Algebraic Groups and Number Theory [1, 2 ed.] 9780521113618, 9781139017756

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C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S 2 0 5 Editorial Board J. BERTOIN, B. BOLLOBÁS, W. FULTON, B. KRA, I. MOERDIJK, C. PRAEGER, P. SARNAK, B. SIMON, B. TOTARO

ALGEBRAIC GROUPS AND NUMBER THEORY The first edition of this book provided the first systematic exposition of the arithmetic theory of algebraic groups. This revised second edition, now published in two volumes, retains the same goals, while incorporating corrections and improvements as well as new material covering more recent developments. Volume I begins with chapters covering background material on number theory, algebraic groups, and cohomology (both abelian and non-abelian), and then turns to algebraic groups over locally compact fields. The remaining two chapters provide a detailed treatment of arithmetic subgroups and reduction theory in both the real and adelic settings. Volume I includes new material on groups with bounded generation and abstract arithmetic groups. With minimal prerequisites and complete proofs given whenever possible, this book is suitable for self-study for graduate students wishing to learn the subject as well as a reference for researchers in number theory, algebraic geometry, and related areas. Vladimir Platonov is Principal Research Fellow at the Steklov Mathematical Institute and the Scientific Research Institute for System Analysis of the Russian Academy of Sciences. He has made fundamental contributions to the theory of algebraic groups, including the resolution of the Kneser–Tits problem, a criterion for strong approximation in algebraic groups, and the analysis of the rationality of group varieties. A recipient of the Lenin Prize (1978) and the Chebyshev Gold Medal for outstanding results in mathematics (2022), he is currently an academician of the Russian Academy of Sciences and of the National Academy of Sciences of Belarus, and a member of the Indian National Academy of Sciences. Andrei Rapinchuk is McConnell-Bernard Professor of Mathematics at the University of Virginia. His contributions to the arithmetic theory of algebraic groups include a variety of results concerning the normal subgroup structure of the groups of rational points of algebraic groups, the congruence subgroup and metaplectic problems, and different aspects of the local-global principle. He has also applied the theory of arithmetic groups to investigate isospectral locally symmetric spaces. Igor Rapinchuk is Associate Professor of Mathematics at Michigan State University. His current research deals mainly with the emerging arithmetic theory of algebraic groups over higher-dimensional fields, focusing on finiteness properties of groups with good reduction, local-global principles, and abstract homomorphisms.

C A M B R I D G E S T U D I E S I N A DVA N C E D M AT H E M AT I C S Editorial Board J. Bertoin, B. Bollobás, W. Fulton, B. Kra, I. Moerdijk, C. Praeger, P. Sarnak, B. Simon, B. Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing, visit www.cambridge.org/mathematics. Already Published 167 D. Li & H. Queffelec Introduction to Banach Spaces, II 168 J. Carlson, S. Müller-Stach & C. Peters Period Mappings and Period Domains (2nd Edition) 169 J. M. Landsberg Geometry and Complexity Theory 170 J. S. Milne Algebraic Groups 171 J. Gough & J. Kupsch Quantum Fields and Processes 172 T. Ceccherini-Silberstein, F. Scarabotti & F. Tolli Discrete Harmonic Analysis 173 P. Garrett Modern Analysis of Automorphic Forms by Example, I 174 P. Garrett Modern Analysis of Automorphic Forms by Example, II 175 G. Navarro Character Theory and the McKay Conjecture 176 P. Fleig, H. P. A. Gustafsson, A. Kleinschmidt & D. Persson Eisenstein Series and Automorphic Representations 177 E. Peterson Formal Geometry and Bordism Operators 178 A. Ogus Lectures on Logarithmic Algebraic Geometry 179 N. Nikolski Hardy Spaces 180 D.-C. Cisinski Higher Categories and Homotopical Algebra 181 A. Agrachev, D. Barilari & U. Boscain A Comprehensive Introduction to Sub-Riemannian Geometry 182 N. Nikolski Toeplitz Matrices and Operators 183 A. Yekutieli Derived Categories 184 C. Demeter Fourier Restriction, Decoupling and Applications 185 D. Barnes & C. Roitzheim Foundations of Stable Homotopy Theory 186 V. Vasyunin & A. Volberg The Bellman Function Technique in Harmonic Analysis 187 M. Geck & G. Malle The Character Theory of Finite Groups of Lie Type 188 B. Richter Category Theory for Homotopy Theory 189 R. Willett & G. Yu Higher Index Theory 190 A. Bobrowski Generators of Markov Chains 191 D. Cao, S. Peng & S. Yan Singularly Perturbed Methods for Nonlinear Elliptic Problems 192 E. Kowalski An Introduction to Probabilistic Number Theory 193 V. Gorin Lectures on Random Lozenge Tilings 194 E. Riehl & D. Verity Elements of ∞-Category Theory 195 H. Krause Homological Theory of Representations 196 F. Durand & D. Perrin Dimension Groups and Dynamical Systems 197 A. Sheffer Polynomial Methods and Incidence Theory 198 T. Dobson, A. Malniˇc & D. Marušiˇc Symmetry in Graphs 199 K. S. Kedlaya p-adic Differential Equations 200 R. L. Frank, A. Laptev & T. Weidl Schrödinger Operators:Eigenvalues and Lieb–Thirring Inequalities 201 J. van Neerven Functional Analysis 202 A. Schmeding An Introduction to Infinite-Dimensional Differential Geometry 203 F. Cabello Sánchez & J. M. F. Castillo Homological Methods in Banach Space Theory 204 G. P. Paternain, M. Salo & G. Uhlmann Geometric Inverse Problems 205 V. Platonov, A. Rapinchuk & I. Rapinchuk Algebraic Groups and Number Theory, I (2nd Edition) 206 D. Huybrechts The Geometry of Cubic Hypersurfaces

Algebraic Groups and Number Theory Volume I Second Edition V L A D I M I R P L ATO N OV Steklov Institute of Mathematics, Moscow ANDREI RAPINCHUK University of Virginia IGOR RAPINCHUK Michigan State University

The translation of the first Russian edition was prepared by Rachel Rowen.

Shaftesbury Road, Cambridge CB2 8EA, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 103 Penang Road, #05–06/07, Visioncrest Commercial, Singapore 238467 Cambridge University Press is part of Cambridge University Press & Assessment, a department of the University of Cambridge. We share the University’s mission to contribute to society through the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521113618 DOI: 10.1017/9781139017756 Originally published in Russian as Алгебраические группы и теория чисел by Nauka c Nauka, 1991 Publishing House in 1991 First published in English by Academic Press, Inc. 1994 c 1994 by Academic Press, Inc. English translation c Vladimir Platonov, Andrei Rapinchuk, and Igor Rapinchuk 2023 Second Edition This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press & Assessment. Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. A Cataloging-in-Publication data record for this book is available from the Library of Congress. ISBN 978-0-521-11361-8 Hardback Cambridge University Press & Assessment has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface to the Second English Edition Preface to the First English Edition (1994) Preface to the Russian Edition (1991) List of Notations

page vii ix x xiii 1 1 11 17 29 42

1

Algebraic Number Theory 1.1 Algebraic Number Fields, Valuations, and Completions 1.2 Adeles and Ideles; Approximation; Local-Global Principle 1.3 Cohomology 1.4 Simple Algebras over Local Fields 1.5 Simple Algebras over Number Fields

2

Algebraic Groups 2.1 Structure of Algebraic Groups 2.2 Classification K-Forms Using Galois Cohomology 2.3 Classical Groups 2.4 Some Results from Algebraic Geometry

53 53 77 90 110

3

Algebraic Groups over Locally Compact Fields 3.1 Topology and Analytic Structure 3.2 The Archimedean Case 3.3 The Non-Archimedean Case 3.4 Elements of Bruhat–Tits Theory 3.5 Basic Results from Measure Theory

124 125 137 155 172 185

4

Arithmetic Groups and Reduction Theory 4.1 Arithmetic Groups 4.2 An Overview of Reduction Theory. Reduction in GLn (R) 4.3 Reduction in Arbitrary Groups 4.4 Group-Theoretic Properties of Arithmetic Groups

198 198 203 220 227

v

Contents

vi

4.5 4.6 4.7 4.8 4.9 5

Compactness of GR /GZ The Finiteness of the Volume of GR /GZ Concluding Remarks on Reduction Theory Finite Arithmetic Groups Abstract Arithmetic Groups

246 253 265 271 284

Adeles 5.1 Basic Definitions 5.2 Reduction Theory for GA Relative to GK 5.3 Compactness and the Finiteness of the Volume of GA /GK 5.4 Reduction Theory for S-Arithmetic Subgroups

293 294 305 314 321

Bibliography Index

338 358

Preface to the Second English Edition

The first edition of Algebraic Groups and Number Theory [AGNT] was published about 30 years ago, first in Russian and then in English, and the book quickly became a standard reference in the arithmetic theory of algebraic groups. Some time ago, Burt Totaro, in his capacity as one of the editors of the series Cambridge Studies in Advanced Mathematics, suggested to prepare and publish a second edition. Following up on our preliminary agreement, Diana Gillooly, a senior editor for mathematics at Cambridge University Press at the time, did a marvelous job of getting the project on track by first resolving some legal issues with Academic Press, which had published the first English edition as volume 139 of their Pure and Applied Mathematics series, and then by arranging for the text of the present edition to be retyped in LATEX since the TEX file used by AP had not survived. At the outset, the decision was made to implement a couple of structural alterations in the second edition: the new edition will be published in two volumes, and we have opted to omit Chapter 8 of the original version in order to make room for various additions without significantly increasing the overall size of the book. While Diana and her staff were very efficient in completing the work on their end, we were much less efficient in doing our part. For a variety of reasons, ranging from our involvement in other projects to some personal circumstances, the revision process was moving forward rather slowly, despite the interest and encouragement of many colleagues. The turnaround occurred when the third-named author joined the team, and now we are very pleased to present Volume I of the second edition of [AGNT], which comprises the material of Chapters 1–5 of the first edition. A new Section 4.9 was written for this edition to reflect the joint work of the late Fritz Grunewald, an outstanding mathematician and a dear friend, with the first-named author on finite extensions of arithmetic groups. Furthermore, numerous edits, corrections of typos and certain notations as well as of some mathematical inaccuracies, and also updates have been made to the text. vii

viii

Preface to the Second English Edition

We hope that these changes have improved the clarity of the exposition and the readability of the book. During the entire revision process, we have benefitted greatly from correspondence with numerous colleagues, who have generously shared with us their comments, remarks, and suggestions – we thank them all for their enthusiasm and help. Special thanks are due to Dave Witte Morris, who carefully read the entire manuscript and suggested many corrections and improvements. We are also grateful to Brian Conrad for pointing out an inaccuracy in Section 5.1, which has been corrected in the present edition. We thankfully acknowledge the comments we have received from Skip Garibaldi, Alex Lubotzky, Lam Pham, Gopal Prasad, Jinbo Ren, and many others. Let us also point out that the original English translation was done by Rachel Rowen, and we express our thanks for her work. Last but not least, we thank Kaitlin Leach and her staff at Cambridge University Press for their assistance in the final stretch of the project. Finally, since Volume II of the second edition (comprising Chapters 6–7 and 9 of [AGNT]) will appear somewhat later, the cross-references in the current book to the material in Chapters 6–9 of [AGNT] have not been changed. Vladimir P. Platonov Andrei S. Rapinchuk Igor A. Rapinchuk

Preface to the First English Edition (1994)

After the publication of the Russian edition of this book, some new results were obtained in the area; however, we decided not to make any changes or add appendices to the original text, since that would have affected the book’s balanced structure without contributing much to its main contents. As the editor for the translation, A. Borel took considerable interest in the book. He read the first version of the translation and made many helpful comments. We also received a number of useful suggestions from G. Prasad. We are grateful to them for their help. We would also like to thank the translator and the publisher for their cooperation. V. Platonov A. Rapinchuk

ix

Preface to the Russian Edition (1991)

This book provides the first systematic exposition in the mathematical literature of the theory that developed at the meeting ground of group theory, algebraic geometry, and number theory. This line of research emerged fairly recently as an independent area of mathematics, often called the arithmetic theory of (linear) algebraic groups. In 1967, A. Weil wrote in the foreword to Basic Number Theory: “In charting my course, I have been careful to steer clear of the arithmetical theory of algebraic groups; this is a topic of deep interest, but obviously not yet ripe for book treatment.” The sources of the arithmetic theory of linear algebraic groups lie in classical research on the arithmetic of quadratic forms (Gauss, Hermite, Minkowski, Hasse, Siegel), the structure of the group of units in algebraic number fields (Dirichlet), and discrete subgroups of Lie groups in connection with the theory of automorphic functions, topology, and crystallography (Riemann, Klein, Poincaré, and others). Its most intensive development, however, has taken place over the past 20 to 25 years. During this period, reduction theory for arithmetic groups was developed, properties of adele groups were studied, and the problem of strong approximation was solved, important results on the structure of groups of rational points over local and global fields were obtained, various versions of the local-global principle for algebraic groups were investigated, and the congruence problem for isotropic groups was essentially solved. It is clear from this far from exhaustive list of major accomplishments in the arithmetic theory of linear algebraic groups that a wealth of important material of particular interest to mathematicians in a variety of areas has been amassed. Unfortunately, to this day, the major results in this area have appeared only in journal articles, despite the long-standing need for a book presenting a thorough and unified exposition of the subject. The publication of such a book, however, has been delayed largely due to the difficulty inherent in unifying the

x

Preface to the Russian Edition (1991)

xi

exposition of a theory built on an abundance of far-reaching results and a synthesis of methods from algebra, algebraic geometry, number theory, analysis, and topology. Nevertheless, we finally present such a book to the reader. The first two chapters are introductory and review major results of algebraic number theory and the theory of algebraic groups, which are used extensively in later chapters. Chapter 3 presents basic facts about the structure of algebraic groups over locally compact fields. Some of these facts also hold for any field complete relative to a discrete valuation. The fourth chapter presents the most basic material about arithmetic groups, based on results of A. Borel and Harish-Chandra. One of the primary research tools for the arithmetic theory of algebraic groups is adele groups, whose properties are studied in Chapter 5. The primary focus of Chapter 6 is a complete proof of the Hasse principle for simply connected algebraic groups, published here in definitive form for the first time. Chapter 7 deals with strong and weak approximations in algebraic groups. Specifically, it presents a solution of the problem of strong approximation and a new proof of the Kneser–Tits conjecture over local fields. The classical problems of the number of classes in the genus of quadratic forms and of the class numbers of algebraic number fields influenced the study of class numbers of arbitrary algebraic groups defined over a number field. The major results achieved to date are set forth in Chapter 8. Most of these are due to the authors. The results presented in Chapter 9 for the most part are new and rather intricate. Recently, substantial progress has been made in the study of groups of rational points of algebraic groups over global fields. In this regard, one should mention the works of Kneser, Margulis, Platonov, Rapinchuk, Prasad, Raghunathan, and others on the normal subgroup structure of groups of rational points of anisotropic groups and the multiplicative arithmetic of skew fields, which use most of the machinery developed in the arithmetic theory of algebraic groups. Several results appear here for the first time. The final section of this chapter presents a survey of the most recent results on the congruence subgroup problem. Thus, this book touches on almost all the major results of the arithmetic theory of linear algebraic groups obtained to date. The questions related to the congruence subgroup problem merit exposition in a separate book, to which the authors plan to turn in the near future. It should be noted that many wellknown assertions (especially in Chapters 5, 6, 7, and 9) are presented with new proofs that tend to be more conceptual. In many instances, a geometric approach to the representation theory of finitely generated groups is efficiently used.

xii

Preface to the Russian Edition (1991)

In the course of our exposition, we formulate a considerable number of unresolved questions and conjectures, which may give impetus to further research in this actively developing area of contemporary mathematics. The structure of this book, and exposition of many of its results, was strongly influenced by V. P. Platonov’s survey article, “Arithmetic theory of algebraic groups,” published in Uspekhi matematicheskikh nauk (1982, No. 3, pp. 3– 54). Much assistance in preparing the manuscript for print was rendered by O. I. Tavgen, Y. A. Drakhokhrust, V. V. Benyashch-Krivetz, V. V. Kursov, and I. I. Voronovich. Special mention must be made of the contribution of V. I. Chernousov, who furnished us with a complete proof of the Hasse principle for simply connected groups and devoted considerable time to polishing the exposition of Chapter 6. To all of them we extend our most sincere thanks. V. P. Platonov A. S. Rapinchuk

Notations

K ∗ (resp., K + ) – multiplicative (resp., additive) group of a field K V K – set of (equivalence classes of) valuations of a number field K K (resp. V K ) – subset of Archimedean (resp., non-Archimedean) V∞ f valuations Kv – completion of K with respect to a valuation v ∈ V K Ov – valuation ring of Kv (ring of v-adic integers), for v non-Archimedean pv – valuation ideal of Ov Uv – multiplicative group of Ov (group of v-adic units) V (a) = {v ∈ VfK : a ∈ / Uv } for a ∈ K × OK or O – ring of integers of a number field K K) O(S) – ring of S-integers of K (for S ⊂ V K containing V∞ w|v – extension of valuations AK or A – ring of adeles of a number field K A(S) – subring of S-integral adeles A(∞) – subring of integral adeles AS – ring of S-adeles Af – ring of finite adeles AS (T) – the ring of T-integral S-adeles (for T ⊃ S) JK or J – group of ideles J (S) – subgroup of S-integral ideles J (∞) – subgroup of integral ideles hK – the class number of K Br(K) – the Brauer group of K NL/K (resp., TrL/K ) – norm (resp., trace) in a finite field extension L/K NrdA/K (resp., TrdA/K ) – reduced norm (resp., reduced trace) for a central simple algebra A over K Fq – field with q elements

xiii

List of Notations

xiv

Z (resp., Q, R, C, Qp ) – integers (resp., rational, real, complex, and p-adic numbers) An (resp., Pn ) – n-dimensional affine (resp., projective) space Gm – 1-dimensional split torus Ga – 1-dimensional split connected unipotent group SLn (D), SUm (D, f ) etc. – classical groups over division algebras SLn (D), SUm (D, f ) etc. – corresponding algebraic groups RL/K – restriction of scalars X(G) – group of characters of an algebraic group G X∗ (G) – group of cocharacters R(T, G) – root system of a connected algebraic group G with respect to a maximal torus T W (T, G) – corresponding Weyl group L(G) or g – Lie algebra of an algebraic group G R(0, G) – variety of representations of a finitely generated group 0 into an algebraic group G Rn (0) – the variety of n-dimensional representations of 0 Tx (X ) – tangent space to a variety X at a point x dx f – differential of a morphism f at a point x rank G or rk G – absolute rank of an algebraic group G rankK G or rkK G – rank of G over K (K-rank) P rankS G or rkS G – S-rank of G, i.e., rankKv G (for finite S ⊂ V K ) v∈S

GK – group of K-points of an algebraic K-group G GO – group of integral points GO(S) – group of S-integral points GA – group of adeles of an algebraic group G defined over a number field K GA(∞) – subgroup of integral adeles GA(S) – subgroup of S-integral adeles GAS – group of S-adeles GAS (T) – group of T-integral S-adeles (for T ⊃ S) Q GS = GKv (for finite S); in particular G∞ = GV∞ K v∈S

cl(G) – the class number of G H i (G, A) – ith cohomology group/set of a G-module/group/set A H i (L/K, G) = H i (Gal(L/K), GL ) – ith Galois cohomology group/set of an algebraic K-group G with respect to a Galois extension L/K ¯ H i (K, G) = H i (Gal(K/K), GK¯ ), where K¯ is a separable closure of K res – restriction map inf – inflation map cor – corestriction map

List of Notations

lim – direct/inductive limit −→ lim – inverse/projective limit ←−

AG – G-fixed elements of a G-set A G(a) – G-stabilizer of an element a Ga – G-orbit of a

xv

1 Algebraic Number Theory

The first two sections of this introductory chapter provide a brief overview of several concepts and results from number theory. A detailed exposition of this material can be found in the books of Lang (1994) and Weil (1995) (cf. also Chapters 1–3 of [ANT]). It should be noted that, unlike Weil, we state the results here only for algebraic number fields, although the overwhelming majority of them also hold for global fields of positive characteristic, i.e., fields of algebraic functions over a finite field. In §1.3, we present results about group cohomology, including definitions and statements of the basic properties of noncommutative cohomology, that are necessary for understanding the rest of the book. Sections 1.4–1.5 contain basic results on simple algebras over local and global fields. Special attention is given to the investigation of the multiplicative structure of division algebras over such fields, particularly the triviality of the reduced Whitehead group. Moreover, in §1.5, we collect useful results on lattices in vector spaces and orders in semisimple algebras. Throughout the book, we assume familiarity with field theory, particularly Galois theory (finite and infinite), as well as with elements of topological algebra, including the theory of profinite groups.

1.1 Algebraic Number Fields, Valuations, and Completions 1.1.1 Arithmetic of Algebraic Number Fields Let K be an algebraic number field, i.e., a finite extension of the field Q of rational numbers, and let OK be the ring of integers of K. The ring OK is a classical object of interest in algebraic number theory. The analysis of its structural and arithmetic properties, which was initiated by Gauss, Dedekind, Dirichlet, and others in the nineteenth century, remains an active area of research. 1

2

Algebraic Number Theory

From a purely algebraic point of view, the ring O = OK is easy to describe: if [K : Q] = n, then O is a free Z-module of rank n. Furthermore, for any nonzero ideal a ⊂ O, the quotient ring O/a is finite; in particular, any nonzero prime ideal is maximal. Rings with such properties (i.e., integral domains that are noetherian, integrally closed, and in which all nonzero prime ideals are maximal) are known as Dedekind rings. In such a ring, any nonzero ideal a ⊂ O can be written uniquely as the product of prime ideals: a = pα1 1 . . . pαr r . This property generalizes the fundamental theorem of arithmetic on the uniqueness of factorization of any positive integer into a product of primes. Nevertheless, the analogy here is only partial: unique factorization of elements of O into prime elements, generally speaking, does not hold. This fact, which already suggests that the arithmetic of O can differ significantly from the arithmetic of Z, has been crucial in shaping algebraic number theory. The precise degree to which O fails to be a unique factorization domain is measured by the ideal class group of K, which is defined as follows. Recall that the fractional ideals of K are O-submodules a of K such that xa ⊂ O for a suitable nonzero x in O. Define the product of two fractional ideals a, b ⊂ O to be the O-submodule in K generated by the products xy for all x ∈ a, y ∈ b. Then, with respect to this operation, the set of fractional ideals becomes a group, called the group of (fractional) ideals of K, which we denote by Id(O). The principal fractional ideals, i.e., ideals xO where x ∈ K ∗ , form the subgroup P(O) ⊂ Id(O), and the quotient group Cl(O) = Id(O)/P(O) is called the ideal class group of K. A classical result of algebraic number theory is that the group Cl(O) is always finite; its order, denoted by hK , is the class number of K. Moreover, the factorization of elements of O into primes is unique if and only if hK = 1. Another classical result (the Dirichlet Unit Theorem) states that the group of invertible elements O∗ is finitely generated. These two facts are the starting point for the arithmetic theory of algebraic groups (cf. Preface to the Russian edition). However, in generalizing classical arithmetic to algebraic groups, we cannot appeal to ring-theoretic concepts, but rather need to develop such number-theoretic constructions as valuations and completions, as well as adeles, ideles, and others.

1.1.2 Valuations and Completions of Algebraic Number Fields We define a valuation of a field K to be a function | |v : K → R satisfying the following conditions for all x, y in K: (1) |x|v ≥ 0, with |x|v = 0 if and only if x = 0; (2) |xy|v = |x|v |y|v ; (3) |x + y|v ≤ |x|v + |y|v .

1.1 Algebraic Number Fields, Valuations, and Completions

3

If, instead of (3), the following stronger condition holds: (30 ) |x + y|v ≤ max{|x|v , |y|v }, the valuation is called non-Archimedean; otherwise, it is called Archimedean. As an example of a valuation of an arbitrary field K, one can consider the trivial valuation, which is defined by setting |x|v = 1 for all x in K ∗ , and |0|v = 0. We next consider examples of nontrivial valuations of the field K = Q. The ordinary absolute value | |∞ is obviously an archimedean valuation. Furthermore, to each prime p we can associate a non-Archimedean valuation | |p called the p-adic valuation. Namely, given any α ∈ Q∗ , we write it in the form α = pr · β/γ , where r, β, γ ∈ Z and β and γ are not divisible by p, and then set |α|p = p−r ; we also let |0|p = 0. Sometimes, instead of the p-adic valuation | |p , it is convenient to use the corresponding logarithmic valuation v = vp , defined by the formula v(α) = r and v(0) = + ∞, so that |α|p = p−v(α) . Axiomatically v is given by the following conditions: (1) v(x) is an element of the additive group Z of integers (or more generally any ordered abelian group) for x 6= 0, and v(0) = ∞; (2) v(xy) = v(x) + v(y); (3) v(x + y) ≥ min {v(x), v(y)}. We shall use both ordinary valuations as well as the corresponding logarithmic valuations, and it should be clear from the context to which one we are referring. It is worth noting that the examples given earlier actually exhaust all the nontrivial valuations of Q. Theorem 1.1 (OSTROWSKI) Any nontrivial valuation of Q is equivalent either to the archimedean valuation | |∞ or to a p-adic valuation | |p . (Recall that two valuations | |1 and | |2 on K are called equivalent if they induce the same topology on K; in this case we have | |1 = | |λ2 for a suitable real λ > 0.) Thus, restricting any nontrivial valuation | |v of an algebraic number field K to Q, we obtain (up to equivalence) either an archimedean valuation | |∞ or a p-adic valuation (it can be shown that the restriction of a nontrivial valuation is always nontrivial). This means that any nontrivial valuation of K can be obtained by extending to K one of the (nontrivial) valuations of Q. On the other hand, it is known that for any algebraic extension L/K, any valuation | |v of K can be extended to L, i.e., there exists a valuation | |w of L (denoted w|v)

4

Algebraic Number Theory

such that |x|w = |x|v for all x in K. In particular, starting with the valuations of Q, we can obtain all valuations of an arbitrary number field K. Let us analyze the extension procedure in greater detail. To begin with, it is helpful to introduce the completion Kv of K with respect to a valuation | |v . If we consider K as a metric space with respect to the metric arising from | |v , then its completion Kv is a metric space that, at the same time, is a field under the natural operations, and is complete with respect to the corresponding extension of | |v , for which we will use the same notation. It is well known that if L is an algebraic extension of Kv (and, in general, of any field that is complete with respect to a valuation | |v ), then | |v has a unique extension | |w to L. Using this, we can derive an explicit formula for | |w , which can be taken as the definition of | |w . Indeed, since | |v extends uniquely to a valuation of the algebraic closure K¯ v , it follows that |σ (x)|w = |x|w for any x in K¯ v and any σ in Gal(K¯ v /Kv ). Now let L/Kv be a finite extension of degree n, and let σ1 , . . . , σn be the distinct embeddings of L into K¯ v over Kv . Then for the norm NL/K (a) of an element a ∈ L, we have n n Y Y |NL/K (a)|v = σi (a) = |σi (a)|w = |a|nw . i=1

v

i=1

As a result, we obtain the following explicit description of the extension | |w : |a|w = |NL/K (a)|1/n v

for any a in L.

(1.1)

Now let us discuss the procedure of extending valuations to a finite extension L/K for a number field K. Let | |v be a valuation of K and | |w its unique extension to the algebraic closure K¯ v of Kv . Then for any embedding τ : L → K¯ v over K (and in fact we have n = [L : K] such embeddings), we can define a valuation u on L by |x|u = |τ (x)|w , which clearly extends the original valuation | |v of K. In this case, the completion Lu can be identified with the compositum τ (L)Kv . Moreover, any extension may be obtained in this way, and two embeddings τ1 , τ2 : L → K¯ v give the same extension if they are conjugate over Kv , i.e., if there exists λ in Gal(K¯ v /Kv ) with τ2 = λτ1 . In other words, if L = K(α) and f (t) is the irreducible polynomial of α over K, then the extensions | |u1 , . . . , | |ur of | |v over L are in one-to-one correspondence with the irreducible factors of f over Kv , viz. | |ui corresponds to the embedding τi : L → K¯ v that sends α to a root of fi . Further, the completion Lui is the finite extension of Kv generated by a root of fi . It follows that L

O K

Kv '

r Y

Lui ;

(1.2)

i=1

in particular, the degree [L : K] equals the sum of the local degrees [Lui : Kv ].

1.1 Algebraic Number Fields, Valuations, and Completions

5

Moreover, one has the following formulas for the norm and the trace of an element α in L: Y NL/K (a) = NLu /Kv (a), u|v

(1.3) TrL/K (a) =

X

TrLu /Kv (a).

u|v

Thus, the set V K of all pairwise inequivalent valuations of K (or, to put it more precisely, of the equivalence classes of valuations of K) is the union of K of the archimedean valuations, which are the extensions to K the finite set V∞ of the ordinary absolute value | |∞ on Q, and the set VfK of non-Archimedean valuations, obtained as extensions of the p-adic valuation | |p of Q, for each prime number p. The archimedean valuations correspond to the embeddings of K into either R or C, in which case they are respectively called real or complex valuations and the corresponding completions can be identified with K is a real valuation, then an element α in K is said to R or C. If v ∈ V∞ be positive with respect to v if its image under v is a positive number. Let s (respectively t) denote the number of real (respectively pairwise nonconjugate complex) embeddings of K. Then s + 2t = n is the degree of L over K. Non-Archimedean valuations lead to more complicated completions. More specifically, if v ∈ VfK is an extension of a p-adic valuation, then the completion Kv is a finite extension of the field Qp of p-adic numbers. Since Qp is a locally compact field, it follows that Kv is locally compact (with respect to the topology determined by the valuation).1 The closure of the ring of integers O in Kv is the valuation ring Ov = {a ∈ Kv : |a|v ≤ 1}, sometimes called the ring of v-adic integers. Then Ov is a local ring with maximal ideal pv = {a ∈ Kv : |a|v < 1}, called the valuation ideal, and group of invertible elements Uv = Ov \ pv = {a ∈ Kv : |a|v = 1}. It is easy to see that the valuation ring of Qp is the ring of p-adic integers Zp , and the corresponding valuation ideal is pZp . In general, Ov is a free module over Zp , whose rank equals the degree [Kv : Qp ], making Ov an open compact subring of Kv . Moreover, the powers piv of pv form a fundamental system of 1 Henceforth, completions of a number field with respect to nontrivial valuations are called local

fields. It can be shown that the class of local fields thus defined coincides with the class of nondiscrete locally compact fields of characteristic zero. We note also that we shall use the term local field primarily in connection with non-Archimedean completions, and to emphasize this we will use the term non-Archimedean local field.

Algebraic Number Theory

6

neighborhoods of zero in Ov . The quotient ring kv = Ov /pv is a finite field and is called the residue field of v. The ideal pv ⊂ Ov is principal; any of its generators π is called a uniformizer and is characterized by the property that v(π ) is the (positive) generator of the value group 0 = v(Kv∗ ) ' Z. Once we have fixed a uniformizer π, we can write any a in Kv∗ as a = π r u, for a suitable u ∈ Uv ; this yields a continuous isomorphism Kv∗ ' Z × Uv , given by a 7→ (r, u), where Z is endowed with the discrete topology. Thus, to determine the structure of Kv∗ , we need only describe Uv . It can be shown quite easily that Uv is a compact group, locally isomorphic to Ov . It follows that Uv ' F × Znp , where n = [Kv : Qp ], and F is the group of all roots of unity in Kv . Thus Kv∗ ' Z × F × Znp . Two important concepts associated with field extensions are the ramification index and the residual degree. We introduce these concepts first for the local case. Let Lw /Kv be a finite extension of degree n. Then the value group 0v = v(Kv∗ ) has finite index in 0w = w(L∗w ), and the corresponding index e(w|v) = [0w : 0v ] is called the ramification index. The residue field `w = OLw /PLw for Lw is a finite extension of the residue field kv , and f (w|v) = [`w : kv ] is the residual degree. Moreover, e(w|v)f (w|v) = n. An extension for which e(w|v) = 1 is called unramified, while an extension for which f (w|v) = 1 is called totally ramified. Now let L/K be an extension of degree n of number fields. Then for any valuation v in VfK and any extension w to L, the ramification index e(w|v) and residual degree f (w|v) are defined respectively as the ramification index and residual degree for the extension of the completions Lw /Kv . (One can also give an intrinsic definition based on the value groups 0˜ v = v(K ∗ ), 0˜ w = w(L∗ ), and the residue fields k˜ w = OK (v)/pK (v),

`˜w = OL (w)/PL (w),

where OK (v), OL (w) are the valuation rings of v and w in K and L, and pK (v), PL (w) are the respective valuation ideals, but in fact 0˜ v = 0v , 0˜ w = 0w , k˜ v = kv , and `˜w = `w .) As earlier, [Lw : Kv ] = e(w|v)f (w|v). Thus, if w1 , . . . , wr are all the extensions of v to L, then r X i=1

e(wi |v) f (wi |v) =

r X

[Lwi : Kv ] = n.

i=1

Generally speaking, e(wi |v) and f (wi |v) do not have to be equal for different i, but in the important case of a Galois extension L/K, they are indeed the same for all i. To see this, we let G denote the Galois group of L/K. Then all extensions w1 , . . . , wr of v to L are conjugate under G, i.e., for any i = 1, . . . , r, there exists σi in G such that wi (x) = w1 (σi (x)) for all x in L. It follows that

1.1 Algebraic Number Fields, Valuations, and Completions

7

e(wi |v) and f (wi |v) are independent of i (we will denote them simply by e and f ); moreover, the number of different extensions r is the index [G : G(w1 )] of the decomposition group G(w1 ) = {σ ∈ G : w1 (σ x) = w1 (x) for all x in L}. Consequently, efr = n, and G(w1 ) is the Galois group of the corresponding extension Lw1 /Kv of the completions.

1.1.3 Unramified and Totally Ramified Field Extensions Let v ∈ VfK and assume that the corresponding residue field kv is the finite field Fq with q elements. Proposition 1.2 For any integer n ≥ 1, there exists a unique unramified extension L/Kv of degree n. It is generated over Kv by all the (qn − 1)-roots of unity, and therefore is a Galois extension. The correspondence that sends σ ∈ Gal(L/Kv ) to its reduction σ¯ ∈ Gal(`/kv ), where ` ' Fqn is the residue field of L, yields an isomorphism of Galois groups Gal(L/Kv ) ' Gal(`/kv ). In order to define the reduction σ¯ of a given automorphism σ ∈ Gal(L/Kv ), we note that the valuation ring OL and its valuation ideal PL are invariant under σ . So, σ induces an automorphism of the residue field ` = OL /PL which we call σ¯ . Furthermore, we observe that Gal(`/kv ) is a cyclic group generated by the Frobenius automorphism ϕ(x) = xq for all x in `; the corresponding element of Gal(L/Kv ) will also be called the Frobenius automorphism (of the extension L/Kv ) and will be denoted by Fr(L/Kv ). The following proposition describes the properties of norms in unramified extensions. Proposition 1.3 Let L/Kv be an unramified extension, and let Uv and UL denote the groups of units in Kv and L, respectively. Then Uv = NL/K (UL ); in particular, Uv ⊂ NL/Kv (L∗ ). PROOF: Our argument utilizes the canonical filtration on the group of units, which is useful in other situations as well. Namely, for any integer i ≥ 1, we (i) (i) let Uv = 1 + piv and UL = 1 + PiL . It is easy to see that these sets are open subgroups which actually form bases of the neighborhoods of the identity in Uv and UL , respectively. We have the following isomorphisms: Uv /Uv(1) ' kv∗ ,

Uv(i) /Uv(i+1) ' kv+ ,

for i ≥ 1,

(1.4)

Algebraic Number Theory

8

where the first one is induced by the reduction map a 7→ a (mod pv ), and the second is obtained by fixing a uniformizer π of Kv and then mapping 1 + π i a 7→ a (mod pv ). Similarly, (1)

UL /UL ' `∗ ,

(i)

(i+1)

UL /UL

' `+ ,

for i ≥ 1.

(1.5)

Since L/Kv is unramified, π is also a uniformizer of L, so in the rest of the proof we will assume (as we may) that the second isomorphism in (1.5) is defined by means of π. For a in UL , we have NL/Kv (a) =

Y

Y

σ (a) =

σ ∈Gal(L/Kv )

τ (¯a) = N`/kv (¯a),

τ ∈Gal(`/kv )

where the bar denotes reduction modulo PL . (1) (1) Thus the norm map induces a homomorphism UL /UL → Uv /Uv , which in terms of the identifications in (1.4) and (1.5) coincides with N`/kv . Further, for any i ≥ 1 and any a in OL , we have Y NL/Kv (1 + π i a) = σ (1 + π i a) ≡ 1 + π i TrL/Kv (a) (mod P(i+1) ). v σ ∈Gal(L/Kv )

(i)

(i+1)

(i)

(i+1)

It follows that NL/Kv induces homomorphisms UL /UL → Uv /Uv , which with the identifications in (1.4) and (1.5) become the trace map Tr`/kv . But the norm and trace maps are surjective for extensions of finite fields; there(i) (i) fore the group W = NL/Kv (UL ) satisfies Uv = WUv for all i ≥ 1. Since Uv form a base of neighborhoods of identity, the latter condition means that W is dense in Uv . On the other hand, since UL is compact and the norm map is continuous, the subgroup W is closed, and therefore W = Uv . The proof of Proposition 1.3 also yields (i)

(i)

Corollary 1.4 If L/Kv is an unramified extension, then NL/Kv (UL ) = Uv for any integer i ≥ 1. We will need one additional statement about the compatibility of the norm map in arbitrary extensions with the above filtration. Proposition 1.5 For any finite extension L/Kv , we have the following: (1)

(1)

(1) Uv ∩ NL/Kv (L∗ ) = NL/Kv (UL ); (2) if e is the ramification index of L/Kv , then for any integer i ≥ 1, we have ( j) (i) NL/Kv (UL ) ⊂ Uv , where j is the smallest integer ≥ i/e.

1.1 Algebraic Number Fields, Valuations, and Completions

9

PROOF: We begin with the second assertion. Let M be a Galois extension Q of Kv containing L. Then for a in L, NL/K (a) = σ σ (a), where the product is taken over all embeddings, σ : L ,→ M over Kv . As we noted earlier, v uniquely extends to a valuation w of M, and consequently w(a) = w(σ (a)) for any a in L and any σ . In particular, if we choose a uniformizer πL in L, we have (i) σ (πL ) = πL bσ for suitable bσ in UM . It follows that for a = 1 + πLi c ∈ UL , we have NL/Kv (a) =

Y σ

σ (1 + πLi c) =

Y (1 + πLi biσ σ (c)) ∈ (1 + πLi OM ) ∩ Kv . σ

But according to the definition of the ramification index, we have pv OL = PeL , j so that πLi OM ∩ Kv = πLi OL ∩ Kv = PiL ∩ Ov ⊂ pv (where j is chosen as ( j) indicated in the statement of the proposition) and NL/Kv (a) ∈ Uv . In par(1) (1) ticular, NL/Kv (UL ) ⊂ Uv , so to prove the first assertion, it suffices to show (1) (1) (1) that Uv ∩ NL/Kv (L∗ ) ⊂ NL/Kv (UL ). Let a ∈ L∗ be such that NL/Kv (a) ∈ Uv . (1) Then (1.1) implies that a ∈ UL . The isomorphism in (1.5) shows that UL is a maximal pro-p-subgroup in UL for the prime p corresponding to the valuation (1) (1) v, from which it follows that UL ' UL /UL ×UL . In particular, a = bc where (1) c ∈ UL and b is an element of finite order coprime to p. We have d = NL/Kv (b) = NL/Kv (a)NL/Kv (c)−1 ∈ Uv(1) . (1)

We now observe that the order of any torsion element in Uv is a power of p while the order of d divides that of b, hence is prime to p. It follows that d = 1 (1) and therefore NL/Kv (a) = NL/Kv (c) ∈ NL/Kv (UL ). Let us now return to unramified extensions of Kv . It can be shown that the composite of unramified extensions is unramified; hence, there exists a maximal unramified extension Kvnr of Kv , which is Galois, with Gal(Kvnr /Kv ) isomorphic to the Galois group Gal(k¯ v /kv ) of the algebraic closure of the resiˆ the profinite completion of the infinite due field kv . Thus, it is isomorphic to Z, cyclic group with generator the Frobenius automorphism. Now, let L/K be a finite extension of a number field K. It is known that almost all valuations v in VfK are unramified in L/K, i.e., the corresponding extension of the completions Lw /Kv is unramified for any w|v; in particular, the Frobenius automorphism Fr(Lw /Kv ) is defined. If L/K is a Galois extension, then, as we noted earlier, Gal(Lw /Kv ) can be identified with the decomposition subgroup G(w) of the valuation w in the Galois group G = Gal(L/K), so Fr(Lw /Kv ) may be viewed as an element of G.

10

Algebraic Number Theory

We know that any two valuations w1 , w2 extending v are conjugate under G, from which it follows that the Frobenius automorphisms Fr(Lw /Kv ) corresponding to all extensions of v form a conjugacy class F(v) in G. The natural question arises if all conjugacy classes in G can be obtained in this way. In other words, for a given σ in G, does there exist a valuation v in VfK such that for a suitable w|v, the extension Lw /Kv is unramified with Fr(Lw /Kv ) = σ ? Theorem 1.6 (CHEBOTAREV) Let L/K be a finite Galois extension with Galois group G. Then, for any σ in G, there are infinitely many v in VfK such that for suitable w|v, the extension Lw /Kv is unramified and Fr(Lw /Kv ) = σ . In particular, there exist infinitely many v such that Lw = Kv , i.e., L ⊂ Kv . In fact, Chebotarev determined a quantitative measure (density) of the set of v in VfK such that the conjugacy class F(v) coincides with a given conjugacy class C ⊂ G. The density turned out to be |C|/|G| (while the density of the set VfK itself is 1). Therefore, Theorem 1.6 (or, more precisely, the corresponding assertion about the density) is called the Chebotarev Density Theorem. For cyclotomic extensions of K = Q, it is equivalent to Dirichlet’s theorem on prime numbers in arithmetic progression. We note that the last part of Theorem 1.6 can in fact be proved without using any analytic techniques. Next, using the geometry of numbers, one proves Theorem 1.7 (HERMITE) If K/Q is a finite extension that is unramified at all primes p (i.e., Kv /Qp is unramified for all p and all v|p), then K = Q. We will not present here a detailed analysis of totally ramified extensions (in particular, we will not define tamely and wildly ramified extensions), but rather will limit ourselves to describing them using Eisenstein polynomials. Recall that a monic polynomial e(t) = tn + an−1 tn−1 + · · · + a0 ∈ Kv [t] is called an Eisenstein polynomial if ai ∈ pv for all i = 0, . . . , n − 1 and a0 ∈ / p2v . It is well known that an Eisenstein polynomial is irreducible in Kv [t]. Proposition 1.8 If 5 is the root of an Eisenstein polynomial e(t), then L = Kv [5] is a totally ramified extension of Kv with uniformizer 5. Conversely, if L/Kv is totally ramified and 5 is a uniformizer of L, then L = Kv [5] and the minimal polynomial of 5 over Kv is an Eisenstein polynomial. Corollary 1.9 If L/Kv is totally ramified, then NL/Kv (L∗ ) contains a uniformizer of Kv .

1.2 Adeles and Ideles; Approximation; Local-Global Principle

11

To study ramification in a Galois extension L/K with Galois group G, one defines certain subgroups Gi for i ≥ −1, called the ramification groups. Given v ∈ VfK and w|v, we define G−1 to be the decomposition group G(w) of w, which can be identified with the local Galois group Gal(Lw /Kv ). Next, G0 = {σ ∈ G−1 : σ (a) ≡ a(mod PLw ) for all a ∈ OLw } is the inertia group. It is clear that G0 is precisely the kernel of the homomorphism Gal(Lw /Kv ) → Gal(`w /kv ) that sends each automorphism of Lw to its reduction. Therefore, G0 is a normal subgroup of G−1 and by the surjectivity of the reduction homomorphism, we have G−1 /G0 ' Gal(`w /kv ). Moreover, G the fixed field E = Lw0 is the maximal unramified extension of Kv contained in Lw , and Lw /E is totally ramified. The higher ramification groups are defined as follows: Gi = {σ ∈ G−1 : σ (a) ≡ a(mod Pi+1 Lw ) for all a ∈ OLw }. They are normal in G−1 , and Gi = {e} for sufficiently large i. Furthermore, for i ≥ 1, the quotients Gi /Gi+1 are p-groups, where p is the prime corresponding to v. We note that the groups Gi = Gi (v) defined above depend on the choice of an extension w|v, and for a different choice of w they would be replaced by suitable conjugates. In particular, the fixed field LH of the subgroup H ⊂ G generated by the inertia groups G0 (w) for all extensions w|v, is the maximal normal subextension in L that is unramified with respect to all valuations extending v.

1.2 Adeles and Ideles; Strong and Weak Approximation; the Local-Global Principle To gain insights into the arithmetic properties of a number field K, rather than looking at individual valuations, it is often useful to work with families of valuations (e.g., with the entire set V K ) and the corresponding completions simultaneously. In this section, we introduce constructions that enable us to do that.

1.2.1 Adeles and Ideles The set of adeles AK of a number field K is defined to be the subset of the direct Q product v∈V K Kv consisting of x = (xv ) such that xv ∈ Ov for almost all v in VfK . Clearly, AK is a ring with respect to the natural componentwise operations. Furthermore, AK can be endowed with a topology, called the adele topology,

12

Algebraic Number Theory

Q Q by taking sets of the form v∈S Wv × v∈V K \S Ov , where S ⊂ V K is a finite K and W ⊂ K are open subsets for each v in S, as a base subset containing V∞ v v of open sets (observe that this topology is stronger than the topology induced Q from the direct product v∈V K Kv ). It is easy to see that with respect to the adele topology, AK is a locally compact topological ring. Next, for any finite K , one defines the subring of S-integral adeles subset S ⊂ V K containing V∞ as AK (S) =

Y v∈S

Kv ×

Y

Ov ;

v ∈S /

K , then the corresponding ring is called the ring of integral adeles and if S = V∞ S denoted AK (∞). It is clear that AK = S AK (S), where the union is taken over K . It is easy to show that for any a in K all finite subsets S ⊂ V K containing V∞ K and almost all v ∈ Vf , we have |a|v ≤ 1, i.e., a ∈ Ov . Moreover, if a ∈ K ∗ , then, applying this inequality to a−1 , we obtain that, in fact, a ∈ Uv for almost all v ∈ VfK , i.e., the set V (a) := {v ∈ VfK : a ∈ / Uv } is finite. It follows that we have a diagonal embedding K → AK , given by x 7→ (x, x, . . .), whose image, called the ring of principal adeles, will usually be identified with K.

Proposition 1.10 The ring of principal adeles is discrete in AK . T Note that since O = v∈V K (K ∩ Ov ), the intersection K ∩ AK (∞) is the f ring of integers O ⊂ K; thus to prove our proposition it suffices to establish Q the discreteness of O in v∈V∞ K Kv = K ⊗Q R. Let x1 , . . . , xn be a Z-basis of O that is also a Q-basis of K, and consequently also an R-basis of K ⊗Q R. Thus, O can be identified with a Z-lattice in the space K ⊗Q R, and the desired discreteness follows from the discreteness of Z in R. (Incidentally, we note that K ) is the ring of S-integers K ∩ AK (S) (where S ⊃ V∞ O(S) = {x ∈ K : |x|v ≤ 1 for all v ∈ V K \ S}, K ) is the usual ring of integers O.) and moreover O(V∞

The multiplicative analog of the ring AK of adeles of K is the group JK Q of ideles, which, by definition, consists of x = (xv ) ∈ v∈V K Kv∗ , such that xv ∈ Uv for almost all v in VfK . It is clear that JK is a subgroup of the direct product and in fact is precisely the group of invertible elements of AK . Observe, however, that JK is not a topological group with respect to the topology induced from AK (taking inverses is not a continuous operation for this topology). The

1.2 Adeles and Ideles; Approximation; Local-Global Principle

13

“correct” topology on JK is the pullback of the product topology on AK × AK by means of the embedding JK → AK × AK , x 7→ (x, x−1 ). Explicitly, this topology can be described by taking for a base of open sets all sets of Q Q the form v∈S Wv × v∈V K \S Uv where S ⊂ V K is a finite subset containing K and W ⊂ K ∗ are open subsets for v in S. This topology, called the idele V∞ v v topology, is stronger than the topology induced by the adele topology, and with respect to the former, JK is a locally compact topological group. (One cannot help but note the analogy between adeles and ideles. Indeed, both concepts are special cases of the notion of the group of adeles of an algebraic group and of the more general construction of a restricted topological product, which we will consider in Chapter 5.) Continuing the analogy between adeles and ideles, K , the subgroup of we can define, for any finite subset S ⊂ V K containing V∞ S-integral ideles by Y Y JK (S) = Kv∗ × Uv ; v∈S

v ∈S /

K, V∞

in the case where S = this subgroup is called the subgroup of integral ideles and is denoted by JK (∞). As we noted earlier, if a ∈ K ∗ , then a ∈ Uv for almost all v, and consequently we have the diagonal embedding K ∗ → JK , whose image is called the group of principal ideles. Proposition 1.11 The group of principal ideles is discrete in JK . The assertion follows from Proposition 1.10 and the fact that the induced adele topology on JK is weaker than the idele topology. An alternate proof can be given by using the product formula, which Q states that v∈V K |a|nvv = 1 for any a in K ∗ , where V K consists of the extensions of the valuations | |p and | |∞ of Q, and nv = [Kv : Qp ] (respectively nv = [Kv : R]) is the local degree. The product formula can be stated more Q elegantly as v∈V K kakv = 1, where kakv = |a|nvv is the so-called normalized valuation. The function k kv defines the same topology on K as the original valuation | |v , and is actually a valuation equivalent to | |v , except for the case where v is complex. For a non-Archimedean v, the normalized valuation has the following intrinsic description: if π ∈ Kv is a uniformizer, then kπkv = q−1 , where q is the number of elements of the residue field kv . Now let us return to the proof of Proposition 1.11. For archimedean v, we set 1 ∗ Wv = x ∈ Kv : kx−1kv < 2 . We claim that the neighborhood of the identity Q Q ∗ ∗  = v∈V∞ K Wv × v∈VfK Uv satisfies  ∩ K = {1}. Indeed, if a ∈  ∩ K and a 6= 1, then we would have

Algebraic Number Theory

14

Y v∈V K

ka − 1kv
2, then B consists precisely of all maps of the form b(λ)(x, y) = Tr`/Fp (λxσ (y)) with λ ∈ `

(1.22)

in the case where `/k is different from F64 /F4 , and of maps of the form b(λ, µ)(x, y) = Tr`/Fp (λxσ (y) + µxσ (y)8 ) with λ, µ ∈ `

(1.23)

in the case where `/k ' F64 /F4 . (The appearance of the trace map in (1.22) and (1.23) is not accidental. Indeed, for any finite separable field extension P/M, one has the nondegenerate bilinear form f (x, y) = TrP/M (xy), so any M-linear functional ϕ : P → M can be written in the form ϕ(x) = TrP/M (ax) for a suitable a ∈ P.) PROOF: Let r, s > 0 be such that r + s ≡ 0(mod n). Then, for any λ ∈ `, the bilinear form given by br (λ)(x, y) = Tr`/Fp (λxσ r (y))

(1.24)

is 1-invariant. Indeed, by (1.21), for any δ in 1 we have br (λ)(δ · x, δ · y) = Tr`/Fp (λ(δσ r (δ)−1 )xσ r (δσ s (δ)−1 y)) = Tr`/Fp (λ(δσ r+s (δ)−1 )xσ r (y)) = br (λ)(x, y), since r + s = 0(mod n). If, moreover, r 6≡ 0(mod n), then both F(r) and F(s) can be identified with ` and br (1) yields a nondegenerate bilinear pairing F(r) × F(s) → Fp , hence defines an isomorphism between F(r) and the d = Hom(F(s), Fp ). In the case where r ≡ 0(mod n), both dual module F(s) F(r) and F(s) are trivial 1-modules, and therefore F(r) ' F(s). Since clearly d to prove the first assertion of the theorem, B(F(r), F(s)) = Hom1 (F(r), F(s)), it suffices to show that Hom1 (F(r), F(s)) = 0 if r 6≡ s(mod n).

1.4 Simple Algebras over Local Fields

41

Suppose ϕ ∈ Hom1 (F(r), F(s)), ϕ 6≡ 0. Then, for any a in F(r) and any δ in 1, we have ϕ(δ(σ r (δ))−1 a) = δ(σ s (δ))−1 ϕ(a).

(1.25)

Let F1 and F2 denote the additive subgroups of ` generated by elements of the form δ(σ r (δ))−1 and δ(σ s (δ))−1 respectively. Picking a ∈ F(r) so that ϕ(a) 6= 0 P and using (1.25), we obtain that for δi ∈ 1, the condition δi (σ r (δi ))−1 = 0 P implies δi (σ s (δi ))−1 = 0. It follows that the correspondence ψ : δ(σ r (δ))−1 7→ δ(σ s (δ))−1 extends to an additive homomorphism from F1 to F2 . Moreover, F1 and F2 are clearly closed under multiplication, i.e., in effect are finite fields, and the extenl sion of ψ is actually an isomorphism of F1 onto F2 . It follows that ψ(x) = xp for a suitable integer l. So, (1.25) becomes l

(δ(σ r (δ))−1 )p = δ(σ s (δ))−1

(1.26) a −1

for any δ in 1. Suppose that k = Fpa . Then 1 = {xp ab σ (x) = xp for a suitable integer b. Then (1.26) yields l

abr −1)(pa −1)

x−p (p

: x ∈ `∗ } and

abs −1)

= x−(p

for all x in `∗ , whence pl (pabr − l)(pa − 1) ≡ pabs − 1 (mod pan − 1). It was shown in Prasad and Raghunathan (1983, appendix to §7), that the last equation implies that br ≡ bs(mod n), hence r ≡ s(mod n) as gcd(b, n) = 1. This proves the first assertion. To prove the second assertion, we first suppose that `/k is different from F64 /F4 , so that F(r) is a simple 1-module as n > 2. Let b = b(x, y) ∈ B. Then x 7→ b(x, 1) is an Fp -linear map from ` to Fp , and hence b(x, 1) = Tr`/Fp (λx) for a suitable λ ∈ `. Consider b0 = b − b1 (λ), where b1 (λ) is given by (1.24). Since b and b1 (λ) are 1-invariant, the set F(1)⊥ := {y ∈ F(r) : b0 (x, y) = 0 for all x ∈ F(1)} is a 1-submodule of F(r) containing 1. Thus, F(1)⊥ = F(r), hence b0 = 0 and b = b1 (λ), as required. It remains to consider the case where ` = F64 and k = F4 . Here the 1submodules of F(r) correspond to vector subspaces of ` over F8 , and the only nontrivial automorphism of F64 /F8 has the form x 7→ x8 . Let z ∈ ` \ F8 . Then, arguing as above, we obtain that there exist θ, ω ∈ ` such that

Algebraic Number Theory

42

b(x, 1) = Tr`/Fp (θx), b(x, z) = Tr`/Fp (ωx) for all x in `. Since z8 6= z, one can find λ, µ in ` satisfying the equations λ + µ = θ, λσ (z) + µσ (z)8 = ω. In our situation, δ(σ r (δ))−1 ∈ F8 for all δ ∈ `(1) , which implies that the bilinear map b(λ, µ) defined by (1.23) is 1-invariant. Then b0 := b − b(λ, µ) is also 1-invariant. It follows that the subspace F(1)⊥ introduced earlier is a 1-submodule of F(r), containing 1 and z, hence F(1)⊥ = F(r). Thus b0 = 0 and b = b(λ, µ).

1.5 Simple Algebras over Number Fields 1.5.1 The Brauer Group Let A be a central simple algebra over a number field K. Then, for any v ∈ V K , the algebra Av := A ⊗K Kv remains simple, so in the notations of §1.4.1, the correspondence [A] → [Av ] defines a homomorphism of Brauer groups θv

Br(K) −→ Br(Kv ). To describe Br(K), we consider the product Y Y θ= θv : Br(K) → Br(Kv ). v∈V K

v∈V K

In §1.4.2 we saw that, for v ∈ VfK , there is a canonical isomorphism invKv : Br(Kv ) → Q/Z. To treat all the valuations in a unified manner, we will define the invariant of the algebra of Hamiltonian quaternions over Kv = R to be the class 12 + Z ∈ Q/Z. Then, the homomorphism invKv : Br(Kv ) → Q/Z is defined for all v and is injective. Theorem 1.33 (ALBERT, BRAUER, HASSE, NOETHER) The map θ is an injecQ tive homomorphism, and its image consists of a = (av ) ∈ v Br(Kv ) such that P av = 0 for almost all v and v invKv av = 0. Thus, any finite-dimensional central division algebra D over K is determined up to isomorphism by the invariants invKv [Dv ] of the algebras Dv = D ⊗K Kv , which, for simplicity, we will denote by invv D. Conversely, for any choice of

1.5 Simple Algebras over Number Fields

43

invariants, almost all of which equal 0 and the sum of which also equals 0, there is a central division algebra over K having these invariants. The injectivity of θ has several important consequences. First, it follows from §1.4.1 that, given a central division algebra D over K of index n, a field extension P/K of degree n is isomorphic to a maximal subfield of D if and only if Dv ⊗Kv Pw is the matrix algebra for all v ∈ V K and all w|v (which is equivalent to the condition that the local degrees [Pw : Kv ] are divisible by the index of Dv for all v in V K and all w|v). Then, by applying the Grunwald– Wang theorem from class field theory (cf., for example, Artin and Tate [2009]), one can conclude that D contains a maximal subfield L ⊂ D, which is a cyclic extension of K. Taking into account the structure of division algebras over local fields, it is natural to ask the more subtle question of whether there always exists a maximal subfield L ⊂ D, which is a cyclic extension of K and for which the local extensions Lv /Kv are unramified extensions for all v in VfK such that Dv is a division algebra. Unfortunately, an extension L with these properties does not always exist, and in fact one can construct counterexamples even over Q. However, such an L does exist if D satisfies some minor additional restrictions (cf. Platonov and Rapinchuk [1984]). Theorem 1.33 also enables one to show that over number fields, just as over local fields, the exponent of a simple algebra coincides with is its index, and, in particular, the only division algebras of exponent 2 are the algebras of generalized quaternions.

1.5.2 Multiplicative Structure Let D be a central division algebra of index n over a number field K. In this section, we will describe the image of the reduced norm NrdD/K (D∗ ) and also show that the group SL1 (D) = {x ∈ D∗ : NrdD/K (x) = 1} coincides with the commutator group [D∗ , D∗ ] of the multiplicative group D∗ . Theorem 1.34 (EICHLER) The group NrdD/K (D∗ ) coincides with the set of K elements of K ∗ that are positive with respect to all real valuations v ∈ V∞ such that Dv 6' Mn (Kv ). PROOF: See Weil (1995), pp. 279–284 (cf. also [AGNT, §6.7]).

Algebraic Number Theory

44

Theorem 1.35 (WANG [1950]) SL1 (D) = [D∗ , D∗ ]. Wang’s original proof of this theorem is quite complicated and relied on deep results from number theory. We will present a modified argument (cf. Platonov [1976a], Yanchevski˘ı [1975]) that uses only Eichler’s theorem. First, we will reduce the proof of Theorem 1.35 to division algebras of prime power index. For this, we will need some results about the Dieudonné determinant (cf. Artin [1988], Dieudonné [1971]). Let GLm (D) be the group of invertible elements of a matrix algebra A = Mm (D). Then there exists a surjective group homomorphism δ : GLm (D) −→ D∗ /[D∗ , D∗ ], called the Dieudonné determinant, for which   a1 0   ∗ ∗ .. δ   = a1 · · · am [D , D ]. . 0

am

Furthermore, it is well known that in all cases except when m = 2 and D = F2 , the kernel of δ coincides with the commutator subgroup [GLm (D), GLm (D)]. In particular, δ induces an isomorphism SK1 (A) ' SK1 (D), and therefore for any field P (different from F2 when m = 2), the group SLm (P) is precisely the commutator subgroup of the group GLm (P). Lemma 1.36 Let a ∈ SL1 (D) and suppose that a ∈ [(D ⊗K B)∗ , (D ⊗K B)∗ ], where B is an associative m-dimensional K-algebra with identity. Then am ∈ [D∗ , D∗ ]. PROOF: The regular representation B → Mm (K) induces an embedding D ⊗ B → Mm (D), under which an element x ∈ D is mapped to the matrix   x 0   ..  . . 0

x

Now, if a ∈ SL1 (D) and a ∈ [(D ⊗K B)∗ , (D ⊗K B)∗ ], then clearly   a 0   ..   ∈ [GLm (D), GLm (D)]. . 0

a

1.5 Simple Algebras over Number Fields

45

Applying the Dieudonné determinant, we obtain that am ∈ [D∗ , D∗ ], as required. Lemma 1.36 yields Corollary 1.37 For a central division algebra D of index n, the group SK1 (D) has exponent dividing n. Indeed, pick a maximal subfield L ⊂ D. Then [L : K] = n and D ⊗K L = Mn (L). Applying Lemma 1.36 to B = L and using the preceding remark that SLn (L) = [GLn (L), GLn (L)], we obtain our claim. Furthermore, it is well known (cf. Herstein [1994], Theorem 4.4.6) that, if n = pα1 1 . . . pαr r , then D = D1 ⊗K · · · ⊗K Dr , where Di is a division algebra of index pαi i . In these notations, we have Corollary 1.38 If SK1 (Di ) = 1 for all i = 1, . . . , r, then SK1 (D) = 1. N For the proof, we let Bi denote the tensor product j6=i D◦j of the corresponding opposite algebras. Then Bi is a K-algebra of dimension n2i , where ni = n/pαi i ; moreover, D ⊗K Bi ' Mn2 (Di ) for all i = 1, . . . , r. It follows from i the properties of the Dieudonné determinant and the triviality of SK1 (Di ) that SLn2 (Di ) = [GLn2 (Di ), GLn2 (Di )]. Invoking Lemma 1.36, we see that for any i

i

2

i

a in SL1 (D) we have ani ∈ [D∗ , D∗ ] for all i = 1, . . . , r. But the numbers n2i (i = 1, . . . , r) are relatively prime, so we have ui n2i + · · · + ur n2r = 1 for suitable integers ui , whence 2

2

a = (an1 )u1 · · · (anr )ur ∈ [D∗ , D∗ ], as required. Thus, we only need prove Theorem 1.35 in the case of a division algebra D having index pα , where p is a prime and α ≥ 0. We will do this by induction on α, noting that the assertion clearly holds for α = 0. Now, let us assume that SK1 (1) = 1 for any central division algebra 1 of index pα−1 over any number field. We will then show that SK1 (D) = 1 for a central division algebra D of index pα over any number field as well. Let D be a central division algebra of index pα with center K, and let a ∈ SL1 (D). It suffices to find an extension F/K of degree coprime to p such that a ∈ [(D ⊗K F)∗ , (D ⊗K F)∗ ].

