D c .N / x .N / C    , with .N /  N 2 and c .N / ¤ 0. P Let us write F .x/ D i 0 ci x i . In ultrametric analysis, Newton polygons suffice to locate the P absolute values of the zeroes of Taylor series. The Newton polygons of the series i 0 . k ci /x i 2 KŒŒx for k  0 are all equal by (5.3.9). By [23, P k Propositions 2.9, 2.11], we have i 0 . k ci /˛ q i ¤ 0 for k big enough. Now, since for k  0, k

. k FN /.x/ D . k c .N / /x .N /q C    ; we find, when the logarithm is well defined and by (5.3.11), that  1 < log j. k FN /.˛/j  .N /q k log j˛j C log jc .N / j C ":

(5.3.15)

On the other side, by (5.3.14), h.. k FN /.˛//  c8 .N / C c9 N q k ;

(5.3.16)

where c8 is a constant depending on f; ˛; N and c9 is a constant depending on f; ˛. A good choice of N (big) and inequality (5.3.7) with k big enough depending on N give a contradiction (7 ). 7 It is likely that Nishioka’s proof of Theorem 5.2.2 can be adapted to strengthen Theorem 5.3.3, but we did not enter into the details of this verification.

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5.3.2.1 Applications of Theorem 5.3.3. We look at solutions f 2 KŒŒx of  difference equations f D af C b; a; b 2 K.x/: (5.3.17) Theorem 5.3.3 allows to give some information about the arithmetic properties of their values. First application. Assume that in (5.3.17), a; b 2 Fq .t/. Then, since Fq .t/ is contained in the field of constants of  , solutions of this difference equation are related to the variant of Mahler’s method of Section 5.3. If a D .1  t 1 x/1 ; b D 0, the equation above has the solution fDe2 .x/ D

1 Y

n

.1  t 1 x q /;

nD1

which converges for x 2 K, jxj < 1. Q1 If x D t, Theorem 5.3.3 yields the transcendence of fDe2 .t/ D nD1 .1  n q 1 / 2 Fq ŒŒt over Fq .t/ and we get (again) the transcendence of e  over K (we t also notice the result of the paper [2], which allows some other applications). More generally, all the examples of functions in [40, Section 3.1] have a connection with this example. Second application. Theorem 5.3.3 also has some application which does not seem to immediately follow from results such as Theorem 5.2.2. Consider equation (5.3.17) with b D 0 and a D .1 C #x/1 , where # 2 Fq .t; #/ is non-zero. We have the following solution of (5.3.17) in KŒŒx: '.x/ D

1 Y

n

.1 C . n #/x q /:

nD0

P It is easy to show that ' D j 0 cj x j is a formal series of KŒŒx converging for x 2 K, jxj < 1. The coefficients cj can be computed in the following way. We have cj D 0 if the q-ary expansion of j has its set of digits not contained in f0; 1g. Otherwise, if j D j0 C j1 q C    C jn q n with j0 ; : : : ; jn 2 f0; 1g, we have, writing #i for  i r, P 1CqC Cq k , we have cj D #0j0 #1j1P   #njn . Therefore, if .x/ D 1 kD0 #0 #1    #k x 1 j '.x/  1 D j D0  .x/. n For # D t 1 .1 C t=#/1 , the series '.x/ vanishes at every xn D t 1=q .1 C 1=q n t =#/. The xn ’s are elements of K which are distinct with absolute value < 1 (we recall that we are using the t-adic valuation). Having thus infinitely many zeros in the domain of convergence and not being identically zero, ' is transcendental. The series ' converges at x D t. Theorem 5.3.3 implies that the formal series '.t/ 2 C..t// is transcendental over K. We notice that the arguments of 5.3.1.1 can be probably extended to investigate the transcendence of the series ' associated to, say, # D .1 C t=#/1 , case in which we do not necessarily have infinitely many

An introduction to Mahler’s method for transcendence and algebraic independence

317

zeros. The reason is that, over K..x//, the Fq -linear Frobenius twist F W a 7! aq (for all a) splits as F D  D  where  is Anderson’s Fq ..t//-linear twist and  is Mahler’s C..x//-linear twist, and most of the arguments of 5.3.1.1 can be generalised to this setting. 5.3.2.2 Some -difference equations in KŒŒx. The arguments of the previous subsection deal with formal series in KŒŒx D C.t/ŒŒx. We have another important ring of formal series, also embedded in KŒŒx, which is C.x/ŒŒt. Although the arithmetic of values of these series seems to be not deducible from Theorem 5.3.3, we discuss here about some examples because solutions f 2 C.x/ŒŒt of  -difference equations such as f D af C b; a; b 2 C.x/ (5.3.18) are often related to Anderson–Brownawell–Papanikolas linear independence criterion in [6] (see the corresponding contribution in this volume and the refinement [15]). With # a fixed .q  1/-th root of #, the transcendental formal series .t/ WD

q #

1  Y i D1

1

t # qi

 D

1 X

di t i 2 K alg. ŒŒt

(5.3.19)

i D0

 1 and satisfies the functional is convergent for all t 2 C , such that .#/ 2 F qe equation .t q / D .t q  # q /.t/q (5.3.20) (see [6]). By a direct inspection it turns out that there is no finite extension of K containing all the coefficients di of the t-expansion of  (8 ). Hence, there is no variant of Mahler’s method which seems to apply to prove the transcendence of at algebraic elements (and a suitable variant of Theorem 5.2.5 is not yet available). The map  W C..t// ! C..t// acts on in the following way: X X q cD ci t i 7!  c WD ci t i : i

i

Let s.t/ be the formal series  1 1 2 C ŒŒt (where  1 is the reciprocal map of  ). After (5.3.20) this function is solution of the  -difference equation: . s/.t/ D .t  #/s.t/;

(5.3.21)

hence it is a solution of (5.3.18) with a D t  # and b D 0. Transcendence of values of this kind of function does not seem to follow from Theorem 5.3.3, but can be obtained with [6, Theorem 1.3.2]. 8

They generate an infinite tower of Artin-Schreier extensions.

318

Federico Pellarin

More generally, let ƒ be an A-lattice of C of rank r and let Y  z 1 eƒ .z/ D z 

(5.3.22)

2ƒXf0g

be its exponential function, in Weierstrass product form. The function eƒ is an entire, surjective, Fq -linear function. Let ' W A ! EndFq lin. .Ga .C // D C Œ  (9 ) be the associated Drinfeld module. We have, for a 2 A, 'ƒ .a/eƒ .z/ D eƒ .az/. Let us extend 'ƒ over K by means of the endomorphism  ( t D t). After having chosen an element ! 2 ƒ X f0g, define the function sƒ;! .t/ WD

1 X nD0

eƒ

 !  t n 2 C ŒŒt; # nC1

(5.3.23)

convergent for jtj < q. We have, for all a 2 A, 'ƒ .a/sƒ;! D asƒ;! ;

(5.3.24)

where, if a D a.#/ 2 Fq Œ#, we have defined a WD a.t/ 2 Fq Œt. This means that sƒ;! , as a formal series of C ŒŒt, is an eigenfunction for all the Fq ..t//-linear operators 'ƒ .a/, with eigenvalue a, for all a 2 A. If ƒ D e  A, 'ƒ is Carlitz’s module, and equation (5.3.24) implies the  -difference equation (5.3.21).

5.4 Algebraic independence In [28], Mahler proved his first result of algebraic independence obtained modifying and generalising the methods of his paper [27]. The result obtained involved m formal series in several variables but we describe its consequences on the one variable theory only. Let L be a number field embedded in C and let d > 1 be an integer. Theorem 5.4.1 (Mahler). Given m formal series f1 ; : : : ; fm 2 LŒŒx, satisfying functional equations fi .x d / D ai fi .x/ C bi .x/;

1i m

with ai 2 L, bi 2 L.x/ for all i , converging in the open unit disk. If ˛ is algebraic such that 0 < j˛j < 1, then the transcendence degree over L.x/ of the field L.x; f1 .x/; : : : ; fm .x// is equal to the transcendence degree over Q of the field Q.f1 .˛/; : : : ; fm .˛//.

9

Polynomial expressions in  with the product satisfying  c D c q  , for c 2 C .

An introduction to Mahler’s method for transcendence and algebraic independence

319

5.4.1 Criteria of algebraic independence and applications. Mahler’s result remained nearly unobserved for several years. It came back to surface notably thanks to the work of Loxton and van der Poorten in the seventies, and then by Nishioka and several other authors. At the beginning, these authors developed criteria for algebraic independence tailored for application to algebraic independence of Mahler’s values. Later, criteria for algebraic independence evolved in very general results, especially in the hands of Philippon. Here follows a particular case of a criterion of algebraic independence by Philippon The statement that follows merges the results [44, Theorem 2] and [43, Theorem 2.11] and uses the data K; L; j  j; K; A; : : : where K is a complete algebraically closed field in two cases. We examine only the cases in which K is either C or C , but it is likely that the principles of the criterion extend to several other fields, like that of Section 5.3.2. In the case K D C, L is a number field embedded in C, j  j is the usual absolute value, h is the absolute logarithmic height of projective points defined over Qalg. , K denotes Q and A denotes Z. In the case K D C , L is a finite extension of K D Fq .#/ embedded in C , j  j is an absolute value associated to the # 1 -adic valuation, h is the absolute logarithmic height of projective points defined over K alg. , A denotes the ring A D Fq Œ# and we write K D K. In both cases, if P is a polynomial with coefficients in L, we associate to it a projective point whose coordinates are 1 and its coefficients, and we write h.P / for the logarithmic height of this point (which depends on P up to permutation of the coefficients). Theorem 5.4.2 (Philippon). Let .˛1 ; : : : ; ˛m / be an element of Km and k an integer with 1  k  m. Let us suppose that there exist three increasing functions Z1 ! R1 ı .degree/  .height/ .magnitude/ and five positive real numbers c2 ; c3 ; cı ; c ; c satisfying the following properties. 1. limn!1 ı.n/ D 1, 2. limn!1

s.nC1/ s.n/

D cs for s D ı; ; ,

3.  .n/  ı.n/, for all n big enough, 4. The sequence n 7!

.n/ ı.n/kC1 .n/

is ultimately increasing,

5. For all n big enough, .n/kC1 >  .n/ı.n/k1..n/k C ı.n/k /: Let us suppose that there exists a sequence of polynomials .Qn /n0 with Qn 2 LŒX1 ; : : : ; Xm ;

320

Federico Pellarin

with degXi Qn  ı.n/ for all i and n, with h.Qn /   .n/ for all n, with coefficients integral over A, such that, for all n big enough, c2 .n/ < log jQn .˛1 ; : : : ; ˛m /j < c3 .n/: Then, the transcendence degree over L of the field L.˛1 ; : : : ; ˛m / is  k. Theorem 5.4.2 can be applied to prove the following result. Theorem 5.4.3. Let us assume that we are again in one of the cases above; K D C or K D C , let L be as above. Let f1 ; : : : ; fm be formal series of LŒŒx an d > 1 and integer, satisfying the following properties: 1. f1 ; : : : ; fm converge for jxj < 1, 2. f1 ; : : : ; fm are algebraically independent over K.x/, 3. For all i D 1; : : : ; m, there exist ai ; bi 2 L.x/ such that fi .x d / D ai .x/fi .x/ C bi .x/;

i D 1; : : : ; m: n

n

Let ˛ 2 L be such that 0 < j˛j < 1. Then, for all n big enough, f1 .˛ d /; : : : ; fm .˛ d / are algebraically independent over L. This result is, for K D C, a corollary of Kubota’s result [25, Theorem p. 10]. For K D C , it is due to Denis [18, Theorem 2]. See also [9, 17]. Sketch of proof of Theorem 5.4.3 in the case K D C . To simplify the exposition, we assume that L D K. Let N > 0 be an integer; there exists at least one nonzero polynomial PN 2 KŒx; X1 ; : : : ; Xn  (that we choose) of degree  N in each indeterminate, such that the order .N / of vanishing at x D 0 of the function FN .x/ D PN .x; f1 .x/; : : : ; fm .x// (not identically zero because of the hypothesis of algebraic independence of the functions fi over C.x/), satisfies .N /  N mC1 . The choice of the parameter N will be made later. If c.x/ 2 AŒx is a non-zero polynomial such that cai ; cbi 2 AŒx for i D 1; : : : ; m, then we define, inductively, R0 D PN and Rn D c.x/N Rn1 .x d ; a1 X1 C b1 ; : : : ; am Xm C bm / 2 AŒx; X1 ; : : : ; Xm : Elementary inductive computations lead to the following estimates, holding for n big enough depending on N; f1 ; : : : ; fm , where c4 ; c5 ; c6 are integer constants depending on f1 ; : : : ; fm only: degXi .Rn /  N;

.i D 1; : : : ; m/; n

degx .Rn /  c4 d N; deg# .Rn /  c5 d n N; h.Rn /  c7 C c6 d n N; where we wrote c7 .N / D h.R0 /; it is a real number depending on f1 ; : : : ; fm and N .

An introduction to Mahler’s method for transcendence and algebraic independence

Since Rn .x; f1 .x/; : : : ; fn .x// D

n1 Y

321

! di

n

n

n

c.x /N R0 .x d ; f1 .x d /; : : : ; fm .x d //;

i D0

one verifies the existence of two constants c2 > c3 > 0 such that c2 d n .N /  deg# .Rn .˛; f1 .˛/; : : : ; fm .˛///  c3 d n .N / for all n big enough, depending on N; f1 ; : : : ; fm and ˛. Let us define: Qn .X1 ; : : : ; Xn / D D c1 d

nN

Rn .˛; X1 ; : : : ; Xm / 2 AŒ˛ŒX1 ; : : : ; Xm ;

where D 2 A X f0g is such that D˛ is integral over K. The estimate above implies at once that, for n big enough: degXi Qn  N; h.Qn /  c8 .N / C c9 d n N; where c8 .N / is a constant depending on f1 ; : : : ; fm ; N , and ˛ (it can be computed with an explicit dependence on c7 .N /) and c9 is a constant depending on f1 ; : : : ; fm , and ˛ but not on N . Finally, Theorem 5.4.2 applies with the choices: ˛i D fi .˛/; i D 1; : : : ; m kDm .n/ D d n .N / ı.n/ D N  .n/ D c9 d n .N /; provided that we choose N large enough depending on the constants c1 ; : : : introduced so far. Then, one chooses n big enough (depending on the good choice of N ). 5.4.1.1 An example with complex numbers. Theorem 5.4.3 furnishes algebraic independence of Mahler’s values if we are able to check algebraic independence of Mahler’s functions but it does not say anything on the latter problem; this is not an easy task in general. With the following example, we would like to sensitise the reader to this problem which, the more we get involved in the subtleties of Mahler’s method, the more it takes a preponderant place. In the case K D C, we consider the formal series in ZŒŒx: L0 D

1 Y i D0

i

.1  x 2 /1 ;

Lr D

1 X i D0

Qi 1

x2

ir

j D0 .1

 x 2j /

;

r 1

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converging, on the open unit disk jxj < 1, to functions satisfying : L0 .x 2 / D .1  x/L0 .x/;

Lr .x 2 / D .1  x/.Lr .x/  x r /;

r  1:

By a result of Kubota [25, Theorem 2], (see also Töpfer, [51, Lemma 6]), if .Li /i 2I (with I Z) are algebraically dependent, then they also are C-linearly dependent modulo C.x/ in the following sense. There exist complex numbers .ci /i 2I not all zero, such that: m X

ci Li .x/ D f .x/

i D1

with f .x/ 2 C.x/. P. Bundschuh pointed out that L0 ; L2 ; L4 ; : : : are algebraically independent. To obtain this property, he studied the behaviour of these functions near the unit circle. For a long time the author was convinced of the algebraic independence of the functions L0 ; L1 ; : : : until very recently, when T. Tanaka and H. Kaneko exhibited non-trivial linear relations modulo C.x/ involving L0 ; : : : ; Ls for all s  1, some of which looking very simple, such as the relation: L0 .x/  2L1 .x/ D 1  x: 5.4.2 Measuring algebraic independence. Beyond transcendence and algebraic independence, the next step in the study of the arithmetic of Mahler’s numbers is that of quantitative results such as measures of algebraic independence. Very often, such estimates are not mere technical refinements of well known results but deep information on the diophantine behaviour of classical constants; everyone knows the important impact that Baker’s theory on quantitative minorations of linear forms in logarithms on algebraic groups had in arithmetic geometry. Rather sharp estimates are known for complex Mahler’s values. We quote here a result of Nishioka [36, 37] and [38, Chapter 12] (it has been generalised by Philippon: [45, Theorem 6]). Theorem 5.4.4 (Nishioka). Let us assume that, in the notations previously introduced, K D C. Let L be a number field embedded in C. Let f1 ; : : : ; fm be formal series of LŒŒx, let us write f 2 Matn1 .LŒŒx/ for the column matrix whose entries are the fi ’s. Let A 2 Matnn .L.x//; b 2 Matn1 .L.x// be matrices. Let us assume that: 1. f1 ; : : : ; fm are algebraically independent over C.x/, 2. For all i , the formal series fi .x/ converges for x complex such that jxj < 1, 3. f .x d / D A.x/  f .x/ C b.x/. Let ˛ 2 L be such that 0 < j˛j < 1, not a zero or a pole of A and not a pole of b. Then, there exists a constant c1 > 0 effectively computable depending on ˛; f , with the following property.

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For any H; N  1 integers and any non-zero polynomial P 2 ZŒX1 ; : : : ; Xm  whose partial degrees in every indeterminate do not exceed N and whose coefficients are not greater that H in absolute value, the number P .f1 .˛/; : : : ; fm .˛// is nonzero and the inequality below holds: log jP .f1 .˛/; : : : ; fm .˛//j  c1 N m .log H C N mC2 /:

(5.4.1)

We sketch how Theorem 5.4.4 implies the algebraic independence of f1 .˛/, : : :, fm .˛/ and g.˛/ with f1 ; : : : ; fm ; g 2 QŒŒx algebraically independent over QŒŒx satisfying linear functional equations as in Theorem 5.4.3 and g satisfying g.x d / D a.x/g.x/ C b.x/; with a; b 2 Q.x/, A; b with rational coefficients, and ˛ not a pole of all these rational functions. Of course, this is a simple corollary of Theorem 5.4.4. However, we believe that the proof is instructive; it follows closely Philippon’s ideas in [45]. The result is reached because the estimates of Theorem 5.4.4 are precise enough. In particular, the separation of the quantities H and N in (5.4.1) is crucial. 5.4.2.1 Algebraic independence from measures of algebraic independence. For the purpose indicated at the end of the last subsection, we assume that ˛ 2 Q . This hypothesis in not strictly necessary and is assumed only to simplify the exposition of the proof; by the way, the reader will remark that several other hypotheses we assume are avoidable. Step (AP). For all N  1, we choose a non-zero polynomial PN 2 ZŒx; X1 , : : :, Xm ; Y  of partial degrees  N in each indeterminate, such that, writing FN .x/ WD PN .x; f1 .x/; : : : ; fm .x/; g.x// D c .N / x .N / C    2 QŒŒx;

c .N / ¤ 0;

we have .N /  N mC2 (we have already justified why such a kind of polynomial exists). Just as in the proof of Theorem 5.4.3 we construct, for each N  1, a sequence of polynomials .PN;k /k0 in ZŒx; X1 ; : : : ; Xm ; Y  recursively in the following way: PN;0 WD PN ; PN;k WD c.x/N PN;k1 .x d ; a1 .x/X1 C b1 .x/; : : : ; am .x/Xm C bm .x/; a.x/Y C b.x//; where c.x/ 2 ZŒx X f0g is chosen so that cai ; cbj ; ca; cb belong to ZŒx. The following estimates are easily obtained: degZ PN;k  N;

for Z D X1 ; : : : ; Xm ; Y; k

degx PN;k  c2 d N; h.PN;k /  c3 .N / C c4 d k N;

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where c2 ; c4 are positive real numbers effectively computable depending on ˛; f and g, and c3 .N / > 0 depends on these data as well as on N (it depends on the choice of the polynomials PN ). Let us assume by contradiction that g.˛/ is algebraic over the field

F WD Q.f1 .˛/; : : : ; fm .˛//; of transcendence degree m over Q. We observe that, after the identity principle of analytic functions we have, for k big enough depending on ˛; f ; g and N : PN;k .˛; f1 .˛/; : : : ; fm .˛/; g.˛// 2 F  :

(5.4.2)

e 2 F ŒX  X f0g be the minimal polynomial of g.˛/, algebraic over F . We can Let Q e D a0 C a1 X C    C ar1 X r1 C X r with the ai ’s in F . Multiplying by write Q a common denominator, we obtain a non-zero polynomial Q 2 ZŒX1 ; : : : ; Xm ; Y  such that Q.f1 .˛/; : : : ; fm .˛/; g.˛// D 0, with the property that the polynomial Q D Q.f1 .˛/; : : : ; fm .˛/; Y / 2 F ŒY  is irreducible. Step (NV). Let us denote by k the resultant ResY .PN;k ; Q/ 2 ZŒx; X1 ; : : : ; Xm . If ık WD k .˛; f1 .˛/; : : : ; fm .˛// 2 F vanishes for a certain k, then Q and  PN;k WD PN;k .˛; f1 .˛/; : : : ; fm .˛/; Y / 2 F ŒY   have a common zero. Since Q is irreducible, we have that Q divides PN;k in F ŒY  and

PN;k .˛; f1 .˛/; : : : ; fm .˛/; g.˛// D 0I this cannot happen for k big enough by the identity principle of analytic functions (5.4.2) so that we can assume that for k big enough, ık ¤ 0, ensuring that k is not identically zero; the estimates of the height and the degree of k quoted below are simple exercises and we do not give details of their proofs: degZ k  c5 N;

Z D X1 ; : : : ; Xm ;

k

degx k  c6 d N; h.k /  c7 .N / C c8 d k N; where c5 ; c6 ; c8 are positive numbers effectively computable depending on ˛; f and g, while the constant c7 .N / depends on these data and on N . Let D be a non-zero positive integer such that D˛ 2 Z. Then, writing k

k WD D Nd k .˛; X1 ; : : : ; Xm /; we have k 2 ZŒX1 ; : : : ; Xm  X f0g and degXi k  c5 N; h.k /  c7 .N / C c8 d k N:

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Step (LB). By Nishioka’s Theorem 5.4.4, we have the inequality (for k big enough): log jk .f1 .˛/; : : : ; fm .˛//j  c1 N m .d k N C c9 .N //;

(5.4.3)

where c9 .N / is a constant depending on N . To finish our proof, we need to find an upper bound contradictory with (5.4.3); it will be obtained by analytic estimates as usual. Step (UB). Looking at the proof of Lemma 5.3.1 of [53], and using in particular inequality (1.2.7) of loc. cit., we verify the existence of constants c12 ; c15 depending on ˛; f ; g, c14 .˛; "/ depending on ˛ and ", and c13 .N / depending on ˛; f ; g and N , such that:  log jk .f1 .˛/; : : : ; fm .˛//j  log.N C c10 / C c11 h.PN;k / C .N C 1/h.Q/C

log jc N j C .N /d k log j˛j C "  c12 .log N C d k N / C c13 .N /  c14 .˛; "/d k .N /  c15 d k N C c13 .N /  c14 .˛; "/d k N mC2 :

(5.4.4)

Finally, it is easy to choose N big enough, depending on c1 ; c15 ; c14 but not on c9 ; c13 so that, for k big enough, the estimates (5.4.3) and (5.4.4) are not compatible: this is due to the particular shape of (5.4.1), with the linear dependence in log H . 5.4.2.2 Further remarks, comparisons with Nesterenko’s theorem. In the sketch of proof of the previous subsection, the reader probably observed a kind of induction structure; a measure of algebraic independence for m numbers delivers algebraic independence for m C 1 numbers. The question is then natural: is it possible to obtain a measure of algebraic independence for m C 1 numbers allowing continue the process and consider m C 2 numbers? In fact yes, there always is an inductive structure of proof, but no, it is not just a measure for m numbers which alone implies a measure for m C 1 numbers. Things are more difficult than they look at first sight and the inductive process one has to follow concerns other parameters as well. For instance, the reader can verify that it is unclear how to generalise the arguments of 5.4.2.1 and work directly with a polynomial Q which has a very small value at ! D .f1 .˛/; : : : ; fm .˛/; g.˛//. Algebraic independence theory usually appeals to transfer techniques, as an alternative to direct estimates at !. A detour on a theorem of Nesterenko might be useful to understand what is going on so our discussion now temporarily leaves Mahler’s values, that will be reconsidered in a little while. Precise multiplicity estimates in differential rings generated by Eisenstein’s series obtained by Nesterenko, the criteria for algebraic independence by Philippon already mentioned in this paper and a trick of the stephanese team (cf. 5.2.2.1) allowed Nesterenko, in 1996, to prove the following theorem (see [33, 34]): Theorem 5.4.5 (Nesterenko). Let E2 ; E4 ; E6 the classical Eisenstein’s series of weights 2; 4; 6 respectively, normalised so that lim=.z/!1 E2i .z/ D 1 (for 0 and whose coefficients are not greater that H > 0 in absolute value, then log jP .e  ; ; .1=4//j  c1 ."/.N C log H /4C" , where c1 is an absolute constant depending on " only. The dependence in " is completely explicit.

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with c ¤ 0. There exists a constant c1 > 0, depending on f1 ; : : : ; fm only, with the following property. If P is as above and N1 WD maxf1; degx P g and N2 WD maxf1; degX1 P; : : : ; degXm P g, then   c1 N1 N2m : This result is very similar to Nesterenko’s multiplicity estimate [34, Chapter 10, Theorem 1.1] and again, its proof essentially follows Nesterenko’s ideas. The ideal J previously mentioned is defined as the ideal generated by p and a polynomial obtained from PN;k by homogenisation, substitution x D ˛, and a good choice of N; k taking into account the magnitude of the coefficients of the series fi ; g. Indeed, one proves that such a polynomial cannot belong to p. The closest point principle is necessary in this kind of proof. The arguments of the above discussion can be modified to obtain the analog of Theorem 5.4.5 for values of Mahler’s functions at general complex numbers, obtained by Philippon (cf. [45, Theorem 4]). Here, L is again a number field embedded in C and d > 1 is an integer. Theorem 5.4.7 (Philippon). Under the same hypotheses and notations of Theorem 5.4.4, if ˛ is a complex number with 0 < j˛j < 1, then, for n big enough, the complex n n numbers ˛; f1 .˛ d /; : : : ; fm .˛ d / 2 C generate a subfield of C of transcendence degree  m. This result is a corollary of a more general quantitative result [45, Theorem 6] which follows from Philippon’s criterion for measures of algebraic independence (loc. cit. p. 5). 5.4.2.3 Commentaries on the case of positive characteristic. Similar, although simpler arguments are in fact commonly used to obtain measures of transcendence. Several authors deduce them from measures of linear algebraic approximation; see for example Amou, Galochkin and Miller [3, 21, 32] (12 ). These results often imply that Mahler’s values are Mahler’s S -numbers. In positive characteristic, it is well known that separability difficulties occur preventing to deduce good measures of transcendence from measures of linear algebraic approximation (13 ). In [20], Denis proves the following result, where d > 1 is an integer. Theorem 5.4.8 (Denis). Let us consider a finite extension L of K D Fq .#/, f 2 LŒŒx, ˛ 2 L such that 0 < j˛j < 1. Let us assume that f is transcendental over 12 Some

results hold for series which satisfy functional equations which are not necessarily linear. by Lang, for complex numbers, there is equivalence between measures of transcendence and measures of linear algebraic approximation in the sense that, from a measure of linear algebraic approximation one can get a measure of transcendence and then again, a measure of linear algebraic approximation which is essentially that of the beginning, with a controllable degradation of the constants. 13 As first remarked

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C.x/, convergent for x 2 C such that jxj < 1, and satisfying the linear functional equation f .x q / D a.x/f .x/ C b.x/; with a; b 2 L.x/. n For all n big enough, we have the following property. Let ˇ D ˛ d . Then, there exists an effectively computable constant c1 > 0 depending on ˇ; f only, such that, given any non-constant polynomial P 2 AŒX , log jP .f .ˇ//j   degX .P /4 .degX .P / C deg# .P //:

(5.4.5)

This result yields completely explicit measures of transcendence of e  , of Carlitz’s logarithms of elements of K, and of certain Carlitz–Goss’s zeta values (see Section 5.4.3 for definitions). To prove Theorem 5.4.8, Denis uses the following multiplicity estimate. Theorem 5.4.9. Let K be any (commutative) field and f 2 K..x// be transcendental satisfying the functional equation f .x d / D R.x; f .x// with R 2 K.X; Y / and hY .R/ < d . Then, if P is a polynomial in KŒX; Y  X f0g such that degX P  N and degY P  M , we have ordxD0 P .x; f .x//  N.2Md C N hX .Q//: It would be interesting to generalise such a multiplicity estimate for several algebraically independent formal series and obtain a variant of Töpfer’s [52, Theorem 1] (see [11, 12, 41] to check the difficulty involved in the research of an analogue of Nesterenko’s multiplicity estimate for Drinfeld quasi-modular forms). This could be helpful to obtain analogues of Theorem 5.4.4 for Mahler’s values in fields of positive characteristic. 5.4.3 Algebraic independence of Carlitz’s logarithms. For the rest of this chapter, we will give some application of Mahler’s method and of Anderson–Brownawell– Papanikolas method to algebraic independence of Carlitz’s logarithms of algebraic elements of C and of some special values of Carlitz–Goss zeta function at rational integers. Results of this part are not original since they are all contained in the papers [39] by Papanikolas and [14] by Chieh-Yu Chang and Jing Yu. But the methods we use here are slightly different and self-contained. Both proofs of the main results in [39, 14] make use of a general statement [39, Theorem 5.2.2] which can be considered as a variant of Grothendieck period conjecture for a certain generalisation of Anderson’s t-motives, also due Papanikolas. To apply this result, the computation of motivic Galois groups associated to certain t-motives is required. Particular cases of these results are also contained in Denis work [19], where he applies Mahler’s method and without appealing to Galois theory. Hence, we follow the ideas of the example in 5.4.1.1 and the main worry here is to develop analogous

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proofs in the Drinfeldian framework. In 5.4.1.1 the explicit computation of the transcendence degree of the field generated by L0 ; L1 ; : : : was pointed out as a problem. But we have already remarked there, that if L0 ; L1 ; : : : are algebraic dependent, then they also are C-linearly dependent modulo C.x/. This property, consequence of a result by Kubota, is easy to obtain because the matrix of the linear difference system of equations satisfied by the Li ’s has the matrix of its associated homogeneous system which is diagonal. For ƒ D e  A with e  as in (5.3.8), the exponential function eCar WD eƒ (5.3.22) can be explicitly written as follows: eCar .z/ D

X

zq

i 0

Œi Œi  1q    Œ1q i 1

i

;

i

where Œi  WD # q  # (i  1/. This series converges uniformly on every open ball with center in 0 to an Fq -linear surjective function eCar W C ! C . The formal series logCar , reciprocal of eCar in 0, converges for jzj < q q=.q1/ D je  j. Its series expansion can be computed explicitly: logCar .z/ D

X i 0

i

.1/i z q : Œi Œi  1    Œ1

The first Theorem we shall prove in a simpler way is the following (cf. [39, Theorem 1.2.6]): Theorem 5.4.10 (Papanikolas). Let `1 ; : : : ; `m 2 C be such that eCar .`i / 2 K alg. (i D 1; : : : ; m). If `1 ; : : : ; `m are linearly independent over K, then they also are algebraically independent over K. 5.4.3.1 Carlitz–Goss polylogarithms and zeta functions. Let us write AC D fa 2 A; a monicg. In [22], Goss introduced a function , defined over C  Zp with values in C , such that for n  1 integer, .# n ; n/ D

X 1 2 K1 : an

a2AC

In the following, we will write .n/ for .# n ; n/. For n 2 N, let us also write .n/ WD Q ni s s i D0 Di 2 K, n0 C n1 q C    C ns q being the expansion of n  1 in base q and i 1 Di being the polynomial Œi Œi  1q    Œ1q . It can be proved that z=eCar .z/ D P1 n z nD0 Bn .nC1/ for certain Bn 2 K. The so-called Bernoulli–Carlitz relations can be obtained by a computation involving the logarithmic derivative of eCar .z/: for all m  1, .m.q  1// Bm 2 K: (5.4.6) D .m.q  1/ C 1/ e  m.q1/

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In particular, one sees that e  q1 D .# q  #/.q  1/ 2 K1 : We also have the obvious relations: k

.mpk / D .m/p ;

m; k  1:

(5.4.7)

The second theorem we are going to prove directly is: Theorem 5.4.11 (Chang, Yu). The algebraic dependence relations over K between the numbers .1/; .2/; : : : are generated by Bernoulli–Carlitz’s relations (5.4.6) and the relations (5.4.7). 5.4.3.2 Two propositions. In this subsection we develop an analogue of [51, Lemma 6], for the same purpose we needed it in 5.4.1.1. We consider here a perfect field U of characteristic p > 0 containing Fq and a Fq -automorphism  W U ! U . Let U0 be the subfield of constants of  , namely, the subset of U whose elements s are such S that  s D s. n For example, we can consider U D n0 C.x 1=p / with  defined as the identity S n over C , with  x D x q . Another choice is to consider U D n0 C.t 1=p /, with  defined by  c D c 1=q for allSc 2 C and  t D t. In the first example we have U0 D C n while in the second, U0 D n0 Fq ..t 1=p //. S n More generally, after 5.3.2, we can take either U D n0 K.x 1=p / or the field S n m 1=p ; t 1=p / (which is contained in the previous field) with the corren;m0 C.x sponding automorphism  (these settings will essentially include the two examples above). In the first case, we have U0 D Fq hhtii, and in the second case, we have S n U0 D n0 Fq ..t 1=p //. Let us also consider the ring R D U ŒX1 ; : : : ; XN  and write, for a polynomial P P P D  c X  2 R, P  as the polynomial  . c /X  . Let D1 ; : : : ; DN be elements of U  , B1 ; : : : ; BN be elements of U and, for a polynomial P 2 R, let us write e D P  .D1 X1 C B1 ; : : : ; DN XN C BN /: P We prove the following two Propositions, which provide together the analogue in positive characteristic of Kubota [25, Theorem 2]. e Proposition 5.4.12. Let P 2 R be a non-constant polynomial P such that pP =P 2 R. Then, there exists a polynomial G 2 R of the form G D i ci Xi C B such that e G=G 2 R, where c1 ; : : : ; cN 2 U are not all vanishing and B 2 R. If W is the subfield generated by Fq and the coefficients of P , then there exists M  1 such that M for each coefficient c of G, c p 2 W .

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331

e D QP for Q 2 R one sees, comparing the degrees Proof. If P 2 R is such that P e and P , that Q 2 U and if P is non-zero, Q 6D 0. The subset of R of these of P e D QP , then polynomials is a semigroup S containing U . If P 2 S satisfies P e D D 1 QF . Similarly, if P D F p 2 S with F WD @P =@Xi belongs to S since F i e D Q1=p F . F 2 S then F 2 S as one sees easily that in this case, F By hypothesis, S contains a non-constant polynomial P . We now show that the polynomial G 2 S as in the Proposition can be constructed by iterated applications of partial derivatives @1 D @=@X1 ; : : : ; @N D @=@XN and p-root extrations starting from P . Let P be as in the hypotheses. We can assume that P is not a p-th power. We can write: X P D c X  ; c 2 Rp : D.1 ;:::;N /2f0;:::;p1gN

Let M WD maxf1 C    C N ; c 6D 0g. We can write P D P1 C P2 with X c X  : P1 WD 1 C CN DM

There exists .ˇ1 ; : : : ; ˇN / 2 f0; : : : ; p  1gN with ˇ1 C    C ˇN D M  1 and P 0 WD @11    @NN P D ˇ

ˇ

N X

ci0 Xi C c00 2 S X f0g;

0 c00 ; c10 ; : : : ; cN 2 Rp ;

i D1

where @ˇ11    @ˇNN P1 D

N X

ci0 Xi ;

@ˇ11    @ˇNN P2 D c00 :

i D1 0 are all in U , then we are done. Otherwise, (case 1) the polynomials c10 ; : : : ; cN 0 2), there exists i such that ci is non-constant (its degree in Xj is then  p for j ). Now, ci0 D @i P 0 belongs to .Rp \ S / X f0g and there exists s > 0 with 00p s 00

If (case some ci0 D P with P 2 S which is not a p-th power. We have constructed an element P 00 of S which is not a p-th power, whose degrees in Xj are all strictly smaller than those of P for all j (if the polynomial depends on Xj ). We can repeat this process with P 00 at the place of P and so on. Since at each stage we get a polynomial P 00 with partial degrees in the Xj strictly smaller than those of P for all j (if P 00 depends on Xj ), we eventually terminate with a polynomial P which has all the partial degrees < p in the indeterminates on which it depends, for which the case 1 holds. As for the statement on the field W , we remark that we have applied to P an algorithm which constructs G from P applying finitely many partial derivatives and p-th roots extractions successively, the only operations bringing out of the field W being p-root extractions. Hence, the existence of the integer M is guaranteed.