(1.27)

46

Algebraic Number Theory

Indeed, by Lemma 1.36, we would then have a[F : K] ∈ [D∗ , D∗ ]. At the same α time, ap ∈ [D∗ , D∗ ] by Corollary 1.37. Since p and [F : K] are relatively prime, we can find s, t ∈ Z such that s[F : K] + tpα = 1, and then α

a = (a[F:K] )s (ap )t ∈ [D∗ , D∗ ], as required. In order to construct such an F, let us consider a maximal subfield L ⊂ D containing a. Let P be the normal closure of L over K, and let G = Gal(P/K) be the corresponding Galois group. Fix a Sylow p-subgroup Gp ⊂ G, and set F = PGp . Then the degree [F : K] is clearly coprime to p, and it remains to establish (1.27). The Galois group Gal(P/F) is Gp . Let H ⊂ Gp be the subgroup corresponding to the subfield LF ⊂ P. It follows from the properties of p-groups that there exists a normal subgroup N ⊂ Gp of index p, containing H. Then the corresponding fixed field M = PN is a cyclic extension of F of degree p, which is contained in LF. If α = 1, then M = LF is itself a cyclic extension of F of degree p. The fact that a ∈ SL1 (D) implies that NM/F (a) = 1, so that by Hilbert’s Theorem 90, we can write a = σ (b)/b for a suitable b ∈ (LF)∗ , where σ is a generator of Gal(M/F). But by the Skolem–Noether Theorem, there exists an element g ∈ (D ⊗K F)∗ such that σ (b) = gbg−1 (where we identify LF with L ⊗K F ⊂ D ⊗K F), and consequently a = gbg−1 b−1 ∈ [(D ⊗K F)∗ , (D ⊗K F)∗ ], as required. (Note that in this argument, we never used the hypothesis that K is a number field, and thus SK1 (D) = 1 for any division algebra D of a prime index p over an arbitrary field K.) If α > 1, we let 1 denote the centralizer of M in D ⊗K F. By the Double Centralizer Theorem (cf. Herstein [1994], Theorem 4.3.2), 1 is a central division algebra of index pα−1 over M. Clearly a ∈ 1, and moreover 1 = Nrd(D⊗K F)/F (a) = NLF/F (a) = NM/F (NLF/M (a)) = NM/F (Nrd1/M (a)). So, by Hilbert’s Theorem 90, the element t := Nrd1/M (a) can be written in the form t = σ (s)/s (1.28) for some s ∈ M ∗ , where σ is a generator of Gal(M/F). Again, by the Skolem– Noether Theorem there exists g ∈ (D ⊗K F)∗ such that σ (b) = gbg−1 for all b in M. Since 1 is the centralizer of M, we easily see that g1g−1 = 1, and moreover

1.5 Simple Algebras over Number Fields

47

Nrd1/M (gxg−1 ) = g Nrd1/M (x)g−1 for all x ∈ 1. Now assume that we have been able to choose an element s in (1.28), which is defined up to multiplication by an element of F ∗ , from the image Nrd1/M (1∗ ) of the reduced norm. Then, writing s = Nrd1/M (z), with z ∈ 1∗ , we obtain Nrd1/M (gzg−1 z−1 ) = σ (s)/s = Nrd1/M (a), and therefore a0 := a(gzg−1 z−1 )−1 ∈ SL1 (1). By induction, SL1 (1) = [1∗ , 1∗ ] ⊂ [(D ⊗K F)∗ , (D ⊗K F)∗ ], and (1.27) follows. It remains to show that s in (1.28) can indeed be found in Nrd1/M (1∗ ). For this we will use Theorem 1.34. If p is odd, then 1w = 1 ⊗M Mw is the matrix M , so Nrd ∗ ∗ algebra for all w in V∞ 1/M (1 ) = M , and there is nothing to prove. Now let p = 2. In this case, M is a quadratic extension of F, and Nrd1/M (1∗ ) M such consists of those m ∈ M that are positive with respect to all real w in V∞ that 1w is not the matrix algebra. We let S denote the set of all such w’s, and let S0 be the set of restrictions of the valuations w ∈ S to F. Then each v ∈ S0 has two extensions w0 , w00 ∈ S with w00 = w0 ◦ σ and Mw0 = Mw00 = Fv . If s ∈ M ∗ is an arbitrary element satisfying (1.28), then since t = σ (s)/s ∈ Nrd1/M (1∗ ), the element s has the same sign with respect to w0 and w00 , and therefore there exists fv ∈ Kv∗ such that sfv is positive with respect to both w0 and w00 . Using Theorem 1.12 on weak approximation, we can choose an element f ∈ K ∗ so that f and fv have the same sign in Kv for all v in S0 . Now, setting s1 = sf , we obtain t = σ (s)/s = σ (s1 )/s1 , i.e., (1.28) holds with s replaced by s1 . At the same time, it follows from our construction that s1 ∈ Nrd1/M (1∗ ), as required. This completes the proof of Theorem 1.35.

1.5.3 Lattices and Orders Let K be a number field with ring of integers O. A lattice (or, more precisely, an O-lattice) in a finite-dimensional vector space V over K is a finitely generated O-submodule L ⊂ V that contains a basis of V over K. A lattice L ⊂ V is said to be free if it is a free O-module, i.e., possesses an O-basis. When O is a principal ideal domain or, equivalently, the class number of K equals 1, any lattice is free. In general, any lattice L ⊂ V has a pseudobasis, i.e., there exist x1 , . . . , xn ∈ V , where n = dimK V , such that L = Ox1 ⊕ · · · ⊕ Oxn−1 ⊕ axn for some ideal a ⊂ O (cf. O’Meara [2000]).

Algebraic Number Theory

48

An order in a finite-dimensional K-algebra A is an O-lattice B ⊂ A that is simultaneously a subring containing the identity element of A. An order is said to be maximal if it is not properly contained in any larger order. The study of lattices and orders essentially reduces to the study of their local counterparts. More precisely, by a (local) lattice in a finite-dimensional Kv vector space VKv , where v ∈ VfK , we mean a finitely generated Ov -submodule Lv ⊂ VKv containing a basis of VKv . Since Ov is a principal ideal domain, any local lattice has an Ov -basis. One defines orders and maximal orders in the obvious way. Clearly, if L is a lattice in a finite-dimensional vector space V over K (respectively, if B is an order in a finite-dimensional K-algebra A), then Lv := L ⊗O Ov (respectively, Bv := B ⊗O Ov ) is a lattice in the space VKv := V ⊗K Kv (respectively, in the algebra AKv := A ⊗K Kv ). Thus, to each lattice L ⊂ V one can associate the set of localizations {Lv ⊂ VKv : v ∈ VfK }. So, a natural question to ask is the extent to which L is determined by its localizations Lv . T Theorem 1.39 (1) L = v (V ∩ Lv ), in particular a lattice is uniquely determined by its localizations; (2) for any two lattices L, M ⊂ V , we have Lv = Mv for almost all v; (3) if L ⊂ V is a lattice and {Nv ⊂ VKv } is an arbitrary set of local lattices such that Nv = Lv for almost all v, then there exists a lattice M ⊂ V such that Mv = Nv for all v ∈ VfK . PROOF: Let L, M be two lattices, let x1 , . . . , xn be a basis of V contained in L, and let y1 , . . . , yr be a finite system of generators of M as an O-module. Then P we can write yi = nj = 1 aij xj for some aij ∈ K. If we choose an integer m 6= 0 so that maij ∈ O for all i, j, then mM ⊂ L. By interchanging L and M, we can likewise find an integer l 6= 0 so that lL ⊂ M, hence L ⊂ 1l M. If now v ∈ / V (lm) (notation as in §1.2.1), then Lv = Mv , proving the second assertion. To prove assertions (1) and (3), we embed V into the associated adele space VAf = V ⊗K Af , where Af is the ring of finite adeles of K. It follows from the strong approximation theorem (cf. Theorem 1.13) that Y LAf (∞) := L ⊗O Af (∞) = Lv v∈VfK

Q

(where Af (∞) = v∈V K Ov is the ring of integral finite adeles) coincides with f the closure of L in VAf . Therefore \ L0 := Lv v∈VfK

1.5 Simple Algebras over Number Fields

49

is the closure of L in V in the topology induced from VAf . So, to prove the first assertion, we only need to establish that L is closed. To do this, let us take a basis x1 , . . . , xn of V in contained L, and set M = Ox1 + · · · + Oxn . T T Since O = v∈V K (K ∩ Ov ), we see that M = v∈V K (V ∩ Mv ). But f f Q Q v∈VfK Mv , just as v∈VfK Lv , is open in VAf , so M is open in V , and consequently L ⊂ V is open and closed. Finally, if a collection of local lattices Nv ⊂ VKv satisfies Nv = Lv for almost Q all v, then v∈V K Nv is an open compact subgroup in VAf and therefore is f Q commensurable with v∈V K Lv (i.e., their intersection has finite index in each f of them). It follows that M := is commensurable with L = needed.

T

\

v∈VfK

v∈VfK

(V ∩ Nv )

(V ∩ Lv ), implying that M is a lattice, as

We will now review some facts about orders in algebras. Our account will only include results about the existence of maximal orders and embedding an arbitrary order into a maximal one, as these are precisely the questions that arise in the study of maximal arithmetic and maximal compact subgroups of algebraic groups. First, we note the following consequence of Theorem 1.39. Proposition 1.40 An order B ⊂ A is maximal if and only if for each v in VfK , the order Bv ⊂ AKv is maximal. Elementary examples show that an arbitrary algebra may not contain any maximal orders. Our goal is to prove that maximal orders always exist in finite-dimensional semisimple algebras. Recall that a semisimple K-algebra is the direct sum of a finite number of simple (but not necessarily central) K-algebras. Thus, by the Artin–Wedderburn Theorem, a finite-dimensional Lr semisimple algebra can be written in the form A = i = 1 Mni (Di ), where Di is a finite-dimensional division algebra over K. In characteristic zero, a finiteL ¯ dimensional K-algebra A is semisimple if and only if A⊗K K¯ ' ri = 1 Mmi (K) for some integers mi (cf. Pierce [1982]). We therefore begin by considering maximal orders in the matrix algebra A = Mn (Kv ). Our treatment will

50

Algebraic Number Theory

be based on the study of the natural action of A on V = Kvn , in conjunction with some elementary topological considerations involving compactness. For a lattice L ⊂ V , we let AL = { g ∈ Mn (Kv ) : g(L) ⊂ L } denote the stabilizer of L. We note that by choosing a basis of L, we may identify the stabilizer AL with Mn (Ov ), which implies in particular that AL is an order and an open compact subring (we note that these characterizations are in fact equivalent). Proposition 1.41 (1) For any compact subring B ⊂ A, there is a lattice L ⊂ V such that B ⊂ AL ; (2) the ring AL is a maximal order in A, for any lattice L ⊂ V ; (3) any order B ⊂ A is contained in some maximal order, and there exist only finitely many (maximal) orders containing B. PROOF: Let L0 = Ovn be the lattice spanned by the standard basis vectors of V = Kvn . Since AL0 is open and B is compact, there exists a finite collection S of elements x1 , . . . , xr ∈ A such that B ⊂ ri = 1 (xi + AL0 ). It follows that the S Ov -submodule L ⊂ V generated by B(L0 ) = x∈B x(L0 ) is actually generated by L0 ∪ x1 (L0 ) ∪ · · · ∪ xr (L0 ), hence is a lattice. On the other hand, it is clear that B(L) ⊂ L, which proves the first assertion. Now assume that AL is contained in some order B ⊂ A. Since any order is clearly an open compact subring, by part (1) we have B ⊂ AM for a suitable lattice M ⊂ V . Thus, AL ⊂ AM and our goal is to show that AL = AM . Since replacing M with a lattice of the form αM for α ∈ Kv∗ does not change the stabilizer AM , we may assume that M ⊂ L, but M 6⊂ πL, where π is a uniformizer of Kv . We can then choose a basis e1 , . . . , en of L so that M has a basis of the form e1 , π α2 e2 , . . . , π αn en for some nonnegative integers α2 , . . . , αn . For i > 1, consider the transformation gi ∈ AL that interchanges the vectors e1 and ei and fixes all ej , for j 6= 1, i. Since AL ⊂ AM we have gi ∈ AM , whence gi (e1 ) = ei ∈ M and αi = 0. Consequently L = M, so AL = AM , proving the second assertion. It follows from parts (1) and (2) that any order B ⊂ A is contained in some maximal order C = AL , so it remains to show that the set {Cl } of maximal orders in A containing B is finite. We can pick a lattice Ml and a nonnegative integer α so that Cl = AMl and B ⊃ π α C. Then for any l we have Cl ⊃ B ⊃ π α C. Let us show that in this case, we have the inclusion π α Cl ⊂ C. Without loss of generality, as in the proof of (2), we may assume that the lattices L and Ml have bases of the form e1 , e2 , . . . , en and e1 , π α2 e2 , . . . , π αn en for αi ≥ 0,

1.5 Simple Algebras over Number Fields

51

respectively. Since Cl ⊃ π α C, we have C(Ml ) ⊂ π −α Cl (Ml ) = π −α Ml . Again, using the transformations gi ∈ C introduced earlier, we obtain αi ≤ α, hence π α L ⊂ Ml . Then π α Cl (L) ⊂ Cl (Ml ) = Ml ⊂ L, so π α Cl ⊂ C. Thus, for any l we have the inclusions π α C ⊂ Cl ⊂ π −α C. Since the index [π −α C : π α C] is finite, the number of distinct Cl ’s is also finite. This completes the proof of the proposition. Remark The description of maximal orders in Mn (Kv ) as stabilizers of lattices L ⊂ V implies that any two maximal orders in A = Mn (Kv ) are conjugate. The techniques employed in the proof of the proposition easily yield analogous assertions about maximal compact subgroups of G = GLn (Kv ). For a lattice L ⊂ V , we let GL denote the group of automorphisms of L, i.e., GL = {g ∈ G : g(L) = L} (more generally, for any subgroup 0 ⊂ G we set 0 L = {g ∈ 0 : g(L) = L} and call 0 L the stabilizer of L in 0). Clearly, using a basis of L, one can identify GL = (AL )∗ with GLn (Ov ), so GL is an open compact subgroup of G and det g ∈ Uv for any g ∈ GL . Proposition 1.42 (1) Given a compact subgroup B ⊂ G, there is a lattice L ⊂ V such that B ⊂ GL ; (2) GL is a maximal compact subgroup of G for any lattice L ⊂ V ; in particular, any compact subgroup is contained in a maximal compact subgroup; (3) all maximal compact subgroups of G are conjugate. The proof follows easily from Proposition 1.41. One also derives from Proposition 1.41 the following fundamental result about orders in semisimple algebras over local fields. Theorem 1.43 Let A be a semisimple algebra over Kv . Then any order B ⊂ A is contained in a maximal order, and there exist only finitely many (maximal) orders containing B. PROOF: Writing A as the direct sum of simple algebras, one reduces the proof to the case where A is simple. Let F be the center of A and let OF be the corresponding valuation ring. Then, for any Ov -order B ⊂ A, the product OF B (of Ov -submodules) is simultaneously an Ov -order and an OF -order in A. So, it follows that we only need consider the case where F = Kv . Clearly, to prove the theorem it suffices to show that the set {Bi } of all orders in A containing B is

52

Algebraic Number Theory

finite. For this, we pick a finite extension P of Kv such that A ⊗Kv P ' Mn (P), and set B˜ = B ⊗Ov OP , B˜ i = Bi ⊗Ov OP . Then B˜ and B˜ i are orders in Mn (P), and B˜ ⊂ B˜ i . But by Proposition 1.41, among the orders B˜ i there are only finitely many distinct orders. So, it suffices to show that B˜ i = B˜ j can hold only if Bi = Bj . Indeed, pick Ov -bases x1 , . . . , xn2 P2 and y1 , . . . , yn2 of Bi and Bj respectively. Then xl = nm = 1 alm ym and P2 yl = nm = 1 blm xm for suitable alm , blm ∈ Kv . Since x1 , . . . , xn2 and y1 , . . . , yn2 are also OP -bases of B˜ i = B˜ j , then actually alm , blm ∈ OP ∩ Kv = Ov , whence Bi = Bj . Combining Theorem 1.43 with Proposition 1.41, we obtain the existence of maximal orders in semisimple algebras over number fields. Theorem 1.44 Let A be a semisimple algebra over a number field K. Then any order B ⊂ A is contained in some maximal order. PROOF: As above, the proof reduces to the case of a central simple K-algebra A. It suffices to show that the set {Bi } of orders in A containing B is finite. We first prove this assertion for a matrix algebra A = Mn (K). It follows from Propositions 1.40 and 1.41 that the order C = Mn (O) is maximal in A. Then according to Theorem 1.39(2), Bv = Cv is a maximal order in AKv = Mn (Kv ) for almost all v ∈ VfK . On the other hand, by Theorem 1.43, for the remaining v, the number of orders in AKv containing Bv is finite. Combining this with Theorem 1.39(1), we obtain the required result. To reduce the general case to that of a matrix algebra, we choose a finite extension P/K satisfying A ⊗K P ' Mn (P), and replace the orders B and Bi with the orders B˜ = B ⊗O OP and B˜ i = Bi ⊗O OP in Mn (P). Then there exist only finitely many distinct B˜ i ’s, and therefore only finitely many distinct Bi ’s (as by considering localizations and arguing as in the proof of Theorem 1.43 we see that B˜ i = B˜ j implies Bi = Bj ). Remark Even though it can be shown that over Kv all maximal orders are conjugate for any semisimple algebra, over K there may, in general, exist nonconjugate maximal orders.

2 Algebraic Groups

This chapter contains some background material from the theory of algebraic groups that will be needed in later chapters. In §2.1, we give an account (mostly without proofs) of the basic results on the structure of algebraic groups, including the classification of semisimple groups over algebraically closed as well as arbitrary fields. In §2.2, we treat some aspects of the classification of Kforms of algebraic groups via Galois cohomology. This approach is used in §2.3 to give an explicit description of groups of classical types. Furthermore, §2.3 contains some other material related to classical groups, including the absolute and the relative versions of Witt’s Theorem. Finally, in §2.4, we summarize some basic facts from algebraic geometry, including the construction of certain algebraic varieties, that we will need later.

2.1 Structure of Algebraic Groups In this section, we have assembled some basic definitions and results dealing with algebraic groups over algebraically closed as well as arbitrary fields that we will be using repeatedly throughout the book. Most proofs will be omitted, as our primary goal is to fix the terminology and notations; we do, however, give precise references for the key results. The reader can find detailed expositions of the theory of algebraic groups in the books by Borel (1991), Humphreys (1975), Milne (2017), and Springer (1998); while the first two books focus primarily on the case of an algebraically closed base field, the third and fourth contain accounts of the theory over an arbitrary field that was originally developed by Borel and Tits (1965). While familiarity with the results contained in this section should in principle be sufficient for understanding the rest of the book, some prior systematic exposure to the abovementioned sources as well as to some basic facts about algebraic varieties, 53

54

Algebraic Groups

Lie algebras, and root systems (as presented, for example, in Shafarevich [2013] and Bourbaki [2002]) is highly recommended.

2.1.1 Algebraic Groups For the most part, the “naïve” definition of a linear algebraic group as a Zariskiclosed subgroup of the general linear group GLn (), where  is a universal domain (an algebraically closed field having infinite transcendence degree over its prime subfield), will be sufficient for our purposes. This is, for example, the case in the treatment of reduction theory in Chapter 4, where one can even assume that  = C. In some instances, however, particularly when working with the groups of adeles and/or the groups of rational points over different completions, it is natural to adopt the more abstract approach of treating an algebraic group G as an algebraic variety equipped with two morphisms, µ

G × G−→G, (x, y) 7→ xy, i

(2.1)

G−→G, x 7→ x−1 , that satisfy the usual axioms of a group. Actually, in dealing with the groups of points over different rings, the schematic point of view is most natural; however, in the present book, we have made an effort not to use this approach extensively. It should be noted that the different approaches ultimately lead to the same class of objects, since any affine algebraic group (under the second definition) is linear, i.e., is isomorphic to a Zariski-closed subgroup of a suitable GLn (). (A morphism of algebraic groups is defined to be a morphism of algebraic varieties that is also a group homomorphism, and an isomorphism is an invertible morphism.) As algebraic groups more general than linear will not be considered in this book, the word “linear” will frequently be omitted. Sometimes it is convenient to think of an algebraic group G as a Zariskiclosed subset of not just GLn () but of the entire matrix algebra Mn (). This can always be achieved by increasing n (called the degree of G) by 1. Indeed, it suffices to realize GLn () itself as a Zariski-closed subset of Mn+1 (). The desired embedding is given by   0  g  0   g 7→  , ..   . −1 0 0 . . . (det g) with the matrix entries of the image defined by the following equations for y = (yij ) ∈ Mn+1 ():

2.1 Structure of Algebraic Groups

yi,n+1 = yn+1,i = 0,

55

i = 1, . . . , n,

yn+1,n+1 · det((yij )i,j = 1,...,n ) − 1 = 0. It follows that the coordinate ring of GLn () is A = [x11 , x12 , . . . , xnn , det(xij )−1 ], and the coordinate ring of an algebraic group G ⊂ GLn () is A/a, where a is the ideal of all functions in A that vanish on G (see §2.4 for the standard notions of algebraic geometry). Consequently, if f : G → H is a morphism between algebraic groups G ⊂ GLn () and H ⊂ GLm (), then there exist polynomials fkl = fkl (x11 , . . . , xnn , det(xij )−1 ) for k, l = 1, . . . , m

(2.2)

such that f (g) = ( fkl (g))k,l = 1,...,m for all g ∈ G. In this book, we will deal with algebraic groups defined over a certain subfield K of , which will typically be either a number field or its completion. In this regard, we recall that an algebraic group G ⊂ GLn () is said to be defined over K (or simply a K-group) if a, the ideal of the coordinate ring A of GLn () consisting of functions that vanish on G, is generated by aK := a ∩ AK , where AK = K[x11 , . . . , xnn , det(xij )−1 ]. (Henceforth, notations like AK , aK , and so on, will be used in various situations to denote K-rational elements. For example, GK will always denote the group of K-rational points of an algebraic K-group G ⊂ GLn (), thus GK := G ∩ GLn (K).) A morphism f : G → H of two Kgroups, G ⊂ GLn () and H ⊂ GLm (), is defined over K (in other words, is a K-morphism) if it has a description of the form (2.2) with the polynomials fkl in AK . In the sequel, we will deal only with algebraic groups defined over perfect fields. So, unless stated otherwise, K will denote a perfect field. (In fact, in the context of the theory presented here, K will be either a finite field or a field of characteristic zero.) In this case, the Galois criterion for a given object (such as an algebraic variety) to be defined over K (see §2.2.4) becomes particularly easy to use. We also note that a K-group can be defined abstractly as an algebraic K-variety with two K-morphisms (2.1) that satisfy the axioms of a group. However, it can be shown (cf. Borel [1991]) that under this definition, an affine K-group is K-isomorphic to a linear algebraic group defined over K.

Algebraic Groups

56

2.1.2 Restriction of Scalars Let G ⊂ GLn () be an algebraic group defined over a finite (separable) extension L of K. We wish to construct an algebraic K-group G0 whose group of K-points GK0 is naturally isomorphic to GL . This objective is accomplished by a group, denoted RL/K (G), which is said to be obtained from G by restriction of scalars (from L to K). To construct G0 = RL/K (G), we fix a basis w1 , . . . , wd of L over K and consider the corresponding regular representation ρ : L → Md (K), which maps an arbitrary x ∈ L to the matrix of left multiplication y 7→ xy (with respect to the given basis). Let Fk (yαβ ) = 0, where α, β = 1, . . . , d,

k = 1, . . . , r,

be a system of linear equations for the matrix entries of y = (yαβ ) ∈ Md (K) that defines the image ρ(L) ⊂ Md (K). Furthermore, let Pl (xij ), for l = 1, . . . , m, be a finite set of generators of the ideal aL , where a ⊂ [x11 , . . . , xnn , det(xij )−1 ] is the ideal of the functions vanishing on G. Identifying Mn (Md (K)) P γ γ with Mnd (K), we may associate to each Pl (xij ) = aγ11 ···γnn x1111 · · · xnnnn the “matrix” polynomial X αβ αβ αβ γnn P˜ l (yij ) = ρ(aγ11 ···γnn )(y11 )γ11 · · · (yαβ ∈ Md (K[yij ]) nn ) αβ

in the n2 d 2 variables yij , where α, β = 1, . . . , d and i, j = 1, . . . , n. Then the image of GL in Mnd (K) under the map induced by ρ is defined by the equations αβ

Fk (yij ) = 0 αβ P˜ l (yij ) = 0

∀i, j = 1, . . . , n; l = 1, . . . , m

k = 1, . . . , r,

(2.3)

(where 0 in the last equation denotes the zero matrix in Md (K)). Let G0 denote the solution set of the system (2.3) in GLnd (). Then G0 is the desired algebraic K-group. Note that G0 = RL/K (G) is independent (up to K-isomorphism) of the choice of a K-basis of L. The set of equations in (2.3) defining G0 shows that G0 can be interpreted as ¯ the group of points of G in the K-algebra L ⊗K K¯ (or rather in the -algebra d ¯ ¯ L ⊗K ). We note that L ⊗K K ' K via the map that acts on L as follows: x 7→ (σ1 (x), . . . , σd (x)),

2.1 Structure of Algebraic Groups

57

where σ1 , . . . , σd are the distinct embeddings of L in K¯ over K. It follows that ¯ there exists a K-isomorphism G0 ' Gσ1 × · · · × Gσd ,

(2.4)

where Gσi is the subgroup of GLn () defined by the equations from aσLi , which is obtained by applying σi to all functions in aL . To any L-morphism f : G → H of algebraic L-groups G and H, there corresponds a K-morphism f˜ = RL/K ( f ) : RL/K (G) → RL/K (H), whose construction is analogous to the above procedure of obtaining P˜ l from Pl . Thus, RL/K is a functor from the category of L-groups and L-morphisms to the category of K-groups and K-morphisms. We note that not every Kmorphism f˜ : RL/K (G) → RL/K (H) is necessarily of the form f˜ = RL/K ( f ) for a suitable L-morphism f : G → H. (For example, if L/K is a Galois extension, then RL/K (G) has K-defined automorphisms induced by the automorphisms of L/K that cannot be written in the form RL/K ( f ).) Nevertheless, using (2.4) it is easy to show the following for the groups of rational characters: X(RL/K (G))K = X(G)L (see §2.2.7 for the definition of characters, and Borel [1963, Proposition 1.6]). We will now point out two properties of restriction of scalars that are useful in arithmetic considerations. Let L/K be a finite extension of number fields, and let v ∈ V K . Then, using the identification Y L ⊗K Kv = Lw w|v

(cf. (1.2) in §1.1), we obtain RL/K (G)Kv '

Y

GLw

w|v

for any L-group G. Now let K = Q and let w1 , . . . , wd be a Z-basis of the ring of integers OL . Considering the regular representation ρ with respect to this basis, we obtain that RL/K (G)Z ' GOL and Y RL/K (G)Zp ' GOv v|p

for all primes p.

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Algebraic Groups

2.1.3 The Lie Algebra of an Algebraic Group The variety of any algebraic group G is homogeneous, as for any g1 , g2 ∈ G, the translation map x 7→ g2 g1−1 x is a morphism of G as an algebraic variety that maps g1 to g2 . Since a variety always has a simple (smooth) point, one concludes that all the points of G are simple, in other words, G is a smooth variety (see §2.4 for the basic notions and results about tangent spaces and simple points). The tangent space Te (G) of G at the identity is called the Lie algebra of G and denoted L(G). We have dim L(G) = dim G. If G ⊂ GLn (), then L(G) ⊂ Mn () = L(GLn ()), and the Lie bracket is given by the standard formula [X , Y ] = XY − YX . If G ⊂ GLn () is defined over K, then L(G) is an algebra with a K-structure, which means that for L(G)K := L(G)∩Mn (K), we have L(G)K ⊗K  = L(G). For computations with Lie algebras of algebraic groups, one can use the method of dual numbers (cf. Borel [1991], Humphreys [1975], and Milne [2017]). If G ⊂ GLn (), then for any g in G we have gL(G)g−1 = L(G). So, one can define a morphism of algebraic groups G → GL(L(G)), g 7→ ϕg , where ϕg (X ) = gXg−1 for all X ∈ L(G). This morphism is called the adjoint representation of G and is denoted by Ad. Furthermore, one can consider the map ad : L(G) → End(L(G)), ad X (Y ) = [X , Y ], called the adjoint representation of the Lie algebra L(G). Using dual numbers, it is easy to show the differential of Ad at the identity coincides with ad. The Killing form is the symmetric bilinear form f on L(G) given by f (X , Y ) = tr(ad X ad Y ) for X , Y ∈ L(G), where tr denotes the trace function on the matrix algebra End(L(G)); we note that f is invariant under the adjoint action of G.

2.1.4 The Connected Component of the Identity Since G is a smooth variety, its irreducible and connected components coincide. The connected component G0 of the identity is an open-and-closed normal subgroup of G of finite index. Moreover, dim G = dim G0 and L(G) = L(G0 ). If G is defined over K, then G0 is also defined over K. We note that most

2.1 Structure of Algebraic Groups

59

groups considered in this book will be connected. In particular, all reductive or semisimple groups will be assumed to be connected (unless stated otherwise).

2.1.5 The Jordan Decomposition Let g ∈ GLn (). Then g can be uniquely written in the form g = gs gu , with gs a semisimple matrix (in the sense that gs is conjugate to a diagonal matrix) and gu a unipotent matrix (in the sense that all the eigenvalues of gu are equal to 1) such that gs gu = gu gs . The factorization g = gs gu is then called the Jordan decomposition of g. If g ∈ G, where G ⊂ GLn () is an algebraic group, then gs , gu ∈ G. Moreover, if f : G → H is a morphism of algebraic groups, then f (g)s = f (gs ) and f (g)u = f (gu ). This, in particular, implies that the Jordan decomposition is independent of the matrix realization of G. Furthermore, if g ∈ GK , then gs , gu ∈ GK (recall that K is assumed to be perfect). Similarly, any matrix X ∈ Mn () can be uniquely written in the form X = Xs + Xn , where Xs and Xn are respectively semisimple and nilpotent matrices such that Xs Xn = Xn Xs . This decomposition is called the additive Jordan decomposition. If X ∈ L(G), then Xs , Xn ∈ L(G), and the projections X 7→ Xs and X 7→ Xn are functorial, i.e., they commute with the differentials of morphisms of algebraic groups. Moreover, if X ∈ L(G)K , then Xs , Xn ∈ L(G)K .

2.1.6 Quotient Varieties Given an algebraic K-group G and a K-subgroup H, the coset space G/H can be endowed with a structure of a quasi-projective K-variety so that the canonical map G → G/H is a K-morphism of algebraic K-varieties (see §2.4.5 for more details). If, moreover, H is a normal subgroup of G, the quotient G/H actually has the structure of an affine K-variety, which is consistent with the natural group operations on G/H. This makes G/H into an algebraic K-group, and the canonical map G → G/H into a K-morphism of algebraic K-groups.

2.1.7 Diagonalizable Groups and Algebraic Tori An algebraic group G is said to be diagonalizable if, for a suitable faithful representation f : G → GLm () (as an algebraic group), the image f (G) is diagonalizable, i.e., is conjugate to a subgroup of the group Dn of diagonal matrices. In this case, the image of any representation h : G → GL` () is also diagonalizable. It can be shown that the diagonalizable groups coincide with those commutative algebraic groups that consist entirely of semisimple elements. Of special importance are the connected diagonalizable groups, known

60

Algebraic Groups

as algebraic tori. Alternatively, algebraic tori can be defined as those algebraic groups G for which there is an isomorphism G ' (Gm )d , where Gm := GL1 () is the multiplicative group of  and d = dim G. A character of an algebraic group G is a morphism of algebraic groups χ : G → Gm . The characters of G form a commutative (abstract) group, denoted X(G), under the operation (χ1 + χ2 )(g) = χ1 (g)χ2 (g). It is easy to see that for a d-dimensional torus G, the group X(G) is isomorphic to Zd , hence is a finitely generated torsion-free Z-module. In general, for a K-torus G, there may be no isomorphism G ' (Gm )d defined over K; in the case where such an isomorphism does exist, G is said to be K-split. The following conditions on a K-torus G are equivalent: (1) G is K-split; (2) all characters of G are defined over K, i.e., X(G)K = X(G); (3) f (G) is diagonalizable over K, i.e., is conjugate to a subgroup of Dn by a matrix from GLn (K), for some (equivalently, any) faithful K-defined representation f : G → GLn (). We note that conditions (2) and (3) are in fact equivalent for any diagonalizable K-group, which enables us to define K-split diagonalizable groups. In general, given a diagonalizable K-group G, one can always find a finite (separable) field extension L of K over which G splits, and any field extension with this property is called a splitting field of G. According to condition (2), an extension L of K is a splitting field of G if and only if X(G) = X(G)L . One can consider the ¯ natural action of the absolute Galois group G = Gal(K/K) (recall that K is assumed to be perfect) on the character group X(G), and this makes the latter into a discrete module over the profinite group G. Then a field extension L of K contained in K¯ is a splitting field of G if and only if the subgroup H ⊂ G corresponding to L acts on X(G) trivially, i.e., X(G) = X(G)H . It follows that a diagonalizable K-group G has a minimal splitting field, which is automatically a finite Galois extension of K and is contained in every splitting field of G. The opposite of K-split tori, for which we have X(G)K = X(G), are Kanisotropic tori, which are characterized by the condition X(G)K = 0. It is well known that in any K-torus G, one can find two K-subtori Gd and Ga , which are respectively K-split and K-anisotropic, such that G = Gd Ga and Gd ∩ Ga is finite (i.e., G is an almost direct product of Gd and Ga ). Now fix a finite Galois extension L/K with Galois group F, and let A be the category whose objects are diagonalizable K-groups that split over L and

2.1 Structure of Algebraic Groups

61

whose morphisms are the K-defined morphisms of such groups. On the other hand, let 0 = Z[F] be the integral group ring of F, and B be the category of finitely generated 0-modules and 0-module homomorphisms. Then the corre8

spondence G − → X(G) yields a contravariant functor from A to B. Moreover, we have the following. Theorem 2.1 The functor 8 is a contravariant equivalence between the categories A and B under which the ( full) subcategory A0 ⊂ A of K-tori corresponds to the subcategory B0 ⊂ B of Z-torsion-free finitely generated 0-modules. Theorem 2.1 is fundamental for the theory of algebraic tori, which is the subject matter of Voskresenski˘ı’s book (1977) and its expanded English version (1998). This theorem makes it possible to describe an algebraic torus by giving the corresponding character module. Moreover, Voskresenski˘ı discovered that many geometric and arithmetic properties of an algebraic torus can be computed in terms of its character module. In this book, however, we will not provide a detailed treatment of the theory of algebraic tori, referring the interested reader instead to the books by Voskresenski˘ı and limiting our exposition to a few typical examples and constructions that we will need later on. It should be noted that there is also a covariant equivalence between A0 and 9

B0 given by G7→X∗ (G), where X∗ (G) := Hom(Gm , G) is the group of cocharacters or one-parameter subgroups of G endowed with a natural 0-module structure. Furthermore, there is a bilinear pairing X∗ (G) × X(G) → Z that is defined as follows: if ϕ ∈ X∗ (G) and χ ∈ X(G), then χ ◦ϕ is a morphism Gm → Gm . It follows that there exists m ∈ Z such that (χ ◦ ϕ)(t) = tm for all t ∈ ∗ , and we define hχ, ϕi = m. This pairing enables us to identify X∗ (G) with the 0module HomZ (X(G), Z) dual to X(G). It follows, in particular, that if G is a Ksplit torus, and hence X(G) = X(G)K , then also X∗ (G) = X∗ (G)K . On the other hand, if G is K-anisotropic, then X∗ (G)K = 0. Conversely, if X∗ (G) = X∗ (G)K (resp. X∗ (G)K = 0), then G is K-split (resp. K-anisotropic). EXAMPLE Let L/K be a finite (separable) extension of degree d. Set ¯ G = RL/K (Gm ). Then according to (2.4), there exists a K-isomorphism G ' (Gm )d , implying that G is a d-dimensional K-torus. The explicit description of the restriction of scalars in terms of the regular representation of L over K

62

Algebraic Groups

allows G to be realized as a K-subgroup of GLd (). Let ϕ denote the restriction to G of the determinant map. Then ϕ is a K-morphism G → Gm , i.e., an element of X(G)K . Furthermore, the restriction of ϕ to GK = L∗ , being the determinant of the regular representation, coincides with the norm map NL/K : L∗ → K ∗ from field theory. So, the kernel of ϕ, which is a K-torus, is usually called the norm torus associated with the field extension L/K and is (1) denoted by RL/K (Gm ). The minimal splitting field of G is the normal closure P of L over K. Set F = Gal(P/K) and H = Gal(P/L). Then X(G) as a module over 0 = Z[F] is isomorphic to Z[F/H], the free Z-module on the set F/H of left cosets on which F acts by left translations. The norm map ϕ : G → Gm corresponds to the homomorphism of 0-modules X Z → Z[F/H], z 7→ zσ where σ = gH (the sum taken over all cosets). Then the character module X(H) of the norm (1) torus H = RL/K (Gm ) is the quotient Z[F/H]/Zσ . Using the fact that the module of fixed points Z[F/H]F is Zσ , it is easy to see that X(H)K = 0, i.e., H is anisotropic. The same result can be obtained by working with cocharacters instead of characters. Indeed, X∗ (G) is isomorphic to Z[F/H], and X∗ (H) is the kernel of the augmentation map X X Z[F/H] → Z, ag gH → ag . Clearly, X∗ (H)F = X∗ (H) ∩ Zσ = (0), which again proves that H is Kanisotropic. The preceding example can be generalized as follows. Let L1 , . . . , Lr be finite (separable) field extensions of K, and for each i = 1, . . . , r, let ϕi : RLi /K (Gm ) → Gm be the corresponding norm map. Then {(x1 , . . . , xr ) ∈ RL1 /K (Gm ) × · · · × RLr /K (Gm ) : ϕ1 (x1 ) · · · ϕr (xr ) = 1} is a torus, which we call the multinorm torus associated with the extensions L1 , . . . , Lr (cf. Hürlimann [1984]). Tori of the form RL/K (Gm ) and their finite direct products are called quasisplit (over K). They are precisely the K-tori whose character groups are permutation modules, i.e., free finitely generated Z-modules possessing a basis that is permuted by the absolute Galois group. Quasi-split tori are the easiest to study, and sometimes when dealing with arbitrary tori, it may be helpful to present the torus under consideration as either a subtorus or a quotient of a suitable quasi-split torus. We will now describe a few typical constructions of this kind that we will need later on.

2.1 Structure of Algebraic Groups

63

Proposition 2.2 Let F be a diagonalizable K-group that splits over an extension P/K. Then F can be included in an exact sequence, 1 → F → T → S → 1, where T and S are K-tori that split over P, with T quasi-split. ¯ PROOF: Let H denote the kernel of the natural action of G = Gal(K/K) on H ¯ the character group X(F), and let L = K be the corresponding fixed field. Then L is a finite Galois extension of K with Galois group F = G/H; clearly L ⊂ P. We can now consider X(F) as a module over the group ring 0 = Z[F] and present it as a quotient module of a free module 0 ` . This leads to an exact sequence of the form 0 → 1 → 0 ` → X(F) → 0. We now use the contravariant equivalence described in Theorem 2.1 to pass to the corresponding exact sequence of diagonalizable K-groups that split over P: 1 → F → T → S → 1. Since 1 and 0 ` have no Z-torsion, S and T are K-tori, and besides by our construction T = RL/K (Gm )` . If we take F to be a torus, then an argument dual to the proof of Proposition 2.2 yields the following. Proposition 2.3 Any K-torus F can be included into an exact sequence 1 → S → T → F → 1, where S and T are K-tori with T quasi-split. We will also need this proposition. Proposition 2.4 (ONO [1961]) For any K-torus F, there exists an integer m > 0 and a quasi-split torus T 0 such that F m × T 0 is isogenous to some quasi-split K-torus T. Recall that by an isogeny of algebraic groups, we mean a surjective homomorphism with a finite kernel. Two groups are said to be isogenous if there is an isogeny between them. Isogenies between semisimple groups are discussed in §2.1.13. For tori, however, isogenies exhibit a different type of behavior; in particular, they lead to an equivalence relation. In terms of character groups, this relation is expressed as follows: two tori T1 , T2 from the category A0 described

Algebraic Groups

64

in Theorem 2.1 are isogeneous if and only if the Q[F]-modules X(T1 ) ⊗Z Q and X(T2 ) ⊗Z Q are isomorphic. Proposition 2.4 is actually a restatement of Artin’s theorem on induced characters in terms of tori. We omit the proof here and refer the reader to Ono (1961; 1966).

2.1.8 Solvable and Unipotent Groups Throughout this subsection, we assume the base field has characteristic 0. An algebraic group G is said to be unipotent if all of its elements are unipotent. An example of a unipotent group is the additive group of :    1 x Ga = ∈ GL2 () : x ∈  . 0 1 If G ⊂ GLn () is unipotent, then (g−In )n = 0 for any g in G, and the truncated logarithmic map l : G → Mn (), l(g) = (g − In ) −

(g − In )2 (g − In )n−1 + · · · + (−1)n , 2 n−1

defines a polynomial isomorphism of varieties between G and its Lie algebra L(G). The inverse map is given by the truncated exponential map e : L(G) → G, e(X ) = In + X +

X2 X n−1 + ··· + . 2 (n − 1)!

In particular, G is always connected. Now let G ⊂ GLn () be a unipotent Kgroup. Then G is trigonalizable over K; i.e., there exists a matrix g in GLn (K) such that gGg−1 is contained in the group Un of upper unitriangular matrices. It follows, in particular, that G is nilpotent. Moreover, it can be shown that there is a central series G = G0 ⊃ G1 ⊃ · · · ⊃ Gm = {In }

(2.5)

in G such that Gi /Gi+1 ' Ga for i = 0, . . . , m − 1. Note that most of the preceding statements do not carry over to positive characteristic. We will need one technical fact about unipotent groups. Lemma 2.5 Suppose a K-split torus T acts by automorphisms on a (connected) unipotent K-group U. Then for any T-invariant K-subgroup V ⊂ U,

2.1 Structure of Algebraic Groups

65

there exists a T-invariant K-defined Zariski-closed subset P ⊂ U such that the product map induces K-isomorphisms of varieties ∼

∼

P × V →U and V × P→U. Moreover, if U is abelian, then for P one can choose a suitable K-defined subgroup of U. Indeed, if U is abelian, then the map l : U → L(U) introduced above is a group isomorphism. So, it suffices to pick a T-invariant K-defined vector subspace W ⊂ L(U) such that L(U) = L(V ) ⊕ W and then set P = e(W ). The general case can be easily reduced (e.g., by using a T-equivariant version of the central series (2.5)) to the situation where dim U/V = 1, and then one can again pick a 1-dimensional T-invariant K-defined complement W for L(V ) in L(U) and set P = e(W ). Now, let G ⊂ GLn () be a connected solvable group. Then G is conjugate to a group of upper triangular matrices (by the Lie–Kolchin Theorem). This yields the following structure theorem for solvable groups: the set Gu of unipotent elements in G constitutes a normal subgroup of G, and G is a semidirect product of Gu and an (arbitrary) maximal torus T ⊂ G. If G is a K-group, then Gu is also a K-group and there exists a maximal K-torus T ⊂ G; moreover, in this case, the semidirect product decomposition G = TGu is defined over K. There is a composition series (over ) G = G0 ⊃ G1 ⊃ · · · ⊃ Gm = {In }

(2.6)

with the quotients Gi /Gi+1 isomorphic to Gm or Ga . If there is a series (2.6) of K-subgroups such that the quotients Gi /Gi+1 are all K-isomorphic to Gm or Ga , then G is said to be K-split. In fact, this is equivalent to the existence of a maximal K-split torus T ⊂ G, and then any K-torus in G is also K-split. In particular, any unipotent K-group is K-split, and in this case all the quotients in the corresponding series (2.6) are K-isomorphic to Ga .

2.1.9 Connected Groups The analysis of a connected group G typically employs two important classes of subgroups, namely maximal tori and Borel subgroups (i.e., maximal connected solvable subgroups). Since the dimension of G is finite, maximal tori and Borel subgroups always exist. Furthermore, all maximal tori (resp. Borel subgroups) in G are conjugate. (In particular, the dimension r = dim T of a maximal torus T ⊂ G is independent of the choice of T; it is called the (absolute) rank of G and denoted rank G or rk G.) It is also known that any Borel

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Algebraic Groups

subgroup is its own normalizer in G. Consequently, if we fix a maximal torus T ⊂ G (respectively, a Borel subgroup B ⊂ G), then the set of all maximal tori in G (respectively, of Borel subgroups) can be identified with the coset space G/N (respectively G/B), where N = NG (T) is the normalizer of T in G (cf. §§ 2.4.8 and 2.4.9). Subgroups P ⊂ G containing some Borel subgroup B are called parabolic. They are connected and are characterized by the condition that the quotient G/P is a projective variety (cf. Borel [1991]). Any K-defined connected group G always contains a maximal torus that is defined over K. On the other hand, as a rule, G need not have a K-defined Borel subgroup; those groups that do are called quasi-split over K. Furthermore, G is said to be K-split if it contains a maximal K-torus that is K-split. (We note that for connected solvable groups, this notion coincides with the one considered in the previous subsection.) Theorem 2.6 Let G be a connected algebraic group over an infinite perfect field K. Then the group of K-rational points GK is Zariski-dense in G. The maximal connected solvable normal subgroup R(G) of G is called the radical of G, and the maximal connected unipotent normal subgroup Ru (G) of G is called the unipotent radical. (Obviously, Ru (G) coincides with the unipotent part R(G)u of R(G).) A connected group G is said to be reductive (respectively semisimple) if Ru (G) = {e} (respectively R(G) = {e}). It is easy to see that for G connected, the quotient G/R(G) is semisimple and the quotient G/Ru (G) is reductive. For a K-group G, both R(G) and Ru (G) are K-defined. Theorem 2.7 (MOSTOW [1956]) Let G be a connected algebraic group over a field K of characteristic zero. Then there exists a reductive K-subgroup H ⊂ G for which we have a semidirect product decomposition G = HRu (G). Moreover, any reductive K-subgroup H 0 ⊂ G is conjugate to a subgroup of H by an element of Ru (G)K . The decomposition G = HRu (G) described in the theorem is called a Levi decomposition. It is often used to reduce the argument to reductive groups. Theorem 2.7 is an analog of the theorem for Lie groups obtained by Levi and Malcev (cf. Malcev [1944; 1945]). On the other hand, we would like to point out that connected groups in positive characteristic may fail to have a Levi decomposition, and we refer the reader to Conrad, Gabber, and Prasad (2015) and Conrad and Prasad (2016; 2017) for an exposition of the recently developed theory of pseudo-reductive groups that addresses this and many

2.1 Structure of Algebraic Groups

67

other delicate issues of the theory of algebraic groups over fields of positive characteristic; arithmetic applications of this theory are given in Conrad (2012b).

2.1.10 Reductive Groups The following theorem summarizes some useful properties of reductive groups. Theorem 2.8 Let G be a reductive K-group. Then (1) R(G) coincides with the connected component S = Z(G)0 of the center and is a torus; (2) the commutator subgroup H = [G, G] is a semisimple K-group; (3) G = HS is an almost direct product (i.e., H ∩ S is finite); (4) if char K = 0, then any algebraic representation f : G → GLn () is completely reducible. A more thorough analysis of reductive, and particularly of semisimple, groups uses root systems. To attach a root system to a given reductive algebraic group G, we need to fix a maximal torus T ⊂ G. Let g = L(G) be the Lie algebra of G and let Ad : G → GL(g) be the adjoint representation. It follows from §2.1.7 that Ad(T) is a diagonalizable subgroup of GL(g). Thus, if for a character α ∈ X(T) we let gα denote the corresponding weight subspace: gα = {X ∈ g : Ad(t)X = α(t)X , ∀t ∈ T}, and set R(T, G) = {α ∈ X(T) : α 6= 0 and gα 6= 0}; then  g = L(T) ⊕ 

 M

gα  ,

(2.7)

α∈R(T,G)

where L(T) is the Lie algebra of T that coincides with the weight 0 subspace. A fundamental fact is that R = R(T, G) is an abstract root system in the space V = X(T/S)⊗Z R (cf. Bourbaki [2002, Chapter 6] for the definition), called the root system of G relative to T. Note that if G is semisimple, then S = {e} and we obtain a root system in X(T) ⊗Z R. Each root subspace gα is 1-dimensional and in fact is the Lie algebra of a connected 1-dimensional unipotent subgroup Uα ⊂ G. The subgroup Gα ⊂ G generated by Uα and U−α is a semisimple

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Algebraic Groups

group of rank 1, hence isomorphic to SL2 or PGL2 . Alternatively, Gα can be described as the commutator subgroup of the centralizer in G of (ker α)0 ⊂ T. Let 5 ⊂ R be a system of simple roots and R5 + be the corresponding system of positive roots (cf. Bourbaki [2002, Chapter 6]). Then the group U(5) generated by Uα for all α ∈ R5 + is normalized by T, and the semidirect product B(5) := TU(5) is a Borel subgroup of G. Moreover, 5 → B(5) defines a bijection between the systems of simple roots in R and the Borel subgroups of G containing T. Thus, a given Borel subgroup B ⊂ G uniquely determines a system of simple roots 5 ⊂ R, and one can choose an ordering V+ in V so that R5 + = R ∩ V+ . Given a root system R, one can consider the associated Weyl group W = W (R) (Bourbaki [2002]). We recall that W is generated by the set S of reflections with respect to all simple roots α ∈ 5 for a fixed system of simple roots 5 ⊂ R, and moreover the pair (W , S) is a Coxeter group (cf. Bourbaki [2002, Chapters 4 and 6]). Furthermore, W has a unique element w of 5 maximal length (with respect to S); it is characterized by w(R5 + ) = − R+ , and its length actually equals the number of positive roots. It is a fundamental fact that for the root system R = R(T, G) introduced above, the Weyl group W (R) can be naturally identified with the Weyl group W (T, G) of G with respect to T, defined as NG (T)/T, where NG (T) is the normalizer of T. We briefly recall how this identification is implemented. The action of NG (T) on T by conjugation gives rise to a homomorphism W (T, G) → Aut(R). For any α ∈ R, we let Tα = T ∩ Gα . Then W (Tα , Gα ) has order 2, and any element nα in NGα \Tα induces on R the corresponding reflection wα . It follows that the image of W (T, G) in Aut(R) contains W (R). But W (T, G) and W (R) have the same order, since the first group acts simply transitively on the set of Borel subgroups containing T, and the second on the systems of simple roots in R. The Weyl group W (T, G) also arises in the Bruhat decomposition. Fix a Borel subgroup B of G containing T, and for each w ∈ W (T, G) pick a representative nw ∈ NG (T). Then the double coset Bnw B depends only on w but not on the choice of nw . Theorem 2.9 (BRUHAT

DECOMPOSITION )

G=

[

For a reductive group G we have

Bnw B,

(2.8)

w∈W

where the double cosets in the right-hand side are disjoint. Corollary 2.10 The intersection of any two Borel subgroups of G contains a maximal torus.

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69

Indeed, let B ⊂ G be an arbitrary Borel subgroup of G. Fix a maximal torus T ⊂ B, and let (2.8) be the associated Bruhat decomposition. By the conjugacy theorem, any other Borel subgroup is of the form gBg−1 for some g ∈ G. Using the Bruhat decomposition we can write g = b1 nb2 with bi ∈ B (i = 1, 2) and n ∈ NG (T). Then −1 B ∩ gBg−1 = b1 (B ∩ nBn−1 )b−1 1 ⊃ b1 Tb1 ,

and so the torus T1 = b1 Tb−1 1 is as desired. As we already noted, Bnw B is independent of the choice of nw , so we will often write BwB instead of Bnw B, emphasizing thereby that the double cosets of G modulo B in the Bruhat decomposition are parametrized by the elements of W . Of special significance is the double coset BwB corresponding to the element w ∈ W of maximal length, which is called the big cell (of the decomposition (2.8)). More precisely, suppose that B = B(5), where 5 ⊂ R is the corresponding system of simple roots, and let w0 ∈ W be the element of maximal length with respect to the generating set S = {wα : α ∈ −1 5 − 5}. Then w0 (R5 + ) = − R+ (the set of negative roots), and w0 Bw0 = B , − where B = TU(−5) and U(−5) is the subgroup generated by Uα for all − α ∈ (−R5 + ). For the sake of brevity, set U = U(5) and U = U(−5); then ϕ

Bw0 B = UTU − w0 . Furthermore, consider the product map U × T × U − − → G. Computing its differential at the identity and taking into account the decomposition (2.7), we see ϕ is dominant, hence the “big cell” is a Zariski-open subset of G. Moreover, it is easy to verify that ϕ is injective, hence is a birational isomorphism, implying that G is a rational variety (Theorem 2.46). Finally, we have dim G = dim T + |R| = dim T + 2`(w0 ), where `(w0 ) is the length of w0 . EXAMPLE. Let G = GLn (). Then g = Mn (). The group T of all diagonal matrices is a maximal torus of G. Let εi denote the character of T given by εi : diag(t1 , . . . , tn ) 7→ ti .

(2.9)

Clearly, for any matrix X = (xij ) ∈ Mn () and any t = diag(t1 , . . . , tn ) in T, we have Ad(t)(X ) = (ti tj−1 xij ) = ((εi − εj )(t)xij ), hence R(T, G) = {εi − εj : i 6= j}. One can take 5 = {εi − εi+1 : i = 1, . . . , n − 1}

(2.10)

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Algebraic Groups

for a system of simple roots, and then R5 + = {εi − εj : i < j}. Now, it is easy to show that the Borel subgroup B(5) coincides with the group of upper triangular matrices. The normalizer of the torus NG (T) is the group of monomial matrices, and consequently W (T, G) is isomorphic to the symmetric group Sn . In turn, W (R) is also isomorphic to Sn , and the action of the former on roots is consistent with the action of the latter on the corresponding indices. The canonical system S of generators of W (R) = Sn corresponding to the roots α ∈ 5 consists of the transpositions (i, i + 1) for i = 1, . . . , n − 1. The element of maximal length w0 ∈ W (R) sends any i to n − i + 1. It follows that B− = w0 Bw−1 0 is the group of lower triangular matrices, and the product UTU − consists of those matrices for which all the principal minors are nonzero.

2.1.11 Regular Semisimple Elements Let G be a reductive algebraic group, T ⊂ G be a maximal torus, and R = R(T, G) be the associated root system. A semisimple element g in G is regular if the dimension of its centralizer ZG (g) equals the rank of G (i.e., dim T), in which case the connected component ZG (g)0 is a torus. Regular elements always exist; in fact, an element t ∈ T is regular if and only if α(t) 6= 1 for all α ∈ R. It follows that the regular semisimple elements contained in T form a Zariski-open and dense subset V ⊂ T. If we now consider the regular map ϕ : G × V −→ G, (g, v) 7→ gvg−1 , then a simple computation shows that the differential d(g,v) ϕ is surjective for any (g, v) ∈ G × V . This implies that ϕ is an open map, hence the set of regular semisimple elements is Zariski-open in G. A semisimple element X ∈ L(G) is regular if its centralizer is the Lie algebra of a torus. Properties of regular semisimple elements in the Lie algebra of a reductive algebraic group are analogous to those of regular semisimple elements in the group; in particular, these form a nonempty Zariski-open subset of L(G).

2.1.12 Parabolic Subgroups With the same notations and conventions as in the previous subsection, fix a system of simple roots 5 ⊂ R, and let B(5) be the corresponding Borel subgroup. It follows from the Bruhat decomposition and properties of the Weyl group W that any subgroup P ⊂ G containing B coincides with a subgroup of the form P1 = BW1 B for some subset 1 ⊂ 5, where W1 is the subgroup of W generated by the reflections wα for α ∈ 1. Moreover,

2.1 Structure of Algebraic Groups

71

! L(P1 ) = L(T) ⊕

M

gα ,

α∈2

where 2 is the union of the set of positive roots R5 + and the set of the negative roots that are linear combinations of roots from 1. Subgroups of the form P1 are called standard parabolic subgroups. Since the Borel subgroups are all conjugate, any parabolic subgroup of G (i.e., a subgroup containing some Borel subgroup; see §2.1.9) is conjugate to some standard parabolic subgroup.

2.1.13 Semisimple Groups Semisimple algebraic groups are classified up to isogeny. We recall that an isogeny is a surjective morphism f : G → H of algebraic groups with finite kernel. (In positive characteristic, the class of isogenies used in the classification needs to be restricted to central isogenies, which are characterized by the condition that for any -algebra A, the kernel of the induced homomorphism fA : GA → HA of the groups of A-points is contained in the center of GA . We will not discuss noncentral isogenies here as in characteristic zero, which is the main case of interest for us, any isogeny is automatically central. However, the reader who wishes to get an adequate picture of the general case should replace “isogeny” by “central isogeny” throughout this subsection.) A connected group G is said to be an almost direct product of its connected subgroups G1 , . . . , Gr if the product map G1 × · · · × Gr → G is an isogeny of algebraic groups. We call a connected noncommutative algebraic group G (absolutely) almost simple if it has no nontrivial connected normal subgroups. Proposition 2.11 Let G be a semisimple group and let Gi for i ∈ I be the minimal connected normal subgroups of G. Then I is a finite set, say I = {1, . . . , r}, and G is an almost direct product of G1 , . . . , Gr . In particular, G is an almost direct product of almost simple groups. In fact, there is a natural bijection between the almost simple factors Gi , i = 1, . . . , r, of G and the irreducible components Ri in the decomposition S R = ri = 1 Ri of the root system R of G (cf. Bourbaki [2002, Chapter 6]); more precisely, after possible renumbering, Gi is generated by Uα for α ∈ Ri . In general, one cannot replace “almost direct product” by “direct product”; however, we will now describe two cases where this is possible. A connected group G is simply connected if any (central) isogeny f : H → G, with H connected, is necessarily an isomorphism; similarly, G is adjoint if any (central) isogeny f : G → H is an isomorphism.

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Algebraic Groups

Theorem 2.12 Let G be a semisimple group. ˜ → G and ϕ : G → G ¯ with G ˜ simply (1) There exist (central) isogenies π : G ¯ connected and G adjoint. (2) Any simply connected (resp., adjoint) group is a direct product of its minimal connected normal subgroups, all of which are simply connected (resp., adjoint). (3) Let R = R(T, G) be the root system of G with respect to some maximal torus T, and let 5 ⊂ R be a system of simple roots. Then G is simply connected (resp., adjoint) if X(T) has a Z-basis {λα : α ∈ 5} such that wα λβ = λβ − δαβ α, where δαβ is the Kronecker delta (resp., 5 spans X(T)). EXAMPLE. Let G = SLn (). As in the example we considered in §2.1.10, the diagonal matrices in G form a maximal torus T, and then the corresponding root system R(T, G) consists of εi − εj where i, j ∈ {1, . . . , n}, i 6= j, and εi is given by (2.9). Furthermore, let 5 be the system of simple roots (2.10). For each j = 1, . . . , n − 1, consider λj : diag(t1 , . . . , tn ) 7→ t1 · · · tj . Then wαi (λj ) = λj − δij αi , and consequently G is simply connected. ˜ → G in Theorem 2.12(1) is called a universal cover The isogeny π : G and F := ker π is the fundamental group of G. Thus, any semisimple group possesses a universal cover that is a direct product of simply connected almost simple groups. So, the classification, up to (central) isogeny, of semisimple groups is completed by the following. Theorem 2.13 An almost simple simply connected algebraic group is uniquely determined up to isomorphism by its root system. The root system of an almost simple group is irreducible and reduced, and therefore either belongs to one of the four classical series An , Bn , Cn , Dn , or is one of the five exceptional systems E6 , E7 , E8 , F4 , G2 . As usual, root systems are described by the corresponding Dynkin diagrams; here is the list of irreducible ones:

2.1 Structure of Algebraic Groups

An

E6

Bn

E7

Cn

73

E8

Dn

F4 G2

The Dynkin diagram of a semisimple group is the union of the Dynkin diagrams of its (almost) simple components (see Bourbaki [2002, Chapter 6] for the details). We also present a table that contains a description of the simply connected groups of classical types (cf. §2.2.3 for greater detail) and the structure of the center of the almost simple simply connected groups of all types, thus providing a complete description of the simple groups. Type An Bn Cn Dn

Realization SLn+1 Spin2n+1 Sp2n Spin2n

E6 E7 E8 F4 G2

— — — — —

Structure of the Center Z/(n + 1)/Z Z/2Z Z/2Z Z/2Z × Z/2Z, n even Z/4Z, n odd Z/3Z Z/2Z {e} {e} {e}

The Lie algebra g = L(G) of a semisimple group G admits a canonical basis, called the Chevalley basis. More precisely, one can pick elements Xα ∈ gα for each α ∈ R, and Hα ∈ L(T) for each α ∈ 5 so that {Xα }α∈R ∪ {Hα }α∈5 is a basis of g that satisfies the following relations: [Hα , Hβ ] = 0 [Hα , Xβ ] = cαβ Xβ [Xα , X−α ] = Hα [Xα , Xβ ] = dαβ Xα+β [Xα , Xβ ] = 0

for all α, β ∈ 5, with cαβ ∈ Z for all α ∈ 5, β ∈ R, for all α ∈ 5, with dαβ ∈ Z if α + β ∈ R, if β 6= −α and α + β 6∈ R.

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Algebraic Groups

A basis satisfying these relations (where cαβ and dαβ assume values depending only on α, β and R – cf. Steinberg [2016, Theorem 1] for more details) is uniquely determined up to a change of signs of the Xα ’s and an automorphism of g. In order to describe K-forms of a semisimple group G, one needs to know the structure of the automorphism group Aut G. It turns out that Aut G is a semidirect product of the group of inner automorphisms Int G (which can be ¯ by a certain finite group identified with the corresponding adjoint group G) which we will now describe. First, let us assume that G is simply connected. Then any symmetry σ of the Dynkin diagram of the root system R = R(T, G) extends to an automorphism fσ ∈ Aut G such that fσ (T) = T, fσ (B) = B and de fσ (Xα ) = Xσ α for all α ∈ 5, where B = B(5) is the Borel subgroup of G corresponding to the fixed system of simple roots 5, Xα is an element of the Chevalley basis of the Lie algebra g, and de is the differential at the identity (cf. §2.4.3). Moreover, the correspondence σ 7→ fσ is an injective homomorphism of the group Sym(R) of symmetries of the Dynkin diagram of R into Aut G, whose image we will also denote by Sym(R). Theorem 2.14 For a semisimple simply connected algebraic group G, the ¯ by Sym(R). automorphism group Aut G is the semidirect product of Int G ' G π ˜ →G be a universal cover. Then If G is an arbitrary semisimple group, let G ˜ of those automorphisms that Aut G is isomorphic to the subgroup of Aut G leave the fundamental group F = ker π invariant. Having reviewed the fundamentals of the theory of semisimple algebraic groups in the absolute case, i.e., over an algebraically closed field, we would like to observe that the corresponding theory over arbitrary fields is more complicated and less definitive (see §2.1.14 for a brief account). Nevertheless, for semisimple split groups, the theory can be developed almost in parallel with the absolute case. In particular, for an arbitrary field K and any given root system R, there exists a semisimple simply connected K-split group G with a maximal K-split torus T ⊂ G such that the corresponding root system R(T, G) coincides with R. This group is obtained via a construction due to Chevalley (cf. Steinberg [2016]). By and large, the theory of semisimple Ksplit groups merges with the theory of Chevalley groups, which is the focus of Steinberg’s book. We mention only that the corresponding Lie algebra g admits a Chevalley basis lying in gK . All automorphisms in Sym(R) are defined over K, and then Aut G can be identified with the K-defined semidirect product

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75

¯ o Sym(R). Any semisimple K-split group G possesses a K-defined universal G ˜ → G. cover π : G

2.1.14 Relative Root Systems Let G be a semisimple K-group and let S ⊂ G be a maximal K-split torus. The dimension dim S is called the K-rank of G and is denoted rankK G or rkK G. All maximal K-split tori in G are conjugate under GK , so the K-rank is well defined. Groups with K-rank > 0 (respectively K-rank = 0) are called K-isotropic (respectively K-anisotropic). It can be shown that if char K = 0, then G is K-anisotropic if and only if GK contains no unipotent elements other than the identity. Borel and Tits (1965) developed a structure theory for isotropic groups which, although analogous to the absolute case, leads to more modest results; viz., it determines the structure of a given isotropic group modulo information about the structure of its anisotropic kernel. As in the absolute case, the theory is based on associating to the group under consideration a certain root system. For this, fix a maximal K-split torus S ⊂ G and consider the adjoint action of S on g = L(G). For α ∈ X(S), we let gα = {X ∈ g : Ad(s)X = α(s)X , ∀s ∈ S} denote the corresponding weight subspace, and define R(S, G) = {α ∈ X(S) : α 6= 0 and gα 6= 0}. Then we can write  g = L(Z(S)) ⊕ 

 M

gα  ,

α∈R(S,G)

where L(Z(S)) is the Lie algebra of the centralizer Z(S) of S, which coincides with the weight 0 subspace; moreover, all the weight subspaces gα are defined over K. It turns out that RK := R(S, G) is a root system in the vector space V = X (S)⊗Z R, called the relative root system or the system of K-roots. The difference with the absolute case is that the spaces gα for α ∈ RK are generally not 1-dimensional, and the root system RK may not be reduced. The Weyl group W (RK ) of RK can be identified with the Weyl group W (S, G) of G relative to S, defined as the quotient group N(S)/Z(S) of the normalizer of S modulo its centralizer; moreover, any element of W (S, G) has a representative in N(S)K . Let 5 ⊂ RK be a system of simple roots and R5 +K be the corresponding system

76

Algebraic Groups

of positive roots. To each root α ∈ RK , there corresponds a K-defined connected unipotent subgroup Uα with Lie algebra gα if 2α ∈ / RK , and gα ⊕ g2α if 2α ∈ RK . Let U(5) be the group generated by Uα for all α ∈ R5 +K . Then U(5) is a connected unipotent group normalized by Z(S), and the semidirect product P(5) = Z(S)U(5) is a minimal parabolic K-subgroup. Moreover, the correspondence 5 → P(5) yields a bijection between the systems of simple roots in RK and the minimal parabolic K-subgroups of G containing S. Now set P = P(5) and U = U(5), and for each w ∈ W (S, G) pick a representative nw ∈ N(S)K . Then we have the Bruhat decomposition GK =

[

PK nw PK ,

w∈W (S,G)

and moreover, PK nw PK = UK nw PK . A group G is almost K-simple if it does not have nontrivial connected normal K-subgroups. While in the absolute case, a semisimple group G is absolutely almost simple if and only if the corresponding root system R is irreducible, in the relative situation a similar connection works only in one direction: for an almost K-simple group G, the root system RK is irreducible. One can give a graphical representation of the important information about a semisimple K-group G in the form of a Dynkin diagram equipped with some additional data, which is then called the Tits index. The procedure for constructing the Tits index is as follows (cf. Tits [1966], Borel and Tits [1965]). Consider a maximal K-split torus S ⊂ G and a maximal K-torus T ⊂ G containing S, and choose coherent orderings on the vector spaces X(S) ⊗Z R and X(T) ⊗Z R. Let R = R(T, G) be the root system of G relative to T, and let 5 ⊂ R be the system of simple roots resulting from the ordering on X(T)⊗Z R. Since G and T are defined over K, the natural action of the Galois group ¯ G = Gal(K/K) on X(T) gives rise to a permutation action on R. Let us now define the induced action, called the ∗-action, of G on the Dynkin diagram; since the vertices of the latter bijectively correspond to the roots in 5, it is enough to describe the action on 5. Given σ ∈ G, the image σ (5) is a system of simple roots in R = σ (R), and therefore there is a unique w in W (R) such that w(σ (5)) = 5; set σ ∗ = w ◦ σ : 5 → 5. The K-group G is called an inner (respectively, outer) form if the ∗-action is trivial (respectively, nontrivial). Furthermore, we call a vertex of the Dynkin diagram distinguished (and circle it) if the restriction of the corresponding

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77

simple root to S is nontrivial. Vertices of the diagram that belong to the same orbit of the ∗-action of G are placed “close” to each other, and in case they are distinguished a common circle is drawn around them. A Dynkin diagram with marked (circled) distinguished vertices and a specified ∗-action is called a Tits index. For example: Bn

E6

It is also customary to indicate the order of the homomorphic image of G that acts faithfully on 5. Thus, the second diagram has type 2 E6 , whereas all inner forms have type 1 X (where X is the symbol for the corresponding Dynkin diagram). Note that if a diagram has no symmetries (for example, it is of type Bn ), then any K-group of this type is automatically an inner form. Using Tits indexes, it is easy to determine the diagram of the anisotropic kernel of G, which is defined to be the commutator group of the centralizer Z(S) of a maximal K-split torus and is a semisimple K-anisotropic group. For this, one needs to discard the distinguished vertices and the edges in which at least one end is distinguished. (If all the vertices are distinguished, then G is quasisplit, hence Z(S) is a maximal torus T of G, and the anisotropic kernel is trivial). One can also find a maximal K-split torus S and its corresponding relative root system. Namely, S is defined in T by the equations α(x) = 1, where α runs through all non-distinguished roots, and also by the equations α1 (x) = · · · = α` (x) if α1 , . . . , α` lie in the same orbit of the ∗ -action. It follows that if a quasisplit group is an inner form (in particular, if the corresponding diagram has no symmetries), then it is split. The relative roots are obtained as the restriction to S of the roots in R for which this restriction is nontrivial. (We refer the reader to [AGNT, Chapter 6] for some examples of concrete computations.)

2.2 Classification K-Forms Using Galois Cohomology 2.2.1 L/K-Forms Suppose K is a field, and let X be an object with a K-structure (examples include a K-defined algebraic group or, more generally, a K-defined algebraic variety); for the sake of brevity, we will refer to such an X simply as a “Kobject.” Given a finite Galois extension L/K, a K-object Y is said to be an L/Kform of X if there is an L-defined isomorphism f : X → Y . The Galois group F = Gal(L/K) naturally acts on the L-morphisms between two K-objects, so for any σ ∈ F, one can consider

78

Algebraic Groups aσ := f −1 · σf ,

which lies in the group AutL (X ) of L-defined automorphisms of X . It is easy to check now that the map σ 7→ aσ is a (noncommutative) 1-cocycle on F with values in AutL (X ) (cf. §1.3.2). This leads to a map, ϕ : F(L/K, X ) −→ H 1 (F, AutL (X )), from the set of K-isomorphism classes of L/K-forms of X to the 1-cohomology set. Theorem 2.15 If X is an affine K-variety or an algebraic K-group, then ϕ is a bijection. Here is a sketch of the proof (cf. Serre [1997], Voskresenski˘ı [1998, Chapter 1, §3]). First, one shows that ϕ is well defined (i.e., is independent of the choice of a representative Y in a K-isomorphism class of L/K-forms and of the choice of an L-isomorphism f : X → Y ), and is injective. This part of the proof is purely formal and holds in much greater generality. The proof of the surjectivity of ϕ requires a more subtle argument based on twisting, which we have already encountered. Twisting was used in §1.3.2 to study exact sequences in noncommutative cohomology, but it can also be used to prove that ϕ is surjective. Namely, as in §1.3.2, consider a group G, a G-group A, and a G-set F, and assume that A acts on F in a manner compatible with the action of G. Then, given a cocycle a ∈ Z 1 (G, A), one can define the “twisted G-set” a F, which, up to G-isomorphism, depends only on the equivalence class of a in H 1 (G, A). Set H = a F and let f : F → H be the map induced by the identity map of F. Then it follows from the definition of a F that the cocycle {f −1 · sf }s∈G ∈ Z 1 (G, A) coincides with the original cocycle a. We note, however, that if F supports some additional structure (like that of an algebraic variety), then this “abstract” argument requires further refinement in order to show that the twisted object a F also supports the structure in question. In the case described in Theorem 2.15, this is obtained by considering the algebra of regular functions, also called the coordinate ring. Any affine algebraic variety is determined by its coordinate ring, and assigning the structure of an algebraic group to it is equivalent to assigning the structure of a Hopf algebra to the corresponding coordinate ring (cf. Borel [1991], Milne [2017]). So, to construct an L/K-form of an affine algebraic K-variety (resp., an affine algebraic K-group) X corresponding to a = {aσ } ∈ H 1 (F, AutL (G)), we consider the coordinate ring A = L[X ] of L-defined regular functions, and introduce a new action of F on it given by

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σ 0 ( f ) = (a∗σ )−1 (σ ( f )), where a∗σ is the automorphism of A (comorphism) corresponding to aσ . The L-algebra B thus obtained will serve as the ring of L-defined regular functions on the desired affine variety Y . Moreover, this variety Y will have the structure of an algebraic K-group if X has this structure. We will say that Y is obtained from X by twisting using a and write Y = a X . Remark Theorem 2.15 also holds for projective varieties. EXAMPLE 1. Let X = Gm be a 1-dimensional K-split torus, where char K6=2. √ Fix a quadratic extension L = K( c) of K, and let τ denote the generator of F = Gal(L/K). Now, consider the cocycle a = {aσ } ∈ Z 1 (F, AutL X ) given by ae = idX and aτ = θ, where θ(x) = x−1 for all x ∈ X . Then A := L[X ] is L[t, t−1 ] and AK = K[t, t−1 ]. Furthermore, the automorphisms θ and τ act on A as follows: θ ∗ : f (t) + g(t−1 ) 7→ f (t−1 ) + g(t), τ ∗ : f (t) + g(t−1 ) 7→ f τ (t) + gτ (t−1 ). It follows that the action of τ on the twisted algebra B = L[t, t−1 ] is given by τ 0 : f (t) + g(t−1 ) 7→ f τ (t−1 ) + gτ (t). Direct computation shows that the K-algebra BK = BF can be identified with C = K[u, v]/(u2 − cv 2 − 1) (where u and v are indeterminates) by means of the isomorphism given by u 7→

t + t−1 , 2

v 7→

t − t−1 √ . 2 c

(1)

But then C coincides with K[Y ], where Y = RL/K (Gm ) is the norm torus associated with the extension L/K (cf. §2.1.7); moreover, the isomorphism C ' BK respects the Hopf algebra structures on C and BK . Thus a X = Y . In many situations, X as a K-object is determined by the set of K-points XK (this is the case, for example, for vector spaces; vector spaces with equipped with tensors, such as symmetric bilinear/quadratic forms, algebras, etc.). Then, by abuse of terminology, we often do not differentiate between a K-object X and the corresponding set XK , and also between a twisted object Y and the set YK . Here some caution needs to be exercised; for example, this loosening of terminology may not be applied to all algebraic varieties, as it is possible to have XK = YK = ∅ without X and Y being K-isomorphic. Nevertheless, in

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cases where it does not lead to confusion, this approach is convenient, so we will make use of it. EXAMPLE 2. Let V = K 2 be a 2-dimensional space over K with char K6=2, equipped with a quadratic form f given by f (x1 , x2 ) = x1 x2 in terms of the coordinates x1 , x2 with respect to the standard basis e1 , e2 of V . As in Example √ 1, consider a quadratic extension L = K( c), and let τ be the generator of F = Gal(L/K). Let O2 ( f ) be the orthogonal group of the quadratic form f , and let b = {bσ } denote the cocycle in Z 1 (F, O2 ( f )L ) given by be = id and bτ = g, where g ∈ O2 ( f ) switches e1 and e2 . Consider the space V ⊗K L, then twist it using b and set W = b (V ⊗K L)K . Direct computation shows that the vectors 1 1√ u1 = (e1 + e2 ) and u2 = c(e1 − e2 ) 2 2 constitute a K-basis of W , and moreover, in terms of the coordinates y1 , y2 associated with the basis u1 , u2 , the quadratic form f (or, more precisely, its extension to V ⊗K L) looks as follows: f (y1 , y2 ) = y21 − cy22 . Thus, twisting (V , f ) by means of b yields (W , h), where h is given by h(y1 , y2 ) = y21 − cy22 . Note that this example is directly related to Example 1 since (1)

SO2 ( f ) = Gm and SO2 (h) = RL/K (Gm ). We recommend that the reader work out the details of this connection. The second example allows the following generalization.

2.2.2 Spaces with Tensors Consider a pair (V , x), where V is a finite-dimensional vector space over K and q x is a tensor on V of type (p, q), i.e., an element of Tp (V ) := T p (V ) ⊗K T q (V ∗ ) in the standard notations. (The reader not familiar with tensors may assume that x is a bilinear form on V , i.e., a tensor of type (0, 2); with the exception of algebras, which are related to tensors of type (1, 2), we will not work with any other types of tensors in this book.) Given a Galois extension L/K with Galois p p group F, we can consider VL = V ⊗K L and xL = x⊗1 ∈ Tq (VL ) = Tq (V )⊗K L. A pair (W , y), where W is a space over K of the same dimension as V and y ∈

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p

Tq (W ), is called an L/K-form of (V , x) if there is an isomorphism (VL , xL ) ' (WL , yL ). As in §2.2.1, there is a map ϕ : F(L/K, (V , x)) −→ H 1 (F, AutL (VL , xL )). Proposition 2.16 ϕ is a bijection. We need only prove the surjectivity of ϕ, for which we use Lemma 2.17 H 1 (F, GLn (L)) = 1 for any n ≥ 1. In particular, H 1 (F, L∗ ) = 1. The latter assertion is known as Hilbert’s Theorem 90. If L/K is a cyclic extension and σ is a generator of its Galois group F, then, using the wellknown description of H 1 (F, L∗ ) in this case, one can give an equivalent reformulation, which was used repeatedly in §§1.4–1.5: any element a ∈ L∗ such that NL/K (a) = 1, is of the form a = σ (b)/b for some b ∈ L∗ . PROOF: Consider the space V = K n . Then VL = Ln with F = Gal(L/K) acting componentwise. Now let a = {aσ } be a 1-cocycle on F with values in GLn (L). Define a new action of F on VL by σ 0 (v) = aσ σ (v) for σ ∈ F, v ∈ VL . Let U denote the K-vector space of fixed points for the new action. For any P v ∈ VL , the vector b(v) = σ ∈F aσ σ (v) clearly lies in U. We will now show that the b(v) for all v ∈ VL generate VL over L, hence, in particular, U ⊗K L ' VL . Indeed, let u be a linear form on VL that vanishes on all the b(v). Then for any ` ∈ L and any v ∈ VL , we have X 0 = u(b(`v)) = σ (`)u(aσ σ (v)), σ ∈F

which by Artin’s theorem on linear independence of characters (see Lang [2002, Chapter VI, §4]) implies that u(aσ σ (v)) = 0 for all σ , hence u = 0 and our claim follows. Thus we can choose vectors v1 , . . . , vn ∈ VL such that b(v1 ), . . . , b(vn ) are linearly independent over L. Let c be the matrix of the linear transformation of VL that maps the canonical basis to v1 , . . . , vn . Then by P our construction the matrix b := σ aσ σ (c) is nonsingular, and direct calculation shows that aσ = bσ (b)−1 for all σ ∈ F, proving that a is equivalent to the trivial cocycle as required. Now, let a = {aσ } be an arbitrary cocycle on F with values in AutL (VL , xL ). Since the latter group is a subgroup of GL(VL ), it follows from Lemma 2.17

Algebraic Groups

82

that there exists b ∈ GL(VL ) such that aσ = bσ (b)−1 . Then b extends to an p automorphism of Tq (VL ), and this extension will also be denoted by b. We p p claim that the tensor x0 := b−1 (x) ∈ Tq (VL ) in fact lies in Tq (V ) (here, by abuse of notations, we do not differentiate between x and xL etc.). For this, it suffices to show that x0 is fixed by F. We have σ (x0 ) = σ (b)−1 (σ (x)) = σ (b)−1 (x) = b−1 (bσ (b)−1 )(x) = b−1 aσ (x) = b−1 x = x0 since aσ ∈ AutL (VL , xL ), as required. Now, using the map f : (VL , xL ) → (VL , x0L ) given by b−1 , we easily ascertain that the pair (V , x0 ) corresponds to the original cocycle a. There is another way to look at this construction. Consider the Kspace W = b(V ) and let y denote the tensor of W obtained as the pullback of x0 . Then (W , y) ' (V , x0 ) over K, so (W , y) also corresponds to a. On the other hand, it follows, for example, from the proof of Lemma 2.17 that W actually coincides with the space U of fixed points under the twisted Galois action introduced therein, and y coincides with the “restriction” of xL to W (here “restriction” is understood in terms of interpreting xL as an L-linear map ⊗q ⊗p VL → VL ). Applying Proposition 2.16 to nondegenerate symmetric bilinear (or, equivalently, quadratic) forms on V , we obtain Proposition 2.18 Let f be a nondegenerate quadratic form defined on an n-dimensional vector space V over a field K, and let On ( f ) be the orthogonal group of f (cf. §2.3). Then for any Galois extension L/K with Galois group F, there is a natural bijective correspondence between the elements of H 1 (F, On ( f )L ) and the K-equivalence classes of those quadratic forms on V that are L-equivalent to f . Next, applying the proposition to nondegenerate alternating bilinear forms on V and taking into account that any two such forms are equivalent over K (Bourbaki [1998, Chapter 9, §5]), we obtain the following. Proposition 2.19 Let f be a nondegenerate alternating bilinear form on an n-dimensional vector space V over a field K. Then for any Galois extension L/K, we have H 1 (F, Spn ( f )L ) = 1, where Spn ( f ) is the symplectic group of f (cf. §2.3).

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There are other applications of Proposition 2.16. In particular, the structure of an algebra (i.e., multiplication) on a vector space is given by a tensor of type (1, 2). So, the L/K-forms of a K-algebra A are in one-to-one correspondence with the elements of H 1 (F, AutL (AL )), where AutL (AL ) is the group of L-automorphisms of the L-algebra AL = A ⊗K L. Applying this to A = Mn (K) and using the fact that all automorphisms of the L-algebra AL are inner (the Skolem–Noether Theorem, cf. §1.4.1), hence AutL (AL ) = PGLn (L), we see that the elements of H 1 (F, PGLn (L)) are in one-to-one correspondence with the L/K-forms of A, i.e., with the central simple K-algebras of dimension n2 that split over L. In the preceding examples, the groups of L-automorphisms are the groups of L-points of certain algebraic groups (this is always the case when dealing with the automorphism group of a vector space equipped with a tensor). This leads us to our next topic.

2.2.3 Cohomology of Algebraic Groups Let G be an algebraic K-group, and let L/K be a finite Galois extension with Galois group F. Then F acts on the group of L-points GL , and we can define the set H 1 (F, GL ), which in the sequel will be denoted by H 1 (L/K, G). If M ⊃ L are two finite Galois extensions of K, then there is the inflation map ρLM : H 1 (L/K, G) −→ H 1 (M/K, G). This allows us to extend the definition of H 1 (L/K, G) to an infinite Galois extensions L/K. Namely, it is well known that the Galois group Gal(L/K) can be identified with the inverse limit lim Gal(Li /K) of the Galois groups of finite Galois subextensions, and then we ←− set H 1 (L/K, G) = lim H 1 (Li /K, G), −→

where the direct limit is taken with respect to the inflation maps ρLLji for Li ⊃ Lj . Equivalently, H 1 (L/K, G) can be defined as the continuous 1-cohomology of the profinite group Gal(L/K) with coefficients in the discrete group GL . We ¯ will write H 1 (K, G) instead of H 1 (K/K, G). Let X be a K-object having an algebraic K-group G as its automorphism group. Taking direct limits in Theorem 2.15 and Proposition 2.16, we conclude that in the situation considered therein, the set H 1 (K, G) parametrizes ¯ the classes of K-isomorphic K/K-forms of X , i.e., K-isomorphism classes of ¯ We also note that for such K-objects Y that become isomorphic to X over K. any algebraic K-group G, we have H 0 (K, G) = GK .

Algebraic Groups

84

The exact sequences of noncommutative cohomology described in §1.3.2 yield, as a special case, analogous exact sequences for the Galois cohomology of algebraic groups. In particular, any exact sequence 1→F→G→H →1 of K-groups and K-homomorphisms gives rise to the following exact sequence of pointed sets: ψK

ϕ

1 → FK → GK → HK → H 1 (K, F) → H 1 (K, G) → H 1 (K, H),

(2.11)

where ψK is the coboundary map. In addition, if F lies in the center of G, then ψK is a group homomorphism, and there is a map ∂K : H 1 (K, H) → H 2 (K, F) extending (2.11) by one more term: ∂K

· · · → H 1 (K, H) → H 2 (K, F). We will now present some sample computations of the cohomology of algebraic groups. Consider the exact sequence d

1 → SLn → GLn → Gm → 1, where d is the determinant map. The corresponding cohomological exact sequence (2.11) in this case looks as follows: d

GLn (K) → K ∗ → H 1 (K, SLn ) → H 1 (K, GLn ).

(2.12)

According to Lemma 2.17, we have H 1 (K, GLn ) = 1. On the other hand, the determinant map det : GLn (K) → K ∗ is surjective. So, (2.12) yields the following. Lemma 2.20 H 1 (K, SLn ) = 1. Furthermore, the special case of Lemma 2.17 for n = 1 (Hilbert’s Theorem 90) asserts that H 1 (K, Gm ) = 1. Then H 1 (K, RL/K (Gm )) = H 1 (L, Gm ) = 1 for any finite (separable) extension L/K, by Shapiro’s Lemma. Invoking the definition of a quasi-split K-torus, we obtain Lemma 2.21 Let T be a quasi-split K-torus. Then H 1 (K, T) = 1.

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Next, consider the exact sequence ϕ

(1)

1 → RL/K (Gm ) → RL/K (Gm )−→Gm → 1, where ϕ is the norm map. Passing to cohomology, we obtain the exact sequence NL/K

(1)

L∗ −→K ∗ → H 1 (K, RL/K (Gm )) → H 1 (K, RL/K (Gm )) → 1, which implies the following. (1)

Lemma 2.22 H 1 (K, RL/K (Gm )) ' K ∗ /NL/K (L∗ ). Now let us consider the exact sequence [n]

1 → µn → Gm −→Gm → 1,

(2.13)

where [n] denotes the morphism of raising to the nth power, and µn := ker [n] is the group of nth roots of unity. Then (2.13) yields the exact sequences [n]

K ∗ −→K ∗ → H 1 (K, µn ) → H 1 (K, Gm ) = 1, and [n]

1 = H 1 (K, Gm ) → H 2 (K, µn ) → H 2 (K, Gm )−→H 2 (K, Gm ).

(2.14)

Since H 1 (K, Gm ) = 1 and H 2 (K, Gm ) can be identified with the Brauer group Br(K), the sequences (2.14) yield Lemma 2.23 H 1 (K, µn ) ' K ∗ /K ∗n , and H 2 (K, µn ) = Br(K)n , the subgroup of Br(K) annihilated by n. Finally, replacing Lemma 2.17 by Lemma 2.20 in the proof of Proposition 2.16, we obtain the following interpretation of the set of 1-cohomology H 1 (K, SOn ( f )) of the special orthogonal group SOn ( f ) of a nonsingular quadratic form f (cf. also [AGNT, §6.6]). Proposition 2.24 The elements of H 1 (K, SOn ( f )) are in one-to-one correspondence with the K-equivalence classes of n-dimensional quadratic forms over K having the same discriminant as f . More examples of cohomology computations can be found in [AGNT, Chapter 6], which is specifically focused on the Galois cohomology of algebraic groups. For now, however, we conclude our brief survey of this topic by reducing the computation of the cohomology of connected groups to that of reductive groups.

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Lemma 2.25 Let K be a field of characteristic 0. Then for any unipotent Kgroup U, we have H 1 (K, U) = 1. PROOF: We begin by establishing the additive form of Hilbert’s Theorem 90, which asserts that H 1 (K, Ga ) = 1, and amounts to showing that H 1 (F, L) = 1 for any finite Galois extension L/K with Galois group F. Let c ∈ L be an element such that TrL/K (c) 6= 0. Given a 1-cocycle a = {aσ } ∈ Z 1 (F, L), we put X 1 b= aτ τ (c). TrL/K (c) τ ∈F

Direct computation then shows that aσ = b − σ (b) for all σ ∈ F, yielding that the cocycle a is trivial. In fact, it follows from the Normal Basis Theorem (cf. Lang [2002, Chapter VI, §13]) that L is an induced F-module, and therefore H i (F, L) = 1 for all i ≥ 1, by Shapiro’s Lemma. For an arbitrary unipotent K-group U, the proof goes by induction on dim U. It follows from §2.1.8 that we can find a normal K-subgroup W ⊂ U isomorphic to Ga . Then the exact sequence 1 → W → U → U/W → 1 yields the exact cohomological sequence H 1 (K, W ) → H 1 (K, U) → H 1 (K, U/W ). We proved above that H 1 (K, W ) is trivial; on the other hand, H 1 (K, U/W ) is trivial by the induction hypothesis, so H 1 (K, U) is trivial, as desired. We note that the lemma remains valid for a connected unipotent group U over any perfect field K (with basically the same proof). However, if either U is not connected or K is not perfect, H 1 (K, U) may be nontrivial (cf. Serre [1997, Chapter 3]). Proposition 2.26 Let G be a connected algebraic group defined over a field K of characteristic 0, and let H be a maximal reductive K-subgroup (cf. Theorem 2.7). Then the embedding ϕ : H ,→ G induces a bijection H 1 (K, H) ' H 1 (K, G). PROOF: Let U = Ru (G) be the unipotent radical of G, and let π : G → G/U ' H be the canonical map composed with the isomorphism given by the Levi decomposition G = HU (cf. §2.1.9). Since ϕ ◦ π = idH , the composition of the corresponding maps on cohomology

2.2 Classification K-Forms Using Galois Cohomology ϕ∗

87

π∗

H 1 (K, H)−→H 1 (K, G)−→H 1 (K, H) is the identity map as well. So, to prove the proposition it suffices to show that π∗ is injective. For this, we consider the exact sequence of cohomology π∗

H 1 (K, U) −→ H 1 (K, G)−→H 1 (K, H),

(2.15)

arising from the exact sequence 1 → U → G → H → 1. By Lemma 2.25 we have H 1 (K, U) = 1, so we immediately conclude from (2.15) that ker π∗ is trivial. Unfortunately, in noncommutative cohomology, the triviality of ker π∗ may not imply the injectivity of π∗ . To establish the injectivity, one uses a standard trick based on twisting. Namely, suppose that for g, h ∈ Z 1 (K, G) we have π∗ (g) = π∗ (h). (Here we use the same notations for the cohomology classes as for the original cocycles.) Let g G (resp., g U) denote the algebraic Kgroup obtained from G (respectively U) by twisting using g (more precisely, Int g), and let τg : H 1 (K, g G) → H 1 (K, G) be the corresponding bijection (cf. Lemma 1.18). Set F = g G/g U and consider the sequence g π∗

H 1 (K, g U) −→ H 1 (K, g G)−→H 1 (K, F),

(2.16)

which is similar to (2.15). Obviously f := τg−1 (h) ∈ ker g π∗ . On the other hand, ¯ and hence is unipotent, so H 1 (K, g U) = 1 by g U is isomorphic to U over K Lemma 2.25. As (2.16) implies that ker g π∗ is trivial, we conclude that f = 1, hence g = h.

2.2.4 Classification of K-Forms of Algebraic Groups We will consider two special cases: algebraic tori and semisimple groups. (The reader is referred to Knus et al. [1998] for a more detailed treatment.) Let T be an algebraic K-torus of dimension d with splitting field L, and let F = Gal(L/K). Then there is an L-isomorphism T ' Gdm , making T an L/Kform of the d-dimensional K-split torus Gdm . According to Theorem 2.15, the K-isomorphism classes of such tori correspond bijectively to the elements of H 1 (F, AutL (Gdm )). But Theorem 2.1 implies that AutL (Gdm ) = AutK (Gdm ) ' GLd (Z), and therefore the K-isomorphism classes in question are in one-to-one correspondence with the equivalence classes of d-dimensional integral representations of F.

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For example, if L/K is a quadratic extension, then any finitely generated Z[F]-module M without Z-torsion can be written in the form M = Z` ⊕ Z[F]m ⊕ I n , where I is the kernel of the augmentation map Z[F] → Z, with `, m, and n uniquely determined. Therefore any K-torus T that splits over L can be written in the form T = G`m × RL/K (Gm )m × RL/K (Gm )n (1)

for some uniquely determined nonnegative integers `, m, and n; in particular, (1) any K-anisotropic torus in this class is of the form T = RL/K (Gm )n . We now proceed to the semisimple case. First, one shows that for any semisimple K-group G, there exists a K-split group G0 such that G ' G0 π ˜ − ¯ To see this, one considers a universal K-defined ¯ over K. cover G → G (cf. ˜ ' G ˜ 0 , where G ˜ 0 is a K-split ¯ §2.1.13). Then there is a K-isomorphism ϕ: G simply connected group of the same type as G, and to prove the existence of ˜ 0 . But the G0 it suffices to show that ϕ(ker π) is a K-defined subgroup of G ˜ 0 is contained in a maximal K-split torus, hence the Galois group center Z of G acts trivially on the character group X(Z), so any subgroup of Z, in particular ϕ(ker π), is K-defined. Thus, any semisimple K-group G can be obtained from a suitable K-split group G0 by twisting using a cocycle from Z 1 (K, AutK¯ (G0 )). ˜ 0 ) of automorphisms that Since AutK¯ (G0 ) is precisely the subgroup of AutK¯ (G ˜ 0 → G0 leave ker π invariant (Theorem 2.14), the universal K-defined cover G 1 can be twisted using any element of Z (K, AutK¯ (G0 )). This yields Proposition 2.27 Let G be a semisimple K-group. Then there exists a universal ˜ → G. K-defined cover π : G We will see later that the fact that every semisimple K-group possesses a universal K-defined cover plays an important role in the arithmetic theory of algebraic groups. Unfortunately, there is no canonical analog of the universal cover for arbitrary reductive groups; in certain situations, however, special covers (cf. Sansuc [1981]) provide a suitable replacement. A K-isogeny f : H → G of reductive K-groups is called a special cover if H is a direct product of a simply connected semisimple K-group D and a K-quasisplit torus S. Although straightforward examples show that, in general, a reductive group need not have a special covering, we do have Proposition 2.28 For an arbitrary reductive K-group G, there exist an integer m > 0 and a quasisplit K-torus T such that Gm × T possesses a special cover.

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PROOF: By Theorem 2.8(3), the group G is an almost direct product (over K) of the semisimple group D1 = [G, G] and the maximal central torus S1 . Using Proposition 2.4, we can find m > 0 and two quasisplit K-tori S and T such that there exists a K-defined isogeny ϕ : S → S1m × T. Let π : D → D1 be a universal K-defined cover. Set H = Dm × S. Then the composite map π m ×ϕ

m m m H = Dm × S −→ Dm 1 × S1 × T = (D1 × S1 ) × T −→ G × T

is the desired special cover. Proposition 2.27 enables one to reduce the classification of semisimple Kgroups to that of semisimple simply connected K-groups. It follows from Theorem 2.12 that a semisimple simply connected K-group is a direct product of simply connected almost K-simple groups (i.e., groups containing no proper nontrivial connected normal K-subgroups), and, moreover, any simply connected almost K-simple group can be written in the form RL/K (G), where L/K is a finite (separable) field extension and G is an absolutely almost simple L-group (cf. Tits [1966]). So, it suffices to consider K-forms of simply connected absolutely almost simple groups. Let G be an absolutely almost simple simply connected K-split group of a ¯ with the group given type. We will identify the corresponding adjoint group G of inner automorphisms IntG. Then the full automorphism group AutK¯ G is the ¯ ¯ by Sym(R), the group of symmetries of the Dynkin semidirect product of G K ¯ diagram of the root system R of G (cf. §2.1.13); moreover, G = Gal(K/K) acts on Sym(R) trivially. Thus, we have a split exact sequence of K-groups ϕ ¯ ¯ → Aut ¯ G −→ 1→G ←− Sym(R) → 1, K K ψ

(2.17)

which yields the exact sequence of cohomology α 1 ¯ → H 1 (K, Aut ¯ G) −→ H 1 (K, G) ←− H (K, Sym(R)). K β Since Sym(R) = Sym(R)K , a cocycle a on G with values in Sym(R) is just a continuous homomorphism G → Sym(R), and H 1 (K, Sym(R)) is the set of conjugacy classes of such homomorphisms. It is known that ψ(Sym(R)) ⊂ AutK¯ G consists of automorphisms that leave invariant the fixed maximal K-split torus T ⊂ G and the Borel K-subgroup B ⊂ G containing it. Therefore, for any a ∈ H 1 (K, Sym(R)), the K-form corresponding to β(a) ∈ H 1 (K, AutK¯ G) possesses a K-defined Borel subgroup, in other words, is quasisplit over K. Thus, for any a ∈ H 1 (K, Sym(R)), the fiber α −1 (a) contains a quasisplit K-group a G, which, moreover, is unique up to K-isomorphism. Groups that correspond

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Algebraic Groups

to the elements of α −1 (a) are said to be of the same inner type. This term is justified by the fact that α −1 (a) coincides with the image of the map ¯ → H 1 (K, Aut(a G)) ' H 1 (K, Aut ¯ G), H 1 (K, a G) K where the last isomorphism is a “translation” by β(a) (cf. Lemma 1.18). Thus, groups of the same inner type are obtained from the corresponding quasisplit ¯ i.e., using inner group by twisting with the help of an element of H 1 (K, a G), automorphisms. The fiber of α over the trivial cocycle in H 1 (K, Sym(R)) consists of what we call inner forms of G, this definition being consistent with the definition of inner forms given in §2.1.14. Inner forms, and only these, are ¯ In the next section, obtained from G by twisting using elements of H 1 (K, G). we will show how this classification in terms of Galois cohomology can be converted into an explicit classification of groups of classical type.

2.3 Classical Groups The goal of this section is to introduce algebraic groups whose groups of rational points are classical groups over division algebras; this class includes special linear, symplectic, special orthogonal, and special unitary groups. These groups, with few exceptions, are absolutely almost simple algebraic groups of one of the classical types An , Bn , Cn , and Dn . Remarkably, the converse is also true: every group of a classical type, with the exception of the triality forms 3D4 and 6D4 , is isogenous to one of the classical groups. Prior to the first edition of the current book, the details of this result, which is due to A. Weil (1961a), were available only in Kneser’s lecture notes (1969), so we gave (virtually) complete proofs. Subsequently, in Knus et al. (1998), these issues have been treated in a greater generality, including in many cases also over fields of characteristic two. Nevertheless, since our constructions, notations, and so on are routinely used in the rest of the book, we left this material of the first edition unchanged. The arguments here are based on the classification of K-forms in terms of Galois cohomology and the notion of twisting.

2.3.1 The Special Linear Group Let D be a finite-dimensional central division algebra of index d over K, and let n ≥ 1. Then A := Mn (D) is a central simple K-algebra, and in particular one can define the reduced norm map NrdA/K : A∗ → K ∗

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91

(cf. §1.4.1). Our goal is to show that the special linear group SLn (D) := {x ∈ A∗ : NrdA/K (x) = 1} can be realized as the group of K-rational points of a certain algebraic group G, which we will denote by SLn (D). Let ρ : D → Md 2 (K) be the regular representation of D.1 Then the image ρ(D), being a subspace of Md 2 (K), can be described by a system of linear equations fk (xij ) = 0,

i, j = 1, . . . , d 2 ; k = 1, . . . , `,

for the entries xij of a matrix x = (xij ), with coefficients in K. Identifying Mnd 2 (K) with Mn (Md 2 (K)), we let A˜ denote the subset of Mnd 2 (K) consisting αβ of the elements x = (xij ) for i, j = 1, . . . d 2 , and α, β = 1, . . . , n such that αβ

fk (xij ) = 0

for all α, β = 1, . . . , n and k = 1, . . . , `.

(2.18)

˜ Furthermore, it is well known (cf. §1.4.1) that Clearly, ρ identifies A with A. the reduced norm of x ∈ A can be expressed as a polynomial with coefficients in K in terms of the coordinates of x with respect to an arbitrary fixed basis of αβ A/K. It follows that there exists a polynomial g(xij ) over K such that NrdA/K ((xαβ )) = g(ρ(xαβ )) where xαβ ∈ D for α, β = 1, . . . , n. αβ

Then obviously the set of matrices x = (xij ) ∈ Mnd 2 (K) satisfying (2.18) and the equation αβ

g(xij ) = 1

(2.19)

is naturally identified with SLn (D). Now take G to be the set of solutions of (2.18) and (2.19) in Mnd 2 (). Then G is an algebraic K-group whose set of K-points is SLn (D). In addition, using the isomorphisms D ⊗K  ' Md () and A ⊗K  ' Mnd (), it is easy to construct an -isomorphism G ' SLnd , which in particular implies that G is an absolutely almost simple simply connected K-group of type And−1 . Proposition 2.29 For G = SLn (D), we have rankK G = n − 1. In particular, M = SL1 (D) is K-anisotropic. 1 We recall that ρ sends x ∈ D to the matrix (with respect to a fixed K-basis of D) of the K-linear

transformation y 7→ xy of D considered as a K-vector space.

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92

αβ

PROOF: Let T denote the set of matrices x = (xij ) ∈ G such that αβ

xij = 0 αβ xαβ = (xij )i, j = 1,...,d 2 is a scalar matrix

for α 6= β, for α = β.

It is easy to see that T is a K-split torus in G of dimension n − 1. Furthermore, αβ αβ its centralizer ZG (T) consists of all matrices x = (xij ) ∈ G such that xij = 0 for α 6= β. It follows that M n = M × · · · × M, where M = SL1 (D), is naturally embedded in ZG (T), and the restriction to M n of the canonical morphism ZG (T) → ZG (T)/T is an isogeny. Therefore it suffices to establish that M is (1) K-anisotropic. But any maximal K-torus in M is the form RL/K (Gm ), where L ⊂ D is a maximal subfield, hence K-anisotropic (cf. §2.1.7). We will now compute the cohomology of G = SLn (D). For this, we first consider the algebraic K-group H = GLn (D), which is defined to be the subgroup of GLnd 2 () consisting of those matrices which satisfy (2.18). Then H ¯ is K-isomorphic to GLnd and has the group GLn (D) of invertible elements of A = Mn (D) as the group of K-rational points. Using twisting, it is not difficult to establish the following generalization of Lemma 2.17 (cf. Kneser [1969], §1.7). Lemma 2.30 H 1 (K, GLn (D)) = 1 for any n ≥ 1. The cohomology of G is computed using the exact sequence ϕ

1 → G −→ H −→Gm → 1,

(2.20)

where ϕ is induced by NrdA/K . Corresponding to (2.20), we have the exact sequence of cohomology NrdA/K

GLn (D) −→ K ∗ −→ H 1 (K, G) −→ H 1 (K, H) = 1. Analyzing this sequence, one obtains Lemma 2.31 H 1 (K, SLn (D)) can be identified with K ∗ /NrdA/K (GLn (D)) = K ∗ /NrdD/K D∗ .

2.3.2 The Symplectic and Orthogonal Groups Let f (x, y) be a nondegenerate alternating (respectively, symmetric) bilinear form on the vector space V = K n over a field K of characteristic 6= 2. (For the definitions and basic properties of bilinear and sesquilinear forms we refer the reader to one of the standard sources such as Bourbaki [1998],

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93

Dieudonné [1971], Knus et al. [1998], or Scharlau [1985].) We recall, in particular, that if f is a nondegenerate alternating form, then n is necessarily even, say n = 2m. The group of automorphisms of f consisting of those linear transformations σ : V → V such that f (σ (x), σ (y)) = f (x, y) for all x, y ∈ V is called the symplectic group and denoted Sp2m ( f ) in the case where f is alternating, and the orthogonal group, denoted On ( f ), in the case where f is symmetric. (It is well known that a symmetric bilinear form f is uniquely determined by the corresponding quadratic form q(x) = f (x, x), and instead of On ( f ), we will often write On (q).) The determinant of any transformation in Sp2m ( f ) is always 1, while transformations in On ( f ) can have determinant +1 or −1, implying that the special orthogonal group SOn ( f ) := {σ ∈ On ( f ) : det σ = 1} is a subgroup of On ( f ) of index two. Let e1 , . . . , en be a basis of V , and let F = ( f (ei , ej )) be the matrix of f in this basis (the Gram matrix). Then, representing linear transformations by the corresponding matrices in the chosen basis, we get the following matrix descriptions Sp2m (F) = {g ∈ GL2m (K) : tgFg = F} where tF = − F, On (F) = {g ∈ GLn (K) : tgFg = F} where tF = F,

(2.21)

SOn ( F) = {g ∈ On (F) : det g = 1}, where the superscript t denotes the matrix transpose. Now let Sp2m (F), On (F), and SOn (F) denote the sets of matrices g ∈ GLn () satisfying the respective conditions in (2.21). Then each of these sets is an algebraic K-group, whose group of K-points coincides with the corresponding group Sp2m (F), On (F) or SOn (F). (Sometimes, for convenience of notation, we will write On ( f ) instead of On (F), etc.) For a different basis e01 , . . . , e0n of V , the corresponding matrix F 0 = ( f (e0i , e0j )) is equivalent to F, i.e., is related to F by the equation F 0 = xtFx, where x is the transition matrix from e1 , . . . , en to e01 , . . . , e0n , and then Sp2m (F 0 ) = x−1 Sp2m (F)x. On the other hand, it is well known (cf. Bourbaki [1998]) that

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94

any nonsingular skew-symmetric matrix F ∈ M2m (K) is equivalent over K to the standard skew-symmetric matrix   0 Im J= , (2.22) −Im 0 where Im is the m × m-identity matrix, so there is a K-isomorphism Sp2m (F) ' Sp2m (J ). Take T to be {t = diag(α1 , . . . , αm , β1 , . . . , βm ) ∈ GL2m () : αi βi = 1, i = 1, . . . , m}. It is easy to see that T is a K-split torus in G = Sp2m (J ); moreover, direct computation shows that ZG (T) = T. Thus T is a maximal K-torus of G, and therefore G splits over K. Analyzing the root system R = R(T, G) as described in Bourbaki [2002], we see that R is an irreducible root system of type Cm . Moreover, using the criterion for the group to be simply connected (cf. Theorem 2.12(3)), it can be shown that G is simply connected. Thus we obtain Proposition 2.32 Let G = Sp2m (F) with m ≥ 1, where F is a nonsingular skew-symmetric matrix. Then G is an absolutely almost simple simply connected K-split group of type Cm . Similarly, any nonsingular symmetric matrix is equivalent over K¯ to one of the matrices   0 Im Q1 = if n = 2m Im 0 or  Q2

Im

0

  = Im 0

···

0 0

 0 ..  .  if n = 2m + 1. 0

(2.23)

1

Then T = {diag(α1 , . . . , αm , β1 , . . . , βm ) : αi βi = 1, i = 1, . . . , m} (resp., T = {diag(α1 , . . . , αm , β1 , . . . , βm , 1) : αi βi = 1, i = 1, . . . , m}) is a maximal torus in G = SOn (Q1 ) (resp., G = SOn (Q2 )) and the corresponding root system R(T, G) is of type Dm (m ≥ 2) in the first case (under the convention that D2 = A1 + A1 and D3 = A3 ) and type Bm (m ≥ 1) in the second (cf. Bourbaki [2002]); we note that G = SO2 (Q1 ) is a 1-dimensional torus. Thus, G = SOn (F) for n ≥ 3 is a semisimple group of type B n−1 for n odd, and of type D n2 for n even. (Actu2 ally, G is absolutely almost simple except for the case n = 4, where the root system is D2 = A1 + A1 .) The group G is not simply connected; its universal

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95

˜ = Spinn (F), constructed using Clifford algebras K-cover is the spinor group G (cf. Bourbaki [1998], Dieudonné [1971], Knus et al. [1998]). The kernel 8 of ˜ → G has order 2 and in fact is the subgroup {±1} the universal K-cover π : G of the corresponding Clifford group. Using this, it can be shown that for ` ≤ n, we have the following commutative diagram of universal covers: Spinn ↑ Spin`

−→ −→

SOn ↑ SO`

in which the vertical arrows are embeddings. Proposition 2.33 Let G = SOn (F) for n ≥ 3, where F is a nonsingular symmetric matrix. Then G is a nonsimply connected semisimple K-group of type B n−1 if n is odd, and of type D n2 if n is even. Moreover, rankK G equals the Witt 2 index of the corresponding quadratic form q. In particular, G is K-anisotropic if and only if q is anisotropic. We only need to prove the assertion about the K-rank of G. For this, recall that the Witt index of f is the dimension of a maximal totally isotropic subspace W ⊂ V = K n (which means that f (x, y) = 0 for all x, y ∈ W , or equivalently that q(x) = 0 for all x ∈ W where q(x) = f (x, x) is the associated quadratic form). Set ` = dim W . Then q is equivalent over K to a form x1 x`+1 + · · · + x` x2` + q0 (x2`+1 , . . . , xn ), where q0 is K-anisotropic. Therefore, without loss of generality, we may assume that q itself is such a form. Let T = {t = diag(α1 , . . . , α` , β1 , . . . , β` , 1, . . . , 1) : αi βi = 1 for 1 ≤ i ≤ `} be a split `-dimensional torus in GLn . It is easy to see that T ⊂ SOn (q), and direct computation shows that ZG (T)◦ ' T × SOn−2` (q0 ). So, it suffices to show that H = SOn−2l (q0 ) is K-anisotropic. But if S ⊂ H is a nontrivial Ksplit torus, then there exists a nonzero eigenvector v ∈ K n−2` for S with a nontrivial character χ : S → K. Then for any s ∈ S we have q0 (v) = q0 (sv) = q0 (χ(s)v) = χ(s)2 q0 (v). It follows that q0 (v) = 0, contradicting the assumption that q0 is K-anisotropic. This completes the proof of Proposition 2.33.

2.3.3 Unitary Groups We begin with a few remarks concerning algebras with involutions. Let A be a finite-dimensional (associative) algebra over a field K with char K 6= 2, and

96

Algebraic Groups

let L = Z(A) be the center of A. By an involution of A, we mean an arbitrary K-linear map τ : A → A of order 2 such that τ (ab) = τ (b)τ (a) for all a, b ∈ A (thus, τ is an anti-automorphism of A). Furthermore, τ is said to be of the first kind if its restriction to the center is trivial, and of the second kind otherwise. An algebra A with an involution τ will often be denoted by (A, τ ). Here are some examples of algebras with involution: (1) A = Mn (K), τ (x) = tx, the matrix transpose; (2) A = Mn (K), τ (x) = J −1txJ , where J is given by (2.22); (3) A = A1 ⊕ A2 , Ai = Mn (K), τ (x, y) = (ty, tx). Our objective is to show that any involution on a simple algebra, after extending scalars to an algebraically closed field, can be identified with one of the involutions just listed. So, let (A, τ ) be an arbitrary simple K-algebra with involution. Then its center L is a field. Throughout the remainder of the section, we will assume that K coincides with the subfield Lτ of τ -fixed elements. Let σ be another involution of A such that τ |L = σ |L . Then ϕ := σ τ −1 is an automorphism of A that acts trivially on the center, so by the Skolem–Noether Theorem, we have ϕ = Int g for a suitable g ∈ A∗ . Then σ (x) = gτ (x)g−1 for x ∈ A and the condition σ 2 = id yields that gτ (g)−1 ∈ L∗ . If τ is an involution of the first kind, i.e., L = K, then we immediately obtain that τ (g) = ± g. On the other hand, if τ is of the second kind, then since NL/K (gτ (g)−1 ) = gτ (g)−1 τ (gτ (g)−1 ) = 1, we can use Hilbert’s Theorem 90 to find a ∈ L satisfying gτ (g)−1 = aτ (a)−1 . Then replacing g by ga−1 , we may assume that τ (g) = g. Thus, we obtain Lemma 2.34 Let τ and σ be two involutions of a simple algebra A having the same restrictions to the center of A. Then there exists g ∈ A∗ such that σ (x) = gτ (x)g−1

for all x ∈ A;

(2.24)

moreover, g can be chosen to satisfy τ (g) = ± g if τ is of the first kind, and τ (g) = g if τ is of the second kind. Conversely, for any involution τ and any g ∈ A∗ as above, the map σ given by (2.24) is an involution of A. Now let (A, τ ) be a simple K-algebra with involution. If τ is of the first kind, then the center of A coincides with K, hence there exists an isomorphism ϕ

¯ A ⊗K K¯ ' Mn (K).

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97

¯ We will now show that ϕ can be chosen so that the K-linear extension of τ (which we will also denote by τ ) corresponds to one of the involutions (1) or (2). Let ν = ϕτ ϕ −1 , and let σ be the involution given by the matrix transpose. ¯ we see that there Applying Lemma 2.34 to the involutions σ and ν of Mn (K), ¯ such that tF = ± F and exists F ∈ GLn (K) ¯ ν(x) = F −1txF for all x ∈ Mn (K). ¯ such that F = tBB if F is symFurthermore, there exists a matrix B in GLn (K) metric and F = tBJB if F is skew-symmetric (where J is the same as in (2.22)). Then a direct computation shows that for the isomorphism ψ : A ⊗K K¯ ' ¯ given by ψ = Int B ◦ ϕ, the involution ψ ◦ τ ◦ ψ −1 coincides with (1) if Mn (K) F is symmetric, and with (2) if it is skew-symmetric, as required. If τ is of the second kind, then [L : K] = 2, so ϕ

¯ ⊕ Mn (K). ¯ A ⊗K K¯ = (A ⊗K L) ⊗L K¯ ' Mn (K) Thus, here the algebra A ⊗K K¯ is semisimple, rather than simple; nevertheless the proof of Lemma 2.34 goes through without any changes. Arguing as above, we can construct an isomorphism ¯ ⊕ Mn (K) ¯ ψ : A ⊗K K¯ ' Mn (K) that identifies τ with the involution described in (3). Now let D be a finite-dimensional division algebra with an involution τ , let L be the center of D, and let K = Lτ be the subfield of τ -fixed elements. Furthermore, let f (x, y) be a nondegenerate Hermitian or skew-Hermitian sesqui-linear form on the m-dimensional vector space V = Dm . The group of D-linear automorphisms of V preserving f is called the unitary group, denoted by Um (D, f ); its subgroup consisting of automorphisms having reduced norm 1 is called the special unitary group SUm (D, f ). Let e1 , . . . , em be a basis of V and let F = ( f (ei , ej )) be the matrix of f . For g = (gαβ ) ∈ Mm (D), we let g = (τ (gβα )),

∗

noting that the correspondence g 7→ ∗g is an involution of A = Mm (D) having the same type as τ . Then ∗F = ± F, and writing the matrices of linear transformations in the basis e1 , . . . , em , we obtain the following descriptions: Um (D, f ) = {g ∈ GLm (D) : ∗gFg = F}, and SUm (D, f ) = {g ∈ Um (D, f ) : NrdMm (D)/L (g) = 1}.

Algebraic Groups

98

In order to realize Um (D, f ) and SUm (D, f ) as groups of K-rational points of certain algebraic groups, we consider, as in §2.3.1, the regular representation ρ : D → M`n2 (K) of D over K (here n is the index of D and ` = [L : K]), and the corresponding equations of the form (2.18) defining D as a subspace of M`n2 (K). Furthermore, let τ˜ : M`n2 (K) → M`n2 (K) be an invertible linear map extending the involution ρτρ −1 on ρ(D), and let 8 = (ρ( f αβ )) be the matrix in M`mn2 (K) corresponding to F = ( f αβ ). Then the image of Um (D, f ) in Mlmn2 (K) under the homomorphism induced by ρ consists of matrices αβ

g = (gij )

for α, β = 1, . . . , m and i, j = 1, . . . , ln2

satisfying (2.18) and αβ

αβ

(τ˜ (gij ))8(gij ) = 8.

(2.25)

Similarly, the image of SU m (D, f ) is given by (2.18), (2.25), and an equation of the form (2.19). The solutions of these equations over K¯ yield the algebraic groups Um (D, f ) and SUm (D, f ), respectively. To understand the structure of these groups, we set σ (x) = F −1∗xF. By Lemma 2.34, σ is an involution of A = Mm (D); moreover Um (D, f ) = {g ∈ GLm (D) : σ (g)g = Im }, SU m (D, f ) = {g ∈ Um (D, f ) : NrdA/L (g) = 1}. We showed earlier that if τ is an involution of the first kind, then one can ¯ so that the natural extension of σ choose an isomorphism A ⊗K K¯ ' Mmn (K) corresponds to one of the involutions ν in (1) or (2). Then the groups ¯ ∗ : σ (g)g = Im }, Um (D, f ) = {g ∈ (A ⊗K K) SUm (D, f ) = {g ∈ Um (D, f ) : NrdA⊗K K/ ¯ K¯ (g) = 1} are identified with ¯ : ν(g)g = Imn } and G = {g ∈ GLmn (K) ¯ : ν(g)g = Imn }, H = {g ∈ SLmn (K) respectively. However, these groups are precisely the orthogonal and the special orthogonal groups if ν is as in (1) (an involution of the orthogonal type), and both coincide with the symplectic group if ν is as in (2) (an involution of the symplectic type). We note that there is an intrinsic description of orthogonal and symplectic involutions: τ is orthogonal (resp., symplectic) if dimK Dτ = n(n+1) 2 (resp., dimK Dτ = n(n−1) 2 ), and σ (or ν) has the same type as τ if F is Hermitian, and the opposite type if F is skew-Hermitian.

2.3 Classical Groups

99

Now let τ be an involution of the second kind. Then we can choose an iso¯ ⊕ Mmn (K) ¯ so that the natural extension of σ morphism A ⊗K K¯ ' Mmn (K) corresponds to the involution ν as in (3). Then Um (D, f ) and SUm (D, f ) get identified with the groups ¯ × GLmn (K) ¯ : (X , Y )(tY , tX ) = (Imn , Imn )} G = {(X , Y ) ∈ GLmn (K) and ¯ × SLmn (K) ¯ : (X , Y )(tY , tX ) = (Imn , Imn )}. H = {(X , Y ) ∈ SLmn (K) Now it is evident that ¯ G = {(X , tX −1 ) : X ∈ GLmn (K)} and ¯ H = {(X , tX −1 ) : X ∈ SLmn (K)}, ¯ so we conclude that Um (D, f ) and SUm (D, f ) are K-isomorphic to GLmn and SLmn , respectively. Proposition 2.35 Let G = SUm (D, f ), where D is a division algebra over K of index n with an involution τ , and let f be a nondegenerate Hermitian or ¯ skew-Hermitian form on Dm . Then we have the following K-isomorphisms: (1) G ' Spmn , and thus G is a simple simply connected group of type C mn , 2 if τ is an involution of the first kind of the orthogonal type and f is skewHermitian, or if τ is an involution of the first kind of symplectic type and f is Hermitian; (2) G ' SOmn , and thus G is a semisimple non-simply connected group of type B mn−1 or D mn (note that type B occurs only when n = 1, i.e., D = K) if τ 2 2 is an involution of the first kind of the orthogonal type and f is Hermitian, or if τ is an involution of the first kind of the symplectic type and f is skew-Hermitian; (3) G ' SLmn , and thus G is a simply connected simple group of type Amn−1 if τ is an involution of the second kind. In all these cases, rankK G coincides with the Witt index of f , i.e., with the dimension of a maximal totally isotropic subspace of Dm . We note that the groups SOn (F) and Sp2m (F) considered in §2.3.2 can be treated as unitary groups with respect to the identity involution on D = K. The Galois cohomology of the unitary groups can be computed in exactly the same way as in the case of the orthogonal groups (see Proposition 2.18). More precisely, by using Lemma 2.30 instead of Lemma 2.17, we obtain the following result.

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Algebraic Groups

Proposition 2.36 The elements of H 1 (K, Um (D, f )) are in one-to-one correspondence with the equivalence classes of m-dimensional nondegenerate forms over D having the same type as f . Moreover, the proper equivalence classes (i.e., equivalence classes relative to SLm (D)) of those forms having the same discriminant as f are in one-to-one correspondence with the elements of ker(H 1 (K, SUm (D, f )) → H 1 (K, SLm (D)). Rather than giving a proof of Proposition 2.36 in the spirit of Propositions 2.18–2.19 and 2.24, we would like to point out that all of these assertions stem from the following general principle, based on (1.11) in §1.3.2: Let X is a Kdefined homogeneous space of an algebraic K-group G (i.e., there is a transitive K-defined action G × X → X ), let x ∈ XK , and let H = G(x) be the stabilizer of x (so that X can be identified with G/H); then the orbits of GK on XK are in one-to-one correspondence with the elements of ker(H 1 (K, H) → H 1 (K, G)).

2.3.4 Groups of Classical Types Our goal is to establish a converse of Proposition 2.35, i.e., to show that, up to isogeny, any absolutely almost simple K-group of one of the classical types (except for 3D4 and 6D4 ) is either SLm (D) or one of the unitary groups (which includes the symplectic and orthogonal groups). We first consider inner forms of type An−1 . We know that simply connected groups of this type are obtained from G = SLn by twisting using cocycles from ¯ where G ¯ = PSLn . But G ¯ is also the group of automorphisms of the H 1 (K, G), ¯ we ¯ and for any cocycle a = {aσ } in H 1 (K, G), full matrix algebra Mn over K, ¯ Gal( K/K) can consider the twisted algebra A = a Mn . Let B = A be the K-algebra of fixed points. Then ¯ B ⊗K K¯ ' Mn (K), hence B is a central simple K-algebra, and therefore B = Mm (D) for some central division K-algebra D of index d with md = n. As in the description of the special linear group, we consider the regular representation ρ : D → Md 2 (K) and the corresponding representation ψ : B → Mmd 2 (K). Then we have a chain of isomorphisms ϕ

ψ

¯ ' B ⊗K K¯ ' ψ(B) ⊗K K. ¯ Mn (K) Since ψ is a K-isomorphism, we have (ψϕ)−1 · σ(ψϕ) = ϕ −1 · σϕ = aσ .

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101

On the other hand, since by definition for b ∈ B we have NrdB/K (b) = det(ϕ −1 (b)), ¯ onto the group SLn (D) as the restriction of ψϕ isomorphically maps SLn (K) defined in §2.2.1. Thus, we obtain Proposition 2.37 Simply connected inner forms of type An−1 are precisely the groups of the form SLm (D), where D is a central division K-algebra of index d and n = md. Next, we will analyze outer forms of type An−1 . These forms are obtained from G = SLn by twisting using a cocycle a = {aσ } from H 1 (K, Aut G) that ¯ and consequently has nontrivial image does not lie in the image of H 1 (K, G) 1 in H (K, Sym(R)). But the group Sym(R) has order 2 for any root system R of type An−1 with n > 1. Moreover, the automorphism of R given by α 7→ −α for all α ∈ R, does not lie in the Weyl group, and therefore the automorphism ¯ and the automorphism of order 2 given by group Aut G is generated by G t −1 x 7→ x . One can realize Aut G as a group of automorphisms of an algebra A with ¯ ⊕ Mn (K) ¯ with the involution τ given involution. More precisely, let A = Mn (K) t t by τ (X , Y ) = ( Y , X ), and consider the embedding GLn → A defined by X 7→ (X , tX −1 ). We have seen that GLn is thereby identified with U = {Z ∈ A : Zτ (Z) = I}; let SU denote the image of SLn under this embedding. We claim that the group Aut G can be naturally identified with the group of all algebra automorphisms of A that commute with τ . Indeed, it follows from the Skolem–Noether Theorem that the group of all automorphisms of A contains the group of inner automorphisms as a subgroup of index 2, the nontrivial coset being represented by the automorphism (X , Y ) 7→ (Y , X ). This automorphism clearly commutes with τ , and on the other hand, it is easy to see that every inner automorphism that commutes with τ is induced by an element of SU. It follows that restricting the automorphisms of A to SU, we obtain all the automorphisms of SU ' G; moreover, every automorphism of A is uniquely determined by its restriction to SU. Now let a = {aσ } ∈ H 1 (K, AutK¯ G) be an arbitrary cocycle not lying in ¯ Consider a as a cocycle in H 1 (K, Aut ¯ A) and conthe image of H 1 (K, G). K struct the twisted algebra B = a A. Since the aσ commute with τ , the latter gives ¯ rise to an involution ν of B that commutes with the action of Gal(K/K). Set

Algebraic Groups

102 ¯

C = BGal(K/K) . Then the restriction of ν to C induces an involution θ of C, and there exists an isomorphism of algebras with involution ϕ

¯ θ). (A, τ ) ' (C ⊗ K, ¯ ⊕ Mn (K), ¯ we see Let us describe the structure of C. Since C ⊗ K¯ ' Mn (K) that C is either a direct sum of two central simple algebras over K or a central simple algebra over some quadratic extension L of K. We will now show that in our case, the latter holds. Indeed, ¯

¯ Gal(K/K) . Z(C) = a (K¯ ⊕ K) By assumption, the image of a in H 1 (K, Sym(R)) is nontrivial, providing ¯ thereby a nontrivial homomorphism of Gal(K/K) to Sym(R); the kernel of this homomorphism corresponds to a quadratic extension L/K. Then it is easy ¯ ¯ coincides with its to see that the action of the group Gal(K/K) on a (K¯ ⊕ K) ¯ action on L ⊗ K (via the second factor); hence Z(C) = L. Moreover, since τ acts on K¯ ⊕ K¯ by switching the components, the restriction of θ to L is nontrivial. Thus, C is a simple algebra over L with an involution θ of the second kind. Furthermore, it is well known (cf. Albert [1961]) that C = Mm (D) for some division algebra D over L of index d with md = n, equipped with an involution δ of the second kind having the same restriction to L as θ. Then, by Lemma 2.34, we have θ(x) = F ∗x F −1 for x ∈ Mn (D), where ∗(xij ) = (δ(xji )) and ∗F = F. Consider the Hermitian form f on the space V = Dm having matrix F with respect to the canonical base e1 , . . . , em . We claim that a G ' SUm (D, f ). Indeed consider the regular representation ρ : D → M2d 2 (K) over K and the corresponding representation ψ : C → M2md 2 (K). We have a chain of isomorphisms: ϕ

ψ

¯ θ) ' (ψ(C) ⊗ K, ¯ ψ ◦ θ ◦ ψ −1 ). (A, τ ) ' (C ⊗K K, Then (ψϕ)−1 · σ(ψϕ) = aσ since ψ is defined over K. Thus ψϕ and the above ¯ embedding G → A yield the K-isomorphism a G ' SUm (D, f ), as desired. Thus, we have obtained the following. Proposition 2.38 Simply connected outer forms of type 2An−1 over K are the groups of the form SUm (D, f ), where D is a central division algebra of index d = n/m over L equipped with an involution τ of the second kind such that the τ -fixed subfield Lτ coincides with K, and f is a nondegenerate Hermitian form on Dm .

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We will now proceed to describe the K-forms of type Cn . A simply connected split group of this type is ¯ = {X ∈ GL2n (K) ¯ : tXJX = J }, G = Sp2n (K) ¯ defined where J is given by (2.22). Consider the involution τ of A = M2n (K) −1t by τ (X ) = J X J . (This involution is of the first kind and symplectic type.) Any automorphism of G is inner (cf. Theorem 2.14), so any K-form of G ¯ Clearly, each is obtained by twisting using a cocycle a = {aσ } ∈ H 1 (K, G). aσ can be regarded as an automorphism of A that commutes with τ . Arguing as above, we conclude τ induces an involution ν of the twisted algebra ¯ B = a A that commutes with the action of Gal(K/K). It follows that the restric¯ Gal( K/K) tion of ν to C = B induces an involution θ of the latter, and we have an isomorphism of algebras with involution ¯ θ). (A, τ ) ' (C ⊗K K, Since C is a central simple algebra over K, we have C = Mm (D) for some central division algebra D of index d = 2n m over K which possesses an involution of the first kind (cf. Albert [1961]). The last assertion is a straightforward consequence of the following fact: a central simple algebra E over K has an involution of the first kind if and only if it is isomorphic to its opposite algebra E◦ , in other words the corresponding class in the Brauer group has order ≤ 2 (see Beelen and Gramlich [2002] for a simple proof). In our case, C has an involution of the first kind, so C, and hence D, represents an element of order ≤ 2 in the Brauer group. Thus, D has an involution of the first kind, which will be denoted by δ. We may assume δ to be either of orthogonal or of symplectic type, as we wish. (Indeed, if, say, δ is of orthogonal type, then δ 0 given by δ 0 (x) = c−1 δ(x)c, where c is an arbitrary invertible δ-skew-symmetric element, is of symplectic type; in addition, the map x 7→ xc gives a bijection between the δ-symmetric and δ 0 -skew-symmetric elements of D.) Then Lemma 2.34 yields that θ(x) = F −1∗X F, where ∗ (xij ) = (δ(xji )), and ∗F = − F if δ is of orthogonal type, and ∗F = F if δ is of symplectic type. Now, introducing, respectively, a skew-Hermitian or a Hermitian form f on V = Dm having matrix F, and arguing as above, we conclude that a G = SUm (D, f ).