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We recall that U0 is the subfield of U whose elements are the s 2 U such that  s D s. Let V be a subgroup of U  such that V X V p 6D ;. Proposition 5.4.13. Under the hypotheses of Proposition 5.4.12, let us assume that for all D 2 V X f1g, the only solution s 2 U of  s D Ds is zero and that D1 ; : : : ; DN 2 V XPV p . Then, the polynomial G 2 R given by this Proposition is of the form G D i ci Xi C c0 with c1 ; : : : ; cN 2 U0 and c0 2 U . Moreover, if ci ; cj 6D 0 for 1  i < j  N , then Di D Dj . Let I be the non-empty subset of f1; : : : ; N g whose elements i are such that ci 6D 0, let Di D D for all i 2 I . Then, c0 D

 .c0 / 1 X ci Bi : C D D i 2I

e Proof. Proposition 5.4.12 gives us a polynomial G with G=G 2 R, of the form P p c X C B with c 2 U not all vanishing and B 2 R . Let sX p be a monoi i i i p  e D QG with Q 2 U , we have  s D mial of maximal degree in B . Since G 1 N p .D1    DN / Qs. Moreover,  .ci / D Di1 Qci for all i . Hence, if i is such that N p N p / r. Now, Di .D11    DN / 6D ci 6D 0, r WD s=ci satisfies  r D Di .D11    DN p 1 (because Di 2 V XV ) and r D 0, that is s D 0. This shows that B 2 U . Let us suppose that 1  i; j  N are such that i 6D j and ci ; cj 6D 0. Let us write r D ci =cj ; we have  r D Dj =Di r, from which we deduce r 2 UP 0 in case Dj =Di D 1 and r D 0 otherwise. The Proposition is proved dividing i ci Xi C B p by cj with e D QP , once observed that Q D D. j 6D 0 and by considering the relation P We proceed, in the next two subsections, to prove Theorems 5.4.10 and 5.4.11. We will the first theorem applying Propositions 5.4.12 and 5.4.13 to the field U D S prove 1=p n C.t / and then by using the criterion [6, Theorem 1.3.2] and we will prove n0 S n the second theorem applying these propositions to the field U D n0 K alg. .x 1=p / and then by using Theorem 5.4.3. 5.4.3.3 Direct proof of Theorem 5.4.10. For ˇ 2 K alg. such that jˇj < q q=.q1/ , we will use the formal series in K alg. ..t// Lˇ .t/ D ˇ C

1 X

.1/i ˇ q

i D1

.# q  t/    .# q i  t/

i

;

defining holomorphic functions for jtj < q q with Lˇ .#/ D logCar ˇ (14 ). P alg. We denote by W one of the following fields: K alg. ; K1 ; C . For f D i ci t i 2 n P P q 1=q W ..t// and n 2 Z we write f .n/ WD i ci t i 2 W ..t//, so that f .1/ D i ci t i . 14 these series P1 Papanikolas uses i C1 /t i . i D0 eCar ..log Car ˇ /=#

in

[39].

It

is

also

possible

to

work

with

the

series

333

An introduction to Mahler’s method for transcendence and algebraic independence L .t /

.1/

We have the functional equation Lˇ .t/ D ˇ 1=q C t ˇ# . The function Lˇ allows meromorphic continuation to the whole C , with simple poles at the points 2 n # q ; # q ; : : : ; # q ; : : : of residue 2

n

.logCar ˇ/q .logCar ˇ/q ;:::; :::: .logCar ˇ/ ; q q D1 Dn1 q

(5.4.8)

Let ˇ1 ; : : : ; ˇm be algebraic numbers with jˇj < q q=.q1/ , let us write Li D Lˇi for i D 1; : : : ; m. Let us also consider the infinite product  in (5.3.19), converging 2 everywhere to an entire holomorphic function with zeros at # q ; # q ; : : :, and write L0 D 1 , which satisfies the functional equation .1/

L0

.t/ D

L0 .t/ ; t # 2

n

 , meromorphic with simple poles at the points # q ; # q ; : : : ; # q ; : : :, with L0 .#/ D e with residues 2 n e q e q (5.4.9) e q; q ; : : : ; q ; : : : D1 Dn1 We now prove the following Proposition. Proposition 5.4.14. If the functions L0 ; L1 ; : : : ; Lm are algebraically dependent  ; logCar ˇ1 ; : : : ; logCar ˇm are linearly dependent over K. over K alg. .t/, then e Proof. The functions Li are transcendental, since they have infinitely many poles. Without loss of generality, we may assume that m  1 is minimal so that for all 0  n  m the functions obtained from the family .L0 ; L1 ; : : : ; Lm / discarding Ln are algebraically independent over K alg. .t/. S n We now apply Propositions 5.4.12 and 5.4.13. We take U WD n0 C.t 1=p /, which is perfect, and  W U ! U the q-th root map on C (inverse of the Frobenius map), such that  .t/ D t; this is an Fq -automorphism. Moreover, we take N D mC1, D1 D    D DN D .t  #/1 , 1=q

1=q .B1 ; : : : ; BN / D .0; ˇ1 ; : : : ; ˇm /;

and V D .t  #/Z . Let T C ŒŒt be the subring of formal series converging for all t 2 C with jtj  1, let L be its fraction field. Let f 2 L be non-zero. A variant of Weierstrass preparation theorem (see [4, Lemma 2.9.1]) yields a unique factorisation:    Y 1 X (5.4.10) f D .t  a/orda .f / 1 C bi t i ; jaj1 1

i D1

where 0 ¤  2 C , supi jbi j < 1, and jbi j ! 0, the product being over a finite index set. Taking into account (5.4.10), it is a little exercise to show that U0 D

334 S

Federico Pellarin i

Fq .t 1=p / and that for D 2 V X f1g, the solutions in U of f .1/ D Df are identically zero (for this last statement, use the transcendence over U of ). Let P 2 R be an irreducible polynomial such that P .L0 ; L1 ; : : : ; Lm / D 0; we e D QP with Q 2 U and Propositions 5.4.12 and 5.4.13 apply to give clearly have P c1 .t/; : : : ; cm .t/ 2 U0 not all zero and c.t/ 2 U such that i 0

c.t/ D .t  #/c

.1/

.t/ C .t  #/

m X

ci .t/ˇi1=q :

(5.4.11)

i D1

We get, for all k  0: c.t/ D 

m X

ci .t/ ˇi C

i D1

C

k X

qh

.1/h ˇi

.# q  t/.# q 2  t/    .# q h  t/ hD1

c .kC1/ .t/ .# q  t/.# q 2  t/    .# q kC1  t/

! (5.4.12)

:

We endow L with a norm k  k in the following way: if f 2 L factorises as in (5.4.10), then kf k WD jj. Let g be a positive integer. Then k  k extends in a unique g way to the subfield Lg WD ff W f p 2 Lg. If .fi /i 2N is a uniformly convergent sequence in Lg (on a certain closed ball centered at 0) such that kfi k ! 0, then fi ! 0 uniformly. We observe that there exists g  0 such that c.t/; c1.t/; : : : ; cm .t/ 2 Lg . Hence g c1 .t/; : : : ; cm .t/ 2 Fq .t 1=p / and kci k D 1 if ci 6D 0. This implies that  m  X  1=q  1=q ci ˇi   maxfjˇi jg < q 1=.q1/ :    i i D1

By (5.4.11), kck  q q=.q1/ . Indeed, two cases occur. The first case when kc .1/ k  maxi fjˇi1=q jg; here we have kck < q q=.q1/ because kc .1/ k D kck1=q by (5.4.10) and maxi fjˇi jg < q q=.q1/ by hypothesis. The second case occurs when the inequal1=q jg holds. In this case, maxfkc .1/ .t  #/k; k.t  ity kc .1/ k > maxfjˇ11=q j; : : : ; jˇm Pm #/ i D1 ci .t/ˇi1=q kg D kc .1/ .t  #/k which yields kck D q q=.q1/ by (5.4.11). Going back to (5.4.12) we see that the sequence of functions Eh .t/ D

c .hC1/ .t/ .# q  t/.# q 2  t/    .# q hC1  t/

converges uniformly in every closed ball included in ft W jtj < q q g, as the series defining the functions Li (i D 1; : : : ; m) do. We want to compute the limit of this sequence: we have two cases.

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First case. If kck < q q=.q1/ , then, there exists " > 0 such that kck D q .q"/=.q1/ . hC1 hC2 hC1 D q .q "q /=.q1/ . On the other side: Then, for all h  0, kc .hC1/ k D kckq 2

k.# q  t/.# q  t/    .# q

hC1

 t/k D j#jqC Cq D q q.q

hC1

hC1 1/=.q1/

:

Hence, kEh k D q

hC2 q hC2 "q hC1  q q1q q1

Dq

q"q hC1 q1

! 0;

which implies Eh !P0 (uniformly on every ball as above). This means that m i D1 ci .t/Li .t/ C c.t/ D 0. Let g be minimal such that there exists a non-trivial linear relation as above, with c1 ; : : : ; cm 2 U0 \ Lg ; we claim that g D 0. Indeed, if g > 0, c1 ; : : : ; cm 62 Fq and there exists a non-trivial relation Pm pg Cd.t/ D 0 with d1 ; : : : ; dm 2 Fq Œt not all zero, d.t/ 2 C.t/ and i D1 di .t/Li .t/ maxi fdegt di g minimal, non-zero. But letting the operator d=dt act on this relation we get a non-trivial relation with strictly lower degree because dF p =dt D 0, leading to a contradiction. Hence, g D 0 and c1 ; : : : ; cm 2 Fq .t/. This also implies P that c 2 C ; multiplying by a common denominator, we get a non-trivial relation m i D1 ci .t/Li .t/ C c.t/ D 0 with c1 ; : : : ; cm 2 Fq Œt and c 2 C.t/. The function c being algebraic, it has finitely many poles. This means that m X

ci .t/Li .t/

i D1 2

has finitely many poles but for all i , Li has poles at # q ; # q ; : : : with residues as P q q2 in (5.4.8), which implies that m i D1 ci .t/Li .t/ has poles in # ; # ; : : :. Since the functions ci belong to Fq Œt, they vanish only at points of absolute value 1, and the P qk qk residues of the poles are multiples of m (k  1) by non-zero i D0 ci .#/ .logCar ˇi / factors in A. They all must vanish: this happens if and only if m X

ci .#/ logCar ˇi D 0;

i D1

where we also observe that ci .#/ 2 K; the Proposition follows in this case. Second case. Here we know that the sequence Eh converges, but not to 0 and we must alg. compute its limit. Let  be in C with jj D 1. Then, there exists  2 Fq  , unique such that j  j < 1. Hence, if  2 C is such that jj D q q=.q1/ , there exists alg.  2 Fq  unique with j  .#/q=.q1/ j < q q=.q1/ :

(5.4.13)

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We have: c.t/ D 

Y 

g

t 1=p  a

orda c

0 @1 C

X

1 g bi t i=p A ;

i 1

jaj1

with  2 C  , the product being finite and jbi j < 1 for all i so that kck D jj. alg. Let  2 Fq  be such that (5.4.13) holds, and write: 0 1 orda c X Y  g g @1 C bi t i=p A ; t 1=p  a c1 .t/ D .  .#/q=.q1/ / jaj1

i 1

1  Y  X ord c a g g @1 C bi t i=p A ; t 1=p  a

c2 .t/ D .#/q=.q1/

0

(5.4.14)

i 1

jaj1

so that c.t/ D c1 .t/ C c2 .t/, kc1 k < q q=.q1/ and kc2 k D q q=.q1/ . For all h, we also write: E1;h .t/ D E2;h .t/ D

c1.hC1/ .t/ .# q  t/.# q 2  t/    .# q hC1  t/ c2.hC1/ .t/ .# q  t/.# q 2  t/    .# q hC1  t/

; :

Following the first case, we easily check that E1;h .t/ ! 0 on every closed ball of center 0 included in ft W jtj < q q g. It remains to compute the limit of E2;h .t/. We look at the asymptotic behaviour of the images of the factorsP in (5.4.14) under g the operators f 7! f .n/ , n ! 1. The sequence of functions .1 C i 1 bi t i=p /.n/ converges to 1 for n ! 1 uniformly on every closed ball as above. Let E be the finite set of the a’s involved in the finite product (5.4.14), take a 2 E . If jaj < 1, then g g alg. a.n/ ! 0 and .t 1=p  a/.n/ ! t 1=p . If jaj D 1, there exists a 2 Fq  such that ja  a j < 1 and we can find na > 0 integer such that lims!1 a.sna / D a , whence g g lims!1 .t 1=p  a/.sna / D t 1=p  a . n Q Let us also denote by nQ > 0 the smallest positive integer such that q D . Let N be the lowest common multiple of nQ and the na ’s with a varying in E . Then the sequence of functions: 00

10 11.N s/  Y  X ord c a g g @@ A @1 C bi t i=p AA ; t 1=p  a i 1

jaj1 alg.

g

converges to a non-zero element Z 2 Fq .t 1=p /.

s2N

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For n 2 N, let us write: Vn .t/ WD q

.#/q

n

nC1 =.q1/

.# q  t/.# q 2  t/    .# q nC1  t/

:

We have: .#/q=.q1/

nC1 Y i D1

1

1

t #

qi

D .1/q=.q1/ # q=.q1/ # .qC Cq

nC1 /

nC1 Y

i

.# q  t/1

i D1

D .1/q=.q1/ # q

nC2 =.q1/

nC1 Y

i

.# q  t/1 :

i D1 2

nC1

Hence, limn!1 # q=.q1/ =..# q t/.# q t/    .# q t//1 D .t/1 from which g alg. we deduce that lims!1 E2;sN .t/ D c0 .t/L0 .t/ with c0 2 Fq  .t 1=p /. We have g g g proved that for some c1 ; : : : ; cm 2 Fq .t 1=p /; c0 2 Fq .t 1=p / and c 2 K alg. .t 1=p /, P m i D0 ci Li Cc D 0. Applying the same tool used in the first case we can further prove that in fact, g D 0. If c0 is not defined over the operator f 7! f .1/ PFmq , then applying 0 0 we get another non-trivial relation c0 C i D1 ci Li D c with c 0 2 K alg. .t/ and c00 2 alg. .t/ F q .t/ not equal to c0 ; subtracting it from the former relation yields L0 2 K which is impossible since  is transcendental over C.t/. Hence c0 belongs to Fq .t/ too. Multiplying by a common denominator in Fq Œt and applying arguments of the first case again (by using the explicit computation of the residues of the poles of L0 P 2 at # q ; # q ; : : :), we find a non-trivial relation c0 .#/e C m i D1 ci .#/ logCar ˇi D 0. Proof of Theorem 5.4.10. If ` 2 C is such that eCar .`/ 2 K alg. , then there exist a; b 2 A, ˇ 2 K alg. with jˇj < q q=.q1/ such that ` D a logCar ˇ C be  . This well known property (also used in [39], see Lemma 7.4.1), together with Theorem 3.1.1 of [6], implies Theorem 5.4.10. 5.4.3.4 Direct proof of Theorem 5.4.11. Let s  1 be an integer and let Lin denote the s-th Carlitz’s polylogarithm by: Lis .z/ D

1 X kD0

k

.1/ks z q ; .ŒkŒk  1    Œ1/s

so that Li1 .z/ D logCar .z/ (the series Lis .z/ converges for jzj < q sq=.q1/ ). For ˇ 2 K alg. \ K1 such that jˇj < q sq=.q1/ (a discussion about this hypothesis follows in 5.4.3.5), we will use as in [19] the series Fs;ˇ .x/ D e ˇ.x/ C

1 X

.1/i se ˇ.x/q

i D1

.x q  #/s    .x q i  #/s

i

;

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where e ˇ.x/ is the formal series in Fq ..1=x// obtained from the formal series of ˇ 2 Fq ..1=#// by replacing # with x, an independent indeterminate. Let us assume that x 2 C , with jxj > 1. We have, for i big enough, ˇ ˇ i ˇ ˇ e q i 1 ˇ.x/q i ˇ ˇ D jxjq deg# ˇ q q1 ; ˇ q ˇ i ˇ .x  #/s    .x q  #/s ˇ q

so that the series Fs;ˇ .x/ converges for jxj > 1 provided that jˇj < q q1 and x is i not of the form # 1=q . We have the functional equations: ˇ.x//; Fs;ˇ .x q / D .x q  #/s .Fs;ˇ .x/  e moreover, Fs;ˇ .#/ D ˇj C

1 X

.1/i ˇ q

i D1

.# q  #/s    .# q i  #/s

i

D Lis .ˇ/:

Therefore, these series define holomorphic functions for jxj > q 1=q an allow meromorphic continuations to the open set fx 2 C; jxj > 1g, with poles at the points i # 1=q . We have “deformed” certain Carlitz’s logarithms and got in this way Mahler’s functions (except that the open unit disk is replaced with the complementary of the closed unit disk, but changing x to x 1 allows us to work in the neightbourhood of the origin). Let J be a finite non-empty subset of f1; 2; : : :g such that if n 2 J , p does not divide n. Let us consider, for all s 2 J , an integer ls  1 and elements ˇs;1 ; : : : ; ˇs;ls 2 K alg. \ K1 with jˇs;i j < q qs=.q1/ (i D 1; : : : ; ls ). We remark that if s is divisible by q  1 then, for all r > q 1=q the product: .x/

sq=.q1/

1  Y i D1

1

#

s

x qi

converges uniformly in the region fx 2 C; jxj  rg to a holomorphic function Fs;0 .x/, which is the .q  1/-th power of a formal series in K..1=.x/1=.q1///, s. hence in K..1=x// (compare with the function of 5.3). Moreover, Fs;0 .#/ D e Proposition 5.4.15. If the functions .Fs;ˇs;1 ; : : : ; Fs;ˇs;ls /s2J are algebraically dependent over K alg. .x/, there exists s 2 J and a non-trivial relation ls X i D1

ci Fs;ˇs;i .x/ D f .x/ 2 K alg. .x/

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with c1 ; : : : ; cls 2 K alg. if q  1 does not divide s, or a non-trivial relation: ls X

ci Fs;ˇs;i .x/ C Fs;0 .x/ D f .x/ 2 K alg. .x/

i D1

with c1 ; : : : ; cls ;  2 K alg. if q  1 divides s. In both cases, non-trivial relations can be found with c1 ; : : : ; cls ;  2 A. Proof. Without loss of generality, we may assume that J is minimal so that for all n 2 J and i 2 f1; : : : ; ln g the functions obtained from the family .Fs;ˇs;1 ; : : : ; Fs;ˇs;ls /s2J discarding Fn;ˇn;1 are algebraically independent over K alg. .x/. S We want to apply Propositions 5.4.12 and 5.4.13. We take U WD n0 K alg. n .x 1=p /, and  W U ! U the identity map on K alg. extended to U so that  .x/ D x q . We also take: .X1 ; : : : ; XN / D .Ys;1 ; : : : ; Ys;ls /s2J ; .D1 ; : : : ; DN / D ..x q  #/s ; : : : ; .x q  #/s /s2J ; „ ƒ‚ … ls times

.B1 ; : : : ; BN

/ D .ˇe .x/; : : : ; ˇB .x// s;1

s;ls

s2J :

We take V D .x q  #/Z . We have U0 D K alg. and for D 2 V X f1g, the solutions of f .x q / D Df .x/ are identically zero as one sees easily writing down a formal power series for a solution f 2 U . Let P 2 R be an irreducible polynomial such that P ..Fs;ˇs;1 ; : : : ; Fs;ˇs;ls /s2J / D e D QP with Q 2 U and Propositions 5.4.12 and 5.4.13 apply. 0; we clearly have P They give s 2 J , c1 ; : : : ; cls 2 K alg. not all zero and c0 2 U such that c0 .x/ D

e

ls X c0 .x q / 1  ci ˇs;i .x/: .x q  #/s .x q  #/s

(5.4.15)

i D1

We now inspect this relation in more detail. To ease the notations, we write ls D m and Fi .x/ WD Fs;ˇs;i (i D 1; : : : ; m). Since e ˇ.x/q D e ˇ.x q / for all ˇ 2 K, from (5.4.15) we get, for all k  0: ! h m k X X .1/hs ˇei .x/q e c0 .x/ D  ci ˇi .x/ C (5.4.16) ..x q  #/.x q 2  #/    .x q hC1  #//s i D1 hD1 C

c0 .x q ..x q  #/.x

q2

kC1

/

 #/    .x q kC1  #//s

: M

By Proposition 5.4.12, there exists M > 0 such that c0 .x/q 2 K alg. .x/, which M implies that c0 .x q / 2 K alg. .x/. By equation (5.4.16) we see that c0 .x/ 2 K alg. .x/.

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P We write c0 .x/ D i i0 di x i with di 2 K alg. . The sequence .jdi j/i is bounded; let be an upper bound. If x 2 C is such that jxj  r > q 1=q with r independent on x, then jc0 .x/j D supi jdi jjxji  supi jxji  jxjdegx c0 . Moreover, for jxj > r s s s with r as above, jxjq > j#j D q for all s  1 so that jx q  #j D maxfjxjq ; j#jg D qs jxj . Hence we get: 2

j.x q  #/.x q  #/    .x q

kC1

 #/j D jxjqCq

2 C Cq hC1

D jxj

q.q kC1 1/ q1

:

Let us write: Rk .x/ WD

c0 .x q ..x q

#/.x q 2



kC1

/

 #/    .x q kC1  #//s

:

We have, for jxj  r > q 1=q and for all k: jRk .x/j  jxjq

kC1

kC1

degx c0  sq.qq1 1/

:

(5.4.17)

ei .x/ < sq=.q 1/ for all i . In (5.4.15) we have two Since jˇi j < q sq=.q1/ , degx ˇ q ei g, one if deg .c0 .x q /=.#  cases: one if degx .c0 .x /=.#  x q /s /  maxi fdegx ˇ x q s ei g. In the first case we easily see that deg c0 < sq=.q  1/ x / / > maxi fdegx ˇ x (notice that degx c0 .x q / D q degx c0 ). In the second case, degx c0 D q degx c0  sq which implies degx c0 D sq=.q  1/. First case. Here, there exists " > 0 such that degx c0 D .sq  "/=.q  1/. We easily check (assuming that jxj  r > q 1=q ): sq"

jRk .x/j  jxj q1 q  jxj

kC1 1/ kC1  sq.q q1

sq"q kC1 q1

and the sequence of functions .Rk .x//k converges uniformly to zero in P the domain fx; jxj  rg for all r > q 1=q . Letting k tend to infinity in (5.4.16), we find i ci Fi .x/ C c0 .x/ D 0; that is what we expected. Second case. In this case, the sequence jRk .x/j is bounded but does not tend to 0. Notice that this case does not occur if q  1 does not divide s, because c0 2 K alg. .x/ and its degree is a rational integer. Hence we suppose that q  1 divides s. Let us write: X c0 .x/ D x sq=.q1/ C di x i ; i >sq=.q1/

with  2 K alg. . We have P lim

k!1

i >sq=.q1/ di x

qk i

..x q  #/.x q 2  #/    .x q kC1  #//s

D0

An introduction to Mahler’s method for transcendence and algebraic independence

341

(uniformly on jxj > r > q 1=q ), as one verifies following the first case. For all k  0 we have:  kC1 Y # s sq=.q1/ .x/ 1 i xq i D1 D .1/sq=.q1/ x sq=.q1/ x s.qC Cq

kC1 /

kC1 Y

i

.x q  #/s

i D1

D .1/sq=.q1/ x sq

kC2 =.q1/

kC1 Y

i

.x q  #/s :

i D1 2

kC1

sq=.q1/ Hence =..x q #/.x q #/    .x q #//s D Fs;0 .x/ Pwe have limk!1 x and i ci Fi .x/ C Fs;0 .x/ C c0 .x/ D 0. We now prove the last statement of the Proposition: this follows from an idea of Denis. The proof is the same in both cases and we work with the first only. There exists a  0 minimal such that the pa -th powers of c1 ; : : : ; cls are defined over the separable closure K sep of K. The trace K sep ! K can be extended to formal series K sep ..1=x// ! K..1=x//; its image does not vanish. We easily get, multiplying by a denominator in A, a non-trivial relation X a bi Fi .x/q C b0 .x/ D 0 i

with bi 2 A and b0 .x/ 2 K alg. .x/. If the coefficients bi are all in Fq , this relation is the pa -th power of a linear relation as we are looking for. If every relation has at least one of the coefficients bi not belonging to Fq , the one with maxi fdeg# bi g and a minimal has in fact a D 0 (otherwise, we apply the operator d=d # to find one with a smaller degree, because dg p =d # D 0 if a > 0). The following proposition reproduces Denis’ criterion of algebraic independence in [17, 18]. It follows immediately from Theorem 5.4.3. Proposition 5.4.16. Let L K alg. be a finite extension of K. We consider f1 ; : : : ; fm holomorphic functions in a domain jxj > r  1 with Taylor’s expansions in L..1=x//. Let us assume that there exist elements ai ; bi 2 L.x/ (i D 1; : : : ; m) such that fi .x/ D ai .x/fi .x q / C bi .x/; n

1  i  m:

Let ˛ be in L, j˛j > r, such that for all n, ˛ q is not a zero nor a pole of any of the functions ai ; bj . If the series f1 ; : : : ; fm are algebraically independent over K alg. .x/, then the values f1 .˛/; : : : ; fm .˛/ are algebraically independent over K.

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The next step is the following Proposition. Proposition 5.4.17. If the numbers .Lis .ˇs;1/; : : : ; Lis .ˇs;ls //s2J are algebraically dependent over K alg. , there exists s 2 J and a non-trivial linear relation ls X

ci Lis .ˇs;i / D 0

i D1

with c1 ; : : : ; cls 2 A. If q  1 does not divide s, or a non-trivial relation: ls X

ci Lis .ˇs;i / C e s D 0

i D1

with c1 ; : : : ; cls ;  2 A if q  1 divides s. Proof. By Proposition 5.4.16, the functions Fs;i (s 2 J ; 1  i  ls ) are algebraically dependent over K alg. .t/. Proposition 5.4.15 applies and gives s 2 J as well as a nontrivial linear dependence relation. If q  1 does not divide s, by Proposition 5.4.15 there exists a non-trivial relation ls X

ci Fs;ˇs;i .x/ D f .x/ 2 K alg. .x/

i D1

with c1 ; : : : ; cls 2 A. We substitute x D # in this relation: ls X

ci Lis .ˇs;i / D f 2 K alg. :

i D1

After [5] pp. 172–176, for all x 2 C such that jxj < q qs=.q1/ , there exist v1 .x/; : : : ; vs1 .x/ 2 C such that

0

1 0 0 v1 .x/ : :: B : C B : B : C D exp B s @ 0 A @ v .x/ s1 x Lis .x/

1 C C; A

exps being the exponential function associated with the s-th twist of Carlitz’s module. Moreover: 0 1 1 0 cj v1 .ˇs;j / 0 :: B C B : C : C D ' ˝s .cj / B :: C 2 .K alg. /s ; j D 1; : : : ; ns ; exps B Car @ c v .ˇ / A @ 0 A j s1

s;j

cj Lis .ˇs;j /

ˇs;j

An introduction to Mahler’s method for transcendence and algebraic independence

343

˝s where 'Car .cj / denotes the action of the s-th tensor power of Carlitz’s module. By Fq -linearity, there exist numbers w1 ; : : : ; ws1 2 C such that

0

1 w1 B :: C : C 2 .K alg. /s : exps B @ w A s1

c Yu’s sub-t-module Theorem (in [55]) implies the following analogue of HermiteLindemann’s Theorem. Let G D .Gsa ; '/ be a regular t-module with exponential function e' , with '.g/ D a0 .g/ 0 C    , for all g 2 A. Let u 2 C s be such that e' .u/ 2 Gsa .K alg. /. Let V the smallest vector subspace of C s containing u, defined over K alg. , stable by multiplication by a0 .g/ for all g 2 A. Then the Fq -subspace e' .V / of C s equals H.C / with H sub-t-module of G. This result with G the s-th twist of Carlitz’s module and e' D exps implies the vanishing of c and the K-linear dependence of the numbers Lis .ˇs;1/; : : : ; Lis .ˇs;ls /: If q  1 divides s then by Proposition 5.4.15 there exists a non-trivial relation ls X

ci Fs;ˇs;i .x/ C Fs;0 .x/ D f .x/ 2 K alg. .x/

i D1

with c1 ; : : : ; cls ;  2 K. We substitute x D # in this relation: ls X

ci Lis .ˇs;i / C e  s D f 2 K alg. :

i D1

The Proposition follows easily remarking that, after [5] again, there exist v1 ; : : : ; vs1 2 C such that 0 1 v1 0 B :: B :: C B : C D exp B : s@ @ 0 A vs1 e s 0 0

1 C C: A

Proof of Theorem 5.4.11. To deduce Theorem 5.4.11 from Proposition 5.4.17 we quote Theorem 3.8.3 p. 187 of Anderson–Thakur in [5] and proceed as in [14]. For

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Federico Pellarin

all i  nq=.q  1/ there exists hn;i 2 A such that if we set 1 0 X B :: C ˝n : C Pn WD 'Car .hn;i / B @ 0 A; 0

i

#i

then the last coordinate Pn is equal to .n/.n/ (where .n/ denotes Carlitz’s arith˝n metic Gamma function). Moreover, there exists a 2 A X f0g with 'Car .a/Pn D 0 if and only if q  1 divides n. This implies that X

Œnq=.q1/

.n/.n/ D

hn;i Lin .# i /:

i D0

The numbers hn;i are explicitly determined in [5]. In particular, one has .s/ D Lis .1/;

s D 1; : : : ; q  1:

We apply Proposition 5.4.16 and Proposition 5.4.17 with J D J ] [ fq  1g, J being the set of all the integers n  1 with p; q  1 not dividing n, lq1 D 1, ˇq1;1 D 1, and for s 2 J ] , ]

.ˇ1 ; : : : ; ˇls / D .# i0 ; : : : ; # ims /; where the exponents 0  i0 <    < ims  sq=.q  1/ are chosen so that .s/ 2 KLis .1/ C    C KLis .# Œsq=.q1/ / D KLis .# i0 / ˚    ˚ KLis .# ims /:

Remark 5.4.18. With ˇ 2 K as above, we can identify, replacing # with t 1 , the formal series Fˇ 2 C..x// with a formal series Fˇ 2 Fq ŒtŒŒx KŒŒx (as in 5.3.2), over which the operator  defined there acts. Carlitz’s module 'Car W # 7! # 0 C  1 acts on Fq ŒtŒŒx and it is easy to compute the image of Fˇ under this action. from this we get: FˆCar .#/ˇ .x/ D #Fˇ .x/ C .x  #/ˇ.x/; which implies that, for all a 2 A, FˆCar .a/ˇ .x/ 2 a Fˇ .x/ C Fq .#; x/. In some sense, the functions Fˇ are “eigenfunctions” of the Carlitz module (a similar property holds for the functions Lˇ and the functions (5.3.23), which also have Mahler’s functions as counterparts.

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5.4.3.5 Final remarks. The fact that we could obtain Theorems 5.4.10 and 5.4.11 in a direct way should not induce a false optimism about Mahler’s approach to algebraic independence; the matrices of the linear  -difference equation systems involved are diagonal and we benefitted of this very special situation. In the general case, it seems more difficult to compute the transcendence degree of the field generated by solutions f D t .f1 ; : : : ; fm / 2 K..x// of a system like: f .x d / D A.x/  f .x/ C b.x/ (see for example, the difficulties encountered in [37, Section 5.2]). One of the reasons is that tannakian approach to this kind of equation is, the time being, not yet explored. This point of view should be considered since it has been very successful in the context of t-motives as in Papanikolas work [39], which is fully compatible with Galois’ approach. We could expect, once the tannakian theory of Mahler’s functions is developed enough, to reach more general results by computing dimensions of motivic Galois groups (noticing the advantageous fact that the field of constants is here algebraically closed). However, there is an important question we shall deal with: is any “period” of a trivially analytic t-motive (in the sense of Papanikolas in [39]) a Mahler’s value? For example, in 5.4.3.4, we made strong restrictions on the ˇ 2 K alg. so that finally, Denis Theorem in [19] is weaker than Papanikolas Theorem 5.4.10. Is it possible to avoid these restrictions in some way? We presently do not have a completely satisfactory answer to this question, but there seem to be some elements in favour of a positive answer. We will explain this in the next few lines. The method in 5.4.3.4 of deforming Carlitz logarithms logCar .ˇ/ into Mahler’s functions that the ˇ’s in K alg. correspond to analyticP functions at infinity. If P requires i ˇ D i ci # lies in K1 D Fq ..1=#//, the series e ˇ.x/ WD i ci x i converges for x 2 C such that jxj > 1 and ˇ D e ˇ.#/. This construction still works in the perfect closure of the maximal tamely ramified extension F of Fq ..x 1// but cannot be followed easily for general ˇ 2 K alg. X K1 . Artin-Schreier’s polynomial X p  X  # does not split over F . Hence, if  2 K alg. is a root of this polynomial (it has absolute value jj D q 1=p ), the construction fails with the presence of divergent series. Let us consider `1 D logCar ./ and `2 D logCar .#/ D logCar . p /  `1 . It is easy to show that `1 ; `2 are K-linearly independent. By Theorem 5.4.10, `1 ; `2 are algebraically independent over K. The discussion above shows that it is virtually impossible to apply Proposition 5.4.16 with the base point ˛ D #. We now show that it is possible to modify the arguments of 5.4.3.4 and apply Proposition 5.4.16 with the base point ˛ D . To do so, let us consider, for ˇ 2 K, the formal series e ˇ .x/ D e F ˇ.x/ C

1 X nD1

.1/n Qn

j D1 .x

n e ˇ.x q /

q j C1

 x q j  #/

:

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Federico Pellarin q2

j

If jˇj < q q1 and if jxj > 1, x 62 f 1=q C ; j  1;  2 Fq g, these series converge. In particular, under the condition on jˇj above, they all converge at x D  since they define holomorphic functions on the domain fx 2 C; jxj > q 1=pq g, which contains . More precisely, the value at x D  is: e ˇ ./ D logCar .ˇ.//: F We have the functional equations eˇ .x/  ˇ.x// e ˇ .x q / D .#  x q 2 C x q /.F F e ˇ define meromorphic functions in the open set which tells us that the functions F fx 2 C; jxj > 1g. With all these observations, it is a simple exercise to apply Proposition 5.4.16 with ˛ D  and show the algebraic independence of `1 ; `2 . The reader can extend these computations and show the algebraic independence of other logarithms of elements of K alg. . However, the choice of the base point ˛ has to be made cleverly, and there is no general recipe yet. Here, the occurrence of Artin-Schreier extensions is particularly meaningful since it is commonly observed that every finite normal extension of Fq ..1=#// is contained in a finite tower of ArtinSchreier extensions of Fq ..1=# 1=n // for some n [24, Lemma 3]. Remark 5.4.19. Just as in Remark 5.4.18, the action of Carlitz’s module yields the following formula e .# q #/ˇ Cˇ q .x/ D # F e ˇ .x/ C .x q  x  #/e F ˇ.x/: Since there are natural ring isomorphisms KŒX =.X p  X  #/ Š Fq Œ Š A, it could be interesting to see if there is some Fq -algebra homomorphism A ! C ŒŒ  e ˇ are “eigenfunctions”. of which the functions F

References [1] M. Ably, L. Denis, and F. Recher. Transcendance de l’invariant modulaire en caractéristique finie. Math. Z. 231(1) (1999), 75–89. [2] J.-P. Allouche, Sur la transcendance de la série formelle …. Séminaire de Théorie des Nombres de Bordeaux 2 (1990), pp. 103–117. [3] M. Amou, An improvement of the transcendence measure of Galochkin and Mahler’s Snumbers. J. Austral. Math. Soc. (Series A), 52 (1992), 130–140. [4] G. W. Anderson, t-motives, Duke Math. J. 53 (1986), 457–502. [5] G. W. Anderson and D. S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. Math. 132(2) (1990), 159–191. [6] G. W. Anderson, W. D. Brownawell, and M. A. Papanikolas, Determination of the algebraic relations among special -values in positive characteristic, Ann. Math. 160(2) (2004), 237– 313.