Thus, we have proved Proposition 2.39 Simply connected K-forms of type Cn are precisely the groups SUm (D, f ), where D is a central division algebra of index d = 2n m over K

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endowed with involution τ of the first kind, and f is a nondegenerate sesquilinear form that is Hermitian if τ is of symplectic type, and skew-Hermitian if τ is of orthogonal type. Finally, we will consider the K-forms of types Bn and Dn , other than 3D4 and 6D4 . A simply connected K-split group of either type is the spin group ˜ = Spinm ( f ), where f is a quadratic form of maximal Witt index over K of G appropriate dimension (the matrix Q of f coincides with either Q1 or Q2 in (2.23), depending on whether m is even or odd). It is more convenient for us, however, to work with the corresponding orthogonal group G = SOm ( f ). Let ˜ → G denote the universal cover. π: G ˜ According to Theorem 2.14, First, let us relate the groups Aut G and Aut G. ˜ of those autothe group Aut G can be identified with the subgroup of Aut G morphisms that leave ker π invariant. If G has type Bn , then ker π = Z(G), and ˜ coincide. (This also follows from the fact that the therefore Aut G and Aut G Dynkin diagram of a root system of type Bn has no nontrivial symmetries, and ˜ are inner.) therefore all automorphisms of G Now suppose G is of type Dn , in which case m is even. Then there is an outer automorphism of G, induced by conjugation using a matrix from Om ( f )\SOm ( f ), so [Aut G : Int G] ≥ 2. On the other hand, for n 6= 4, the cor˜ : Int G] ˜ = 2. responding Dynkin diagram has exactly two symmetries, so [Aut G Thus, in this case we again obtain that ˜ Aut G = Aut G; moreover, all the automorphisms of G are obtained as conjugation by elements of Om ( f ). In the remaining case where n = 4, the group of symmetries of the Dynkin diagram is isomorphic to the symmetric group S3 . Then, considering ˜ it is easy to show that Aut G is isomorphic the action of S3 on the center of G, ˜ and all such subgroups are conjugate in to a subgroup of index 3 in Aut G, ˜ So, again all the automorphisms of G are induced by conjugation by Aut G. elements of Om ( f ). ˜ that a KIt follows from the preceding discussion of Aut G and Aut G 3 6 ˜ ˜ form H of G of type other than D4 and D4 is obtained from G by twisting ˜ is a universal cover using a cocycle a = {aσ } ∈ H 1 (K, Aut G). Then H = a G of a G. Thus, it suffices to describe the K-forms of G. Repeating verbatim the argument used to describe the K-forms of type Cn , we conclude that a G is SU` (D, f ), where D is a finite-dimensional central division algebra of index d = m/` over K with involution τ of the first kind, and f is a nondegenerate Hermitian (resp., skew-Hermitian) form on V = D` depending on whether τ is orthogonal (resp., symplectic). Now, for groups of type Bn , the dimension m

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is odd, but the index d must be a power of 2, as D has exponent ≤ 2 in the Brauer group Br(K). So, in this case we necessarily have d = 1, hence D = K and τ = id, which means that f is an ordinary quadratic form. Proposition 2.40 (1) Simply connected K-forms of type Bn are the spin groups of nondegenerate quadratic forms over K of dimension m = 2n + 1. (2) Simply connected K-forms of type Dn , other than 3D4 and 6D4 , are the universal covers of the special unitary groups SU` (D, f ), where D is a finite-dimensional central division algebra over K of index d = 2n ` with an involution τ of the first kind (hence d must be a power of 2), and f is a nondegenerate sesquilinear form that is Hermitian if τ is of the orthogonal type, and skew-Hermitian if τ is of the symplectic type. (For d = 1 and τ = id we obtain the spin groups Spin2n ( f ) of nondegenerate quadratic forms.) In the case where K is a local field or a number field, the above results can be made significantly more precise. It is well known (cf. Albert [1961, Theorem 10.22]) that over a local field, there is no division algebra of index d > 1 with an involution of the second kind. Therefore, in this case the simply connected K-groups of type 2An−1 are just the special unitary groups SUn (L, f ), where L is a quadratic extension of K and f is a nondegenerate Hermitian form on V = Ln relative to the nontrivial automorphism σ ∈ Gal(L/K). Picking an orthogonal basis e1 , . . . , en of V , we can write f in terms of the corresponding coordinates as follows: f (x1 , . . . , xn ; y1 , . . . , yn ) = a1 σ (x1 )y1 + · · · + an σ (xn )yn , where the coefficients ai satisfy σ (ai ) = ai , hence belong to K. In particular, q(x) = f (x, x) has the following coordinate expression: q(x1 , . . . , xn ) = a1 NL/K (x1 ) + · · · + an NL/K (xn ). Furthermore, it is known (cf. §§1.4–1.5) that the exponent of a central simple algebra over a local or number field in the corresponding Brauer group equals its index. So, division algebras with an involution of the first kind are all quaternion division algebras. A quaternion division algebra D over K has a standard basis 1, i, j, k, and the canonical involution τ is given by τ (a0 + a1 i + a2 j + a3 k) = a0 − a1 i − a2 j − a3 k (cf. Pierce [1982]). This involution is symplectic, and moreover, the set of symmetric elements coincides with the center K of D. So, a τ -Hermitian form f on V = Dn in terms of the coordinates associated with an orthogonal basis of V assumes the form

106

Algebraic Groups f (x1 , . . . , xn ; y1 , . . . , yn ) = τ (x1 )a1 y1 + · · · + τ (xn )an yn .

(2.26)

The coefficients ai must satisfy τ (ai ) = ai , hence belong to K. Then q(x) = f (x, x) can be written in the form q(x1 , . . . , xn ) = a1 NrdD/K (x1 ) + · · · + an NrdD/K (xn ). Thus, for any simply connected nonsplit K-group of type Cn (where K is either a local field or a number field), we have m = n in the notations of Proposition 2.39, so the group is the special unitary group SUn (D, f ) of a Hermitian form f as in (2.26). Finally, simply connected K-groups of type Dn that are not isomorphic to spin groups of quadratic forms are universal covers of the unitary groups SUn (D, f ), where D is a quaternion division algebra over K with the canonical involution τ and f is a nondegenerate skew-Hermitian form on Dn . For convenience of reference, we will now put together a list of simply connected absolutely almost simple K-groups of classical types for the case where K is either a local field or a number field. First of all, we have the groups G = SLm (D), where D is a central division algebra over K of index d, and these are inner forms of the type An with n = md − 1. All the remaining groups, with the exception of 3D4 and 6D4 , are obtained from the special unitary groups SUm (D, f ). To be more precise, any such simply connected K-group G is the universal cover of some special unitary group H = SUm (D, f ), where D is a division algebra with center L and of index d equipped with an involution τ such that Lτ = K, and f is a nondegenerate Hermitian or skew-Hermitian form (relative to τ ) on an m-dimensional space W over D. Here is the list of possibilities for D, τ and f , in which we have also included the value of a certain number m0 for each type that will come up later: (1) [L : K] = 2, hence τ is an involution of the second kind, and f is Hermitian; in this case G = H is of type 2An with n = md − 1 (m0 = 2). In the remaining cases L = K, i.e., τ is an involution of the first kind. Then either D = K or D is a central quaternion division algebra over K, and the list of classical groups continues as follows: (2) D = K and f is symmetric. Then H = SOm ( f ) and G = Spinm ( f ); moreover, these groups are of type B m−1 for m odd and of type D m2 for m even 2 (m0 = 4). (3) D = K and f is alternating. Then m is even and G = H is Spm ( f ), which is of type C m2 (m0 = 2). (4) D is a quaternion division algebra over K with the canonical involution τ , and f is Hermitian. Then G = H is of type Cm (m0 = 1).

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107

(5) D and τ the same as in (4), but f is skew-Hermitian. Then H is a nonsimply connected group of type Dm and H ' SO2m ( f¯ ) over K¯ (m0 = 3).

2.3.5 Witt’s Theorem We keep the notations G, H, f , W , . . . introduced earlier. The group H = ¯ := W ⊗K K¯ preserving the natural extension of f . This SUm (D, f ) acts on W realization of H induces a realization of G (the universal cover of H), which will be freely used without any additional comments; the integer m will be called the degree of G. In the sequel, we will need Witt’s Theorem, which describes the orbits of the action of Um (D, f ) on W (relative version) and the ¯ (absolute version). orbits of G on W Theorem 2.41 (WITT’S THEOREM, RELATIVE VERSION) Let a, b ∈ W be two nonzero vectors such that f (a, a) = f (b, b). Then there exists an element g in Um (D, f ) such that b = ga. We refer the reader to Bourbaki [1974, Chapter 9] for the proof. Note that the name “Witt’s Theorem” is usually reserved for the more general statement that any metric isomorphism σ : U1 → U2 between two subspaces U1 , U2 of W extends to an isometry of the entire space W , in other words, is induced by an element of Um (D, f ). With the exception of the situation described in item (3) of the above list, W always has an orthogonal basis with respect to f . In particular, there always exists a vector a ∈ W such that f (a, a) 6= 0 (anisotropic vector). Theorem 2.42 (WITT’S THEOREM, ABSOLUTE VERSION) Let m ≥ 2 and let ¯ such that f (b, b) = f (a, a), a ∈ W be an anisotropic vector. Then for any b ∈ W there exists g ∈ G satisfying b = ga. PROOF: It suffices to find h ∈ H satisfying b = ha. In case (2), the existence ¯ of such an h follows immediately from Theorem 2.41, applied to the K-vector ¯ space W , and the assumption that m ≥ 2. Now let us consider the cases (4) and (5). For this, we include a in an orthogonal basis e1 = a, e2 , . . . , em of W and henceforth will work with the coordinates relative to this basis. Let q(x) = f (x, x). If q(ei ) = di , then q(x1 , . . . , xm ) = τ (x1 )d1 x1 + · · · + τ (xm )dm xm . Next, we choose a skew-symmetric element c ∈ D∗ , and pass from τ to the involution σ defined by σ (d) = cτ (d)c−1 . Then σ is of orthogonal type, and therefore we can find an isomorphism

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108

¯ := D ⊗K K¯ ' M2 (K) ¯ D so that σ corresponds to the matrix transpose. It is easy to see that for d ∈ D, the condition τ (d) = ± d is equivalent to σ (cd) = ∓ cd. So, the elements cdi correspond to symmetric (in case (5)) or skew-symmetric (in case (4)) matrices ¯ If b = (b1 , . . . , bm ) with bi ∈ D, ¯ and Bi ∈ M2 (K) ¯ is the matrix Di ∈ GL2 (K). corresponding to bi , then t

B1 D1 B1 + · · · + tBm Dm Bm = D1 .

(2.27)

¯ ' M2 (K) ¯ induces an identification of As we have seen, the isomorphism D H with H˜ = SO2m (f˜ ) or H˜ = Sp2m (f˜ ) respectively, where the matrix of f˜ is diag(D1 , . . . , Dm )). Let ! (i) (i) b11 b12 Bi = . (i) (i) b21 b22 Then, it follows from (2.27) that the subspace spanned by the vectors u1 = (1, 0, . . . , 0) and u2 = (0, 1, 0, . . . , 0) in K¯ 2m is isometric (relative to f˜ ) to the subspace spanned by the vectors (1)

(1)

(m)

(m)

(1)

(1)

(m)

(m)

w1 = (b11 , b21 , . . . , b11 , b21 ) and w2 = (b12 , b22 , . . . , b12 , b22 ). Therefore, by Witt’s Theorem for subspaces, there exists h˜ ∈ H˜ such that ˜ i ) = wi for i = 1, 2. Then the element h ∈ H corresponding to h˜ satisfies h(u ha = b, as required. The argument in case (1), although similar, is of somewhat different nature. Again, let e1 = a, e2 , . . . , em be an orthogonal basis of W . Set q(x) = f (x, x) and let q(ei ) = di , so that q(x1 , . . . , xm ) = τ (x1 )d1 x1 + · · · + τ (xm )dm xm . Next, we choose an isomorphism ¯ ⊕ Md (K) ¯ D ⊗K K¯ ' Md (K) so that the natural extension of τ (which we will also denote by τ ) corresponds to the involution (X , Y ) → (t Y , t X ) (cf. §2.3.3). Let di = (Ci , t Ci ) and (1) (2) b = (b1 , . . . , bm ) with bi = (Bi , Bi ). Then the condition q(b) = q(a) amounts to either of the two equivalent matrix equations: tB(1) C B(2) + · · · + tB(1) C B(2) = C , m m m 1 1 1 1 tB(2) tC B(1) + · · · + tB(2) tC B(1) = tC . m 1 1 m m 1 1

(2.28)

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109

Furthermore, H corresponds to ¯ H˜ = {(X , tC −1 (tX −1 ) tC) : X ∈ SLn (K)}, ¯ is naturally identified where n = md, C = diag(C1 , . . . , Cm ), and Mm (Md (K)) ¯ In these notations, what we need to show is that if B(1) , B(2) satisfy with Mn (K). i i ¯ such that (2.28), then there exists X ∈ SLn (K)     Ed (1) B  0   1    ..  X . =  ..   .  (1) Bm 0    (2.29)  Ed (2) B  0   1    ..  . (tC −1 (tX −1 )tC)  .  =   ..   .  (2) Bm 0 The second equation in (2.29) can be rewritten in the form    tC  1 tC B(2) 1 1  0     .. t  = X . . .    ..  tC B(2) m m 0

(2.30)

The first equation in (2.29) is satisfied precisely by matrices of the form  (1)  B1 X12 . . . X1m   X =  ......................  (1) Bm Xm2 . . . Xmm with arbitrary Xij , 2 ≤ i ≤ m, 1 ≤ j ≤ m. In view of (2.28), such a matrix satisfies (2.30) if and only if (2)

X1i t C1 B1 + · · · + Xmi t Cm B(2) m = 0,

i = 2, . . . , m.

(2.31)

Clearly, (2.31) reduces to m linear equations on each column of X beginning with the (m+1)-st, so it is easy to see that there is a solution to (2.31) satisfying   X12 . . . X1m rank  . . . . . . . . . . . . . . . . .  = n − m. Xm2 . . . Xmm Using the fact that C1 is nonsingular, we conclude X is also nonsingular. Moreover, since the equations (2.31) are homogeneous and m ≥ 2, we can actually ¯ find X ∈ SLn (K).

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Algebraic Groups

Remark Since Um (D, f ) = SUm (D, f ) in case (4) for all m ≥ 1, Theorem 2.42 holds in this case also for m = 1. Thus, in all the cases considered, the “sphere” ¯ : f (x) = f (a)} X ( f , a) := {x ∈ W is a homogeneous space of G, and therefore can be identified with G/G(a) where G(a) is the stabilizer of a. In the sequel, we will need some information about the stabilizers G(a) and G(a, b) of vectors a ∈ W and of pairs of vectors a, b ∈ W . Proposition 2.43 If m ≥ m0 , then for any anisotropic vector a ∈ W , the stabilizer G(a) belongs to the same item in the above description of classical groups as G itself and is a simply connected semisimple group (possibly, G(a) = {e}). Consequently, if m ≥ m0 + 1, then the same conclusion holds for the stabilizer G(a, b) of an arbitrary pair of vectors a, b ∈ W that span a nondegenerate 2-dimensional subspace. PROOF: It is easy to see that for any anisotropic a ∈ W , the stabilizer H(a) is SUm−1 (D, g), where g is the restriction of f to the orthogonal complement of a. On the other hand, G(a) is the preimage of H(a) under the universal cover π : G → H. It follows from the above analysis of classical groups that for m ≥ m0 , the group H(a) is semisimple. Thus it remains to show that the restriction π|G(a) : G(a) → H(a) is a universal cover. This is obvious for the classical groups described in items (1) and (4) of the list given in the end of §2.3.4 as then H and H(a) are simply connected since we are excluding case (3). In the remaining cases (2) and (5), the required fact follows from the compatibility of the universal covers of the special orthogonal groups of a quadratic space and of its nondegenerate subspace (cf. §2.3.2). The second assertion of the proposition concerning the stabilizers of pairs of vectors immediately follows from the first.

2.4 Some Results from Algebraic Geometry Most of the varieties that we will work with are either affine or projective, in other words, isomorphic to a Zariski-closed subset of n-dimensional affine space An or n-dimensional projective space Pn , respectively. We assume that the reader is familiar with the standard notions of algebraic geometry, including regular and rational functions, regular and rational maps of algebraic varieties, the concept of dimension, etc. (cf. Hartshorne [1977],

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111

Shafarevich [2013], and also Borel [1991, Chapter AG]). Some more specialized material is presented below.

2.4.1 The Field of Definition of an Algebraic Variety See Borel (1991, Chapter AG, §§11–14). Let K be a subfield of the universal domain . A closed subvariety X ⊂ An is said to be defined over K if the ideal a ⊂ [x1 , . . . , xn ] of polynomials that vanish on X is generated by a ∩ K[x1 , . . . , xn ], where the xi are the standard coordinate functions on An . A regular (respectively, rational) map f : X → Y between two K-defined subvarieties X ⊂ An and Y ⊂ Am is defined over K if there exist polynomials fi ∈ K[x1 , . . . , xn ] (respectively, rational functions fi ∈ K(x1 , . . . , xn )) for i = 1, . . . , m such that f (x) = ( f1 (x), . . . , fm (x)) for all x ∈ X . For a perfect field K, one can use the following Galois criterion: a closed subvariety X ⊂ An is defined over K if and only if X is defined over K¯ and X = σX ¯ ¯ 1 , . . . , xn ]) for all σ ∈ Gal(K/K), where σX is defined by the ideal σ (a ∩ K[x ¯ 1 , . . . , xn ]. A similar criterion is available also for arbitrary varieties and of K[x for regular (rational) maps.

2.4.2 Dominant Morphisms A morphism ϕ : X → Y is said to be dominant if ϕ(X ) is Zariski-dense in Y . For such morphisms, we have the following: Theorem 2.44 (DIMENSION OF THE FIBERS) Let ϕ : X → Y be a dominant morphism of irreducible algebraic varieties, and let r = dim X − dim Y . Then (1) for any point y ∈ ϕ(X ) we have dim ϕ −1 (y) ≥ r; (2) {y ∈ Y : dim ϕ −1 (y) = r} is a nonempty Zariski-open set. PROOF: See Shafarevich (2013, Chapter 1, §6).

2.4.3 Tangent Spaces: Simple and Singular Points See Shafarevich (2013, Chapter 2). These concepts are of a local nature, i.e., in treating them, one can replace a given variety X by an open neighborhood of a given point. In particular, using affine open neighborhoods, we can always assume X to be affine. So, let X ⊂ An be a closed subvariety, and let a ⊂ [x1 , . . . , xn ] be the ideal of all polynomials that vanish on X . For an arbitrary

112

Algebraic Groups

polynomial f (x1 , . . . , xn ) ∈ [x1 , . . . , xn ] and a point x in An , we introduce a linear form n X ∂f dx f (X1 , . . . , Xn ) = (x)Xi , ∂xi i=1

where Xi (i = 1, . . . , n) are the coordinates in the n-dimensional vector space associated with An . The tangent space of X at a point x ∈ X is the subspace Tx (X ) of the n-dimensional vector space, given by the linear equations dx f (X1 , . . . , Xn ) = 0 for all f ∈ a.

(2.32)

By Hilbert’s Basis Theorem, a is generated by a finite number of polynomials f1 , . . . , fr , and then instead of (2.32) one may consider the equivalent finite linear system dx fi (X1 , . . . , Xn ) = 0,

i = 1, . . . , r.

(2.33)

Now, if X is defined over K and x ∈ XK , then choosing fi ∈ a ∩ K[x1 , . . . , xn ], we see that Tx (X ) is also defined over K. Given a regular map ϕ : X → Y of algebraic varieties and a point x ∈ X , one naturally defines the linear map dx ϕ : Tx (X ) → Tϕ(x) (Y ) of the corresponding tangent spaces, called the differential of ϕ at x; moreover, dx ϕ is defined over K if ϕ is defined over K and x ∈ XK . In (2.32) and (2.33), the point x was fixed. If we let x vary in X , i.e., if we consider the set of all points (x, t) ∈ An × An such that x ∈ X and t ∈ Tx (X ), then we obtain the tangent bundle T(X ) of X . The equations in (2.33) show that T(X ) is a closed subvariety. Assuming X to be irreducible, and applying Theorem 2.44 to the canonical projection T(X ) → X , we see that dim Tx (X ) has a constant value, say d, for all points x of some Zariski-open subset U ⊂ X , and moreover, dim Tx (X ) ≥ d for any x ∈ X . Furthermore, d turns out to be dim X . Thus, dim Tx (X ) ≥ dim X for all x ∈ X , and the points for which equality holds (these are called simple (or nonsingular), while all other points are called singular) form a nonempty Zariski-open subset. For a reducible variety X , a point x ∈ X is simple if and only if it lies on a single irreducible component Y ⊂ X and is a simple point on Y .

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Proposition 2.45 A point x of a variety X ⊂ An is simple if and only if there exist polynomials f1 , . . . , fr ∈ [x1 , . . . , xn ], where r = n − dimx X , and a Zariski-open subset U ⊂ An such that x ∈ Y := {y ∈ U : fi (y) = 0, i = 1, . . . , r} ⊂ X and the Jacobian matrix 

∂fi (x) ∂xj

 i = 1, . . . , r j = 1, . . . , n

has rank r. If X is defined over K and x is a simple point belonging to XK , then f1 , . . . , fr can be chosen to have coefficients in K. If all points of a variety are simple, the variety is called smooth. Since simple points always exist, any homogeneous variety is smooth. In particular, the variety of an arbitrary algebraic group is smooth.

2.4.4 Birational Isomorphisms A rational map ϕ : X → Y of irreducible varieties is called a birational isomorphism if there exists an inverse rational map ϕ −1 : Y → X . In this case, ϕ induces a biregular isomorphism between Zariski-open sets U ⊂ X and V ⊂ Y . Varieties that are birationally isomorphic to an affine space are said to be rational. A dominant morphism ϕ : X → Y is a birational isomorphism if its comorphism ϕ ∗ induces an isomorphism between the fields of rational functions (X ) and (Y ). In particular, X is rational if and only if the field of rational function (X ) is a purely transcendental extension of . Note that all these definitions and properties have obvious analogs for K-defined varieties. There is one useful sufficient condition for a dominant morphism ϕ : X → Y to be a birational isomorphism. To formulate this condition, recall that ϕ is said to be separable if the comorphism ϕ ∗ gives rise to a separable field extension (X )/ϕ ∗ (Y ) (which, of course, holds automatically if char  = 0). Theorem 2.46 Let ϕ : X → Y be an injective dominant separable morphism of irreducible varieties. Then ϕ is a birational isomorphism. If, moreover, ϕ is a K-morphism of K-varieties, then it is a birational isomorphism over K. PROOF: See Humphreys (1975, Chapter 1, §4.6).

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It follows from Theorem 2.46 that to prove that a certain variety X is Krational, it suffices to construct an injective dominant separable K-morphism ϕ : U → X of some open subset U ⊂ An . A K-variety X for which there exists a dominant (but not necessarily injective) K-morphism ϕ : U → X of a Zariski-open subset U ⊂ An is called unirational. The question of whether every unirational variety is rational is known as the Lüroth problem. The answer to Lüroth’s problem turns out to be negative, in general, in both the absolute (i.e., over an algebraically closed base field) and the relative (i.e., over an arbitrary base field) cases, but even the unirationality of a variety has important consequences. For example, the variety of a connected (linear) algebraic K-group G is always unirational over K if either K is perfect or G is reductive (cf. Borel [1991, §18]), and this implies the assertion of Theorem 2.6. On the other hand, the question of whether the variety of a connected algebraic K-group is K-rational can be regarded as a particular case of the relative version of Lüroth’s problem (see [AGNT, §7.3] for a discussion of this question). The following theorem clarifies the distinction between birational and biregular isomorphisms. Theorem 2.47 Let ϕ : X → Y be a regular map of irreducible varieties, which is a birational isomorphism, and let x ∈ X . Assume that y = ϕ(x) is a simple point of Y . If the inverse rational map ψ = ϕ −1 is not regular at y, then dim ϕ −1 (y) ≥ 1. PROOF: See Shafarevich (2013, Chapter 2, §4). It follows from Theorems 2.46 and 2.47 that if ϕ : X → Y is a bijective regular map of irreducible varieties in characteristic zero and Y is smooth, then ϕ is a biregular isomorphism.

2.4.5 Actions of Algebraic Groups on Varieties An action of an algebraic group G on an algebraic variety X is a morphism µ : G × X → X such that (1) µ(e, x) = x, (2) µ(gh, x) = µ(g, µ(h, x)). (This is the definition of a left action. Sometimes, however, we will need to consider a right action µ : X × G → X , which is characterized by µ(x, e) = x and µ(x, gh) = µ(µ(x, g), h).) As usual, we will often write gx instead of µ(g, x)

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if the action is clear from the context. The action is defined over K if G, X , and µ are defined over K. Of the general results on actions of algebraic groups, we will only need the following statement, known as the Closed Orbit Lemma (cf. Borel [1991, §1.8]). Proposition 2.48 Let G be an algebraic group acting on a variety X . Then each orbit is a smooth variety that is open in its closure in X . Its boundary is a union of orbits of strictly lower dimension. In particular, the orbits of minimal dimension are closed. A variety X is called a homogeneous G-space if there exists a transitive action G×X → X . Fixing a point x ∈ X , we will have a bijection G/G(x) ↔ X between the left cosets modulo the stabilizer G(x) and the points of X , which can be used to endow G/G(x) with the structure of an algebraic variety. (Note: it follows from the smoothness of homogeneous spaces and the results of §2.4.3 that this structure is uniquely defined, at least in characteristic 0.) One may wonder if for any (closed) subgroup H ⊂ G, there exists an action of G on a suitable algebraic variety X and a point x ∈ X such that G(x) = H. The affirmative answer follows from Chevalley’s theorem (cf. Borel [1991, §5.1]; Humphreys [1975]): Let G be an algebraic K-group, and H be a closed K-subgroup of G. Then there exists a faithful K-representation ρ : G → GL(V ) and a 1-dimensional K-subspace D ⊂ V such that H = {g ∈ G : gD = D}. Considering now the induced action of G on the projective space P(V ) and the point x ∈ P(V ) corresponding to D, we obtain a geometric realization of G/H as the orbit Gx, which is a quasi-projective variety, i.e., an open subset of a projective variety. To develop the reduction theory in Chapters 4 and 5, we will need the following result, which elaborates on Chevalley’s theorem. Theorem 2.49 (A STRONGER VERSION OF CHEVALLEY’S THEOREM) Let G be a connected algebraic group, and H ⊂ G be a reductive subgroup, both defined over a field K of characteristic 0. Then there exists a K-representation ρ : G → GL(V ) and a vector x ∈ VK such that G(x) = H and Gx is closed in V. Since this result is not as widely known as Chevalley’s theorem, we will sketch its proof. We consider the action of H on G by right translations, and ¯ the associated representation of H in K[G]. It is well known that this representation is locally finite, and the classical argument of Nagata (1964) from

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the invariant theory of reductive groups shows that the algebra of invariants H is finitely generated and its elements separate the disjoint closed ¯ A = K[G] H-invariant subsets of G (in particular, the distinct left cosets). These results enable us to choose a finite set of generators x1 , . . . , xr of AK , and we let Vi denote a finite-dimensional G-invariant K-defined subspace of A containing L xi . We will show that the natural representation of G in V = ri = 1 Vi and the vector x = (x1 , . . . , xr ) in VK are as required. Indeed, by our construction H , hence G(x) ⊃ H. On the other hand, if gx = x, then, in par¯ xi ∈ A = K[G] ticular, xi (g) = xi (e). But since the functions xi generate A, they must separate distinct cosets; hence gH = H and g ∈ H. It remains to show that the orbit of ¯ ] → K[G] ¯ Gx is closed. Set X = Gx and consider the comorphism η : K[X for the morphism G → X given by g 7→ gx. It follows from our construction that ¯ ] and A. Using the natural bijection η induces an isomorphism between K[X between the points of an affine variety and the maximal ideals of its coordinate ring, it is easy to see that the fact that X = Gx is equivalent to the following ¯ ¯ assertion: for any proper maximal ideal m of A, the ideal mK[G] ⊂ K[G] is ¯ proper. But since H is reductive, there exists an A-linear projection K[G] → A; ¯ ¯ so mK[G] = K[G] would imply that mA = A, a contradiction. When applying Theorem 2.49 in Chapters 4 and 5, we will use instead of ρ the corresponding right representation ρ ∗ given by ρ ∗ (g) : = ρ(g−1 ); then the result of the action will be denoted by xρ ∗ (g), so that xρ ∗ (gh) = (xρ ∗ (g))ρ ∗ (h)). In the sequel, we will need the construction and properties of some concrete algebraic varieties, which we will now describe.

2.4.6 Multidimensional Conjugacy Classes The orbits of the adjoint action G × G → G given by (g, h) 7→ ghg−1 are the conjugacy classes of G. The following fact is well known (cf. Borel et al. [1970]): The conjugacy class of an element h of a reductive group G is closed if and only if h is semisimple. Similarly, one can consider the adjoint action of G on the Cartesian product Gd given by (g, (h1 , . . . , hd )) 7→ (gh1 g−1 , . . . , ghd g−1 ). The orbits of this action will be called multidimensional conjugacy classes. We will need a sufficient condition for a multidimensional conjugacy class to be closed. We say that a subgroup H ⊂ G (not necessarily closed) is reduced

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if, for some faithful representation ρ : G → GLn (), the image of H is a completely reducible linear group. Theorem 2.50 Let G be a reductive group and suppose that h1 , . . . , hd ∈ G generate a reduced subgroup. Then the multidimensional conjugacy class of (h1 , . . . , hd ) in G is closed in Gd . PROOF: First, we consider the special case where G = GLn () and the subgroup H ⊂ G, generated by h1 , . . . , hd , is completely reducible. Then the -hull A of H, i.e., the subalgebra of Mn () generated by H, is a semisimple algebra. Let u1 , . . . , um be a basis of A contained in H, and let wi (i = 1, . . . , m) be the words on d letters such that ui = wi (h1 , . . . , hd ). We can write m X ui uj = ckij uk k=1

for some ckij ∈ , which are usually called the structure constants of A. On the other hand, there are dij ∈ , where i = 1, . . . , d and j = 1, . . . , m, such that m X hi = dij uj . j=1

Let F denote the subvariety of (x1 , . . . , xd ) ∈ Gd satisfying m X wi (x1 , . . . , xd )wj (x1 , . . . , xd ) = ckij wk (x1 , . . . , xd ) (i, j = 1, . . . , m), k=1

and xi =

m X

dij wj (x1 , . . . , xd )

(2.34) (i = 1, . . . , d).

(2.35)

j=1

We will now show that F coincides with the multidimensional conjugacy class C of h = (h1 , . . . , hd ). The inclusion C ⊂ F is obvious. Now, let f = ( f1 , . . . , fd ) ∈ F. Let B denote the subspace spanned by vi = wi ( f1 , . . . , fd ) for i = 1, . . . , m. It follows from (2.34) that B is a subalgebra of Mn () and the correspondence ui → vi , where i = 1, . . . , m, extends to a surjective algebra homomorphism ϕ : A → B. Moreover, (2.35) implies that ϕ(hj ) = fj for j = 1, . . . , d. Let t M A= Ai i=1

be the decomposition of A as a direct sum of simple subalgebras. Then for each i ∈ {1, . . . , t} we have that either ϕ(Ai ) = 0 or the restriction of ϕ to

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Ai is an isomorphism onto its image. Then, by the Skolem–Noether Theorem (cf. §1.4.1), there exists g in GLn () such that the homomorphism ψϕ, where ψ = Int g, has the property that for each i = 1, . . . , t, the restriction of ψϕ to Ai is either the zero homomorphism or the identity. If ψϕ|Ai = 0 for some i, then B = ϕ(A) consists entirely of degenerate matrices, which contradicts the fact that fi ∈ B. Thus, ψϕ|A = idA , hence fi = ϕ(hi ) = ψ −1 (hi ) = g−1 hi g for all i = 1, . . . , d, implying that f ∈ C, as required. We will now reduce the general case in Theorem 2.50 to the special case that we have just considered. Pick a faithful representation ρ : G → GLn () such that ρ(H) is a completely reducible linear group, and then view G as a subgroup of G0 := GLn (). Let C0 (resp., C) denote the multidimensional conjugacy class of h = (h1 , . . . , hd ) in G0d (resp., in Gd ). Clearly C ⊂ C0 ∩ Gd . Lemma 2.51 C is an irreducible component of C0 ∩ Gd . PROOF: We will use a suitable generalization of the argument given by Richardson (1967). Let W denote an irreducible component of C0 ∩Gd that contains C; we will show that W = C. Since G is reductive and char K = 0, all representations of G are completely reducible (cf. Theorem 2.8(4)), so we can choose a G-invariant complement m ⊂ g0 := L(G0 ) to g := L(G). Consider the map π : G0 → D0 , where D0 = C0 h−1 , given by −1 −1 π(g) = (gh1 g−1 h−1 1 , . . . , ghd g hd ).

It is easy to see that π(G0 ) = D0 and de π(X ) = (X − Ad(h1 )(X ), . . . , X − Ad(hd )(X )) for all X ∈ g0 , and moreover, de π(g0 ) coincides with Te (D0 ), the tangent space of D0 at the identity. Let us show that gd ∩ de π(g0 ) = de π(g).

(2.36)

Indeed, suppose that for X ∈ g0 , we have de π(X ) ∈ gd . We can write X = Y +Z with Y ∈ g and Z ∈ m. Then for any i = 1, . . . , d, we have X − Ad(hi )(X ) = (Y − Ad(hi )(Y )) + (Z − Ad(hi )(Z)), which implies that Z − Ad(hi )(Z) ∈ m ∩ g = {0} as m is G-invariant. Thus, de π(X ) = de π(Y ), proving (2.36). Since C is a smooth variety which is open in

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its closure (Proposition 2.48), we may consider the tangent space Te (D), where D = Ch−1 . Then, from (2.36) and the inclusions de π(g) ⊂ Te (D) ⊂ Te (Wh−1 ) ⊂ gd ∩ de π(g0 ), we conclude that T(Ch−1 )e = T(Wh−1 )e , and hence C is open in W . This argument can be applied to any multidimensional conjugacy class of elements of G contained in W . Since W is irreducible, there can exist only one such class, hence C = W , proving the lemma. Since by the lemma the multidimensional conjugacy classes are the irreducible components of the closed subset C0 ∩ Gd , they are closed, and the proof of Theorem 2.50 is now complete. We will use Theorem 2.50 in the following two cases over a base field of characteristic zero: the Zariski closure of the subgroup H generated by h1 , · · · , hd is a connected reductive group, or H is finite. We note that for d = 1, the subgroup generated by an element h in G is reduced if and only if h is semisimple, so we obtain one direction of the above criterion for the conjugacy class of h to be closed. In this respect, it would be interesting to see whether the converse of Theorem 2.50 also holds.

2.4.7 Representation Varieties Let 0 be an arbitrary finitely generated group, and let G be some algebraic K-group. We will see shortly that the homomorphisms 0 → G are in a natural one-to-one correspondence with the points of a certain K-variety R(0, G) called the representation variety of 0 in G. For this, fix a finite generating set γ1 , . . . , γd of 0, and consider the associated surjective homomorphism π : Fd → 0 of the free group Fd of rank d with generators x1 , . . . , xd that sends xi to γi for all i = 1, . . . , d. Let N := ker π be the set of all relations for γ1 , . . . , γd in 0. Then set R(0, G) = {(g1 , . . . , gd ) ∈ Gd : w(g1 , . . . , gd ) = e, ∀w = w(x1 , . . . , xd ) ∈ N}. Since the algebraic operations on G are given by regular K-defined maps, for any word w = w(x1 , . . . , xd ), the equation w(g1 , . . . , gd ) = e defines a K-closed subset of Gd . It follows that R(0, G) is a K-closed subset (subvariety) of Gd . On the other hand, given (g1 , . . . , gd ) ∈ Gd , the necessary and sufficient condition for the existence of a group homomorphism 0 → G such that γi 7→ gi is that (g1 , . . . , gd ) ∈ R(0, G). Moreover, any homomorphism of 0 is uniquely determined by the images of the generators, so the points of R(0, G) yield a bijective parametrization of the representations of 0 in G.

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Next, we note that up to biregular isomorphism, the variety R(0, G) is independent of the choice of generators γ1 , . . . , γd . Indeed, if δ1 , . . . , δ` is another system of generators, and δi = wi (γ1 , . . . , γd ),

i = 1, . . . , `,

γi = θj (δ1 , . . . , δ` ),

j = 1, . . . , d,

then the maps (g1 , . . . , gd ) 7→ (w1 (g1 , . . . , gd ), . . . , wl (g1 , . . . , gd )), (h1 , . . . , h` ) 7→ (θ1 (h1 , . . . , h` ), . . . , θd (h1 , . . . , h` )) are mutually inverse K-defined morphisms between the representation varieties Rγ (0, G) and Rδ (0, G) constructed using the generating sets γ1 , . . . , γd and δ1 , . . . , δ` , respectively. The group G acts on R(0, G) by conjugation: g · (g1 , . . . , gd ) = (gg1 g−1 , . . . , ggd g−1 ). In this context, the orbits of G correspond to the classes of G-equivalent representations. We refer the reader to Lubotzky and Magid (1985); Platonov and Benyash-Krivetz (1986); Rapinchuk and Benyash-Krivetz (1993); Rapinchuk, Benyash-Krivetz, and Chernousov (1996); and Benyash-Krivetz and Chernousov (1996a,b, 1997a,b) for results dealing with representation varieties and the associated character varieties. We also note that in fact every Q-defined affine algebraic variety can be realized as the character variety of some finitely generated group minus the point corresponding to the character of the trivial representation (cf. I. Rapinchuk [2015]). Here we limit ourselves to proving the following finiteness result (recall that char K = 0). Theorem 2.52 Let 0 be a finite group, and G be reductive. Then there are only finitely many orbits of the natural action of G on R(0, G), and these orbits are closed. PROOF: The fact that the orbits are closed follows from Theorem 2.50, as in characteristic zero any representation 0 → GLn () is completely reducible. Moreover, according to the classical representation theory of finite groups, 0 has finitely many inequivalent irreducible representations, which implies the finiteness of the number of orbits in the case where G = GLn (). We then use Lemma 2.51 to reduce the general case to this one. We can assume that G ⊂ G0 := GLn (). Now, if C0 is the G0 -orbit of ρ ∈ R(0, G), then by Lemma 2.51, each irreducible components of R(0, G) ∩ C0 is a single G-orbit. So, our claim follows from the fact that there are only finitely many irreducible

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components. (We note that using the Levi decomposition (cf. Theorem 2.7), one can remove the assumption that G be reductive.)

2.4.8 The Variety of Maximal Tori Let G be a reductive K-group, T ⊂ G be a maximal K-torus, and N = NG (T) be its normalizer. It follows from the conjugacy theorem for maximal tori that the map Tg := gTg−1 7→ gN yields a bijection between the maximal tori of G and the points of the quotient variety T = G/N, which is called the variety of maximal tori of G. Moreover, the points of TK correspond to the maximal K-defined tori of G. (Note that up to K-isomorphism, T does not depend on the choice of the original torus T.) Theorem 2.53 (CHEVALLEY [1954], BOREL char K = 0, then T is a rational variety over K.

AND

SPRINGER [1968]) If

PROOF: Let g = L(G) and h = L(T) be the Lie algebras of G and T, respectively. Pick a K-defined subspace m of g such that g = h ⊕ m, and fix a regular element X ∈ hK . Let m0 denote the subset of m consisting of Z such that X + Z is regular semisimple and its centralizer zg (X + Z) has zero intersection with m. We claim that m0 is a Zariski-open subset of m. Since the set of regular semisimple elements is open (cf. §2.1.11), it suffices to show that m1 := {Z ∈ m : zg (X + Z) ∩ m = (0)} is also open. For this, we introduce the variety P = {(Y , Z) ∈ m × m : [X + Z, Y ] = 0} π

and consider the projection P→m given by (Y , Z) → Z. Clearly, for any Z ∈ m, we have (0, Z) ∈ P; in particular, π is surjective and, moreover, π −1 (0) = (0, 0) since by our choice of X we have zg (X ) = h and h ∩ m = (0). By the theorem on the dimension of the fibers (cf. Theorem 2.44), we have dim P = dim m and {Z ∈ m : dim π −1 (Z) = 0} is open in m. However, it is easy to see that this set in fact coincides with m1 . Set W = {(Z, g) ∈ m × G : g−1 (X + Z)g ∈ h}, and let U = (m0 × G) ∩ W . Being the preimage of h under the K-morphism ϕ : m × G → g given by ϕ(z, g) = g−1 (X + Z)g, the set W is a K-closed subset

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of m × G. Furthermore, (0, 1) ∈ U, so U is a nonempty open subset of W . Consider the projections θ : W → m and δ : W → G. Since for any Z ∈ m0 the centralizer zg (X + Z) is the Lie algebra of some maximal torus, it follows from the conjugacy theorem that there exists g ∈ G such that g−1 (X + Z)g ∈ h, which implies that θ(U) = m0 . Moreover, by our construction we have ghg−1 ∩ m = (0),

(2.37)

θ −1 (Z) = (Z, gN).

(2.38)

and Furthermore, if x = (Z, g) ∈ U and x0 = (Z 0 , g0 ) ∈ δ −1 (δ(x)), then g0 ∈ gN and X + Z, X + Z 0 ∈ ghg−1 = g0 h(g0 )−1 , hence Z − Z 0 ∈ m ∩ (ghg−1 ) and therefore Z = Z 0 by (2.37). Let ψ: m×G → m×T be the map induced by the natural morphism G → T = G/N, and let θ 0 : ψ(W ) → m and δ 0 : ψ(W ) → T be the corresponding projections. Then the restriction θ 0 |ψ(U) : ψ(U) → m0 is a bijection, and the restriction δ 0 |ψ(U) is injective. Then by Theorem 2.46, θ 0 has a K-defined rational inverse χ : m0 → ϕ(W ) (recall that char K = 0), and moreover ξ := δ 0 ◦ χ is injective on its domain. Since dim m = dim T , we can use Theorem 2.46 one more time to conclude that ξ is a birational isomorphism between m and T , which proves the theorem. Proposition 2.54 Let G be a connected algebraic group over a field K of characteristic zero, and let W ⊂ G be the set of regular semisimple elements. Then there exists a regular K-defined map ϕ : W → T such that x ∈ Tϕ(x) for all x ∈ W , i.e., each element is mapped into the torus that contains it. PROOF: Fix a maximal K-torus T of G, and consider the corresponding variety of maximal tori T = G/N, where N = NG (T). Furthermore, set Z = {(x, g) ∈ W × G : g−1 xg ∈ T}, which is obviously a closed subset of W × G. Let θ : W × G → W × T denote the regular map induced by the natural morphism ψ : G → T . It is easy to see that Z = θ −1 (θ (Z)), so since ψ is open (cf. Borel [1991, § 6]), we conclude that Y := θ(Z) is closed. Let π1 : W × T → W and π2 : W × T → T be

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the natural projections. We claim that π1 |Y : Y → W is bijective. Indeed, the conjugacy theorem for maximal tori implies that for any x ∈ W , there exists g ∈ G such that g−1 xg ∈ T, and moreover, since x is regular, the element g is unique up to multiplication by an element of N, hence the bijectivity of π1 |Y . Since W is open in G and consequently smooth, by the results of §2.4.3, the restriction π1 |Y has a regular inverse δ : W → Y . Then the map ϕ := π2 ◦ δ is as desired. Note that if we let G act on W by conjugation and on T by translation, then ϕ as constructed in the preceding proposition is G-equivariant.

2.4.9 The Variety of Borel Subgroups In §2.1.9, we noted that the Borel subgroups of a connected algebraic group G are in one-to-one correspondence with the points of the quotient variety B = G/B, where B ⊂ G is a fixed Borel subgroup; for this reason, B is called the variety of Borel subgroups. If a K-group G has a K-defined Borel subgroup, then B receives a K-structure such that the action of G on B by left translation is K-defined. However, typically a K-defined algebraic group may not have a K-defined Borel subgroup, so it is natural to ask if B has a K-structure in the general case. Theorem 2.55 Let G be a connected algebraic K-group. Then the variety B of its Borel subgroups has a K-structure such that points of BK correspond to K-defined Borel subgroups, and the action of G on B is K-defined. PROOF: Set H = G/R(G) and H¯ = H/Z(H); clearly H¯ is a semisimple adjoint group. If we let ϕ : G → H¯ denote the natural map, then for any Borel sub¯ group B of G, the image ϕ(B) is a Borel subgroup of H¯ and G/B ' H/ϕ(B). Thus, it is enough to consider the case of a semisimple adjoint group. Any such group G can be obtained from a quasisplit group G0 by twisting using a suitable cocycle a = {aσ } with aσ = Int gσ ∈ Int G0 = G0 . Let B0 ⊂ G0 be a K-defined Borel subgroup and B0 = G0 /B0 the corresponding K-defined variety of Borel subgroups. Then the left translations rσ by gσ define a cocycle r in the group of K-automorphisms of B0 . Since B0 is a projective variety, there exists a “twisted” variety r B0 (cf. the remark following Theorem 2.15), and then B = r B0 is as required.

3 Algebraic Groups over Locally Compact Fields

In Chapter 2, we considered properties of algebraic groups that are independent of the specific nature of the base field. Starting with this chapter, though, one of our objectives will be to study the effect of properties of the base field on the structure of algebraic groups defined over it. We begin with groups over locally compact fields for several reasons. First, the group of rational points of an algebraic group over such a field is naturally endowed with the additional structure of an analytic (Lie) group, enabling us to use a variety of results from the rich structure theory of Lie groups. Second, arithmetic subgroups and their generalizations, as well as the groups of rational points over number fields, which are the real focus of the arithmetic theory of algebraic groups, can be viewed as discrete subgroups of suitable direct products of groups of rational points over appropriate completions, so properties of the latter have a significant impact on properties of the original groups. In §3.1, we give an account of the basic results dealing with the topological and analytic properties of the sets of rational points of algebraic varieties over locally compact fields, some of which remain valid over any field complete (or Henselian) with respect to a discrete valuation. In §3.2, we examine the classical case where the base field is either R or C. The key result here is the Iwasawa decomposition, which plays an important role in Chapter 4. In §§3.3–3.4, we investigate groups over non-Archimedean locally compact fields. Results obtained by applying the theory of profinite groups or by using techniques involving the reduction of algebraic varieties are presented in §3.3, while §3.4 contains a survey of the results of Bruhat–Tits theory that are needed in §5.4. Finally, in §3.5 we assemble some basic results from measure theory that will be used later in the book on several occasions.

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3.1 Topology and Analytic Structure Throughout this chapter, K will denote a nondiscrete locally compact field of characteristic zero. It is well known (cf., for example, Bourbaki [1998a]) that K is either connected (in which case it is either R or C) or totally disconnected (in which case it is a finite extension of the p-adic field Qp ). In particular, K is complete with respect to a nontrivial valuation | |v , which is either the usual absolute value on the real or the complex numbers, or is discrete, i.e., has value group isomorphic to Z. Then the open balls B(a, ε) = {x ∈ K : |a − x|v < ε}, for a ∈ K and ε > 0, form a base for the topology on K. This topology can be naturally extended to a topology on the set VK of K-rational points of an arbitrary algebraic K-variety V . More precisely, given a Zariski-open K-subset U ⊂ V , a finite set f1 , . . . , fr of regular K-functions on U, and a real ε > 0, we set V (U; f1 , . . . , fr ; ε) = {x ∈ UK : | fi (x)|v < ε, i = 1, . . . , r}. Then it is easy to show that the sets V (U; f1 , . . . , fr ; ε) constitute a base of a topology on VK , which we call the topology defined by the valuation v, or, more concisely, the v-adic topology. We note that this topology is stronger than the Zariski topology. In contrast to the Zariski topology, it has the following natural property: if V = V1 × V2 is a K-defined product of two algebraic K-varieties, then the topological space VK is canonically homeomorphic to V1K × V2K endowed with the direct product topology. If W is an open (respectively, closed) K-subvariety of V , then WK is an open (respectively, closed) subset of VK . It follows that VK is Hausdorff for any variety V . Indeed, the diagonal 1 ⊂ V × V is closed in the Zariski topology, and therefore 1K is closed in (V × V )K ; on the other hand, (V × V )K ' VK × VK with the direct product topology. So, the diagonal 1K ⊂ VK × VK is closed, which implies that VK is Hausdorff. Any regular K-morphism f : V → W induces a continuous map fK : VK → WK . It follows that if G is an algebraic K-group, then GK is a topological group. The topology just introduced has a more explicit description for affine and projective varieties. Namely, if V ⊂ An , it is induced via the inclusion VK ⊂ K n by the natural topology on K n (which is nothing else but the direct product topology on K n = K × · · · × K, having the open n-balls B(a, ε) = {x ∈ K n : ka − xkv < ε} for a ∈ K n and ε > 0 as a base, where for z = (z1 , . . . , zn ) ∈ K n we set kzkv = max |zi |v ). This rather straightforward remark has several consei quences.

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First, for any affine variety V , the space VK is locally compact. Since any point of an arbitrary variety has an open affine neighborhood, this result in fact remains valid for all varieties. Second, for an algebraic K-group G ⊂ GLn (), the topology on GK is induced by the natural topology on GLn (K). In particular, if K is nonArchimedean, then the topology on GK can be described as follows. Let O ⊂ K be the valuation ring; then the group of O-points GO = G ∩ GLn (O) is the “principal” open compact subgroup, and its congruence subgroups


GO pα = G ∩ In + pα Mn O for α ≥ 1, where p ⊂ O is the valuation ideal, constitute a base of neighborhoods of the identity in GK . Now suppose that V ⊂ Pn is a projective variety. Then the topology on VK is induced by the natural topology on PnK , which is defined as the quotient topology resulting from the canonical map K n+1 \{(0)} → PnK . It is well known that PnK is compact with respect to this topology, hence VK , being closed in PnK , is also compact. The question of the compactness of VK will not be considered here in full generality; we note, however, the following compactness criterion for homogeneous spaces. Theorem 3.1 Let G be an algebraic K-group, and let H be a K-defined subgroup of G. The quotient GK /HK is compact if and only if H contains a maximal connected K-split solvable subgroup of the connected component G0 . In particular, GK is a compact group if and only if G0 is reductive and anisotropic over K. PROOF: The proof reduces easily to the case where G is connected. (⇐) Assume that H contains a maximal connected K-split solvable subgroup B of G. We will show that GK /BK is compact, and then GK /HK is compact as well. According to Chevalley’s theorem (cf. §2.4.5), we can choose a K-representation G → GL(V ) and a 1-dimensional subspace V1 ⊂ V whose stabilizer in G is precisely B. Since B is split over K, its image in GL(V /V1 ) is trigonalizable over K, and therefore there exists a B-invariant K-defined flag F = {V1 ⊂ · · · ⊂ Vi ⊂ · · · ⊂ Vn = V , dim Vi = i} that begins with V1 . Let X denote the closure of the orbit GF in the flag variety F(V ) (cf. Borel [1991]). Since F(V ) is projective, X is also projective, and therefore, as noted above, the set XK is compact.

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We will soon see that the Inverse Function Theorem implies (cf. Corollary 3.7) the openness of the GK -orbits in (GF)K . Assume now that we can show that XK = (GF)K .

(3.1)

Then all the GK -orbits in XK are open. So, the orbit GK F, being the complement of the union of the other orbits, is closed, hence compact. Since by construction the stabilizer of F in G is B, the natural map ϕ : G → X given by g 7→ gF induces a continuous bijection ψ : GK /BK → GK F. But ϕK : GK → XK is an open map (Corollary 3.7), so actually we have a homeomorphism GK /BK ' GK F, and the compactness of GK /BK follows. Turning now to the proof of (3.1), we observe that since (GF)K ⊂ XK , we only need to show the inclusion XK ⊂ GF. Assume the contrary, and let L ∈ XK \GF. Then the dimension of the orbit GL is strictly less than dim GF (cf. Proposition 2.48), hence for the stabilizer G(L), we have dim G(L) > dim B. On the other hand, G(L) is clearly trigonalizable over K, making the connected component G(L)0 a solvable K-split subgroup. But being a maximal connected K-split subgroup, B has maximal possible dimension among all such subgroups (which follows from the fact that any two maximal connected solvable K-split subgroups are conjugate, see Borel and Tits [1965]). A contradiction, proving our claim. (⇒) Suppose GK /HK is compact. Choose a maximal connected solvable K-split subgroup B of G that contains a maximal connected solvable K-split subgroup of H. By the first part of the argument, the quotient HK /(H ∩ B)K is compact. Using the fact that the product of a compact subset with a closed subset in a topological group is closed, we see that HK BK , hence also BK HK = (HK BK )−1 , is closed in GK . Therefore, BK /(B ∩ H)K ' BK HK /HK is compact. We will conclude the proof by showing that B = B∩H, hence B ⊂ H. Indeed, it is well known (cf. §2.1.8) that B possesses a normal series B = B0 ⊃ B1 ⊃ · · · ⊃ Bn = {e} defined over K, whose composition factors Bi /Bi+1 are K-isomorphic to Ga or Gm . If B ∩ H 6= B, then there exists i ∈ {1, . . . , n − 1} such that Bi (B ∩ H) = B but F := Bi+1 (B ∩ H) 6= B. Observe that the compactness of BK /(B ∩ H)K implies that of BK /FK . Now consider the action of T = Bi /Bi+1 , which is isomorphic to Ga or Gm , on B/F. Corollary 3.9 (proved later in this section) yields the openness of all the orbits of TK on (B/F)K , from which it follows that the orbit TK e, where e stands

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for the coset F ∈ B/F, is closed. This means TK /F¯ K is compact, where F¯ is the image of F ∩ Bi in T. But F¯ is finite since dim T = 1, hence TK must be compact, a contradiction. Remark The finiteness theorem for Galois cohomology of algebraic groups over local fields (cf. [AGNT, § 6.4]) implies that GK has finitely many orbits in its natural action on (G/H)K . It follows that the spaces GK /HK and (G/H)K are simultaneously either compact or noncompact. We note that if K is totally disconnected, then so is VK . In §3.2, we will examine the connectedness of VK when K = R or C. Most of the preceding remarks on the topology of VK remain valid when K is a locally compact field of characteristic p > 0, where it is isomorphic to the field F((t)) of formal Laurent series over a finite field F. However, our assumption that char K = 0 is essential and cannot be dropped in the study of the analytic structure on VK , to which we now proceed. Our next objective is to introduce on VK (or, more precisely, on a suitable Zariski-open subset) the structure of an analytic manifold. It is convenient to assume from the outset that VK is Zariski-dense in V as in this case, the dimension of VK as an analytic manifold will coincide with dim V (the dimension of V as an algebraic variety). This condition can always be satisfied by replacing V with the Zariski-closure W of VK in V ; note that W is defined over K and satisfies WK = VK . Our goal is to show that each simple point x ∈ VK has a neighborhood that is homeomorphic to an open ball in the space K m , where m = dimx V (the dimension of the unique irreducible component of V that passes through x, which coincides with the dimension of the tangent space at x). Replacing V by a suitable affine neighborhood of x, we may assume V to be affine, and, in fact, a Zariski-closed subset of some An . Then one uses Proposition 2.45 in conjunction with the following version of the Inverse Function Theorem. Theorem 3.2 (INVERSE FUNCTION THEOREM) Let U ⊂ K n be an open subset, x ∈ U, and let f = ( f1 , . . . , fn ) : U → K n be a polynomial (or, more generally, an analytic) map. Set y = f (x) and assume that the Jacobian   ∂fi (x) ∂xj i,j = 1,...,n is nonsingular. Then f is a local analytic isomorphism at x, and thus there exists a neighborhood W ⊂ U of x such that f (W ) is a neighborhood of y and f restricts to an analytic isomorphism from W to f (W ). For the proof, we refer the reader to Serre (1992, Part II, Chapter 3, §9), where one can also find a discussion of the basic concepts and results pertaining to

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analytic functions and manifolds (cf. the paragraph following Proposition 3.3). It should be noted that the majority of applications of the Inverse Function Theorem in this book require just the fact that under the assumptions made in the statement, f is a local homeomorphism at x; in other words, the fact that f is a local analytic isomorphism will not actually be used in most situations. Now, let x = (x01 , . . . , x0n ) ∈ VK be a simple point. (Note that our assumption on the Zariski-density of VK in V implies the existence of such an x as the set of simple points is Zariski-open.) As above, let m = dimx V . Then according to Proposition 2.45, the variety V is defined in a suitable neighborhood of x in An by r = n − m equations. More precisely, there exist a Zariski-open set U0 ⊂ An and polynomials f1 , . . . , fr ∈ K[x1 , . . . , xn ] such that Y = {y ∈ U0 : fi (y) = 0, i = 1, . . . , r} is contained in V and the Jacobian   ∂fi (x) ∂xj

i = 1, . . . , r j = 1, . . . , n

has rank r. Without loss of generality, we may assume that   ∂fi det (x) 6= 0. ∂xj i,j = 1,...,r Consider g = (g1 , . . . , gn ) : K n → K n , where gi = fi for i ≤ r, and gi = xi for i > r. Clearly,     ∂gi ∂fi det (x) = det (x) 6= 0. ∂xj ∂xj i,j = 1,...,n i,j = 1,...,r By Theorem 3.2, there exists a neighborhood U ⊂ K n of x such that W = g(U) is a neighborhood of g(x) = (0, . . . , 0, x0r+1 , . . . , x0n ) and g : U → W is an analytic isomorphism. Let h = (h1 , . . . , hn ) = g−1 : W → U. Then ϕ = (ϕ1 (t1 , . . . , tn−r ), . . . , ϕr (t1 , . . . , tn−r ), t1 , . . . , tn−r ), where ϕi (t1 , . . . , tn−r ) = hi (0, . . . , 0, t1 , . . . , tn−r ) for i = 1, . . . , r parametrizes a neighborhood of x in VK by the points of some open set in K n−r = K m . Moreover, the parametrizing map is actually the inverse map for the projection onto the (last) n−r coordinates. From this, it follows that any two such parametrizations of a neighborhood of a given point differ by an analytic isomorphism. Thus, if V is affine, the set of simple points of VK is endowed

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with a natural structure of an analytic manifold. We note that regular maps of affine varieties respect this structure in the following sense: any K-defined regular map f : V → W of affine K-varieties V and W induces the analytic map ˜ K) → W ˜ K, f˜ : V˜ K ∩ f −1 (W ˜ K ) is the set of simple points of VK (resp., WK ) equipped where V˜ K (resp., W with the structure of an analytic manifold introduced above. Using affine neighborhoods, it is easy to show that the set V˜ K of simple points of an algebraic K-variety V carries the structure of an analytic variety in the general case, and furthermore that any regular K-defined map f of algebraic K-varieties induces an analytic map f˜ in the manner described above. Indeed, the coordinates of any two affine neighborhoods of the same point are related by a birational transformation defined at that point. Therefore, two parametrizations of a neighborhood of a given point, constructed using these two affine neighborhoods, differ by an analytic isomorphism. Furthermore, any regular K-defined map f : V → W induces a regular map of affine neighborhoods, and therefore an analytic map f˜ . In summary, we have proved Proposition 3.3 Let V be an algebraic K-variety. Then the set V˜ K of simple points of VK has a natural structure of an analytic manifold over K. Any regular K-defined map f : V → W of algebraic K-varieties induces an analytic ˜ K) → W ˜ K. map f˜ : V˜ K ∩ f −1 (W We can now analyze VK (or rather V˜ K ) using the theory of analytic manifolds (cf. Serre [1992]). First, we would like to remind the reader of several concepts needed for our discussion. For each point x of an analytic manifold X , one defines the associated tangent space Tx (X )an ,1 which is a vector space over K of dimension equal to dimx X – the dimension of the affine space used to parametrize a suitable neighborhood of x. A morphism f : X → Y of analytic manifolds is a continuous locally analytic map, which means that it induces an analytic map (in the usual sense) between the parametrization domains for the respective points. Any morphism f : X → Y gives rise to a linear map dx fan : Tx (X )an → Tf (x) (Y )an , called the differential of f at x. The morphism f is called an immersion at x if dx fan is injective, and simply an immersion if this condition holds for all points. If a topological subspace X of a manifold Y is equipped with a structure of an analytic manifold and the inclusion map 1 The subscript an will be used to distinguish between the tangent space of an algebraic variety

and the tangent space of the corresponding analytic manifold.

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X ,→ Y is an immersion, then X is said to be a submanifold of Y . To clarify this concept, we would like to point out that the subset in R2 defined by y = |x| is not a submanifold:

Figure 3.1

(This set is “non-smoothly” embedded in R2 . In contrast, a submanifold, after a suitable analytic change of coordinates, can be defined by linear equations in a neighborhood of any point.) We will also need the following criterion for the openness of a morphism. Proposition 3.4 Let f : X → Y be a morphism of analytic manifolds, and let x ∈ X . If dx fan : Tx (X )an → Tf (x) (Y )an is surjective, then f is an open map at x. PROOF: The proof follows easily from the Inverse Function Theorem 3.2 (cf. Serre [1992, Part II, Chapter 3, §9]). Lemma 3.5 Let V be an algebraic K-variety, and let x ∈ V˜ K . Then Tx (V˜ K )an = Tx (V )K , in other words, the “analytic” tangent space coincides with the set of K-defined elements of the “algebraic” tangent space. If f:V →W is a regular K-defined map of algebraic K-varieties, and x ∈ V˜ K is such that ˜ K , then for f˜ : V˜ K ∩ f −1 (W ˜ K) → W ˜ K , we have dx f˜an = (dx f )| ˜ f (x) ∈ W Tx (VK )an . The proof immediately follows from the comparison of relevant definitions. Proposition 3.4 and Lemma 3.5 are applied to algebraic varieties as follows. Proposition 3.6 Let f : V → W be a dominant K-defined morphism of irreducible algebraic K-varieties. If x ∈ VK is a simple point such that f (x) is a simple point on W , and the differential dx f : Tx (V ) → Tf (x) (W ) is surjective, then fK : VK → WK is open at x. Consequently, there exists a Zariski-open subset U ⊂ V such that fK is open at any point x ∈ UK . ˜ K) → W ˜ K induced by f . PROOF: Consider the analytic map f˜ : V˜ K ∩ f −1 (W −1 ˜ ˜ Our assumptions yield that x ∈ VK ∩ f (WK ), and moreover by Lemma 3.5

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the differential dx f˜an is surjective. Then f˜ is open at x by Proposition 3.4, and therefore so is fK . Furthermore, since char K = 0, the morphism f is automatically separable, i.e., the corresponding extension K(V )/f ∗ (K(W )) of fields of rational functions is separable. It follows (cf. Borel [1991, AG §17]) that there exists a Zariski-open subset U ⊂ V such that for any x ∈ U, the image f (x) is a simple point on W and the differential dx f is surjective, and then our second assertion follows from the first. We observe that if VK is not assumed to be dense in V , then UK could be empty and Proposition 3.6 would become meaningless. Therefore, we now describe two cases, of particular interest to us in the sequel, where such degeneracy does not occur. Corollary 3.7 Keep the notations introduced in Proposition 3.6 and assume that V is smooth and VK 6= ∅. Then the image fK ( F) of any nonempty open set F ⊂ VK contains a nonempty open subset of WK . In particular, if f : G → H is a surjective K-defined morphism of algebraic K-groups, then fK (GK ) is an open subgroup of HK . To prove this, we need the following. Lemma 3.8 Let V be a smooth irreducible K-variety such that VK 6= ∅. Then for any nonempty K-defined Zariski-open subset U ⊂ V , the set UK is dense in VK in the v-adic topology. Furthermore, any nonempty v-adically open subset F ⊂ VK is Zariski-dense in V; in particular, VK is Zariski-dense in V . PROOF: It is easy to see that both assertions of the lemma reduce to proving that U ∩ F is nonempty whenever U ⊂ V is Zariski-open and F ⊂ VK is open in the topology defined by the valuation. Set X = V \ U. For any x ∈ F, we can find a K-defined Zariski-open subset W ⊂ V containing x and a nonzero regular function h ∈ K[W ] that vanishes on X ∩ W . By our assumption, V is smooth, so x is simple and therefore, as we have seen, there exists an analytic parametrization of some neighborhood of x in VK . If U ∩ F = ∅, and consequently F ⊂ X , then the analytic Taylor series for h must vanish. Since the algebraic and analytic Taylor series coincide, this contradicts the injectivity of the map associating to a regular function at a simple point its algebraic Taylor series (cf. Shafarevich [2013, Chapter 2]). PROOF OF COROLLARY 3.7: Let U ⊂ V be the Zariski-open subset provided by Proposition 3.6. Then U ∩ F 6= ∅ by Lemma 3.8. Since fK : VK → WK is open at any point x ∈ U ∩ F, the first assertion of Corollary 3.7 easily follows. The second assertion is an immediate consequence of the first one. 