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[7] E. Artin and G. Whaples, Axiomatic Characterization of fields by the product formula for valuations. Bull. AMS 51 (1945) 469–492. [8] K. Barré-Sirieix, G. Diaz, F. Gramain, G. Philibert, Une preuve de la conjecture de MahlerManin. Invent. Math. 124(1–3) (1996), 1–9 [9] P.-G. Becker, Algebraic independence of the values of certain series by Mahler’s method. Monatsh. Math. 4 (1992), 183–198. [10] P.-G. Becker and W. Bergweiler, Transcendency of Local Conjugacies in Complex Dynamics and Transcendency of Their Values, Manuscripta Math. 81 (1993), 329–337. [11] V. Bosser and F. Pellarin, Differential properties of Drinfeld quasi-modular forms. Int. Math. Res. Not. 2008. [12] V. Bosser and F. Pellarin, On certain families of Drinfeld quasi-modular forms. J. Number Theory 129, I. 12, (2009), 2952–2990. [13] L. Carlitz, An analogue of the von Staudt theorem. Duke Math. J. 3 (1937), 503–517. [14] C.-Yu Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic. Adv. Math. 216 (2007), 321–45. [15] Chieh-Yu Chang, A note on a refined version of Anderson-Brownawell-Papanikolas criterion. J. Number Theory 129(3) (March 2009), 729–738 [16] P. Corvaja and U. Zannier, Some New Applications of the Subspace Theorem. Compositio Math. 131 (2002), 319–340. [17] L. Denis, Indépendance algébrique de différents . C. R. Acad. Sci. Paris Math. 327, (1998), 711–714. [18] L. Denis, Indépendance algébrique des dérivées d’une période du module de Carlitz. J. Austral. Math. Soc. 69, (2000), 8–18. [19] L. Denis, Indépendance algébrique de logarithmes en caractéristique p. Bull. Austral. Math. Soc. 74 (2006), 461–470. [20] L. Denis, Approximation algébrique en caractéristique p. Manuscrit. 2010. [21] A. I. Galochkin, Transcendence measure of values of functions satisfying certain functional equations. Mat. Zametki 27 (1980); English transl. in Math. Notes 27 (1980). [22] D. Goss, v-adic zeta functions, L-series and measures for function fields. Invent. Math. 55 (1979), 107–119. [23] D. Goss, Basic Structures of Function Field Arithmetic, Springer [24] K. Kedlaya, The algebraic closure of the power series field in positive characteristic. Proc. Amer. Math. Soc. 129 (2001), 3461–3470. [25] K. Kubota, On the algebraic independence of holomorphic solutions of certain functional equations and their values. Math. Ann. 227 (1977), 9–50. [26] S. Lang, Fundamentals of diophantine geometry, Springer. [27] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101 (1929), 342–366. [28] K. Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen. Math. Z. 32 (1930), 545–585. [29] K. Mahler, Über das Verschwinden von Potenzreihen mehrerer VerŠnderlichen in speziellen Punktfolgen. Math. Ann. 103 (1930), 573–587.

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[30] K. Mahler, Remarks on a paper by W. Schwarz. J. Number Theory 1 (1969), 512–521. [31] D. W. Masser, Heights, Transcendence, and Linear Independence on Commutative Group Varieties In Diophantine approximation: lectures given at the C.I.M.E. summer school held in Cetraro, Italy, Springer Lectures Notes in Math 1819, 2000. [32] W. Miller, Transcendence measures by a method of Mahler. J. Austral. Math. Soc. (Series A) 32 (1982), 68–78. [33] Yu. V. Nesterenko. Modular functions and transcendence questions. Sb. Math. 187(9) (1996), 1319–1348; translation from Mat. Sb. 187(9) (1996), 65–96. [34] Yu. V. Nesterenko, Algebraic independence for values of Ramanujan functions. In Introduction to algebraic independence theory (Yu. V. Nesterenko and P. Philippon, eds.), Chapters 3–10, pp. 27–43 and 149–165, Lecture Notes in Mathematics 1752, Springer, 2001. [35] K. Nishioka, Algebraic function solutions of a certain class of functional equations. Arch. Math. 44 (1985), 330–335. [36] K. Nishioka, Algebraic independence measures of the values of Mahler functions. J. Reine Angew. Math. 420 (1991), 203–214. [37] K. Nishioka, Mahler functions and transcendence. Lecture Notes in Mathematics 1631, Springer, 1996. [38] K. Nishioka, Measures of algebraic independence for Mahler functions. (Yu. V. Nesterenko and P. Philippon, eds.), Chapter 12, pp. 187–197, Lecture Notes in Mathematics 1752, Springer, 2001. [39] M. A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math. 171 (2008), 123–174 [40] F. Pellarin, Aspects de l’indépendance algébrique en caractéristique non nulle. Séminaire N. Bourbaki 973, March 2007. [41] F. Pellarin, Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms. J. Reine Angew. Math. 687 (2014), 1–42. [42] G. Philibert, Une mesure d’indépendance algébrique. Ann. Inst. Fourier 38 (1988), 85–103. [43] P. Philippon, Criteres pour l’independance algebrique. Inst. Hautes Études Sci. Publ. Math. 64 (1986), 5–52. [44] P. Philippon, Criteres pour l’independance algebrique dans les anneaux Diophantiens. C. K. Acad. Sci. Paris 315 (1992), 511–515. [45] P. Philippon, Independance algebrique et K-fonctions. J. Reine Angew. Math. 497 (1998) 1– 15. [46] P. Philippon, Diophantine geometry. In Introduction to algebraic independence theory. (Yu. V. Nesterenko and P. Philippon, eds.), Chapter 6, pp. 83–93, Lecture Notes in Mathematics 1752, Springer (2001). [47] N. Pytheas Fogg, V. Berthé, S. Ferenczi, Ch. Mauduit, and A. Siegel (eds.), Substitutions in dynamics, arithmetics and combinatorics. Springer Lecture Notes in Mathematics 1794, 2002. [48] H. Remmert, Classical topics in complex function theory, Graduate Text in Math. 172, Springer, 1991. [49] H. Sharif and C. F. Woodcock, Algebraic Functions Over a Field of Positive Characteristic and Hadamard Products J. London Math. Soc. s2–37, 395–403.

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[50] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves. Graduate Texts in Mathematics 151. Springer, 1994. [51] T. Töpfer, Algebraic independence of the values of generalized Mahler functions. Acta Arith. 72 (1995), 161–181. [52] T. Töpfer, Zero order estimates for functions satisfying generalized functional equations of Mahler type. Acta Arith. 85 (1998), 1–12. [53] M. Waldschmidt, Nombres Transcendants, Springer Lecture Notes in Mathematics 402, (1974). [54] M. Waldschmidt, Diophantine Approximation on Linear Algebraic Groups Transcendence Properties of the Exponential Function in Several Variables. Grundlehren der mathematischen Wissenschaften, Vol. 326, 2000 [55] J. Yu, Analytic homomorphisms into Drinfeld modules. Ann. Math. 145, (1997), 215–233.

Chapter 6

Automata methods in transcendence Dinesh S. Thakur1 The purpose of this expository article is to explain diverse new tools that automata theory provides to tackle transcendence problems in function field arithmetic. We collect and explain various useful results scattered in computer science, formal languages, logic literature and explain how they can be fruitfully used in number theory, dealing with transcendence, refined transcendence and classification problems.

6.1 Introduction Naturally occurring interesting numbers (say real, complex, p-adic or their function field counterparts) in number theory or algebraic geometry, such as periods, special values of ; ; L or other special functions, are usually defined by analytic processes such as infinite sums, products, limits or integrals. In transcendence theory, we are interested in knowing whether they are linked algebraically or not, i.e. whether they are transcendental, algebraically independent etc. Though in science, the usual way to exhibit a number given by a limiting process is by its decimal (base p, p-adic, Laurent series etc.) expansion, such a description had been usually useless for transcendence purposes, for naturally occurring numbers. For usual numbers such as e and  we do not know good description of expansion, and carry over makes it hard to manipulate the expansions. Work [C79, CKMR80] (also see Furstenberg [F67]) showed that the P of Christol series fn t n 2 Fq ŒŒt for an algebraic function over Fq .t/ (‘numbers’ in function field arithmetic) can be generated by a finite q-automaton (weakest machine model with no memory and which accepts q different inputs only) taking the expansion digits of n base q one by one as input and producing fn as the output at the end. Not only that, but conversely such machines produce algebraic series. This translates the finite algebraic description of the polynomial that function satisfies over Fq .t/ to finite computer science description of patterns of digits. There are similar nice equivalent descriptions given by formal language theory, logic etc. These subjects have been developed a lot and generated various equivalent viewpoints and many more implications, which are enough for transcendence applications and often directly applicable as we will see. Our previous accounts [T94, T96, T98, T04] explained applications in detail while quoting automata results used. Here we explain automata tools in more detail (at least 1 The

author is supported by NSA grants H98230-08-1-0049 and H98230-10-1-0200.

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with definitions and full statements and sketches of proofs of things we use) and only sketch and quote applications. We describe a few more things than we use in the hope that they might be of future use. For more extensive treatments of automata theory, we refer to books [HU79, S85, AS03], and especially the excellent bibliography and bibliographical notes of [AS03].

6.2 Automata: implications, equivalences: definitions and statements 6.2.1 Automata: definition. Let q be a positive integer. A q-automaton is a ‘machine’ which can be in one of finitely many states and takes q possible inputs which act on these states, and finally each state has an output associated with it. Mathematically, we can consider it as a quadruple .S; s0 ; T; O/ where S is a finite set (think of set of ‘states’), s0 2 S (think of ‘initial state’), T W f0; 1;    q  1g  S ! S is a function (called transition function describing how possible inputs marked by digits from 0 to q  1 act on states) and O W S ! F is a function (called output function describing the output in F when the machine is in ‘final state’). Of course, one is interested in infinitely many inputs represented by all integers n P0. This is achieved as follows. The input n is expended into its base q expansion as ni q i , 0  ni < q, and is fed digit by digit, say from left to right, and output on the final state ns0 is read when one is finished. Without loss of generality, F is a finite set, which will in fact be a finite field in our applications. Often the question can be reduced to only two output values: 0 (rejection), and 1 (acceptance or recognition). Hence we call a subset M of ZC qautomatic subset, if for some q-automaton we have O.ns0 / D 1 if and only if n 2 M . Our main interest is of course infinite subsets, and thus we call an infinite increasing sequence of positive integers listing all elements of such a set a q-automatic sequence. 6.2.2 Visualization and representations. Those familiar with Turing machine format can visualize a machine whose control head moves, depending on input and transition function, on a tape representing (outputs on) states and once it stops, one just reads the output at that position. Those familiar with neural network models can visualize active neuron (current state s) firing on input i and making another neuron T .i; s/ active. If N is the number of states, one can describe the automaton by a transition function by giving q  N table and output function; or one can describe it by labeled graph, with vertices representing the states with initial and accepting states specially marked and edges from a state to another labeled by digits giving those transitions. Then accepted words are just ordered lists of labels of edges of all paths from initial to accepted states. For automaton with output 0; 1 and with N states, we have a matrix representation h W f0; 1;    ; q  1g ! MatN N .f0; 1g/ with h.d /ij being 1, if the digit d takes the state i to state j , and being 0 otherwise. This is extended by taking concatenation of digits to multiplication of corresponding matrices. For N -vectors v1 ; v2 with 0; 1

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entries, corresponding to initial and final states, v1 h.w/v2 > 0, if and only if the word w is accepted. 6.2.3 Variants on automata models. The notion of q-automata (also called finite automata in the literature) is quite robust in the sense that many a priori different variants like (i) non-deterministic automata (think of a parallel computer), where the transition function is multi-valued, so that acceptance is through some possible transitions path, (ii) incomplete automata, where transition function is partial i.e. not always defined, (iii) non-deterministic automata with "-moves, that is state can change without input, (iv) two-way automata, (v) with one marker, where the control head can return as memory etc. We refer to [HU79, S85, AS03] for definitions and proofs of equivalences. It is easy to see that at the expense of exponential blow up of the number of states, we can use usual automata simulating non-deterministic one, by using its new states to be all subsets of set of states of the non-deterministic one and keeping track of possibilities at each stage. The non-deterministic ones are quite useful in proofs because of their flexibility. For example, to make a machine accepting n’s accepted by two machines, we just feed the new start state into start states of the two machines on any input, possible in non-deterministic realm. 6.2.4 Variants on digit models. Also, whether (a) we allow leading zeros in base q expansions of n, whether (b) we read from left to right or right to left, or whether (c) we use base q or q k , does not change the final outcome. To see that there is no difference on (a), we just introduce a new start state, which stays the same on input zero and on other input goes to the start state of the new machine. For (b), we just reverse arrows and interchange initial and final states, which is possible in non-deterministic realm (if we insist on one start state, introduce one and feed into (old) terminal states on any input). For (c), given a q-automaton, corresponding q k -automaton would have same states, start and final states with transition on a digit base q k being the transition on corresponding base q word, and conversely given a q k -automaton, from each state make k-length paths with new states according to q-digits of the base q word corresponding to q k base digit. 6.2.5 Christol’s theorem. Automata method is based on the following theorem of Christol [C79, CKMR80]. P Theorem 6.2.1. The series fn t n 2 Fq ŒŒt is algebraic over Fq .t/, if and only if there is a q-automaton which gives output fn on input n. Remark 6.2.2. Note that for algebraicity questions, general (finite tail) Laurent series immediately reduce to corresponding power series part, and also whether a series in t or in 1=t does not make a difference.

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6.2.6 Some simple consequences. The power of the method derives from the nonobvious equivalence, as well as various viewpoints and tools from which automata has been studied over the years by computer scientists, logicians, formal language theory experts. We will describe these and the applications. But first let us see, how such unusual equivalence gives new perspective on algebraic relationships. P P P Theorem 6.2.3. (1)PIf fn t n ; fn0 t n 2 Fq ŒŒt are algebraic, then so is fn fn0 t n . P n (2) The series fn t 2 Fq ŒŒt is algebraic, if and only if each fn Df t n is algebraic for each f 2 Fq . Proof. To see (1), just imagine two corresponding automata running in parallel (take direct product) and at the end combine their outputs by multiplying, which can be achieved by a table. (Hence, the algebraicity of this Hadamard or term-by-term product is obvious from the automata viewpoint, but hard from the definition of algebraicity, whereas for the algebraicity of usual product of two algebraic series, the situation is reversed!) The ‘if’ direction of (2) is clear, by taking linear combinations. The hard converse direction is clear from automata viewpoint: change the output function by sending f to 1 and the other elements to 0.  Here is another direct nice consequence (pointed out to me by Allouche): The P pi P i series ai t 2 Fp ŒŒt is algebraic over Fp .t/ if and only if ai t 2 Fp .t/. This is since both the statements are equivalent to the fact that ai is an eventually periodic sequence. 6.2.7 Examples. Now we give three Pexamples, all with outputs only 0; 1, so that the corresponding series is of form s D m2M t m . Example I: We describe the transition function by the following table, where 2  i < q, and the output function separately by O.s0 / D O.s2 / D 0 and O.s1 / D 1 s0 0 s0 1 s1 i s2

s1 s1 s2 s2

s2 s2 s2 s2

or both by graph, with accepted states overlined and only movements shown: / s0 >1

/ s1 ~ ~ ~~ ~~ 1 ~  ~ ~ 1

s2

To get the final state s1 to end up with output 1, digit expansion of m has to be 1 followed P q n by all zeros, i.e. m is a power of q, so that the corresponding series is sD t , which is algebraic, as you can see by telescoping and noticing that q-th power is linear: s q  s D t.

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Example II: Let q D 2. Consider the output O.s0 / D O.s1 / D 1 and O.s2 / D O.s3 / D 0, and the transition function s0 0 s0 1 s1

s1 s2 s3

s2 s3 s1 s3 s1 s3

Now m 2 M precisely when ms0 is either s0 or s1 . Note 0; 1 2 M and we claim that m 2 M precisely when 4m; 4m C 1 2 M . This is immediately seen by diagram-chase of automata theory: note that base 2 expansions of 4m; 4m C 1 are obtained by appending 00, 01 respectively to the expansion of m and the diagram chase shows that if you are in (out respectively) fs0 ; s1 g you stay in (out respectively) by the action can reformulate this in terms of the generating function P ofm00 or 01. WeP s as s D t D .1 C t/ t 4m D .1 C t/s 4 , so that we have algebraic series s D .1 C t/1=3 . Example III: Let q D 3. Consider the two-state machine whose initial state s0 is the only accepting state and machine stays in the initial state on input 0 or 2, but goes to error (i.e. the other non-accepting) state s1 on input 1, and once it is there, stays there on any input. Here it is immediate that m 2 M precisely when its base 3 expansion does not contain digit 1 (Cantor type description!), so that s D s 3 C t 2 s 3 . Example IV: We leave it as a fun exercise to the reader to build a 2-automaton representing Thue-Morse sequence containing m’s whose base 2 expansion has even number of ones (by keeping track of parity of the number) and get algebraic equation for the corresponding series. The reader should also work out details of other equivalences mentioned below on this example. We also mention a simple non-example: f1n 0n g cannot be recognized by automata. 6.2.8 Languages and grammars. We saw how digit pattern of algebraic power series is described by machines. Other common ways of describing patterns are to give generating or accepting rule for the language exponents or describe it by logical sentences. In fact, all these a priori different and independent descriptions of classes have converged to the same robust classes. We will exploit this fact later. In this language perspective, the digits are now possible words so that Z0 consists of all possible tries to make sentences, whereas a subset M of the set of nonnegative integers can be considered as a language of particular class of ‘grammatical’ sentences. Grammar teaches us, for example, how a sentence can break up into a noun phrase and a verb phrase; a noun phrase can break up into an adjective and another noun phrase; a noun phrase can be a noun like the word ‘man’, and adjective can be a word like ‘tall’. In a formal language theory, this is abstracted by defining a grammar to be a tuple .V; T; fP ! Qg; N /. Here V is a finite set (think of variables or non-terminals or

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syntactic classes), T is a disjoint finite set (think of terminals or words), P ! Q are finitely many ‘production rules’ (think of ‘sentence’ goes to ‘noun phrase’, ‘verb phrase’ etc.) where P is a string on V [T containing at least one element of V and Q is a string on V [T , and N 2 V (think of a special start symbol). Finally the language generated by this grammar is collections of all possible grammatical sentences, i.e. strings on words (terminals) arising via the allowed productions starting from the start symbol (i.e. N !    ! string on terminals). For our applications to q-automata, T D Dq WD f0; 1;    ; q  1g, so the digits are possible words, non-negative integers are sentences (do not have to worry about leading zeros as we saw) and a language will thus P correspond to a set M of nonnegative integers or equivalently a power series m2M t m . In other words, we are talking about the language of ‘exponents’. The class of a language is defined by the production types allowed. (i) The language is regular language if each production rule is of form X ! YP or X ! P , where X; Y 2 V and P a string on T . (ii) The language is context-free language if each production rule is of form X ! Q, where X 2 V and Q a string on V [ T . (iii) The language is context-sensitive language if each production rule is of form Q1 XQ2 ! Q1 PQ2 , where Qi are strings over V [ T and X and P as above. Warning: though the definition of context-sensitive allows replacement only within some context as you would imagine, the context-free and context-sensitive are not opposite notions. In fact, regular is context-free and context-free is context-sensitive. Our examples I, II, III correspond to regular languages given by production rules (I) N ! 1, N ! N 0; (II) N ! 0, N ! 1, N ! N 00, N ! N 01; (III) N ! 0, N ! 2, N ! N 0, N ! N 2, where in the last two we have allowed leading zeros for simplicity. If we do not want to do that, we use, e.g., for (III), N ! 0, N ! N 0 , N 0 ! 2, N 0 ! N 0 0, N 0 ! N 0 2. 6.2.9 Regular expressions. Regular languages on words T can also be described by regular expressions defined recursively by strings of symbols in T and operations of concatenation (denoted by juxtaposition), or (denoted by C), and repetition (denoted by  ). Rather than giving a formal definition [HU79], we note that our examples I, II, III correspond to regular expressions (I) 10 on T D Dq , (II) 0 C 1.00 C 01/ on T D D2 , (III) 0 C 2.0 C 2/ (or simply .0 C 2/ , if we allow leading zeros) on T D D3 . 6.2.10 Fixed point of a q-substitution. A q-substitution over S is just a function f W S ! S q . If q > 1, writing f .s/ D sw and extending by juxtaposition, we see that f .1/ .s/ D swf .w/f .2/ .w/    D s1 s2    is a fixed point. If I W S ! F is some function, then we say that fn D I.sn / is a sequence which is image of fixed point (starting at s) of q-substitution. Again, we also call the subset of integers where the sequence is 1 by the same name.

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The Example I above corresponds to S D fa; b; cg, f .a/ D abc    c; f .b/ D bc    c; f .c/ D c    c and I.a/ D I.c/ D 0; I.b/ D 1, with starting point a, the n-th position of I.f 1 .a// being one, exactly when n is a power of q. The Example II above corresponds to S D fa; b; c; d g. f .a/ D ab; f .b/ D cd; f .c/ D bb; f .d / D d d , and I.a/ D I.b/ D 1; I.c/ D I.d / D 0, with starting point a. We give a little more details to help understand the notation. The fixed point is abcdbbd dcdcd d d d dbbd dbbd d d d    , its image is 11001100000000001100110 000    , with 1’s at (base 10) positions 0; 1; 4; 5; 16; 17; 20; 21;    . The Example III above corresponds to S D fa; bg, f .a/ D aba; f .b/ D bbb and I.a/ D 1; I.b/ D 0, with starting point a. 6.2.11 q-Definability. Let vq .x/ denote the largest power of q dividing x, for a nonzero x, and vq .0/ D 1. Again, rather than giving a formal definition [BHMV94], we describe q-definable set informally as a subset of Z0 which can be defined by a first order formula over the structure .Z0 ; C; vq /. Roughly, this means that the subset is defined by a statement involving sentential logical operations and quantifiers running over non-negative integers (but not, for example, over subsets or functions), and involving only C, vq (but not, say multiplication, or individual non-negative integers, unless they are defined by such properties). The following examples should make this clearer. The Example I above corresponds to vq .x/ D x, which isolates powers of q. Let us see, following [BHMV94], how to build a formula for example III. Note that ‘0’ can be defined as x with ‘for all y, x Cy D y’ and ‘x  y; x > y’ etc. can be defined by ‘there is z such that x D y C z’ and so on. Then the formula we need is ‘there is no y such that y is a power of 3 and 1 is a coefficient of y in the base 3 expansion of x’, where the first part is done in Example I above and second part is taken care of by ‘there exist w; z such that x D z C 1  y C w and w < y and (v3 .z/ > y or z D 0)’. We leave (II) to interested readers by saying that the accepted words can be described as those having digit zero at any even numbered position from right. We refer to [BHMV94] for detailed description and references to literature with this viewpoint and for example, work of Büchi in 1960 giving ‘second order arithmetic’ description of automata. 6.2.12 q D 1. There is a way to define things, so that for q D 1, the notions give periodic sequences. We ignore this, except for pointing out this terminology. 6.2.13 Equivalent notions to automata. Theorem 6.2.4. Let q > 1 be an integer, and F be a finite set. Consider a F -valued sequence fn and for each f 2 F , put Mf WD fm W fm D f g. Then the following are equivalent. (i) The sequence fn is q-automatic. (ii) There are only finitely many distinct subsequences of the form n ! fq k nCr , where k varies through positive integers and r through 0  r < q k .

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(iii) The sequence fn is q-definable. (iv) For each f 2 F , Mf is a regular language. (v) For each f 2 F , Mf is given by a regular expression. (vi) For each f 2 F , Mf is an image of a fixed point of a q-substitution. Further, if q is a power of a prime p, F D Fq then these are equivalent to the following. P (vii) The series s D fn t n is algebraic over Fq .t/.P P n (viii) The series s is (a diagonal) fn n t n , where fn1 nk x1n1    xk k is a rational function in Fq .x1 ;    ; xk /. We have already given several equivalences in our examples. For the Example I, the series is a diagonal of the rational function x1 =.1  x1q1  x2 /, and the only subsequences of the type we consider are clearly fn itself and identically 0 sequence. We leave the other examples to the reader. 6.2.14 Some properties of automatic sequences. Instead of equivalent notions, we now look at implications of automata which are simple to check and thus good tools to prove transcendence results. Language theoretic viewpoint immediately suggests that long enough grammatical sentences should contain some parts, which can be pumped any number of times retaining grammatical property. For example, from ‘he is a friend of mine’ we can have ‘he is a friend of a friend of mine’ etc. Indeed, there are such pumping lemmas for many languages, as we will see and use below. Theorem 6.2.5. (Pumping lemma for finite automata/regular languages) Let S be an q-automatic set. Then there is N such that any word in S of length at least N can be written as juxtaposition xyz, with y being a non-empty word and length of xy not more than N and so that for all i , the juxtaposition xy i z is also in S . Proof. The number N is just (greater than) the number of states in the corresponding automaton. As you keep inputing the digits from left, we will get a repeat of states: the portion between the repeat is y, and can be clearly pumped.  Here are some results [Co72, E74, A87, AS03] about restrictions on densities, gaps and asymptotic behavior of automatic sequences: Theorem 6.2.6. Let S D fni g be a q-automatic set, with ni an increasing sequence. Define its maximum growth rate to be lim sup ni C1 =ni , its natural density P to be lim sn =n, where sx WD jfni  xgj, and its logarithmic density to be lim. 1=ni /= log.n/, where the sum is over ni  n. Then (1) The maximum growth rate is finite. (2) Either the maximum growth rate is more than one, or limsup .ni C1 ni / < 1, and these are mutually exclusive.

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(3) The logarithmic density exists and if the natural density exists, then both are the same and rational. If the natural density is zero, then either there is an integer d  1 and a real number s with 0 < s < 1 such that 0 < lim inf sx =.x s logd 1 x/ < lim sup sx =.x s logd 1 x/ < 1; or there are integers d  1, m  2 and rational c > 0 such that sx is asymptotic to c.logm x/d 1 . (4) Consider the characteristic sequence  of S , namely the sequence .n/ which takes values one or zero according as n is in S or not. There is c such that the number ˇ.n/ of distinct blocks of given length n occurring in this sequence of zero-ones is at most cn. The number ˇ.n/ is at least n, for .n/ not ultimately periodic. (5) There is a subsequence n0i of ni such that n0i C1 =n0i ! q d . More precisely, there are non-negative integers a, b > 0, c, d > 0 such that for all positive integers n, aq nd C b.q nd P  1/=.q d  1/ C c 2 S . (6) The series Nk t k 2 QŒŒt, where Nk is number of elements in S of k digits, is a rational function in Q.t/. (7) If lim sup sn = log.n/ is infinite, then lim inf .ni C1  ni / < 1 (i.e. small gaps infinitely often). If the natural density is zero, then maximum growth rate is more than one (i.e. large gaps infinitely often).

6.3 Sketches of proofs We now sketch proofs of some parts of Theorems 6.2.4 (which implies Theorem 6.2.1), 6.2.6, as well as of some other facts mentioned above. For full proofs or parts we do not cover, we refer to several available treatments such as [HU79, S85, AS03, CKMR80, Co69, Co72, E74, F67, A87, BHMV94]. 6.3.1 Ideas connecting parts of Theorem 6.2.4. (i) implies (ii): If we note that the base q expansion of q k nCr is obtained from that of n by just appending the expansion of r (with leading zeros, if necessary, to make it of size k), this is immediate, since there are only finitely many possible maps ˇ W S ! S and any fq k nCr is of the form O.ˇ.ns0//. (Equivalence of regular languages with (ii) is known as Myhill-Nerode theorem in language theory perspective.) (vii): Let V be the vector space over Fq .t/ generated by monomials in P (ii) implies fq k nCr t n . Then V is finite dimensional with sV V , so s satisfies its characteristic polynomial. (vii) P implies (ii): P For 0  r < q, define Cr (twisted Cartier operators) by Cr . fn x n / D fq nCr x n . Considering the vector space over Fq generated P i by the roots of the polynomial satisfied by s, we can assume that kiD0 ai s q D 0, Pq1 r q with a0 ¤ 0. Using g D and Cr .g q h/ D gCr .h/, rD0 x .Cr .g//

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we see that fh 2 Fq ..x// W h D

k X

i

hi .s=a0 /q ; hi 2 Fq Œx; deg hi  max.deg a0 ; deg ai a0q

i 2

/g

i D0

is a finite set containing s and stable under Cr ’s. (ii) implies (i): If there are m subsequences fn.i / with fn.1/ D fn say, put S WD / D fn.k/ . Define fs0 WD ˛1 ;    ; ˛m g. Define a digit action by, r˛i WD ˛k if fq.inCr O.˛i / WD fn , if n ˛1 D ˛i with n being the base q expansion of n written in the reverse order. Equivalence of (i) and (vi): Given a q-automaton .S; s0 ; T; O/, construct q-substitution with same start state and image map and with substitution function f .s/ D T .0; s/T .1; s/    T .q  1; s/. Conversely, given a q-substitution, we define the corresponding automaton similarly with the transition function T .j; s/ being the .j C1/-th letter in the word f .s/. It is easy to see that the image of the fixed point starting with s0 represents the sequence O.ns0 /. Equivalence of (i) and (v): We just mention that it is done constructively, by induction on the number of operations C;  and concatenation, by constructing an automaton, and in the other direction, by induction on the number of edges in the nondeterministic automaton, by building the regular expression from regular expressions obtained (by induction) by four non-deterministic automata obtained by erasing an edge from p to q say, and by using initial s0 , final state set A replaced by .s0 ; A/, .s0 ; fpg/, .q; fpg/, .q; A/ in the four cases. For equivalence of (vii) and (viii), due to Furstenberg, see [F67] and also [T04, 11.1]. This was one of the first result, and was used in [C79] to prove equivalence of (i) and (vii), whereas [CKMR80] gave a direct proof. For (iii), see [BHMV94]. For (iv), see [HU79, 9.1]. Remark 6.3.1. A given algebraic series can be generated by several different automata. Even if you restrict to ‘smallest’ automaton, there is no simple way to connect basic numerical data on both sides, for example genus, degree, height of a series versus number of states, or invariants of corresponding labeled graph. Following the proofs above, rough bounds on the degree and height in terms of the size of the automaton, and rough bounds on the size in terms of the degree and height (together) can be given. 6.3.2 Parts of Theorem 6.2.6. Part (4): We use the substitution equivalence. Let q m1  n < q m , and j q m  i < .j C 1/q m . Then I.si    si Cn1 / is a sub-block of I.f m .sj sj C1 //, and depends only on sj sj C1 (at most N 2 possibilities where N is the cardinality of S ) and i  j q m (at most q m < q n possibilities). This proves the first part, with c D qN 2 . Now ˇ is non-decreasing function, thus for the least k such that ˇ.k/ < k, we have ˇ.k/ D ˇ.k  1/ D k  1. Thus, for j  k  1, I.sj / is uniquely determined by the .k  1/-block immediately preceding it. This implies eventual periodicity. Part (5) is nothing but the pumping lemma above read in base q. It also implies (1) showing, in fact, that the maximum growth rate  q d , in the notation of (5).

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Alternately, note that if s is algebraic of degree d , then truncation approximation with Liouville theorem shows that growth rate cannot be bigger than d . Now we prove the first part of (7): let u be smallest such that every symbol occurring at least once in sq u sq u C1    occurs infinitely often there. If q u  i < q uC1 , and k  1, such that 1 occurs at least twice in I.f k .si //, then since si occurs infinitely often, this block occurs infinitely often, and hence gaps of size at most q k occur infinitely often. Otherwise, each q uCkC1 - image block can contain 1 at most q k  .k C 1/.q uC1  q u / times and thus sn = log.n/  .q  1/q k = log.q/ < 1. Now we explain ideas and techniques coming in the proof of (3). These density and asymptotics results come through study of powers and eigenvalues of incidence matrix: Let N be the cardinality of S as usual, and consider square matrix .mij / of size N , with mij being the number of occurrences of the j -th entry of S in f applied to the i -th entry, so that corresponding entry in the k-th power of the matrix denotes the same number with f replaced by f k . Now the matrix M obtained by multiplying the above matrix by 1=q, giving proportions is stochastic matrix with all row sums being one and all entries non-negative. Powers and eigenvalues of such matrices are described by Perron–Frobenius theory. In particular, there is h such that M h has 1 as the largest eigenvalue and it is simple. Then M hk tends to a limit, say M1 , which can be described by rational operations and density (if the limits exist) can also be expressed in terms of entries rationally, because of interpretation given above. Hence the density, if it exists, is rational. The other asymptotics results mentioned depend on how entries of powers grow and hence on finer analysis of eigenvalues and Jordan blocks, and we just refer to [Co72] or [AS03, Chap. 8]. For Part (6) Consider automaton with N states and output 1 on subset Sf . If M D .mss 0 / is N  NPmatrix with mss 0 being the number of digits taking s to s 0 . Then .I  tM /1 D t n M n has entries rational functions in t and ss 0 -th entry being number of words of length n taking s to s 0 . Let X be a row vector with 1 at the place s0 and 0 otherwise, Y be a column vector with 1 at places corresponding to Sf and 0 otherwise. Then the series in (6) is just X.I  tM /1 Y , and thus rational. Part (6) is due to Chomsky and Schützenberger, who also gave a general version with series in noncommuting variables (words giving monomials in the alphabet and summing over all the words in the language), in which context rationality is equivalent to regular languages. Note that when we specialize to all variables being equal, we get the series in (6). They also showed that unambiguous (only one way to derive from grammar) context-free languages give algebraic series (in different perspective than we are considering). See [SS78] for details. We see part (2) as follows. If for each n, ns0 is a state from which a final (accepting) state can be reached, then since maximum lengths of such paths are bounded, say by k, for any n, there is 0  r < q k such that nq k C r is accepted, and thus limsup .ni C1  ni /  2q k . Otherwise, let m be such that ms0 is a state from which accepting state cannot be reached. Then there is no ni between mq n and .m C 1/q n , for any n, so that the maximum growth rate is at least .m C 1/=m > 1. It is clear that the two statements are mutually exclusive. For (7), we refer to [Co72].

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6.4 Applications to function field arithmetic We will focus on how and which automata techniques were used in these applications, leaving full detailed definitions of the objects, statements of the theorems and detailed proofs to the references provided. 6.4.1 Special values of gamma function. The gamma function  W Zp ! Fq ..1=t// associated with Fq Œt by Carlitz and Goss is defined by Y X .n C 1/ D ….n/ D .Di =t deg Di /ni ; n D ni q i ; 0  ni < q; where Di WD

iY 1

i

k

.t q  t q / 2 Fq Œt

kD0

is the product of all monic polynomials of degree i . The author had proved functional equations and related some special values to periods of Drinfeld modules, given an analog of Chowla–Selberg formula. Combined with Thiery and Jing Yu’s transcendence results [Thi92, Y92] on periods, the known transcendence results (for fractions with only a few denominators) were exactly parallel [T04, pa. 334] in this case and in the number field case. In [T96], the automata method, namely (ii) of Theorem 6.2.4, was used to show that  value at any proper fraction in Zp , with some restrictions on the numerator is transcendental. Using logarithmic derivatives (which reduce modulo p the exponents in the monomials occurring in the products and causing troubles and restrictions on numerators with their size, and thus removing size problem), Allouche [A96] proved the transcendence of all values at proper fractions. Author [A96, T98, T04] then used the same trick combined with his earlier work to show that all the monomials in gamma values at fractions that were not earlier shown by him to be algebraic were, in fact, transcendental. Soon afterwards, it was shown [Mf-Y97] that any ….n/ is transcendental, for n 2 Zp  Z0 . I will now give a quick sketch of this very nice proof in [Mf-Y97]. Note that if s is algebraic non-zero, s 0 WD ds=dt (and thus s 0 =s) is also algebraic, as one can see directly, or by applying (1) by Theorem 6.2.3. Thus, as mentioned before, using logarithmic derivatives, products are turned into sums and it is enough to prove that P qj if the sequence nj 2 Fq is not ultimately zero, then nj =.tP  t/ is transcendental c.m/t m1 and using over Fq .t/. This is then achieved by writing the series as (ii) of Theorem 6.2.4 to conclude by showing that there are infinitely many distinct subsequences cr .m/ D c.q r m C 1/. In fact, one shows by direct manipulation using elementary number theory of divisibility arguments that ca ¤ cb if a > b and na and nb are non-zero by showing that these sequences differ at their .q h  2/-th term, where h is the least positive integer s dividing a, but not dividing b such that ns ¤ 0. See [T04, Cha.4, 11] and the references quoted above for more details and motivations and comparison with classical case. 6.4.2 Tate multiplicative period for elliptic curves. Mahler–Manin conjecture asserted transcendence (over the field of definition) of the fundamental multiplicative

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period (traditionally denoted by q not to be confused with our usage in this paper) of Tate elliptic curve Gm =q Z ) over (complex or p-adic) local field. Voloch proved [V94] analog over finite characteristic local field of Laurent series using Igusa towers and cohomology. When he lectured on this at University of Arizona, the author could give automata style proof [T94] as follows. By transcendence degree considerations (and algebraic identities between coefficients) the question of transcendence of q over the field of coefficients, which are essentially ‘q-expansions’ of certain Eisenstein series, is equivalent to whether these expansions represent transcendental series over field of rational functions over q. By algebraic identities between modular forms this in turn P 2 can be deduced from the transcendence of theta series q n modular form. This is equivalent to the set of squares being not automatic, well-known fact to computer scientists, in fact, one of the first application of automata due to Richie, Büchi etc. For us it may be the easiest to deduce this from (2) or (7) of Theorem 6.2.6, since ratio of successive squares tends to one, but the gap between them tends to infinity. (See another proof in the next subsection.) The original conjecture was soon proved [BDGP96] by Mahler method. It had a nice application to BSD conjecture case studied by Mazur, Tate and Teitelbaum. See the papers quoted or [T04, Cha. 11] for more details on this. 6.4.3 q-Expansions of Eisenstein and fake Eisenstein series. In [AT99], more direct (more automatic!) proof of the transcendence of q was given, by noticing that coefficients of the expansions of eisenstein series are given by arithmetic functions called higher divisor functions u .n/, and modulo p distribution of their values has been well-studied by number theorists. In particular, Rankin proved some asymptotics which does not fit in classification of automatic sequence asymptotics studied by Cobham, more precisely with (3) of Theorem 6.2.6, so that these series are transcendental. Another application [AT99] was showing that if .p/=.p 1/ is an irrational real number (here  is the Riemann Pzeta function, so this is expected but not known), and if p 1 divides u, then Su D u .n/q n 2 Fp ŒŒq is transcendental over Fp .q/. First note that when, as in this case, u is even, the expansions are no longer connected to Eisenstein series (which have even weight and odd u), thus we call them fake Eisenstein series. Again, Rankin’s result gives in this case the natural density as rational multiple of this zeta values ratio, and thus proof just consists of quoting that together with (3) of Theorem 6.2.6. The result is amusing concluding transcendence of a finite characteristic Laurent series representing (fake) modular form from irrationality of a real number representing (ratio of) zeta value! Soon afterwards, Yazdani [Yaz01], using stronger automata criterion (ii) of Theorem 6.2.4 showed the transcendence unconditionally and dropping the condition on u. 6.4.4 Special values of v-adic gamma. For a monic irreducible polynomial v of Fq Œt, Goss defined v-adic gamma function v W Zp ! Fq Œtv by Y X .Di;v /ni ; n D ni q i ; 0  ni < q; v .n C 1/ D …v .n/ D

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where Di;v WD .Di =Di deg v /v ordv .Di =Di deg v / 2 Fq Œt is the product of monic polynomials not divisible by v and of degree i . The author had proved functional equations and analog of Gross-Koblitz formula and conjectured which monomials in values at fractions would be transcendental. It was proved in [T98] in the special case of fractions and in [WY02] in the general case, with a simpler proof based on results of [Mf-Y97], that for v a prime of degree one, for n 2 Zp , …v .n/ is transcendental, if and only if the digit sequence nj is not ultimately constant. (The only if part was proved earlier by the author.) Using translation automorphisms, it is enough to prove the claim for v D t. Again one turns products into sums by using logarithmic derivatives and is led to prove P j transcendence of the series .nj  nj C1 /=..1=t/q  .1=t// and one applies the result of [Mf-Y97] quoted above. Similar transcendence results are not known for higher degree v. See [T04, Cha.4, 11] and the references quoted above for more details and motivations and comparison with classical case. 6.4.5 Algebraicity criterion for hypergeometric functions. We now quote a generalization (due independently to Sharif and Woodcock, Harase [SW88, Ha88, A89]) of Christol’s theorem, or rather of equivalence (ii) of Theorem 6.2.4, to function fields over any field (not necessarily finite) of characteristic p and just mention its recent application [TWYZ11, TWYZ09] to characterization of all parameters for which the hypergeometric function of [T04, 6.5(a)] is an algebraic function. P Theorem 6.4.1. Let F be a field of characteristic p. The series fn t n 2 F ŒŒt is algebraic over F .t/ if and only if the F -vector space generated by the sequences 1=p k n ! fnCk , as k runs through positive integers, is of finite dimension. 6.4.6 Carlitz periods, logarithm and zeta values. We have only looked at applications which are (or were when introduced) new results. For other automata applications in Drinfeld module context for Carlitz period, Carlitz zeta values and logarithm values, as well as many others outside this context, we refer to papers by Allouche, Berthé and Yao [A87, A90, AS03, B92, B93, B94, B95, Ya97] and references there.