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Corollary 3.9 Let G × X → X be a K-defined action of a connected algebraic K-group G on a K-variety X . If x ∈ XK and Y is the Zariski-closure of the orbit Gx, then for any open set F ⊂ GK , the set Fx is open in YK . PROOF: The morphism ϕ : G → Y , given by ϕ(g) = gx, is dominant. Let U ⊂ G be the Zariski-open set constructed in Proposition 3.6 for this morphism. Then ϕK is open at any point g ∈ UK . Since ϕ(hg) = hϕ(g) for any h in G, ϕK actually is open at any point h ∈ GK . It follows at once that ϕ( F) = Fx is open for any open F ⊂ GK . We are now ready to give an example of how these general results can be used to analyze the structure of groups of rational points over locally compact fields. Theorem 3.10 (RIEHM [1970a,b]) Let G be an almost K-simple algebraic group. Then every noncentral normal subgroup of GK is open. PROOF: For each g in G, we let Wg = {[g, h] = g−1 h−1 gh : h ∈ G}. All the sets Wg (g ∈ G) are the images of G under regular maps, contain the identity, and together generate the commutator subgroup [G, G] of G. In our case [G, G] = G, so using a straightforward dimension argument (cf. Borel [1991, Proposition 2.2]) and taking into account that (Wg )−1 = Wg−1 , we see that there exists a finite collection of elements g1 , . . . , gn ∈ G such that G = Wg1 · · · Wgn . Consider the morphism ψ : X = G × · · · × G → G × · · · × G = Y, | {z } | {z } 2n

n+1

ψ(x1 , . . . , xn , y1 , . . . , yn ) = (x1 , . . . , xn , [x1 , y1 ] · · · [xn , yn ]). By the theorem on the dimension of fibers (cf. §2.4.2), for any y ∈ ψ(X ), we have dim ψ −1 (y) ≥ (n − 1) dim G. We claim there is a y for which dim ψ −1 (y) = (n − 1) dim G. By our construction, the morphism ϕg1 ,...,gn : G × · · · × G → G given by | {z } n

ϕg1 ,...,gn (y1 , . . . , yn ) = [g1 , y1 ] · · · [gn , yn ] is surjective, and therefore there exists a point g ∈ G such that dim ϕg−1 (g) = (n − 1) dim G. 1 ,...,gn Then y = (g1 , . . . , gn , g) is as required.

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Applying the theorem on the dimension of fibers once again, we obtain a Zariski-open subset U ⊂ Y such that dim ψ −1 (x) = (n − 1) dim G for all x ∈ U. Let V denote the projection of U onto the first n components. Then V is open in G × · · · × G, and for any (x1 , . . . , xn ) ∈ V , we can find g ∈ G satisfying | {z } n

dim ϕx−1 (g) = (n − 1) dim G. 1 ,...,xn It follows that ϕx1 ,...,xn is dominant for any (x1 , . . . , xn ) ∈ V . Now, let N ⊂ GK be a noncentral normal subgroup. Since GK is Zariskidense in G (see Theorem 2.6; for a local field this also follows from Lemma 3.8), the Zariski-closure N¯ of N is a noncentral normal K-defined subgroup of G, and consequently N¯ = G since by assumption G is almost K-simple. It now follows from the preceding argument that there are x1 , . . . , xn ∈ N such that the morphism ϕx1 ,...,xn is dominant. Applying Corollary 3.7, we see that ϕx1 ,...,xn (GK × · · · × GK ) contains an open subset of GK . But ϕx1 ,...,xn (GK × · · · × GK ) = {[x1 , h1 ] · · · [xn , hn ] : hi ∈ GK } ⊂ N, so N is open in GK . Remark The proof of Theorem 3.10 relies only on Proposition 3.6, which is a formal corollary of the Inverse Function Theorem, but never uses the local compactness of K. So, the assertion of the theorem in fact holds whenever the Inverse Function Theorem is valid over K, which is, for example, the case when K is complete with respect to a nontrivial discrete valuation. This generalization is used to analyze the deviation from the weak approximation property for simply connected groups over an arbitrary field (cf. [AGNT, § 7.3]). On the other hand, we will see in §§3.2–3.3 that for locally compact fields, Theorem 3.10 yields that any noncentral normal subgroup of GK has finite index. Most analytic manifolds we will encounter in this book carry a group structure. This fact, however, did not play any significant role in the results from the theory of analytic manifolds we have considered so far. So, in the remainder of this section, we will give a brief account of some results about analytic groups, which are analytic manifolds that also carry a group structure such that the group operations are given by analytic maps. Analytic groups, particularly over the fields of real or complex numbers, are commonly known as Lie groups, and we refer the reader to Serre (1992), Bourbaki (1998c), and Helgason (2001) for an exposition of the classical Lie theory. Here, we will limit our attention to some results dealing with the Lie groups that arise from

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algebraic groups. So, let G be an algebraic group defined over K. As we know, G is a smooth variety, and therefore GK has a natural structure of an analytic manifold over K. Moreover, since the group operations on G are given by regular K-defined maps, their restrictions to GK are represented by analytic maps, making GK into a Lie group (cf. Serre [1992]). The Lie algebra gan of the analytic group GK is the tangent space at the identity T(G)an , which by Lemma 3.5 coincides with the subspace of K-defined elements of the algebraic tangent space Te (G), in other words, of the Lie algebra g = L(G) as an algebraic group. Moreover, the Lie bracket on gan is induced by that of g. We can define the exponential and logarithmic maps (cf. Bourbaki [1998c]), which are mutually inverse, local analytic isomorphisms between gan and GK . If G ⊂ GLn () is a K-defined matrix realization of G, then exp and log are given by the usual formulas: X X2 Xm + + ··· + + · · · for X ∈ gan , (3.2) 1! 2! m! (x − In )2 (x − In )m log(x) = (x − In ) − + · · · + (−1)m−1 + · · · for x ∈ GK . 2 m In particular, the exponential map for a group restricts to the exponential map for a subgroup. If X , Y ∈ gan (respectively, x, y ∈ GK ) commute, then exp(X ) = In +

exp(X + Y ) = exp(X ) exp(Y ), log(xy) = log(x) + log(y) (assuming that all the expressions here are defined, i.e., that the corresponding series converge). It follows that GK always has a neighborhood of the identity that does not contain any nontrivial elements of finite order (cf. Serre [1992]). We also note the following formulas: exp(x−1 Xx) = x−1 exp(X )x,

(3.3)

log(x−1 yx) = x−1 log(y)x, implying that the exponential and logarithmic maps commute with the adjoint action of GK . If a subgroup H of GK is also a submanifold of GK , then H is said to be a Lie subgroup of GK . It follows from the definition that given a Lie subgroup H ⊂ GK , we have the inclusion han ⊂ gan for the corresponding Lie algebras. A Lie subgroup H ⊂ GK need not be closed in the topology of GK and consequently it need not be closed in the Zariski topology either. Let B denote the Zariski-closure of H. Then BK is a Lie subgroup of GK containing H, and one may wonder about the possible difference between the two. We will answer this question in terms of the corresponding Lie algebras han and ban .

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Proposition 3.11 In the above notations, han is a Lie ideal of ban . PROOF: Consider the adjoint representation Ad : G → GL(g), where g = L(G) is the Lie algebra of G as an algebraic group; note that g = gan ⊗K , where gan is the Lie algebra of GK as an analytic group. The space han , and consequently the space h := han ⊗K , are clearly H-invariant. On the other hand, S := {g ∈ G : Ad(g)(h) = h} is a Zariski-closed subgroup of G, and so the inclusion H ⊂ S implies that B ⊂ S. Taking into account that the differential of the adjoint representation of an algebraic group is the adjoint representation of the corresponding Lie algebra (cf. Borel [1991, §3]), for b = L(B) we have [b, h] ⊂ h. Since ban = bK and han = hK , we see that [ban , han ] ⊂ han . We conclude our survey of the required results from Lie theory with the statement of a theorem originally proved by E. Cartan for K = R. Theorem 3.12 Let K be either the field of real numbers R or the field of p-adic numbers Qp . Then any closed subgroup of a Lie group over K is a Lie group. Every continuous homomorphism of Lie groups over K is analytic. PROOF: See Serre (1992, pp. 155–157). We close this section with a proposition that illustrates how the techniques of Lie groups and analytic manifolds can be used in group theory. Proposition 3.13 Let G ⊂ GLn be a reductive algebraic group defined over a non-Archimedean local field K. Then the group GO = G ∩ GLn (O) of points over the valuation ring O ⊂ K has finitely many conjugacy classes of finite subgroups. In particular, the number of conjugacy classes of finite subgroups in SLn (Zp ) is finite. PROOF: As we noted earlier, there exists a neighborhood of the identity in GO that does not contain nontrivial elements of finite order. Since the congruence subgroups GO (pα ) = {x ∈ GO : x ≡ In (mod pα )}, 2 2 Two matrices over a ring are congruent modulo its ideal if and only if all their respective entries

are congruent.

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where p ⊂ O is the valuation ideal and α ≥ 1, constitute a base of neighborhoods of the identity, there exists a congruence subgroup (say, GO (pα )) with this property. It follows that any finite subgroup of GO is isomorphic to a subgroup of the finite group GO /GO (pα ). Thus, GO contains only a finite number of non-isomorphic finite subgroups, and it suffices to show that the finite subgroups of GO that are isomorphic to a given finite group 0 split into a finite number of conjugacy classes. For this, we consider the representation variety R = R(0, G) (cf. §2.4.7). We will show that the set RO = Hom(0, GO ) consists of a finite number of orbits under the natural action of GO . It follows from Theorem 2.52 that there are finitely many G-orbits in R(0, G), and these orbits are Zariski-closed. So, it suffices to show that if X is one of those orbits, then XO consists of finitely many GO -orbits. This is obvious if XO = ∅. If XO 6= ∅, then X is clearly defined over K and for any point x ∈ XO , the orbit GO x is open in XO by Corollary 3.9. On the other hand, since X is closed in R and O is compact, the set XO is compact. Thus, the open covering [ XO = GO x x ∈ XO

must have a finite subcovering, yielding the finiteness of the number of GO -orbits in XO .

3.2 The Archimedean Case In the previous section, we discussed a number of elementary topological and analytic properties of the space XK , where X is an algebraic variety defined over a locally compact field K. The proofs therein relied only on the Inverse Function Theorem, which works in both the “classical” case of K = R or C as well as in the non-Archimedean case where K is a finite extension of Qp . In this section, we will present some results that are specific only to the Archimedean case. In this respect, the most salient statements are those treating the connectedness of XK . Theorem 3.14 Let X be an irreducible algebraic variety defined over C. Then the space XC is connected. PROOF: See Shafarevich (2013, Ch. 7, §2). (We will not use this result in our book, however.)

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Theorem 3.15 (WHITNEY [1957]) Let X be an algebraic variety defined over R. Then the space XR has finitely many connected components. PROOF: Replacing X by the Zariski-closure of XR (which has the same Rpoints), we may assume that XR is Zariski-dense in X . Then for any irreducible component X 0 of X , the intersection X 0 ∩ XR is Zariski-dense in X 0 , and therefore X 0 is R-defined. Thus, we may assume X to be irreducible. Suppose that the theorem is not valid, and let X be a counterexample of minimal dimension (clearly dim X > 0). Choose an R-defined open affine subset Y ⊂ X . Then T = X \ Y is an algebraic R-variety with dim T < dim X . By our assumption, TR has finitely many connected components, so YR must have infinitely many of these. So, we may assume that X is affine. Let S be the set of singular points of X (cf. §2.4.3). As we know, S is a proper R-defined Zariski-closed subset of X . Pick a nonzero regular function f ∈ R[X ] that vanishes on S, and let Y ⊂ X be the hypersurface given by the equation f = 0. Then X 0 = X \ Y is an R-defined affine variety. Moreover, since by our construction Y ⊃ S, the variety X 0 is smooth. At the same time, the fact that dim Y < dim X by our assumption implies that the space YR has finitely many connected components, which means that XR0 must have infinitely many of those. Thus, without any loss of generality, we may assume that X is smooth. Then, as we have shown in §3.1, the space V := XR is an analytic manifold and, in particular, a locally connected space. It follows that each of its infinitely many connected components Vj is a closed-and-open (i.e., clopen) subset of V . Suppose that we have been able to find a proper R-defined Zariski-closed S subset Z ⊂ X that intersects all of the Vj ’s. Then ZR = j (ZR ∩ Vj ), where each intersection ZR ∩ Vj is a nonempty clopen subset of ZR . This implies that ZR has infinitely many connected components, which would contradict our construction as dim Z < dim X . It remains to construct such a Z. Suppose X is realized as an R-defined Zariski-closed subset of the affine space An , and endow the corresponding set of real points Rn with the usual metric. Fix an arbitrary point a = (a1 , . . . , an ) in the component V1 . For each j, the subset Vj is closed in Rn , so we can find a point bj ∈ Vj that is the closest to a. We will now construct a proper R-defined Zariski-closed subset Z ⊂ X containing all the bj ’s, which will be as required. The equations for such a Z can be easily obtained from the fact that the points bj yield a local minimum for the function g(X1 , . . . , Xn ) = (X1 − a1 )2 + · · · + (Xn − an )2

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139

on V = XR . More precisely, if r = n − dim X and a is the ideal of polynomials that vanish on X , then for any f1 , . . . , fr ∈ aR and any j, the linear forms dbj f1 , . . . , dbj fr , dbj g (cf. §2.4.3) are linearly dependent, which is equivalent to the conditions   r+1 1i ( f1 , . . . , fr , g)(bj ) = 0, i = 1, . . . , , n where 1i ( f1 , . . . , fr , g)(x) runs through all the (r + 1) × (r + 1) minors of the matrix  ∂f  ∂f1 1 (x) · · · (x)  ∂X1  ∂Xn    ......................     ∂fr . ∂fr    ∂X1 (x) · · · ∂Xn (x)     ∂g  ∂g (x) · · · (x) ∂X1 ∂Xn Let Z be the subset of X given by 1i ( f1 , . . . , fr , g)(x) = 0 r+1
n and all f1 , . . . , fr

for all i = 1, . . . , ∈ a. By construction, Z contains all the bj , so it only remains to show that Z 6= X , for which we will show that in fact V1 ⊂ / Z. Since a is simple on X , there exist polynomials f1 , . . . , fr ∈ aR such that the forms dx f1 , . . . , dx fr are linearly independent at x = a (cf. Proposition 2.45). Then they remain linearly independent at all x that are sufficiently close to a. Furthermore, let (u1 (t1 , . . . , td ), . . . , un (t1 , . . . , td )), where d = dim X > 0, be an analytic parametrization of a neighborhood of a (cf. §3.1). Since g represents the squared distance from a, the analytic function ϕ(t1 , . . . , td ) = g(u1 (t1 , . . . , td ), . . . , un (t1 , . . . , td )) does not reduce to a constant. Therefore,   ∂ϕ ∂ϕ ,··· , ≡ 6 (0, . . . , 0) ∂t1 ∂td

(3.4)

on any open domain of parameters. The equations fi (u1 (t1 , . . . , td ), . . . , un (t1 , . . . , td )) = 0,

i = 1, . . . , r

yield  (dx fi )

∂u1 ∂un ,..., ∂tj ∂tj

 =

n X ∂fi ∂uk · =0 ∂Xk ∂tj

k=1

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for all i = 1, . . . , r and j = 1, . . . , d. Now, if x is sufficiently close to a, then the fact that the forms dx f1 , . . . , dx fr , dx g are linearly dependent really means that dx g is a linear combination of dx f1 , . . . , dx fr , and therefore the preceding equations imply that   n X ∂u1 ∂un ∂g ∂uk (dx g) ,..., = · = 0 for all j = 1, . . . , d. (3.5) ∂tj ∂tj ∂Xk ∂tj k=1

But the left-hand side of (3.5) equals ∂ϕ/∂tj , so (3.5) contradicts (3.4). We conclude that dx f1 , . . . , dx fr , dx g cannot be linearly dependent at all points x ∈ V1 that are sufficiently close to a. Therefore, all the determinants 1i ( f1 , . . . , fr , g)(x) do not vanish identically on V1 , hence V1⊂ / Z. Corollary 3.16 Let G be an algebraic R-group. Then GR has finitely many connected components; in other words, the connected component of the identity (GR )0 is a subgroup of finite index. For G connected, the group GR is connected whenever it is compact. Only the second assertion requires a proof. Being compact, GR consists entirely of semisimple elements, and therefore every element lies in a suitable R-defined torus T ⊂ G (cf. Borel [1991, Theorem 11.10]). The group TR is (1) also compact, so T is isomorphic to a torus of the form (RC/R (Gm ))d , where d = dim T (cf. §2.2.4). It follows that TR can be identified with the product of d copies of the unit circle, which is connected. Thus, the connected component (GR )0 must contain TR for any R-defined subtorus T of G, and therefore coincides with GR . An alternate proof of the connectedness of GR can be derived from the fact that a compact linear group over R is automatically Zariski-closed (cf. Chevalley [1946–55, vol. 3]). Proposition 3.17 Let G be a connected almost R-simple algebraic group. Then any noncentral normal subgroup of GR has finite index. If GR is compact, then it does not have any proper noncentral normal subgroups. PROOF: By Theorem 3.10, any noncentral normal subgroup N of GR is open and therefore must contain the connected component (GR )0 , which by Corollary 3.16 has finite index in GR . If GR is compact, then GR = (GR )0 , and therefore N = GR . We continue with the corollaries of Theorem 3.15. Corollary 3.18 Let G × X → X be a transitive R-defined action of an algebraic R-group G on an R-defined variety X . Then XR is the union of a finite number of GR -orbits. If XR is connected, then there is exactly one orbit.

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PROOF: For any point x ∈ XR , the orbit GR x is open in XR (Corollary 3.9). The complement of XR \ GR x is the union of the remaining orbits, and therefore is also open. Thus, the orbit GR x is a clopen subset of XR , and therefore contains a connected component of the latter. It follows that the number of distinct orbits does not exceed the number of connected components of XR , which is finite, and is equal to 1 if XR is connected. Remark Corollary 3.18 has a useful cohomological interpretation. More precisely, if x ∈ XR , then X can be identified with the homogeneous space G/H, where H = G(x) is the stabilizer of x, and then the GR -orbits on XR are in oneto-one correspondence with the elements of ker(H 1 (R, H) → H 1 (R, G)) (cf. §1.3.2). Thus, by Corollary 3.18, this kernel is finite. Considering an embedding of a given R-group H into some R-group G with trivial cohomology (for example, one can use a faithful R-defined representation H ,→ G = GLn ), we conclude that H 1 (R, H) is finite for any R-group H. We refer the reader to [AGNT, § 6.4] for an alternative proof of this fact that also works for non-Archimedean local fields. Corollary 3.19 Let f : G → H be a surjective R-defined morphism of algebraic R-groups. Then the index [HR : f (GR )] is finite. If HR is connected, in particular if H is unipotent, then fR : GR → HR is surjective. The proof follows from Corollary 3.18 applied to the action G × H → H, (g, h) 7→ f (g)h. The connectedness of the set of R-points of a unipotent group H is a consequence of the fact that the “truncated” logarithmic map defines a homeomorphism between HR and L(H)R , where L(H) is the Lie algebra of H (cf. §2.1.8). Next, we will develop the polar and Iwasawa decompositions for the groups of real and complex points of reductive algebraic groups and will use these decompositions to obtain more precise information about the algebraic and topological structure of such groups. To explain the nature of these decompositions, we begin by considering the simplest case of the group GLn , where the decompositions in question can be easily derived from well-known facts of linear algebra. Let K denote the subgroup of GLn (R) consisting of orthogonal matrices, i.e., of matrices x ∈ GLn (R) satisfying t

xx = In ,

tx

(3.6)

where denotes the matrix transpose of x. Clearly, K coincides with the group of R-points On ( f )R of the orthogonal group of the standard quadratic form f = x21 +· · ·+x2n . This form is anisotropic over R, and therefore the group On ( f ) is also R-anisotropic (cf. Proposition 2.33). Then it follows from Theorem 3.1

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that K = On ( f )R is compact. (Of course, the latter fact can be easily proved directly by writing out the relations on the matrix entries of x arising from (3.6) and showing that these define a closed and bounded subset of Mn (R).) Furthermore, let S denote the set of positive definite symmetric matrices in P GLn (R); thus, a = (aij ) ∈ S if aij = aji and the quadratic form f = ni,j = 1 aij xi xj is positive definite. With these notations, we have Proposition 3.20 GLn (R) = KS, and for any matrix its factorization on the right is unique. Furthermore, the set S is connected and simply connected. PROOF: Let x ∈ GLn (R). Then a = t xx ∈ S, hence the eigenvalues α1 , . . . , αn of a are real and positive. It is well known from linear algebra that there exists b ∈ K such that bab−1 is the diagonal matrix diag(α1 , . . . , αn ). Let c denote the matrix b−1 db, where √ √ d = diag( α1 , . . . , αn ) (positive square roots). Then c ∈ S and tcc = c2 = a. Thus a = txx = tcc, whence t

(xc−1 )(xc−1 ) = In

and z = xc−1 ∈ K. It follows that x = zc ∈ KS. If x = z1 c1 is another such factorization, then taking the matrix transpose of both sides of zc = z1 c1 ,

(3.7)

cz−1 = c1 z1−1 .

(3.8)

we obtain

Multiplying (3.7) and (3.8), we get c2 = c21 . To see that c = c1 , we will first prove the following statement, which will be used repeatedly in the sequel. Lemma 3.21 Let c ∈ S. Then for any integer r 6= 0, the Zariski closure of the cyclic subgroup generated by cr contains c. PROOF: As we have already noted, c is conjugate to a diagonal matrix, so we may assume from the outset that c is diagonal, c = diag(γ1 , . . . , γn ), with γi > 0. If c ∈ / {crn }n ∈ Z , then one can find a character χ of the group of diagonal matrices Dn such that χ(cr ) = 1, but χ(c) 6= 1 (cf. Borel [1991]). However, χ(c) = γ1a1 . . . γnan for suitable integers ai , and therefore χ(c) > 0. Since χ (cr ) = (χ(c))r = 1, it follows that χ (c) = 1, a contradiction.

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143

In view of Lemma 3.21, the fact that c2 = c21 implies that c and c1 commute, and therefore can be diagonalized simultaneously. Assuming that they are already diagonal, we see that d := cc−1 1 is a diagonal matrix with positive eigenvalues satisfying d 2 = In . So, d = In , hence c = c1 and z = z1 , proving the uniqueness of the factorization. It remains to show that S is connected and simply connected. For this, we will use a method that actually applies to an arbitrary reductive group, as we will see later on. More precisely, we will show that the exponential map induces a homeomorphism between the vector space s of real symmetric matrices and S. Indeed, it follows from (3.2) that for any X ∈ s, we have exp(X ) ∈ S. (It is well known that the series (3.2) converges everywhere.) Using (3.3) in conjunction with the fact that every element of S is conjugate to a diagonal matrix with positive eigenvalues, we see that the map exp : s → S is surjective. Furthermore, using the Inverse Function Theorem, it is easy to show that the exponential map actually yields a local analytic isomorphism between s and S. Thus, it remains to show that the restriction of exp to s is injective. m Arguing as above, one proves that if c1 , c2 ∈ S satisfy cm 1 = c2 for some nonzero integer m, then c1 = c2 . It follows that for X , Y ∈ s, the condition exp(X ) = exp(Y ) implies that     1 1 exp X = exp Y m m for any integer m > 0. Choosing m sufficiently large, we can make m1 X and 1 m Y arbitrarily close to 0. Then, since exp is a local homeomorphism, we obtain that m1 X = m1 Y for all such m, whence X = Y as required. Proposition 3.20 also has the following complex analog. Let B denote the subgroup of unitary matrices in GLn (C), which, as usual, are defined by the condition ∗

xx = In ,

where ∗x denotes the conjugate transpose of x. Writing this relation in terms of the matrix entries of x, we immediately see that B is compact. Let E be the set of positive definite Hermitian matrices, thus, a = (aij ) ∈ E if aij = a¯ ji (where P the bar denotes complex conjugation) and the Hermitian form f = aij x¯ i xj is positive definite. Then we have Proposition 3.22 GLn (C) = BE, and for any matrix, its factorization on the right is unique. The set E is connected and simply connected. The proof is similar to the proof of Proposition 3.20 and makes use of the following generalization of Lemma 3.21.

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Lemma 3.23 Let e ∈ E. Then for any integer r 6= 0, the Zariski closure of the subgroup generated by er contains e. The fact that E is connected and simply connected is established as follows. Let e denote the space of the Hermitian matrices in Mn (C). Then one shows that the exponential map yields a homeomorphism between e and E. The decompositions in Propositions 3.20 and 3.22 are called the polar decompositions. We now intend to construct an analog of this decomposition for an arbitrary reductive R-subgroup G ⊂ GLn (C). For this, we will need to “position” G inside GLn (C) in a certain special way. We recall that a subgroup G ⊂ GLn (C) is called self-adjoint if it is invariant under taking the matrix transpose, which means that x ∈ G implies that tx ∈ G. Theorem 3.24 (MOSTOW [1955]) Let G ⊂ GLn (C) be a reductive algebraic R-group. There exists a matrix a ∈ GLn (R) such that a−1 Ga is self-adjoint. The proof is based on the following result. Proposition 3.25 Let G ⊂ GLn (C) be a reductive algebraic R-group. Then there exists a Zariski-dense compact subgroup K ⊂ G that is invariant under complex conjugation. PROOF OF THEOREM 3.24: To derive the theorem from Proposition 3.25, we fix a compact Zariski-dense subgroup K ⊂ G invariant under complex conjugation. Set Z ∗ m= kkdk, (3.9) K

where the “matrix” integral is taken with respect to the Haar measure dk on K (cf. §3.5). Since K is invariant under complex conjugation, so is the measure dk, from which it follows that the matrix m is real. Moreover, for any k, the product ∗kk is a positive definite Hermitian matrix, so m is actually a positive definite symmetric matrix. As we have seen in the proof of Proposition 3.20, we can then write m = a2 for a suitable positive definite symmetric matrix a. It follows from (3.9) that any k ∈ K satisfies ∗kmk = m. Then any x ∈ a−1 Ka satisfies ∗xx = In ; in other words, a−1 Ka ⊂ B. Furthermore, for any such x, we have t

x = ∗x¯ = x¯ −1 ∈ a−1 Ka,

where the bar denotes complex conjugation. Thus, a−1 Ka is invariant under taking the matrix transpose, and consequently its Zariski closure a−1 Ga has the same property, in other words, is self-adjoint.

3.2 The Archimedean Case

145

PROOF OF PROPOSITION 3.25: The group G is an almost direct product of the maximal central R-defined subtorus T and a semisimple R-subgroup D. The existence of a required subgroup in T is established as follows. Pick a Cdefined isomorphism T ' C∗d , and let K be the subgroup of T corresponding under this isomorphism to S d , where S is the group of complex numbers having absolute value 1. Then K is the unique maximal compact subgroup of T. So, it is invariant under all continuous automorphisms of T, and therefore is as required. Now, if we can construct a subgroup K1 ⊂ D having the desired property, then the product K0 := KK1 will be a required subgroup of G. Thus, it remains to consider the case where G is semisimple. Pick a maximal R-torus T of G, and let R = R(T, G) be the root system of G relative to T. Furthermore, let {Xα }α ∈ R be the elements of the corresponding Chevalley basis in the Lie algebra L(G) (cf. §2.1.13). If we let σ denote the involution of L(G) given by complex conjugation, then σ (Xα ) = cα Xα¯ for some nonzero cα ∈ C, where α¯ is the character of T obtained by applying complex conjugation to α. Set τ (Xα ) = |cα |X−α , where |cα | is the absolute value of cα . Using the structural relations for the Chevalley basis listed in §2.1.13, one easily checks that τ extends to an involutive semi-automorphism (relative to the complex conjugation) of L(G) that commutes with σ (cf. Jacobson [1979, Chapter IV, §7]). Let f denote the Killing form on L(G) (cf. §2.1.3). A direct computation shows that f (X , τ (X )) < 0 for any X 6= 0 in L(G). Then the fixed subspace k of L(G) under τ is an R-subalgebra of L(G) such that the restriction of f to k is a negative definite symmetric bilinear form. Let K be the subgroup of those g ∈ G for which Ad g leaves k invariant. Since f is Ad G-invariant and k ⊗R C = L(G) (cf. the proof of Lemma 2.17), the group Ad K can be identified with a closed subgroup of O(g)R , the orthogonal group of the form g = f |k. But g is negative definite, so the group O(g)R is compact. So, K is also compact as Ad has finite kernel since G is semisimple. Furthermore, since k is a subalgebra, for any X ∈ k we have exp X ∈ K. Taking into account that d (exp(tX ))t = 0 = X , dt we see that the Lie algebra of K (as a Lie group, cf. Theorem 3.12) contains k. ¯ of Then it follows from Lemma 3.5 that the Lie algebra of the Zariski-closure K ¯ K as an algebraic group contains k ⊗R C = L(G), whence K = G. Finally, since σ and τ commute, the algebra k is σ -invariant, and therefore K is invariant under conjugation. Remark Proposition 3.25 lays the foundation for the technical tool generally known as Weyl’s unitary trick. More precisely, the analysis of some problems

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involving reductive algebraic groups over a field K of characteristic 0 can often be reduced to the case K = C, and then, instead of dealing with the reductive algebraic group itself, one may be able to work with a compact Zariski-dense subgroup of the group of complex points. For example, this approach easily yields the complete reducibility of representations of reductive algebraic groups in characteristic 0 (cf. Theorem 2.8(4)). We are now in a position to construct the polar decomposition for an arbitrary reductive R-group G. Using Theorem 3.24, we can realize G as a self-adjoint R-defined subgroup of some GLn (C). Then the polar decomposition of GLn (R) (Proposition 3.20) yields the polar decomposition of GR . More precisely, we have the following. Proposition 3.26 (1) Let G be self-adjoint, and let other notations be as in Proposition 3.20. Then GR = (G ∩ K)(G ∩ S). Furthermore, K1 := G ∩ K is a maximal compact subgroup of GR , and the set S1 := G ∩ S is connected and simply connected. Consequently, the quotient GR /K1 is connected and simply connected. (2) Any compact subgroup of GR is contained in a maximal compact subgroup, and all maximal compact subgroups of GR are conjugate. PROOF: (1): Let x ∈ GR , and let x = kc be the polar decomposition of x in GLn (R). We will show that k ∈ K1 and c ∈ S1 . Since G is self-adjoint, we have tx ∈ G, and consequently c2 = txx ∈ G. Applying Lemma 3.21, we obtain c ∈ G, and consequently c ∈ S1 . Then also k ∈ K1 . Any subgroup of GR strictly containing K1 must contain an element of S1 other than the identity. On the other hand, using the fact that any element of S is conjugate to a diagonal matrix with positive entries, one easily shows that an element of S different from the identity cannot be contained in a compact subgroup. Thus, K1 is a maximal compact subgroup. To prove that S1 is connected and simply connected, we will argue as in the proof of Proposition 3.20. Let s1 denote the subspace of symmetric matrices in the Lie algebra gan of GR . We will show that the exponential map induces a homeomorphism between s1 and S1 . In the proof of Proposition 3.20, we established that exp yields a homeomorphism between s to S. So, it suffices to show that exp(s1 ) = S1 . Obviously exp(s1 ) ⊂ S1 . Now, let c = exp(X ) ∈ S1 , where X ∈ s. It follows from Lemma 3.21 that exp( n1 X ) also lies in S1 , for any integer n. Thus, exp(QX ) ⊂ S1 and consequently exp(tX ) ∈ S1 for any t ∈ R. But then

3.2 The Archimedean Case

X=

147

d (exp(tX ))t = 0 ∈ gan ∩ s = s1 , dt

as required. Finally, using the uniqueness of the polar decomposition in GLn (R) and the fact that K1 is compact and S1 is closed in GR , it is easy to show (cf. the proof of Proposition 3.30) that the product map K1 × S1 → GR is a homeomorphism. So, GR /K1 is homeomorphic to S1 , hence connected and simply connected. (2): The proof, the details of which we omit (cf. Helgason [2001]), goes as follows. The space X = GR /K1 can be equipped with a GR -invariant metric so that it becomes a Riemannian manifold of negative curvature. According to a result of Cartan, any compact group of isometries of such a manifold has a fixed point. Thus, given any compact subgroup K0 ⊂ GR , there exists a point x = gK1 ∈ X that is fixed by K0 . This means that g−1 K0 g ⊂ K1 , yielding the desired result. We will also need a complex version of Proposition 3.26. Proposition 3.27 (1) Let G ⊂ GLn (C) be a reductive self-adjoint R-group, and let other notations be as in Proposition 3.22. Then GC = (G ∩ B)(G ∩ E). Furthermore B1 := G ∩ B is a maximal compact subgroup of GC , and the exponential map induces a homeomorphism between the space e1 of Hermitian matrices in the Lie algebra L(G) and E1 := G ∩ E. (2) Any compact subgroup of GC is contained in a maximal compact subgroup, and all maximal compact subgroups are conjugate. The proof is similar to the proof of Proposition 3.26. (In fact, Proposition 3.27 can be derived from Proposition 3.26 using restriction of scalars.) We mention one more useful technical fact. Lemma 3.28 If b ∈ B, e ∈ E are such that e−1 be ∈ B, then eb = be. PROOF: We have ∗x = x−1 for x ∈ B, and ∗x = x for x ∈ E. So, ∗ −1

(e

be)−1 = e−1 be = ebe−1 ,

implying that b and e2 commute. Our claim now follows from Lemma 3.23. After these preparations, we will now prove the following strengthening of Theorem 3.24.

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Theorem 3.29 (MOSTOW [1955]) Let G1 ⊂ · · · ⊂ Gr be a tower of reductive R-defined subgroups of GLn (C). Then there exists a matrix a ∈ GLn (R) such that all the groups aGi a−1 are self-adjoint. PROOF: Examining the proof of Theorem 3.24, we see that it suffices to find Zariski-dense maximal compact subgroups Ki ⊂ Gi that are invariant under complex conjugation, and satisfy K1 ⊂ · · · ⊂ Kr . This, in turn, reduces to proving the following for two R-defined reductive groups H ⊂ G: any maximal compact subgroup B ⊂ H that is invariant under complex conjugation is contained in a maximal compact subgroup C ⊂ G that is also invariant under complex conjugation. (We note that since by Proposition 3.25 a complex reductive group always contains a Zariski-dense maximal compact subgroup, it follows from Proposition 3.27(2) that any maximal compact subgroup is automatically Zariski-dense.) Pick a maximal compact subgroup D ⊂ G containing B. It follows from Proposition 3.27 that D = gK1 g−1 for some t ∈ G, where K1 is the maximal compact subgroup of all unitary matrices in G. Set f = Ad(g)(e1 ), where e1 ⊂ L(G) is the R-subspace of Hermitian matrices. Then for F := exp(f), the product map D × F → G is a bijection. Let θ denote the complex conjugation on G. Then θ(D) is also a maximal compact subgroup of G and therefore θ(D) = a−1 Da for a suitable a = exp(X ), where X ∈ f. Set b = exp(X /2) so that b2 = a. We claim that the maximal compact subgroup C = b−1 Db is as required. Indeed, we have θ(C) = θ(b)−1 θ(D)θ(b) = θ(b)−1 a−1 Daθ (b) = (θ(b)−1 a−1 b)C(b−1 aθ(b)). Thus, to prove that C is θ-invariant, it suffices to show that b−1 aθ(b) lies in the center Z of G, or equivalently, bθ(b)−1 = az with z ∈ Z.

(3.10)

Since θ 2 = id, we have D = θ 2 (D) = θ(a−1 Da) = θ(a)−1 a−1 Daθ(a). Write aθ(a) = df with d ∈ D and f ∈ F. Then f −1 Df = D, so it follows from Lemma 3.28 that f centralizes D. Since D is Zariski-dense in G, we conclude that f ∈ Z. Define an automorphism α of G by α(x) = aθ(x)a−1 . Then D is αinvariant, implying that F is also α-invariant. (Indeed, it is enough to prove that f is invariant under the differential of α. For this, one observes that f is precisely the orthogonal complement of L(D)an in L(G)an , the Lie algebras of

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149

D and G viewed as real Lie groups, under the Killing form, which is invariant under all automorphisms of G.) So, aθ (a)a−1 ∈ F. But aθ (a)a−1 = dfa−1 , and if f = exp(Y ) with Y ∈ f, then fa−1 = exp(Y − X ) ∈ F since f ∈ Z. It follows that d = 1 so aθ(a) = f . We now compute θ(b). Set t = aθ(b)a−1 ∈ F, and write t = exp(T) with T ∈ f. Then t2 = aθ(a)a−1 = exp(2T) = fa−1 = exp(Y − X ), whence T = (Y − X )/2. So, T commutes with X , and therefore t commutes with a and θ(b) = t = exp((Y − X )/2). It follows that b commutes with θ(b) and       X Y −X Y Y −1 bθ (b) = exp − = exp X − = a exp − . 2 2 2 2 So, for the element z in (3.10), we have z = exp(−Y /2) ∈ Z, as required. It remains to show that B ⊂ C. We have B = θ(B) ⊂ θ(D) = a−1 Da, whence aBa−1 ⊂ D. So, by Lemma 3.28, the element a centralizes B. Since b2 = a, invoking Lemma 3.23 we see that b also centralizes B. Thus, B = b−1 Bb ⊂ C. Our final topic in this section is the Iwasawa decomposition, which is crucial for reduction theory (cf. Chapter 4). In our current context, it can be viewed from the following perspective. Let G be a reductive R-group, and let H ⊂ G be a maximal connected solvable R-split subgroup. Then by Theorem 3.1, the quotient GR /HR is compact, so coset representatives can be chosen from a suitable compact subset of GR . The Iwasawa decomposition asserts that coset representatives can in fact be found in a suitable maximal compact subgroup. Moreover, this compact subgroup is precisely a transversal in GR for the connected component of HR . Thus, GR is a product of a compact subgroup and a solvable subgroup. As usual, we will first develop the Iwasawa decomposition in the group GLn (R). To formulate the result, we let K (resp., A and U) denote the subgroup of GLn (R) consisting of the matrices that are orthogonal (resp., diagonal with positive entries and upper triangular unipotent). Proposition 3.30 (IWASAWA map

DECOMPOSITION FOR

GLn (R)) The product

ϕ : K × A × U → GLn (R), ϕ(k, a, u) = kau, is a homeomorphism.

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PROOF: Fix an orthonormal basis e = {e1 , . . . , en } of the space Rn , and let g ∈ GLn (R). Applying the classical Gram–Schmidt orthogonalization process, we obtain an orthonormal basis d = {d1 , . . . , dn } of Rn such that d1 = β11 ge1 , d2 = β12 ge1 + β22 ge2 , . . . , dn = β1n ge1 + · · · + βnn gen with βii > 0. Let  β11  .. b=  . 0

... .. . ...

 β1n ..  ∈ B := AU, .  βnn

and let k be the change of basis matrix from e to d. Then g = kb−1 , where the matrix k is orthogonal, i.e., k ∈ K, while b−1 ∈ B, which shows that ϕ is surjective. Using the fact that K is defined by the matrix equation txx = In , one easily checks that K ∩ B = {In }. Since the group B is the semidirect product of A and U (as topological groups), we conclude that ϕ is a continuous bijection. The continuity of the inverse map easily follows from the compactness of K. Indeed, suppose that gm = km am um −→ g = kau as m → ∞, where k, km ∈ K, a, am ∈ A, and u, um ∈ U. By the compactness of K we may assume that km −→ k 0 ∈ K. Then bm = am um −→ b0 = a0 u0 ∈ B since B is closed. In these notations, we have g = k 0 a0 u0 , and by uniqueness, we see that k 0 = k, a0 = a, u0 = u. Thus, km −→ k, am −→ a, and um −→ u as m → ∞, proving the continuity of ϕ −1 . The presentation of an element g in GLn (R) in the form g = kg ag ug , where kg ∈ K, ag ∈ A, and ug ∈ U, is called the Iwasawa decomposition, and the elements kg , ag and ug are called the K-, A-, and U-components of g, respectively. Our objective is to construct a similar decomposition for an arbitrary reductive R-subgroup G ⊂ GLn (C). More precisely, we will show that by replacing G with a conjugate group, we can ensure that the components of the Iwasawa decomposition in GLn (R) of an arbitrary g ∈ GR actually lie in GR itself. By Theorem 3.24, we may assume from the outset that G is self-adjoint. Then L(G) is invariant under the matrix transpose as well. Let k (respectively, p) denote the subspace of skew-symmetric (resp., symmetric) matrices in g := L(G)R ; clearly g = k ⊕ p. (This is the infinitesimal analog of the polar

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151

decomposition described in Proposition 3.26.) Let a denote a maximal abelian subalgebra of p. Lemma 3.31 There exists an R-split torus T ⊂ G that consists of symmetric matrices and satisfies a = L(T)R . PROOF: Let T denote the connected component of the Zariski-closure of the set exp(a). Since any element of p is diagonalizable over R and a is an abelian subalgebra of g, we see a is diagonalizable over R. It follows that T is an Rsplit subtorus of G. By our construction, a consists of symmetric matrices, so the same is true for exp(a) and T. Thus, L(T)R is contained in p and at the same time centralizes a, so in fact L(T)R = a. Since TR consists of pairwise-commuting symmetric matrices, there exists b ∈ K such that bTb−1 is contained in the group Dn of diagonal matrices. Taking into account that tb = b−1 , we see that the group bGb−1 is still selfadjoint, so replacing G with the latter, we may assume that T ⊂ Dn . Let R = {α} denote the set of nonzero weights of T in the adjoint representation on L(G). Since T is R-split, all α ∈ R are defined over R, so that we obtain the following R-defined decomposition, ! M T L(T) ⊕ uα , (3.11) α∈R

L(G)T

where is the centralizer of T and uα is the eigenspace of T corresponding to the weight α. Choose an ordering on V = X(T) ⊗Z R, where X(T) is the character group of T. It is easy to see that there exists an ordering on V0 = X(Dn ) ⊗Z R such that the natural projection V0 → V , obtained from the homomorphism X(Dn ) → X(T) that corresponds to the embedding T ,→ Dn , takes positive elements to nonnegative elements. Let R0 be the root system of GLn (C) relative to Dn ; recall that R0 = {εi − εj : i, j = 1, . . . , n; i 6= j}, where εi (diag(a1 , . . . , an )) = ai , and let 5 ⊂ R0 be the system of simple roots that corresponds to the chosen ordering on V0 (cf. Bourbaki [2002, Chapter 6]). It is well known that the Weyl group W (R0 ) contains an element w such that w5 coincides with the standard system of simple roots 50 = {εi − εi+1 : i = 1, . . . , n − 1}. But W (R0 ) can be identified with the subgroup W ⊂ GLn (C) of permutation matrices. Thus, there exists c ∈ W for which the torus T 0 = cTc−1 possesses the following property: one can find an ordering on X(T 0 ) ⊗Z R such that the

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Algebraic Groups over Locally Compact Fields

positive roots of GLn (C) with respect to 50 restrict to nonnegative elements of that ordering. Replacing G by cGc−1 (which remains self-adjoint as W ⊂ K), we may assume that such an ordering (which we fix) can be found already on V . Let R+ be the set ofX weights in R that are positive relative to the chosen ordering on V . Set u = uα . α∈R+

Lemma 3.32 u is an R-defined Lie subalgebra of L(G) normalized by T. Furthermore, u is contained in the algebra un of upper triangular nilpotent matrices. PROOF: Clearly, T normalizes u. For α, β ∈ R+ , the commutator [uα , uβ ] is either 0 (if α + β ∈ / R+ ) or is contained in uα+β (if α + β ∈ R+ ), implying that u is a Lie subalgebra. Since the subspaces uα are defined over R for all α ∈ R+ , their sum u also is defined over R. To prove the inclusion u ⊂ un , it suffices to show that uα ⊂ un for each α ∈ R+ . Let X = (xij ) ∈ uα , and suppose that xij 6= 0 for some i ≥ j. Then the restrictions of the character εi − εj ∈ X(Dn ) to T must coincide with α. Since εi − εj is negative with respect to the ordering on V0 associated with 50 , it would follow from our construction that α is negative relative to the chosen ordering on V . A contradiction, proving the lemma. A straightforward argument (cf. Borel [1991, §7]) shows that there exists a unipotent R-subgroup U ⊂ G whose Lie algebra is u. (In fact, U = exp(u), where exp is the “truncated” exponential map from §2.1.8, and then UR = exp(uR ).) Clearly, U is normalized by T and is contained in the group Un of upper triangular unipotent matrices. Let A1 denote the connected component of TR , and set K1 = G ∩ K. Theorem 3.33 (IWASAWA DECOMPOSITION) The product map θ : K1 × A1 × UR → GR is a homeomorphism. PROOF: We will first establish the infinitesimal analog of the Iwasawa decomposition: g = k ⊕ a ⊕ uR

(3.12)

in the notations introduced earlier. Let τ denote the automorphism of gln given by τ (X ) = − tX for X ∈ gln . By our construction, τ induces an automorphism of g, the subalgebra gτ of fixed points coincides with k, and we have τ (X ) = − X for all X ∈ a. It follows

3.2 The Archimedean Case

153

that τ (uα ) = u−α for any α ∈ R. So, given X ∈ (u−α )R , we can write X = τ (Y ) with Y ∈ (uα )R , and then X = (τ (Y ) + Y ) − Y ∈ k ⊕ uR . In view of (3.11), we see that to prove g = k + a + uR ,

(3.13)

we only need to establish that the right-hand side of (3.13) contains the subalgebra c := L(G)TR , which is the centralizer of a in g. Since a is invariant under τ , so is c. Any X ∈ c can be written as X=

X + τ (X ) X − τ (X ) + . 2 2

Clearly, (X + τ (X ))/2 ∈ k. On the other hand, (X − τ (X ))/2 ∈ p centralizes a, hence actually belongs to it. So, X ∈ k ⊕ a, proving (3.13). If for X ∈ k, Y ∈ a, and Z ∈ uR we have X + Y + Z = 0, then applying τ , we obtain X − Y + τ (Z) = 0, whence !

−Z + τ (Z) Y= ∈a ∩ 2

X

uα

= {0}.

α∈R

Then  Z =τZ ∈ 

 X

α ∈ R+



uα  ∩ 

 X

u−α  = {0},

α ∈ R+

proving (3.12). By the Inverse Function Theorem (see Theorem 3.2), the decomposition (3.12) implies the existence of connected neighborhoods of the identity V ⊂ K1 and W ⊂ B := A1 UR such that the product map induces a homeomorphism of V × W to a neighborhood of the identity in GR . We can find connected neighborhoods of the identity V1 ⊂ V , W1 ⊂ W and continuous functions ϕ : V1 × W1 → K1 , ψ : V1 × W1 → B such that bk = ϕ(k, b)ψ(k, b) for all k ∈ V1 and b ∈ W1 . Using induction, it is easy to show that given any finite ordered collection F = {k1 , . . . , kp } of elements ki ∈ V1 , we can find a connected neighborhood of the identity W ( F) ⊂ W1 and continuous functions ϕ F : V1 × W ( F) → K1 and ψ F : V1 × W ( F) → B,

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Algebraic Groups over Locally Compact Fields

such that b5( F) = ϕ F (k, b)ψ F (k, b) for all k ∈ V1 and b ∈ W ( F), where 5( F) = k1 · · · kp . The set V1 generates the connected component K01 of K1 , so using the compactness of K one can find a finite family of the ordered collections F as described earlier S T such that K01 = F 5( F)V1 . Set W2 = F W ( F); then W2 K01 ⊂ K01 B. Being a connected group, B is generated by W2 , and therefore BK01 = K01 B. But K01 B contains a neighborhood of the identity in GR , and therefore generates the connected component (GR )0 . It follows that (GR )0 = K01 B. To prove that 0 = G since according to PropGR = K1 B, we only need to observe that K1 GR R osition 3.26, we have GR = K1 S1 , where S1 is a connected set. This shows that θ is surjective. It follows from our construction that a presentation of g ∈ GR in the form g = kau, where k ∈ K1 , a ∈ A1 , and u ∈ UR , is actually its Iwasawa decomposition in GLn (R). The uniqueness of the latter then implies that θ is bijective. The continuity of the inverse map is established as in the proof of Proposition 3.30. One easily derives from Theorem 3.33 that H = TU is a maximal connected R-split solvable subgroup of G (and, consequently, T is a maximal R-split torus and U is a maximal unipotent R-subgroup). Indeed, for any R-subgroup H 0 ⊃ H, we have HR0 = (H 0 ∩ K1 )HR , so the quotient HR0 /HR must be compact. On the other hand, as we have seen in the proof of Theorem 3.1, this cannot be the case for a connected R-split solvable subgroup H 0 strictly containing H. As another consequence (of the proof of the theorem as well as of the constructions preceding its statement), we note the following. Proposition 3.34 Let G ⊂ GLn (C) be a reductive R-group. There exists a ∈ GLn (R) such that H = aGa−1 has the following properties: (1) H is self-adjoint; (2) the connected component of the intersection of H with Dn is a maximal R-split torus S of H; (3) there exists an ordering on V = X(S) ⊗Z R such that the restriction of the positive roots εi − εj (1 ≤ i < j ≤ n) of GLn (C) to S are nonnegative with respect to this ordering, and the maximal unipotent R-subgroup corresponding to this ordering lies in the group of upper triangular unipotent matrices Un ; (4) the components of the Iwasawa decomposition in GLn (R) of any element of HR lie in HR .

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155

3.3 The Non-Archimedean Case Unless stated otherwise, throughout this section K will denote a nonArchimedean locally compact field of characteristic 0, i.e., a finite extension of the p-adic field Qp . As we have noted in §3.1, for an algebraic K-group G, the group of K-rational points GK is locally compact and totally disconnected in the p-adic topology. It is well known (cf. Bourbaki (1998b, Chapter 3, §4)) that any topological group with these properties has a base of neighborhoods of the identity consisting of subgroups. In the case at hand, one can provide such a base explicitly using congruence subgroups. More precisely, fix a matrix realization G ⊂ GLn (), and let O and p be the valuation ring and the valuation ideal of the natural valuation v on K, respectively. Then the group of O-points GO := G ∩ GLn (O) is the “principal” open compact subgroup of GK . (Its openness is a consequence of the openness of O in K, and its compactness follows from the compactness of GLn (O) since GO is closed in GLn (O).) The congruence subgroups GO (pd ) ⊂ GO , given by GO (pd ) = {g ∈ GO : g ≡ In (mod pd )}, d > 0, constitute a required base of neighborhoods of the identity in GK . It should be noted that Lie theory is less effective in the non-Archimedean setting than in the Archimedean case, so we will have to develop some other tools in order to analyze the structure of GK . Several important results, dealing mostly with the compact subgroups of GK , can be obtained by using facts about lattices and orders in semisimple algebras (cf. §1.5.3). Given an algebraic K-group G ⊂ GLn () and an O-lattice L will denote the stabilizer of L in G, more L ⊂ K n , throughout the book, GO precisely, L GO = {g ∈ GK : g(L) = L}. L consists of transformations in G whose (This notation reflects the fact that GO K matrix with respect to an O-basis of L belongs to GLn (O), which also implies L is an open compact subgroup of G , for any lattice L.) that GO K By Proposition 1.42, any compact subgroup B of GLn (K) is contained in the L of some lattice L ⊂ K n ; in particular, any compact subgroup of stabilizer GO GK is contained in some open compact subgroup. Moreover, if B is a maximal L for some L. Another consequence of Proposicompact subgroup, then B = GO tion 1.42 is that in the case G = GLn , any compact subgroup of GK is contained in some maximal compact subgroup, and all maximal compact subgroups are conjugate, just as in the Archimedean case. In this regard, it is somewhat surprising that this analogy breaks down already if we pass from G = GLn even to H = SLn .

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L Proposition 3.35 Let H = SLn . Then for any lattice L ⊂ K n , the stabilizer HO is a maximal compact subgroup of HK . Any compact subgroup of HK is contained in some maximal compact subgroup, and there are exactly n conjugacy classes of maximal compact subgroups of HK .

PROOF: Minor modifications in the argument used to prove Propositions 1.41 and 1.42 enable one to establish the following two facts: L is a maximal compact subgroup of (a) for any lattice L ⊂ K n , the stabilizer HO HK ; L = H M implies that L and M are (b) for any two lattices L, M ⊂ K n , HO O proportional. (We leave the details to the reader.) Thus, it only remains to prove the last assertion of the proposition, for which we will construct a surjection

ϕ : B → K ∗ /UK ∗n , where B is the set of all maximal compact subgroups of the group HK = SLn (K), and U is the group of v-adic units in K, such that the fibers of ϕ coincide with the conjugacy classes of maximal compact subgroups. Since the order of K ∗ /UK ∗n is n, the desired result will follow. To construct such a ϕ, we need to fix a lattice L ⊂ K n . Given any B ∈ B, we M for some lattice M ⊂ K n , and then using the fact that L and can write B = HO M are free O-modules of rank n, we can find g ∈ GLn (K) such that M = g(L). We define ϕ as follows: ϕ(B) = (det g)UK ∗n . M1 M2 According to (b), the equality HO = HO implies that M2 = tM1 for some L ' GL (O). Taking t ∈ K ∗ , and consequently if Mi = gi (L), then tg2−1 g1 ∈ GO n the determinant, we see that ϕ is well defined. Clearly, ϕ is also surjective. Mi Now assume that ϕ(B1 ) = ϕ(B2 ), where Bi = HO , Mi ⊂ K n . It follows from the definition of ϕ that in this case, M2 = g(M1 ) for some g ∈ GLn (K) such M1 that det g ∈ UK ∗n , i.e., det g = utn for some u ∈ U, t ∈ K ∗ . Choose s ∈ GO so that det s = u, and set h = t−1 gs−1 . Then h ∈ SLn (K) and h(M1 ) = t−1 M2 , so that h(M1 )

hB1 h−1 = HO

M2 = HO = B2 ,

Mi i.e., B1 and B2 are conjugate in HK . Conversely, if Bi = HO , where i = 1, 2, are conjugate, i.e., B2 = hB1 h−1 with h ∈ SLn (K), then it follows from (b) that the lattices h(M1 ) and M2 are proportional, easily yielding that ϕ(B1 ) = ϕ(B2 ).

The detailed analysis of the properties of maximal compact subgroups in the non-Archimedean case, due to Bruhat and Tits (1972, 1984a), shows that

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157

the last assertion of Proposition 3.35 is a special case of the following general result: if G is a simply connected absolutely almost simple K-group of Krank `, then GK has exactly (` + 1) conjugacy classes of maximal compact subgroups (see Theorem 3.54). We will give an account of some results of Bruhat–Tits theory in the next section. Some initial facts, however, such as that if G is reductive, then any compact subgroup of GK is contained in a maximal compact subgroup, can be established by relatively elementary techniques, as we will see shortly. First, we would like to point out that the assumption that G is reductive cannot be omitted. Proposition 3.36 If GK contains a maximal compact subgroup, then G is reductive. PROOF: Consider the Levi decomposition G = HU of G, where U = Ru (G) is the unipotent radical of G and H is reductive (cf. §2.1.9). Assume that U 6= {1}. Then the center Z(U) is also nontrivial, and the “truncated” logarithmic map induces a K-defined isomorphism of algebraic groups ϕ : Z(U) → V , where V = L(Z(U)) is the corresponding Lie algebra (note that dim V > 0). Since U is a normal subgroup of G, the center Z(U) is also a normal subgroup, and we have ϕ(g−1 zg) = (Adg)ϕ(z) for any g ∈ G, z ∈ U.

(3.14)

Let ρ : G → GL(V ) be the representation arising from the adjoint action of G on V . Given a maximal compact subgroup B ⊂ GK , by Proposition 1.42, we can find a lattice L ⊂ VK that is invariant under ρ(B). Set Zi = ϕ −1 (π −i L), where π ∈ K is a uniformizer. Clearly the Zi ’s are compact subgroups of Z(U)K whose union coincides with Z(U)K . Furthermore, it follows from (3.14) that B normalizes each Zi , so the product BZi is a compact subgroup. By the maximality of B, we get B = BZi for all i. But then Z(U)K ⊂ B, a contradiction since Z(U)K is noncompact. We will now show that if G is reductive, then GK does contain maximal compact subgroups, and in fact every compact subgroup of GK is contained in some maximal compact subgroup. Proposition 3.37 Let G be a reductive K-group. Then (1) the number of compact subgroups of GK containing a given open compact subgroup is finite; (2) any compact subgroup of GK is contained in some maximal compact subgroup.

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Moreover, if G is semisimple, then the normalizer in GK of any open compact subgroup is compact. PROOF: Let G ⊂ GLn (). Using the embedding GLn () ,→ GLn+1 (), g →



g 0

0 det−1 g

 ,

we may assume G to be Zariski-closed in Mn (). Let A denote the -span [G] of G in Mn () (i.e., the set of all -linear combinations of the elements of G), and B the K-span K[GK ] of GK in Mn (K). Since G is reductive, it follows from Theorem 2.8 that A and B are semisimple algebras over  and K, respectively. Now, fix an open compact subgroup U ⊂ GK . By Lemma 3.8, U is Zariski-dense in G, implying that [U] = A and K[U] = B. It follows that P := O[U] is an order in B. By Theorem 1.43, P is contained in a finite number of maximal orders P1 , . . . , Pr . For each i, the intersection Pi ∩ G is obviously compact and closed under multiplication, so Ui := (Pi ∩ G) ∩ (Pi ∩ G)−1 is a compact subgroup of GK . Now let W ⊂ GK be a compact subgroup containing U. Then O[W ] is an order in B containing P and therefore O[W ] ⊂ Pi for some i. Consequently, W ⊂ Pi ∩ G and W = W −1 ⊂ (Pi ∩ G)−1 , so that W ⊂ Ui . We have shown that any compact subgroup containing U must be contained in one of the groups Ui . This already yields assertion (1) of the proposition as the index [Ui : U] is finite due to the openness of U, hence the number of intermediate subgroups between U and Ui is also finite. As we mentioned earlier, any compact subgroup of GK is contained in some open compact subgroup, so assertion (1) immediately implies assertion (2). Suppose now that G is semisimple. Consider the adjoint action of G on A, and let ϕ : G → Aut A be the corresponding adjoint representation given by ϕ : g 7→ ig where ig (x) = gxg−1 . Clearly ker ϕ coincides with the center of G, hence is finite. Let U ⊂ GK be an arbitrary open compact subgroup, and let N denote its normalizer in GK . As we have established above, P = O[U] is an order in B = K[GK ], so any O-basis x1 , . . . , xm of P is also an -basis of A. For any g ∈ N, we have g−1 Pg = P, so the matrix of the transformation ϕ(g) with respect to the basis x1 , . . . , xm has all entries in O. Thus, ϕ(N) ⊂ ϕ(G)O . Since ϕ(G)O is compact and ker ϕ is

3.3 The Non-Archimedean Case

159

finite, we conclude that ϕ −1 (ϕ(G)O ) is compact, and consequently N is relatively compact. On the other hand, since U is closed, N is also closed, hence compact. We will return to the properties of maximal compact subgroups in the next section, turning our attention now to some structural results about GK that can be derived from Proposition 3.37. Our arguments will be based on the theory of profinite groups. Since profinite groups will be used repeatedly in this book, we will briefly review their definition and basic properties (a more detailed exposition can be found in Ribes and Zalesskii [2000], Serre [1997], or Wilson [1998]). Let I be a directed set, i.e., a set with a partial order ≤ such that for any i, j ∈ I, there exists k ∈ I satisfying i ≤ k and j ≤ k. (In our discussion, I will typically be the set N of positive integers with its natural ordering.) A j projective (or inverse) system G = (Gi , ϕi ) over I consists of a family of objects (sets, groups, rings, etc.) Gi , indexed by the elements of I, and a family of j morphisms ϕi : Gj → Gi , one for each pair i, j ∈ I satisfying j ≥ i, such that ϕii j is the identity morphism for all i ∈ I and ϕik = ϕi ◦ ϕjk whenever k ≥ j ≥ i. The j

projective (or inverse) limit lim Gi (more precisely, lim(Gi , ϕi )) is the subset of ← ← Q j G consisting of those g = (g ) that satisfy ϕ (g i i∈I i i j ) = gi for all j ≥ i in I. It is easy to see that if all the Gi possess a certain type of algebraic structure (group, j ring, etc.) and the ϕi respect this structure (i.e., are homomorphisms), then G = lim Gi inherits the same type of structure. Furthermore, if all the Gi are ← j Hausdorff topological spaces and the ϕi are continuous maps, then G = lim Gi ← Q is closed in i ∈ I Gi in the product (Tychonoff) topology. We will now specialize to the situation where the Gi are finite groups j endowed with the discrete topology, and the ϕi are group homomorphisms (automatically continuous). Then G = lim Gi is called a profinite group. Being Q← a closed subgroup of the product i∈I Gi , which is a compact topological group for the product topology, G itself is a compact topological group. Furthermore, G is totally disconnected as the kernels of the restriction π` = p` |G Q of canonical projections p` : i ∈ I Gi → G` form a fundamental system of neighborhoods of the identity consisting of subgroups. Conversely, any compact totally disconnected topological group G is profinite, i.e., can be written as a projective limit of finite groups. In fact, such a presentation can be produced given any fundamental system {Ni }i ∈ I of neighborhoods of the identity consisting of open normal subgroups. (It can be shown that such a system does

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exist in any compact totally disconnected group – cf. Koch [1970, §1.2].) More precisely, then each quotient G/Ni is finite and the natural homomorphism G → lim G/Ni , ←

g 7→ (gNi )i ∈ I

is an isomorphism of topological groups. Let us apply this fact to the group of points GO over the valuation ring O ⊂ K of an algebraic K-group G. Since the congruence subgroups GO (pd ) for d > 0 (where p ⊂ O is the valuation ideal) are normal in GO and constitute a base of neighborhoods of the identity, we have GO ' lim GO /GO (pd ). ←

(3.15)

This presentation enables one to draw some conclusions about the structure of GO . Recall that the projective limit of a projective system of finite p-groups is called a pro-p group. Lemma 3.38 GO (p) is a pro-p group. PROOF: The prime p is determined by the either of the equivalent conditions: Qp ⊂ K or p ∈ p. If G = lim G/N is a presentation of a profinite group G as the ← projective limit of its finite quotients, then for any closed subgroup H ⊂ G we have H = lim H/(H ∩ N) ← (cf. [ANT, Chapter V, 1.4, Corollary 2]). Applying this to the subgroup GO (p), we obtain GO (p) ' lim GO (p)/GO (pd ). ← We will now show that GO (p)/GO (pd ) is a p-group. It suffices to show that GO (pd )/GO (pd+1 ) is a p-group for any d ≥ 1. Let x ∈ GO (pd ). Write x = In +y with y ≡ 0(mod pd ). Then     p p p x = In + y + ··· + y p−1 + y p , 1 p−1
where the pi
are the binomial coefficients. For 0 < i < p, the binomial coefficient pi is divisible by p, hence   p i y ≡ 0(mod pd+1 ). i Since clearly y p ≡ 0(mod pd+1 ), we obtain that xp ≡ In (mod pd+1 ). Consequently, the order of any element of GO (pd )/GO (pd+1 ) divides p.

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Corollary 3.39 The order of any element of GO (p) is either infinite or a power of p. PROOF: It is easy to see that any closed subgroup of a pro-p group is again a pro-p group. Therefore if the order of x ∈ GO (p) is finite, the cyclic subgroup H = hxi must be a finite pro-p group, in other words, a usual p-group. It follows that GO is a finite extension of the pro-p group GO (p). In the theory of profinite groups, pro-q subgroups (where q is an arbitrary prime) serve as the profinite analogs of q-subgroups in the theory of finite groups and actually retain many properties of the latter. In particular, any pro-q subgroup is contained in a maximal (Sylow) pro-q subgroup, and any two maximal pro-q subgroups are conjugate (cf. Serre [1968]). In the situation at hand, pro-p subgroups play a special role as their properties yield, in fact, important results about the structure of GK . Our immediate goal is to establish the conjugacy of maximal pro-p subgroups in GK . (In view of the existence of nonconjugate maximal compact subgroups, this result is by no means obvious.) Theorem 3.40 (MATSUMOTO [1966]) Let G be a semisimple algebraic Kgroup, and let H be an open subgroup of GK . Then H contains a maximal open pro-p subgroup S, and any pro-p subgroup of H is contained in an H-conjugate of S. PROOF: Being an open subgroup of GK , the group H contains a suitable congruence subgroup GO (pd ), which, by Lemma 3.38, is a pro-p group. Applying Proposition 3.37(1), we conclude that GO (pd ) is contained in a maximal pro-p subgroup S ⊂ H. Next, let T ⊂ H be an arbitrary pro-p subgroup. We will first show that T is contained in a Sylow pro-p subgroup of H. Invoking Proposition 3.37(1) one more time, we see that it suffices to find an open pro-p subgroup containing T. For this, we note that since T is compact, the index [T : T ∩ S] is finite, and therefore there are only finitely many distinct conjugates t−1 St for t ∈ T. It follows that \ S0 := (t−1 St) t∈T

is open and is normalized by T, so T0 := TS0 is a required open subgroup. Thus, it is enough to prove the conjugacy statement assuming T to be a Sylow pro-p subgroup of H. For technical reasons, it is more convenient for us to prove the following stronger result: there exists x ∈ H such that xTx−1 = S and [S : S ∩ T] = [S : x(S ∩ T)x−1 ].

(3.16)

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(Actually, it follows from the unimodularity of GK (cf. §3.5) that the two indices in (3.16) are automatically equal.) We proceed by induction on n = [S : S ∩ T]. If n = 1, then S = T, and there is nothing to prove. Now, assume that n > 1. Let N denote the normalizer of S ∩ T in H. By Proposition 3.37, N is a compact subgroup of H, so the index [N : S ∩ T] is finite. First, we will show that neither N1 := N ∩ S = NS (S ∩ T) nor N2 := N ∩ T = NT (S ∩ T) reduces to S ∩ T. Since S ∩ T is a proper subgroup of both S and T, the required result follows from the following. Lemma 3.41 Let P be a pro-p group, and let Q be a proper open subgroup of P. Then the normalizer NP (Q) properly contains Q. PROOF: The assertion is well known for finite p-groups. To reduce to the finite case, set \ F= (g−1 Qg). g∈P

Then F is an open normal subgroup of P contained in Q. Clearly NP (Q) = π −1 (NP/F (Q/F)), where π : P → P/F is the canonical homomorphism. But the quotient P/F is finite, so NP/F (Q/F) 6= Q/F and therefore NP (Q) 6= Q. Continuing the proof of Theorem 3.40, we consider the finite group ¯ N = N/S ∩ T, and let ϕ : N → N¯ denote the canonical homomorphism. Then ¯ so by the classical Sylow the images ϕ(N1 ) and ϕ(N2 ) are p-subgroups of N, theorems, there exists a Sylow p-subgroup P of N¯ that contains ϕ(N1 ) and ¯ The pullback ϕ −1 (P) is a pro-p subgroup of H, x¯ ϕ(N2 )¯x−1 for some x¯ ∈ N. hence is contained in some Sylow pro-p subgroup V ⊂ H. Picking x ∈ N so that ϕ(x) = x¯ , we will have N1 , xN2 x−1 ⊂ V . Then [S : S ∩ V ] < n, so by the induction hypothesis, we can write S = yVy−1 for some y in H such that [S : S ∩ V ] = [S : y(S ∩ V )y−1 ]. Consider T 0 = (yx)T(yx)−1 . Clearly, S ∩ T 0 ⊃ (yx)N2 (yx)−1 % (yx)(S ∩ T)(yx)−1 , and moreover, x normalizes S ∩ T, implying [S : (yx)(S ∩ T)(yx)−1 ] = [S : y(S ∩ T)y−1 ] −1

= [S : y(S ∩ V )y

−1

][y(S ∩ V )y

(3.17) −1

: y(S ∩ T)y

= [S : S ∩ V ][S ∩ V : S ∩ T] = [S : S ∩ T].

]

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163

Thus, [S : S ∩ T 0 ] < n, so again by the induction hypothesis, we can find z ∈ H satisfying S = zT 0 z−1 and [S : S ∩ T 0 ] = [S : z(S ∩ T 0 )z−1 ]. Then S = (zyx)T(zyx)−1 and [S : (zyx)(S ∩ T)(zyx)−1 ] = [S : z(S ∩ T 0 )z−1 ][z(S ∩ T 0 )z−1 : (zyx)(S ∩ T)(zyx)−1 ] = [S : S ∩ T 0 ][S ∩ T 0 : (yx)(S ∩ T)(yx)−1 ] = [S : (yx)(S ∩ T)(yx)−1 ] = [S : S ∩ T] by (3.17), completing the argument. Remark The original proof of Theorem 3.40 given by Matsumoto (1966) is incomplete: his induction on the pair of indexes [S : S ∩ T] and [T : S ∩ T] does not work. Our proof is a modified version of Matsumoto’s argument. Theorem 3.40 yields, in particular, the following important structural result. Proposition 3.42 Let G be an almost K-simple algebraic K-group. Then any noncentral normal subgroup of GK has finite index. PROOF: Let H be a noncentral normal subgroup of GK . By Theorem 3.10, H is open in GK , so the quotient GK /H is discrete, and to prove that it is actually finite it is enough to show that it is compact. This follows from Proposition 3.43 Let H be an open normal subgroup of GK , where G is a semisimple K-group. Then there exists a maximal compact subgroup B ⊂ GK such that GK = BH. PROOF: Let S ⊂ H be a Sylow pro-p subgroup, and let g ∈ GK . Then g−1 Sg is also a Sylow pro-p subgroup of g−1 Hg = H, so by Theorem 3.40 we can write g−1 Sg = h−1 Sh for a suitable h ∈ H. Therefore x := gh−1 ∈ N := NGK (S), yielding GK = NH. On the other hand, N is compact by Proposition 3.37 and therefore is contained in a maximal compact subgroup B ⊂ GK by Proposition 3.37(1), which is as required. Remark Propositions 3.17 and 3.42 together allow one to draw the following general conclusion: if K is a locally compact field and G is a K-simple algebraic K-group, then any noncentral normal subgroup of GK has finite index. (Strictly speaking, we have not yet examined explicitly the case K = C, but it is well known that in this situation GC does not have any proper noncentral normal subgroups. As shown in [AGNT, §7.2], this immediately follows

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from general structural results. At the same time, one can give the following simple topological argument: any non-central normal subgroup of GC is open and therefore necessarily contains the connected component (GC )0 . However, according to Theorem 3.14, the group GC is connected, implying in effect that GC = (GC )0 has no proper noncentral normal subgroups.) We note that the property that every noncentral normal subgroup has finite index is valid also for the groups of rational points of absolutely almost simple simply connected groups over global fields (cf. [AGNT, §9.1]), but this is considerably more difficult to prove. Besides the theory of profinite groups, the analysis of algebraic groups over non-Archimedean local fields uses such techniques as the reduction of algebraic varieties modulo the maximal ideal p. This procedure associates to a given algebraic K-variety X an algebraic variety X defined over the residue field k = O/p. Furthermore, if certain smoothness conditions are satisfied, then the points of X k bijectively correspond to the congruence classes of points in XO modulo p. To avoid some tricky technical details, we will present the basic definitions and results for the case of affine varieties. In a couple of instances later in the book, we will have to deal also with projective varieties, but for these, the general framework of the reduction techniques is similar. In fact, by and large, the consideration of general varieties can be reduced to the affine case using a finite affine cover, cf. Weil (1961b and 1982). It is convenient to introduce reduction in the more general setting of an affine algebraic variety X ⊂ An defined over a field P that is the field of fractions of an integral domain R. Let a ⊂ P[x1 , . . . , xn ] be the ideal of polynomials that vanish on X . By the reduction of X modulo a maximal ideal m ⊂ R, we mean the subvariety X (m) of An defined by the ideal a(m) ⊂ k[x1 , . . . , xn ], where k = R/m, obtained by reducing all the polynomials in a ∩ R[x1 , . . . , xn ] modulo m. (Note that, in general, X (m) is only k-closed in An , but not necessarily defined over k. However, in this book, k will be a finite field, hence perfect, in which case the notions of k-closed and k-defined subvarieties coincide.) Even though this definition is quite straightforward, certain properties of the reduction procedure are rather delicate, and therefore some care needs to be exercised in dealing with it. For example, let P = Q, R = Z, and let X ⊂ A1 be the singleton {p−1 }. Then a is generated by px − 1, so a ∩ Z[x] = (px − 1)Z[x], and the reduction of X modulo p is given by 0 · x − 1 = 0, implying that X (p) = ∅. A detailed treatment of the theory of reduction of algebraic varieties goes far beyond the scope of this book. So, we will limit our exposition to the basic definitions and fundamental results such as Hensel’s Lemma. Most proofs will be omitted, but some useful elementary tricks will be outlined. (The interested reader can use these to reconstruct some of the proofs.)

3.3 The Non-Archimedean Case

165

We will now define the notion of smooth reduction. As above, let X ⊂ An be an affine P-variety with all irreducible components having the same dimension m, and let a be the ideal of all polynomials in P[x1 , . . . , xn ] that vanish on X . Furthermore, let R ⊂ P be a subring such that P is the field of fractions of R, and let X (m) be the reduction of X modulo a maximal ideal m of R. We say that a point x ∈ X (m) is a simple point of reduction if there exist polynomials f1 , . . . , fr ∈ a ∩ R[x1 , . . . , xn ], with r = n − m, such that the rank of the Jacobian ! ∂ f¯i (x) ∂xj i = 1, . . . , r j = 1, . . . , n

equals r (where the bar denotes reduction modulo m). The reduction is said to be smooth if all the points of X (m) are simple. (We note that the smoothness of reduction is a stronger condition than just the smoothness of X (m) .) Given x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) ∈ XR , we write x ≡ y(mod m) if xi ≡ yi (mod m) for all i = 1, . . . , n. For x ∈ XR , the (m) reduction modulo m gives a point x¯ ∈ X k , defining thereby the reduction map (m)

ρ : XR → X k , x 7→ x¯ , whose nonempty fibers are the congruence classes of points in XR modulo m. The natural question that arises here is how to identify the image of the reduction map. We will not discuss this problem in detail since the following classical result, stated for the situation where K is a finite extension of Qp , O is the valuation ring in K, and p ⊂ O is the valuation ideal, is sufficient for our purposes. (p)

Theorem 3.44 (HENSEL’S LEMMA) Any simple point of reduction x ∈ X k lies in the image of the reduction map. In particular, if the reduction X (p) is smooth, then the reduction map is surjective. Note that if the reduction X (p) is smooth, then distinct K-defined irreducible components of X remain distinct in the reduction. In particular, if X is finite and the reduction X (p) is smooth, then the reduction map is injective. Obviously, the reduction of a smooth variety does not need to be smooth. However, if X is a smooth variety defined over a number field K, then for almost all non-Archimedean valuations v of K, the reduction X (v) modulo the corresponding maximal ideal p(v) of the ring of integers O ⊂ K is smooth. Furthermore, if X is a K-defined absolutely irreducible variety, then the reduction

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Algebraic Groups over Locally Compact Fields

X (v) is absolutely irreducible for almost all v. Last but not least, if X is an algebraic K-group, then X (v) is an algebraic group for all non-Archimedean v with the corresponding reduction map being a group homomorphism. The remainder of this section is devoted to the precise formulations and partial proofs of these facts. Theorem 3.45 (NOETHER) Let X be an (absolutely) irreducible affine variety of dimension m over a number field K. Then the reduction X (v) is also an irreducible variety of dimension m for almost all v ∈ VfK . Let us make one remark regarding the definition of X (v) . We define X (v) as the reduction of X modulo the ideal p(v) of the ring of integers O ⊂ K. At the same time, we can take for the base ring the ring O0 = O(S) of S-integers for K , and consider the reduction modulo any subset S ⊂ V K such that v ∈ / S ∪ V∞ the corresponding ideal p0 (v) ⊂ O0 . It turns out, however, that we end up with the same variety. To see this, we need to consider the ideals b = a ∩ O[x1 , . . . , xn ] and b0 = a ∩ O0 [x1 , . . . , xn ]. Since the ring O0 is Noetherian, by Hilbert’s Basis Theorem, the ideal b0 admits a finite set of generators f1 , . . . , fr . Since v ∈ / S, one can find a ∈ O\p(v) such that all af1 , . . . , afr lie in O[x1 , . . . , xn ], hence in b. Bearing in mind that O/p(v) = O0 /p0 (v) =: k, we see that the images of the ideals b and b0 in k[x1 , . . . , xn ] coincide and therefore 0

X (p(v)) = X (p (v)) . Thus the reduction of a K-variety actually depends only on v rather than on the pair (O, p(v)), justifying thereby the notation X (v) . We also remark that given x ∈ XK , for a sufficiently large S, we will have x ∈ XO0 , where O0 is the ring of S-integers, and then for v ∈ / S, we can consider 0 0 the reduction x¯ ∈ X (p (v)) . However, as we have seen, X (p (v)) can be identified with X (p(v)) , enabling us to view x¯ as a point of X (p(v)) . In this case, we say that for v ∈ / S, the point x can be reduced modulo p(v), and the result of its reduction is the point x¯ ∈ X (p(v)) . Similar terminology will be applied to polynomials, regular maps, and so on. In what follows, we will often use the fact that the reduction X (v) coincides with the reduction X (pv ) modulo the valuation ideal pv of the valuation ring Ov in the corresponding completion Kv (regarding X as a Kv -variety).

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Lemma 3.46 In the above notations, we have X (v) = X (pv ) . PROOF: Set b = a ∩ O[x1 , . . . , xn ] and b0 = av ∩ Ov [x1 , . . . , xn ], where av is the ideal of polynomials in Kv [x1 , . . . , xn ] that vanish on X . Let f1 , . . . , fr and g1 , . . . , gs be finite sets of generators of the ideals b and b0 , respectively. Since X is defined over K, there exist hij in Kv [x1 , . . . , xn ] P such that gi = rj = 1 hij fj . Choosing tij ∈ K[x1 , . . . , xn ] with coefficients sufficiently close to the respective coefficients of hij , we obtain polynomials P gi0 := rj = 1 tij fj ∈ b0 satisfying gi ≡ gi0 (mod pv ). Then the reduction X (pv ) is defined by the equations gi0 = 0, i = 1, . . . , s. We now can choose a finite subset S ⊂ VfK with v ∈ / S such that the coefficients 0

of g0 lie in the ring of S 0 -integers O0 . Then it is easy to see that X (pv ) = X (p (v)) 0 for the corresponding ideal p0 (v) ⊂ O0 , while X (p (v)) is identical to X (p(v)) as was noted earlier. The same method can be used to prove the following statement, which shows that the reduction is independent of the base field for almost all valuations. Lemma 3.47 Let L/K be a finite extension of number fields. Given a K-defined affine variety X , for almost all v ∈ VfK , the reduction X (v) coincides with the reduction X (w) for any extension w|v, where in the latter case, X is considered as an L-variety. The following assertion is often useful in working with reductions of algebraic varieties. Lemma 3.48 Let f1 , . . . , , fr ∈ K[x1 , . . . , xn ]. Then if the system fi = 0,

i = 1, . . . , r

(3.18)

¯ is inconsistent in the sense that it has no solution over an algebraic closure K, K then for almost all v ∈ Vf the system f¯i = 0,

i = 1, . . . , r

(3.19)

obtained by reduction modulo v is also inconsistent. PROOF: Since (3.18) is inconsistent, by Hilbert’s Nullstellensatz, there exist polynomials g1 , . . . , gr ∈ K[x1 , . . . , xn ] such that f1 g1 + · · · + fr gr = 1.

(3.20)

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Algebraic Groups over Locally Compact Fields

We can find a finite set S ⊂ VfK such that the coefficients of all the f1 , . . . , fr , g1 , . . . , gr lie in the ring of S-integers O0 . Then for v ∈ / S, the relation (3.20) can be reduced modulo v, yielding f¯1 g¯ 1 + · · · + f¯r g¯ r = 1. This immediately implies that the system (3.19) is inconsistent. Exercise Using Lemma 3.48, prove Theorem 3.45 for a hypersurface in An (which is the basic case from the birational point of view). In other words, show that if f ∈ K[x1 , . . . , xn ] is an absolutely irreducible polynomial, then its reduction f¯ modulo v is also absolutely irreducible for almost all v ∈ VfK . (Hint. Note that the existence of a factorization f¯ = gh can be interpreted as the existence of a solution to a certain system of polynomial equations in terms of the coefficients of g and h. On the other hand, use Lemma 3.48 to show that for almost all v, the reduction of the polynomial system in question is inconsistent.) Proposition 3.49 Let X be a smooth affine variety defined over an algebraic number field K with all irreducible components having the same dimension m. Then for almost all v ∈ VfK , the reduction X (v) is smooth. PROOF: Suppose that X ⊂ An , and let a ⊂ K[x1 , . . . , xn ] be the ideal of polynomials vanishing on X . Set b = a ∩ O[x1 , . . . , xn ], where O is the ring of integers of K, and pick a finite set of generators f1 , . . . , f` of b. Let r = n − m, and let {Dj }dj= 1 be all the r × r minors of the Jacobian matrix  ∂f ∂f1  1 ···  ∂x1 ∂xn     ................ .  ∂f ∂f`  ` ··· ∂x1 ∂xn Since X is smooth, the system of equations fi = 0,

i = 1, . . . , `,

Dj = 0,

j = 1, . . . , d

is inconsistent. Then, by Lemma 3.48, for almost all v ∈ VfK , the reduction of this system is also inconsistent. For those v, the reduction X (v) contains no points where the reduction of the preceding Jacobian matrix has rank < r. This means that the reduction X (v) is smooth, as required.

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169

Exercise Give a projective analog of Proposition 3.49. Before proving that the reduction of an algebraic group is an algebraic group, we will first discuss some basic results dealing with the reduction of morphisms of general affine varieties. Let P be an arbitrary field, and let f : X → Y be a P-defined regular map between two P-defined varieties X ⊂ An and Y ⊂ Am . Then f is given by an m-tuple of “coordinate” polynomials f1 , . . . , fm ∈ P[x1 , . . . , xn ]. We say that f is defined over a subring R ⊂ P if in fact it can be given by polynomials f1 , . . . , fm ∈ R[x1 , . . . , xn ]. In this case, given any maximal ideal m ⊂ R, we can reduce the polynomials modulo m, and then view f (m) = (f¯1 , . . . , f¯m ) as a regular map from An to Am . We will now show that this map descends to a regular map f (m) : X (m) → Y (m) . Lemma 3.50 f (m) (X (m) ) ⊂ Y (m) . PROOF: Let aX (resp., aY ) be the ideal of polynomials in P[x1 , . . . , xn ] (resp., in P[y1 , . . . , ym ]) that vanish on X (resp., on Y ). Set bX = aX ∩ O[x1 , . . . , xn ] and bY = aY ∩ O[y1 , . . . , ym ]. Let f ∗ : P[y1 , . . . , ym ] → P[x1 , . . . , xn ], yj 7→ fj (x1 , . . . , xn ) be the comorphism associated with f . Then f (X ) ⊂ Y implies that f ∗ (aY ) ⊂ aX . Since f is defined over R, we in fact have f ∗ (bY ) ⊂ bX . Reducing modulo m, (m) (m) we obtain ( f (m) )∗ (bY ) ⊂ bX , hence f (m) (X (m) ) ⊂ Y (m) . (We note that a similar argument shows that the regular map f (m) : X (m) → Y (m) does not depend on the choice of polynomials f1 , . . . , fm ∈ R[x1 , . . . , xn ] representing the given R-defined regular map f : X → Y .) While there are easy criteria to determine whether a morphism of algebraic varieties is defined over a given field (cf. §2.4), there seem to be no explicit criteria for a morphism to be defined over a given ring. Thus, given a P-defined morphism f : X → Y and a subring R ⊂ P, it may be difficult to determine whether f is defined over R and hence can be reduced modulo a specific maximal ideal m ⊂ R. However, we will focus mainly on morphisms f : X → Y defined over a number field K, in which case we can guarantee that f can be reduced modulo almost all v ∈ VfK , yielding a regular map f (v) : X (v) → Y (v) (although it may be hard to identify the finite set of exceptional valuations explicitly). This observation implies that the reduction X (v) is independent of

Algebraic Groups over Locally Compact Fields

170

the geometric realization of X as a Zariski-closed subset of an affine space, for almost all v. More precisely, if X and Y are biregularly isomorphic over K, then X (v) and Y (v) are biregularly isomorphic over the corresponding residue field, for almost all v ∈ VfK . Using reductions of morphisms, one can define the reduction of arbitrary varieties via affine covers. We will only give a sketch of this construction, referring the reader to Weil (1961b) for more details. Let X=

d [

Xi

i=1

be a finite affine cover of an arbitrary K-variety X . We fix geometric realizations of the Xi as Zariski-closed subsets of affine spaces by fixing for each i = 1, . . . , d a K-defined biregular isomorphism fi : Xi → Xi0 with a K-defined Zariski-closed subvariety of some affine space Ani . Set Yij = fi (Xi ∩ Xj ) ⊂ Ani . Then for i, j = 1, . . . , d, there are natural K-defined isomorphisms gij : Yij → Yij , and our original variety X can be viewed as the result of gluing the Xi0 along the {Yij , gij } (cf. Shafarevich [2013]). Note that the varieties Yij are affine. Then it follows from Lemma 3.50 and subsequent remarks that for almost all v ∈ VfK , (v)

the morphisms gij admit reductions g(v) that are isomorphisms between Yij ij (v)

and Yji . Moreover, the isomorphisms g(v) satisfy the conditions required to ij (v)

implement the gluing of the (X 0i )(v) along the {Y ij , g(v) } (this follows from the ij fact that the gij satisfy these conditions). The result of the gluing procedure is the required reduction X (v) . Proposition 3.51 Let G ⊂ GLn be an algebraic group defined over a number field K. Then for all v ∈ VfK , the reduction G(v) is an algebraic group defined (v)

over the residue field kv , and the reduction map GOv → Gkv , where Ov is the valuation ring of the completion Kv , is a group homomorphism. Moreover, the reduction G(v) is smooth for almost all v ∈ VfK , and then the reduction map (v)

GOv → Gkv is surjective. PROOF: Let µ : GLn × GLn → GLn , µ(x, y) = xy and let i : G → G, i(x) = x−1 be the multiplication and the inversion maps, respectively. Since G is an algebraic group, we have the inclusions µ(G ×G) ⊂ G and i(G) ⊂ G. Clearly, µ and

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i are defined over Z, hence can be reduced for any v ∈ VfK . Moreover, for the corresponding reductions, by Lemma 3.50, we have µ(v) (G(v) × G(v) ) ⊂ G(v) and i(v) (G(v) ) ⊂ G(v) , proving that G(v) is an algebraic group. Furthermore, the reduction map is the ρ restriction to GOv of the homomorphism GLn (Ov ) → GLn (kv ) induced by the canonical ring homomorphism O → kv = O/p(v) = Ov /pv , hence is itself a group homomorphism. It remains to observe that since G is a smooth variety, by Proposition 3.49, the reduction G(v) is smooth for almost (v) all v ∈ VfK , and then the reduction map GOv → Gkv is surjective by Hensel’s lemma. Proposition 3.52 Let f : G → H be a K-defined morphism of algebraic Kgroups. Then for almost all v ∈ VfK , the reduction f (v) : G(v) → H (v) exists and is a morphism of algebraic groups. PROOF: As we noted earlier, the reduction f (v) : G(v) → H (v) indeed exists for almost all v ∈ VfK , and we only need to check that f (v) is multiplicative. However, the multiplicativity of f can be expressed as the relation f ◦ µG = µH ◦ ( f , f ), where µG and µH are the multiplication maps on G and H, respectively. Reducing it, we obtain the relation f (v) ◦ µG(v) = µH (v) ◦ ( f (v) , f (v) ) in self-explanatory notations, which means that f (v) is multiplicative. We conclude this section with Proposition 3.53 Let G be a connected algebraic K-group, and let H ⊂ G be a K-defined subgroup. Set X = G/H. Then for almost all v ∈ VfK , the reduction X (v) coincides with the quotient G(v) /H (v) . PROOF: The natural action f : G × X → X , (g, y) 7→ gy gives rise to an action f (v) : G(v) × X (v) → X (v) for almost all v. To prove our claim, we need to verify the following two properties for almost all v:

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(1) f (v) is transitive; (2) let x = eH ∈ X ; then the stabilizer G(v) (¯x) of the corresponding point x¯ ∈ X (v) coincides with H (v) . Excluding finitely many v’s, we may assume that the reductions G(v) , H (v) , and X (v) are smooth, the reduced varieties have dimensions equal to those of G, H, and X , respectively, and moreover, the reduction X (v) is irreducible. For (g, y) ∈ G × X , we consider the differential d(g,y) f : Tg (G) × Ty (X ) −→ Tgy (X ). Let t = dim X . Then the condition that dim d(g,y) f (Tg (G) × {0}) < t can be expressed as a system of polynomial conditions hi (g, y) = 0, i = 1, . . . , d.

(3.21)

It is easy to see, however, that in our situation we have dim d(g,y) f (Tg (G) × {0}) = t for all (g, y) ∈ G × X , which means that (3.21) is inconsistent on G ×X . Now, it follows from Lemma 3.48 that the reduction (v)

hi (g, y) = 0, i = 1, . . . , d, is inconsistent on G(v) × X (v) for almost all v. For any such v and any y ∈ X (v) , the map ϕ : G(v) → X (v) , g 7→ gy, has the property that the differential de ϕ : Te (G(v) ) → Ty (X (v) ) is surjective, implying that the orbit G(v) y is Zariski-open in X (v) . Since X (v) is irreducible, the fact that every orbit is open means that in fact there is only one orbit, verifying property (1) above. As for property (2), one easily shows using Hilbert’s Nullstellensatz that since the condition gx = x defines H inside G, the corresponding reduced condition g¯x = x¯ defines H (v) inside G(v) , for almost all v. Then G(v) (¯x) = H (v) , as required.

3.4 Elements of Bruhat–Tits Theory A fundamental theory for analyzing the groups of rational points of semisimple algebraic groups over local fields was developed by F. Bruhat and J. Tits (1972, 1984a, 1987); for a new approach to Bruhat–Tits theory, see Kaletha and Prasad (2023). The centerpiece of the theory is the construction, in the group

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of rational points of an absolutely simple simply connected algebraic group G over a local field K, of a BN-pair associated to an affine root system. Furthermore, this BN-pair leads to a simplicial complex A, called a building, equipped with a natural action of the group GK . This complex turns out to be contractible, and its properties translate into information about the group GK and its subgroups. (Note that the building is actually a natural non-Archimedean analog of the symmetric space GR /K, where K is a maximal compact subgroup of GR in the Archimedean case; cf. Proposition 3.26). In particular, let B ⊂ GK be a compact subgroup. One shows that the natural action of B on A has a fixed vertex. But the stabilizers of vertices (known as maximal parahoric subgroups) are themselves compact, so the set of maximal compact subgroups of GK coincides with the set of maximal parahoric subgroups. On the other hand, parahoric subgroups admit a description in terms of the affine root system that is very similar to the one given in §2.1.12 for parabolic subgroups, which enables one to describe the conjugacy classes of maximal compact subgroups of GK , and, in particular, to determine the precise number of these classes (cf. Theorem 3.54). This example demonstrates how Bruhat–Tits theory provides elegant solutions to various questions about the group GK . Unfortunately, in the present book, we are unable to go into the details of Bruhat–Tits theory and refer the interested reader to the original works cited above, as well as to Iwahori and Matsumoto (1965), Macdonald (1971), Satake (1963), and Hijikata (1975). (We recommend Abramenko and Brown [2008] as a general reference on buildings.) A detailed exposition of this theory would be a separate project, comparable in size to this book, that would involve a series of new concepts and techniques that will not be used here. For this reason, we will limit our account in this section to the description of the basic objects (in some instances, this can be achieved more efficiently by taking the results of the theory as definitions) and the formulation of several key theorems. So, let G be an absolutely simple simply connected algebraic group defined over a finite extension K of Qp . An Iwahori subgroup B ⊂ GK is the normalizer of a Sylow pro-p subgroup of GK . Note that since Sylow pro-p subgroups are all conjugate (Theorem 3.40), the Iwahori subgroups are also conjugate. A subgroup P ⊂ GK is called parahoric if it contains an Iwahori subgroup. The building A associated to GK (or simply of G over K) is a simplicial complex whose vertices are maximal proper parahoric subgroups of GK , and in which a collection {P0 , P1 , . . . , Ps } of such subgroups defines an s-simplex if the interT section si = 0 Pi is also a parahoric subgroup. The group GK acts on A by conjugation; this action clearly preserves the simplicial structure, and, moreover, the stabilizers of simplices are proper parahoric subgroups. Sometimes, by

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the building, we also mean the geometric realization of this simplicial complex, which is contractible (by the Solomon–Tits theorem) of dimension equal to the K-rank of G. In particular, if rankK G = 1, then the complex is a tree, enabling one to use the structure theory of groups acting on trees (cf. Serre [2003]). It follows, for example, that any torsion-free discrete subgroup 0 of GK is free. (For G = SL2 , this was obtained by Ihara [1966]; note that this case is also treated in detail in Humphreys [1980].) We will now proceed to the construction of a BN-pair in GK . First, we recall (cf. Bourbaki (2002, Chapter 4) for more details) that a BN-pair (or a Tits system) in an abstract group G is a pair of subgroups B, N ⊂ G such that for a suitable subset C ⊂ N/(B ∩ N), the following axioms are satisfied: (1) (2) (3) (4)

B ∪ N generates G and H = B ∩ N is a normal subgroup of N; C generates W = N/H and consists of elements of order 2; rBw ⊂ BwB ∪ BrwB for any r ∈ C and w ∈ W ; rBr ⊂ 6 B for any r ∈ C.

The quotient W = N/H is called the Weyl group of (B, N). We note that given w = gH ∈ W , the double coset BgB is independent of the choice of a representative g ∈ N for w, and therefore is typically denoted by BwB; the relations described in items (3) and (4) should be interpreted in accordance with this convention. We also recall that the set C is in fact uniquely determined by B and N, namely one shows that C consists of those w ∈ W for which B ∪ BwB is a subgroup of G. Henceforth, we again let G denote an absolutely simple simply connected algebraic group over a finite extension K of Qp , and let S ⊂ G be a maximal K-split torus of G. We will write NG (S) and ZG (S) to denote, respectively, the normalizer and the centralizer of S in G. We set N := NG (S)K and H := {x ∈ ZG (S)K : χ (x) ∈ U for all χ ∈ X(ZG (S))K }, where U is the group of v-adic units in K. Then there exists an Iwahori subgroup B of GK satisfying B∩N = H and such that B and N constitute a BN-pair in GK . The corresponding Weyl group W = N/H turns out to be the Weyl group of some affine root system R of rank ` = dim S (cf. Bourbaki [2002, Chapter 6, §2]) for a discussion of affine root systems and their Weyl groups). In particular, the distinguished set of generators C for W consists of (` + 1) elements r1 , r2 , . . . , r`+1 , which can be labeled so that the subgroup W0 ⊂ W generated by r1 , . . . , r` is isomorphic to the Weyl group of a certain (usual) reduced root R0 system associated with R. It should be pointed out that W0 is isomorphic to the relative Weyl group W (S, G) = NG (S)/ZG (S); however, in general, R0 may be different from the relative root system R(S, G), even if the

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latter is reduced. Nevertheless, every root of R0 is proportional to some root of R(S, G) and vice versa. If G is K-split, hence S is a maximal torus of G, we always have R0 = R(S, G). It follows from the general theory of groups with a BN-pair that any subgroup P ⊂ GK that contains B is of the form PX = BWX B for suitable X ⊂ C, where WX is the subgroup of W generated by X . In addition, if PX1 and PX2 are conjugate in GK , then X1 = X2 . It follows that the parahorics Pi = PXi , where Xi = C \ {ri } for i = 1, . . . , ` + 1, form a complete system of representatives of the conjugacy classes of maximal proper parahorics. As we have already mentioned, the maximal parahorics coincide with the maximal compact subgroups, so we obtain the following. Theorem 3.54 Let G be an absolutely simple simply connected algebraic Kgroup, and let ` = rankK G. Then GK has (` + 1) conjugacy classes of maximal compact subgroups. It follows that for a semisimple simply connected K-group G, the group GK has only finitely many conjugacy classes of maximal compact subgroups. In fact, it is established in Bruhat–Tits theory that this result remains valid for any reductive K-group G. There are several decompositions for the groups of rational points over local fields, some of which are non-Archimedean analogs of the decompositions developed in §3.2. From these results, we will only need the non-Archimedean analog of the Cartan decomposition, which we will now discuss in greater detail. Let K be the maximal compact subgroup BW0 B in the preceding notations. It is well known that W is a semidirect product W0 T, where T is the free abelian group of rank ` = dim S generated by the roots from R0 . Furthermore, let 50 ⊂ R be the system of simple roots associated with B, and let T + ⊂ T be the subsemigroup of elements t ∈ T such that ht, αi ≥ 0 for all α ∈ 50 , where h , i is a positive definite symmetric bilinear form on T ⊗Z R invariant under W0 . Letting ν : N → W = N/H denote the natural homomorphism, we set Z + = ν −1 (T + ). Note that if z1 , z2 ∈ ZG (S)K and ν(z1 ) = ν(z2 ), then z1 z2−1 ∈ H and therefore Kz1 K = Kz2 K. Thus, for z ∈ ZG (S)K , the double coset KzK depends only on t := ν(z), hence will be denoted by Kν −1 (t)K. With this notation, we have Theorem 3.55 (CARTAN DECOMPOSITION) We have GK = KZ + K, and the map t 7→ Kν −1 (t)K yields a bijection between T + and the set of double cosets K\GK /K.

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Example 3.56 Let G = SLn and K = Qp . It is easy to see that the matrices x = (xij ) in SLn (Zp ) such that xii ≡ 1(mod p) and xij ≡ 0(mod p) for all i, j = 1, . . . , n, i > j, form a Sylow pro-p subgroup of SLn (Qp ). The corresponding Iwahori subgroup (i.e., the normalizer of this Sylow subgroup in SLn (Qp )) is B = {x = (xij ) ∈ SLn (Zp ) : xij ≡ 0(mod p) for i > j}. This subgroup, together with the group N of K-rational points of the normalizer of the diagonal torus S ⊂ G, constitute a BN-pair as above. The distinguished set of generators of C ⊂ W := N/(B∩N) consists of the following matrices:   1 0     ..   0 1 0 .   −1 0    0 1         1 r1 =  −1 0  , . . . , ri =  ,     ..     1 .     . .. 0 1   0 1   −1 0 0 p  ..   . 1 0    ..  . .. . . . , rn =    . .   0  1 −p 0 Let W0 be the subgroup of W generated by r1 , . . . , rn−1 ; then the corresponding parahoric subgroup P = BW0 B coincides with SLn (Zp ). Other maximal proper parahoric subgroups are of the form Pi = BWi B for i = 1, . . . , n − 1, where Wi is generated by C \ {ri }. It is easy to show that Pi coincides with the stabilizer in SLn (Qp ) of the lattice with the basis e1 , . . . , en−i , pen−i+1 , . . . , pen , where e1 , . . . , en is the standard base of Qnp . The reader can verify that this leads to the same classification of conjugacy classes of maximal compact subgroups of SLn (Qp ) as Proposition 3.35. Finally, let us illustrate the Cartan decomposition for SLn (Qp ). As we have seen, in this case K coincides with SLn (Zp ). Furthermore, the group T is isomorphic to

3.4 Elements of Bruhat–Tits Theory ( (a1 , . . . , an ) ∈ Z : n

n X

177

) ai = 0 ;

i=1

and the elements αi = (0, . . . , 0, 1, −1, 0, . . . , 0) for i = 1, . . . , n − 1 constitute the system of simple roots 50 (cf. the description of the root system of G = SLn in §2.1.3). The Weyl group W0 ' Sn acts on T by permutations of the coordinates, so the usual dot product is W0 -invariant. It follows that in the situation at hand, T + consists of (a1 , . . . , an ) ∈ T such that a1 ≤ a2 ≤ · · · ≤ an . Theorem 3.55 then asserts that any double coset KgK with g ∈ SLn (Qp ) contains a representative g of the form diag(pa1 , pa2 , . . . , pan ) where a1 ≤ a2 ≤ · · · ≤ an are uniquely determined. Thus, in this case, Theorem 3.55 essentially reduces to the well-known Invariant Factor Theorem for matrices: if A is a principal ideal domain, then for any matrix X ∈ Mn (A), there exist matrices Y1 , Y2 ∈ SLn (A) such that Y1 XY2 = diag(d1 , . . . , dr , 0, . . . , 0), where di ∈ A, di 6= 0, and di divides di+1 for all i = 1, . . . , r − 1; moreover, the elements di (known as the invariant factors of X ) are uniquely determined up to units of A (cf. Curtis and Reiner [2006, Chapter 3, §16.5]). In the rest of this section, using Theorem 3.55, we will show that the group GK is compactly presented (this result will be used in §5.4 to prove that Sarithmetic subgroups are finitely presented). To give the precise statement, let us fix the following terminology: a subset C of an abstract group 0 is called defining if it generates 0 and if any relation in 0 between the elements of C is a consequence of relations of the form ab = c with a, b, c ∈ C. In other words, this means that the natural homomorphism f : F(C) → 0 of the free group F(C) on C is surjective, and its kernel is generated as a normal subgroup of F(C) by elements of the form abc−1 , where a, b, c ∈ C are such that ab = c in 0. A topological group 0 is said to be compactly presented if it admits a compact defining subset C ⊂ 0 as an abstract group. Theorem 3.57 (BEHR [1967]) Let G be a reductive algebraic group defined over a non-Archimedean local field K. Then the group GK is compactly presented. PROOF: We begin with the following remark that will be used repeatedly below. To prove that a topological group 0 is compactly presented it suffices to find a compact subset C of 0 that generates 0 as an abstract group and is such that

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all the relations in 0 for the elements of C follow from relations of a bounded length. First, we will consider the case where G is an absolutely simple simply connected K-group. Here the proof of compact presentation of GK is derived from the following statement. Lemma 3.58 Suppose a topological group 0 is equipped with an integervalued function | | : 0 → Z (“absolute value”) that has the following properties: (a) (b) (c) (d)

|g| ≥ 0, |e| = 0 (where e is the identity element); |g1 g2 | ≤ |g1 | + |g2 | for all g1 , g2 ∈ 0; |g−1 | = |g| for all g ∈ 0; 0n := {g ∈ 0 : |g| ≤ n} is compact for each n.

Assume, furthermore, that there exist positive integers c, d, and b satisfying the following: (i) if |g| > c, then there are g1 , g2 in 0 such that g = g1 g2 and |g1 | < c, |g2 | < |g| − d; (ii) if f , g, h ∈ 0 satisfy fgh = 1, then there are g1 , g2 , . . . , gt ∈ 0 such that g = g1 g2 · · · gt with |gi | ≤ c for all i = 1, . . . , t where t ≤ |g| + b and | fg1 · · · gj | ≤ max{| f |, |h|} + d for all j = 1, . . . , t − 1. Then 0 is compactly presented. PROOF: It follows from (d) and (i) that 0c is a compact generating set for 0, so it suffices to show that all the relations for the elements of 0c are consequences of relations of length ` with     c ` ≤ `0 := max 3c + b + 3 , 2 +5 , d+1 where [ ] denotes the integral part. Since 0c contains 1 and is closed under taking inverses, it suffices to consider relations r of the form g1 g2 · · · gn = e with gi ∈ 0c . Set pj = gj gj+1 · · · gn and define the norm krk = max {|pj |}. 1≤j≤n

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First, we will show that any relation r as above is a consequence of relations r0 of norm kr0 k ≤ 2c. This is established by induction on krk. Suppose krk > 2c, and pick an index j so that max{|pj |, |pj+1 |} = krk. Then min{|pj | |pj+1 |} > c 0 since |gj | ≤ c. So, by (i) we can find gj0 , gj+1 in 0c such that 0−1 |gj0−1 pj | < |pj | − d and |gj+1 pj+1 | < |pj+1 | − d.

Set 0−1 0−1 0 0 f = p−1 j gj , g = gj gj gj+1 , h = gj+1 pj+1 .

Then fgh = 1 and max{| f |, |h|} < krk − d. Applying (ii), we can find g¯ 1 , . . . , g¯ t ∈ 0c so that g = g¯ 1 · · · g¯ t with t ≤ |g| + b and | fg1 · · · gk | ≤ max{| f |, |h|} + d for all k = 1, · · · , t − 1. Since |g| ≤ 3c, we have t ≤ 3c + b. Consider the relations 0−1 rj : gj = gj0 g¯ 1 · · · g¯ t gj+1 ,

whose length does not exceed `0 . Let us now replace gj in r with the right-hand side of rj for all j such that max{|pj |, |pj+1 |} = krk (where, of course, we choose the same gj0 for the pairs (j − 1, j) and (j, j + 1) if |pj | = krk). Then for any k ≤ t, we have 0−1 |¯gk · · · g¯ t gj+1 pj | = | f g¯ 1 · · · g¯ k−1 | < krk.

Thus we have obtained a relation r0 that is equivalent to r modulo the relations rj , and for which kr0 k < krk. Repeating this procedure, we will eventually arrive at a relation r0 of norm kr0 k ≤ 2c, which is equivalent to the original relation r modulo some relations of length 6 `0 . Thus, it remains to treat the relations r : g1 g2 · · · gn = e satisfying krk ≤ 2c. Then for any 1 ≤ j ≤ n, h we have i |pj |≤ 2c and therefore c by (i), we can write pj as a product of at most d+1 + 2 elements from 0c . Let us write p−1 as a product of the inverses. Substituting these expressions j into gj = pj p−1 j+1 for 1 ≤ j ≤ n − 1 (respectively, into gn = pn for j = n), we arrive at relations whose lengths are bounded by   c 2 + 5 ≤ `0 . d+1

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On the other hand, it is clear that r is a consequence of these relations. This completes the proof of the lemma. To construct a function | | on 0 = GK with the properties described in Lemma 3.58, we proceed as follows. Fix a maximal K-split torus S ⊂ G (note that if S = {e}, then by Theorem 3.1, the group GK is compact, and we can take C = GK as the compact defining set). Let R(S, G) denote the corresponding relative root system. Then the Lie algebra g = L(G) has the following decomposition:   M g = g0 ⊕  gα  , α∈R(S,G)

where g0 (resp., gα ) is the weight space for Ad S of weight 0 (resp., of weight α ∈ R(S, G)). Pick lattices L0 ⊂ (g0 )K , Lα ⊂ (gα )K and set   M L = L0 ⊕  Lα  . α ∈ R(S,G)

Furthermore, we define the distance between two arbitrary lattices L1 , L2 ⊂ gK as follows: d(L1 , L2 ) := min{n : π n L1 ⊂ L2 ⊂ π −n L1 }, where π ∈ K is a uniformizer. Our goal is to show that all the requirements of Lemma 3.58 are true for the (absolute value) function on 0 = GK given by |g| = d(L , Ad(g)L). Properties (a)–(c) are immediate – one only needs to observe that the distance function on lattices is symmetric and invariant under Ad(g) for all g ∈ 0 and also satisfies the triangle inequality. To prove that 0n is compact for any n > 0, we first note that the adjoint representation α : G → Ad(G) induces an open map αK : GK → Ad(G)K (Corollary 3.7); in particular, the subgroup α(GK ) is open, hence closed, in Ad(G)K . Then it follows from our construction that Ad(0n ) is bounded and closed, hence compact. Furthermore, since α has finite kernel, the map αK is proper, implying that 0n , being the preimage of the compact set Ad(0n ), is itself compact. To verify (i) and (ii), we need a decomposition of GK that is a consequence of the Cartan decomposition GK = KZ + K, where Z + is the preimage under the canonical homomorphism ν : N → W = N/H, of the semigroup T + ⊂ W , which, we recall, is defined as the set of all t in the abelian group T generated by all the roots in R0 that satisfy ht, αi ≥ 0 for all α in the system of simple

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roots 50 ⊂ R0 as above. The explicit construction, given in Bruhat–Tits theory, yields that ν(SK ) ⊂ T. But SK ' (K ∗ )` ' Z` × U ` , where ` = dim S and U is the group of units in K. On the other hand, since H = ker ν is compact, ker (ν|SK ) is also compact, hence contained in U ` . Thus, ν(SK ) contains a free abelian subgroup of rank `. But T itself is a free abelian group of rank `, so the index m = [T : ν(SK )] is finite. Then ν(SK ) ⊃ mT and ν(SK ) ∩ T + ⊃ mT + . We leave it as an exercise for the reader to show that T + is a finitely generated semigroup, and consequently ν(SK ) ∩ T + is such as well. Set S + = Z + ∩ SK . We have ν(S + ) = ν(SK )∩T + , implying that ν(S + ) ⊃ mT + . It follows that there exists a finite subset E ⊂ Z + such that ν(ES + ) = T + ; then Z + = HES + , hence GK = KZ + K = KES + K.

(3.22)

Furthermore, the form h, i can be scaled so that for s in SK and α in R0 , we have hν(s), αi = v(α(s)), where v is the valuation of K. Since the roots in R0 and R(S, G) are proportional, we conclude that v(α(s)) ≥ 0 for all s ∈ S + and all roots α ∈ R(S, G) that are positive relative to the order defined by the system of simple roots 50 ⊂ R0 . Now it is clear that for s ∈ S + , we have |s| = v(α0 (s)), where α0 ∈ R(S, G) is the maximal root; in particular, for s, s1 , s2 ∈ S + such that s = s1 s2 , |s| = |s1 | + |s2 |. As we have seen above, ν(S + ) is a finitely generated semigroup, so there exists an integer r > 0 such that the elements s ∈ S + satisfying |s| ≤ r generate S + as a semigroup. Let us introduce two more integers that will enable us to produce the desired constants c, d, and b with the properties described in Lemma 3.58. More precisely, since K is compact and E finite, there exist integers c1 , c2 such that |k| ≤ c1 for all k ∈ K and |e| ≤ c2 for all e ∈ E. Fix an integer d > 0 and set c = 5c1 + 2c2 + r + d. We will first establish that (i) holds for these c and d, and will then specify d to ensure that (ii) holds

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as well. Let g ∈ GK be such that |g| > c, and pick a factorization g = k1 esk2 as in (3.22). Then |s| = |e−1 k1−1 gk2−1 | > 3c1 + c2 + r + d. There exist s1 , s2 ∈ S + such that s = s1 s2 and 3c1 + c2 + d < |s1 | ≤ 3c1 + c2 + r + d. Then for g1 := k1 es1 and g2 := s2 k2 , we have |g1 | ≤ |s1 | + c1 + c2 ≤ 4c1 + 2c2 + r + d ≤ c and |g2 | ≤ |s2 | + c1 = |s| − |s1 | + c1 < |g| + 2c1 + c2 + c1 − (3c1 + c2 + d) = |g| − d. Before proceeding to (ii), we need to make one observation. For any s ∈ SK and any α ∈ R(S, G), we have Ad(s)Lα = α(s)Lα and Ad(s)L−α = α(s)−1 L−α . It now follows from our construction of the lattice L that for s ∈ SK , the inclusion π n Ad(s)L ⊂ L implies the inclusion π n Ad(s−1 )L ⊂ L, hence π n L ⊂ Ad(s)L. The point that we want to make is that a somewhat weaker implication holds for any g ∈ GK . Namely, suppose that for g ∈ GK , we have π n Ad(g)L ⊂ L. Choose a factorization g = k1 esk2 as in (3.22). Then using properties (b) and (c) of the absolute value function, we obtain π n+2c1 +c2 Ad(s)L ⊂ L. So, π n+2c1 +c2 L ⊂ Ad(s)L, and consequently, π n+4c1 +2c2 L ⊂ Ad(g)L. Now, suppose that f , g, h ∈ GK satisfy fgh = 1. Write g as g = k1 esk2 and factor s into a product s = s1 . . . st , where si ∈ S + and |si | ≤ r. Then, letting g1 = k1 es1 , g2 = s2 , . . . , gt−1 = st−1 , gt = st k2 , we will have |gi | ≤ c1 + c2 + r ≤ 5c1 + 2c2 + r + d = c for any choice of d. Moreover, we may assume t ≤ |s| ≤ |g| + b, where b := 2c1 + c2 . Given j ∈ {1, . . . , t − 1}, we consider the segments u = s1 · · · sj and v = sj+1 · · · st . Then it follows from the definition of L that Ad(u)L ⊂ Ad(s)L + L,

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whence Ad( fk1 eu)L ⊂ Ad( fk1 es)L + Ad( fk1 e)L, and therefore, π n Ad( fk1 eu)L ⊂ L for n = max{| fk1 es|, | fk1 e|}. So, using the preceding remark, we conclude that | fk1 eu| ≤ max{| fk1 es|, | fk1 e|} + 4c1 + 2c2 . On the other hand, | fk1 es| ≤ | fk1 esk2 | + |k2 | ≤ | fg| + c1 = |h| + c1 and | fk1 e| ≤ | f | + |k1 e| ≤ | f | + c1 + c2 . We see that | fg1 · · · gj | ≤ max{| f |, |h|} + (5c1 + 3c2 ). Thus, letting d = 5c1 + 3c2 (so that c = 10c1 + 5c2 + r), we obtain the constants c, d, and b with the required properties. This concludes the proof of the fact that GK is compactly presented in the case where G is an absolutely simple simply connected K-group. The rest of the proof of Theorem 3.57 is a relatively easy reduction of the general case to this one. First, let G be a semisimple simply connected K-group. Then Q G = di= 1 RL/K (Gi ), where each Gi is an absolutely simple simply connected group defined over a finite extension Li of K for i = 1, . . . , d, so that Q GK ' di= 1 (Gi )Li . It is easy to see that a finite product of compactly presented groups is compactly presented; on the other hand, as we have established, all the groups (Gi )Li are compactly presented. Thus, the group GK is also compactly presented. Next, let G be an arbitrary reductive K-group. Then G = DT, an almost direct product of a semisimple K-group D and the maximal central K-torus ˜ → D denote a K-defined universal cover. Furthermore, we can T. Let π : D write T as an almost direct product T = T1 T2 , where T1 is K-split and T2 is K-anisotropic. Set ˜ × T1 × T2 , H =D and let ϕ : H → G be the isogeny induced by π and the product map. Let ϕ F = ker ϕ. Then the exact sequence 1 → F −→ H −→ G → 1 yields the exact cohomological sequence ϕ

HK −→ GK −→ H 1 (K, F),

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which in conjunction with Proposition 3.6 and the finiteness theorem for Galois cohomology over local fields (cf. [AGNT, § 6.4]) implies that ϕ(HK ) is an open subgroup of GK of finite index. (We note that for semisimple G, this fact also follows from Proposition 3.42.) We claim that ϕ(HK ) is a compactly presented group. Indeed, by our ˜ K × (T1 )K × (T2 )K . We showed that D ˜ K is compactly construction, HK = D presented. It follows from Theorem 3.1 that (T2 )K is compact, and hence compactly presented. Finally, (T1 )K ' K ∗` ' Z` × U ` , where ` = dim T1 and U is the group of units in K, immediately implying that (T1 )K is compactly presented. Without any loss of generality, we may assume that the compact defining set C ⊂ HK contains FK . Then it is easy to show that ϕ(C) is a compact defining set for ϕ(HK ). So, the proof of Theorem 3.57 is completed by Lemma 3.59 Let 0 be a locally compact topological group, and 1 an open normal subgroup of finite index. If 1 is compactly presented, then so is 0. PROOF: Let D ⊂ 1 be a compact defining set. From the outset, we may assume that D contains the identity. Moreover, passing from D to DD−1 , we may assume that D = D−1 . It suffices to construct a compact subset C ⊂ 0 generating 0 and having the property that all the relations in 0 for the elements of C are consequences of relations of a bounded length. Let {xi }ni= 1 be a set S of representatives of cosets 0/1, containing the identity. Set C = ni= 1 xi D, and construct a system of defining relations for the elements of C. First of all, since D ⊂ C, we need to consider the relations of the form ab = c

(3.23)

for a, b, c ∈ D that hold in 1. (We recall that by our assumption, these relations define 1.) Furthermore, for any two indexes i, j ∈ {1, . . . , n}, there exists a unique index k = k(i, j) such that Eij := x−1 k ((xi D)(xj D)) ⊂ 1 (clearly Eij is compact). By our assumption 1=

∞ [

Dm where Dm = D · · · D (m times),

m=1

so by Baire’s theorem (cf. Bourbaki [1998b, Chapter 9, §5]), there exists t ≥ 1 such that Dt contains an open subset U (containing the identity e) of 1, and

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S m then 1 = ∞ m = 1 UD is an open cover of 1. Since Eij is compact, there exists an integer `(i, j) ≥ 1 such that Eij ⊂ D`(i,j) , hence (xi D)(xj D) ⊂ xk(i,j) D`(i,j) . Set ` =

max `(i, j), and noting that Dk1 ⊂ Dk2 for k1 ≤ k2 , consider all

i,j = 1,...,n

relations of the form (xi a)(xj b) = xk(i,j) d1 . . . d` for i, j = 1, . . . , n,

(3.24)

with a, b, d1 , . . . , d` ∈ D, that hold in 0. To complete the proof of the lemma, we will show that the relations (3.23) and (3.24) define 0. Indeed, let N be the normal subgroup of the free group F(C) generated by the elements corresponding to the relations (3.23) and (3.24). Set H = F(C)/N, and let κ : F(C) → H and δ : H → 0 be the corresponding homomorphisms. Let L = κ( f (D)). Since the relations (3.23) are satisfied in L, δ restricts to an isomorphism of L to 1, and therefore ker δ ∩ L = {e}. On the other hand, the fact that the relations (3.24) hold in H implies that any element in H can be written in the form xk y with k ∈ {1, . . . , n} and y ∈ L. So, [H : L] ≤ n = [0 : 1], and therefore [H : L] = [0 : 1]. It follows that ker δ = {e}, proving the lemma.

3.5 Basic Results from Measure Theory This section assembles some notions and results from the theory of integration that will be needed later on. While these concepts play a key role in many results on lattices and arithmetic groups (cf. Margulis [1991]) and also in the analytic aspects of the arithmetic theory of algebraic groups (Tamagawa numbers, applications to the theory of automorphic forms), the material we treat in this book relies on them only in a rather limited way (although they are indispensable for the proofs of such central results as the strong approximation theorem [AGNT, Chapter 7]). So, the goal of this section is to remind the reader of the basic definitions of measure theory, to state the results that will be used in the sequel, and to discuss some examples. A systematic exposition of measure theory and the theory of integration on (locally compact) topological groups (including the proofs of the results stated below) can be found in Bourbaki (2004). Let X be a locally compact topological space. Recall that B ⊂ X is a Borel subset if it can be formed from open (equivalently, from closed) sets through the operations of countable union, countable intersection, and complement; in

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other words, it is an element of the σ -algebra of subsets of X generated by the open (equivalently, closed) subsets. A nonzero measure µ on X is a (locally finite) Borel measure if all the Borel subsets are µ-measurable and µ(C) < ∞ for every compact C ⊂ X . Suppose that a group 0 acts on X by homeomorphisms. Then µ is said to be 0-invariant if for every measurable subset M ⊂ X and every γ ∈ 0, the set γ (M) is also measurable and µ(γ (M)) = µ(M). One of the important examples for us is a locally compact topological group G acting on itself by left (resp., right) translations. In this case a (nonzero) invariant Borel measure on G is called a left (resp., right) Haar measure. Theorem 3.60 Let G be a locally compact topological group. Then there is a left (resp., right) Haar measure on G. This measure is unique up to multiplication by a positive constant. (Note that if µ is a left Haar measure on G, then µ, ˆ given by µ(X ˆ ) = µ(X −1 ) for all X ⊂ G such that X −1 is µ-measurable, is a right Haar measure on G. So, the existence and the uniqueness of the left Haar measure automatically yield the corresponding assertions for the right Haar measure, and vice versa.) Somewhat later, we will describe an explicit construction of the Haar measure in certain cases of special interest to us, but first we would like to list a few general properties of the Haar measure. Proposition 3.61 Let G be a locally compact topological group, and let µ be a (left) Haar measure on G. Then (1) G is discrete if and only if µ({e}) > 0; (2) G is compact if and only if µ(G) < ∞. In particular, if G is compact, then there is a unique (left) Haar measure µ on G such that µ(G) = 1. The uniqueness of the Haar measure in Theorem 3.60 can be used to assign to a (topological) automorphism ϕ of a locally compact topological group G a positive number, called the modulus of ϕ and denoted modG ϕ. More precisely, fix a left Haar measure µ on G, and set ν(X ) = µ(ϕ(X )) for any subset X ⊂ G such that ϕ(X ) is µ-measurable. Since ϕ takes Borel sets to Borel sets and compact sets to compact sets, ν will also be a left Haar measure on G. Thus, by uniqueness, we have ν = cµ for some constant c ∈ R, c > 0. It is easy to see that c is in fact independent (in the obvious sense) of the choice of the Haar measure µ, and we define modG ϕ = c. (Example: if Kv is a locally compact field and a ∈ Kv∗ , then the modulus of the left multiplication x 7→ ax, viewed as

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an automorphism of the additive group Kv+ , is equal to the value kakv of the normalized valuation; cf. §1.2.1.) For x ∈ G, we let 1G (x) denote the modulus of the corresponding inner automorphism ϕx : g 7→ xgx−1 . Then the function 1G : G → R+ is called the modular function of G; it is obviously a continuous group homomorphism. If 1G ≡ 1, then the group G is said to be unimodular. Every left Haar measure µ on a unimodular group G is also a right Haar measure; moreover, in this case, µ(X ) = µ(X −1 ) for every measurable subset X ⊂ G. Proposition 3.62 (1) Any abelian locally compact topological group is unimodular. (2) The modulus of any automorphism of a discrete or compact group equals 1; in particular, such groups are unimodular. Next, we will describe how to obtain a Haar measure on various grouptheoretic constructions from the Haar measures on the groups involved in these constructions. To treat finite direct products, it is clearly sufficient to consider the case of two factors. So, let G = G1 × G2 , where Gi for i = 1, 2 is a locally compact topological group with a Haar measure µi . Then G has a unique measure µ = µ1 × µ2 such that for any µi -measurable subsets Mi of Gi for i = 1, 2, the set M = M1 × M2 is µ-measurable and µ(M) = µ1 (M1 )µ2 (M2 );

(3.25)

moreover, µ is in fact a Haar measure. More generally, the product measure µ = µ1 ×µ2 can be defined by (3.25) on any product X = X1 ×X2 , where Xi is a locally compact topological space equipped with a measure µi for i = 1, 2. One can reformulate (3.25) in terms of integrals with respect to the corresponding measures. Namely, let fi be a µi -integrable function on Xi with the integral R Xi fi (xi )dµ(xi ), where i = 1, 2. Then the function f on X = X1 × X2 defined by f (x1 , x2 ) := f1 (x1 )f2 (x2 ) is µ-integrable and Z Z Z f (x1 , x2 )dµ(x1 , x2 ) = f1 (x1 )dµ1 (x1 ) f2 (x2 )dµ2 (x2 ). X

X1

X2

More generally, for any integrable function f on X , we have Fubini’s Theorem:  Z Z Z f (x1 , x2 )dµ(x1 , x2 ) = f (x1 , x2 )dµ2 (x2 ) dµ1 (x1 ) X X1 X2  Z Z = f (x1 , x2 )dµ1 (x1 ) dµ2 (x2 ). X2

X1

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Note that, in general, one may not be able to extend the definition of the product of measures to an infinite number of factors as the product of infinitely many locally compact non-compact groups is not a locally compact group (for the product topology). To deal with this problem, one needs to use some modifications of the direct product. One such modification, which is important for us, is the restricted topological product that formalizes the construction used to define adeles (cf. §1.2). Definition Let {Xλ }λ ∈ 3 be a family of locally compact topological spaces, indexed by a countable set of indices 3. Assume that for almost all λ ∈ 3, we are given compact open sets Kλ ⊂ Xλ . Then consider the subset X of the direct Q product λ ∈ 3 Xλ consisting of x = (xλ )λ ∈ 3 such that xλ ∈ Kλ for almost all λ ∈ 3. Furthermore, endow X with the topology that has all sets of the Q form λ Uλ , where Uλ ⊂ Xλ is open for all λ ∈ 3 and Uλ = Kλ for almost all λ ∈ 3, as a basis. The topological space X thus obtained is called the restricted topological product of Xλ with respect to the distinguished subsets Kλ (λ ∈ 3). Let us point out some straightforward properties of this construction. Lemma 3.63 (1) Given a finite subset S ⊂ 3 such that Kλ is defined for all λ ∈ 3\S, set Y Y XS = Xλ × Kλ . λ∈S

λ ∈ 3\S

Then XS is open in X in the restricted product topology, and the latter induces the direct product topology on XS . S (2) Each XS is locally compact and X = S XS , where the union is taken over all finite subsets S ⊂ 3 as above. Consequently, X is locally compact. (3) If {Gλ }λ ∈ 3 is a family of locally compact topological groups and the distinguished subsets Kλ ⊂ Gλ are open compact subgroups for almost all λ ∈ 3, then the restricted topological product G of the Gλ with respect to the Kλ is a locally compact topological group. In view of Theorem 3.60, it follows from (3) that G possesses a Haar measure. We will now show how this measure can be constructed explicitly from the Haar measures µλ on Gλ . First, let us assume that the measures µλ are normalized so that µλ (Kλ ) = 1 for all λ ∈ 3 such that Kλ is defined. Then, given a finite subset S ⊂ 3 as in Lemma 3.63(1), one can define a measure µS on Q Q GS = λ ∈ S Gλ × λ ∈ 3\S Kλ , which should be thought of as the “infinite” product of the µλ . More precisely, we set µS = µ1 × µ2 , where µ1 is the usual

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Q finite product of the µλ for λ ∈ S on λ ∈ S Gλ and µ2 is the Haar measure Q on the compact group KS := λ ∈ 3\S Kλ normalized so that µ2 (KS ) = 1. It is easy to see that if S1 ⊂ S2 , then GS1 ⊂ GS2 , and the restriction of µS2 to GS1 S coincides with µS1 . Since G = S GS , we can now use countable additivity to extend the measures µS to a measure µ on G, which will be the desired Haar measure. We note, however, that sometimes in defining a measure on G, it is convenient not to impose the condition µλ (Kλ ) = 1 but rather to require that the Q product λ µλ (Kλ ) be absolutely convergent. Then an obvious modification of the preceding construction yields a Haar measure µ0 on G that is related to the measure µ by Y µ0 = cµ, where c = µλ (Kλ ). λ

Next, we will consider the problem of constructing an invariant measure on the homogeneous space X = G/H of a locally compact topological group G modulo a closed subgroup H, on which G acts by left multiplication. Theorem 3.64 The homogeneous space X = G/H supports a nonzero Ginvariant Borel measure β if and only if the restriction of the modular function 1G to H coincides with the modular function 1H , in which case β is unique up to a positive scalar. In particular, if 0 is a discrete subgroup of G (which will be the case of principal interest to us in the sequel), then an invariant measure on G/ 0 exists if and only if 0 lies in the kernel of 1G . Furthermore, the phrase “G/ 0 has finite invariant measure (or volume)” means that G/ 0 supports an invariant measure and its volume with respect to this measure is finite. We note that if G possesses a lattice, which is a discrete subgroup 0 ⊂ G such that G/ 0 has finite invariant measure, then G is necessarily unimodular. Returning to the general situation considered in Theorem 3.64, we would like to recall how the “quotient measure” β (if it exists) is related to the Haar measures µ and ν on G and H, respectively. This is most easily done in terms of the integrals over these measures. So, let f be an integrable function on G. For g ∈ G, set Z ϕ(g) = f (gh)dν(h). H

Then ϕ is a function on G, which is constant on cosets modulo H, hence can be regarded as a function on X = G/H. Under this convention, we have the following formula:

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 Z f (gh)dν(h) dβ(gH) = f (g)dµ(g). H

(3.26)

G

In the general case, this equation may not yield an explicit description of β. However, in the case of a discrete subgroup 0 ⊂ G (which is of primary interest to us), integration over G/ 0 with respect to β can be replaced by integration over a certain subset of G with respect to the original measure µ. (To make this transition work, one needs to assume, in addition, that G is second-countable, which holds in all situations relevant for us.) A subset F ⊂ G is called a fundamental domain for a discrete subgroup 0 ⊂ G if the restriction to F of the quotient map π : G → G/ 0 is bijective. This obviously amounts to the following two conditions: (1) G = F0, (2) F ∩ Fγ = ∅ for any γ 6= e in 0. In the situation at hand, there always exists a µ-measurable fundamental domain F ⊂ G; in fact, a fundamental domain can be found inside an arbitrary measurable subset  ⊂ G satisfying π() = G/ 0. Then (3.26) yields Z Z ϕ(x)dβ(x) = ϕ(π(g))dµ(g) (3.27) X

F

for any function ϕ integrable over X . Sometimes it is convenient to use a slightly less restrictive definition of a fundamental domain in which condition (1) remains the same, but condition (2) is replaced by the following: (20 ) F ∩ Fγ has measure zero for any γ 6= e in 0. This more general definition allows us to work with fundamental domains F ⊂ G, which are closed sets that satisfy π( F) = G/ 0, have boundary of measure zero, and possess the property that their translates under different elements of 0 intersect only in the boundary – cf. the classical fundamental domain for SL2 (Z) in SL2 (R) described in §4.2. Using the fact that a discrete subgroup of a second-countable topological group is necessarily countable, it is easy to show that (3.27) remains valid for this more general definition of a fundamental domain. Taking ϕ ≡ 1 in (3.27), we see that if X = G/ 0 has an invariant measure, then the measure of X is finite if and only if there exists a measurable fundamental domain F ⊂ G for 0 of finite measure, and then any other measurable fundamental domain also has finite measure. Since a measurable fundamental domain can be found inside any measurable set  ⊂ G with the property π() = X , we conclude that X has finite measure if and only if there exists a measurable set  ⊂ G satisfying π() = X and µ() < ∞.