6.5 Comparison with other tools When one tries to prove transcendence results for naturally occurring quantities in function field arithmetic, say from the theory of Drinfeld modules, t-motives or varieties, there are several tools available. When automata method applies, which is surprisingly often, it leads to quick direct proofs. But when the other methods such

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as functional equations, Mahler method, period methods using algebraic groups and motives etc. apply, they usually give more general and structurally more satisfying results. For example, the transcendence result for appropriate gamma monomials at fractions mentioned above has been now generalized [CPTY10] to complete algebraic independence results between appropriate gamma as well as Carlitz zeta values, by using transcendence techniques of Anderson, Brownawell, Papanikolas covered in Brownawell’s and other lectures of this Banff workshop. Using author’s generalizations for gamma functions to other function fields and his results connecting them to periods of Drinfeld modules together with Jing Yu’s results about transcendence of such periods prove transcendence of some special gamma values but for any function field. As for the method of Diophantine approximation is considered, it is interesting to note that very often, in function field case, it is much easier to obtain faster approximations than for the number field counterparts of the quantities, but at the same time, because of the failure of Roth theorem analog, it is harder in function field case to conclude transcendence from such fast approximations! As we have seen above, automata leads to equations of type handled in Mahler’s P i method, such as ai .x/f .x p / D 0, which becomes algebraic equation in charactersitic p, as we can take the exponent out to get powers of f . We refer to Pellarin’s lectures at this workshop for more on Mahler method. The applicability of the automata method is somewhat limited so that we do not know how to generalize it to get similar results for quantities occurring in the context of higher genus function fields, or to get strong algebraic independence results. For example, period methods have finally, not only caught up with automata methods for transcendence of gamma values at fractions, but have also provided much stronger full independence results. On the other hand, some results such as Mendès France-Yao result on transcendence of gamma values at p-adic integers which are non-fractions and v-adic gamma results, as well as refined transcendence results are still only obtained by automata method and not by other methods. Same can be said for transcendence results for quantities not strongly related to geometry, such as ‘wrong weight’ Eisenstein series discussed above. Many of these function field results were proved by different methods by Denis, Hellagoauarch, de Mathan, Yao etc. References and comparison of these methods can be found in [FKdM00, Ya09]. In the next section, we discuss another big strength of automata methods. We end this section with a challenge: it is well-known that the power series for the Artin–Hasse exponential ! n 1 X Y xp exp .1  x n /.n/=n D n p 0 n6 0 mod p

has coefficients in Zp . I wonder which method will settle the open question about transcendence over Fp .x/ of its reduction modulo p.

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6.6 Refined transcendence classification based on strength of computers 6.6.1 Computational classification with algebraic properties. We now briefly explain computational classification [BT98] with good algebraic properties and give an illustration of refined transcendence classification of some important Laurent series. We saw above how various ways of thinking of automata have helped giving transcendence proofs by completely different methods when usual methods do not apply. Now usual numbers/Laurent series coming up in number theory and geometry are computable (already a small countable subclass) and like automata, computability has various incarnations studied by various viewpoints, such as Turing machines, languages generated by unrestricted grammar, recursive function theory, Post systems, Church’s lambda calculus etc. Computer scientists, logicians, linguists have also studied intermediate strength classes (e.g., Chomsky hierarchy) and often converged to the same notions. So we examined [BT98] these robust classes from computational, series perspective as we have been studying above and found that many of these have good algebraic properties, such as forming a field, a field algebraically closed in Laurent series etc.; in addition to closure, logical properties, such as closure under union, concatenation, complementation etc., explored before. The algebraic properties allow you to move by algebraic operations the problem about one series to another series which might be more convenient to deal by these generalized automata tools. We will give an example below. For example, the context-sensitive languages form a field, and are equivalent to Turing machine which uses work space at most linear in input size. Context-free languages are equivalent to ‘pushdown automata’, but have very weak algebraic properties. On the other hand, Turing machines which take polynomial space to operate etc. form fields algebraically closed in Laurent series. For proofs, precise definitions, various equivalences and algebraic and closure properties, we refer to [BT98] and only sketch here some applications. 6.6.2 Refined transcendence of  by language tools. Let Q be the fundamental period of Carlitz module for Fq Œt, so that  WD t q=.q1/ Q is a Laurent series. By result of Wade in 1941, it is known to be transcendental, thus non-automatic. (For a direct automata proof, see [A90]). We show that it (or rather its reciprocal) is not contextfree (which gives, in particular, a language theoretic proof of its transcendence), but is context-sensitive. The tools here are language theoretic closure properties, moving to convenient series by algebraic properties and getting contradiction by pumping lemma for context-free languages. Here Pis a part of the argument illustrating this dealing with series with coefficients fn D q j 1jn 1 2 Fq . Just dividing n by q j  1’s, one at a time, in a linear space, and reusing it, we see the series to be context-sensitive, but if it were context-free, u intersection with regular $ .q  1/    .q  1/ is also context-free. Qlanguage fq  1gQ u Now cq u 1 D d.u/ D .ui C 1/ for u D pi i . Hence, the subset of these q n  1’s where d.n/ D 2 (odd respectively), is also context-free. By pumping, the only digits

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being q  1, we get same value (modulo p) of d on some arithmetic progression and are led to a contradiction by elementary number theoretic arguments. 6.6.3 Refined transcendence of e, # . Using computational and language tools, we [BT98] show that Carlitz analog of e, (known to be non-automatic by Wade 1941) is context-sensitive and theta series or set of squares (explained to be non-automatic above) is context-sensitive (even in logarithmic space under GRH), but (for q D 2) not context-free. 6.6.4 Algebraic dependence. We also give a computational criterion for algebraic independence, but have not found any application to natural quantities. Algebraic properties mentioned above and theta or Eisenstein calculations above show that modular forms expansions are in polynomial space class. It was suggested in [T04, pa. 369] that construction of non-periods can probably be done by computational methods, because periods are probably in polynomial or exponential space class. Mahler put all complex numbers in four classes of A; S; T; U numbers, with property that algebraically dependent transcendental numbers belong to the same class. We have divided countable class of computable numbers into infinitely many finer classes with this property, but do not have a good diophantine approximation theoretic description yet. For some applications of automata tools in diophantine approximation questions, we refer to [T03]. Remark 6.6.1. In these equivalences, the role of generalization in substitution viewpoint is not yet clear. For example, what do context-free and context-sensitive correspond to on substitution side, and what do non-uniform substitutions (different letters can go to strings with different lengths) correspond to in other viewpoints? For some other interesting open questions, see [BT98, pa. 816].

6.7 Beyond function field real numbers Automata method immediately applies to Puiseux series, which are Laurent series in t 1=k (for some k), which is an algebraic quantity. But unlike the characteristic zero case, the Puiseux series field no longer gives the algebraic closure of the Laurent series field, for example, as Chevalley pointed out, in characteristic p, the polynomial x p  x  t 1 has no root in Puiseux series field. But as Abhyankar noticed, it has P 1=pi a root t in the (algebraically closed) field of ‘generalized fractional power series’ (considered first by Hahn and studied in detail in this context by Huang) of P the form i 2S fi t i , where S Q is a well-ordered subset such that for some m the elements of mS have denominators powers of p. Kedlaya [K01] described the algebraic closure of Fq ..t// as subfield of this field and described [K06] the algebraic closure of Fq .t/ in it by adapting Christol’s algebraicity criterion to these type of series by considering automata with radix point, a special symbol sk , so that one

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considers a string of symbols s1    sk1 sk skC1    sn , with s1    sk1 is integral part and skC1    sn is the fractional part. We refer to [K01, K06] and corrections published by Kedlaya in 2017 for the detailed definitions, statements and proofs. I do not know yet any naturally occurring such generalized series, which is not a Puiseux series, to have a nice application of this generalization.

6.8 Strong characteristic dependence for algebraicity and real numbers 6.8.1 Finite characteristic. No carry over of power series expansions, finite possibilities of coefficients and Fq -linearity of q-power map makes function field case amenable to combinatorial description of automata. But in fact, even in function field case, this algebraicity description is very strongly dependent on characteristic as the following theorem by Cobham (whose combinatorial proof is a ‘challenge to algebraists’ according to Eilenberg [E74]) shows. Theorem 6.8.1. Non-periodic sequence cannot be m-automatic and n-automatic, if m; n > 1 are multiplicatively independent (e.g., if m; n are powers of distinct primes). P Corollary 6.8.2. If s D m2M t m is irrational algebraic in one finite characteristic, then it is transcendental in all other finite characteristics. 6.8.2 Characteristic zero. Natural question is what happens for similar expansions in characteristic zero. Corresponding power series is transcendental over Q.t/ as reduction shows. But how about related p-adic, real numbers? Despite earlier claimed proofs in the literature, the following theorem was only recently proved [ABL04] by Adamczewski, Bugeaud, Luca, making very nice use of Schmidt’s subspace theorem and automata equivalences. (Note that naive analogs of this theorem and of its consequence Roth’s theorem, fail in finite characteristic.) We sketch an important special case. For words W in alphabet, say S or F , we denote by w D jW j its length, and for x > 0, we put W x WD W bxc W 0 , where W 0 prefix of W of length d.x  bxc/we. p-automatic non-periodic sequence taking Theorem 6.8.3. Let p be a prime. If fn isP n values in F D f0; 1;    ; p  1g (so that P P fn t n is algebraic over Fp .t/), then  WD n fn =p 2 R and p-adic number  WD fn p are transcendental over Q. Proof. In the notation of 6.2.10, fn corresponds to p-substitution .f; I /, with S of cardinality N , and a fixed point u D u1 u2    whose image is the sequence fn . Write a prefix of length N C 1 of u as W1 aW2 aW3 , with a 2 S and Wi words (possibly empty) over S . Put Un D I.f n .W1 //, Vn D I.f n .aW2 //. Put r D 1 C 1=N . Then (1) Un Vnr is a prefix of non-eventually periodic sequence fn , (2) un =vn  c WD N 1, (3) vn is increasing sequence.

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.n/

Put fk WD fk , if k  un C rvn and WD fkCvn , if k  un C vn . For every n, it gives ultimately periodic sequence of preperiod Un and period Vn . Put n WD P .n/ k fk =p , so there is pn such that n D pn =.pun .pvn  1// and j  n j < p.un Crvn / , by (1). Consider six linear forms in x; y; z with algebraic coefficients: L1 D x C y C z; L2 D y; L3 D z; L01 D x; L02 D y; L03 D z: The forms Li (L0i respectively) are linearly independent over Q. Put Xn D .pun Cvn , pun ; pn /. Evaluated on Xn , we have jL1 j1
0 such that Y

jLi .Xn /j1 jL0i .Xn /jp 0. It should have been mentioned that ni is increasing sequence and thus S is infinite. (iii) Pa. 349, last para. It should be added to ‘turns out to be a trivial monomial’ that ‘after a preliminary reduction as in Thm. 4.6.4, which changes it by a rational function’. (iv) Pa. 353, first paragraph of 11.4 is misplaced and should be dropped.

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Acknowledgements. This article is based on the mini-course the author gave at ETH, Zurich in 2006 and the lecture at BIRS, Banff in September 2009. I would like to thank both the institutions, Richard Pink and the organisers of this conference. I would also like to thank my collaborators Jean-Paul Allouche, and Robert Beals, from whom I learned many things. I am grateful to the referee for a very careful reading and suggestions.

References [ABL04] B. Adamczewski, Y. Bugeaud, and F. Luca, Sur la complexité des nombres algébriques, C. R. Acad. Sci. Paris. Sér. I 339 (2004), 11-14. [A87]

J.-P. Allouche, Automates finis en théorie des nombres, Expo. Math. 5 (1987), 239-266.

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J.-P. Allouche, Note sur un article de Sharif et Woodcock, Sém. Théor. Nombres Bordeaux (2) 1(1) (1989), 163-187.

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J.-P. Allouche, Sur la transcendance de la série formelle …, Sém. Théor. Nombres Bordeaux (2) 2(1) (1990), 103-117.

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J.-P. Allouche, Transcendence of the Carlitz-Goss gamma function at rational arguments. J. Number Theory 60 (1996), 318-328.

[AS03] J.-P. Allouche and J. Shallit, Automatic sequences. Theory, Applications, Generalizations, Cambridge University Press, Cambridge, 2003. [AT99] J.-P. Allouche and D. S. Thakur, Automata and transcendence of the Tate period in finite characteristic. Proc. Amer. Math. Soc. 127 (1999), 1309–1312. [BDGP96] K. Barre-Sirieix, G. Diaz, F. Gramain, and G. Philibert, Une preuve de la conjecture de Mahler-Manin. Invent. Math. 124 (1996), 1-9. [BT98] R. Beals and D. S. Thakur, Computational classification of numbers and algebraic properties. Internat. Math. Res. Notices 15 (1998), 799–818. [B92]

V. Berthé, De nouvelles preuves “automatiques” de transcendance pour la fonction zêta de Carlitz. Journées Arithmétiques, 1991 (Genève), Astérisque 209 (1992), 159–168.

[B93]

V. Berthé, Fonction  de Carlitz et automates. J. Théor. Nombres Bordeaux 5 (1993), 53– 77.

[B94]

V. Berthé, Automates et valeurs de transcendance du logarithme de Carlitz. Acta Arith. 66 (1994), 369–390.

[B95]

V. Berthé, Combinaisons linéaires de .s/=…s sur Fq .x/, pour 1  s  q  2. J. Number Theory 53 (1995), 272–299.

[BHMV94] V. Bruyére, G. Hansel, C. Michaux, and R. Villemaire, Logic and p-recognizable sets of integers. Bull. Belg. Math. Soc. 1 (1994) 191-238. [CPTY10] C.-Y. Chang, M. Papanikolas, D. Thakur, and J. Yu, Algebraic independence of arithmetic gamma values and Carlitz zeta values. Adv. Math. 223 (2010) 1137–1154. [C79]

G. Christol, Ensembles presque périodiques k-reconnaissables. Theoret. Comput. Sci. 9 (1979), 141-145.

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[CKMR80] G. Christol, T. Kamae, M. Mendès France, and G. Rauzy, Suites algebriques, automates et substitutions. Bull. Soc. Math. France 108 (1980), 401-419. [Co69] A. Cobham, On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3 (1969), 186-192. [Co72] A. Cobham, Uniform tag sequences. Math. Systems Theory 6 (1972), 164-192. [E74]

S. Eilenberg, Automata, Languages and machines, vol. A, B, Academic Press, New York, 1974.

[FKdM00] J. Fresnel, M. Koskas, and B. de Mathan, Automata and transcendence in positive characteristic. J. Number Theory 80 (2000) 1-24. [F67]

H. Furstenberg, Algebraic functions over finite fields. J. Algebra 7 (1967), 271-277.

[G96]

D. Goss, Basic structures of function field arithmetic, Springer, Berlin, 1996.

[GHR92] D. Goss, D. Hayes, and M. Rosen (eds.), The Arithmetic of Function Fields, de Gruyter, Berlin, NY, 1992. [Ha88] T. Harase, Algebraic elements in formal power series rings, Israel J. Math. 63 (1988), 281-288. [H95]

Y. Hellegouarch, Une généralisation d’un critére de De Mathan. C. R. Acad. Sci. Paris, Sér I, 321 (1995), 677-680.

[HU79] J. Hopcroft and J. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley, Reading, Mass., 1979. [K01]

K. Kedlaya, The algebraic closure of the power series field in positive characteristic. Proc. Amer. Math. Soc. 129 (2001), 3461–3470.

[K06]

K. Kedlaya, Finite automata and algebraic extensions of function fields. J. Theor. Numbers Bordeaux 18(2) (2006), 379-420.

[Mf-Y97] M. Mendès France and J.-Y. Yao, Transcendence and the Carlitz-Goss gamma function. J. Number Theory 63 (1997), 396–402. [Ri63]

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[SW88] H. Sharif and C. Woodcock, Algebraic functions over a field of positive characteristic and Hadamard products. J. Lond. Math. Soc. 37 (1988), 395-403. [SS78] A. Salomaa and M. Soittola, Automata theoretic aspects of formal power series, Springer, New York, 1978. [T94]

D. Thakur, Automata-style proof of Voloch’s result on Transcendence. J. Number Theory 58 (1996), 60-63.

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D. Thakur, Transcendence of Gamma values for Fq ŒT . Ann. Math. 144 (1996), 181-188.

[T98]

D. Thakur, Automata and Transcendence, Number theory (Tiruchirapalli, 1996), Contemp. Math. 210 Amer. Math. Soc. Providence, RI (1998), 387–399.

[T03]

D. Thakur, Diophantine approximation in finite characteristic, Algebra, Arithmetic and Geometry with applications (Christensen et al., eds.), Springer, New York, 2003, 757-765.

[T04]

D. Thakur, Function Field Arithmetic, World Scientific, New Jersey, 2004.

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[TWYZ11] D. Thakur, Z. Wen, J.-Y. Yao, and L. Zhao, Transcendence in positive characteristic and special values of Hypergeometric functions. J. Reine Angew. Math. 657 (2011), 135171. [TWYZ09] D. Thakur, Z. Wen, J.-Y. Yao, and L. Zhao, Hypergeometric functions for function fields and transcendence, Comptes Rendus Acad. Sci. Paris, Ser. I 347 (2009), 467-472. [Thi92] A. Thiery, Indépendance algébrique des périodes et quasi-périodes d’un module de Drinfeld, in [GHR] (1992), 265-284. [V94]

J. Voloch, Transcendence of elliptic modular functions in characteristic p. J. Number Theory 58 (1996), 55-59.

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L. Wade, Certain quantities transcendental over GF .p n ; x/. Duke Math. J. 8 (1941), 701720.

[WY02] Z.-Y. Wen and J.-Y. Yao, Transcendence, automata and gamma functions for polynomial rings. Acta Arith. 101 (2002), 31-51. [Ya97]

J.-Y. Yao, Critères de non-automaticité et leurs applications. Acta Arith. 80 (1997), 237248.

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J.-Y. Yao, Some transcendental functions over function fields with positive characteristic. C. R. Acad. Sci. Paris, Sér. I 334 (2002), 939-943.

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J.-Y. Yao, Carlitz-Goss gamma function, Wade’s method, and transcendence. J. Reine Angew. Math. 579 (2005), 175-193.

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J.-Y. Yao, A transcendence criterion in positive characteristic and applications. C. R. Acad. Sci. Paris, Sér. I 343 (2006), 699-704.

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J. Yu, Transcendence in finite characteristic, in [GHR] (1992), 253-264.

Part B

Research articles

Chapter 7

Aspects of Iwasawa theory over function fields Andrea Bandini, Francesc Bars1 , and Ignazio Longhi2 We consider ZN p -extensions F of a global function field F and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety A defined over F , we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the p-adic L-function associated to A and F . We do the same, with characteristic ideals and p-adic L-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for Zdp -extensions). The final section provides more details for the cyclotomic ZN p -extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.

7.1 Introduction The main theme of number theory (and, in particular, of arithmetic geometry) is probably the study of representations of the Galois group Gal.Q=Q/ – or, more generally, of the absolute Galois group GF WD Gal.F sep =F / of some global field F . A basic philosophy (basically, part of the yoga of motives) is that any object of arithmetic interest is associated with a p-adic realization, which is a p-adic representation of GF with precise concrete properties (and to any p-adic representation with such properties should correspond an arithmetic object). Moreover from this p-adic representation one defines the L-function associated with the arithmetic object. Notice that the image of is isomorphic to a compact subgroup of GLn .Zp / for some n, hence it is a p-adic Lie group and the representation factors through Gal.F 0 =F /, where F 0 contains subextensions F and F 0 such that F =F 0 is a pro-p extension and F 0 =F and F 0 =F are finite. Iwasawa theory offers an effective way of dealing with various issues arising in this context, such as the variation of arithmetic structures in p-adic towers, and is one of the main tools currently available for the knowledge (and interpretation) of zeta values associated with an arithmetic object when F is a number field [35]. This theory constructs some elements, called p-adic L-functions, which provide a good understanding of both the zeta values and the properties of the arithmetic object. In 1 Supported 2 Supported

by MTM-2016-75980-P and MDM-2014-0445 by NSC 099-2811-M-002-096

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particular, the various forms of Iwasawa Main Conjecture provide a link between the zeta side and the arithmetic side. The prototype is given by the study of class groups in the cyclotomic extensions Q.pn /=Q. In this case the arithmetic side corresponds to a torsion ƒ-module X , where ƒ is an Iwasawa algebra related to Gal.Q.p1 /=Q/ and X measures the ppart of a limit of the class groups of the subfields Q.pn /. As for the zeta side, it is represented by a p-adic version of the Riemann zeta function, that is, an element  2 ƒ interpolating the zeta values. One finds that  generates the characteristic ideal of X . For another example of Iwasawa Main Conjecture, take E an elliptic curve over Q and p a prime of good ordinary reduction (in terms of arithmetic objects, here we deal with the Chow motive h1 .E/, as before with h0 .Q/). Then on the arithmetic side the torsion Iwasawa module X corresponds to the Pontrjagin dual of the Selmer group associated to E and the p-adic L-function of interest here is an element Lp .E/ in an Iwasawa algebra ƒ (that, as before, is Zp ŒŒGal.Q.p1 /=Q/) which interpolates twists of the L-function of E by Dirichlet characters of .Z=pn / . As before, conjecturally Lp .E/ should be the generator of the characteristic ideal of X . In both these cases, we had F D Q and F 0 D Q.p1 /. Of course there is no need for such a limitation and one can take as F 0 any p-adic extension of the global field F : for example one can deal with Znp -extensions of F . A more recent creation is non-commutative Iwasawa theory, which allows to deal with non-commutative padic Lie groups, as the ones appearing from non-CM elliptic curves (in particular, this may include the extensions where the p-adic realization of the arithmetic object factorizes). In most of these developments, the global field F was assumed to be a number field. The well-known analogy with function fields suggests that one should have an interesting Iwasawa theory also in the characteristic p side of arithmetic. So in the rest of this paper F will be a global function field, with char.F / D p and constant field FF . Observe that there is a rich and well-developed theory of cyclotomic extension for such an F , arising from Drinfeld modules: for a survey on its analogy with the cyclotomic theory over Q see [63]. We shall limit our discussion to abelian Galois extension of F . One has to notice that already with this assumption, an interesting new phenomenon appears: there are many more p-adic abelian extensions than in the number field case, since local groups of units are Zp -modules of infinite rank. So the natural analogue of the Zp -extension of Q is the maximal p-adic abelian extension F =F unramified outside a fixed place and we have  D Gal.F =F / ' ZN p . It follows that the ring Zp ŒŒ is not noetherian; consequently, there are some additional difficulties in dealing with ƒ-modules in this case. Our proposal is to see ƒ as a limit of noetherian rings and replace characteristic ideals by Fitting ideals when necessary. As for the motives originating the Iwasawa modules we want to study, we start considering abelian varieties over F and ask the same questions as in the number field case. Here the theory seems to be rich enough. In particular, various control theorems allow to define the algebraic side of the Iwasawa Main conjecture. As for

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the analytic part, we will sketch how a p-adic L-function can be defined for modular abelian varieties. Then we consider the Iwasawa theory of class groups of abelian extensions of F . This subject is older and more developed: the Iwasawa Main Conjecture for Znp extension was already proved by Crew in the 1980’s, by geometric techniques. We concentrate on ZN p -extensions, because they are the ones arising naturally in the cyclotomic theory; besides they are more naturally related to characteristic p L-functions (a brave new world where zeta values have found another, yet quite mysterious, life). The final section, which should be taken as a report on work in progress, provides some material for a more cyclotomic approach to the Main Conjecture. 7.1.1 Contents of the paper. In Sect. 7.2 we study the structure of Selmer groups associated with elliptic curves (and, more in general, with abelian varieties) and Zdp extensions of a global function field F . We use the different versions of control theorems available at present to show that the Pontrjagin duals of such groups are finitely generated (sometimes torsion) modules over the appropriate Iwasawa algebra. These results allow us to define characteristic and Fitting ideals for those duals. In Sect. 7.3, taking the Zdp -extensions as a filtration of a ZN p -extension F , we can use a limit argument to define a (pro-)Fitting ideal for the Pontrjagin dual of the Selmer group associated with F . This (pro-)Fitting ideal (or, better, one of its generators) can be considered as a worthy candidate for an algebraic L-function in this setting. In Sect. 7.4 we deal with the analytic counterpart, giving a brief description of the p-adic L-functions which have been defined (by various authors) for abelian varieties and the extensions F =F . Sections 7.3 and 7.4 should provide the ingredients for the formulation of an Iwasawa Main Conjecture in this setting. In Sect. 7.5 we move to the problem of class groups. We use some techniques of an (almost) unpublished work of Kueh, Lai and Tan to show that the characteristic ideals of the class groups of Zdp -subextensions of a cyclotomic ZN p -extension are generated by some Stickelberger element. Such a result can be extended to the whole ZN p -extension via a limit process because, at least under a certain assumption, the characteristic ideals behave well with respect to the inverse limit (as Stickelberger elements do). This provides a new approach to the Iwasawa Main Conjecture for class groups. At the end of Sect. 7.5 we briefly recall some results on what is known about class groups and characteristic p zeta values. Section 7.6 is perhaps the closest to the spirit of function field arithmetic. For simplicity we deal only with the Carlitz module. We study the Galois module of cyclotomic units by means of Coleman power series and show how it fits in an Iwasawa Main Conjecture. Finally we compute the image of cyclotomic units by Coates–Wiles homomorphisms: one gets special values of the Carlitz–Goss zeta function, a result which might provide some hints towards its interpolation3. This paper was first written in 2010 so it reflects the situation at the time. We have added a few references to more recent developments related to the theory presented 3 A different approach using a version of Iwasawa Main Conjecture for the cyclotomic Carlitz extension and leading to information on special values of the Carlitz–Goss zeta function is carried out in [1].

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here but have not attempted to include detailed descriptions of new results. A recent excellent source for the arithmetic of function fields is the book [14], in particular (since here we focus on Iwasawa theory) we mention the paper [18] which also covers the non-commutative approach. 7.1.2 Some notations. Given a field L, L will denote an algebraic closure and Lsep a separable closure; we shall use the shortening GL WD Gal.Lsep =L/. When L is (an algebraic extension of) a global field, Lv will be its completion at the place v, Ov the ring of integers of Lv and Fv the residue field. We are going to deal only with global fields of positive characteristic: so FL shall denote the constant field of L. As mentioned before, let F be a global field of characteristic p > 0, with field of constants FF of cardinality q. We also fix algebraic closures F and Fv for any place v of F , together with embeddings F ,! Fv , so to get restriction maps GFv ,! GF . All algebraic extensions of F (resp. Fv ) will be assumed to be contained in F (resp. Fv ). Script letters will denote infinite extensions of F . In particular, F shall always denote a Galois extension of F , ramified only at a finite set of places S and such that  WD Gal.F =H / is a free Zp -module, with H=F a finite subextension (to ease notations, in some sections we will just put H D F ); the associated Iwasawa algebra Q WD Zp ŒŒ. Q is ƒ WD Zp ŒŒ. We also put Q WD Gal.F =F / and ƒ The Pontrjagin dual of an abelian group A shall be denoted as A_ . Remark 7.1.1. Class field theory shows that, in contrast with the number field case, in the characteristic p setting Gal.F =F / (and hence ) can be very large indeed.  Actually, it is well known that for every place v the group of 1-units Ov;1 Fv (which is identified with the inertia subgroup of the maximal abelian extension unramified outside v) is isomorphic to a countable product of copies of Zp : hence there is no bound on the dimension of . Furthermore, the only Zfp i ni t e -extension of F which arises somewhat naturally is the arithmetic one F arit , i.e., the compositum of F with the maximal pro-p-extension of FF . This justifies our choice to concentrate on the case of a  of infinite rank: F shall mostly be the maximal abelian extension unramified outside S (often imposing some additional condition to make it disjoint from F arit ). We also recall that a Zp -extension of F can be ramified at infinitely many places [26, Remark 4]: hence our condition on S is a quite meaningful restriction.

7.2 Control theorems for abelian varieties 7.2.1 Selmer groups. Let A=F be an abelian variety, let AŒpn  be the group scheme of pn -torsion points and put AŒp1  WD lim AŒpn . Since we work in characteristic ! p we define the Selmer groups via flat cohomology of group schemes. For any finite algebraic extension L=F let XL WD Spec L and for any place v of L let Lv be the

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completion of L at v and XLv WD Spec Lv . Consider the local Kummer embedding

Lv W A.Lv / ˝ Qp =Zp ,! lim Hf1 l .XLv ; AŒpn / DW Hf1 l .XLv ; AŒp1 / : !

n

Definition 7.2.1. The p part of the Selmer group of A over L is defined as ( ) Y 1 1 1 1 SelA .L/p WD Ker Hf l .XL ; AŒp / ! Hf l .XLv ; AŒp /=I m Lv v

where the map is the product of the natural restrictions at all primes v of L. For an infinite algebraic extension L=F we define, as usual, the Selmer group SelA .L/p via the direct limit of the SelA .L/p for all the finite subextensions L of L. In this section we let Fd =F be a Zdp -extension (d < 1) with Galois group d and associated Iwasawa algebra ƒd . Our goal is to describe the structure of SelA .Fd /p (actually of its Pontrjagin dual) as a ƒd -module. The main step is a control theorem proved in [9] for the case of elliptic curves and in [59] in general, which will enable us to prove that S .Fd / WD SelA .Fd /_ p is a finitely generated (in some cases torsion) ƒd -module. The proof of the control theorem requires semi-stable reduction for A at the places which ramify in Fd =F : this is not a restrictive hypothesis thanks to the following (see [46, Lemma 2.1]) Lemma 7.2.2. Let F 0 =F be a finite Galois extension. Let Fd0 WD Fd F 0 and ƒ0d WD Zp ŒŒGal.Fd0 =F 0 /. Put A0 for the base change of A to F 0 . If S 0 WD SelA0 .Fd0 /_ p is a finitely generated (torsion) ƒ0d -module, then S is a finitely generated (torsion) ƒd -module. Proof. From the natural embeddings SelA .L/p ,! Hf1 l .XL ; AŒp1 / (any L) one gets a diagram between duals

H

1

Hf1 l .XFd0 ; AŒp1 /_

/ / S0

 Hf1 l .XFd ; AŒp1 /_

 // S



 / / S =I m S 0

.Gal.Fd0 =Fd /; AŒp1 .Fd0 //_

(where in the lower left corner one has the dual of a Galois cohomology group and the whole left side comes from the dual of the Hochschild–Serre spectral sequence). Obviously Fd0 =Fd is finite (since F 0 =F is) and AŒp1 .Fd0 / is cofinitely generated, hence H 1 .Gal.Fd0 =Fd /; AŒp1 .Fd0 //_ and S =I m S 0 are finite as well. Therefore S is a finitely generated (torsion) ƒ0d -module and the lemma follows from the fact that Gal.Fd0 =F 0 / is open in d . 

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7.2.2 Elliptic curves. Let E=F be an elliptic curve, non-isotrivial (i.e., j.E/ 62 FF ) and having good ordinary or split multiplicative reduction at all the places which ramify in Fd =F (assuming there is no ramified prime of supersingular reduction one just needs a finite extension of F to achieve this). We remark that for such curves EŒp1 .F sep / is finite (it is an easy consequence of the fact that the pn -torsion points for n 0 generate inseparable extensions of F , see for example [11, Proposition 3.8]). For any finite subextension F  L  Fd we put L WD Gal.Fd =L/ and consider the natural restriction map aL W SelE .L/p ! SelE .Fd /pL : The following theorem summarizes results of [9] and [59]. Theorem 7.2.3. In the above setting assume that Fd =F is unramified outside a finite set of places of F and that E has good ordinary or split multiplicative reduction at all ramified places. Then Ker aL is finite (of order bounded independently of L) and Coker aL is a cofinitely generated Zp -module (of bounded corank if d D 1). Moreover if all places of bad reduction for E are unramified in Fd =F , then Coker aL is finite as well (of bounded order if d D 1). Proof. Let Fw be the completion of Fd at w and, to shorten notations, let ) ( Y 1 1 1 1 G .XL / WD I m Hf l .XL ; EŒp / ! Hf l .XLv ; EŒp /=I m Lv v

(analogous definition for G .XFd /). Consider the diagram SelE .L/p  



/ H 1 .XL ; EŒp 1 / fl

aL

SelE .Fd /pL 





bL

/ H 1 .XF ; EŒp 1 /L d fl

/ / G .XL / cL



/ G .XF /L : d

Since XFd =XL is a Galois covering the Hochschild–Serre spectral sequence (see [44, III.2.21 a),b) and III.1.17 d)]) yields Ker bL D H 1 .L ; EŒp1 .Fd // and Coker bL  H 2 .L ; EŒp1 .Fd // (where the H i ’s are Galois cohomology groups). Since EŒp1 .Fd / is finite it is easy to see that jKer bL j 6 jEŒp1 .Fd /jd

and jCoker bL j 6 jEŒp1 .Fd /j

d.d 1/ 2

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([9, Lemma 4.1]). By the snake lemma the inequality on the left is enough to prove the first statement of the theorem, i.e., jKer aL j 6 jKer bL j 6 jEŒp1 .Fd /jd which is finite and bounded independently of L (actually one can also use the upper bound jEŒp1 .F sep /jd which makes it independent of Fd as well). We are left with Ker cL : for any place w of Fd dividing v define dw W Hf1 l .XLv ; EŒp1 /=I m Lv ! Hf1 l .XFw ; EŒp1 /=I m w : Then Ker cL ,!