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Example 3.65 Let G = Rn . Then the usual Lebesgue measure on G is a Haar measure (both left and right). Any x ∈ GLn (R) can be viewed as a topological automorphism of G, and then it follows from the formula for the change of variables for multiple integrals that modG x = |det x|. In particular, the transformations from SLn (R) have modulus 1, hence are measure-preserving. Now let e1 , . . . , en be a basis of Rn , and let 0 denote the lattice Ze1 ⊕ · · · ⊕ Zen . Then 0 is a discrete subgroup of Rn , and F = {t1 e1 + · · · + tn en : 0 ≤ ti < 1} is a fundamental domain (neither open nor closed) satisfying (1) and (2), and F 0 = {t1 e1 + · · · + tn en : 0 ≤ ti ≤ 1} is a closed fundamental domain satisfying (1) and (20 ). The preceding example is atypical in the sense that, in general, it is very difficult to give an explicit construction of a fundamental domain satisfying (1)-(2) or (1)-(20 ). For this reason, in Chapters 4 and 5, we will use the following more general concept: a (closed) subset 8 ⊂ G is called a fundamental set for a discrete subgroup 0 ⊂ G if G = 80 and 8−1 8 ∩ 0 is finite. Such a set is sufficient to decide whether or not the volume of X = G/ 0 is finite (although it can hardly be used to compute this volume precisely). Indeed, if there is a fundamental set 8 of finite measure, then it follows from the preceding remark that X has finite measure. Conversely, given any fundamental set S 8, we can find a measurable fundamental domain F ⊂ 8, and then 8 ⊂ Fγ , where γ runs through a finite subset of 0. So, if X has finite measure, then so does F, and hence 8. Thus, X has finite measure if and only if there exists a fundamental set 8 ⊂ G of finite measure. Let us point out one straightforward property of quotient measures (cf. Raghunathan [1972], Lemma 1.6): let H1 ⊂ H2 be two closed subgroups of a locally compact group G; if G/H1 has finite G-invariant measure, then so does G/H2 . Using this fact, we obtain the following. Lemma 3.66 Let G = G1 × G2 be the direct product of two locally compact topological groups, with G1 noncompact, and let pi : G → Gi be the respective projections. Let H ⊂ G be a closed subgroup for which the intersection H ∩ (G1 × {e}) is trivial and the quotient G/H has finite invariant measure. Then p2 (H) is nondiscrete and G2 /p2 (H) (where the bar denotes closure) has finite invariant measure. Indeed, assuming that p2 (H) is discrete, we can find a neighborhood of the identity U ⊂ G2 such that U −1 U ∩ p2 (H) = {e}. Then our assumption that the

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intersection H ∩ (G1 × {e}) is trivial implies that the restriction to G1 × U of the canonical homomorphism G → G/H is one-to-one, and therefore G1 × U has finite measure. Let µi denote the Haar measure on Gi ( for i = 1, 2), and let µ = µ1 × µ2 . Then µ(G1 × U) = µ1 (G1 )µ2 (U), and since µ2 (U) 6= 0 by the openness of U, we see that µ1 (G1 ) is finite. But then G1 is compact (Proposition 3.61), a contradiction. The assertion about the existence of a finite invariant measure for G2 /p2 (H) follows from the preceding remark applied to the closed subgroup H ⊂ G1 × p2 (H) of G since G2 /p2 (H) ' G/(G1 × p2 (H)). Next, we will construct the Haar measure explicitly in some cases of interest. First, we mention the following result, which, in conjunction with the Iwasawa decomposition (cf. §3.2), yields a Haar measure on the group of real points of a real reductive algebraic group. Proposition 3.67 Let G be a unimodular locally compact topological group and let H, A, and U be closed subgroups of G such that the product map H × A × U → G is a homeomorphism. Assume that A and U are unimodular and that A normalizes U. If ν, θ, and ω are left Haar measures on H, A, and U, respectively, then µ(g) = ρ(a)ν(h) × θ(a) × ω(u), where g = hau and ρ(a) for a ∈ A is the modulus of the automorphism of U given by u 7→ aua−1 , is a left Haar measure on G. The notation for µ in the proposition means that the measure of E ⊂ G is given by Z µ(E) = ρ(a)dν(h)dθ(a)dω(u). E

Other explicit examples of Haar measures can be obtained by integrating differential forms. (The necessary background on differential forms, though in the context of real manifolds, can be found in any book on differential geometry, such as Helgason [2001].) Let X be an n-dimensional analytic manifold over a nondiscrete locally compact field Kv , let x0 ∈ X , and let x1 , . . . , xn be local coordinates in a neighborhood of x0 . This means that the xi are analytic functions such that the map ϕ : x 7→ (x1 (x), . . . , xn (x))

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193

is an analytic isomorphism of a neighborhood of x0 onto an open subset of Kvn (in other words, ϕ is the inverse map of some parametrization of a neighborhood of x0 ), or, equivalently, the differential dx0 ϕ is a linear isomorphism between the tangent space Tx0 (X ) and Kvn . A differential form of degree n in a neighborhood of x0 is an expression of the form ω = f (x)dx1 ∧ · · · ∧ dxn , where f is an analytic function in the neighborhood of x0 . Let F : Y → X be an analytic map between two n-dimensional analytic manifolds, let x0 ∈ X and y0 ∈ Y be such that F(y0 ) = x0 , and let x1 , . . . , xn and y1 , . . . , yn be local coordinates in suitable neighborhoods of x0 and y0 , respectively. In terms of these coordinates, we can write F in the form (y1 , . . . , yn ) 7→ ( F1 (y1 , . . . , yn ), . . . , Fn (y1 , . . . , yn )), and then the pullback of the differential form ω = f (x)dx1 ∧ · · · ∧ dxn is defined to be F ∗ (ω) = f ( F(y))dF1 (y1 , . . . , yn ) ∧ · · · ∧ dFn (y1 , . . . , yn ), where, as usual, dFi (y1 , . . . , yn ) =

n X ∂Fi dyj . ∂yj

j=1

This equation, in particular, describes how a local differential form transforms under a change of local coordinates. We can now define a (global) n-dimensional differential form on a manifold X as a family of n-dimensional local differential forms with domains covering X that agree in a neighborhood of each point under the above transition rule for differential forms in different coordinate systems. We say that a differential form ω on X is invariant under an analytic automorphism F : X → X if F ∗ (ω) = ω. In the rest of this section, we will deal exclusively with analytic manifolds that arise from algebraic varieties, so we will now briefly describe the interface between the “analytic” and “algebraic” notions. Let X be a smooth algebraic variety defined over a field K such that all irreducible components of X have the same dimension n. Then a K-defined system of local parameters in a neighborhood of x0 ∈ XK is a system of K-rational functions x1 , . . . , xn defined at x0 such that the differential dx0 ϕ of the rational map ϕ : X → An given by ϕ : x 7→ (x1 (x), . . . , xn (x)), is an isomorphism of tangent spaces. Then an ndimensional differential form over K in a neighborhood of x0 is defined as an expression of the form ω = f (x)dx1 ∧ · · · ∧ dxn , where f is a K-rational function on X defined at x0 . Just as in our earlier discussion, one can describe

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the transformation rule for differential forms under rational maps, and then define global and invariant differential forms. Note that if X is defined over a nondiscrete locally compact field Kv and x0 ∈ XKv , then any rational differential Kv -form in a Zariski-neighborhood of x0 can also be regarded as an analytic differential form on XKv in a (usual) neighborhood of x0 . Now let G be a connected algebraic K-group and let n = dim G. It is well known that there exists a nonzero n-dimensional rational differential K-form ω on G that is left-invariant (i.e., invariant under left translations), and moreover this form is unique up to multiplication by a nonzero element of K. Here are some examples. Example 3.68 Let G = GLn . The natural coordinate functions xij for i, j = 1, . . . , n form a system of local parameters. Let ω = f (X )dx11 ∧ · · · ∧ dxnn , where X = (xij ), be a left-invariant differential form. Fix A = (aij ) ∈ GLn . Then the left translation λA : X 7→ AX can be written in coordinates as X x0ij = aik xkj . k

λ∗A

of f (X 0 )dx011 ∧ · · · ∧ dx0nn is ! ! X X f (AX ) d a1k xk1 ∧ · · · ∧ d ank xkn

It follows that the pullback under

k

k

= f (AX ) (det A) dx11 ∧ · · · ∧ dxnn . n

Using that ω is left-invariant, we obtain f (X ) = f (AX )(det A)n . Setting X = In yields f (A) = c(detA)−n , where c = f (In ), hence ω=

c dx11 ∧ · · · ∧ dxnn
n . det(xij )

In particular, for n = 1, we get ω = c xdx . Example 3.69 Let G = SL2 . Writing X ∈ G in the form   x y X= , z t we observe that the coordinate functions x, y, and z form a system of local parameters in a neighborhood of the identity, and that t = 1+yz x . Let

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195

ω = f (X )dx ∧ dy ∧ dz be a left-invariant differential form. In terms of these   a b coordinates, left translation by A = ∈ G is given by c d x0 = ax + bz, 1 + yz y0 = ay + b , x z0 = cx + dz. So, the pullback of f (X 0 )dx0 ∧ dy0 ∧ dz0 under λ∗A is   1 + yz f (AX )d(ax + bz) ∧ d ay + b ∧ d(cx + dz). x Then an easy computation shows that the left-invariance of ω amounts to the equation ax + bz f (AX ) = f (X ). x Noting that ax + bz is the (11)-entry of AX , we see that f (AX )(AX )11 = f (X )(X )11 , hence ω is of the form h ω = dx ∧ dy ∧ dz. x We will later need an expression for 1 ω = dx ∧ dy ∧ dz x in a different coordinate system. More precisely, the Iwasawa decomposition (cf. §3.2) asserts that any matrix GR = SL2 (R) can be uniquely written as a product of three matrices having, respectively, the following shape       cos ϕ − sin ϕ α 0 1 u , with α > 0, and . sin ϕ cos ϕ 0 α −1 0 1 Then we can take ϕ, α, and u as (analytic) coordinates on GR . A direct computation shows that x = α cos ϕ, y = αu cos ϕ − α −1 sin ϕ, and z = α sin ϕ, so ω = α dϕ ∧ dα ∧ du. Example 3.70 (Exercise) Let Un be the group of upper triangular unipotent V (n × n)-matrices. Show that the differential form dxij is a left-invariant form i 2/ 3 and v > 1/2 is Indeed, the interior 6t,v easily seen to be a fundamental set for 0 = GLn (Z). Starting with this set and using the above procedure, one can construct an open fundamental set for an arbitrary 0.

4.3 Reduction in Arbitrary Groups In this section, we will implement the second step of the plan outlined in §4.2, and will thereby establish the existence of fundamental sets for arithmetic subgroups of an arbitrary connected algebraic Q-group G. The following assertion reduces our task to the case of a reductive group. Lemma 4.22 (1) Let N be a unipotent Q-group. Then there exists an open relatively compact subset U ⊂ NR such that NR = UNZ and the intersection U −1 U ∩ (nNZ m) is finite for all n, m ∈ NQ . (2) Let G = HN be a Levi decomposition of a connected Q-group G, where H is a maximal reductive Q-subgroup of G and N = Ru (G) is the unipotent radical. Given a subset 6 ⊂ HR that satisfies (a) HR = 6HZ , and (b) 6 −1 6 ∩ (gHZ h) is finite for all g, h ∈ HQ ,

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the subset  = 6U, where U is as in (1), satisfies (α) GR = GZ , and (β) −1  ∩ (xGZ y) is finite for all x, y ∈ GQ . PROOF: (1) It suffices to show that NR /NZ is compact as then an elementary topological argument (cf., for example, Bourbaki [1998b, Chapter 3]) yields an open relatively compact subset U ⊂ NR such that NR = UNZ . The finiteness of U −1 U ∩ (nNZ m) for such a U follows from the fact that nNZ m is discrete and closed. The compactness of NR /NZ is easily established by induction on r = dim N (cf. the proof of Lemma 4.13). For r = 1, there is a Q-isomorphism N ' C, which induces the identifications NR ' R and NZ ' aZ for some nonzero a ∈ Q. So, NR /NZ ' R/aZ is the 1-dimensional compact torus. Now let r > 1. Since N/[N, N] is abelian and unipotent, the truncated log∼ arithmic map (see §2.1.8) yields a Q-isomorphism N/[N, N] → C` , where ` = dim N/[N, N]. (We note that ` ≥ 1 as N is nilpotent.) It follows that there is an (r − 1)-dimensional normal Q-subgroup M C N, and then one can find a 1-dimensional Q-subgroup L ⊂ N such that N = LM, a semidirect product over Q. By induction, the quotients LR /LZ and MR /MZ are compact, and therefore (as above), there exist compact subsets A ⊂ LR and B ⊂ MR such that LR = ALZ and MR = BMZ . Let us show that the compact set C = AB satisfies NR = CNZ . Take n = lm ∈ NR = LR MR . Then l = az for some a ∈ A and z ∈ LZ , and zmz−1 = bx for some b ∈ B and x ∈ MZ . Then n = lm = azm = azmz−1 z = abxz with xz ∈ NZ . This completes the proof of (1). (2) The proof of (α) is similar to the concluding part of the argument used in item (1). So, we will only give an argument for (β). According to Corollary 4.4, the subgroup HZ NZ is of finite index in GZ . Therefore, decomposing GZ into right or left cosets modulo HZ NZ , we see that (β) is equivalent to −1  ∩ (xHZ NZ y) being finite for arbitrary x, y ∈ GQ . Since GQ = HQ NQ , we can write x = ab and y = cd with a, c ∈ HQ and b, d ∈ NQ . Let h ∈ HZ and n ∈ NZ . Then 6Uxhny = 6Uabhncd = 6(ahc)U 0 g,

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where U 0 = (ahc)−1 Uabhc is a compact subset of NR and g = c−1 ncd ∈ NR . So, 6U ∩ 6U(xhny) 6= ∅ holds if and only if 6 ∩ 6(ahc) 6= ∅, and 0

U g ∩ U 6= ∅.

(4.11) (4.12)

According to (b), the number of h’s satisfying (4.11) is finite. On the other hand, the number of possible n’s of the form n = cgd −1 c−1 with g satisfying (4.12) is also finite as the set (U 0 )−1 U is relatively compact and g in (4.12) belongs to the closed discrete set c−1 NZ cd. Thus, we will assume henceforth that G is a reductive group. The strategy for obtaining a fundamental set in this case has already been discussed (cf. Lemma 4.10): if G ⊂ GLn (C), then we need: (1) to define a (right) action of GLn on some set X , such that G is the stabilizer of a suitable point x in X ; (2) to find a in GLn (R) for which xa6 ∩ xGLn (Z) is finite, where 6 is a Siegel set in GLn (R). Then we will have a fundamental set for G of the form ! r [ −1 = a6bi ∩ G, where bi ∈ GLn (Z). i=1

We will take the required X to be the vector space V of a Q-defined representation % : GLn (C) → GL(V ) such that G = {g ∈ GLn (C) : v%(g) = v} is the stabilizer of some v ∈ VQ (see the remark at the end of §2.4.5 about right actions). The existence of such a representation is guaranteed by the stronger version of Chevalley’s theorem (cf. Theorem 2.49), which also asserts that for v as above, the orbit v%(GLn (C)) is Zariski-closed. Then, if we choose a ∈ GLn (R) such that a−1 Ga is self-adjoint, i.e., invariant under taking the transpose (cf. Theorem 3.24), and let v0 = va, the finiteness of the required intersection follows from Proposition 4.23 Let % : GLn (C) → GL(V ) be a Q-representation and let L be a lattice in VQ . If v0 ∈ VR is such that the stabilizer G = {g ∈ GLn (C) : v0 %(g) = v0 }

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223

is a self-adjoint group and the orbit v0 %(GLn (C)) is Zariski-closed in V , then the intersection v0 %(6) ∩ L is finite for every Siegel set 6 ⊂ GLn (R). PROOF: Choose a basis in VQ consisting of eigenvectors with respect to %(Dn ), where Dn is the group of diagonal matrices, and equip V with an inner product for which this basis is orthonormal. Let k k be the Euclidean norm on VR associated with this inner product. For a character µ ∈ X(%(Dn )), we set Vµ = {v ∈ V : v%(g) = µ(g)v ∀g ∈ Dn } to be the corresponding weight space; in the sequel, we will deal only with those µ for which Vµ is nonzero. Then Vµ1 is orthogonal to Vµ2 for µ1 6= µ2 , L so V = µ Vµ is an orthogonal direct sum. Let πµ denote the orthogonal projection of V onto Vµ . Since Vµ is defined over Q, the Z-submodule generated by all intersections L ∩ Vµ has finite index in L. Thus πµ (mL) ⊂ L for a suitable integer m, and consequently πµ (L) ⊂ m1 L is a lattice in Vµ . Therefore, there exists a constant c1 > 0 such that kπµ (w)k ≥ c1 for all w ∈ L and all µ such that πµ (w) 6= 0. −2 Now, let x = kx ax ux ∈ GLn (R). Set yx = xa−1 x and zx = xax and write vx −1 instead of v%(x). The set of elements of the form ax ux ax , where x ∈ 6, is relatively compact by Lemma 4.11. Since yx = kx ax ux a−1 x , it follows that there exists c2 > 0 such that kv0 yx k ≤ c2 for all x in 6. We claim that the set 1 = {v0 zx : x ∈ 6, vx ∈ L} 2 is bounded. Indeed, set c = c−1 1 c2 . We have −1 πµ (v0 yx ) = πµ (v0 xa−1 x ) = µ(ax ) πµ (v0 x),

and similarly πµ (v0 zx ) = µ(ax )−2 πµ (v0 x), implying that kπµ (vzx )k =

c2 kπµ (vyx )k2 ≤ 2 = c. kπµ (vx)k c1

Since the orbit v0 %(GLn (C)) is Zariski-closed in V , the orbit v%(GLn (R)) is closed in VR in the Euclidean topology (cf. proof of Corollary 3.18). Therefore, the set W = {w ∈ v%(GLn (R)) : kπµ (w)k ≤ c for all µ}

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is compact. Using the fact that the map GLn (R) → v0 %(GLn (R)), g 7→ v0 %(g) is open (cf. Corollary 3.9), one easily finds a compact subset U ⊂ GLn (R) such that W = v%(U). Since 1 ⊂ W , we have the inclusion {zx : vzx ∈ 1} ⊂ GR U. But −1 2 −2 zx = kx ax ux a−2 x = kx ax ax ux ax ,

and therefore for x ∈ 6 = 6t,b , we have a2x ∈ At2 , implying that the set {a2x ux a−2 x : x ∈ 6} is relatively compact. Thus, kx a−1 x ∈ GR U1 for a suitable compact subset U1 of GLn (R). Applying the automorphism θ : g 7→ tg−1 and bearing in mind that kx is an orthogonal matrix and ax a diagonal matrix, we see that kx ax ∈ GR θ(U1 ) since G is self-adjoint. Since ux lies in a compact subset U2 ⊂ GLn (R), we deduce that x = kx ax ux ∈ GR U3 , where U3 := θ(U1 )U2 is compact. Thus, v0 %(6) ∩ L is contained in vU3 , and consequently is both compact and discrete, hence finite, as needed. This completes the construction of a fundamental set in a reductive group. The results obtained so far are summarized in Theorem 4.24 (BOREL AND HARISH-CHANDRA [1962]) Let G ⊂ GLn (C) be a reductive algebraic Q-group, and let 6 = 6t,v (t ≥ √2 , v ≥ 12 ) be a Siegel 3 set of GLn (R). Then we can find a ∈ GLn (R) and b1 , . . . , br ∈ GLn (Z) such that ! r [ = a6bi ∩ G i=1

has the following properties: (0) K =  for a suitable maximal compact subgroup K of GR ; (1) GZ = GR ; (2) −1  ∩ xGZ y is finite for all x, y in GQ . It remains only to prove (0). Recall that a ∈ GLn (R) has been chosen to satisfy the requirement that a−1 Ga be self-adjoint. But then a−1 Ga ∩ On (R)

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225

is a maximal compact subgroup of a−1 GR a (cf. Proposition 3.26), so K = G ∩ (aOn (R)a−1 ) is a maximal compact subgroup of GR . By construction 6 satisfies On (R)6 = 6, from which (0) clearly follows. As in §4.2, the results of this section can be restated more concisely using the concept of a fundamental set. Its definition for the general case is analogous to the definition given in §4.2 for the case of GLn (C) and is as follows: Definition Let G be an algebraic Q-group, and let 0 ⊂ GQ be an arithmetic subgroup. A subset  ⊂ GR is a fundamental set for 0 if ( F0) K =  for a suitable maximal compact subgroup K ⊂ GR ; ( F1) 0 = GR ; ( F2) −1  ∩ (xGZ y) is finite for all x, y in GQ . In view of Lemma 4.22 and the absence of compact subgroups in unipotent groups, Theorem 4.24 yields Corollary 4.25 Let G be a connected Q-group, and let 0 ⊂ GQ be an arithmetic subgroup. Then there exists an open fundamental set for 0 in GR . This corollary will serve as the basis for the structure theorems for arithmetic group that we will establish in §4.4, and, in fact, each of the three conditions ( F0)−( F2) will play a role. More precisely, conditions ( Fl) and ( F2) alone guarantee that 0 is finitely generated. Condition ( F0) means that the image of  in the symmetric space X = K\GR is a fundamental set for the induced action of 0 on X , which, in conjunction with the fact that X is simply connected, implies that 0 can be defined by a finite number of relations. Another application of reduction theory is the following finiteness theorem for the orbits of arithmetic groups. Theorem 4.26 Let % : G → GL(V ) be a Q-representation of a reductive Qgroup G, let 0 ⊂ GQ be an arithmetic subgroup, and let L ⊂ VQ be a 0invariant lattice. If the orbit X = v%(G) of v ∈ VQ is Zariski-closed, then X ∩ L is the union of a finite number of orbits of 0. PROOF: Let G ⊂ GLn (C). First, we consider the special case where % is the restriction of a Q-representation π : GLn (C) → GL(V ), such that the orbit

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Y = vπ(GLn (C)) is Zariski-closed and the stabilizer of v under π lies in G, i.e., equals the stabilizer H of v under %. According to Corollary 3.18, the set XR is S the union of a finite number of orbits of GR , i.e., XR = i vi %(GR ). To prove the theorem, one needs to consider only those i for which vi %(GR ) ∩ L 6= ∅, and then we may assume that vi ∈ L. So, it suffices to show that v%(GR ) ∩ L consists of a finite number of orbits of 0. Moreover, without loss of generality, we may assume that 0 = GZ , and that L is invariant under %(GLn (Z)) (cf. the remark after Proposition 4.5). By Theorem 4.24, we can find a ∈ GLn (R) and b1 , . . . , br ∈ GLn (Z) such that ! GR =

[ (a6bi )

! ∩ G GZ

(4.13)

i

for a suitable Siegel set 6 ⊂ GLn (R). Moreover, for a we can take any element of GLn (R) for which a−1 Ga is self-adjoint. To proceed with the argument, we need to observe that the stabilizer H is a reductive group, which is a consequence of the Zariski-closedness of the orbit X = v%(G) and the following well-known result often referred to as the Matsushima criterion: For a complex reductive algebraic group G and a subgroup H, the quotient G/H is affine if and only if H is reductive. (Of course, the if part is an immediate corollary of the strong version of Chevalley’s Theorem 2.49.) The Matsushima criterion was originally established independently by Matsushima (1961) and Onishchik (1960); for other proofs see Bialynicki-Birula (1963), Haboush (1978), and Alper (2014) – the last article contains the most general form of the result and also some historical remarks. Since H is reductive, using Theorem 3.29, we can choose a ∈ GLn (R) so that both groups a−1 Ga and a−1 Ha are self-adjoint. It follows from (4.13) that it suffices to establish that for any b ∈ GLn (Z), the intersection v%(a6b ∩ GR ) ∩ L is finite. As L is invariant under %(GLn (Z)), for this it is enough to show that for w = v%(a), the intersection w%(6) ∩ L is finite. However, since the stabilizer of w is the self-adjoint group a−1 Ha, the required finiteness follows from Proposition 4.23. We will now reduce the general case to the special case considered above. Since H is reductive, we can apply the strong version of Chevalley’s theorem (Theorem 2.49) to find a Q-representation π : GLn (C) → GL(W ) such that the stabilizer of a suitable vector w in WQ is H and the orbit Y = wπ(GLn (C)) is Zariski-closed. By the remark following Proposition 4.5, we can find a %(GLn (Z))-invariant lattice M ⊂ WQ containing w. Since the canonical map GLn (C) → Y (given by g 7→ wπ(g)) is open, the orbit X 0 = wπ(G) is Zariskiclosed. Thus, the set X 0 ∩ M is the union of a finite number of orbits of 0.

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227

Moreover, passing from M to d1 M (d ∈ Z), which is also %(GLn (Z))-invariant, we see that X 0 ∩ ( d1 M) is the union of a finite number of orbits of 0, for each d in Z. It remains to pass from X 0 to X . To do so, we note that both X and X 0 are realizations of the homogeneous space G/H, so there exists a G-equivariant Q-defined isomorphism ϕ : X → X 0 . Since X is Zariski-closed in V , the isomorphism ϕ in terms of the coordinates with respect to some bases of L and M, respectively, is given by polynomials Pi (x1 , . . . , xr ) (1 ≤ i ≤ s) with rational coefficients. If d is the common denominator of these coefficients, then 1  ϕ(X ∩ L) ⊂ X 0 ∩ M . d Therefore, since the number of orbits of GZ in X 0 ∩ ( d1 M) is finite and ϕ is Gequivariant and injective, the number of orbits of GZ in X ∩ L is also finite. To conclude this section, we present a restatement of condition ( F2) from the definition of a fundamental set, which we will use in the next chapter to develop reduction theory for the groups of adeles. Lemma 4.27 For  ⊂ GR , condition ( F2) is equivalent to the following: ( F2)0 −1  ∩ xGr y is finite, for all x, y in GQ and all nonzero r ∈ Z, where Gr = {g ∈ GQ : rg, rg−1 ∈ Mn (Z)}. PROOF: To prove the implication ( F2) ⇒ ( F2)0 , it suffices to show that Gr is contained in a finite union of cosets of GZ . But if g ∈ Gr , then rZn ⊂ g(Zn ) ⊂ r−1 Zn . Since there are only finitely many lattices between rZn and r−1 Zn , there are only a finite number of possibilities for g(Zn ) (g ∈ Gr ). Noting that S g(Zn ) = h(Zn ) implies h−1 g ∈ GZ , we obtain Gr ⊂ i gi GZ , as desired, where the gi are chosen in such a way that gi (Zn ) run through all possible intermediate lattices of the form g(Zn ) for g in Gr , between rZn and r−1 Zn . The converse implication ( F2)0 ⇒ ( F2) is obvious.

4.4 Group-Theoretic Properties of Arithmetic Groups In this section, we will prove several fundamental results (stated in §4.1) on the abstract properties of arithmetic groups. The elegance and relative brevity of the proofs may be viewed as compensation for the effort needed to develop reduction theory. We will return to this subject in §4.9.

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We begin with Theorem 4.8 Let 0 be an arithmetic subgroup of an algebraic Q-group G. Then 0 is finitely presented as an abstract group, i.e., can be defined by a finite number of generators and defining relations. PROOF OF THEOREM 4.8: It suffices to establish that 0 has a finitely presented subgroup of finite index. Therefore, we may assume that G is connected and that 0 ⊂ GR . We will work in the space X = K\GR , where K is a maximal compact subgroup of GR . By Proposition 3.26, X is connected and simply connected. If 6 is an open fundamental set for 0 in GR (see Corollary 4.25), then since K6 = 6, the following two conditions are satisfied for the image  of 6 in X : (i) 0 = X ; (ii) 1 := {δ ∈ 0 : δ ∩  6= ∅} is finite. (We consider the natural action of 0 on X by right translations.) We will show that the finite presentability of 0 is a formal consequence of the connectedness, local connectedness, and simple-connectedness of X , of the openness of , and of (i) and (ii). Therefore the same result holds for an arbitrary group of transformations of any topological space X satisfying these hypotheses. (We note that Behr [1962] proves this under somewhat weaker hypotheses, i.e., instead of requiring that  be open, he requires only that  lie in the interior of 1.) Lemma 4.28 The set 1 generates the group 0. PROOF: Let 00 be the subgroup generated by 1. Then (i) implies that X = (00 ) ∪ ((0 \ 00 )). Moreover, if γ ∩ δ 6= ∅, where γ ∈ 00 , then δγ −1 ∈ 1, implying δ ∈ 00 ; thus 00 and (0\00 ) are disjoint. Since both are open sets and X is a connected space, we conclude (0\00 ) = ∅, i.e., 0 = 00 , proving the lemma. We now proceed to construct the defining relations for 0. Let F denote the free group on a replica 1 of the set 1, whose elements are in one-to-one correspondence with those of 1 via a bijection δ¯ 7→ δ, and let ϕ : F → 0 denote the homomorphism determined by this bijection. To obtain the defining relations for 0, first of all we consider the local relations, i.e., those of the form δ¯1 δ¯2 (δ1 δ2 )−1 = e, where δ1 , δ2 run through the elements of 1 for which δ1 δ2 ∈ 1.

(4.14)

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229

To understand their role, let L denote the normal subgroup of F generated by the left-hand sides of the local relations. Clearly, ϕ(δ¯1 δ¯2 (δ1 δ2 )−1 ) = 1, so ϕ(L) = 1. Let N be a normal subgroup of F containing L and contained in K = ker ϕ. Consider H = F/N and the natural homomorphisms σ : F → H, θ : H → 0, satisfying θ ◦ σ = ϕ. Also, endowing H with the discrete topology, we introduce the quotient space S of  × H under the following relation: ¯ 2. (x1 , h1 ) ∼ (x2 , h2 ) if there is δ in 1 such that x2 = x1 δ and h1 = σ (δ)h Note that it is precisely relations (4.14) that ensure that ∼ is an equivalence relation, thereby enabling us to define S. Indeed, the reflexivity of ∼ is obvious (since 1 ∈ 1) and symmetry follows from the fact that if δ ∈ 1, then δ −1 ∈ ¯ −1 = σ (δ¯−1 ) by 1 (since δ ∩  6= ∅ ⇔  ∩ δ −1 6= ∅); moreover, σ (δ) virtue of the local relations. Let us prove transitivity. If (x1 , h1 ) ∼ (x2 , h2 ) and (x2 , h2 ) ∼ (x3 , h3 ), then there are δ1 , δ2 in 1 such that x2 = x1 δ1 , x3 = x2 δ2 ,

h1 = σ (δ¯1 )h2 , h2 = σ (δ¯2 )h3 .

Then x3 = x1 δ1 δ2 , so δ1 δ2 ∈ 1, and h1 = σ (δ¯1 )σ (δ¯2 )h3 = σ (δ1 δ2 )h3 , due to the local relations. Now let α :  × H → S denote the canonical map and β :  × 0 → X the ¯ 2 for some “product” map. If (x1 , h1 ) ∼ (x2 , h2 ), then x2 = x1 δ and h1 = σ (δ)h δ in 1, implying ¯ −1 h1 ) = x2 θ(h2 ) = β(x2 , θ(h2 )), β(x1 , θ(h1 )) = x1 θ(h1 ) = (x1 δ)θ (σ (δ) so there exists a unique continuous map p : S → X making the following diagram commutative: ×H  S

(id,θ)

α

/ ×0

(4.15)

β

p

 / X.

Lemma 4.29 The map p is a covering map with fibers of cardinality |ker θ|. PROOF: Put 9 = α( × {e}). Then the preimage [ ¯ −1 ) α −1 (9) = ( ∩ δ , σ (δ) δ∈1

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is an open subset of  × H and hence 9 is open in S. The restriction ∼ of p to 9 is injective and yields a homeomorphism 9 → ; indeed, if p(α(x1 , e)) = p(α(x2 , e)), then x1 = x2 by the commutativity of (4.15). Next, we show that [ p−1 () = 9h, h

where the union is taken over all h ∈ ker θ, and all the 9h are disjoint. (Henceforth, we consider the induced action of H on S, for which it is useful to note the relation p(xh) = p(x)θ(h).) If p(α(x, h)) = y ∈ , then xθ(h) = y ∈ , so ¯ −1 h ∈ ker θ. But then by definition (x, h) ∼ (y, g), θ(h) = δ ∈ 1, i.e., g = σ (δ) as required. Lastly, if α(x, e) = α(y, h), where h ∈ ker θ, then h ∈ 1; thus h = e, since the restriction ϕ|1 is injective. This completes the proof of the lemma. If  is connected, then S is also connected. Indeed, we have [ S= 9h. h∈H

Now, since 9 is connected, it suffices to observe that every translate 9h can be connected to 9 by a chain of pairwise overlapping translates 90 = 9, 91 = 9h1 , . . . , 9m = 9hm , where hm = h. To create such a chain for h = σ (δ¯1 · · · δ¯d ), we set h1 = σ (δ¯d ), h2 = σ (δ¯d−1 δ¯d ), . . . , hm−1 = σ (δ¯2 · · · δ¯d ), hm = σ (δ¯1 · · · δ¯d ). As X is simply connected, we conclude that p is bijective. It follows that ker θ is trivial and hence N = L. Thus, for  connected, the local relations suffice to define 0. In general, we work with the connected components of , which we will denote by {i }i∈I . Since X is locally connected and  is open, all the i are also open in X . Fix a component X 0 = 0 , and let X (1) denote the union [ i γ i,γ

taken over all pairs (i, γ ) in I × 0 for which i γ ∩ 0 6= ∅. Inductively, if X (1) , . . . , X (k) have already been constructed, we set [ X (k+1) = i γ , i,γ

where the union is taken over all pairs (i, γ ) for which i γ ∩ X (k) 6= ∅. Clearly, X 0 :=

∞ [ k=0

X (k)

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231

is open in X . Its complement is a union of sets of the form i γ , and therefore is also open. Indeed, we have [ X = 0 = i γ , i∈I γ ∈0

and if i1 γ1 ∩ i2 γ2 6= ∅ for i γ1 ⊂ X (k) , then i2 γ2 ⊂ X (k+1) ⊂ X 0 . Since X is connected, it follows that X 0 = X , i.e., each i γ can be connected to 0 by a chain of pairwise overlapping translates of the connected components of . Now take 0 δ for δ ∈ 1. There is a sequence {ij γj }m j = 0 such that i0 = im = 0, γ0 = 1, γm = δ, and ij γj ∩ ij+1 γj+1 6= ∅ for all j = 0, . . . , m − 1. By induction, define wj in F with the property that ϕ(wj ) = γj for all j = 0, . . . , m. For this, set w0 = e. If w0 , . . . , wk have already been defined, we let wk+1 = δ¯k wk , where δk = γk+1 γk−1 ∈ 1. (Thus, wm depends on δ.) Let N denote the normal subgroup of F generated by the left-hand sides of the relations (4.14) as well as the relations δ −1 wm = e

(4.16)

for all δ in 1. Lemma 4.30 The normal subgroup N coincides with K = ker ϕ, implying that 0 is finitely presented. PROOF: Set 9i = α(i × {e}). We claim that for every δ in 1 and for the corresponding elements wj constructed above, we have 9ij σ (wj ) ∩ 9ij+1 σ (wj+1 ) 6= ∅.

(4.17)

Indeed, suppose xj γj = xj+1 γj+1 ∈ ij γj ∩ ij+1 γj+1 . Then, δj = γj+1 γj−1 ∈ 1 and xj = xj+1 δj , σ (wj+1 ) = σ (δ¯j )σ (wj+1 ), from which (4.17) follows. Let 8 denote the union of the 9ij σ (wj ) constructed for all δ ∈ 1. Since each chain of 9ij σ (wj ) begins with 90 , we see 8 is connected. Let Y be the connected component of S containing 8. For every δ ∈ 1, we have ¯ = 9im σ (wm ) ⊂ 8, so 8 ∩ 8σ (δ) ¯ 6= ∅ and therefore Y σ (δ) ¯ = Y . Since 90 σ (δ) ¯ (δ ∈ 1) generate H, we have Yh = Y for every h ∈ H. By the local the σ (δ) connectedness of S it is easy to show that the restriction p|Y : Y → X

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D ρ1

–2

–1

D′ ρ2

0

1

2

Figure 4.2

is also a covering map. Since X is simply connected, we conclude that p|Y is a bijection. On the other hand, all sets 90 h for h ∈ ker θ lie in Y , are disjoint (cf. proof of Lemma 4.29), and have 0 as their image. So, ker θ = {e}, proving the lemma.  This completes the proof of Theorem 4.8. In principle, the proof of Theorem 4.8 enables us to give an explicit presentation of an arithmetic group in terms of generators and relations if a “good” fundamental set is known. In this regard, let us work out the classical example of SL2 (Z). Example 4.31 (generators and relations for SL2 (Z)): As we have seen (cf. §4.2), X = SO2 (R)\SL2 (R) can be identified with the complex upper halfplane P. The group 0 = PSL2 (Z) = SL2 (Z)/{±I2 } acts on the right on P with fundamental domain D = {z ∈ P : |z| > 1, |0



−1 x−1 α x−α xα

4

= 1.

Here, in the first equation α , β range over all roots satisfying α + β 6= 0, α,β the symbol [xα , xβ ] denotes the commutator of xα and xβ , and Ni,j are certain integers described in Lemma 15 of Steinberg (2016), while in the second equation, α is some long root. In the case where R = A1 , i.e., G = SL2 , the first of the above relations should be replaced by −1 −1 xα x−1 −α xα = x−α xα x−α .

For G = SLn , Behr’s result, in particular, yields the well-known generation of SLn (Z) by elementary matrices. In this regard, we should mention a result of Carter and Keller (1984) that every element of SLn (Z) (n ≥ 3) can be in fact written as a product of elementary matrices, the number of which is bounded by a constant depending only on n (see also Adian and Mennicke [1992]). Furthermore, it was shown in Carter and Keller (1983) that the result remains valid for SLn (O) (n ≥ 3), where O is the ring of integers of an arbitrary number field K, with a bound depending on n and K. These results were generalized by Tavgen (1990) to arbitrary Chevalley groups of rank > 1 over the rings of algebraic integers and S-integers using one-parameter root subgroups over the corresponding rings in place of elementary matrices. Moreover, Tavgen (1991) was also able to handle some twisted Chevalley groups. In general, we say that an abstract group 0 is boundedly generated by a subset X ⊂ 0 (or has finite width with respect to X ) if there exists N > 0 such that any g ∈ 0 can be written in the form g = xε11 · · · xεr r , with xi ∈ X , εi = ± 1, and r ≤ N. Thus, the above-mentioned results tell us that groups such as SLn (O) (n ≥ 3) and, more generally, Chevalley groups of rank > 1 as well as some of their twisted analogs over the rings of algebraic integers and S-integers are boundedly generated by their natural generating sets that consist of elementary matrices for the case of SLn and the one-parameter root subgroups for the case of Chevalley groups. For general groups, the implications of such boundedness properties very much depend on the choice of a generating set. For example, if 0 is perfect, i.e., 0 = [0, 0], then finite width with respect to the generating set consisting of all commutators is equivalent to bounded commutator length.

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In our present discussion, though, we would like to focus on a different choice of generating sets, namely finite unions of cyclic groups. We then arrive at the following notion. An abstract group 0 is said to be a group with bounded generation if there exist elements γ1 , . . . , γd ∈ 0 such that 0 = hγ1 i · · · hγd i,

(BG)

where hγi i denotes the cyclic subgroup generated by γi . It is not difficult to see, for example, that the results of Tavgen imply that arithmetic and S-arithmetic (for finite S) subgroups of absolutely almost simple split (and some quasisplit) algebraic groups of relative rank > 1 over number fields have bounded generation. These results prompt the following general problem. Problem Which arithmetic and S-arithmetic subgroups have bounded generation? Interest in groups with bounded generation (particularly, S-arithmetic ones) has been stimulated over the years by numerous important consequences and applications of this property. First, the fact that Carter and Keller (1983) employed some of the same techniques that had been developed in Bass, Milnor, and Serre (1967) and Mennicke (1965) to settle the congruence subgroup problem for G = SLn (n ≥ 3) suggested that there should be a connection between these two properties. This expectation was confirmed by Lubotzky (1995) and Platonov and Rapinchuk (1993), who proved that if G is an absolutely almost simple simply connected algebraic group over a number field K such that G(K) satisfies the Platonov–Margulis conjecture (cf. [AGNT, §9.1]), and S is a finite set of places of K containing all Archimedean ones but not containing any non-Archimedean v for which G is Kv -anisotropic, then whenever GO(S) has bounded generation, it also has the congruence subgroup property, i.e., the corresponding congruence kernel C S (G) is finite (see [AGNT, §9.5] and the survey of Prasad and Rapinchuk [2010] for the relevant definitions). So, bounded generation appeared to be a property that might lead to a uniform approach to Serre’s Congruence Subgroup Conjecture (cf. Serre [1970] and [AGNT, §9.5]). Next, it was shown in Rapinchuk (1991) (see also [AGNT, appendix A.2]) that an abstract group 0 having bounded generation and possessing the following additional property: (Fab ) for every subgroup 1 ⊂ 0 of finite index, the abelianization 1ab = 1/[1, 1] is finite, is SS-rigid, i.e., has only finitely many inequivalent completely reducible complex representations in each dimension (we note that for a finitely generated

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0, condition (Fab ) is necessary for SS-rigidity). To put this result in context, we recall that, as follows from the Margulis Superrigidity Theorem (cf. Margulis [1991, chapter VII, §5]), an arbitrary lattice 0 ⊂ GR , where G is an absolutely almost simple R-group of R-rank > 1, is SS-rigid. In Lifschitz and Morris (2004, 2008), bounded generation was used to show that many (and conjecturally all) S-arithmetic subgroups of isotropic absolutely almost simple K-groups G whose S-rank X rkS G := rkKv G v∈S

is ≥ 2 cannot act on the real line. In Shalom (1999) and Kassabov (2005), bounded generation was an essential ingredient for obtaining explicit estimates of the Kazhdan constant for SLn (Z) (n ≥ 3) – we note that in the initial work on this subject, Burger (1991) used the results of Tits (1976) instead. Last but not least, Shalom and Willis (2013) used bounded generation to establish the Margulis–Zimmer conjecture for commensurated subgroups of S-arithmetic groups with rkS G ≥ 2 in the case of split groups. We will now survey the available results on bounded generation for arithmetic and some other linear groups. Let us start with non-examples. Observing that a non-abelian free group cannot possibly have bounded generation and combining this observation with the fact that SL2 (Z) has a subgroup of index 12  that isa free group ofrank 2 (this subgroup is generated by the matrices 1 2 1 0 and and coincides with the commutator subgroup of 0 1 2 1 SL2 (Z)), we conclude that SL2 (Z) does not have bounded generation. Furthermore, according to a result of Grunewald and Schwermer (1981), for every square-free integer d > 0, the group SL2 (O √ d ), where Od is the ring of integers in the imaginary quadratic field Kd = Q( −d), possesses a subgroup of finite index that admits a free non-abelian quotient, which again prevents it from having bounded generation. Similarly, using results of Lubotzky (1996a) (generalizing an earlier result of Millson [1976]) and Margulis and Vinberg (2000), one shows, for example, that if f is a nondegenerate n-dimensional integral quadratic form having signature (1, n − 1) over R, then for G = SOn ( f ), the group GZ does not have bounded generation (this approach applies also to other discrete subgroups acting on hyperbolic space). The following general result in the negative direction was obtained by Fujiwara (2005) using bounded cohomology: Let 0 be a discrete subgroup of a rank-one simple Lie group. If 0 does not contain a nilpotent subgroup of finite index (which is automatically the case if 0 is a lattice, in particular, an arithmetic subgroup), then 0 does not have bounded generation. (We note that this applies not only to real, but

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also to p-adic Lie groups, as in the latter case, any (finitely generated) discrete subgroup is virtually free – cf. Lubotzky [1991], Serre [2003].) Thus, if G is an absolutely almost simple algebraic group over a number field K, S is a finite set of places of K containing all the Archimedean ones, and 0 is an S-arithmetic subgroup (see §5.4 for the definition and basic properties of S-arithmetic subgroups), then 0 can have bounded generation only when rkS G ≥ 2. There is an expectation, with substantial evidence to support it, that the converse should also be true in the isotropic situation, namely S-arithmetic subgroups of an absolutely almost simple algebraic K-group G should be boundedly generated provided that rkK G > 0 and rkS G ≥ 2. The results of Tavgen confirm this expectation/conjecture for split and some quasi-split Kgroups of K-rank ≥ 2. The first examples of boundedly generated S-arithmetic subgroups in isotropic, but not necessarily split or quasi-split groups, were found by Erovenko and Rapinchuk (2001, 2006). They proved that given a nondegenerate quadratic form f over a number field K in n ≥ 5 variables and a finite set S of places of K containing all Archimedean ones such that either the Witt index of f is ≥ 2 or that it is 1 and that S contains a non-Archimedean place, every S-arithmetic subgroup of SOn ( f ) is boundedly generated. This result was extended to S-arithmetic subgroups of some other isotropic groups of classical types by A. Heald (2013). In all previously mentioned examples of boundedly generated S-arithmetic subgroups, the (absolute) rank of the ambient absolutely almost simple algebraic group was > 1. So, it is worth mentioning that a direct proof of bounded generation of SL2 (O) by elementary matrices (and hence bounded generation in the abstract sense), where O is the ring of S-integers of a number field K such that the group of units O∗ is infinite, was found only recently. First, we note that O∗ is finite only in the following two cases: (1) O = Z; √ and (2) O = Od , the ring of integers of the imaginary quadratic field K = Q( −d). As we have already noted, in both cases SL2 (O) does not have bounded generation, so the condition that O∗ be infinite is necessary – by the Dirichlet Unit Theorem (see Corollary 5.39) it is equivalent to the condition that the S-rank of SL2 (which is equal to |S|) is ≥ 2. In this case, bounded generation of SL2 (O) was established by Cooke and Weinberger (1975) assuming a suitable form of the Generalized Riemann Hypothesis, which still remains unproven. Later, it was shown in Loukanidis and Murty (1994) (see also Murty [1995]) by analytic tools that the argument can be made unconditional if |S| ≥ max(5, 2[K : Q] − 3). On the other hand, Liehl (1984) proved bounded generation of SL2 (O) by algebraic methods for some special fields K. The first unconditional proof in full generality was given by D. Carter, G. Keller, and E. Paige in an unpublished

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preprint; their argument was streamlined and made available to the public by D. Morris (Witte) (2007). This argument is based on model theory and provides no explicit bound on the number of elementary matrices required; besides, it uses difficult results of additive number theory related to the Goldbach problem. Vsemirnov (2014) proved bounded generation of SL2 (Z[1/p]) (p a prime) using the results of D. R. Heath-Brown (1986) on Artin’s Primitive Root Conjecture. (Thus, in a broad sense, this proof develops the initial approach of Cooke and Weinberger.) Finally, Morgan, Rapinchuk, and Sury (2018) showed using only standard results of algebraic number theory that if the group of units O∗ of a ring of S-integers O is infinite, then every matrix in SL2 (O) is a product of at most nine elementary matrices. This enables one to make the bound on the number of elementaries required to factor every matrix in SLn (O) in Carter and Keller (1983) dependent only on n and not on K, over all number fields K other than imaginary quadratic extensions of Q. We also note that bounded generation of SL2 (O) was used in Erovenko and Rapinchuk (2006), so now these results are unconditional. See also Jordan and Zaytman (2019) for further results improving the bound. Despite rather active investigation into bounded generation, particularly of S-arithmetic subgroups, no examples of boundedly generated infinite S-arithmetic subgroups of absolutely almost simple anisotropic algebraic Kgroups have been found in the course of more than 30 years. In fact, even for those S-arithmetic groups 0 that are known to have bounded generation, there were no examples of a bounded factorization (BG) with all elements γi semisimple (in which case we say that 0 is boundedly generated by semisimple elements). This led to a number of questions ranging from Do there exist infinite S-arithmetic subgroups of anisotropic algebraic K-groups that have bounded generation? to Which linear groups are boundedly generated by semisimple elements? Recently these questions were resolved in Corvaja et al. (2022). In order to formulate some of their results, we recall that a reductive algebraic group G over a field K of characteristic zero is K-anisotropic if and only if the group GK consists entirely of semisimple elements (cf. §2.1.14). In analogy, we say that a linear group 0 ⊂ GLn (K) is anisotropic if it consists only of semisimple elements. Theorem 4.33 (CORVAJA, RAPINCHUK, REN, AND ZANNIER [2022]) An anisotropic linear group 0 ⊂ GLn (K) over a field of characteristic zero is boundedly generated if and only if it is virtually abelian. The proof relies on Laurent’s Theorem from Diophantine geometry (cf. Corvaja and Zannier [2018, §2.3]) and the existence of generic elements in

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Zariski-dense subgroups (see Prasad and Rapinchuk [2003, 2017]). One of the consequences of the theorem is that infinite S-arithmetic subgroups of absolutely almost simple K-anisotropic algebraic groups are never boundedly generated. (The quantitative aspect of the lack of bounded generation in this situation was recently investigated in Corvaja, Demeio et al. [2022]). Current work in this direction continues along two lines. First, one would like to complete the proof of the conjecture that S-arithmetic subgroups of absolutely almost simple isotropic algebraic groups G over number fields K with rkS G ≥ 2 are boundedly generated. Second, one should investigate conditions that have many of the same implications as bounded generation (such as the congruence subgroup property, SS-rigidity etc.) and are likely to hold also for S-arithmetic subgroups of absolutely almost simple anisotropic algebraic groups G with rkS G ≥ 2. Among such conditions are (PG) and (PG)0 introduced in Platonov and Rapinchuk (1993) (cf. also Corvaja1 [2022, §6]). To complete our discussion of generators and relations of arithmetic groups, we must call the reader’s attention to the fact that this subject can be treated in terms of the general theory of discrete transformation groups (cf. the survey of Vinberg and Schwarzman [1988]). It should also be emphasized that many groups studied in this theory – in particular, discrete groups generated by reflections and groups with a simplicial fundamental domain (cf. the example above) – turn out to be non-arithmetic. Our next result can be traced back to the classical work of Jordan, where it was established that SLn (Z) has only a finite number of non-conjugate finite subgroups (cf. Delone et al [1973]). Theorem 4.9 Let G be an algebraic Q-group. Then GZ has finitely many conjugacy classes of finite subgroups. PROOF: We set 0 = GZ and continue using the notations introduced in the proof of Theorem 4.8. In particular, K will denote a maximal compact subgroup of GR , X = K\GR , and  ⊂ X is a subset such that (1) X = 0, and (2) 1 := {δ ∈ 0 : δ ∩  6= ∅} is finite. Let 2 ⊂ 0 be a finite subgroup. By Proposition 3.26, every compact subgroup of GR is conjugate to a subgroup of K, so there exists g in GR such that g2g−1 ⊂ K. This means that x = Kg ∈ X is a fixed point for 2. Write x = x0 γ with x0 ∈  and γ ∈ 0. Then x0 is fixed by γ 2γ −1 , and in particular x0 ∈  ∩ δ for every δ in γ 2γ −1 . Thus, γ 2γ −1 ⊂ 1, proving that

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every finite subgroup of 0 is conjugate to a subgroup contained in the finite set 1. Remark Theorem 4.9 can also be proved by using Theorem 4.26 and proceeding as in the proof of Proposition 3.13 (cf. also the proof of Theorem 5.33). Now we are ready to prove the invariance of the class of arithmetic subgroups under arbitrary surjective morphisms. Theorem 4.7 Let ϕ : G → H be a surjective Q-morphism of algebraic Qgroups. If 0 is an arithmetic subgroup of G, then ϕ(0) is an arithmetic subgroup of H. PROOF: It suffices to show that ϕ(GZ ) is arithmetic in H. Moreover, we can choose a matrix realization of H so that ϕ(GZ ) ⊂ HZ (cf. remark following Proposition 4.5). We need to prove that the index [HZ : ϕ(GZ )] is finite. We can obviously assume that G is connected, and will now reduce the argument to the case where G is either reductive or unipotent. Let G = CU be the Levi decomposition of G, where U = Ru (G) is the unipotent radical of G and C is a reductive group. Let D = ϕ(C) and V = ϕ(U); then H = DV is the Levi decomposition of H. Assume that both indices [DZ : ϕ(CZ )] and [VZ : ϕ(UZ )] are finite. Then an elementary argument (cf. proof of Lemma 4.22) shows that the index [DZ VZ : ϕ(CZ UZ )] is also finite. Furthermore, by Corollary 4.4, the index [HZ : DZ VZ ] is finite, so the index [HZ : ϕ(CZ UZ )] is also finite, and the finiteness of [HZ : ϕ(GZ )] follows. Now suppose that G is unipotent. Then GR /GZ is compact (Lemma 4.22). Since HR = ϕ(GR ) by Corollary 3.19, HR /ϕ(GZ ) is compact. On the other hand, HZ /ϕ(GZ ) is closed and discrete in HR /ϕ(GZ ), and hence [HZ : ϕ(GZ )] is finite, as required. It remains to consider the case where G is reductive. Here we appeal to Theorem 4.26. So, let H ⊂ GLn (C). Using the embedding   A 0 GLn (C) → GLn+1 (C), A 7→ , 0 (det A)−1 we may assume that H is Zariski-closed in Mn (C). Let us define an action of G on V = Mn (C) by Ag = Aϕ(g), the usual matrix product. Then H = In ϕ(G) is a closed orbit of G, so HZ is the union of a finite number of orbits of GZ , by

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Theorem 4.26. But these orbits are precisely the cosets of ϕ(GZ ) in HZ , so the finiteness of the index [HZ : ϕ(GZ )] follows immediately. The preceding results represent the first steps in the analysis of abstract properties of arithmetic groups. Deeper results, some of which will be presented in §4.9 as well as in later parts of the book, require more sophisticated machinery. Before formulating the Borel Density Theorem, we will introduce important classes of semisimple groups whose arithmetic properties are fundamentally different. Definition An algebraic Q-group G is said to have compact type if the group i of real points GR is compact. For G semisimple, G has noncompact type if GR is noncompact for each Q-simple factor Gi of G. If G does not belong to either of those types, it is said to have mixed type. Theorem 4.34 (BOREL DENSITY THEOREM; BOREL [1966]) Let G be a (connected) semisimple Q-group of noncompact type. Then every arithmetic subgroup 0 ⊂ G is Zariski-dense in G. PROOF: Suppose the theorem has already been proven for Q-simple groups. Letting the bar denote the Zariski closure, we will then have 0¯ ⊃ 0 ∩ Gi = Gi Q for each Q-simple factor Gi of G. So, 0¯ ⊃ i Gi = G, as desired. This reduces the proof to the case where G is Q-simple, which we will treat in the rest of the argument. In order to continue, we need the following fact, which will be verified in the next section (cf. Corollary 4.41): For G a semisimple Q-group, if GR is noncompact,

(4.20)

then GZ is infinite. (We note that the converse is also true, since if GR is compact, then GZ , being a discrete subgroup, must be finite.) We now observe that if 01 ⊂ 02 are subgroups of G and the index [02 : 01 ] is finite, then for the Zariski closures the index [0¯ 2 : 0¯ 1 ] is also finite. Indeed, S S if 02 = di= 1 01 γi , then 0¯ 2 = di= 1 0¯ 1 γi , hence [0¯ 2 : 0¯ 1 ] ≤ d. It follows that the connected components (0¯ 1 )0 and (0¯ 2 )0 of the Zariski closures coincide. More generally, if 01 and 02 are commensurable, then (0¯ 1 )0 = (01 ∩ 02 )0 = (0¯ 2 )0 . We can now complete the proof of the theorem. Without loss of generality, we may assume that 0 ⊂ GZ . Then it follows from (4.20) that H := 0¯ is an algebraic Q-group of positive dimension. For any g in GQ , the group g0g−1

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is arithmetic (see Corollary 4.3); thus g0g−1 is commensurable with 0. So, it follows from the above remark that ¯ 0 = (g0g−1 )0 = gH 0 g−1 , H 0 = (0) and therefore H 0 is normalized by GQ . Since the normalizer NG (H 0 ) is a closed subgroup, and GQ is Zariski-dense in G (Theorem 2.6), we see that NG (H 0 ) = G, in other words, H 0 is a normal subgroup of G. Since G does not have any proper normal Q-subgroups of positive dimension, we conclude that H 0 = G, and our claim follows. Remark Although the proof of the Borel Density Theorem presented above uses the fact that 0 is arithmetic and does not carry over to other types of subgroups of G, the result itself holds in much greater generality. For example, every closed subgroup 0 ⊂ GR (in the real topology) for which GR / 0 has finite invariant volume is Zariski dense in G (cf. Raghunathan [1972, Chapter 5]). In particular, every lattice 0 ⊂ GR i.e., a discrete subgroup for which GR / 0 has finite volume, is Zariski-dense. (The fact that an arithmetic subgroup of a semisimple Q-group is a lattice will be established in §4.6.) We conclude this section with a partial converse of Corollary 4.3. For a semisimple Q-group G and its arithmetic subgroup 0, we define C(0) = {g ∈ GC : 0 is commensurable with g−1 0g}. Since commensurability is an equivalence relation, it is easy to check that C(0) is a subgroup of G = GC , which is called the commensurability subgroup (or the commensurator) of 0. It should also be noted that C(0) is actually independent of 0, i.e., it is the same for any other arithmetic group. Since clearly 0 ⊂ C(0), we see that C(0) can be viewed as the universal repository of all the arithmetic subgroups of G. A description of C(0) is given in the following: Proposition 4.35 Let G be a (connected) semisimple algebraic Q-group. Denote by N the largest normal Q-subgroup of G that has compact type, and let π : G → G/N be the corresponding quotient map. Then C(0) = π −1 ((G/N)Q ). PROOF: Let G1 , . . . , Gr be the Q-simple factors of G. Then N is generated i is compact and by the centers of the remaining by those Gi for which GR i is factors. Let H be the normal subgroup generated by those Gi for which GR noncompact. Clearly, G = H · N and H ∩ N is finite, making the product map H × N → G an isogeny. Since H ∩ 0 and N ∩ 0 are arithmetic in H and N respectively, Theorem 4.7 implies (H ∩ 0)(N ∩ 0) is arithmetic in G. Since NR

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is compact, N ∩ 0 is finite, so H ∩ 0 is a subgroup of finite index in 0. As noted earlier, it follows that C(0) = C(0 ∩ H). In view of the fact that the restriction of π to H is an isogeny, it is easy to see that C(0 ∩ H) is the preimage under π of the commensurability subgroup of π(0 ∩ H) in G/N. Thus, we have reduced the proof to the case where N = {1} and, in particular, Z(G) = {1}. We will show that in this case C(0) = GQ . By the Borel Density Theorem, the Zariski closure 0¯ is G; moreover, without loss of generality, we may assume that 0 ⊂ GZ . Fix a Q-embedding G ,→ GLn (C), and for a subset A ⊂ G let C[A] (resp., Q[A]) denote the C- (resp., Q)-subspace of Mn (C) spanned by A. Then, since 0¯ = G, we have C[0] = C[G]. Moreover, for g ∈ C(0) the intersection 0 ∩ g−1 0g is of finite index in 0, which implies that 0 ∩ g−1 0g = G, hence Q[0] = Q[0 ∩ g−1 0g]. It follows that for every g ∈ C(0), we have gQ[0]g−1 = gQ[0 ∩ g−1 0g]g−1 = Q[0].

(4.21)

Consider the adjoint representation % : G → GL(V ) on V = C[G], given by %(g)v = gvg−1 . Since VQ = Q[0], the computation (4.21) yields the inclusion %(C(0)) ⊂ %(G)Q . But Z(G) = {1} by assumption, so % is faithful, allowing us to conclude that C(0) ⊂ GQ . The opposite inclusion follows from Corollary 4.3. It follows from Proposition 4.35 that every arithmetic subgroup of a semisimple adjoint group G of noncompact type is contained in GQ . On the other hand, if  G = SLn (C), then the subgroup 0 generated by SLn (Z) and  s 0   ..  , where s is a primitive nth root of unity, is an arithmetic sub. 0 s group of G. Clearly 0 ⊂ / GQ , for n > 2. Here the commensurability subgroup is   1 A : A ∈ GL (Q) . n (detA)1/n We note that if G is a connected algebraic Q-group, then every arithmetic subgroup 0 of G has infinite index in its commensurator C(0). An important theorem due to Margulis states that, for example, when G is an absolutely almost simple group, this property actually characterizes arithmetic lattices 0 ⊂ GR – see Margulis (1991, Chapter IX, Theorem B) for a much more general result.