Y\

Ker dw

v wjv

(note also that Ker dw really depends on v and not on w). If v totally splits in Fd =L then dw is obviously an isomorphism. Therefore from now on we study the Ker dw ’s only for primes which are not totally split. Moreover, because of the following diagram coming from the Kummer exact sequence Hf1 l .XLv ; EŒp1 /=I m Lv  



/ H 1 .XL ; E/ v fl

dw

Hf1 l .XFw ; EŒp1 /=I m Fw 





hw

/ H 1 .XF ; E/ ; w fl

one has an injection Ker dw ,! Ker hw ' H 1 .Lv ; E.Fw // which allows us to focus on H 1 .Lv ; E.Fw //. be the maximal un7.2.2.1 Places of good reduction. If v is unramified let Lunr v ramified extension of Lv . Then using the inflation map and [45, Proposition I.3.8] one has unr H 1 .Lv ; E.Fw // ,! H 1 .Gal.Lunr v =Lv /; E.Lv // D 0 :

b be the formal group associated with E and, for any place v, let Ev be the Let E reduced curve. From the exact sequence b O / ,! E.Lv /  Ev .Fv / E. Lv and the surjectivity of E.Lv /  Ev .Fv / (see [44, Exercise I.4.13]), one gets b Ow // ,! H 1 .Lv ; E.Fw // ! H 1 .Lv ; Ev .Fw // : H 1 .Lv ; E.

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Using the Tate local duality (see [45, Theorem III.7.8 and Appendix C]) and a careful study of the pn -torsion points in inseparable extensions of Lv , Tan proves that b Ow // is isomorphic to the Pontrjagin dual of Ev Œp1 .Fv / (see [59, TheH 1 .Lv ; E. orem 2]). Hence jH 1 .Lv ; E.Fw //j 6 jH 1 .Lv ; Ev .Fw //j jEv Œp1 .Fv /j : Finally let L0v be the maximal unramified extension of Lv contained in Fw (so that, in particular, Fw D FL0v ) and let L0v WD Gal.Fw =L0v / be the inertia subgroup of Lv . The Hochschild–Serre sequence reads as H 1 .Lv = L0v ; Ev .FL0v // ,!H 1 .Lv ; Ev .Fw // # H 1 .L0v ; Ev .Fw //

Lv =L0

v

! H 2 .Lv = L0v ; Ev .FL0v // : Now Lv = L0v can be trivial, finite cyclic or Zp and in any case Lang’s theorem yields H i .Lv = L0v ; Ev .FL0v // D 0

i D 1; 2 :

Therefore 0

H 1 .Lv ; Ev .Fw // ' H 1 .L0v ; Ev .FL0v //Gal.Lv =Lv / ' H 1 .L0v ; Ev .Fv // and eventually jH 1 .Lv ; E.Fw //j 6 jH 1 .L0v ; Ev .Fv //j jEv Œp1 .Fv /j 0

6 jEv Œp1 .Fv /jd.Lv /C1 6 jEv Œp1 .Fv /jd C1 where d.L0v / WD rankZp L0v 6 d and the middle bound (independent from Fd ) comes from [9, Lemma 4.1]. We are left with the finitely many primes of bad reduction. 7.2.2.2 Places of bad reduction. By our hypothesis at these primes we have the Tate curve exact sequence Z ,! Fw  E.Fw / : qE;v

For any subfield K of Fw =Lv one has a Galois equivariant isomorphism  Z qE;v ' TK K  =OK

(coming from E0 .Fw / ,! E.Fw /  TFw ), where TK is a finite cyclic group of order ordK .j.E// arising from the group of connected components (see, for example, [9, Lemma 4.9 and Remark 4.10]). Therefore H 1 .Lv ; E.Fw // D lim H 1 .K ; E.K// ,! lim H 1 .K ; TK / ' lim.TK /d.K/ ; p !

K

!

K

!

K

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where .TK /p is the p-part of TK and d.K/ D rankZp K . If v is unramified then all TK ’s are isomorphic to TLv and d.K/ D d.Lv / D 1; hence jH 1 .Lv ; E.Fw //j D j.TLv /p j D j.TF /p j ; where  is the prime of F lying below v (note that the bound is again independent of Fd ). If v is ramified then taking Galois cohomology in the Tate curve exact sequence, one finds Z Ker hw D H 1 .Lv ; E.Fw // ,! H 2 .Lv ; qE;v /

where the injectivity comes from Hilbert 90. Z Since Lv acts trivially on qE;v , one finds that Z / ' H 2 .Lv ; Z/ ' .Labv /_ Ker hw ,! H 2 .Lv ; qE;v

' .Qp =Zp /d.Lv / ,! .Qp =Zp /d ; i.e., Ker hw is a cofinitely generated Zp -module. This completes the proof of Theorem 7.2.3 for the general case. If all ramified primes are of good reduction, then the Ker dw ’s are finite so Coker aL is finite as well. In particular its order is bounded by Y Y jEv Œp1 .FLv /jd C1  pordp .ordv .j.E ///  jCoker bL j : v ram; good

v inert; bad

If d D 1 the last term is trivial and, in a Zp -extension, the (finitely many) places which are ramified or inert of bad reduction admit only a finite number of places above them. If d > 2 such bound is not independent of L because the number of terms in the products is unbounded. In the case of ramified primes of bad reduction the bound for the corank is similar.  Both SelE .Fd /p and its Pontrjagin dual are modules over the ring ƒd in a natural way. An easy consequence of the previous theorem and of Nakayama’s Lemma (see [6]) is the following (see for example [9, Corollaries 4.8 and 4.13]) Corollary 7.2.4. In the setting of the previous theorem, let S .Fd / be the Pontrjagin dual of SelE .Fd /p . Then S .Fd / is a finitely generated ƒd -module. Moreover if all ramified primes are of good reduction and SelE .F /p is finite, then S .Fd / is ƒd torsion. Remarks 7.2.5. 1. We recall that, thanks to Lemma 7.2.2, the last corollary holds when there are no ramified primes of supersingular reduction for E (when such a prime is present the finitely generated statement does not hold anymore, see [60, Theorem 3.10]).

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2. The ramified primes of split multiplicative reduction are the only obstacle to the finiteness of Coker aL and this somehow reflects the number field situation as described in [42, section II.6], where the authors defined an extended MordellWeil group whose rank is rank E.F / C N (where N is the number of primes of split multiplicative reduction and dividing p, i.e., totally ramified in the cyclotomic Zp -extension they work with) to deal with the phenomenon of exceptional zeroes. 3. A different way of having finite kernels and cokernels (and then, at least in some cases, torsion modules S .Fd /) consists in a modified version of the Selmer groups. Examples with trivial or no conditions at all at the ramified primes of bad reduction are described in [9, Theorem 4.12]. 4. The available constructions of a p-adic L-function associated to ZN p -extensions require the presence of a totally ramified prime p of split multiplicative reduction for E. Thus the theorem applies to that setting but, unfortunately, it only provides finitely generated ƒd -modules S .Fd /. 5. The paper [38] describes an example of an elliptic curve E=F and a Zp -extension F1 such that S .F1 / is a non-torsion ƒ1 -module (the last section of [38] verifies the vanishing of the p-adic L-function attached to these E and F , in accordance with the Iwasawa Main Conjecture). 7.2.3 Higher dimensional abelian varieties. We go back to the general case of an abelian variety A=F . For any finite subextension L=F of Fd we put L WD Gal.Fd =L/ and consider the natural restriction map aL W SelA .L/p ! SelA .Fd /pL : The following theorem summarizes results of [59]. Theorem 7.2.6. In the above setting assume that Fd =F is unramified outside a finite set of places of F and that A has good ordinary or split multiplicative reduction at all ramified places. Then Ker aL is finite (of bounded order if d D 1) and Coker aL is a cofinitely generated Zp -module. Moreover if all places of bad reduction for A are unramified in Fd =F , then Coker aL is finite as well (of bounded order if d D 1). Proof. We use the same notations and diagrams as in Theorem 7.2.3, substituting the abelian variety A for the elliptic curve E. The Hochschild–Serre spectral sequence yields Ker bL D H 1 .L ; AŒp1 .Fd // and Coker bL  H 2 .L ; AŒp1 .Fd // : Let L0  Fd be the extension generated by AŒp1 .Fd /. The extension L0 =L is everywhere unramified (for the places of good reduction see [59, Lemma 2.5.1 (b)], for the other places note that the pn -torsion points come from the pn -th roots of the periods provided by the Mumford parametrization so they generate an inseparable extension while Fd =F is separable): hence Gal.L0 =L/ '   Zep where  is finite

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and e D 0 or 1. Let  be a topological generator of Zep in Gal.L0 =L/ (if e D 0 then  D 1) and let L1 be its fixed field. Then AŒp1 .Fd /< > D AŒp1 .L1 / is finite and we can apply [10, Lemma 3.4] (with b the maximum between jAŒp1 .L1 /j and jAŒp1 .Fd /=.AŒp1 .Fd //d iv j) to get jKer bL j 6 b d

and jCoker bL j 6 b

d.d 1/ 2

:

By the snake lemma the inequality on the left is enough to prove that Ker aL is finite (for the bounded order in the case d D 1 see [59, Corollary 3.2.4]). The bounds for the Ker dw ’s are a direct generalization of the ones provided for the case of the elliptic curve so we give just a few details. Recall the embedding Ker dw ,! Ker hw ' H 1 .Lv ; E.Fw // : 7.2.3.1 Places of good reduction. If v is unramified then unr H 1 .Lv ; A.Fw // ,! H 1 .Gal.Lunr v =Lv /; A.Lv // D 0 :

If v is ramified one has an exact sequence (as above) A.OFw // ,! H 1 .Lv ; A.Fw // ! H 1 .Lv ; Av .FFw // : H 1 .Lv ; b By [59, Theorem 2] A.OFw // ' Bv Œp1 .FLv / H 1 .Lv ; b (where B is the dual variety of A) and the last group has the same order of Av Œp1 .FLv /. Using Lang’s theorem as in 7.2.2.1, one finds jH 1 .Lv ; A.Fw //j 6 jAv Œp1 .FLv /jd C1 : 7.2.3.2 Places of bad reduction. If v is unramified let 0;v .A/ be the group of connected components of the Néron model of A at v. Then, again by [45, Proposition I.3.8], unr H 1 .Lv ; A.Fw // ,! H 1 .Gal.Lunr v =Lv /; A.Lv //

' H 1 .Gal.Lunr v =Lv /; 0;v .A// unr

and the last group has order bounded by j0;v .A/Gal.Lv =Lv / j. If v is ramified one just uses Mumford’s parametrization with a period lattice v Lv      Lv (genus A times) to prove that H 1 .Lv ; A.Fw // is cofinitely generated as in 7.2.2.2.  We end this section with the analogue of Corollary 7.2.4.

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Corollary 7.2.7. In the setting of the previous theorem, let S .Fd / be the Pontrjagin dual of SelA .Fd /p . Then S .Fd / is a finitely generated ƒd -module. Moreover if all ramified primes are of good reduction and SelA .F /p is finite, then S .Fd / is ƒd torsion. Remark 7.2.8. In [46, Theorem 1.7], by means of crystalline and syntomic cohomology, Ochiai and Trihan prove a stronger result. Indeed they can show that the dual of the Selmer group is always torsion, with no restriction on the abelian variety A=F , but only in the case of the arithmetic extension F arit =F , which lies outside the scope of the present paper. Moreover in the case of a (not necessarily commutative) prop-extension containing F arit , they prove that the dual of the Selmer group is finitely generated (for a precise statement, see [46], in particular Theorem 1.9) 4 .

7.3 ƒ-Modules and Fitting ideals We need a few more notations. For any Zdp -extension Fd , let .Fd / WD Gal.Fd =F / and ƒ.Fd / WD Zp ŒŒ.Fd / (the Iwasawa algebra) with augmentation ideal I Fd (or simply d , ƒd and Id if the extension Fd is clearly fixed). For any d > e and any Zdpe -extension Fd =Fe , we put .Fd =Fe / WD Gal.Fd =Fe /, F ƒ.Fd =Fe / WD Zp ŒŒ.Fd =Fe / and IFed as the augmentation ideal of ƒ.Fd =Fe /, F i.e., the kernel of the canonical projection Fed W ƒ.Fd / ! ƒ.Fe / (whenever possible all these will be abbreviated to ed , ƒde , Ied and ed respectively). Recall that ƒ.Fd / is (noncanonically) isomorphic to Zp ŒŒT1 ; : : : ; Td . A finitely generated torsion ƒ.Fd /-module is said to be pseudo-null if its annihilator ideal has height at least 2. If M is a finitely generated torsion ƒ.Fd /-module, then there is a pseudo-isomorphism (i.e., a morphism with pseudo-null kernel and cokernel) M ƒ.Fd /

n M

e

ƒ.Fd /=.gi i / ;

i D1

where the gi ’s are irreducible elements of ƒ.Fd / (determined up to an element of ƒ.Fd / ) and n and the ei ’s are uniquely determined by M (see e.g. [15, VII.4.4 Theorem 5]). Definition 7.3.1. In the above setting the characteristic ideal of M is 8 ˆ ! if M is not torsion < 0 n Y : Chƒ.Fd / .M / WD e otherwise gi i ˆ : i D1

arit can be found in [60] and [8] generalization of the torsion statement for Zd p -extensions containing F (where one also finds an approach to the results of Section 7.3 in terms of characteristic ideals) and in [12] for a generic p-adic Lie extension. 4A

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Let Z be a finitely generated ƒ.Fd /-module and let '

ƒ.Fd /a !ƒ.Fd /b  Z be a presentation where the map ' can be represented by a b a matrix ˆ with entries in ƒ.Fd /. Definition 7.3.2. In the above setting the Fitting ideal of Z is 8 0 if a < b ˆ < the ideal generated by all the Fitt ƒ.Fd / .Z/ WD if a > b ˆ : determinants of the b  b minors of the matrix ˆ

:

Let F =F be a ZN p -extension with Galois group . Our goal is to define an ideal in ƒ WD Zp ŒŒ associated with S , the Pontrjagin dual of SelA .F /p . For this we consider all the Zdp -extensions Fd =F (d 2 N) contained in F (which we call Zp -finite extensions). Then F D [Fd and ƒ D lim ƒ.Fd / WD lim Zp ŒŒGal.Fd =F /. The classical characteristic ideal does not behave well (in general) with respect to inverse limits (because the inverse limit of pseudo-null modules is not necessarily pseudonull). For the Fitting ideal, using the basic properties described in the Appendix of [43], we have the following Lemma 7.3.3. Let Fd Fe be an inclusion of multiple Zp -extensions, e > d . Assume that AŒp1 .F / D 0 or that Fitt ƒ.Fd / .S .Fd // is principal. Then FFde .Fitt ƒ.Fe / .S .Fe ///  Fittƒ.Fd / .S .Fd // : e

Proof. Consider the natural map ade W SelA .Fd /p ! SelA .Fe /pd and dualize to get

S .Fe /=Ide S .Fe / ! S .Fd /  .Ker ade /_ where (as in Theorem 7.2.6) Ker ade ,! H 1 .de ; AŒp1 .Fe // is finite. If AŒp1 .F / D 0 then .Ker ade /_ D 0 and de .Fittƒe .S .Fe /// D Fittƒd .S .Fe /=Ide S .Fe //  Fittƒd .S .Fd // : If .Ker ade /_ ¤ 0 one has Fittƒd .S .Fe /=Ide S .Fe //Fittƒd ..Ker ade /_ /  Fitt ƒd .S .Fd // : The Fitting ideal of a finitely generated torsion module contains a power of its annihilator, so let 1 ; 2 be two relatively prime elements of Fitt ƒd ..Ker ade /_ / and #d

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a generator of Fitt ƒd .S .Fd //. Then #d divides 1 ˛ and 2 ˛ for any ˛ 2 Fitt ƒd .S .Fe /=Ide S .Fe // (it holds, in the obvious sense, even for #d D 0). Hence de .Fittƒe .S .Fe /// D Fitt ƒd .S .Fe /=Ide S .Fe //  Fitt ƒd .S .Fd // :



Remark 7.3.4. In the case A D E an elliptic curve, the hypothesis EŒp1 .F / D 0 is satisfied if j.E/ 62 .F  /p , i.e., when the curve is admissible (in the sense of [11]); n nC1 n and one can work over the field F p . The otherwise j.E/ 2 .F  /p  .F  /p other hypothesis is satisfied in general by elementary ƒ.Fd /-modules or by modules having a presentation with the same number of generators and relations. Let Fd be the canonical projection from ƒ to ƒ.Fd / with kernel IFd . Then the previous lemma shows that, as Fd varies, the .Fd /1 .Fitt ƒ.Fd / .S .Fd /// form an inverse system of ideals in ƒ. Definition 7.3.5. Assume that AŒp1 .F / D 0 or that Fittƒ.Fd / .S .Fd // is principal for every Fd . Define

e

Fitt ƒ .S .F // WD lim.Fd /1 .Fittƒ.Fd / .S .Fd /// 

Fd

to be the pro-Fitting ideal of S .F / (the Pontrjagin dual of SelE .F /p ). Proposition 7.3.6. Assume that AŒp1 .F / D 0 or that Fitt ƒ.Fd / .S .Fd // is princi.F / pal for every Fd . If rankZp SelA .F1 /p 1 > 1 for any Zp -extension F1 =F contained in F , then Fittƒ .S .F // I (where I is the augmentation ideal of ƒ).

e

Proof. Recall that I Fd is the augmentation ideal of ƒ.Fd /, that is, the kernel of .F /  Fd W ƒ.Fd / ! Zp . By hypothesis Fitt Zp ..SelA .F1 /p 1 /_ / D 0. Thus, since F1 / _ / D S .F1 /=I F1 S .F1 /, Zp D ƒ.F1 /=I F1 and .SelA .F1 /. p F1 / _ 0 D Fitt Zp ..SelA .F1 /. / / D  F1 .Fittƒ.F1 / .S .F1 /// ; p

i.e., Fitt ƒ.F1 / .S .F1 // Ker  F1 D I F1 . For any Zdp -extension Fd take a Zp -extension F1 contained in Fd . Then, by Lemma 7.3.3, F

F1d .Fitt ƒ.Fd / .S .Fd ///  Fitt ƒ.F1 / .S .F1 // I F1 : F

Note that  Fd D  F1 ı F1d . Therefore F

Fittƒ.Fd / .S .Fd // I Fd ” F1d .Fittƒ.Fd / .S .Fd /// I F1 ; i.e., Fitt ƒ.Fd / .S .Fd // I Fd for any Zp -finite extension Fd . Finally \ \ Fitt ƒ .S .F // WD .Fd /1 .Fittƒ.Fd / .S .Fd /// .Fd /1 .I Fd / I : 

e

Fd

Fd

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389

Remark 7.3.7. From the exact sequence aF1

F1 / Ker aF1 ,! SelA .F /p ! SelA .F1 /.  Coker aF1 p

and the fact that Ker aF1 is finite one immediately finds out that the hypothesis on F1 / rankZp SelA .F1 /. is satisfied if rankZ A.F / > 1 or rankZp Coker aF1 > 1. As p already noted, when there is a totally ramified prime of split multiplicative reduction, the second option is very likely to happen. In the number field case, when F is the cyclotomic Zp -extension and, in some cases, SelE .F /_ p is known to be a torsion module, this is equivalent to saying that T divides a generator of the characteristic ideal of SelE .F /_ p (i.e., there is an exceptional zero). Note that all the available constructions of p-adic L-function for our setting require a ramified place of split multiplicative reduction and they are all known to belong to I .

7.4 Modular abelian varieties of GL2 -type The previous sections show how to define the algebraic (p-adic) L-function associated with F =F and an abelian variety A=F under quite general conditions. On the analytic side there is, of course, the complex Hasse–Weil L-function L.A=F; s/, so the problem becomes to relate it to some element in an Iwasawa algebra. In this section we will sketch how this can be done at least in some cases; in order to keep the paper to a reasonable length, the treatment here will be very brief. We say that the abelian variety A=F is of GL2 -type if there is a number field K such that ŒK W Q D dim A and K embeds into EndF .A/ ˝ Q. In particular, this implies that for any l ¤ p the Tate module Tl A yields a representation of GF in GL2 .K ˝ Ql ). The analogous definition for A=Q can be found in [53], where it is proved that Serre’s conjecture implies that every simple abelian variety of GL2 type is isogenous to a simple factor of a modular Jacobian. We are going to see that a similar result holds at least partially in our function field setting. 7.4.1 Automorphic forms. Let AF denote the ring of adeles of F . By automorphic form for GL2 we shall mean a function f W GL2 .AF / ! C which factors through GL2 .F /nGL2 .AF /=K, where K is some open compact subgroup of GL2 .AF /; furthermore, f is cuspidal if it satisfies some additional technical condition (essentially, the annihilation of some Fourier coefficients). A classical procedure associates with such an f a Dirichlet sum L.f; s/: see e.g. [65, Chapters II and III]. The C-vector spaces of automorphic and cuspidal forms provide representations of GL2 .AF /. Besides, they have a natural Q-structure: in particular, the decomposition of the space of cuspidal forms in irreducible representations of GL2 .AF / holds over Q (and hence over any algebraically closed field of characteristic zero); see e.g. the discussion in [52, page 218]. We also recall that every irreducible automorphic representation  of GL2 .AF / is a restricted tensor product ˝0v v , v varying

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over the places of F : the v ’s are representations of GL2 .Fv / and they are called local factors of . Let WF denote the Weil group of F : it is the subgroup of GF consisting of elements whose restriction to FF is an integer power of the Frobenius. By a fundamental result of Jacquet and Langlands [32, Theorem 12.2], a two-dimensional representation of WF corresponds to a cuspidal representation if the associated L-function and its twists by characters of WF are entire functions bounded in vertical strips (see also [65]). Let A=F be an abelian variety of GL2 -type. Recall that L.A=F; s/ is the Lfunction associated with the compatible system of l-adic representations of GF arising from the Tate modules Tl A, as l varies among primes different from p. Theorems of Grothendieck and Deligne show that under certain assumptions L.A=F; s/ and all its twists are polynomials in q s satisfying the conditions of [32, Theorem 12.2] (see [23, §9] for precise statements). In particular all elliptic curves are obviously of GL2 -type and one finds that L.A=F; s/ D L.f; s/ for some cusp form f when A is a non-isotrivial elliptic curve. 7.4.2 Drinfeld modular curves. From now on we fix a place 1. The main source for this section is Drinfeld’s original paper [24]. Here we just recall that for any divisor n of F with support disjoint from 1 there exists a projective curve M.n/ (the Drinfeld modular curve) and that these curves form a projective system. Hence one can consider the Galois representation H WD lim Het1 .M.n/  F sep ; Ql /: !

Besides, the moduli interpretation of the curves M.n/ allows to define an action of GL2 .Af / on H (where Af denotes the adeles of F without the component at 1). Let …1 be the set of those cuspidal representations having the special representation of GL2 .F1 / (i.e., the Steinberg representation) as local factor at 1. Drinfeld’s reciprocity law [24, Theorem 2] (which realizes part of the Langlands correspondence for GL2 over F ) attaches to any  2 …1 a compatible system of two-dimensional Galois representations  ./l W GF ! GL2 .Ql / by establishing an isomorphism of GL2 .Af /  GF -modules M .˝0v¤1 v / ˝  ./l : (7.4.1) H ' 2…1

As  ./l one obtains all l-adic representations of GF satisfying certain properties: for a precise list (and a thorough introduction to all this subject) see [52]. Here we just remark the following requirement: the restriction of  ./l to GF1 has to be the special l-adic Galois representation sp1 . For example, the representation originated from the Tate module Tl E of an elliptic curve E=F satisfies this condition if and only if E has split multiplicative reduction at 1. The Galois representations appearing in (7.4.1) are exactly those arising from the Tate module of the Jacobian of some M.n/. We call modular those abelian varieties isogenous to some factor of Jac.M.n//. Hence we see that a necessary condition

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391

for abelian varieties of GL2 -type to be modular is that their reduction at 1 is a split torus. The paper [25] provides a careful construction of Jacobians of Drinfeld modular curves by means of rigid analytic geometry. 7.4.3 The p-adic L-functions. For any ring R let Meas.P1 .Fv /; R/ denote the Rvalued measures on the topological space P1 .Fv / (that is, finitely additive functions on compact open subsets of P1 .Fv /) and Meas0 .P1 .Fv /; R/ the subset of measures of total mass 0. A key ingredient in the proof of (7.4.1) is the identification of the space of R-valued cusp forms with direct sums of certain subspaces of Meas0 .P1 .F1 /; R/ (for more precise statements, see [52, §2] and [25, §4]). Therefore we can associate with any modular abelian variety A some measure A on P1 .F1 /; this fact can be exploited to construct elements (our p-adic L-functions) in Iwasawa algebras in the following way. Let K be a quadratic algebra over F : an embedding of K into the F -algebra of  2  2 matrices M2 .F / gives rise to an action of the group G WD .K ˝ F1 / =F1 on 1 the P GL2 .F1 /-homogeneous space P .F1 /. Class field theory permits to relate G to a subgroup  of Q D Gal.F =F /, where F is a certain extension of F (depending on K) ramified only above 1. Then the pull-back of A to G yields a measure on ; this is enough because Meas.; R/ is canonically identified with R ˝ ƒ (and Meas0 .; R/ with the augmentation ideal). Various instances of the construction just sketched are explained in [41] for the case when A is an elliptic curve: here one can take R D Z. Similar ideas were used in Pál’s thesis [48], where there is also an interpolation formula relating the element in ZŒŒ so obtained to special values of the complex L-function. One should also mention [50] for another construction of p-adic L-function, providing an interpolation property for one of the cases studied in [41]. Notice that in all these cases the p-adic L-function is, more or less tautologically, in the augmentation ideal. A different approach had been previously suggested by Tan [58]: starting with cuspidal automorphic forms, he defines elements in certain group algebras and proves an interpolation formula [58, Proposition 2]. Furthermore, if the cusp form is “wellbehaved” his modular elements form a projective system and originate an element in an Iwasawa algebra of the kind considered in the present paper: in particular, this holds for non-isotrivial elliptic curves having split multiplicative reduction. In the case of an elliptic curve over Fq .T / Teitelbaum [62] re-expressed Tan’s work in terms of modular symbols (along the lines of [42]); in [30] it is shown how this last method can be related to the “quadratic algebra” techniques sketched above. Thus for a modular abelian variety A=F we can define both a Fitting ideal and a p-adic L-function: it is natural to expect that an Iwasawa Main Conjecture should hold, i.e., that the Fitting ideal should be generated by the p-adic L-function. Remark 7.4.1. In the cases considered in this paper (involving a modular abelian variety and a geometric extension of the function field) the Iwasawa Main Conjecture

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is still wide open. However, recently there has been some interesting progress in two related settings. First, one can take A to be an isotrivial abelian variety (notice that [58, page 308] defines modular elements also for an isotrivial elliptic curve). Thanks to an observation of Ki-Seng Tan, in this setting one can use an algebraic functional equation (proved in [38]) to reduce the Main Conjecture to the one for class groups, which is already known to hold (as it will be explained in the next section). On this basis, the Iwasawa Main Conjecture for constant ordinary abelian varieties is proved in [39] when F is a Zdp -extension. From here, it is easy to deal with the case of a ZN pextension as well (see [8]). On the subject of the algebraic functional equation, we also mention [49] (treating the `-adic case). Second, one can take as F the maximal arithmetic pro-p-extension of F , i.e., .p/ .p/ F D F arit D F F.p/ F , where FF is the subfield of FF defined by Gal.FF =FF / ' Zp (note that this is the setting of [46, Theorem 1.7]). In this case Trihan has obtained a proof of the Iwasawa Main Conjecture, by techniques of syntomic cohomology. No assumption on the abelian variety A=F is needed: the relevant p-adic L-function is defined by means of cohomology and it interpolates the Hasse–Weil L-function (see [40]).

7.5 Class groups For any finite extension L=F , A.L/ will denote the p-part of the group of degree zero divisor classes of L; for any F 0 between F and F , we put A.F 0 / WD lim A.L/ as L runs among finite subextensions of F 0 =F (the limit being taken with respect to norm maps). The study of similar objects and their relations with zeta functions is an old subject and was the starting point for Iwasawa himself (see [63] for a quick summary). The goal of this section is to say something on what is known about Iwasawa Main Conjectures for class groups in our setting. 7.5.1 Crew’s work. A version of the Iwasawa Main Conjecture over global function fields was proved by R. Crew in [22]. His main tools are geometric: so he considers an affine curve X over a finite field of characteristic p (in the language of the present paper, F is the function field of X ) and a p-adic character of 1 .X /, that is, a continuous homomorphism W 1 .X / ! R , where R is a complete local noetherian ring of mixed characteristic, with maximal ideal m (notice that the Iwasawa algebras ƒd introduced in Sect. 7.2.1 above are rings of this kind). To such a are attached H. ; x/ 2 RŒx (the characteristic polynomial of the cohomology of a certain étale sheaf - see [21] for more explanation) and the L-function L. ; x/ 2 RŒŒx. The main theorem of [22] proves, by means of étale and crystalline cohomology, that the ratio L. ; x/=H. ; x/ is a unit in the m-adic completion of RŒx. An account of the geometric significance of this result (together with some of the necessary background) is provided by Crew himself in [21]; in [22, §3] he shows the following application

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to Iwasawa theory. Letting (in our notations) R be the Iwasawa algebra ƒ.Fd /, the special value L. ; 1/ can be seen to be a Stickelberger element (the definition will be recalled in Sect. 7.5.3 below). As for H. ; 1/, [21, Proposition 3.1] implies that it generates the characteristic ideal of the torsion ƒd -module lim A.L/_ , L varying among finite subextensions of Fd =F 5 . The Iwasawa Main Conjecture follows. Crew’s cohomological techniques are quite sophisticated. A more elementary approach was suggested by Kueh, Lai and Tan in [36] and [37] (and refined, with Burns’s contribution and different cohomological tools from [16], in [17]). In the next two sections we will give a brief account of this approach (and its consequences) in a particularly simple setting, related to Drinfeld–Hayes cyclotomic extensions (which will be the main topic of Sect. 7.6). 7.5.2 Characteristic ideals for class groups. In this section (which somehow parallels Sect. 7.3) we describe an algebraic object which can be associated to the inverse limit of class groups in a ZN p -extension F of a global function field F . Since our first goal is to use this “algebraic L-function” for the cyclotomic extension which will appear in Sect. 7.6.1, we make the following simplifying assumption. Assumption 7.5.1. There is only one ramified prime in F =F (call it p) and it is totally ramified (in particular this implies that F is disjoint from F arit ). We shall use some ideas of [36] which, in our setting, provide a quite elementary approach to the problem. We maintain the notations of Sect. 7.3: F =F is a ZN pextension with Galois group  and associated Iwasawa algebra ƒ with augmentation ideal I . For any d > 0 let Fd be a Zdp -extension of F contained in F , taken so that S Fd D F . For any finite extension L=F let M.L/ be the p-adic completion of the group of divisor classes D iv.L/=PL of L, i.e.,

M.L/ WD .L nIL =…v Ov / ˝ Zp where IL is the group of ideles of L. As before, when L=F is an infinite extension, we put M.L/ WD lim M.K/ as K runs among finite subextensions of L=F (the limit being taken with respect to norm maps). For two finite extensions L L0 F , the degree maps degL and degL0 fit into the commutative diagram (with exact rows)

A.L/  



/ M.L/

L NL 0

A.L0 / 





degL

L NL 0

/ M.L0 /

degL0

/ / Zp

(7.5.1)

 / / Zp ;

5 Note that in [21] our A.L/’s appear as Picard groups, so the natural functoriality yields A.L/ ! A.L0 / if L  L0 - that is, arrows are opposite to the ones we consider in this paper: hence Crew takes Pontrjagin duals and we don’t.

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where NLL0 denotes the norm and the vertical map on the right is multiplication by ŒFL W FL0  (the degree of the extension between the fields of constants). For an infinite extension L=F contained in F , taking projective limits (and recalling Assumption 7.5.1 above), one gets an exact sequence

A. L/ 



degL

/ M.L/

/ / Zp :

(7.5.2)

Remark 7.5.2. If one allows non-geometric extensions, then the degL map above becomes the zero map exactly when the Zp -extension F arit is contained in L. It is well known that M.Fd / is a finitely generated torsion ƒ.Fd /-module (see e.g. [28, Theorem 1]), so the same holds for A.Fd / as well. Moreover take any Zdp extension Fd of F contained in F : since our extension F =F is totally ramified at the prime p, for any Fd 1 Fd one has

M.Fd /=IFFdd1 M.Fd / ' M.Fd 1 /

(7.5.3)

(see for example [64, Lemma 13.15]). As in Sect. 7.3, to ease notations we will often F erase the F from the indices (for example IFdd1 will be denoted by Idd1 ), hoping that no confusion will arise. Consider the following diagram 

/ M.Fd /





A.Fd /  

1

A.Fd /

deg

1

/ M.Fd /

deg

/ / Zp 

(7.5.4)

1

/ / Zp

(where h i D Gal.Fd =Fd 1 / DW dd1 ; note also that the vertical map on the right is 0) and its snake lemma sequence

A.Fd /d 1  d

Zp o o



/ M.Fd /dd1

deg

M.Fd /=Idd1 M.Fd / o

deg

/ Zp

(7.5.5)



A.Fd /=Idd1 A.Fd / :

For d > 2 the ƒd -module Zp is pseudo-null, hence (7.5.2) yields Chƒd .M.Fd // D Chƒd .A.Fd //, and, using (7.5.3) and (7.5.5), one finds (for d > 3) Chƒd 1 .A.Fd /=Idd1 A.Fd // D Chƒd 1 .M.Fd /=Idd1 M.Fd // D Chƒd 1 .M.Fd 1 // D Chƒd 1 .A.Fd 1 // (7.5.6) (where all the modules involved are ƒd 1 -torsion modules).

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Let



N.Fd / ,! A.Fd / ! E.Fd /  R.Fd /

(7.5.7)

be the exact sequence coming from the structure theorem for ƒd -modules (see Sect. 7.3), where M E.Fd / WD ƒd =.fi;d / i

is an elementary module and N.Fd /, R.Fd / are pseudo-null. Let Chƒd .A.Fd // be the characteristic ideal of A.Fd /: we want to compare Chƒd 1 .A.Fd 1 // with F Fdd1 .Chƒd .A.Fd /// for some Fd 1 Fd and show that these characteristic ideals form an inverse system (in ƒ). Consider the module B.Fd / WD N.Fd / ˚ R.Fd /. We need the following hypothesis. Assumption 7.5.3. There is a choice of the pseudo-isomorphism in (7.5.7) and a splitting of the projection d  d 1 so that i) d D hd i ˚ d 1 ; ii) B.Fd / is a finitely generated torsion Zp ŒŒd 1 -module. As explained in [29] (see the remarks just before Lemma 3), for any Fd and one can find a subfield Fd 1 so that Assumption 7.5.3 holds. In order to ease notations, we put  D d , so that dd1 D h i. Lemma 7.5.4. With the above notations, one has   d Chƒd 1 .A.Fd /=Idd1 A.Fd // D dd1 Chƒd .A.Fd //  Chƒd 1 .A.Fd /d 1 / : Proof. We split the previous sequence in two by N.Fd / ,! A.Fd /  C.Fd / ; C.Fd / ,! E.Fd /  R.Fd / and consider the snake lemma sequences coming from the following diagrams   / A.Fd / / E.Fd / / / C.Fd / / / R.Fd / C.Fd /  N.Fd /  

1

N.Fd / 





1

/ A.Fd /



1

/ / C.Fd /



1

C.Fd / 





1

/ E.Fd /



1

/ / R.Fd / ;

(7.5.8) i.e., N.Fd /d 1  d



C.Fd /=Idd1 C.Fd / o o

/ A.Fd /dd1

A.Fd /=Idd1 A.Fd / o

/ C.Fd /dd1  N.Fd /=Idd1 N.Fd / (7.5.9)

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and C.Fd /d 1  d



R.Fd /=Idd1 R.Fd / o o

/ R.Fd /dd1

/ E.Fd /dd1

E.Fd /=Idd1 E.Fd / o

 C.Fd /=Idd1 C.Fd / : (7.5.10)

From (7.5.7) we get an exact sequence M A.Fd /=Idd1 A.Fd / ! ƒd =.  1; fi;d / ! R.Fd /=Idd1 R.Fd / i

where the last term is a torsion ƒd 1 -module. So is A.Fd 1  / for d > 3 and, by  (7.5.6), Chƒd 1 .A.Fd 1 // D Chƒd 1 A.Fd /=Idd1 A.Fd / . It follows that none of the fi;d ’s belongs to Idd1 . Therefore: d

1. the map   1 W E.Fd / ! E.Fd / has trivial kernel, i.e., E.Fd /d 1 D 0 so d that C.Fd /d 1 D 0 as well; 2. the characteristic ideal of the ƒd 1 -module E.Fd /=Idd1 E.Fd / is generated by the product of the fi;d ’s modulo the ideal Idd1 , hence it is obviously equal to dd1 .Chƒd .A.Fd ///. Moreover, from the fact that N.Fd / and R.Fd / are finitely generated torsion ƒd 1 modules6 and the multiplicativity of characteristic ideals, looking at the left (resp. right) vertical sequence of the first (resp. second) diagram in (7.5.8), one finds d

Chƒd 1 .N.Fd /d 1 / D Chƒd 1 .N.Fd /=Idd1 N.Fd // and d

Chƒd 1 .R.Fd /d 1 / D Chƒd 1 .R.Fd /=Idd1 R.Fd // : Hence from (7.5.9) one has Chƒd 1 .A.Fd /=Idd1 A.Fd //   d D Chƒd 1 C.Fd /=Idd1 C.Fd /  Chƒd 1 .N.Fd /d 1 /   d D Chƒd 1 C.Fd /=Idd1 C.Fd /  Chƒd 1 .A.Fd /d 1 / d

d

(where the last line comes from the isomorphism A.Fd /d 1 ' N.Fd /d 1 ). The sequence (7.5.10) provides the equality Chƒd 1 .C.Fd /=Idd1 C.Fd // D Chƒd 1 .E.Fd /=Idd1 E.Fd // D dd1 .Chƒd .A.Fd /// : 6 It might be worth to notice that this is the only point where we use the hypothesis that Assumption 7.5.3 holds.

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397

Therefore one concludes that   d Chƒd 1 .A.Fd /=Idd1 A.Fd // D dd1 Chƒd .A.Fd //  Chƒd 1 .A.Fd /d 1 / : (7.5.11)  d

Our next step is to prove that A.Fd /d 1 D 0 (actually it would be enough to prove that it is pseudo-null as a ƒd 1 -module). For this we need first a few lemmas. Lemma 7.5.5. Let G be a finite group and endow Zp with the trivial G-action. Then for any G-module M we have H i .G; M ˝ Zp / D H i .G; M / ˝ Zp for all i > 0. This result should be well-known. Since we were not able to find a suitable reference, here is a sketch of the proof. Proof. Let X be a G-module which has no torsion as an abelian group and put Y WD X ˝ Q. It is not hard to prove that Y G ˝ Zp D .Y ˝ Zp /G and it follows that the same holds for X , since X G ˝ Zp is a saturated submodule of X ˝ Zp . Applying this to the standard complex by means of which the H i .G; M / are defined, one can prove the equality in the case M has no torsion as an abelian group. The general case follows because any G-module is the quotient of two such modules.  Up to now we have mainly considered M.L/ as an Iwasawa module (for various L), now we focus on its interpretation as a group of divisor classes. Let L be a finite extension of F and recall that we defined M.L/ D .D iv.L/=PL / ˝ Zp . From the exact sequence FL ,! L  PL and the fact that jFL j is prime with p, one finds an isomorphism between L ˝ Zp and PL ˝ Zp . Hence we can (and will) identify the two. Lemma 7.5.6. For any finite Galois extension L=F , the map

D iv.L/Gal.L=F / ˝ Zp ! M.L/Gal.L=F / is surjective. Proof. The sequence L ˝ Zp ,! D iv.L/ ˝ Zp  M.L/

(7.5.12)

is exact because Zp is flat and jFL j is prime with p. The claim follows by taking the Gal.L=F /-cohomology of (7.5.12) and applying Lemma 7.5.5 and Hilbert 90. 

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For any finite subextension L of F =F , let pL be the unique prime lying above p. In the following lemma, we identify pL with its image in D iv.L/ ˝ Zp . Moreover for any element x 2 M.F / we let xL denote its image in M.L/ via the canonical norm map. Lemma 7.5.7. Let x 2 M.F / : then, for any L as above, xL is represented by a -invariant divisor supported in pL . Proof. For any L, let yL be the image of pL in M.L/. Since Zp yL is aclosed subset of M.L/, to prove the lemma it is enough to show that xL Cpn M.L/ \Zp yL ¤ ; for any n. For any finite Galois extension K=L we have the maps

K L W D iv.L/ ˝ Zp ! D iv.K/ ˝ Zp and NLK W D iv.K/ ˝ Zp ! D iv.L/ ˝ Zp induced (respectively) by the inclusion and the norm. For any divisor whose support is unramifed in K=L we have NLK . K L .D// D ŒK W LD : Also, Lemma 7.5.5 yields .D iv.K/ ˝ Zp /Gal.K=L/ D D iv.K/Gal.K=L/ ˝ Zp D K L .D iv.L/ ˝ Zp / (since in a Gal.K=L/-invariant divisor all places of K above a same place of L occur with the same multiplicity). Choose n and let K F be such that ŒK W L > pn . By Lemma 7.5.6, there exists a Gal.K=L/-invariant EK 2 D iv.K/ ˝ Zp having image xK . Write EK D DK C aK pK , where aK 2 Zp and DK has support disjoint from pK . Then DK is Galois invariant, so DK D K L .DL / and (using Assumption 7.5.1) NLK .EK / D ŒK W LDL C aK pL : Projecting into M.L/ we get xL 2 aK yL C pn M.L/ :



d

Corollary 7.5.8. A.Fd /d 1 D 0. Proof. Taking dd1 -invariants in (7.5.2) (with L D Fd ), one finds a similar sequence degF  d d / Zp : / M.Fd /dd1 A.Fd /d 1  (7.5.13) Lemma 7.5.7 holds, with exactly the same proof, also replacing F and  with Fd d and d . Therefore any x D .xL /L 2 M.Fd /d 1 can be represented by a sequence

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Iwasawa theory over function fields

.aL pL /L . Furthermore NLK .aK pK / D aL pL implies that the value aL is independent of L: call it a. Then   degFd .x/ D lim aL degL .pL / D a degF .p/ : d

Hence x 2 Ker.degFd / D A.Fd /d 1 only if a D 0.



Remark 7.5.9. The image of the degree map appearing in (7.5.13) is .deg p/Zp , so d degFd always provides an isomorphism between M.Fd /d 1 and Zp . Moreover, if p does not divide deg p, one has surjectivity as well. In this case, looking back at the sequence (7.5.5), one finds a short exact sequence

A.Fd /=Idd1 A.Fd / 



/ M.Fd /=I d M.Fd / deg d 1

/ / Zp :

From (7.5.1), by taking the limit with L and L0 varying respectively among subextensions of Fd and Fd 1 , one obtains a commutative diagram

M.Fd /=Idd1 M.Fd / 

deg

/ / Zp

N

M.Fd 1 /

degF

d 1

/ / Zp

where the map N is the isomorphism induced by the norm, i.e., the one appearing in (7.5.3). Together with the exact sequence (7.5.2) for L D Fd 1 , this shows that A.Fd /=Idd1 A.Fd / ' A.Fd 1 / (for any d > 1). From (7.5.11) one finally obtains Chƒd 1 .A.Fd 1 // D Chƒd 1 .A.Fd /=Idd1 A.Fd // D dd1 .Chƒd .A.Fd /// : (7.5.14) We remark that this equation holds for any Zp -extension Fd =Fd 1 satisfying Assumption 7.5.3. If the filtration fFd W d 2 Ng verifies that Assumption at any level d , then the inverse images of the Chƒ.Fd / .A.Fd // in ƒ (with respect to the canonical projections Fd W ƒ ! ƒ.Fd /) form an inverse system and we can define Definition 7.5.10. The pro-characteristic ideal of A.F / is f ƒ .A.F // WD lim.F /1 .Chƒ.F / .A.Fd /// : Ch d d 

Fd

Remark 7.5.11. Two questions naturally arise from the above definition: a. is there a filtration verifying Assumption 7.5.3 at any level d ? b. (assuming a has a positive answer) is the limit independent from the chosen filtration?

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In the next section we are going to show (in particular in (7.5.17) and Corollary 7.5.16) that there is an element # 2 ƒ, independent of the filtration and such that, for all Fd , its image in ƒ.Fd / generates Chƒ.Fd / .A.Fd //. Hence Question b has a positive answer (and presumably so does Question a) and we only needed (7.5.14) as a first step and a natural analogue of (7.5.16). Nevertheless we believe that these questions have some interest on their own and it would be nice to have a direct construction of a “good” filtration fFd W d 2 Ng based on a generalization of [29, Lemma 2]. Since our goal here is the Main Conjecture we do not pursue this subject further, but we hope to get back to it in a future paper. We also observe that Assumption 7.5.3 was used only in one passage in the proof of Lemma 7.5.4, as we evidentiated in a footnote. It might be easier to show that in that passage one does not need the finitely generated hypothesis: if so, Definition 7.5.10 would makes sense for all filtrations fFd gd 7 . Remark 7.5.12. We could have used Fitting ideals, just as we did in Sect. 7.3, to provide a more straightforward construction (there would have been no need for preparatory lemmas). But, since the goal is a Main Conjecture, the characteristic ideals, being principal, provide a better formulation. We indeed expect equality between Fitting and characteristic ideals in all the cases studied in this paper but, at present, are forced to distinguish between them (but see Remark 7.5.17). 7.5.3 Stickelberger elements. We shall briefly describe a relation between the characteristic ideal of the previous section and Stickelberger elements. The main results on those elements are due to A. Weil, P. Deligne and J. Tate and for all the details the reader can consult [61, Ch. V]. Let S be a finite set of places of F containing all places where the extension F =F ramifies; since we are interested in the case where F is substantially bigger than the arithmetic extension, we assume S ¤ ;. We consider also another non-empty finite set T of places of F such that S \ T D ;. For any place outside S let F rv be the Frobenius of v in  D Gal.F =F /. Let Y Y ‚F =F ;S;T .u/ WD .1  F rv q deg.v/ udeg.v/ / .1  F rv udeg.v/ /1 : (7.5.15) v2T

v62S

For any n 2 N there are only finitely many places of F withP degree n: hence we can expand (7.5.15) and consider ‚F =F ;S;T .u/ as a power series cn un 2 ZŒŒŒu. Moreover, it is clear that for any continuous character W  ! C the image .‚F =F ;S;T .q s // is the L-function of , relative to S and modified at T . For any subextension F L F , let LF W ZŒŒŒu ! ZŒGal.L=F /ŒŒu be the natural projection and define ‚L=F ;S;T .u/ WD LF .‚F =F ;S;T .u//: For L=F finite it is known (essentially by Weil’s work) that ‚L=F ;S;T .q s / is an element in the ring CŒGal.L=F /Œq s  (see [61, Ch. V, Proposition 2.15] for a proof): 7

This is exactly the approach taken in [7], providing a positive answer to question b.

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401

hence ‚L=F ;S;T .u/ 2 ZŒGal.L=F /Œu: It follows that the coefficients cn of ‚F =F ;S;T .u/ tend to zero in lim ZŒGal.L=F / DW ZŒŒ ƒ : Therefore we can define #F =F ;S;T WD ‚F =F ;S;T .1/ 2 ƒ : We also observe that the factors .1  F rv q deg.v/ udeg.v/ / in (7.5.15) are units in the ring ƒŒŒu. Hence the ideal generated by #F =F ;S;T is independent of the auxiliary set T and we can define the Stickelberger element Y .1  F rv q deg.v/ /1 : #F =F ;S WD #F =F ;S;T v2T

We also define, for F L F , #L=F ;S;T WD LF .#F =F ;S;T / D ‚L=F ;S;T .1/: It is clear that these form a projective system: in particular, for any Zp -extension Fd =Fd 1 the relation dd1 .#Fd =F ;S;T / D #Fd 1 =F ;S;T

(7.5.16)

clearly recalls the one satisfied by characteristic ideals (equation (7.5.14)). Also, to define #L=F ;S there is no need of F : one can take for a finite extension L=F the analogue of product (7.5.15) and reason as above. Theorem 7.5.13 (Tate, Deligne). For any finite extension L=F , jFL j#L=F ;S is in the annihilator ideal of the class group of L (considered as a ZŒGal.L=F /-module). Proof. This is [61, Ch. V, Théorème 1.2].



Remark 7.5.14. Another proof of this result was given by Hayes [31], by means of Drinfeld modules. Corollary 7.5.15. Let Fd =F be a Zdp -extension as before and S D fpg, the unique (totally) ramified prime in F =F : then 1. #Fd =F ;S A.Fd / D 0; 2. if #Fd =F ;S is irreducible in ƒ.Fd /, then Chƒ.Fd / .A.Fd // D .#Fd =F ;S /m for some m > 1; f ƒ .A.F // D .#F =F ;S /m 3. if #Fd =F ;S is irreducible in ƒ.Fd / for all Fd’s, then Ch for some m > 1.

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Proof. For 1 one just notes that jFL j is prime with p. Part 2 follows from the structure theorem for torsion ƒ.Fd /-modules. Part 3 follows from 2 by taking limits (as in Definition 7.5.10) and noting that the m is constant through the Fd ’s because of equations (7.5.14) and (7.5.16).  The exponent in 2 and 3 of the corollary above is actually m D 1. A proof of this fact is based on the following technical result of [37] (generalized in [17, Theorem A.1]). Once Fd is fixed it is always possible to find a Zcp -extension of F containing Fd , call it Ld , such that: S containing S a. the extension Ld =F is ramified at all primes of a finite set e (moreover e S can be chosen arbitrarily large); b. the Stickelberger element #Ld =F ;e S is irreducible in the Iwasawa algebra ƒ.Ld /; c. there is a Zp -extension L0 of F contained in Ld which is ramified at all primes of e S and such that the Stickelberger element #L0 =F ;e S is monomial, i.e., conr rC1 gruent to u.  1/ modulo .  1/ (where  is a topological generator of Gal.L0 =F / and u 2 Zp ). With condition b and an iteration of equation (7.5.14) one proves that Chƒ.Fd / .A.Fd // D .#Fd =F ;S /m for some m > 1 : The monomiality condition c (using L0 as a first layer in a tower of Zp -extensions) leads to m D 1 (see [36, section 4] or [17, section A.1] which uses the possibility of varying the set e S , provided by a, more directly). We remark that the proof only uses the irreducibility of #Ld =F ;e S , i.e., Chƒ.Fd / .A.Fd // D .#Fd =F ;S /

(7.5.17)

holds in general for any Fd . Corollary 7.5.16 (Iwasawa Main Conjecture). In the previous setting we have f ƒ .A.F // D .#F =F ;p / : Ch Proof. From the main result of [36], one has that Chƒ.Fd / .A.Fd // D .#Fd =F ;p / and we take the limit in both sides.



Remark 7.5.17. The equality between characteristic ideals and ideals generated by Stickelberger elements has been proved by K.-L. Kueh, K. F. Lai and K.-S. Tan ([36]) and by D. Burns ([16] and the Appendix coauthored with K.F. Lai and K-S. Tan [17]) in a more general situation. The Zdp -extension they consider has to be unramified outside a finite set S of primes of F (but there is no need for the primes to be totally

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ramified). Moreover they require that none of the primes in S is totally split (otherwise #Fd =F ;S D 0). The strategy of the proof is basically the same but, of course, many technical details are simplified by our choice of having just one (totally) ramified prime (just compare, for example, Lemma 7.5.7 with [36, Lemma 3.3 and 3.4]). Moreover, going back to the Fitting vs. characteristic ideal situation, it is worth noticing that Burns proves that the first cohomology group of certain complexes (strictly related to class groups, see [16, Proposition 4.4] and [17, section A.1]) are of projective dimension 1 ([16, Proposition 4.1]). In this case the Fitting and characteristic ideals are known to be equal to the inverse of the Knudsen-Mumford determinant (the ideal by which all the results of [16] are formulated). 7.5.4 Characteristic p L-functions. One of the most fascinating aspects of function field arithmetic is the existence, next to complex and p-adic L-functions, of their characteristic p avatars. For a thorough introduction the reader is referred to [27, Chapter 8]: here we just provide a minimal background. Recall our fixed place 1 and let C1 denote the completion of an algebraic closure of F1 . Already Carlitz had studied a characteristic p version of the Riemann zeta function, defined on N and taking values in C1 (we will say more about it in Sect. 7.6.6). More recently Goss had the intuition that, like complex and p-adic L-functions have as their natural domains respectively the complex and the p-adic (quasi-)characters of the Weil group, so one could consider C1 -valued characters. In particular, the analogue of the complex plane as domain for the characteristic p L-functions is S1 WD C1  Zp , that can be seen as a group of C1 -valued homo morphisms on F1 , just as for s 2 C one defines x 7! x s on RC . The additive group Z embeds discretely in S1 . Similarly to the classical case, one can define L. ; s/ for a compatible system of v-adic representation of GF (v varying among places different from 1) by Euler products converging on some “half-plane” of S1 . The theory of zeta values in characteristic p is still quite mysterious and at the moment we can at best speculate that there are links with the Iwasawa theoretical questions considered in this paper 8 . To the best of our knowledge, the main results available in this direction are the following. Let F .p/=F be the extension obtained from the p-torsion of a Drinfeld–Hayes module (in the simplest case, F .p/ is the F1 we are going to introduce in Sect. 7.6.1). Goss and Sinnott have studied the isotypic components of A.F .p// and shown that they are non-zero if and only if p divides certain characteristic p zeta values: see [27, Theorem 8.14.4] for a precise statement. Note that the proof given in [27], based on a comparison between the reductions of a p-adic and a characteristic p L-function respectively mod p and mod p ([27, Theorem 8.13.3]), makes use of Crew’s result. Okada [47] obtained a result of similar flavor for the class group of the ring of “integers” of F .p/ when F is the rational function field, and Shu [55] extended it to any F ; since Okada’s result is strictly related to the subject of Sect. 7.6.6 below, we will say more about it there.

8 This field is in rapid evolution. After this paper was written, L. Taelman introduced some important new ideas: see [56] and [57]. Further recent developments can be found in [1, 2, 3, 4, 5] and [51].

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7.6 Cyclotomy by the Carlitz module 7.6.1 Setting. From now on we take F D Fq .T / and let 1 be the usual place at infinity, so that the ring of elements regular outside 1 is A WD Fq ŒT ; this allows a number of simplifications, leaving intact the main aspects of the theory. The “cyclotomic” theory of function fields is obtained via Drinfeld–Hayes modules: in the setting of the rational function field the only one is the Carlitz module ˆW A ! Af g, T 7! ˆT WD T C  (here  denotes the operator x 7! x q and, if R is an Fp -algebra, Rf g is the ring of skew polynomials with coefficients in R: multiplication in Rf g is given by composition). We also fix a prime p A and let  2 A be its monic generator. In order to underline the fact that A and its completion at p play the role of Z and Zp in the Drinfeld–Hayes cyclotomic theory, we will often use the alternative notation Ap for the ring of local integers Op Fp . Let Cp be the completion of an algebraic closure of Fp . As usual, if I is an ideal of A, ˆŒI  will denote the I -torsion of ˆ (i.e., the common zeroes of all ˆa , a 2 I ). One checks immediately that if is the unique monic generator of I , then Y ˆ .x/ D .x  u/ : u2ˆŒI 

We put Fn WD F .ˆŒpn / and Kn WD Fp .ˆŒpn / : As stated in Sect. 7.1.2, we think of the Fn ’s as subfields of Cp , so that the Kn ’s are their topological closures. We shall denote the ring of A-integers in Fn by Bn and its closure in Kn by On , and write Un for the 1-units in On . Let F WD [Fn and Q WD Gal.F =F /. Consider the ring of formal skew power series Ap ff gg: it is a complete local ring, with maximal ideal Ap C Ap ff gg . The map ˆ extends to a continuous homomorphism ˆW Ap ! Ap ff gg (i.e., a formal Drinfeld module) and this allows to define  a “cyclotomic" character W Q ! Ap . More precisely, let Tp ˆ WD lim ˆŒpn  (the limit is taken with respect to x 7! ˆ .x/) be the Tate module of ˆ. The ring Ap acts on Tp ˆ via ˆ, i.e., a  .u/n WD .ˆa .un //n , and the character  is defined by  u DW . /  u, i.e., . / is the unique element in Ap such that ˆ./ .un / D  un for all n. From this it follows immediately that Q D   , where  ' Fp is a finite group of order prime to p and  is the inverse image of the 1-units. Since ˆ has rank 1, Tp ˆ is a free Ap -module of rank 1. As in [13], we fix a generator ! D .!n /n>1 : this means that the sequence f!n g satisfies ˆ n .!n / D 0 ¤ ˆ n1 .!n /

and ˆ .!nC1 / D !n :

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By definition Kn D Fp .!n /. By Hayes’ theory, the minimal polynomial of !n over F is Eisenstein: it follows that the extensions Fn =F and Kn =Fp are totally ramified, !n is a uniformizer for the field Kn , On D Ap ŒŒ!n  D Ap Œ!n . The extension Fn =F is unramified at all other finite places: this can be seen directly by observing that ˆ n has constant coefficient  n . Furthermore Fn =F is tamely ramified at 1 with inertia group I1 .Fn =F / ' Fq . The similarity with the classical properties of Q.pn /=Q is striking. The formula NFnC1 =Fn .!nC1 / D !n shows that the !n ’s form a compatible system under the norm maps (the proof is extremely easy; it can be found in [13, Lemma 2]). This and the observation that ŒFnC1 W Fn  D q deg.p/ for n > 1 imply lim Kn D ! Z  Fp  lim Un :

(7.6.1)

Q Note that lim Un is a ƒ-module. 7.6.2 Coleman’s theory. A more complete discussion and proofs of results in this section can be found in [13, §3]. Let R be a subring of Cp : then, as usual, R..x// WD RŒŒx.x 1/ is the ring of formal Laurent series with coefficients in R. Moreover, following [20] we define RŒŒx1 and R..x//1 as the subrings consisting of those (Laurent) power series which converge on the punctured open ball B 0 WD B.0; 1/  f0g Cp : The rings RŒŒx1 and R..x//1 are endowed with a structure of topological R-algebras, induced by the family of seminorms fk  kr g, where r varies in jCp j \ .0; 1/ and kf kr WD supfjf .z/j W jzj D rg. All essential ideas for the following two theorems are due to Coleman [20]. Theorem 7.6.1. There exists a unique continuous homomorphism

N W Fp ..x//1 ! Fp ..x//1 such that Y

f .x C u/ D .N f / ı ˆ :

u2ˆŒp

Theorem 7.6.2. The evaluation map ev W f 7! ff .!n /g gives an isomorphism .Ap ..x///N Did ' lim Kn where the inverse limit is taken with respect to the norm maps.

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We shall write Colu for the power series in Ap ..x// associated with u 2 lim Kn by Coleman’s isomorphism of Theorem 7.6.2. Remark 7.6.3. An easily obtained family of N -invariant power series is the following. Let a 2 Ap : then Y

ˆa .x C u/ D

u2ˆŒp

Y

.ˆa .x/ C ˆa .u// D ˆ .ˆa .x//

u2ˆŒp

(since ˆa permutes elements in ˆŒp) and, from ˆ ˆa D ˆa ˆ in Ap ff gg, it follows that ˆa .x/ is invariant under the Coleman norm operator N (as observed in [13, page 797], this just amounts to replacing ! with a  ! as generator of the Tate module). Following [20], we define an action of  on Fp ŒŒx1 by . f /.x/ WD f .ˆ./ .x//. Then Colu D .  Colu /, as one sees from .  Colu /.!n / D Colu .ˆ./ .!n // D Colu .!n / D  .Colu .!n // D  .un/ : (7.6.2) 7.6.3 The Coates–Wiles homomorphisms. We introduce some operators on power series. Let dlog W Fp ..x//1 ! Fp ..x//1 be the logarithmic derivative, i.e., dlog .g/ WD g0 . Also, for any j 2 N let j W Fp ..x// ! Fp ..x// be the j -th Hasse-Teichmüller g derivative, defined by the formula !  1 1  X X nCj n WD cnCj x n cn x j j nD0

nD0 j

d (i.e., j “is” the differential operator j1Š dx j ). A number of properties of the Hasse– Teichmüller derivatives can be found in [33]; here we just recall that the operators j are Fp -linear and that 1 X f .x/ D j .f /jxD0 x j : (7.6.3) j D0

The last operator we need to introduce is composition with the Carlitz exponential eC .x/ D x C : : : , i.e., f 7! f .eC .x//. Definition 7.6.4. For any integer k > 1, define the k-th Coates–Wiles homomorphism ık W lim On ! Fp by     ık .u/ WD k1 .dlog Colu /.eC .x// jxD0 D k1 ..dlog Colu / ı eC / .0/ :

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Notice that by (7.6.3) this is equivalent to putting .dlog Colu /.eC .x// D

1 X

ık .u/x k1 :

(7.6.4)

kD1

Lemma 7.6.5. The Coates–Wiles homomorphisms satisfy ık . u/ D . /k ık .u/ : Proof. Recall that

d ˆ .x/ dx a

D a for any a 2 Ap . Then from (7.6.2) it follows

dlog Colu D dlog .Colu ı ˆ./ / D . /.dlog Colu / ı ˆ./ ; 0

since dlog .f ıg/ D g 0 . ff ıg/. Composing with eC and using ˆa .eC .x// D eC .ax/, one gets, by (7.6.4), .dlog Colu /.eC .x// D . /.dlog Colu /.eC .. /x// 1 X D . / ık .u/. /k1x k1 : kD1



The result follows. 7.6.4 Cyclotomic units.

Definition 7.6.6. The group Cn of cyclotomic units in Fn is the intersection of Bn with the subgroup of Fn generated by  .!n/,  2 Gal.Fn =F /. By the explicit description of the Galois action via ˆ, one sees immediately that this is the same as Bn \ hˆa .!n /ia2Ap . P Lemma 7.6.7. Let c  be an element in ZŒGal.Fn =F /: then Y X  .!n /c 2 Cn ” c D 0 : 2Gal.Fn =F /

Proof. Obvious from the observation that !n is a uniformizer for the place above p  and a unit at every other finite place of Fn . Let Cn and Cn1 denote the closure respectively of Cn \ On and of Cn1 WD Cn \ Un . Let a 2 Ap . By Remark 7.6.3 .ˆa .!n //n is a norm compatible system: hence one can define a homomorphism Q ! lim Kn ‡ W ZŒ X Y Y   c  7!  .!n /c n D ˆ./ .!n /c n :

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2 l2 im K ' !

Let lim Kn be the p-adic completion of lim Kn . By (7.6.1) one gets the isomorphism  n

Zp

 lim Un .

2

Lemma 7.6.8. The restriction of ‡ to ZŒ can be extended to ‡ W ƒ ! lim Kn . Proof. If a 2 Ap is a 1-unit, then ˆa .!n / D !n un

(7.6.5)

with un 2 Un . Since by definition  D 1 .1 C Ap /, it follows that ‡ sends ZŒ into ! Z  lim Un . To complete the proof it suffices to check that ‡ is continuous with

2

respect to the natural topologies on ƒ D lim.Z=pn Z/ŒGal.Fn =F / and lim Kn . But a  a0 .mod  n / in Ap implies ˆa .!j / D ˆa0 .!j / for any j 6 n and the result follows from the continuity of .  Proposition 7.6.9. Let I ƒ denote the augmentation ideal; then ‡ induces a surjective homomorphism of ƒ-modules I ! lim Cn1 . The kernel has empty interior. Proof. From Lemma 7.6.7 and (7.6.5) it is clear that ‡ .˛/ 2 lim Cn1 if and only if ˛ 2 I . This map is surjective because I is compact and already the restriction to the augmentation ideal of ZŒ is onto Cn1 for all n. A straightforward computation shows that it is a homomorphism of ƒ-algebras: for  2    Y X   Y  ˆ./ .!n /c

c  D  ˆ. / .!n /c

(7.6.6) ‡ D n

n

because .ˆa .!n // D ˆa ..!n // D ˆa .ˆ. / .!n //. For the statement about the kernel, let AC A be the subset of monic polynoC mials and consider any Q function A na ! Z, a 7! na , such that na D 0 for almost all a. We claim that a2AC ˆa .x/ D 1 only if na D 0 for all a. To see it, let ua denote a generator of the cyclic A-module ˆŒ.a/. Then x Q  ua divides ˆb .x/ if and of u as root of ˆa .x/na D 1 is exactly only if b 2 .a/: hence the multiplicity m a a P b2.a/\AC nb . For b 2 A, let ".b/ denote the number of primes of A dividing b (counted with multiplicities): then a simple combinatorial argument shows that X X .1/".b/ nabc : na D b2AC

c2AC

It follows that ma D 0 for all a 2 AC if and only if na D 0 for all a. As in Sect. 7.5.3, for v ¤ p; 1 let F rv 2 Q be its Frobenius. By [27, Proposition 7.5.4] one finds that .F rv / is the monic generator of the ideal in A corresponding Q Thus to the place v: hence, by Chebotarev density theorem, 1 .AC / is dense in . the isomorphism of Theorem 7.6.2 shows that we have proved that ‡ W I ! lim On is injective on a dense subset; the kernel must have empty interior. 

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Remark 7.6.10. Since I ˝Z ZŒ D ˚ı2 I ı, formula (7.6.6) shows that ‡ can be Q extended to a homomorphism of ƒ-modules I ˝Z ZŒ ! lim Cn . Proposition 7.6.11. We have: lim On = lim Cn ' lim Un = lim Cn1 . Proof. Consider the commutative diagram 1 ! lim Cn1 ! lim Un ! lim Un = lim Cn1 ! 1 ? ? ? ?˛ ?˛ ?˛ y 2 y 3 y 1 1 ! lim Cn ! lim On ! lim On = lim Cn ! 1 : All vertical maps are injective and by (7.6.1) the cokernel of ˛2 is Fp . For ı 2  one has ı!n D ˆ.ı/ .!n / D .ı/!n un for some un 2 Un . By the injectivity part of the proof of Proposition 7.6.9, Cn D Cn1  ‡ .ZŒ/ and it follows that the cokernel of ˛1 is also Fp .  7.6.5 Cyclotomic units and class groups. Let FnC Fn be the fixed field of the inertia group I1 .Fn =F /. The extension FnC =F is totally split at 1 and ramified only above the prime p. We shall denote the ring of A-integers of FnC by BnC . Also, define En and En1 to be the closure respectively of Bn \ On and Bn \ Un . We need to introduce a slight modification of the groups A.L/ of Sect. 7.5. For any finite extension L=F , A1 .L/ will be the p-part of the class group of A-integers of L, so that, by class field theory, A1 .L/ ' Gal.H.L/=L/, where H.L/ is the maximal abelian unramified p-extension of L which is totally split at places dividing 1. We shall use the shortening An WD A1 .FnC /. Also, let Xn WD Gal.M.FnC /=FnC /, where M.L/ is the maximal abelian pextension of L unramified outside p and totally split above 1. As in the number field case, one has an exact sequence 1 ! En1 =Cn1 ! Un =Cn1 ! Xn ! An ! 1

(7.6.7)

coming from the following Proposition 7.6.12. There is an isomorphism of Galois modules

Un =En1 ' Gal.M.FnC /=H.FnC // : Proof. This is a consequence of class field theory in characteristic p > 0, as the analogous statement in the number field case: just recall that the role of archimedean places is now played by the valuations above 1. Under the class field theoretic identification of idele classes IF C =.FnC / with a dense subgroup of Gal..FnC /ab =FnC /, n one finds a surjection Y  OP  Gal.M.FnC /=H.FnC // Pjp

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whose kernel contains the closure of Y Y Y   C  OP \ .FnC / Ow .Fn;w / D p ..BnC / / Pjp

w−p

wj1

(where p denotes the diagonal inclusion). Reasoning as in [64, Lemma 13.5] one proves the proposition.  Taking the projective limit of the sequence (7.6.7), we get 1 1 1 1 ! E1 = C1 ! U1 =C1 ! X1 ! A1 ! 1 : (7.6.8)

Lemma 7.6.13. The sequence (7.6.8) is exact. Proof. Taking the projective limit of the short exact sequence 1 ! En1 ! Un ! Gal.M.FnC /=H.FnC // ! 1 we obtain 1 ! U1 ! Gal.M.F C /=H.F C// ! lim1 En1 ; 1 ! E1

where M.F C / and H.F C / are the maximal abelian p-extensions of F C totally split above 1 and unramified respectively outside the place above p and everywhere. To prove the lemma it is enough to show that lim1 .En1 / D 1. By a well-known result in homological algebra, the functor lim1 is trivial on projective systems satisfying the Mittag-Leffler condition. We recall that an inverse system .Bn ; dn / enjoys such property if for any n the images of the transition maps BnCm ! Bn are the same for all large m. So we are reduced to check that this holds for the En1 ’s with the norm maps. Observe first that En1 is a finitely generated ƒn -module, thus noetherian because 1 so is ƒn . Consider now \k Im.NnCk;n /, where NnCk;n W EnCk ! En1 is the norm 1 map. This intersection is a ƒn -submodule of En , non-trivial because it contains the cyclotomic units. By noetherianity it is finitely generated, hence there exists l such that Im.NnCk;n / is the same for all k > l. Therefore .En1 / satisfies the Mittag-Leffler property.  The exact sequence (7.6.8) lies at the heart of Iwasawa theory. Its terms are all ƒ-modules and, in Sect. 7.5.2, we have shown how to associate a characteristic ideal with A1 and its close relation with Stickelberger elements. In a similar way, i.e., working on Zdp -subextensions, one might approach a description of X1 , while, for the first two terms of the sequence, the filtration of the FnC ’s seems more natural (as the previous sections show). Assume for example that the class number of F is prime with p, then it is easy to see that An D 1 for all n. Moreover, using the fact that, by a theorem of Galovich

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and Rosen, the index of the cyclotomic units is equal to the class number (see [54, Theorem 16.12]), one can prove that En1 =Cn1 D 1 as well. This provides isomorphisms

Un =Cn1 ' Xn and 1 U1 = C1 ' X1 :

In general one expects a relation (at least at the level of Zdp -subextensions, then a limit procedure should apply) between the pro-characteristic ideal of A1 and the 1 1 (yet to be defined) analogous ideal for E1 = C1 (the Stickelberger element might be a first hint for the study of this relation). Consequently (because of the multiplicativity of characteristic ideals) an equality of (yet to be defined) characteristic ideals of X1 1 and of U1 =C1 is expected as well. Any of those two equalities can be considered as an instance of Iwasawa Main Conjecture for the setting we are working in. 7.6.6 Bernoulli–Carlitz numbers. We go back to the subject of characteristic p Lfunction. Let AC A be the subset of monic polynomials. The Carlitz zeta function is defined X 1 A .k/ WD ak C a2A

for k 2 N. Recall that the Carlitz module corresponds to aQlattice A C1 and can be constructed via the Carlitz exponential eC .z/ WD z a2A0 .1  z 1 a1 / (where A0 denotes A  f0g). Rearranging summands in the equality 1 X 1 1 X X z k1 1 D dlog .eC .z// D D  eC .z/ z  a z .a/k 0 a2A

0

(and using A D

Fq

a2A kD1

C

 A ) one gets the well-known formula 1

1 1 X A .n.q  1// n.q1/1 z : D C eC .z/ z nD1  n.q1/ From Sect. 7.6.4 it follows that for any a; b 2 A  p, the function N -invariant power series, associated with   ˆa .!n / ˆa .!/ D 2 lim On : c.a; b/ WD ˆb .!/ ˆb .!n / n

(7.6.9) ˆa .x/ ˆb .x/

is an

(7.6.10)

Theorem 7.6.14. The k-th Coates–Wiles homomorphism applied to c.a; b/ is equal to: ( 0 if k 6 0 .mod q  1/ ık .c.a; b// D : if k  0 .mod q  1/ .ak  b k / A .k/ k

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We remark that the condition k D n.q  1/ here is the analogue of k being an even integer in the classical setting (since q  1 D jFq j just as 2 D jZ j). Proof. Observe that (7.6.10) amounts to giving the Coleman power series Colc.a;b/ . Let  be the Carlitz logarithm, i.e.,  2 F ff gg is the element uniquely determined by eC ı  D 1. Then ˆa .x/ D eC .a.x// and by (7.6.10) and (7.6.9) one gets a b  ˆa .x/ ˆb .x/ X A .n.q  1// .an.q1/  b n.q1/ / .x/n.q1/1 : D n.q1/  n>1

dlog Colc.a;b/ .x/ D

Since .eC .x// D x, we get X A .n.q  1// n.q1/1 .dlog Colc.a;b/ /.eC .x// D .an.q1/  b n.q1/ / x (7.6.11)  n.q1/ n>1 and the theorem follows comparing (7.6.11) with (7.6.4).



Remark 7.6.15. As already known to Carlitz, A .k/ k is in F when q  1 divides k. Note that by a theorem of Wade,  2 F1 is transcendental over F . Furthermore, Jing Yu [66] proved that A .k/ for all k 2 N and A .k/ k for k “odd” (i.e., not divisible by q  1) are transcendental over F . Theorem 7.6.14 can be restated in terms of the Bernoulli–Carlitz numbers BCk [27, Definition 9.2.1]. They can be defined by X BCn 1 D z n1 eC .z/ ….n/ n>0 (where ….n/ is a function field analogue of the classical factorial nŠ); in particular BCn D 0 when n 6 0 .mod q  1/. Then Theorem 7.6.14 becomes ık .c.a; b// D .ak  b k /

BCk : ….k/

(7.6.12)

Theorem 7.6.14 and formula (7.6.12) can be seen as extending a result by Okada, BCk who in [47] obtained the ratios ….k/ (for k D 1; : : : ; q deg.p/  2) as images of cyclotomic units under the Kummer homomorphisms (which are essentially a less refined version of the Coates–Wiles homomorphisms). From here one proves that the non-triviality of an isotypic component of A1 implies the divisibility of the corresponding “even” Bernoulli–Carlitz number by p: we refer to [27, §8.20] for an account. As already mentioned, Shu [55] generalized Okada’s work to any F (but with the assumption deg.1/ D 1): it might be interesting to extend Theorem 7.6.14 to a “Coates–Wiles homomorphism” version of her results.

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7.6.7 Interpolation? In the classical setting of cyclotomic number fields, the analogue of the formula in Theorem 7.6.14 can be used as a key step in the construction of the Kubota-Leopoldt zeta function (see e.g. [19]). Hence it is natural to wonder if something like it holds in our function field case. Up to now we have no answer and can only offer some vague speculation. As mentioned in Sect. 7.5.4, Goss found a way to extend the domain of A from N to S1 . He also considered the analogue of the p-adic domain and defined it to be Cp  Sp , with Sp WD Zp  Z=.q deg.p/  1/ (observe that Cp  Sp is the Cp -valued dual of Fp ). Then functions like A enjoy also a p-adic life: for example, letting v 2 AC be a uniformizer for a place v, A;p is defined on Cp  Sp by A;p .s/ WD

Y

.1  vs /1 ;

v−p1

at least where the product converges. The ring Z embeds discretely in S1 and has dense image in 1  Sp . So Theorem 7.6.14 seems to suggest interpolation of A;p on 1  Sp . Another clue in this direction is the fact that Sp is the “dual” of , just as Zp is the “dual” of Gal.Q.p1 /=Q/ (a strengthening of this interpretation has been recently provided by the main result of [34]).

Acknowledgements. We thank Sangtae Jeong, King Fai Lai, Jing Long Hoelscher, Ki-Seng Tan, Dinesh Thakur, Fabien Trihan for useful conversations and Bruno Anglès for pointing out a mistake in the first version of this paper. F. Bars and I. Longhi thank CRM for providing a nice environment to complete work on this paper.

References [1] B. Anglès, A. Bandini, F. Bars, and I. Longhi, Iwasawa main conjecture for the Carlitz cyclotomic extension and applications, to appear in Math. Ann. (2020). https://doi.org/10.1007/s00208-019-01875-8. [2] B. Anglès and F. Pellarin, Functional identities for L-series values in positive characteristic. J. Number Theory 142 (2014), 223–251. [3] B. Anglès and F. Pellarin, Universal Gauss-Thakur sums and L-series. Invent. Math. 200(2) (2015), 653–669. [4] B. Anglès and L. Taelman, On a problem à la Kummer-Vandiver for function fields. J. Number Theory 133 (2013), 830–841. [5] B. Anglès and L. Taelman, Arithmetic of characteristic p special L-values (with an appendix by V. Bosser). Proc. Lond. Math. Soc. 110(4) (2015), 1000–1032. [6] P. N. Balister and S. Howson, Note on Nakayama’s lemma for compact ƒ-modules. Asian J. Math. 1 (1997), 224–229.

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[7] A. Bandini, F. Bars and I. Longhi, Characteristic ideals and Iwasawa theory. New York J. Math. 20 (2014), 759–778. [8] A. Bandini, F. Bars, and I. Longhi, Characteristic ideals and Selmer groups. J. Number Theory 157 (2015), 530–546. [9] A. Bandini and I. Longhi, Control theorems for elliptic curves over function fields. Int. J. Number Theory 5 (2009), 229–256. [10] A. Bandini and I. Longhi, Selmer groups for elliptic curves in Zdl -extensions of function fields of characteristic p. Ann. Inst. Fourier 59(6) (2009), 2301–2327. [11] A. Bandini, I. Longhi, and S. Vigni, Torsion points on elliptic curves over function fields and a theorem of Igusa. Expo. Math. 27 (2009), 175–209. [12] A. Bandini and M. Valentino, Control theorems for `-adic Lie extensions of global function fields. Ann. Sc. Norm. Super. Pisa Cl. Sci. XIV(4) (2015), 1065–1092. [13] F. Bars and I. Longhi, Coleman’s power series and Wiles’ reciprocity for rank 1 Drinfeld modules. J. Number Theory 129 (2009), 789–805. [14] G. Böckle, D. Burns, D. Goss, D. Thakur, F. Trihan, and D. Ulmer, Arithmetic geometry over global function fields. Lecture notes from the Arithmetic Geometry Research Program held at the Centre de Recerca Matemàtica (CRM), Barcelona, February 22–March 5 and April 6– 16, 2010. (F. Bars, I. Longhi, and F. Trihan, eds.) Advanced Courses in Mathematics – CRM Barcelona, Basel, Birkhäuser, Springer, 2014. [15] N. Bourbaki, Commutative algebra – Chapters 1–7. Elements of Mathematics, Springer, Berlin, 1998. [16] D. Burns, Congruences between derivatives of geometric L-functions. Invent. Math. 184(2) (2011), 221–256. [17] D. Burns, K.F. Lai, and K.-S. Tan, On geometric main conjectures. Appendix to [16]. [18] D. Burns and F. Trihan, On geometric Iwasawa theory and special values of zeta functions (with an appendix by F. Bars). In [14], 119–181. [19] J. Coates and R. Sujatha, Cyclotomic fields and zeta value. SMM, Springer, 2006. [20] R. Coleman, Division values in local fields. Invent. Math. 53 (1979), 91–116. [21] R. Crew, Geometric Iwasawa theory and a conjecture of Katz. In Number theory (Montreal, Que., 1985), CMS Conf. Proc. 7, Amer. Math. Soc., Providence, RI, 1987, 37–53. [22] R. Crew, L-functions of p-adic characters and geometric Iwasawa theory. Invent. Math. 88(2) (1987), 395–403. [23] P. Deligne, Les constantes des équations fonctionnelles des fonctions L. In Modular functions of one variable II, Lecture Notes in Mathematics 349, Springer 1973, 501–597. [24] V. G. Drinfeld, Elliptic modules. (Russian) Mat. Sb. (N.S.) 94 (136) (1974), 594–627, 656; translated Math. USSR-Sb. 23(4) (1974), 561–592. [25] E.-U. Gekeler and M. Reversat, Jacobians of Drinfeld modular curves. J. Reine Angew. Math. 476 (1996), 27–93. [26] R. Gold and H. Kisilevsky, On geometric Zp -extensions of function fields. Manuscr. Math. 62 (1988), 145–161. [27] D. Goss, Basic structures of function field arithmetic, Springer, New York, 1996.

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[50] A. Pál, Proof of an exceptional zero conjecture for elliptic curves over function fields. Math. Z. 254(3) (2006), 461–483. [51] F. Pellarin, Values of certain L-series in positive characteristic. Ann. Math. 176 (2012), 2055– 2093. [52] M. van der Put and M. Reversat, Automorphic forms and Drinfeld’s reciprocity law. In Drinfeld modules, modular schemes and applications. Proceedings of the workshop held in AldenBiesen, September 9–14, 1996 (E.-U. Gekeler, M. van der Put, M. Reversat, and J. Van Geel, eds.), World Scientific, River Edge, NJ, 1997. [53] K. A. Ribet, Abelian varieties over Q and modular forms. In Algebra and topology 1992 (Taejon), Korea Adv. Inst. Sci. Tech., Taejon, 1992, 53–79. Reprinted in Modular curves and abelian varieties, Progr. Math., 224, Birkhäuser, Basel 2004, 241–261. [54] M. Rosen, Number theory in function fields. GTM 210, Springer, New York, 2002. [55] L. Shu, Kummer’s criterion over global function fields. J. Number Theory 49(3) (1994), 319– 359. [56] L. Taelman, Special L-values of Drinfeld modules. Ann. Math. 175 (2012), 369–391. [57] L. Taelman, A Herbrand-Ribet theorem for function fields. Invent. Math. 188 (2012), 253– 275. [58] K.-S. Tan, Modular elements over function fields. J. Number Theory 45(3) (1993), 295–311. [59] K.-S. Tan, A generalized Mazur’s theorem and its applications. Trans. Amer. Math. Soc. 362 (2010), 4433–4450. [60] K.-S. Tan, Selmer groups over Zdp -extensions. Math. Ann. 359 (2014), 1025–1075. [61] J. Tate, Les conjectures de Stark sur les fonctions L d’Artin en s D 0. Progr. Math., 47, Birkhäuser, 1984. [62] J. Teitelbaum, Modular symbols for Fq ŒT . Duke Math. J. 68 (1992), 271–295. [63] D. Thakur, Iwasawa theory and cyclotomic function fields. In Arithmetic geometry (Tempe, AZ, 1993), Contemp. Math. 174, Amer. Math. Soc., Providence, RI, 1994, 157–165. [64] L. Washington, Introduction to cyclotomic fields. Second edition. GTM 83, Springer, New York, 1997. [65] A. Weil, Dirichlet series and automorphic forms. Lecture Notes in Mathematics 189, Springer. [66] J. Yu, Transcendence and special zeta values in characteristic p. Ann. Math. (2) 134(1) (1991), 1–23.

Chapter 8

1-t-Motifs Lenny Taelman

We show that the module of rational points on an abelian t-module E is canonically isomorphic with the module Ext1 .ME ; KŒt/ of extensions of the trivial t-motif KŒt by the t-motif ME associated with E. This generalizes prior results of Anderson and Thakur, Papanikolas and Ramachandran, and Woo. In case E is uniformizable we show that this extension module is canonically isomorphic with the corresponding extension module of Pink-Hodge structures. This situation is formally very similar to Deligne’s theory of 1-motifs and we have tried to build up the theory in a way that makes this analogy as clear as possible.

8.1 Introduction & statement of the main results Anderson and Thakur [3] have shown that the group of K-rational points on the n-th tensor power of the Carlitz module is canonically isomorphic with the group of extensions of the corresponding t-motif C ˝n over K by the trivial t-motif KŒt, and Papanikolas and Ramachandran [9] and Woo [15] have a similar result for points on Drinfeld modules. We provide a generalization of these results. The present treatment is less dependent on calculations and more visibly functorial than the prior ones in [3] and [9]. In fact, this paper grew out of an attempt to understand these results, and it was written with the belief that with an approach which is sufficiently intrinsic and functorial the results would follow in their full generality at once, without extra effort. We now summarize the main results of this paper. Fix a finite field k of q elements and a field K containing k. Consider the skew P polynomial ring KŒ  whose elements are polynomials i xi  i and where multiplication is defined by the rule x D x q  for all x 2 K. Note that KŒ  is the ring of endomorphisms of the k-vector space scheme Ga;K , in particular, if q is a prime number then KŒ  is the endomorphism ring of the additive group scheme over K. Note that k is central in KŒ . Denote by KŒ; t the ring kŒt ˝k KŒ . Elements of this ring can be identified with polynomials in t and  over K and multiplication satisfies t D  t, tx D xt and x D x q  for all x in K. The field K is naturally a left KŒ -module through x D x q and similarly the ring KŒt is naturally a left KŒt;  -module. Theorem 8.1.1. Let M be a left KŒt;  -module which is free and finitely generated both as a KŒt-module and as a KŒ -module. There is an isomorphism of kŒt-

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modules HomKŒ .M; K/ ! Ext1KŒt; .M; KŒt/; functorially in M and compatible with field extensions K ! K 0 . The proof of this theorem will be given in § 8.4. Our main interest in this theorem lies in the case where M is an (effective) t-motif. Assume that a k-algebra homomorphism W kŒt ! K has been fixed (the “structure homomorphism”), and that relative to this homomorphism M is an an effective tmotif which is finitely generated over KŒ  (see Sect. 8.3.1 for the definition). Then HomKŒ .M; K/ is the module E.K/ of K-valued points of the abelian t-module E associated to M (see Sect. 8.3.2), M satisfies the hypothesis of Theorem 8.1.1 and we obtain a canonical isomorphism E.K/ D Ext1KŒt; .M; KŒt/:

(8.1.1)

For M a tensor power of the Carlitz module this was proven by Anderson and Thakur [3], and for M a Drinfeld module a similar result was shown by Papanikolas and Ramachandran [9] (see the end of § 8.4 for the precise relation between their results and ours.) The isomorphism (8.1.1) should be compared with the canonical isomorphism A_ .L/ D Ext1 .A; Gm / for an abelian variety A over a field L. Note however that the collection of t-motifs M to which (8.1.1) applies is closed under tensor product, very much unlike the analogous situation with Abelian varieties. The isomorphism (8.1.1) allows us to work with extensions of t-motifs like M by Artin t-motifs (t-motifs that are trivialised by a finite separable extension of K, in other words: t-motifs corresponding to finite image Galois representations) in much the same way as one uses Deligne’s theory of 1-motifs to work with mixed motifs of weights 1 and 0. We call a 1-t-module a triple .X; E; u/ consisting of 1. a free and finitely generated kŒt-module X equipped with a continuous action of Gal.K sep =K/; 2. an abelian t-module E over K; 3. a Gal.K sep =K/-equivariant homomorphism of kŒt-modules u W X ! E.K sep /. A 1-t-motif is an effective t-motif MQ that is an extension of an effective t-motif M which is finitely generated as a KŒ -module by an Artin t-motif V . In §8.4 we show how the canonical isomorphism (8.1.1) implies that the category of 1-t-modules is anti-equivalent with the category of 1-t-motifs. Under this equivalence M is determined by E, V by X and the extension class of MQ by u, via (8.1.1). The second part of the paper provides an analytic interpretation of the extension group Ext1 .M; KŒt/ in terms of Hodge structures of t-motifs. We need to assume

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that W kŒt ! K is injective (K has “generic characteristic”), that K is a local field with absolute value j  j and that j .t/j > 1 (K has the “1-adic topology”). Denote by C the completion of the algebraic closure of K. Let M be an effective t-motif over K which is finitely generated as a KŒ -module and denote by E the associated abelian t-module. Denote by LieE .C / the tangent space at zero to EC . This is naturally a C -vector space and by functoriality it is also a kŒt-module. There is a unique holomorphic map expE W LieE .C / ! E.C / which is kŒt-linear and tangent to the identity map id W E.C / ! E.C /. The kernel of expE is a lattice, that is, a discrete and finitely generated kŒt-submodule. We say that E is uniformizable if expE is surjective. This is equivalent with M being analytically trivial (see §8.5 for the definition.) Also the “unit” effective t-motif KŒt (which does not correspond to an abelian t-module since it is not finitely generated as a KŒ -module) is analytically trivial and any extension of one analytically trivial effective t-motif by another is analytically trivial (see Proposition 8.5.6.) In §8.5 we define (a primitive version of) the Hodge structure H.M / of an analytically trivial effective t-motif M , following Pink [10]. Also we relate the uniformization of an abelian t-module E to the Hodge structure H.M / of the corresponding t-motif M . In §8.6 we show: Theorem 8.1.2. Let M be an effective t-motif over K, finitely generated over KŒ  and analytically trivial. The natural map Ext1C Œt; .MC ; C Œt/ ! Ext1 .H.MC /; H.C Œt// that maps the class of an extension of t-motifs to the class of its Hodge structure is an isomorphism of kŒt-modules. This is an analogue of Deligne’s theorem [4, §10] on the equivalence between 1-motifs and Hodge structures of type f.0; 0/; .0; 1/; .1; 0/; .1; 1/g, with the exception that we do not say anything about which Hodge structures occur as H.M / for some analytically trivial effective t-motif M , finitely generated over KŒ . This appears to be a difficult problem. The author would like to thank the anonymous referee for pointing out a mistake in a previous version of Proposition 8.4.9.

8.2 Duality for torsion modules over kŒŒz Let k be a field and kŒŒz the ring of power series in a single variable z over k. This section recalls some easy facts about finitely generated torsion modules over kŒŒz.

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Not only will we use some of these facts later on, but we will also carry out a number of constructions that are directly inspired by them. Denote by k..z// the field of Laurent series in z. If ˛ W M ! N is a morphism of kŒŒz-modules, then we write ˛ı for the induced map M ˝ k..z// ! N ˝ k..z//. We denote by ReszD0 the residue map from k..z// to k that maps a Laurent series to its O k..z//=k with k..z// by choosing coefficient of z 1 . (So we have silently identified the generator dz.) Let ˛  (8.2.1) 0 ! F ! G ! T ! 0 be a short exact sequence of finitely generated kŒŒz-modules, with T a torsion module and F and G free. It follows that ˛ı is an isomorphism. This short exact sequence induces a short exact sequence of kŒŒz-modules ˛t

0 ! HomkŒŒz.G; kŒŒz/ ! HomkŒŒz.F; kŒŒz/ ! Homk .T; k/ ! 0 (8.2.2) where the surjection is given by

' 7! .g/ 7! ReszD0 'ı .˛ı1 .g// : The short exact sequence (8.2.2) can be identified with 0 ! HomkŒŒz .G; kŒŒz/ ! HomkŒŒz .F; kŒŒz/ ! Ext1kŒŒz.T; kŒŒz/ ! 0; the long exact sequence of cohomology obtained by applying Hom.; kŒŒz/ to (8.2.1). Also, the residue map yields an isomorphism of kŒŒz-modules 

HomkŒŒz.T; k..z//=kŒŒz/ ! Homk .T; k/W f 7! ReszD0 ıf: Alternatively, one can directly obtain an isomorphism 

HomkŒŒz .T; k..z//=kŒŒz/ ! Ext1kŒŒz .T; kŒŒz/ by applying Hom.T; / to the injective resolution 0 ! kŒŒz ! k..z// ! k..z//=kŒŒz ! 0 of kŒŒz.

8.3 Effective t-motifs and abelian t-modules In this section we define effective t-motifs, abelian t-modules and recall Anderson’s correspondence [1] between certain effective t-motifs and abelian t-modules. We fix a finite field k of q elements.

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8.3.1 Effective t-motifs. Let R be a commutative ring containing k and RŒt the polynomial ring in one variable t over R. Denote by  W RŒt ! RŒt the ring endomorphism that restricts to the q-th power Frobenius endomorphism on R and that fixes t. If M is an RŒt-module then we define the pull-back of M along  as follows: M 0 WD   M D M ˝RŒt ; RŒt: We give M 0 the structure of an RŒt-module through the second factor in the tensor product. There is a canonical kŒt-linear map from M to M 0 , which we denote by  . It is given by  W M ! M 0 W m 7! m ˝ 1: Definition 8.3.1. A  -module over RŒt is a pair .M;  / of an RŒt-module M and an RŒt-linear map  W M 0 ! M . A morphism of  -modules is a morphism f W M1 ! M2 of RŒt-modules such that f ı 1 D 2 ı f 0 . We will usually suppress the  from the notation and write M for a  -module .M;  /. The category of  -modules is abelian and kŒt-linear. We denote the kŒt-module of morphisms M1 ! M2 by   HomRŒt ; M1 ; M2 : P Remark 8.3.2. Let RŒ  denote the ring whose elements are polynomials xi  i in  but where multiplication is defined through x D x q  for all x 2 R. Note that k is central in RŒ . Write RŒt;   for the tensor product kŒt ˝k RŒ . Let .M; M / be a  -module over RŒt. Then M has naturally the structure of a left RŒt;  -module through  m WD M . .m//. In fact, this construction defines an isomorphism of categories between the category of  -modules over RŒt and the category of left modules over RŒt;  . We will use the language of  -modules and the language of RŒt;  -modules interchangeably, each time choosing the most convenient for the task at hand. There is considerable abuse of notation in denoting the linear map  W M 0 ! M and the semi-linear action of  2 RŒt;   on M by the same symbol  . We hope this will not lead to confusion. If .M1 ; 1 / and .M2 ; 2 / are  -modules then we define their tensor product to be the  -module .M1 ˝RŒt  M2 ; 1 ˝ 2 /. Similarly one can define symmetric and exterior powers. In particular, given a  -module M whose underlying module is locally free of some constant rank r then one can consider its determinant det.M / WD ^r M which is a  -module that is locally free of rank one. If R ! S is a k-algebra homomorphism and M a  -module over R then we denote by MS the  -module over S obtained by extension of scalars. Now let K be a field containing k and fix a k-algebra homomorphism W kŒt ! K. We denote the image of t by # 2 K. One should think of as an analogue to the canonical homomorphism from Z to any commutative ring. From now on we

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shall always consider the field K as a kŒt-algebra, so we will silently consider the distinguished element .t/ D # 2 K to be part of the data when referring to “K”. Definition 8.3.3. An effective t-motif 1 of rank r over K is a  -module M over K whose underlying module is free of rank r and such that the cokernel of the linear map  W M 0 ! M is annihilated by some power of t  #. The condition on  is equivalent with the condition that det.M / is isomorphic with the  -module .KŒte;  e 7! ˛.t  #/n e/ for some ˛ 2 K  and n  0. If  is a maximal ideal of kŒt then we write kŒt for the -adic completion of kŒt and k.t/ for its quotient field. For every  the kŒt -module   T .M / WD M ˝KŒt  K sep Œt of   -invariants carries a continuous action of Gal.K sep=K/ and V .M / WD T .M / ˝kŒt  k.t/ is a k.t/ -vector space with a continuous action of Gal.K sep=K/. The following proposition gives some justification for the “motivic” terminology (which is Anderson’s), but we stress that we are merely dealing with an analogy; there is no known direct relation with any kind of algebro-geometric motifs. Proposition 8.3.4 (Thm 3.3 of [7]). Assume that is injective and that K is finite over its subfield k.#/. Let M be an effective t-motif over K of rank r. Then dim V .M / D r for all . Moreover, there exists a finite set S of places of K such that 1. for every place v … S and for all non-zero prime ideals  coprime with  v the representation V .M / is unramified at v; 2. for these  and v the characteristic polynomial of Frobenius at v has coefficients in kŒt and is independent of .  In other words: the V .M / form a “strictly compatible system” of Galois representations à la Serre [11]. Example 8.3.5. Assume that is injective and that K is finite over k.#/. Let C be the Carlitz t-motif over K. This is the rank one effective t-motif given by   C D KŒte;  e 7! .t  #/e : Let v be a finite place of K (i.e. v does not lie above the place # D 1 of k.#/ K.) Let f 2 kŒ# be a monic generator of the ideal in kŒ# corresponding to the norm of v in k.#/ K. One verifies that 1. the representation V .C / is unramified at v for all  coprime with  v; 2. for such  we have that Frobv acts as f .t/1 2 k.t/. So C plays the role of the Lefschetz motif Z.1/. 1 The terminology used here is that of [13]. What is called a “t -motive” in [1] would be an “effective t -motif that is finitely generated as a KŒ-module” in our language.

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8.3.2 Abelian t-modules. Let .M;  / be a  -module over K. Now consider the functor EM W fK-algebrasg ! fkŒt-modulesgW R 7! HomKŒ .M; R/; where R is a left KŒ -module through  r D r q . This functor is representable by an affine kŒt-module scheme over K, which is not necessarily of finite type over K. Conversely, given a kŒt-module scheme E over K define ME WD Homgr.sch.=K .E; Ga /; which is naturally a left KŒt-module. The q-th power Frobenius map  W   Ga ! Ga induces a linear map  W ME0 ! ME which makes ME into a  -module over KŒt. Proposition 8.3.6 (§1 of [1], §10 of [12]). The functors M 7! EM and E 7! ME form a pair of quasi-inverse anti-equivalences between the categories of effective t-motifs M over K that are finitely generated as left KŒ -modules and the category of kŒt-module schemes E over K that satisfy 1. for some d  0 the group schemes EKN and Gda;KN are isomorphic; 2. t  # acts nilpotently on Lie.E/; 3. ME is finitely generated as a KŒt-module. These anti-equivalences commute with base change K ! K 0 .



Definition 8.3.7. A kŒt-module scheme E satisfying the above three conditions is called an abelian t-module of dimension d . An abelian t-module of dimension one is called a Drinfeld module. The tangent space at the identity of E can be expressed in terms of ME as follows: Proposition 8.3.8 (see [1]). LieE .K/ D HomK .ME =ME0 ; K/ as KŒŒt  #-modules. Also the Galois representations associated with ME can be expressed in terms of E. If  D .f / kŒt is a non-zero prime ideal then define the -adic Tate module of E to be V .E/ WD .lim EŒf n .K sep // ˝kŒt  k.t/ :  n

If M is the effective t-motif associated with E then we have Proposition 8.3.9. V .M / D Hom.V .E/; k.t/ /.



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8.3.3 Weights and Dieudonné t-modules. In this section we recall the definition of the weights of an effective t-motif [13]. This will allow for a more useful equivalent description of the “finitely generated over KŒ ” condition. As usual k is a finite field of q elements and K a field containing k. Denote by  the continuous endomorphism of the field of Laurent series K..t 1 // that fixes t 1 and that restricts to the q-th power map on K. Note that K..t 1 // D k..t 1 //. If V is a K..t 1 //-vector space then we denote the pull-back by  of V by V 0 . Definition 8.3.10. A Dieudonné t-module over K is a pair .V;  / consisting of 1. a finite-dimensional K..t 1 //-vector space V and 2. a K..t 1 //-linear isomorphism  W V 0 ! V . A morphism of Dieudonné t-module is a K..t 1 //-linear map compatible with  . Just as with  -modules one can also consider Dieudonné t-module as modules over the skew polynomial ring K..t 1 //Œ  whose elements are polynomials in  and where  t D t and x D x q  . Dieudonné t-modules over a separably closed field admit a simple classification. The main ‘building blocks’ are the following modules: Definition 8.3.11. Let  D s=r be a rational number with .r; s/ D 1 and r > 0. The Dieudonné t-module V is defined to be the pair .V ;  / with 1. V WD K..t 1 //e1 ˚ : : : ˚ K..t 1 //er 2.   ei WD ei C1 (i < r) and   er WD t s e1 Proposition 8.3.12. If K is separably closed then the category of Dieudonné t-modules over K is semi-simple. The simple objects are the V with  2 Q and V Š V if and only if  D . Note that this classification is formally identical to the classification of the classical (p-adic) Dieudonné modules [5]. Proof. This is shown in [8, Appendix B]. Although the statements therein are made only for a particular field K, nowhere do the proofs make use of anything stronger than the separably closedness of K.  Let V =K be a Dieudonné t-module. Then by the above Proposition there exist rational numbers 1 ; : : : ; n such that VK sep Š V1 ˚    ˚ Vn ; O K K sep . We call these rational where VK sep denotes the completed tensor product V ˝ numbers the weights of V . If M=K is a  -module which is finitely generated over KŒt then M..t 1 // WD M ˝KŒt  K..t 1 // is naturally a Dieudonné t-module and we define the weights of M to be the weights of M..t 1 //.

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Proposition 8.3.13. Let M be a  -module, free and finitely generated as KŒt-module. Then M is finitely generated as KŒ -module if and only if M is a  -module and all weights of M are positive. Proof. The “if” part is shown in Theorem 5.3.1 of [13], the “only if” part in Proposition 8 of [14].  Corollary 8.3.14. The functors M 7! EM and E 7! ME form a pair of quasiinverse anti-equivalences between the categories of effective t-motifs M=K whose weights are positive and the category of abelian t-modules over K. Finally we prove a useful fact about extensions of Dieudonné t-modules over arbitrary fields. Proposition 8.3.15. Let V and W be Dieudonné t-modules over K. If V and W have no weights in common then Hom.V; W / D Ext1 .V; W / D 0 in the category of Dieudonné t-modules over K. Proof. A non-zero morphism V ! W would induce a non-zero morphism VK sep ! WK sep after base change, and such a morphism cannot exist by Proposition 8.3.12. If U is an extension of V by W then UK sep splits by Proposition 8.3.12. So U is a form over K of the split extension VK sep ˚ WK sep over K. Such forms are classified by the Galois cohomology group   H1 Gal.K sep =K/; Aut.VK sep ˚ WK sep / but since the weights of V and W are disjoint we have Aut.VK sep ˚ WK sep / D Aut.VK sep /  Aut.WK sep /; and it follows that any extension of V by W splits over K.



8.4 Algebraic theory of 1-t-motifs Theorem 8.4.1 (repeated from §8.1). Let M be a left KŒt;  -module which is free and finitely generated both as a KŒt-module and as a KŒ -module. There is an isomorphism of kŒt-modules HomKŒ .M; K/ ! Ext1KŒt; .M; KŒt/; functorially in M and compatible with field extensions K ! K 0 . Corollary 8.4.2. If M is the effective t-motif associated to an abelian t-module E we have a natural isomorphism of kŒt-modules E.K/ D Ext1KŒt; .M; KŒt/.

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Proof of Theorem 8.4.1. We will produce the desired isomorphism as the composition of two isomorphisms HomKΠ.M; K/

 HomKŒt; .M; K..t 1 //=KŒt/ ! Ext1KŒt; .M; KŒt/:

The residue map (coefficient of t 1 ) defines a natural isomorphism HomKΠ.M; K/

Res

 HomKŒt; .M; K..t 1 //=KŒt/;

which is the first of the two isomorphisms. To obtain the second isomorphism apply HomKŒt; .M; / to the short exact sequence 0 ! KŒt ! K..t 1 // ! K..t 1 //=KŒt ! 0; which yields a connecting homomorphism HomKŒt; .M; K..t 1//=KŒt/ ! Ext1KŒt; .M; KŒt/: To show that it is an isomorphism it suffices to prove that HomKŒt; .M; K..t 1 /// D Ext1KŒt; .M; K..t 1/// D 0: Consider the kŒt-linear map ıW HomKŒt  .M; K..t 1 /// ! HomKŒt  .M 0 ; K..t 1 ///W f 7! f ı    ı f: The kernel of ı is HomKŒt; .M; K..t 1/// and the cokernel is Ext1KŒt; .M; K..t 1 ///. Moreover, ı coincides with the k..t 1 //-linear map HomK..t 1 // .M..t 1 //; K..t 1/// ! HomK..t 1 // .M 0 ..t 1 //; K..t 1 /// f 7! f ı    ı f and its kernel and cokernel are Hom.M..t 1 //; K..t 1 /// and Ext.M..t 1 //; K..t 1/// respectively (both in the category of Dieudonné t-modules), which vanish by Proposition 8.3.15.  Remark 8.4.3. Let  W M ! K be an element of EM .K/ D HomKŒ .M; K/. Then the corresponding extension 0 ! KŒt ! M ! M ! 0 can be described explicitly as ( ) 1 X M WD .m; f / 2 M  K..t 1 // W f  .t i m/t i 1 2 KŒt : (8.4.1) i D0

In case M is a tensor power of the Carlitz t-motif such a description is used in [3].

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Definition 8.4.4. A 1-t-module over K is a triple .X; E; u/ consisting of 1. a free and finitely generated kŒt-module X equipped with a continuous action of Gal.K sep =K/; 2. an abelian t-module E over K; 3. a Gal.K sep =K/-equivariant homomorphism of kŒt-modules u W X ! E.K sep /. We now construct a t-motif MQ associated with a 1-t-module .X; E; u/. From X one builds the t-motif V WD HomkŒt  .X; K sepŒt/Gal.K

sep =K/

:

This is a KŒt-module which is free of rank the kŒt-rank of M and since  W K sep Œt ! K sep Œt commutes with the action of GK it defines a map  W V ! V which makes V into a t-motif over K. This V is an Artin t-motif: Definition 8.4.5. An Artin t-motif over K is an effective t-motif V over K such that VK sep is isomorphic to the effective t-motif K sep Œtr , where r is the rank of V . One easily verifies that X 7! V is an anti-equivalence from the category of continuous Gal.K sep =K/-modules over kŒt that are free of finite rank over kŒt to the category of Artin t-motifs over K. Let M be the effective t-motif associated to E. The map u defines through Theorem 8.4.1 an extension of KŒt;  -modules 0 ! V ! MQ ! M ! 0 which defines the t-motif MQ associated with .X; E; u/. Definition 8.4.6. A 1-t-motif is a t-motif MQ which fits into an exact sequence 0 ! V ! MQ ! M ! 0 with V an Artin t-motif and M a t-motif of strictly positive weights. Note that the exact sequence is uniquely determined by MQ , so from propositions 8.3.6, 8.3.13, and 8.4.1 we obtain: Corollary 8.4.7. The above construction .X; E; u/ 7! MQ defines an anti-equivalence  from the category of 1-t-modules over K to the category of 1-t-motifs over K. Finally, we briefly explain the relationship between Theorem 8.4.1 and the corresponding result of Papanikolas and Ramachandran [9]. Let E be a Drinfeld module of rank r over K and M D ME the corresponding effective t-motif. Define M _ by M _ WD HomKŒt  .M; C /: Note that .M _ /0 D HomKŒt  .M 0 ; C 0 /.

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Definition 8.4.8. We make M _ into an effective t-motif by the rule 1 M _ W HomKŒt  .M 0 ; C 0 / ! HomKŒt  .M; C /W f 7! C ı f ı M :

The map M is not surjective, but the above formula should be read as follows. After extending scalars from KŒt to K.t/, the linear map M becomes an isomor1 maps phism and one has that C ı f M .M _ /0 .M _ /0 ˝KŒt  K.t/ into M _ M _ ˝KŒt  K.t/: Proposition 8.4.9. There is a natural short exact sequence  1 1 _ 0 ! Ext .M; KŒt/ ! Ext .C; M / ! HomKŒt  C 0 ;

M_ M _ .M _ /0

 ! 0:

of kŒt-modules. This recovers the short exact sequence of Theorem 1.1 (b) of [9]. To see that   M_ 0 HomKŒt  C ; M _ .M _ /0 is indeed an .r  1/-dimensional K-vector space, first assume that r > 1. Note that C is free of rank 1 over KŒt and that the quotient M _ =M _ .M _ /0 is the dual of the tangent space of the t-module corresponding to M _ . Since M _ is pure of weight 1  1=r and has rank r, it has dimension r  1. If r D 1 then M _ does not correspond to an abelian t-module, but it is easy to see directly that M _ D M _ .M _ /0 . Proof of the proposition. The module Ext1 .M; KŒt/ is the cokernel of the map ı1 W HomKŒt  .M; KŒt/ ! HomKŒt  .M 0 ; KŒt/W f 7! f ı    ı f: This map is naturally isomorphic with the map ı2 W HomKŒt  .C; M _ / ! HomKŒt  .C 0 ; .M _ /0 / given by 1 0 f 7! M _ ı f ı C  f :

(The double composition is well-defined in the sense of the remark after Definition 8.4.8.) Consider the square ı2

HomKŒt  .C; M _ / ! HomKŒt  .C 0 ; .M _ /0 / ? ? ? ? _ idy y M ı3

HomKŒt  .C; M _ / ! HomKŒt  .C 0 ; M _ /

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where the bottom map is the map ı3 W f 7! f ı C  M _ ı f 0 : Note that the square commutes and that the cokernel of the bottom map is Ext1 .C; M _ /. The right-hand side is injective with cokernel   M_ 0 HomKŒt  C ; ; M _ .M _ /0 which yields the desired short exact sequence.



8.5 Uniformization and Hodge structures For the remainder of this paper we assume that K is a local field and that j#j > 1, so in particular W kŒt ! K W t 7! # is injective. We also fix an algebraic closure KN of N The field C is algebraically closed. K and a completion C of K. Although we will consider effective t-motifs over C , we will need to assume that they are defined over KN C . This is because we will use results of Anderson [1] that use a locally compact field of definition. It is possible that these results could be generalized to include all effective t-motifs over C . 8.5.1 Uniformization of abelian t-modules. N Proposition 8.5.1 (see §2 of [1]). Let E be an abelian t-module over K. 1. There exists a unique entire kŒt-module homomorphism expE W LieE .C / ! E.C / that is tangent to the identity map; 2. The kernel of expE is a finitely generated free discrete sub-kŒt-module in LieE .C /. When expE is surjective we say that E is uniformizable, and in that case we have a short exact sequence of kŒt-modules 0 ! ƒE ! LieE .C / ! E.C / ! 0

(8.5.1)

where ƒE WD ker.expE /. Example 8.5.2. Drinfeld modules are uniformizable [6]. Remark 8.5.3. There exist abelian t-modules that are not uniformizable [1]. Whether or not an abelian t-module E is uniformizable may be read off from the associated t-motif M . To do so, we need the Tate algebra: o nX fi t i 2 C ŒŒt W jfi j ! 0 as i ! 1 : C ftg WD

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This algebra has the following properties: 1. C ftg is a C Œt-algebra; P q P 2.  W fi t i 7! fi t i defines an endomorphism of the k-algebra C ftg; 3. C ftg D kŒt. N Given a KŒt-module M we denote by M ftg the tensor product M ˝KŒt N  C ftg. If M is a  -module then  extends to a C ftg-linear map M 0 ftg ! M ftg. Also the canonical map  extends to a kŒt-linear map  W M ftg ! M 0 ftg. The invariants M ftg form a kŒt-module. Definition 8.5.4 ([1, §2]). An effective t-motif M over KN is said to be analytically trivial if the natural map M ftg ˝kŒt  C ftg ! M ftg is an isomorphism. Proposition 8.5.5 ([1, §2]). Let E=KN be an abelian t-module and let M be the associated effective t-motif. Let r be the rank of M . The following are equivalent: 1. M ftg is free of rank r as kŒt-module; 2. M is analytically trivial; 

3. E is uniformizable.

Proposition 8.5.6. If M1 and M2 are analytically trivial t-motifs then any extension of M1 by M2 is analytically trivial. Proof. It suffices to show that any extension of the “trivial” C ftgŒ -module C ftg by itself splits. Let D be such an extension. Then D has a basis .e1 ; e2 / such that  e1 D e1 and  e2 D e2 C f e1 for some f 2 C ftg. The extension splits if and only if there exists a g 2 C ftg such that f D  .g/  g: Note that if a is an element of C with jaj < 1 then there P is ia (unique) x 2 C such jxj D jaj. If we write f as f D that a D x q  x and i fi t then we can take for g P any power series i gi t i with giq  gi D fi for all i , as long as we choose gi such that jgi j D jfi j for all i sufficiently large.  Next we will define Hodge structures of analytically trivial t-motifs and use them to recover the uniformization sequence (8.5.1) from M . 8.5.2 Hodge structures of t-motifs. Definition 8.5.7. A (function field) Hodge structure is a diagram H1 ! H2 consisting of 1. a kŒt-module H1 ; 2. a C ŒŒt  #-module H2 ; 3. a kŒt-linear map H1 ! H2 .

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A morphism of Hodge structures is a commutative square. The category of Hodge structures is denoted by H. The category H is an abelian category. Remark 8.5.8. These structures were introduced by Pink in [10]. His definition contains both more data and more conditions, but for the purposes of this note the above simple definition suffices. It would be more correct to call our triples pre-Hodge structures. If M=C is a uniformizable effective t-motif then every element of M ftg has an infinite radius of convergence [2, 3.13]. In particular, it makes sense to talk about the power series expansion of such an element around t D #. This gives the Hodge structure HM associated with M , namely

HM WD M ftg ! M ŒŒt  # ; where M ŒŒt  # WD M ˝C Œt  C ŒŒt  #. If M is a t-motif over a subfield of C then we define HM to be Hodge structure of the extension of scalars of M to C . In the remainder of this section we relate the Hodge structure H D HM of an effective t-motif M with strictly positive weights with the uniformization of the associated abelian t-module E D EM . We largely follow Anderson [1], and for most of the part are merely rephrasing him using the language of Hodge structures. A special role will be played by the following Hodge structures:

1 WD kŒt ! C ŒŒt  # ;

A WD kŒt ! C..t  #// ; and

B WD 0 ! C..t  #//=C ŒŒt  # : Note that they naturally sit in a short exact sequence 0 ! 1 ! A ! B ! 0. Proposition 8.5.9. For all uniformizable abelian t-modules E, and functorially in E, there is a commutative square of kŒt-modules    ! HomH H; A ? ? y    LieE .C / ! HomH H; B ; ƒE ? ? y

where H is the Hodge structure of ME . Both horizontal maps are isomorphisms and the right map is induced by the natural map A ! B.

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Before giving the proof we first state the following consequence: Corollary 8.5.10. ƒE generates LieE .C / as a C ŒŒt  #-module. This corollary generalises Corollary 3.3.6 of [1], which employs the additional hypothesis that E be pure. Proof of the corollary. By Proposition 8.5.9 it suffices to check that the map HomH .H; A/ ˝kŒt  C ŒŒt  # ! HomH .H; B/ induced by A ! B is surjective, which is straightforward.



The proof of Proposition 8.5.9 fills the rest of this section. Lemma 8.5.11. The sequence of C ŒŒt  #-modules "

0 ! M ftg ˝kŒt  C ŒŒt  # ! M ŒŒt  # ! M=M 0 ! 0

(8.5.2)

is exact. Proof. Consider the commutative square of C ŒŒt  #-modules M 0 ftg  ˝kŒt  C ŒŒt  # ! M 0 ŒŒt  # ? ? ? ? y y M ftg ˝kŒt  C ŒŒt  # ! M ŒŒt  #: We first show that the top map is an isomorphism. To do so, it suffices to verify that its determinant is an isomorphism. Since the formation of the this map commutes with tensor products and exterior powers, we can reduce to the case where M is the Carlitz t-motif. In that case a simple computation shows that M 0 ftg  is generated by   q Y t 1 i e .#/ q1 #q i >0 whose image is a generator of M 0 ŒŒt  #. Now since the left and upper morphisms are isomorphisms, from which we obtain an isomorphism between the cokernels of the right and lower maps. The cokernel of the right map is isomorphic with M=M 0 , so combining these we get the desired short exact sequence.  Dualizing the sequence (8.5.2) à la §8.2 gives the bottom isomorphism HomH .H; B/



 HomC .MC =MC0 ; C / D LieE .C /

in the commutative square of Proposition 8.5.9.

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Lemma 8.5.12. Let N be a non-negative integer. For all m 2 M and for all " 2 LieE .C / D HomC .MC =MC0 ; C / the Laurent series ˇN .m; "/ WD

X

exp.t i 1"/.m/t i 2 C..t//

(8.5.3)

i >N

defines a function which is meromorphic in t D # and Rest D# ˇN .m; "/ D ".m C M 0 /:

(8.5.4)

Proof. The validity of the statements does not depend on N . For N D 0 a proof is in [1, 3.3.4].  Consider the map X

LieE .C /  M ! C..t// W ."; m/ 7! ˇN .m; "/ D

expE .t i 1 "/.m/t i ;

i N

which is kŒt-linear in the first and C Œ -linear in the second argument. The restriction ƒE  M ! C..t// is independent of N . Also, it is C Œt;  -linear in M so it extends to a map ƒE  M ftg ! C..t//; which is C ftgŒ -linear in its second argument. Restriction to  -invariants now gives a kŒt-bilinear form ƒE  M ftg ! k..t//: Lemma 8.5.13. The above bilinear form takes values in kŒt and the resulting ƒE  M ftg ! kŒt is a perfect pairing. Proof. See Sect. 2.6 of [1].



Lemma 8.5.13 provides the top isomorphism 

ƒE ! HomkŒt  .M ftg ; kŒt/ D HomH .H; A/; of the commutative square whose existence is claimed in Proposition 8.5.9 and Lemma 8.5.12 shows commutativity. This finishes the proof of Proposition 8.5.9.

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8.6 Transcendental theory of 1-t-motifs As in the previous section, M denotes a uniformizable t-motif over KN whose weights are strictly positive and let E D E.M / be the corresponding abelian t-module. The Hodge structure of M is denoted by H . In this section we will show that any extension HQ of H by 1 comes from a unique 1-t-motif MQ , extension of M by 1. Applying Hom.H; / to the short exact sequence 0!1!A!B!0 of Hodge structures we obtain a long exact sequence of kŒt-modules    ! Hom.H; A/ ! Hom.H; B/ ! Ext1 .H; 1/ !    Lemma 8.6.1. The sequence 0 ! Hom.H; A/ ! Hom.H; B/ ! Ext1 .H; 1/ ! 0 is exact. Proof. Injectivity of Hom.H; A/ ! Hom.H; B/ follows from Proposition 8.5.9. The only thing that is new is the surjectivity of Hom.H; B/ ! Ext1 .H; 1/. It suffices to show Ext1 .H; A/ D 0. So let 0 ! A ! HQ ! H ! 0 with

HQ D HQ 1 ! HQ 2 be an extension of Hodge structures. Choose a splitting s1 W HQ 1 ! kŒt of 0 ! kŒt ! HQ 1 ! H1 ! 0: Note that by Lemma 8.5.11 the natural map H1 ˝kŒt  C..t  #// ! H2 ˝C ŒŒt # C..t  #// is an isomorphism, and hence also HQ 1 ˝kŒt  C..t  #// ! HQ 2 ˝C ŒŒt # C..t  #// is an isomorphism. It follows that the splitting s1 induces a compatible splitting s2 W  HQ 2 ! C..t  #// and hence that HQ is a split extension of H by A .

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Consider now the following commutative diagram: 0 ? ? y

0 ? ? y 

! HomH .H; A/ ? ? y

ƒE ? ? y



LieE .C / ! HomH .H; B/ ? ? ? ? expE y y E.C / ? ? y

Ext1 .H; 1/ ? ? y

0

0

The left column is the uniformization short exact sequence. The right column is a short exact sequence from Proposition 8.6.1. The horizontal isomorphisms are given by Proposition 8.5.9. Theorem 8.6.2 (repeated from §8.1). The natural map h W Ext1 .MC ; C Œt/ ! Ext1 .H; 1/ is an isomorphism. In fact we will prove the following stronger statement: Proposition 8.6.3. The unique isomorphism u W E.C / ! Ext1 .H; 1/ that makes the above diagram commute coincides with the natural map h W E.C / D Ext1 .MC ; C Œt/ ! Ext1 .H; 1/: The proof of this proposition takes up the rest of this section. The idea of the proof is to first prove that u and h coincide on the torsion submodule of E.C /, and then to use a kind of density argument to conclude that they coincide on all of E.C /. Lemma 8.6.4. For all integers N > 0 and x 2 E.C / with t N x D 0 we have h.x/ D u.x/. Proof. We need to show that for all N > 0 and for all  2 ƒE the image of " WD t N  2 LieE

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in Ext1 .H; 1/ does not depend on the chosen path in the square ˛

LieE .C / ! HomH .H; B/ ? ? ? ? expE y ıy h

Ext1 .M; 1/ ! Ext1H .H; 1/: We first construct the 1-t-motif MQ corresponding to exp."/. By (8.4.1) it is defined by the short exact sequence 0 ! MQ ! M ˚ t N C Œt

.ˇN .";/;1/ N

!

t

C Œt=C Œt ! 0:

Applying the functor H one finds that h.exp."// is given by the kernel of the following morphism of Hodge structures (from the left column to the right column): M ftg ˚ t N kŒt ? ? y

! t N kŒt=kŒt ? ? y

M ŒŒt  # ˚ C ŒŒt  # !

(8.6.1)

0;

where the upper map is given by .m; f / 7! ˇN ."; m/  f: The kernel of (8.6.1) coincides with the kernel of M ftg ˚ t N kŒt ? ? y

!

t N kŒt=kŒt ? ? y

(8.6.2)

M ŒŒt  # ˚ C..t  #// ! C..t  #//=C ŒŒt  #; where both horizontal maps are given by .m; f / 7! ˇN ."; m/  f . On the other hand, the extension ı˛."/ is the kernel of the following subdiagram of (8.6.2) M ftg ˚ kŒt ? ? y

!

0 ? ? y

(8.6.3)

M ŒŒt  # ˚ C..t  #// ! C..t  #//=C ŒŒt  #: To conclude that the extensions ı˛."/ and h exp."/ coincide it suffices to observe that the quotient of the diagram (8.6.2) by the diagram (8.6.3) is an isomorphism when seen as a map from the left column to the right column.  Lemma 8.6.5. The homomorphism h W E.C / ! Ext1 .H; 1/ is locally analytic.

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Proof of the lemma. Denote the rank of M by r and choose a KŒt-basis of M . This induces a basis of M 0 , so let the r by r matrix A over C Œt represent the linear map  W MC0 ! MC with respect to these bases. Extensions of MC by C Œt are then represented by block matrices of the form   1 B 0 A where B is a length r row vector. Denote MQ .B/ the extension of MC by C Œt given by B by MQ .B/. There is an integer d such that all extension classes are represented by some B whose entries have degree at most d and it suffices to show that the homomorphism h is locally analytic as a function of the coefficients of these entries. The   -invariants of MQ .B/ftg are those pairs .v; w/ with v 2 C ftg and w 2 M ftg such that  .v/  v D B   .w/: Fix a basis w1 ; : : : ; wr of M ftg . Now there exists an " > 0 such that for all B whose entries have degree at most d and whose coefficients have absolute value at most " and for all i the power series B   .wi / 2 C ftg has all coefficients of absolute value smaller than 1. For these B, let vi be the unique element of C ftg such that  .vi /  vi D B   .wi / and which has all coefficients smaller than 1 in absolute value. This vi is given by the formula vi D

1 X

 j .B/

j D0

and clearly is an analytic function of B. The .vi ; wi / together with .1; 0/ form a basis of MQ .B/ftg and using this basis one sees that the resulting extension of Hodge structures depends analytically on the coefficients of B.  Lemma 8.6.6. There exists a neighbourhood U of 0 in LieE .C / on which h ı expE and u ı expE agree. Proof. Since h ı expE is locally analytic and u ı expE is analytic their difference f is a locally analytic function on LieE .C /. Now by Proposition 8.5.9 we have LieE .C / Š C ŒŒt  #r =L with L a free sub C ŒŒt  #-module of rank r, such that ƒE is the image of the natural map kŒtr ! C ŒŒt  #r =L. By Lemma 8.6.4 f vanishes on the image of k..t 1 //r ! C ŒŒt  #r =L: We will show that any locally analytic additive function on C ŒŒt #r =L that vanishes on the image of k..t 1 //r is locally constant, and so in particular that there exists a neighbourhood U of 0 on which f vanishes.

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Clearly enlarging L we may assume that L D .t #/s C ŒŒt #r for some positive integer s. Also, without loss of generality we may assume that r D 1. So it suffices to show that any locally analytic additive function g on the C -vector space V D C ŒŒt  #=.t  #/s C ŒŒt  # that vanishes on the image of k..t 1 // is locally constant. So assume g W V ! C is not locally constant. Then the kernel of g is a smooth analytic subgroup of V whose tangent space T V to the origin is a proper subvector space. For any positive integer i denote the image of t i 2 C ŒŒt  # in V by vi . The vi converge to 0 as i tends to infinity. Let Œvi  denote the image of vi in the projective space P.V /. The sequence .Œvi /i is periodic yet there is no proper subspace T V such that the sequence lies within P.T / P.V /, a contradiction.  Proof of Proposition 8.6.3. Let " be an element of LieE .C /. Note that for sufficiently n n large n we have that the actions of t p and # p on LieE .C / coincide. Possibly makn ing n even larger we may also assume that # p " lies in the U whose existence is asserted in Lemma 8.6.6. Using such n we find n

n

.h  u/ expE ."/ D t p .h  u/ expE .# p "/ D 0: This finishes the proofs of Proposition 8.6.3 and of Theorem 8.6.2.



References [1] Greg W. Anderson, t-motives. Duke Math. J. 53(2) (1986), 457–502. MR850546. [2] Greg W. Anderson, W. Dale Brownawell, and Matthew A. Papanikolas, Determination of the algebraic relations among special -values in positive characteristic. Ann. Math. (2), 160(1) (2004), 237–313. MR2119721. [3] Greg W. Anderson and Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132(1) (1990), 159–191 MR1059938. [4] Pierre Deligne, Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77. MR0498552. [5] Jean Dieudonné, Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique p > 0. VII. Math. Ann. 134 (1957), 114–133. MR0098146. [6] V. G. Drinfeld, Elliptic modules. Mat. Sb. (N.S.) 94(136) (1974), 594–627, 656. MR0384707. [7] Francis Gardeyn, t-Motives and Galois Representations. PhD thesis, Universiteit Gent, 2001. [8] Gérard Laumon, Cohomology of Drinfeld modular varieties. Part I, volume 41 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1996. MR1381898. [9] Matthew A. Papanikolas and Niranjan Ramachandran, A Weil-Barsotti formula for Drinfeld modules. J. Number Theory 98(2) (2003), 407–431. MR1955425.

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[10] Richard Pink, Hodge structures over function fields. pre-print, 1997. [11] Jean-Pierre Serre, Abelian l-adic representations and elliptic curves. McGill University lecture notes written with the collaboration of Willem Kuyk and John Labute, W. A. Benjamin, New York, Amsterdam, 1968. MR0263823. [12] N. R. Stalder, Algebraic Monodromy Groups of A-Motives. PhD thesis, ETH Zürich, 2007. [13] L. Taelman, Artin t-Motifs. J. Number Theory 129 (2009), 142–157. MR2468475. [14] L. Taelman, Special L-values of t-motives: a conjecture. Int. Math. Res. Not. IMRN (16) (2009), 2957–2977. MR2533793. [15] Sung Sik Woo, Extensions of Drinfeld modules of rank 2 by the Carlitz module. Bull. Korean Math. Soc. 32(2) (1995), 251–257. MR1356079.

Chapter 9

Multizeta in function field arithmetic Dinesh S. Thakur1

This is a brief report on recent work of the author (some joint with Greg Anderson) and his student on multizeta values for function fields. This includes definitions, proofs and conjectures on the relations, period interpretation in terms of mixed Carlitz-Tate t-motives and related motivic aspects. We also verify Taelman’s recent conjectures in special cases.

9.1 Introduction Euler’s multizeta values have been pursued recently again with renewed interest because of their emergence, for example in Grothendieck-Ihara program to study the absolute Galois group through the fundamental group of projective line minus three points and related studies of iterated extensions of Tate motives. Two types of multizeta were defined [T04, Sec 5.10] for function fields, one complex valued (generalizing Artin-Weil zeta function) and the other with values in Laurent series over finite fields (generalizing Carlitz zeta values). For the Fq Œt case, the first type was completely evaluated in [T04] (see [M06] for more detailed study in the higher genus case). We focus on the second analog in this report. In contrast to the classical division between the convergent versus the divergent (normalized) values, all the values are convergent in our case. In place of the sum or the integral shuffle relations, we have different kinds of relations: the shuffle type relations with Fp -coefficients and the relations with Fp .t/-coefficients. (Classically, of course, there is no such distinction, the rational number field being the prime field in that case). The first kind of relations have been understood (though not with a satisfying structural description) and show that the product of multizeta values can also be expressed as a sum of some multizeta values, so that the Fp -span of all multizeta values is an algebra. While [T09, Lr09, Lr10] conjectured and proved, in the special case A D Fq Œt, many such interesting relations, combinatorially quite involved to describe unlike the classical case, the proofs [T10] give the existence directly (for general A, defined below) rather than proving those conjectures. We only have examples of second kind of relations so far. As for the analogs of interconnections mentioned in the first paragraph, we can connect to absolute Galois group (through analog of Ihara power series [ATp]) and fundamental group approach in the Grothendieck-Ihara program only through the mixed motives [A86, AT09]. We describe some of these motivic aspects and relation with recent work of V. Lafforgue and L. Taelman. 1

Supported by NSA grants H98230-08-1-0049, H98230-10-1-0200.

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9.2 Multizeta values for function fields: Definitions 9.2.1 Notation Z D fintegersg ZC q

D fpositive integersg D a power of a prime p

Fq K

D a finite field of q elements D a function field of one variable with field of constants Fq

1 K1

D a place of K of degree one D Fq ..1=t// D the completion of K at 1

C1 A

D the completion of an algebraic closure of K1 D the ring of elements of K with no poles outside 1

Ad C D fmonic elements of A of degree d g n n D tq  t Q D .1/n niD1 Œi  `n ‘even’ D multiple of q  1 The simplest case is when A D Fq Œt and K D Fq .t/, with the usual notions of infinite place, degree and sign (in t). 9.2.2 Definition of multizeta values First we define the power sums. Given s 2 ZC and d  0, put Sd .s/ D

X a2Ad C

1 2 K; as

and given integers si 2 ZC and d  0 put X Sd .s1 ;    ; sr / D Sd .s1 /

Sd2 .s2 /    Sdr .sr / 2 K:

d >d2 > >dr 0

For si 2 ZC, we define multizeta value .s1 ;    ; sr / following [T04, Sec. 5.10] (where it was denoted by d to stress the role of the degree) by using the partial order on AC given by the degree, and grouping the terms according to it: .s1 ;    ; sr / D

X d1 > >dr 0

Sd1 .s1 /    Sdr .sr / D

X a1s1

1 2 K1 ;    arsr

where the second sum is over ai 2 Adi C with di ’s satisfying the conditions as in the first sum.

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We P say that this multizeta value (or rather the tuple .s1 ;    ; sr /) has depth r and weight si . Note we do not need s1 > 1 condition for convergence as in the classical case. This definition generalizes, in one way, the r D 1 case corresponding to the Carlitz zeta values. For discussion, references, interpolations and analytic theory, we refer to [G96, T04]. In [T04], we discuss interpolations of multizeta at finite and infinite primes.

9.3 First kind of relations between multizeta Recall that Euler’s multizeta values  (we will use Pthis clashing same notation only in this paragraph) are defined by .s1 ;    ; sr / D .ns11    nsrr /1 , where the sum is over positive integers n1 > n2 >    > nr and si are positive integers, with s1 > 1. We then have ‘sum shuffle relation’ X 1 X 1 .s1 /.s2 / D D .s1 ; s2 / C .s2 ; s1 / C .s1 C s2 /; ns11 ns22 just because n1 > n2 or n1 < n2 or n1 D n2 . Since there are many polynomials of given degree (or norm), this usual proof of the sum shuffle relations fails. Theorem 9.3.3 below shows that in place of the three multizeta on the right of the displayed equation, in out case there can be arbitrarily large number of multizeta, depending on si ’s. In fact, it can be seen that naive analogs of the sum or integral shuffle relations fail. The Euler identity .2; 1/ D .3/ fails in our case, for simple reason that degrees on both sides do not match. 9.3.1 Examples However, the multizeta values satisfy many interesting combinatorially involved new relations [T09] which we now describe first. Theorem 9.3.1. (1) .ps1 ;    ; psk / D .s1 ;    ; sk /p . (2) (Carlitz) If q  1 divides s, .s/=Q s 2 K, where Q is a fundamental period of the Carlitz module, (3) Any classical sum-shuffle relation with fixed si ’s works for q large enough. For example, if s1 C s2  q, we have .s1 /.s2 / D .s1 C s2 / C .s1 ; s2 / C .s2 ; s1 /: Now we describe simplest examples when the hypothesis of part (3) is violated. Theorem 9.3.2. (1) When a; b  q and a C b > q, we have .a/.b/ D .a C b/ C .a; b/ C .b; a/ C .a C b/.a C b  q C 1; q  1/: (2) When 1  b  q, q ¤ 2, we have .b/.2q/ D .2q C b/ C .b; 2q/ C .2q; b/ C b.q C b C 1; q  1/ ! bC1 C .b C 2; 2q  2/: 2

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(3) .q n  1/..q  1/q n / D .q nC1  1/ C .q n  1; .q  1/q n /: Note in the special cases, such as when a or b is q  1 or when p divides a C b, three depth 2 multizetas in part (1) can mix or disappear giving difference appearance to the identities. P Theorem 9.3.3. Let q D 2. We have (1) .1/.a/ D .1 C a/ C a1 i D1 .i; a C 1  i /; P (2) If b is odd, .2/.b/ D .2 C b/ C 1i .b3/=2 .2i C 1; 1 C b  2i /, P and if b is even, .2/.b/ D .2 C b/ C 1i b=21 .2i; b C 2  2i /: 9.3.2 Conjectural recursive recipe In [T09], for q D 2 a full conjectural description of how the product of two zeta values can be described as the sum of multizetas is given. Here we just give an example of the recursive recipe: In the notation of the first theorem of the next subsection, we let q D 2, so that fi D 1. Let a D 19. Then the aj ’s for b replaced by b C 32 are given by those for b and b C 32; b C 31; b C 28; b C 27; b C 24; b C 23; b C 20; b C 19. In other words, at each recursion step of 32, eight new multizeta values described get added. In [Lr09, Lr10] these conjectures were partially generalized, giving full recursive step (but not the initial values) when q is a prime, and a partial description for any q as follows. Again in the notation of the first theorem of the next subsection, we have a recipe giving fi and ai , given .a; b/. For a fixed a it is recursive in b of recursion the smallest integer such that a  pm , and at each length .q  1/pm , where m is Q recursive step one adds ta D .p  j /j new multizeta terms, where j is the number of j ’s in the base p expansion of a  1. David Goss has recently stressed the role of the ‘digit expansion permutation symmetries (  below)’ in the theory of Carlitz–Goss zeta function, with respect to its zeros, orders of vanishing etc. If denotes arbitraryP permutationP of the set of nonnegative integers, then we have resulting action  . ni q i / WD ni q .i / on digit expansions base q. Note that, with q D p, and given any and a, the same recipe works for both a 1 and  .a 1/ for the recursion length and for the number of multizetas to be added, if we do not insist on the smallest recursion length. Thus a strong form of such symmetry shows up in the theory of multizeta values. In [T09], it was also described how the ‘soliton’ technology allows us to prove any such relation (for fixed si ’s). This is much simplified by the next theorem. It seems plusible that the complicated combinatorial recipe above can be deudced from the next theorem, but this has not been done yet. 9.3.3 General theorem In [T10], we proved all relations of the first kind, namely with coefficients in the prime field, bypassing nice explicit or recursive relations conjectured above. This is done as follows. First we consider X Sd .a/Sd .b/  Sd .a C b/ D fi Sd .ai ; a C b  ai /; (*) with fi 2 Fp .

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Theorem 9.3.4. (1) Let A D Fq Œt. Given a; b 2 ZC, there are fi 2 Fp and ai 2 ZC, such that (*) holds for d D 1. (2) Fix q. If (*) holds for some fi 2 Fp and ai 2 ZC for d D 1 and A D Fq Œt, then (*) holds for all d  0 and for all A (corresponding to the given q). In this case, we have the shuffle relation X .a/.b/  .a C b/  .a; b/  .b; a/ D fi .ai ; a C b  ai /: (**) P (3) Sd .a1 ;    ; ar /Sd .b1 ;    ; bk / can be expressed as fi SP  ; ci mi /, d .ci1 ;   P 2 F , c ’s and m ’s being independent of d , and with a C bj D with f i p ij i i P c and m  r C k. ij i j (4) For any A, the product of multizeta values can be expressed as a sum of some multizeta values, such an expression preserving total weight and keeping depth filtration. In particular, the Fp -span of all the multizeta values is an algebra. Note that this theorem gives an effective procedure for expressing a given product of multizeta values as a sum of multizeta values. The resulting proof of such an expression is much simpler than the process mentioned above.

9.4 Second kind of relations between multizeta In place of the Euler identity, we have Theorem 9.4.1. When q D 3, we have .1; 2/ D .3/=`1 D .3/=.t  t 3 /. More generally, for any q, we have .m; m.q  1// D .mq/=`m 1; mq .1; q 2  1/ D .q 2 /.1=`2 C 1=`1 / Remark 9.4.2. When q D 3 (when ‘even’ agrees with even), in comparison with the Euler identity we have an order switch. But, using the sum shuffle identity 2.3 for .1/.2/, we can express .2; 1/ in terms of .3/ D .1/3 and . Q Here are some expressions involving logarithms of algebraic quantities. Theorem 9.4.3. (1) We have   .1; q  1/ D  q 3 3



1 t 1 C C `3 `2 `2

 

1 3 .log.t 1=q //q `2

(2) .1; q n  1/ is (explicit) .q n / times a rational plus linear combination of q-power powers of logarithms of q-power roots of polynomials.

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9.5 Period interpretation and motivic aspects In [AT09], the following theorem was proved. Theorem 9.5.1. Given multizeta value .s1 ;    ; sr /, we can construct explicitly iterated extension of Carlitz–Tate t-motives over FqŒt which has as period matrix entry this multizeta value (suitably normalized). This generalizes result [AT90] connecting .s/ to the logarithm of and explicit algebraic point on Carlitz–Tate t-motive C ˝s , or equivalently to the period of one step extensions of such t-motives. In [T92, A94, A96], these were generalized somewhat to higher genus and L-function situation. In [ATp] Ihara power series theory is developed. (It is meta-abelian etale aspect of the Grothendieck-Ihara program [I91], whereas the multizeta values should be DeRham-Betti aspect at nilpotent level.) In studying the big Galois representation it provides, complicated digit combinatorics [A07], just as what we encountered above in describing the relations between multizeta, enters the picture. The extension giving zeta values is also linked to [ATp] analog of Deligne-Soule cocycles, which have connections with ‘cyclotomic unit module’ of [A96] in addition to zeta values (though no K-theory link yet). While all these connections with analogs of motives are exciting, and concrete, and while the natural constructions here lead to much stronger transcendence results than in the number field case, their larger perspective is not yet fully understood even conjecturally, and we lack a good analogous description to that of Deligne (and others) linking zeta and multizeta values to motivic extensions and K-theory. Recent exciting works by (i) V. Lafforgue [L09] giving an analog of Bloch–Kato, Fontaine–Perrin–Riou work relating p-adic L-values and motivic extensions; and (ii) L. Taelman [Ta10] (which I learned about at this Banff workshop) defining a notion of good extensions (and of class module that we were after [T94, p. 163] for long!) and conjecturing a class number formula for the L-values at the infinite place; represent excellent steps in this direction. We end this short report by giving some calculations, inspired by these results, verifying Taelman’s conjectures (as hinted in remark 1 of [Ta10]) in the special case of higher genus, class number one [T92] results mentioned above. In the notation of [Ta10], we deal with R D A, where A (below) is the base generalizing A D kŒt there and (below) generalizing the Carlitz module E there in conjecture 1. We deal with examples A-D on pages 192-194 [T92] and recalled below. These are known to be all the examples A having class number one, and positive genus. The sign normalized rank one Drinfeld A-modules ’s for these A’s are given there explicitly, for the sign function with the sign of x and y to be 1. Let log .z/ and P qi z =di be the corresponding logarithm and exponential ree.z/ WD exp .z/ D spectively.

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Theorem 9.5.2. With the notation as above, for the examples (A)-(C), 1 X X 1 A .1/ WD D log .g/; a d D0 a2Ad C

where g is the unique generator (with its exponential being one-unit at infinity) of Taelman’s .A/ WD e.K1 / \ A, which is a rank one A-module under . Further, Taelman’s class module is trivial in each case, and thus the zeta value is class module order times the regulator. Proof. Example (A) is A D F3 Œx; y=y 2 D x 3  x  1. We show [T92, Thm VI] that .1/ D log .y  1/. Now we claim that y  1 generates .A/ as A-module under . From the result quoted above, y  1 2 .A/. Using the functional equations given by explicit description of , we calculate the di ’s and see that the degree of di is .i  2/3i , for i > 0, and is 0 for i D 0. If z 2 K1 has degree d , then we thus see that degree of e.z/ is less than 1 for d < 1, 3d C1 for d  0, and 0 for d D 1. So it is immediate, for example, that the degree two elements x; x ˙ 1 do not belong to .A/. Using the Fq -linearity of e, it is enough to show that 1 does not belong to .A/. If e.z/ D 1, z is of the form ˙x=yC terms of the lower degree. Using d1 D 1=y and degrees of di given above, ignoring degrees less than 1, we see that 1 D x=y C x 3 =y 2 D x=y C x 3 =.x 3  x  1/ and thus we have an element of degree 1 in e.K1 / contradicting the estimates above. (Another way to show this is to use the explicit and degree estimates and show that y  1 is not in the module, if it is not a generator). Thus this rank one module is generated by y  1, which is in fact the unique generator whose exponential is a one-unit at infinity. Example (B) is A D F4 Œx; y=y 2 C y D x 3 C 3 . We show [T92, Thm. VIII] that e..1// D x 8 C x 4 C x 2 C x. The calculation as above, shows that degree of di is now .i  3/4i for i > 0 and 0 for i D 0, so that if degree of z 2 K1 is d , then degree of e.z/ is  3 for d  3, D 4d C2 for d  1, and 0 for d D 2. Thus without loss of generality, by degree considerations, the generator has degree 0; 4 or 16. Again, a straight degree (and x .1/ D x 8 C x 2 C x C 1) calculation shows that the first two possibilities make it impossible for the module to contain x 8 C x 4 C x 2 C x. Thus x 8 C x 4 C x 2 C x is generator of .A/, again a unique generator whose exponential is one-unit at infinity. Example (C) is A D F2 Œx; y=y 2 C y D x 3 C x C 1. We show [T92, Thm. X] that e..1// D 0 and in fact that .1/ is the fundamental period (value of logarithm of zero), as d1 D 1. Thus .A/ is the torsion module generated by zero. (Compare [Ta10] with the Carlitz module case, where it is generated by 1, as .1/ D log.1/, which was essentially [AT90, p. 181] proved by Carlitz. In that case, when q D 2, it is torsion module f0; 1; t; t C 1g generated by t 2 C t-torsion point 1). This takes care of the first part of the theorem. As explained in [Ta10] to show that the class module is trivial, it is enough to show the Claim: X WD e.K1 / C A D K1 .

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It is enough to show that K1 X . For example A (B respectively), by the calculation of degree di ’s above (or using that log.z/ converges for z of degree less than 3=2 (8=3 respectively)), we see that e.K1 / contains all elements in K1 of each degree less than 1 (2 respectively) and A contains elements of all non-negative degrees except 1. Hence it remains to show that X contains elements of degree 1; 1 (2; 1; 1 respectively). We take care of the remaining degrees as follows. For example A: e.x=y/ is x=y (of degree 1) plus y.x=y/3 (which is 1 2 A plus an element of degree 4, and thus in X ) plus an element of degree 9 (and thus in X ), so that degree 1 is also taken. Also, e.y=x/ is y=x (of degree 1) plus y.y=x/3 (which is x 3 C x C 1 2 A plus an element of degree 2 and thus in X ) plus .y 9 C y 13 .x 3  x//=.x 9 .x 9  x// (which is y 3 C y plus element of degree 3 and thus in X ) plus (sum of) terms of degree non-positive (and thus in X ). This proves the claim in example A. For example B: Since e.1=x/ is 1=x (of degree 2) plus .x 4 C x/=x 4 2 X plus terms of degree less than 15 (and thus in X ), we get degree 2. The expansions of e.x=y/ (e.y=x/ respectively) consist of degree 1 (1 respectively) plus two (four respectively) terms which are rational functions of x (and thus subtracting an appropriate polynomial in x contribute degree  2 and thus in X ) plus terms of degree  2 (and thus in X ). This proves the claim in example B. For example C: Calculation is similar and even simpler. Now degree of di is .i  1/2i , for i  1, and thus (alternately, logarithm converges for elements of degree less than zero) all elements of degree less than 0 are in the image of exponential. Only degree one is missing in elements of A, but as above we see that e.y=x/ C x is of degree one, proving the claim.  Example (D) is A D F2 Œx; y=y 2 C y D x 5 C x 3 C 1 of genus 2. Here .1/ is x 2 C x times the fundamental period. The only x 2 C x-torsion is zero, thus class module should have order x 2 C x, as we have verified directly.

9.6 Updates added on 23 August 2011 In his doctoral thesis work with the author, Alejandro Lara Rodriguez has now proved [Lr10, Lr11] most of the conjectures mentioned in 3.2 and has also proved [Lr11, Thm. 7.1, Cor. 7.2] the following theorem by making the recipe of Theorem 9.3.4 explicit in the depth 2 special case. Theorem 9.6.1. Let q be a power of prime p, a; b be positive integers and m be the smallest integer such that a C b  pm . Then we have .a/.b/  .a C b/  .a; b/  .b; a/ D

b1 X i D0

fi .b  i; a C i / C

a1 X j D0

gj .a  j; b C j /;

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where, if q D 2, we have ! 2m  a ; fi D i

! 2m  b gj D ; j

and more generally, for q arbitrary, with Ha;b .t/ given by 0 1 X  1 @ q1 m a  1/p a .t C #/q1  1 mod t aCb A ; Ha;b .t/ D a .t t  #2Fq

we have f .t/ WD f0 C f1 t C    C fb1 t b1 D Ha;b .t/, g.t/ WD g0 C g1 t C    C ga1 t a1 D Hb;a .t/, where Hb;a is obtained from Ha;b .t/ by interchanging a and b.

9.7 Updates added on 5 February 2013 Using similar, but better partial fraction decomposition formula, Huei Jeng Chen [Ch15] has simplified the above recipe considerably to .a/.b/  .a C b/  .a; b/  .b; a/ ! !! X a1 j  1 b1 j  1 C .1/ .a C b  j; j /; D .1/ a1 b1 where the sum is over j which are multiples of q  1 and 0 < j < a C b. [LTp] proved and conjectured which shuffle relations survive in Theorem 9.3.4, if we drop the condition on A that the infinite place is of degree one. Chang [C14] and Mishiba [Mip1, Mip2] have proved interesting general transcendence theorems for the multizeta values, making use of Theorem 9.5.1 and the transcendence criterion of Anderson, Brownawell and Papanikolas. In a recent preprint, Kirti Joshi has constructed a neutral, tannakian, F -linear category of mixed t-motives, and also of mixed Carlitz–Tate t-motives containing all those mentioned in Theorem 9.5.1, thus providing a natural playground for multizeta and setting the stage for exploring analogs of various recent motivic works related to multizeta.

9.8 Updates added on 27 April 2015 The shuffle relations of Theorem 4 are thus special quadratic relations whose use reduces the study of the algebraic relations between multizeta (at least in principle) to the study of linear relations between them.

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Results, conjectures and algorithms to decide when the ratio of a multizeta value with a zeta value is algebraic have appeared in [LT14, CPY, KL, Chp]. Good understanding of all the multizeta linear relations with Fp .t/-coefficients is slowly emerging through the extensive numerical calculation based on the lattice reduction method, parallel to the similar one performed by Zagier, by author’s current doctoral student George Todd. Based on this, Todd has conjectured the following formula for the dimension dn of the span of multizetas of a given weight n: dn should be 2n1 , 2n1  1, or dn1 C    C dnq depending on 1  n < q, n D q, n > q respectively. This has led the author to update his earlier speculation that .s1 ;    ; sk / with si < q should be linearly independent, to the speculation that a basis for the linear span for weight n could be obtained by exactly such multizetas for n  q, and for n > q by adding s1 D i in front to tuples in a basis for weights n  i , for i  q. This is, so far, consistent with Todd’s data. Given a linear relation which works at the Sd level (called ‘fixed’, e.g. as in (*) before Theorem (4)) or at certain Sd Ci -levels for some i (e.g. i D 0; 1, called ‘binary’, e.g. as in first relation in Theorem 5), one can generate more relations by multiplying them on the left or right and using the shuffle relations, obtained from Theorem 4, at Sd or S