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4.5 Compactness of GR /GZ The reduction theory developed in §§4.2–4.3 has already enabled us to obtain several structure theorems for arithmetic groups. The construction of fundamental sets implemented there will be used in various situations in the book. This construction, however, does not enable us to answer all the questions that arise in reduction theory; in particular, it does not lead to a criterion for the quotient GR /GZ to be compact. In this section, we will examine this problem by considering first the case of algebraic tori. It turns out that the general case (1) here reduces to norm tori S = RK/Q (Gm ), where K is a finite extension of Q (cf. §2.1.7 for the definition of the norm torus). As an application, we will show that the compactness of SR /SZ for such a torus S yields the classical Dirichlet Unit Theorem (see Corollary 4.39 and subsequent remarks). The method that we will use is of a fairly general nature and is applicable to other groups arising from division algebras. (1)

Proposition 4.36 Let K be a finite field extension of Q, and let S = RK/Q (Gm ) be the corresponding norm torus. Then SR /SZ is compact. PROOF: Set V = K ⊗Q R and let N denote the natural extension of the norm map NK/Q to V that sends an arbitrary a in V to the determinant N(a) of the left multiplication map x 7→ ax (x ∈ V ). It follows that multiplication by elements of SR = {x ∈ V : N(x) = 1} preserves the Haar measure µ on the additive group V (cf. §3.5). Let O be the ring of integers of K. Then O can be viewed as a lattice in V , so the quotient V /O is compact and consequently µ(V /O) < ∞. Choose a compact subset B of V satisfying µ(B) > µ(V /O), and set C = {b1 − b2 : b1 , b2 ∈ B}. For every a ∈ SR , we have µ(aB) = µ(a−1 B) = µ(B) > µ(V /O), which implies that the restrictions of the natural map V → V /O to aB and a−1 B cannot be injective. Consequently, there are c, d ∈ C such that α := ac and β := a−1 d lie in O. Then αβ = cd ∈ C 2 ∩ O. Since this intersection is both compact and discrete, it is finite. We will need the following simple fact. Lemma 4.37 Every γ 6= 0 in O has only finitely many nonassociated factorizations γ = αβ with α, β ∈ O. (We recall that two factorizations γ = α1 β1 = α2 β2 are said to be associated if there exists a unit ε ∈ O∗ such that α2 = εα1 and β1 = εβ2 .

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To prove the lemma, one observes that any two factorizations γ = α1 β1 = α2 β2 for which α1 ≡ α2 (mod γ O) and β1 ≡ β2 (mod γ O) are necessarily associated.) Lemma 4.37 implies that if we consider all possible factorizations γ = αβ of elements γ ∈ C 2 ∩ O with α, β ∈ O, then there are only finitely many non-associated possibilities for β. Since the norm of any unit ε ∈ O∗ is ±1, we conclude that there exist finitely many elements β1 , . . . , βr ∈ O such that any β that occurs in one of the preceding factorizations is of the form β = βi ε for some i = 1, . . . , r and some ε ∈ O∗ ∩ S = SZ . On the other hand, by construction β = a−1 d with d ∈ C. Then a = dβi−1 ε−1 , and therefore

SR =

r  [

Cβi−1 ∩ SR



! SZ .

i=1

Since C is compact, it follows from this decomposition that SR /SZ is compact. The Dirichlet Unit Theorem can be easily derived from Proposition 4.36; however, we will postpone the argument to until after we have established the following compactness criterion for SR /SZ for an arbitrary Q-torus S. Theorem 4.38 For an algebraic Q-torus S, the following conditions are equivalent: (1) S is Q-anisotropic; (2) SR /SZ is compact. PROOF: (2) ⇒ (1). Suppose S is not Q-anisotropic. Then there exists a Qepimorphism ϕ : S → Gm =: T. Since ϕ(SR ) has finite index in TR (Corollary 3.19), the quotient TR /ϕ(SZ ) must be compact if the quotient SR /SZ is such. On the other hand, ϕ(SZ ) is an arithmetic subgroup of T (Theorem 4.7) and thus is a finite group, since TZ ' Z∗ = {±1}. Therefore, TR /ϕ(SZ ) cannot be compact, since TR ∼ = R∗ is not compact. (1) ⇒ (2). It is well known (cf. Proposition 2.3) that there exists a Q-epimorphism ϕ : T → S, where T is a quasi-split torus, i.e., has the form

T=

d Y i=1

RKi /Q (Gm ),

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where the Ki are finite field extensions of Q. Since S is Q-anisotropic, the restriction of ϕ to T0 :=

d Y

(1)

RKi /Q (Gm )

i=1

is still surjective. By Proposition 4.36, the quotient (T0 )R /(T0 )Z is compact, and therefore the quotient ϕ((T0 )R )/ϕ((T0 )Z ) is also compact. Combining this with the facts that the index [SR : ϕ((T0 )R )] is finite and the subgroup ϕ((T0 )Z ) is arithmetic in S, we conclude that the quotient SR /SZ is compact. Corollary 4.39 (DIRICHLET UNIT THEOREM) Let S be an algebraic Q-torus. Then the group SZ is isomorphic to the direct product of a finite group and a free abelian group of rank equal to rankR S − rankQ S. PROOF: Let S1 and S2 be the maximal split and the maximal anisotropic Qsubtori of S, respectively. Since (S1 )Z is a finite group, applying Theorem 4.7 to the isogeny S1 × S2 → S, we see that the index of (S2 )Z in SZ is finite. However, rankR S − rankQ S = (rankR S1 + rankR S2 ) − (rankQ S1 + rankQ S2 ) = rankR S2 since rankR S1 = rankQ S1 = dim S1 . In view of general results on abelian groups, it suffices to establish that (S2 )Z is the direct product of a finite group by Zr , where r = rankR S2 . Thus, we have reduced the proof to the case where S is a Q-anisotropic torus. It follows from the discussion in §2.2.4 that every torus over R is isomorphic (1) to a product of copies of Gm , RC/R (Gm ) and RC/R (Gm ). It follows that there exists an isomorphism of topological groups SR ' Rr × D, where r = rankR S and D is a compact group. On the other hand, by Theorem 4.38 the quotient SR /SZ is compact. So, our claim is a direct consequence of the following well-known fact, which we state in a somewhat more general form that we will need later. Lemma 4.40 Let G be an abelian topological group of the form Za × Rb × D, where D is a compact group. Then every discrete subgroup 0 of G with compact quotient G/ 0 is isomorphic to Za+b × F for a suitable finite group F. The proof easily reduces easily to the case G = Rn , which is treated in Bourbaki (1998b, Chapter 7, §1).

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Now take S to be the torus RK/Q (Gm ), where K is a finite field extension of Q. We have an R-defined isomorphism S ' Gsm × RC/R (Gm )t , where s and t are the numbers of real and pairwise nonconjugate complex valuations of K, respectively; in particular rankR S = s + t. Since rankQ S = 1, we obtain the classical statement of the Dirichlet Unit Theorem on the structure of the group of units U(K) = OK∗ : U(K) ' F × Zs+t−1 , where F is the group of all roots of unity in K. In the next chapter, we will generalize this result to S-units (cf. Corollary 5.39). We can now prove the assertion that we used in the previous section. Corollary 4.41 Let G be a semisimple Q-group. Then GZ is infinite if and only if GR is noncompact. PROOF: If GR is compact, then GZ is a discrete subgroup of a compact group, hence finite. To prove the converse, suppose that GR is noncompact. If G is isotropic over Q, then it contains a 1-dimensional unipotent Q-subgroup U. Since UZ is infinite, the desired fact is obvious. Now suppose that G is Q-anisotropic. Since GR is noncompact, G is R-isotropic (cf. Theorem 3.1), hence contains a maximal R-torus T that is R-isotropic. By [AGNT, Corollary 7.3], there exists a maximal Q-torus S ⊂ G that is also R-isotropic. (The proof of this assertion follows from the rationality of the variety of maximal tori and does not depend on any results of the present chapter.) But then SZ is infinite by Corollary 4.39, so GZ is also infinite. Another proof of Corollary 4.41 can be obtained from the results of the next section (cf. Theorem 4.54 and subsequent remarks). We will now show that conditions (1) and (2) of Theorem 4.38 are equivalent not only for algebraic Q-tori, but in fact for all (reductive) Q-groups. Theorem 4.42 Let G be an algebraic Q-group. Then the following conditions are equivalent: (1) GR /GZ is compact; (2) the reductive part of the connected component of G is anisotropic over Q. (Note that condition (2) can be reformulated as X(G0 )Q = {1} and each unipotent element in GQ belongs to the unipotent radical of G.)

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PROOF: It suffices to consider the case of connected G. Let G = HRu (G) be its Levi decomposition. Then GR /GZ is compact if and only if HR /HZ is compact, which follows from Lemma 4.22 combined with Lemma 4.43 Let H be a reductive Q-subgroup of a connected Q-group G. Then HR /HZ is closed in GR /GZ . PROOF: By the stronger version of Chevalley’s theorem (see Theorem 2.49), there exists a Q-representation % : G → GL(V ) and a vector v in VQ such that the stabilizer of v under % is H. Then the orbit W := v%(GZ ) is contained in a lattice in VQ (cf. remark following Proposition 4.5), and hence is closed in VR . If we define f : GR → V by f (g) = v%(g), then HR GZ = f −1 (W ) hence closed in GR , and the lemma follows. Thus, the proof of Theorem 4.42 reduces to the case of a reductive group G. Then the implication (1) ⇒ (2) readily follows from Lemma 4.43. Indeed, if G is Q-isotropic, then it contains a nontrivial Q-split torus S. Then SR /SZ is noncompact (cf. Theorem 4.38) and is closed in GR /GZ ; so the latter cannot be compact either. To prove the implication (2) ⇒ (1), we first reduce the problem to the case of a semisimple adjoint group. Let Z = Z(G) be the center of G. Then its connected component S is a Q-anisotropic torus, and therefore SR /SZ is compact (Theorem 4.38). Since the index [ZR : SR ] is finite, ZR /ZZ is also compact. Set H = G/Z and let π : G → H be the canonical morphism. Furthermore, let ϕ : GR /GZ → π(GR )/π(GZ ) be the map induced by π. The compact group B = ZR /ZZ acts on the quotient GR /GZ by multiplication, and it is easy to see that the orbits of B are precisely the fibers of ϕ. It follows that ϕ is proper. Since the index [HR : π(GR )] is finite and the group ϕ(GZ ) is arithmetic, we see that the compactness of GR /GZ is equivalent to the compactness of HR /HZ . To complete the argument, we will establish the compactness of HR /HZ , for which we will need the following criterion for the compactness of a subset of GLn (R)/GLn (Z). It should be noted that in its formulation we use the standard left action of GLn (R) on Rn . Proposition 4.44 (MAHLER’S COMPACTNESS CRITERION) A subset  ⊂ GLn (R) is relatively compact modulo GLn (Z) if and only if (a) | det g| for g ∈  is bounded, (b) (Zn \ {0}) ∩ U = ∅ for a suitable neighborhood of zero U in Rn .

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PROOF: If the image of  in GLn (R)/GLn (Z) is relatively compact, then there is a compact D ⊂ GLn (R) such that  ⊂ D · GLn (Z). It clearly follows that | det g| is bounded for all g in . Furthermore, (Zn \ {0}) ⊂ (D · GLn (Z))(Zn \ {0}) = D(Zn \ {0}). Since Zn is discrete and D is compact, the right-hand side is closed in Rn and does not contain zero, implying the existence of the required neighborhood U. Conversely, assume conditions (a) and (b) hold, and let 6 = 6t,v (for t ≥ √2 , v ≥ 1 ) be a Siegel set of GLn (R). We know from Theorem 4.12 that 2 3 6 · GLn (Z) = GLn (R), so there is a subset 2 ⊂ 6 such that 2GLn (Z) = GLn (Z). We have (Zn \ {0}) = 2(Zn \ {0}), which implies that conditions (a) and (b) remain valid for 2, and it suffices to establish the relative compactness of 2. Note that (b) means that there is c > 0 such that kg(x)k ≥ c for all g in 2 and all x in Zn \{0}, where k k is the Euclidean norm on Rn . In particular, c ≤ kg(e1 )k = kag (e1 )k = a1 in the notation used in §4.2, i.e., g = kg ag ug is the Iwasawa decomposition of g in 2, and e1 , . . . , en is the standard basis of Rn . Since ag ∈ At , there exists s > 0 such that ai ≥ sa1 for all i = 1, . . . , n, demonstrating that all ai are bounded from below. On the other hand, | det g| = | det ag | = a1 · · · an is bounded from above; therefore all ai are also bounded from above. Thus we have shown that the a-components ag of the elements g in 2 form a relatively compact set. Since 2 ⊂ 6, it follows that 2 is relatively compact, completing the proof of the proposition. We now have all the ingredients needed to prove that HR /HZ is compact. Consider the Lie algebra h = L(H) and the adjoint representation % : H → GL(h). Without loss of generality, we may assume that HZ = {h ∈ H : %(h)L = L} for a suitable lattice L ⊂ hQ . In view of Lemma 4.43, it suffices to prove that HR is relatively compact modulo GLn (Z) (n = dim h) defined in terms of some basis of L. For this, we will apply Mahler’s criterion. Since det %(H) = 1, we only need to verify condition (b). Consider the characteristic polynomial det(t − ad x) = tn +

n−1 X i=0

fi (x)ti ,

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corresponding to the adjoint action of an element x in L. Since H is Qanisotropic, hQ does not contain nilpotent elements, and consequently the fi (x) P 2 cannot vanish simultaneously, i.e., f (x) = fi (x) > 0. On the other hand, the fi are polynomials with rational coefficients in terms of the coordinates of x with respect to a fixed basis of L. Therefore, we can write 1 f (x) = g(x) d for some nonzero d ∈ Z and some polynomial g(x) with integer coefficients. Since g(x) 6= 0 for all x in L \ {0}, we have 1 . d But fi (x) = fi (%(h)x) for h ∈ H, so for every h ∈ HR , we have |f (L \ {0})| ≥

|f (%(h)(L \ {0}))| = |f (L \ {0})| ≥

1 . d

Thus, the condition 1 d defines a neighborhood of zero U required to apply Mahler’s compactness criterion, which completes the proof of Theorem 4.42. |f (x)|
0 : xi ≤ t for all i}. Furthermore, we can write ρ in terms of the coordinates x1 , . . . , xn as ρ(a) =

n−1 Y

xri i ,

i=1

where the ri are positive integers. Since the (multiplicative) Haar measure on R>0 is dx x , we have Z

ρ(a)da =

t

Z

(A0 )t

0

Z

Z ···

t n−1 Y

0 i=1

xri i

n−1 Y Z t r −1 dx1 dxn−1 ··· = xi i dxi , x1 xn−1 0 i=1

t

1 xr−1 dx = tr is finite for r > 0. r 0 Broadly speaking, the argument for an arbitrary semisimple group G uses a similar idea, but is much more involved technically. More precisely, since GR is unimodular (again by Corollary 3.72), the quotient GR /GZ carries an invariant measure, and to prove that the volume of GR /GZ with respect to this measure is finite, it suffices to find a measurable subset of GR that has finite volume and maps surjectively onto GR /GZ . For this we will show that the fundamental set in GR relative to GZ constructed in §4.3 is contained in the union of a finite number of translates of a suitable relative Siegel set in GR , and then establish that every relative Siegel set has finite volume. (This part of the argument hardly differs from the case of SLn .) So, let G ⊂ GLn (C) be a semisimple Q-group. In § 4.3 we showed that for a fundamental set in GR relative to GZ , we can take a set of the form ! [ = a6b ∩ GR , which is finite since

b

where 6 is a Siegel set in GLn (R), the element a ∈ GLn (R) is such that H := a−1 Ga is self-adjoint, and b runs through a finite set of matrices from GLn (Z). To prove Theorem 4.47, our choice of a must be subject to stricter

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constraints, namely it should satisfy all the conditions listed in Proposition 3.34. In what follows, we will use the notations introduced there. In particular, let S = H ∩ Dn be a maximal R-split torus of H and U = H ∩ Un a maximal unipotent subgroup. Also, let R denote the root system of H relative to S, and let 5 ⊂ R be the system of simple roots corresponding to U. Since a−1 (a6b ∩ GR )a = 6ba ∩ HR , the finiteness of the volume of  follows from Proposition 4.48 For every Siegel set 6 ⊂ GLn (R) and every x ∈ GLn (R), the intersection 6x ∩ HR has finite volume with respect to the Haar measure of HR . The proof is based on the construction of relative Siegel sets for H. Let HR = K∗ A∗ U ∗ be the Iwasawa decomposition of HR (cf. Theorem 3.33), where K∗ is a maximal compact subgroup of HR , A∗ is the connected com∗ in H (for t > 0 and a ponent of SR , and U ∗ = UR . A (relative) Siegel set 6t,ω R ∗ ∗ ∗ compact subset ω ⊂ U ) is the product K At ω, where A∗t = {a ∈ A∗ : α(a) ≤ t for all α ∈ 5}. Clearly, for G = SLn , relative Siegel sets basically amount to intersections G ∩ 6 with Siegel sets 6 in GLn (R). An easy argument shows that in our setup, the same remains valid for H. Indeed, let h ∈ 6t,v ∩ H. Then the Iwasawa decompositions of h in HR and in GLn (R) coincide. According to Proposition 3.34 (iii), simple roots of H are of the form α = d1 α1 + · · · + dn−1 αn−1 , where the αi are simple roots of GLn (C) and di ≥ 0. It follows that we can find a constant s > 0 such that α(ah ) ≤ s for all h ∈ 6t,v ∩ HR and all α ∈ 5; then ∗ 6t,v ∩ HR ⊂ 6s,ω where ω = (UnR )v ∩ HR .

A similar argument also proves the converse (which we will not need): every Siegel set of H is contained in a suitable Siegel set of GLn (R). Proving the analogous assertion for the translates of Siegel sets is the most technical step in the proof of Proposition 4.48. Proposition 4.49 Let 6 be a Siegel set of GLn (R), and let x ∈ GLn (R). Then there is a Siegel set 6 ∗ ⊂ HR and a finite set of elements xi ∈ HR such that [ 6x ∩ HR ⊂ 6 ∗ xi . i

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Given this result, to complete the proof of Proposition 4.48, we only need to prove ∗ has finite volume with respect to Proposition 4.50 Every Siegel set 6 ∗ = 6t,ω the Haar measure on HR .

PROOF: This relies on the formula for the Haar measure of HR , which is analogous to the formula for SLn (R) and is a consequence of Proposition 3.67: dh = ρ(a)dk ∗ da∗ du∗ , where dk ∗ , da∗ , and du∗ are the Haar measures on K∗ , A∗ and U ∗ , respectively, and ρ is the sum of positive roots in R. As in the case of SLn (R), we have Z Z Z Z ∗ ∗ dh = dk ρ(a)da du∗ . 6∗

K∗

A∗t

w

Since the first and third integrals are taken over compact sets, they are finite. To compute the second integral, we let d = |5| and consider the map  d ϕ : A∗ → R>0 , ϕ(a) = (α(a))α∈5 . It is easy to see that ϕ is a group isomorphism; in addition,    d ∗ >0 ϕ(At ) = (x1 , . . . , xd ) ∈ R : xi ≤ t . Furthermore, ρ = 6α∈5 bα α for some positive integers bα . Then we have ! Z YZ t Y bα −1 −1 ρ(a)da = x dx = bα t6bα < ∞, A∗t

α∈5 0

α∈5

proving the proposition. Before proving Proposition 4.49, we must first establish an auxiliary assertion for the translates of Siegel sets in GLn (R). Lemma 4.51 Let 6 be a Siegel set of GLn (R) and let x ∈ GLn (R). Then for every s > 0, the intersection 6x ∩ KAs UnR is contained in a Siegel set of GLn (R). The proof does not involve H and its subgroups; therefore, in order not to complicate the notation, we will return to the notation used in §4.2. In particular, instead of UnR we will write simply U, and let B = AU. Clearly, for any b ∈ B, the translates 6b and (KAs U)b are contained respectively in a suitable

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Siegel set and in a set of the form KAs U. Since every x ∈ GLn (R) has a Bruhat decomposition x = vx− wbx with vx− ∈ U, bx ∈ B, w ∈ W , it suffices to prove the lemma for x = w ∈ W . Let π be the permutation of {1, . . . , n} corresponding to w. Set I = {(i, j) : i < j and πi > π j}, and let S denote the torus {x = diag(x1 , . . . , xn ) : xi = xj for all (i, j) ∈ I}. Let F be the commutator group of the centralizer CGLn (S), and then let T = Dn ∩ F and U 0 = U ∩ F. Furthermore, we set A0 = (TR )0 , A00 = (SR )0 , and U 00 = {u = (uij ) ∈ U : uij = 0 for all (i, j) ∈ I}. We will need the following simple fact, whose proof is left to the reader as an exercise: the product map induces an isomorphism A0 × A00 ' A and a homeomorphism U 0 × U 00 ' U. Let δ 0 and δ 00 denote the projections of A to A0 and A00 , respectively. Lemma 4.52 (1) w centralizes S; (2) the set δ 0 (wAt w−1 ∩ As ) is compact for all s, t > 0. PROOF: (1) Let {1, . . . , n} = J1 ∪· · ·∪Jr be the partitioning into disjoint orbits under π. The set of elements of Dn that commute with w is {x = diag(x1 , . . . , xn ) : xi = xj if i, j lie in the same orbit}. Now, let x = diag (a1 , . . . , an ) ∈ S and let J be an arbitrary orbit. If |J | = l, then in order to simplify notations, we assume that J = {1, . . . , l}. Suppose f ≤ l such that a1 = · · · = af −1 , but af 6= af −1 . Since the subsets {j ∈ J : j < f } and {j ∈ J : j > f } are not invariant under π, we can find j, k ∈ J such that j < f < k and πk ≤ f ≤ πj; in addition, πk < πj. Clearly ( j, k) ∈ I; therefore aj = ak , by the definition of S. If π( f ) > π(k), then ( f , k) ∈ I, hence af = ak = aj ; if π( f ) ≤ π(k) < π ( j), then ( j, f ) ∈ I and again af = aj . Thus, in all cases af = aj , contradicting our assumption since j < f . This proves that a1 = · · · = al , and our claim follows. (2) From (1), we see that A0 and A00 are invariant under w, so we have δ 0 (wAt w−1 ∩ As ) = wδ 0 (At ∩ w−1 As w)w−1 ,

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and it is enough to establish that δ 0 (At ∩ w−1 As w) is bounded. For this, we let d = |I| and consider the map    d ai >0 ϕ: A → R , ϕ(diag(a1 , . . . , an )) = . aj (i,j)∈I Clearly, ker ϕ = A00 , so the restriction ϕ|A0 gives an isomorphism A0 ' ϕ(A0 ). So, it is enough to show that 8 = ϕ(At ∩ w−1 As w) is relatively compact. There are constants t0 , s0 > 0 such that aaji ≤ t0 (resp., s0 ) for all a = diag(a1 , . . . , an ) ∈ At (resp., As ) and all i < j. We claim that −1 d 8 ⊂ [s−1 0 , t0 ] .

Indeed, by our construction ϕ(At ) ⊂ [−∞, t0 ]d . But if a ∈ w−1 As w, then   waw−1 = diag aπ −1 (1) , . . . , aπ −1 (n) ∈ As . Let (i, j) ∈ I, i0 = π(i), and j0 = π( j). Then i0 > j0 , which means that ai aπ −1 (i0 ) = ≥ s−1 0 , aj aπ −1 ( j0 ) as desired, completing the proof of Lemma 4.52. We will now complete the proof of Lemma 4.51. We first note that we only need to show that the u-components of the elements of 6w ∩ KAs U are bounded. Let us take g = kau ∈ 6 and compute the a-component of gw. Set m = aua−1 w and let m = km am um be the corresponding Iwasawa decomposition. Then we have gw = kauw = kmw−1 aw = (kkm )am um w−1 aw. Since w−1 aw normalizes U, we may equate the a-components to obtain agw = aaua−1 w w−1 aw.

(4.22)

Now let gw run through 6w ∩ KAs U. Then it follows from Lemma 4.11 that the aaua−1 w constitute a relatively compact set. Since agw ∈ As , it follows from (4.22) that w−1 aw ∈ As0 for sufficiently large s0 . Applying the second assertion of Lemma 4.52, we conclude that the δ 0 (a) are bounded. Let u = u0 u00 , where u0 ∈ U 0 , u00 ∈ U 00 . Then we have gw = kδ 0 (a)δ 00 (a)u0 u00 w = (kw)(w−1 δ 0 (a)u0 w)δ 00 (a)w−1 u00 w. Note that w−1 Fw = F since F = [CGLn (S), CGLn (S)] and w centralizes S; hence h := w−1 δ 0 (a)u0 w ∈ FR . Then the components of the Iwasawa decomposition h = kh ah uh also lie in FR . Indeed, it is easy to verify that if

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c = diag(ε1 , . . . , εn ), where εi = ± 1, and if Zc is the centralizer of c in GLn and h ∈ (Zc )R , then the components of the Iwasawa decomposition of h also lie T in (Zc )R . On the other hand, F can be written as c Zc , where c runs through a suitable set of diagonal elements of the form we are considering. Taking this into account, we can continue the computations: gw = (kw)hδ 00 (a)w−1 u00 w = (kwkh )(ah δ 00 (a))(uh w−1 u00 w). Since δ 0 (a) and u0 are bounded, it follows from the definition of h that h runs through a relatively compact set. Consequently, its components, and in particular the uh , also run through relatively compact sets. Since g ∈ 6, the component u00 is bounded, so the expression uh w−1 u00 w is bounded as well. It remains to observe that the definition of U 00 implies the inclusion w−1 U 00 w ⊂ U, so uh w−1 u00 w is precisely the u-component of gw. This completes the proof of Lemma 4.51.  We also need to extend the following remark to arbitrary groups. Let a ∈ A. Then for a suitable w in W , we have w−1 aw ∈ A1 ( = At for t = 1); in other words  [  A= w−1 A1 w . w∈W

Indeed, fix a = diag(a1 , . . . , an ) ∈ A and list its entries ai in increasing order:  1 ... n ai1 ≤ · · · ≤ ain . Let π denote the permutation and let w be i1 . . . in the corresponding element of W . Then w−1 aw = diag(ai1 , . . . , ain ) ∈ A1 . To generalize this to H (at this point we return to the notation introduced in the beginning of this section), recall that the relative Weyl group W ∗ of H is defined as NH (S)/CH (S), where NH (S) (resp., CH (S)) is the normalizer (resp., centralizer) of S in H; moreover, representatives of all the classes of W ∗ can be taken from NH (S)R , so that actually W ∗ = NH (S)R /CH (S)R . Also note that W ∗ naturally acts on S by conjugation. Lemma 4.53 (1) All cosets in W ∗ admit a representative from the maximal compact subgroup K∗ . S (2) We have A∗ = w∈W ∗ w−1 A∗1 w. PROOF: (1) Let x ∈ NH (S)R and let x = kau ∈ K∗ A∗ U ∗ be its Iwasawa decomposition. Then for every b ∈ S, we have −1 ¯ k −1 bk = (au)b(au) ,

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where b¯ = x−1 bx ∈ S. But k −1 = t k, so k −1 bk = t kbk is a symmetric matrix. −1 is an upper triangular matrix. Therefore ¯ On the other hand, (au)b(au) −1 ¯ (au)b(au) is actually a diagonal matrix, so au ∈ NH (S). Since the centralizer of every torus in a connected solvable group coincides with its normalizer (cf. Borel [1991, Theorem 10.6, (5) ii]), we in fact have au ∈ CH (S). It follows that x and k represent the same coset in W ∗ . (2) Let a ∈ A∗ . Set P = {α ∈ R : α(a) ≥ 1}. It is easy to see that if α, β ∈ P, and α + β ∈ R, then α + β ∈ P, and moreover P ∪ (−P) = R. Thus, in the terminology of Bourbaki (2002) (cf. Chapter 4, §1.7), P is a parabolic set and therefore contains some system of simple roots 50 of R. Since W ∗ is naturally isomorphic to W (R), and the latter acts simply transitively on systems of simple roots, there is a w˜ in W (R) such that w5 ˜ 0 = 5; ∗ hence wP ˜ ⊃ 5. Then, if w is an element of W corresponding to w, ˜ we have α(w−1 aw) ≤ 1 for all α in 5, i.e., w−1 aw ∈ A1 . Lemma 4.53 is proved. We will now explain the strategy of the concluding part of the proof of Proposition 4.49. Let y = zx ∈ 6x ∩ HR , and let y = k1 a1 u1 be the corresponding Iwasawa decomposition. By Lemma 4.53, we can find an element w in NHR (A∗ ) ∩ K∗ satisfying w−1 a1 w ∈ A∗1 . Then w−1 a1 w ∈ A1 , since by assumption, for each i = 1, . . . , n − 1, the restriction of αi to S is positive, i.e., P αi = α∈5 ciα α, where ciα ≥ 0. Furthermore, in the proof of Lemma 4.51, we established (cf. (4.22)) that ayw = aa1 u1 a−1 w w−1 a1 w. 1

If we prove that the elements of the form a1 u1 a−1 1 constitute a relatively compact set, then this formula gives us that ayw ∈ As for sufficiently large s, i.e., yw ∈ KAs (Un )R ∩ 6xw. It follows from Lemma 4.51 that there exists a Siegel set 61 of GLn (R), such that KAs (Un )R ∩ 6xw ⊂ 61 ; then 61 ∩ HR ⊂ 6 ∗ for a suitable Siegel set 6 ∗ of HR . Finally, we have yw ∈ 6 ∗ , i.e., [ 6x ∩ HR ⊂ 6 ∗ w−1 , w

6∗

where is a sufficiently large Siegel set in H, and w runs through a system of representatives of elements of W ∗ chosen in K∗ . Thus, it remains to show that the set of elements of the form a1 u1 a−1 1 is bounded, for which we first reduce the proof of Proposition 4.49 to elements

4.6 The Finiteness of the Volume of GR /GZ

261

x of a specific form. Let x = bwu be the “inverted” Bruhat decomposition of x, where b is an upper triangular matrix, u is an upper unitriangular matrix, and w ∈ W . Since 6b is contained in some large Siegel set of GLn (R), we may assume that b = 1, i.e., x = wu. Furthermore, by Lemma 2.5, we can find a Zariski-closed R-set P ⊂ Un , invariant under the adjoint action of S such that ∼ ∼ the product maps induce R-isomorphisms P × U → Un and U × P → Pn . Write u = pv, with p ∈ PR and v ∈ UR . If we can show that [ 6wp ∩ HR ⊂ 6 ∗ xi , i

S then 6x ∩ HR ⊂ i 6xi v. Thus, we may assume that x = wp with p ∈ PR . We will prove that for x of this form, the set {a1 u1 a−1 1 : y = k1 a1 u1 ∈ 6x ∩ HR } is bounded, thus completing the proof of Proposition 4.49. So, let y = zx ∈ 6x ∩ HR ; and let z = kau and y = k1 a1 u1 be the corresponding Iwasawa decompositions. We are going to express a1 , u1 in terms of a, u, and then use the fact that z is taken from 6. We have y = (kau)wp = (kw)(w−1 aua−1 w)(w−1 aw)p.

(4.23)

c = w−1 aua−1 w

If we set and then take the Iwasawa decomposition c = kc ac uc and substitute it in (4.23), we obtain y = (kwkc )ac uc w−1 awp = (kwkc )(ac w−1 aw)((w−1 aw)−1 uc (w−1 aw))p, hence a1 = ac w−1 aw, u1 = (w−1 aw)−1 uc (w−1 aw)p. Therefore,    −1 a1 u1 p−1 a−1 a1 p−1 a−1 = ac uc a−1 c . 1 = a1 u1 a1 1 Since z was chosen from a Siegel set, Lemma 4.11 shows that the elements of the form aua−1 constitute a relatively compact set. This implies that the set −1 {c} is bounded, hence so is the set {ac uc a−1 c }. Now, to verify that {a1 u1 a1 } is −1 bounded, we only need to observe that a1 u1 a1 is the projection of ac uc a−1 c on UR under the isomorphism UR × PR ' (Un )R . This completes the proof of Proposition 4.49.  Thus, we have proven Theorem 4.54 Let G be a semisimple Q-group, and let 0 ⊂ GR be an arithmetic subgroup. Then the quotient GR / 0 has finite invariant volume. In other words, 0 is a lattice in GR .

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(Recall that by a lattice in a locally compact topological group G we mean a discrete subgroup 0 ⊂ G such that G/ 0 has finite invariant volume. One of the crowning achievements of the theory of discrete subgroups of Lie groups is the Margulis Arithmeticity Theorem, which, in particular, implies that if G is an absolutely almost simple R-group of R-rank > 1, then every lattice 0 ⊂ GR is arithmetic in an appropriate sense. We refer the reader to Margulis [1991, Chapter IX] for the details and the statement of a much more general result – see Theorem A.) Theorem 4.54 immediately yields another proof of the fact that an arithmetic subgroup of a semisimple algebraic Q-group G with noncompact group GR is infinite (Corollary 4.41). Indeed, if GR is noncompact, then it has infinite volume with respect to a Haar measure. Consequently, GR / 0 also has infinite volume for every finite subgroup 0 of GR . On the other hand, for an arithmetic subgroup 0 this volume must be finite. We can now deduce Theorem 4.47 from Theorem 4.54 by straightforward argument. Namely, we may assume from the start that G is connected. If G is a Q-torus, then the quotient GR /GZ is a group, so it has finite volume if and only if it is compact (Proposition 3.61). The latter holds if and only if G is Q-anisotropic, i.e., X(G)Q = 1 (Theorem 4.38). Thus, Theorem 4.47 holds for this case. Now let G be an arbitrary reductive group. Write G as an almost direct product G = FS, where F is a semisimple Q-group and S is a maximal central Q-torus of G; note that X(G)Q = 1 is equivalent to S being Q-anisotropic. Set H = F × S and consider the isogeny ϕ : H → G. Then HR = FR × SR is clearly unimodular; therefore the unimodularity of GR follows from the finiteness of the index [GR : ϕ(HR )]. Taking into account the finiteness of ker ϕ and the arithmeticity of ϕ(HZ ), one easily shows that GR /GZ and HR /HZ simultaneously have either finite or infinite volume. Since FR /FZ has finite volume by Theorem 4.54, it follows that HR /HZ has finite volume if and only if SR /SZ does, which, as we have seen, is equivalent to S being Q-anisotropic. As usual, the case of an arbitrary Q-connected group G reduces to the reductive case by means of the Levi decomposition G = HU, where U is the unipotent radical of G and H is a Q-reductive group. Then GR = HR UR is a semidirect product, and therefore the Haar measure dg of GR can be written as the direct product dg = dh du of the Haar measures dh and du on HR and UR , respectively (see Proposition 3.67). By Lemma 4.22, GR contains a fundamental set  (relative to GZ ) of the form  = 68, where 6 is a fundamental set in HR relative to HZ and 8 is a compact subset of UR . Clearly,  has finite volume if and only if 6 does. On the other hand, it follows from the results in §3.5 that the existence of a finite invariant measure on

4.6 The Finiteness of the Volume of GR /GZ

263

GR /GZ is equivalent to the unimodularity of GR together with the existence of a fundamental set F ⊂ GR relative to GZ having finite volume; then every fundamental set also has finite volume. If X(G)Q 6= 1, then X(H)Q 6= 1, and from our consideration of the reductive case we conclude that 6 has infinite volume. Hence  also has infinite volume, which means that GR /GZ cannot have finite volume. Conversely, if X(G)Q = 1, then X(H)Q = 1; hence 6 and  both have finite volume. Thus, it remains to be shown that in this case GR is unimodular. But this can be proven in exactly the same way as Corollary 3.72. Indeed, let ω be a left-invariant rational differential Q-form on G of degree n = dim G. We have seen in the proof of Corollary 3.72 that letting %g denote the right multiplication by g, we have %g∗ (ω) = χ(g)ω for some character χ of G. Since ω is defined over Q, it is easy to see that χ is also defined over Q. Therefore, in this case χ = 1, i.e., ω is also right-invariant, and consequently GR is unimodular by Theorem 3.71. This completes the proof of Theorem 4.47. In the cases where GR /GZ has finite volume, we naturally have the problem of computing its exact value with respect to some canonical Haar measure. As we will see in the next chapter, this problem is closely related to computing (1) Tamagawa numbers. For a norm torus S = RK/Q (Gm ), the value of µ(SR /SZ ) can be expressed in terms of the discriminant and regulator of K, if we normalize the Haar measure in such a way that the volume of K∞ /O equals 1, where K∞ = K ⊗Q R and O is the ring of integers in K (cf. Lang [1994]). Langlands (1966) found the volume of GR /GZ for a semisimple simply connected split Q-group G; we note that his result can be written as a product of values of the ζ -function at certain integers – cf. Example 4.55. The general formula for the volume of the quotients of GR by so-called principal arithmetic subgroups for an arbitrary semisimple simply connected Q-group G, as well as of similar quotients in the S-arithmetic situation over arbitrary global fields was found by Prasad (1989). Without going into the details, we would like to point out that the required volume is expressed as a product of three types of factors: those depending on the discriminant of the field of definition and its relevant extensions, those depending on the root system, and certain Euler-type products of local factors that eventually yield the values of some L-functions. We refer the reader to Kaletha and Prasad (2023, Chapter 18) for a detailed exposition. The structure of the Prasad volume formula makes it amenable to analysis by number-theoretic techniques, which has led to numerous important applications. Among these are the finiteness theorems for S-arithmetic subgroups of bounded covolume in semisimple groups (cf. Borel and Prasad [1989] and Appendix A.1 in [AGNT]), determination of (arithmetic) lattices of minimum covolume (cf. Belolipetsky and Emery [2012],

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Golsefidy [2009, 2013]), counting lattices in terms of their covolume (see Belolipetsky and Lubotzky [2012, 2019] for the most recent results), and others. One should also mention the resolution of the problem concerning fake projective planes introduced by D. Mumford, which was obtained by Prasad and Yeung (2010); this included the construction of a series of new examples of fake projective planes, and, eventually, their complete classification (cf. Cartwright and Steger [2010]). Fake versions of some other projective varieties were analyzed using the volume formula in Prasad and Yeung (2009, 2012). Example 4.55 Let G = SL2 . The Iwasawa decomposition determines coordinates ϕ, a, u on GR = SL2 (R), which can be computed for x in GR from the equation     cos ϕ − sin ϕ a 0 1 u x= . sin ϕ cos ϕ 0 a−1 0 1 In §3.5 (cf. Example 3.69), we showed that with respect to these coordinates, the Haar measure on GR can be written as adϕ da du. Therefore, the volume of GR /GZ is given by the integral Z adϕ da du, F

where F ⊂ GR is a measurable fundamental domain relative to GZ . One can easily construct a fundamental domain satisfying conditions (1) and (2) in (3.22) of §3.5 using the considerations of §4.2. We will again use the map ψ : SL2 (R) → SO2 (R)/SL2 (R) = P (the upper half-plane) given by   ti + y x y ψ: → . u t ui + x Furthermore, consider the closed domain   1 ¯ = z ∈ P : |0 ¯  = (a, u) ∈ R × R : ψ ∈D . 0 a−1

4.7 Concluding Remarks on Reduction Theory

265

Therefore, Rπ R the volume of GR /GZ relative to the given measure is equal to 0 dϕ  a da du. Direct computation shows that   1 1 >0  = (a, u) ∈ R × R : |u| ≤ , 0 ≤ a ≤ √ , 4 2 1 − u2 and then vol(GR /GZ ) =

Z

π

Z dϕ

0



Z a da du =

π

Z dϕ

0

1 2

− 12

Z du

1 √ 4

1−u2

a da =

0

π2 . 6

(Note that this value equals ζ (2) – cf. Serre [1973].)

4.7 Concluding Remarks on Reduction Theory The plan for developing reduction theory for arithmetic subgroups of algebraic Q-groups, which we outlined in §4.2, has been completed. However, several interesting concepts and results, not directly related to the topics chosen for detailed exposition in this book, have been neglected so far. To partially rectify the situation, in this section, we include (without proofs) several additional results on reduction theory, such as an alternative construction of fundamental sets, the connection with the reduction theory for quadratic forms, and so on. Moreover, the previously obtained results are reformulated for O-arithmetic subgroups. The construction of fundamental sets in §4.3 relies mainly on properties of GR and depends relatively little on GQ and GZ . A modification of this construction, however, leads to a more subtle construction that uses the Q-structure on G in a very essential way and typically produces fundamental sets with better properties. Its advantages become apparent in the theory of automorphic functions, since the behavior of these fundamental sets at infinity is similar to the behavior of fundamental domains for Fuchsian groups near cusp points. This construction also provides insights needed for the construction of a compactification of GR /GZ , which is fundamental to the study of the cohomology of arithmetic groups. So, let G be a semisimple algebraic Q-group. The situation where G is Qanisotropic should be regarded as the most favorable: here the quotient GR /GZ is compact (Theorem 4.42), and therefore there exists a compact fundamental set in GR with respect to GZ . Now suppose that G is Q-isotropic. We denote by S a maximal Q-split torus of G and by P a minimal parabolic Q-subgroup containing S. It is well known that P is a semidirect product of its unipotent

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radical U and ZG (S), the centralizer of S. In turn, ZG (S) can be written as an almost direct product ZG (S) = M · S,

(4.24)

where M is the largest connected Q-anisotropic subgroup of ZG (S). Let A denote the connected component of SR . One can choose a maximal compact subgroup K of GR so that GR = K · PR , and then (4.24) yields the following decomposition: GR = KMR AUR . (Note that, as a rule, a decomposition of this form is not unique.) Since MR /MZ is compact, it makes sense to look for a fundamental set in GR relative to GZ in the form of a generalized Siegel set 6t,y,ω = KyAt ω,

t > 0,

where y (resp., ω) is a compact subset of MR (resp., UR ), At = {a ∈ A : α(a) ≤ t ∀α ∈ 5}, and 5 is the system of simple roots in the root system of G relative to S associated with P. Theorem 4.56 Let G be a semisimple algebraic Q-group, and let 0 ⊂ GQ be an arithmetic subgroup. (1) There exists a generalized Siegel set 6 = 6t,y,ω and a finite subset C of GQ such that  = 6C is a fundamental set for 0 in GR . The set C contains at least one representative of each double coset PQ \GQ / 0 (in particular, the number of these double cosets is finite). (2) Conversely, if C is a finite subset of GQ containing a representative of each double coset PQ \GQ / 0, then there exists a Siegel set 6 such that  = 6C is a fundamental set in GR relative to 0. We note only that the proof of (1) uses the construction of a fundamental set developed in §4.3, and generally speaking, is similar to the proof of Proposition 4.49. It is easy to show that each generalized Siegel set 6 has finite volume with respect to the Haar measure on GR , so Theorem 4.56 (1) enables one to obtain another proof of Theorem 4.54. Due to Theorem 4.56, it is natural to introduce a new invariant in the theory – the number of double cosets PQ \GQ / 0. It turns out to be the smallest number of translates of 6 whose union can form a fundamental set for 0.

4.7 Concluding Remarks on Reduction Theory

267

n  o a b ∗ , b ∈ C , and then For G = SL2 , one can take P = : a ∈ C −1 0 a PR can be viewed as the stabilizer of the point ∞ at infinity for the natural left action of SL2 (R) on the upper half-plane P (cf. §4.2).2 Then the orbit SL2 (Q)(∞) coincides with the set of cusp points, which is the union of {∞} with the set of points on the real axis having rational coordinates. The number of double cosets PQ \GQ / 0 in this case is the number of equivalence classes of cusp points relative to 0. This number turns out to be equal to the number of points that need to be added to the quotient-space P/ 0 to obtain its compactification, or the number of vertices (cusps) of the corresponding fundamental domain at infinity (cf. Figure 4.4 for the case where 0 is the congruence subgroup SL2 (Z, 2); here there are three cusps). In general, the complements of compact subsets in the Siegel set 6t,y,ω contain Siegel sets 6s,y,ω for s sufficiently small, and may be regarded as analogs of cusps in the case of SL2 . In this manner, the number of double cosets 0\GQ /PQ can also be interpreted as the smallest number of cusps of a fundamental domain for 0. Note that in the next chapter, we will give an adelic proof of the finiteness of the number of double cosets 0\GQ /PQ (cf. Theorem 5.19) and reveal its connection with the class number of P. Generalized Siegel sets are functorial in the sense that given a surjective Qmorphism f : G → H of semisimple Q-groups and a generalized Siegel set 6 of G, the image f (6) is contained in a suitable generalized Siegel set of H. This implies that the fundamental sets  constructed in Theorem 4.56 satisfy the following condition, which is stronger than condition ( F2) in the definition of a fundamental set (cf. §4.3): ( F2)bis The intersection −1  ∩ x0y is finite for all x, y ∈ C(0)R , where C(0) is the commensurability subgroup of 0. Indeed, it suffices to show that 6 −1 6 ∩ x0y is finite, for an arbitrary generalized Siegel set 6. For this, we use the following description of C(0) given in Proposition 4.35: C(0) = π −1 ((G/N)Q ), where N is the largest normal Q-subgroup of G of compact type, and π : 0 → ˜ be a Siegel set of G/N such that π(6) ⊂ 6. ˜ G/N is the canonical map. Let 6 Then ˜ −1 6 ˜ ∩ π(x)π(0)π(y). π(6 −1 6 ∩ x0y) ⊂ 6 2 In §4.2 we denoted the upper half-plane by P. Here we have changed the notation to P to avoid

confusion with the parabolic subgroup.

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Figure 4.4

˜ But π(x), π(y) ∈ (G/N)Q , and π(0) is an arithmetic subgroup of G/N; since 6 satisfies the usual condition (F2), it follows that the latter intersection is finite. It remains to note that 0 ∩ N is finite since NR is compact, and consequently the finiteness of π(6 −1 6 ∩ x0y) implies that of 6 −1 6 ∩ x0y, as desired. The fundamental sets provided by Theorem 4.56 have another noteworthy property. Before introducing its general formulation, let us recall one of the essential steps in the reduction theory for GLn (R) (cf. §4.2). Having fixed an orthonormal base e1 , . . . , en of Rn , we introduced the continuous function 8 : GLn (R) → R+ , 8(g) = kge1 k. Then the set of values of 8 on each coset gGLn (Z) is bounded away from 0, and the minimal value is attained at some point of 6 = 6 √2 , 1 , implying that 3 2

GLn (R) = 6GLn (Z). It turns out that a similar minimum principle holds for an arbitrary semisimple algebraic Q-group G. In order to describe the corresponding function ϕ, we use the above notations. Let π : G → GL(V ) be an absolutely irreducible Q-representation of G for which the parabolic subgroup P has an eigenvector v in VQ . Then set ϕπ (g) = kπ(g)vk, where the norm is taken with respect to an orthonormal basis of VR consisting of eigenvectors for S. A special case of functions of the form ϕπ are the 8i in §4.2 that correspond to the fundamental representations of SLn . Theorem 4.57 Let G be a semisimple algebraic Q-group, ϕ = ϕπ be the function corresponding to an absolutely irreducible Q-representation π : G → GL(V ), and C be a set of representatives of the double cosets 0\GQ /PQ , where 0 ⊂ GQ is an arithmetic subgroup. Then there exists a generalized Siegel set 6 of GR such that for every x ∈ GR , the function ϕ attains its minimum on x0C at some point of x0C ∩ 6, implying that GR = 6C −1 0.

4.7 Concluding Remarks on Reduction Theory

269

Note that any semisimple Q-group G has a sufficient supply of representations π with the properties described above. Moreover, as is well known, any absolutely irreducible representation is defined by its highest weight, and any dominant weight can be realized as the highest weight of some representation (cf. Humphreys [1975]). Furthermore, it turns out that a suitable multiple of any dominant weight can be realized as the highest weight of a Q-representation satisfying the preceding requirements. The functions ϕπ have the same properties as the functions 8i in §4.2. Relying on these properties, one can prove an analog of Harish-Chandra’s theorem in this case, and as a consequence deduce property ( F2) in the definition of a fundamental set. Even though throughout this chapter we have been considering only algebraic groups defined over Q, the results obtained can be extended to algebraic groups defined over an arbitrary algebraic number field K. Let O be the ring of integers of K. Then by an O-arithmetic subgroup of G, we mean a subgroup of G that is commensurable with the group GO of O-points of G (defined in terms of some matrix realization of G). The group GO is a discrete subgroup Q of G∞ = v∈V∞ K GKv , which is the analog of the group of real points for Qgroups. Naturally, we have the problem of developing a reduction theory for G∞ relative to GO . We state the basic results obtained along these lines in the following. Theorem 4.58 Let G be an algebraic group defined over an algebraic number field K. Then the following hold: (1) there exists an open fundamental set  ⊂ G∞ relative to GO so that ( F0) K =  for a suitable maximal compact subgroup K ⊂ G∞ , ( F1) GO = G∞ , ( F2) −1  ∩ xGO y is finite for all x, y ∈ GK ; (2) GO is a group with a finite number of generators and defining relations; (3) G∞ /GO is compact if and only if the reductive part of the connected component G0 is anisotropic over K;
(4) G∞ /GO has finite invariant volume if and only if X G0 K = 1. To prove this, we pick a Z-basis of O and use it to construct H = RK/Q (G) (cf. §2.1.2). Then GO ' HZ and G∞ ' HR , and applying the corresponding results for Q-groups one easily proves (1) and (2). To prove (3), one needs to observe that the reductive part of the connected component of H has the form RK/Q (D), where D is the reductive part of the connected component of G. Moreover, RK/Q (D) is Q-anisotropic if and only if D is K-anisotropic, so

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one can use Theorem (4) follows from Theorem 4.47 in view of
4.42. Finally,
0 0 the fact that X H Q = X G K . Other results (such as the Borel Density Theorem, the description of commensurable subgroups, etc.) can also be extended without much effort to O-arithmetic subgroups (cf. Proposition 5.35 regarding the density theorem). We will not formulate these results, but rather will confine ourselves to generalizing the concepts involved in their statements to the O-arithmetic case. An algebraic group G defined over an algebraic number field K is said to Q have compact type if G∞ = v∈V∞ K GKv is compact. Now let G be semisimi is noncompact for each ple. Then G is said to have noncompact type if G∞ i simple K-subfactor G of G. A semisimple group that has neither compact nor noncompact type is said to have mixed type. To conclude our exposition of reduction theory for arithmetic subgroups, we must note that this theory is rooted in the classical reduction theory of quadratic forms (cf., for example, Cassels [1978]), which goes back to Gauss, Hermite, and Minkowski. In particular, the construction of fundamental sets as the union of a finite number of translates of a suitable Siegel set can be viewed as a generalization of the construction used by Hermite (1905) in the case of indefinite rational quadratic forms. Paying tribute to these distinguished predecessors, we now present a few results on the reduction of positive definite quadratic forms. Let us identify the set of positive definite quadratic forms on Rn with the space H of real symmetric positive definite (n × n)-matrices. Then there is a transitive right action of G = GLn (R) on H given by g : F → F[g] = tgFg. The stabilizer of the unit form is the group K = On (R), so H = K\G, where the natural map π : G → H is given by π(g) = tgg. Clearly, π sends SLn (R) to the set H (1) of matrices in H having determinant 1, and H (1) = SOn (R)\SLn (R). With notations as in §4.2, we will call a set of the form  0 6t,v = tuau : a ∈ At , u ∈ Uv a Siegel set in H. Since tkk = In for every k in K, we have 0

π(6t,v ) = 6t2 ,v . Since π(g1 g2 ) = π(g1 )[g2 ], Theorems 4.12 and 4.47 yield 0

Theorem 4.59 (i) (KORKIN AND ZOLOTAREV) We have H = 6t,v [GLn (Z)] for t ≥ 43 , v ≥ 12 .

4.8 Finite Arithmetic Groups

271

(ii) (HERMITE) If F is a positive definite form on Rn , then   n−1 1 4 2 min F(x) ≤ (det F) n . x∈Zn \{0} 3 0

(iii) (MINKOWSKI) We have H (1) = (6t,v ∩ H (1) )[SLn (Z)] for v ≥ 4 (1) 3 , and H /SLn (Z) has finite invariant volume.

1 2

and t ≥

4.8 Finite Arithmetic Groups The arithmetic theory of algebraic groups is mainly concerned with the analysis of infinite arithmetic groups, since only in this case can one expect a close connection between the properties of an algebraic group G and the properties of its arithmetic subgroups. The case of finite arithmetic groups used to be thought of as being relevant mostly for the theory of finite simple groups. Indeed, the group of automorphisms of the 24-dimensional positive definite Leech lattice (more precisely, the quotient-group modulo its center) turned out to be a new simple sporadic group, discovered by Conway (1968). However, more recently, some interesting arithmetic questions involving finite arithmetic groups have come to light. For the most part, these questions center around the following conjecture. Conjecture 4.60 Let G be an algebraic Q-group of compact type. Then for every totally real finite extension K/Q, we have GO = GZ , where O is the ring of integers of K. (Recall that a finite extension K/Q is said to be totally real if the image of every embedding K ,→ C is contained in R. If G is a Q-group of compact Q type, then for any such extension, the group G∞ = v∈V∞ K GKv is compact, so GO is finite.) In other words, this conjecture asserts that the group of units of a compact algebraic Q-group does not change (is “stable”) under extensions of the base ring from Z to the ring of integers O of a totally real number field K. Since every totally real extension is clearly contained in a totally real Galois extension, we may assume without loss of generality that K/Q is a Galois extension. This conjecture came up in the analysis of properties of positive definite quadratic lattices, and, as we will see shortly, its proof would actually yield a number of interesting corollaries in the theory of lattices. It is worth observing that the general case of the conjecture in fact reduces to the case of orthogonal groups of positive definite quadratic forms. To see this, we need the following.

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Proposition 4.61 Let G ⊂ GLn (C) be an algebraic Q-group whose group of real points GR is compact and Zariski dense in G. Then there exists an ndimensional positive definite quadratic form f with integer coefficients such that G ⊂ On ( f ). PROOF: Let h be an arbitrary n-dimensional positive definite quadratic form. Since GR is compact, Z h0 (v) :=

h(gv)dg GR

is defined for each v in Rn (here dg is the Haar measure on GR ). Elementary verification shows that the map Rn → R, v 7→ h0 (v), is a positive definite GR -invariant quadratic form on Rn . Since GR is Zariskidense in G, the extension of h0 to Cn is invariant under G. Let V denote the space of all G-invariant quadratic forms on Cn , viewed as a subspace of the space of symmetric matrices. Since G is defined over Q, so is V . Moreover, the density of Q in R implies the density of VQ in VR . On the other hand, the subset W of VR of positive definite forms contains h0 , and therefore is a nonempty open subset of VR (we note that the openness of this subset is a consequence of the well-known Silvester criterion). Since VQ is dense in VR , this implies the existence of a positive definite form f in VQ . Multiplying by a scalar, we can assume that f corresponds to an integer matrix, as required. We will now show that Conjecture 4.60 is equivalent to the following. Conjecture 4.60∗ Let f be a positive definite quadratic form of dimension n with integer coefficients. Then On ( f )O = On ( f )Z for the ring of integers O of any totally real finite extension K/Q. Conjecture 4.60∗ is obviously a special case of Conjecture 4.60. In order to see the converse, we fix a totally real finite Galois extension K/Q with ring of integers O, and let H denote the subgroup generated by G0 and GO . Since O is invariant under all automorphisms of C/Q, we see that GO and hence 0 G , so the fact that G0 also H = G0 GO are defined over Q. Clearly, HR = GR O R 0 is Zariski-dense in G (Theorem 2.6) implies that HR is Zariski-dense in H. Applying Proposition 4.61, we see that there exists a positive definite quadratic form f with rational coefficients such that H ⊂ On ( f ), where n is the degree of G as a linear group. Assuming the truth of Conjecture 4.60∗ , we will have GO = HO ⊂ On ( f )O = On ( f )Z , so GO = GZ , as required. In connection with Conjecture 4.60∗ , we should mention the following result:

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273

Proposition 4.62 Let f (x1 , . . . , xn ) = a1 x21 + · · · + an x2n be a diagonal integral positive definite quadratic form. Then On ( f )O = On ( f )Z for the ring of integers O of any totally real extension K/Q. PROOF: We will show that for every element b = (bij ) in On ( f )O , each row and each column contains only one nonzero element, which necessarily equals ±1. Without loss of generality, we may assume that a1 ≤ a2 ≤ · · · ≤ an . The fact that b ∈ On ( f ) means that tbFb = F, where F = diag(a1 , . . . , an ), from which we obtain the following relations: n X

for each j = 1, . . . , n;

ai b2ij = aj

(4.25)

i=1 n X

ai bij bik = 0

for each j 6= k between 1 and n.

(4.26)

i=1

Pick a nonzero entry bnj in the last row of b. Then, due to (4.25), we have an τ (bnj ) ≤ 2

n X

ai τ (bij )2 = aj ≤ an

i=1

for every embedding τ : K ,→ R, and consequently |bnj |v ≤ 1 for every real place v of K. But Y |bnj |v = |NK/Q (bnj )|, K v∈V∞

where NK/Q is the norm map from K to Q. On the other hand, since bnj ∈ O, we have NK/Q (bnj ) ∈ Z, hence |NK/Q (bnj )| ≥ 1. Therefore |bnj |v = 1 for K and consequently b = ± 1. (We note that |b| = |τ (b)|, where every v ∈ V∞ nj v v τv : K ,→ R is the corresponding embedding and | | is the absolute value.) Returning to (4.25) and bearing in mind that aj ≤ an , we obtain aj = an and bij = 0 for i < n. Furthermore, applying (4.26) we see that bnk = 0, for all k 6= j. A similar argument can be applied to the jth row, if aj = an . Thus, we deduce that b has the block structure   b1 0 b= , 0 b2 where b1 is a square l × l matrix, l being the maximal index for which al < an , and b2 is a monomial matrix of size (n − l) × (n − l), all of whose nonzero elements equal ±1. (One can have a situation where all the ai coincide, but then there is nothing left to prove.) The proof of Proposition 4.62 can now be completed by an obvious inductive argument.

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274

Next, we would like to point out one more equivalent formulation of Conjecture 4.60. Conjecture 4.60∗∗ Let K/Q be a totally real finite Galois extension, and let O be the ring of integers in K. If a finite subgroup 0 ⊂ GLn (O) is invariant (as a set, but not necessarily elementwise) under Gal(K/Q), then 0 ⊂ GLn (Z). It is easy to see that Conjectures 4.60 and 4.60∗∗ are equivalent. Indeed, as we have already noted, we may assume that K/Q in Conjecture 4.60 is a Galois extension. If G is an algebraic Q-group of compact type, then GO is a finite Gal(K/Q)-invariant subgroup of GLn (O), so Conjecture 4.60∗∗ implies Conjecture 4.60. To see the converse implication, one observes that a finite Gal(K/Q)-invariant subgroup 0 ⊂ GLn (O) can be regarded as an algebraic Q-group G with compact GR = 0. So far, Conjecture 4.60∗∗ has been established only for some special Galois extensions, namely for those extensions whose Galois group is either nilpotent or has cyclic Sylow subgroups (cf. Bartels and Kitaoka [1980]). The proof for nilpotent extensions contains most of the conceptual features, so we will present this case in detail and refer the reader to the paper by Bartels and Kitaoka for other results. It should be pointed out that the two facts that are used repeatedly in the analysis of Conjecture 4.60∗∗ are Hermite’s theorem on the non-existence of nontrivial extensions of Q unramified at all primes (cf. Theorem 1.7) and the following lemma due to Minkowski (1887a): Lemma 4.63 (MINKOWSKI) For any prime p 6= 2, the principal congruence subgroup GLn (Z, p) is torsion-free. PROOF: Using the embedding Z ,→ Zp and applying Lemma 3.38, we see that the order of any element of GLn (Z, p) is either infinite or is a power of p. Thus, it suffices to show that GLn (Z, p) does not contain any (nontrivial) elements of order p. Let In 6= x ∈ GLn (Z, p) be such that xp = In . Write x = In + pα y with y ∈ Mn (Z) and y 6≡ On (mod p). Then xp = (In + pα y)p = In +

    p α p p y + ··· + pα(p−1) yp−1 + pαp yp = In , 1 p−1

i.e., pα+1 y = −

    p 2α 2 p p y − ··· − pα(p−1) yp−1 − pαp yp . 2 p−1

(4.27)

4.8 Finite Arithmetic Groups

All binomial coefficients

275

  p for 0 < i < p are divisible by p, hence i

  p iα i p y ≡ On (mod p2α+1 ) for all 1 < i < p. i Since p > 2, we have αp ≥ 2α + 1, and therefore pαp yp = On (mod p2α+1 ). It follows that the right-hand side of (4.27) is congruent to On (mod p2α+1 ), while, by our construction, the left-hand side is not, since α + 1 < 2α + 1. A contradiction, proving the lemma. Now let K/Q be a totally real finite Galois extension. If K/Q is unramified at all primes, then by Hermite’s theorem K = Q, and there is nothing to prove. Thus, we may assume that at least one prime ramifies in K/Q. We will now show that to prove Conjecture 4.60∗∗ , it suffices to consider the case where there is only one ramified prime. Proposition 4.64 Let K/Q be a totally real finite Galois extension with Galois group G and let 0 be a finite G-invariant subgroup of GLn (O). Suppose that 0 ∩ GLn (L) ⊂ GLn (Z) for every proper Galois subextension L of K. If 0 ⊂ / GLn (Z), then there is exactly one prime that ramifies in K/Q. PROOF: We will assume that two distinct primes p and q > 2 are ramified in K/Q, and show that then 0 ⊂ GLn (Z). Take some extensions wp |p and wq |q to K, and let p = O ∩ pwp and q = O ∩ qwq denote the corresponding ideals in O. We will use the ramification groups G (i) defined at the end of §1.1. Let x ∈ 0 and suppose σ is in the inertia group G (1) (wq ). Then x ≡ σ (x)(mod q), and therefore, since 0 is G-invariant, y := x−1 σ (x) lies in the congruence subgroup 0(q). Now take an arbitrary τ in G (1) (wp ). Then the group commutator [y, τ (y)] = yτ (y)y−1 τ (y)−1 lies, on the one hand, in 0(q) (since 0(q) is a normal subgroup of 0) and, on the other hand, in 0(p) (since y ≡ τ (y)(mod p)). Embedding O into Owp and Owq and applying Lemma 3.38, we see that the order of every element of 0(p) (respectively, of 0(q)) is a power of p (respectively, q), so in particular, 0(p) ∩ 0(q) = {In }. So, [y, τ (y)] = In , i.e., y and τ (y) commute. Furthermore, since y ∈ 0(q), the order of y, and hence also of z := y−1 τ (y), is a power of

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q. But at the same time z ∈ 0(p), so as previously, we conclude that actually z = In , i.e., τ (y) = y. We have shown that τ (y) = y for all y of the form y = x−1 σ (x) with x ∈ 0 and σ ∈ G (1) (wq ) (we note that here wp is an arbitrary extension of p and τ an arbitrary element of G (1) (wp )). Now let H be the subgroup of G generated by the inertia groups G (1) (wp ) for all extensions wp |p, and let L = K H be the corresponding fixed field. As we noted at the end of §1.1, L can be characterized as the maximal Galois extension of Q contained in K and unramified at p. Since by assumption p is ramified in K, we have L 6= K and therefore 0 ∩ GLn (L) ⊂ GLn (Z). We have already shown that every y ∈ 0 as above belongs to GLn (L), so in effect y ∈ GLn (Z). Recalling that y ∈ 0(q), we see that actually y is an element of GLn (Z, q) having finite order, which means that y = In by Minkowski’s lemma. Invoking the description of y as y = x−1 σ (x) with x ∈ 0 and σ ∈ G (1) (wq ), we see that in fact 0 ⊂ GLn (P), where P = K F is the fixed field of the subgroup F ⊂ G generated by the inertia groups G (1) (wq ) for all extensions wq |q. Again, since q is ramified in K, we conclude that P 6= K, and therefore 0 = 0 ∩ GLn (P) ⊂ GLn (Z), contradicting our assumption. (We note that the fact that the extension K/Q is totally real was not actually used in the proof of this proposition.) Proposition 4.64 shows that if K/Q is a totally real Galois extension of minimal possible degree over which there is a counterexample to Conjecture 4.60∗∗ , then there is only one prime that ramifies in K/Q. We will now use this observation to prove Conjecture 4.60∗∗ for nilpotent extensions. Theorem 4.65 (BARTELS AND KITAOKA) Let K/Q be a totally real finite Galois extension with nilpotent Galois group G. If 0 is a finite G-invariant subgroup of GLn (O), then 0 ⊂ GLn (Z). PROOF: Since every Galois subextension of a nilpotent Galois extension is also nilpotent, it follows from the preceding remark that we only need to consider nilpotent totally real extensions K/Q with a single ramified prime p. First, let us show that every such extension is necessarily cyclic. We will argue by induction on [K : Q]. The center Z of G is nontrivial, and by the induction hypothesis G/Z is cyclic. But then G is abelian, and by the Kronecker–Weber theorem (cf., for example, Iwasawa [1986, §8.1]), the field K is contained in a cyclotomic extension Q(ζpd ) for suitable d, where ζpd is a primitive pd th root of unity. Moreover, since K/Q is totally real, we actually have the inclusion   K ⊂ Q ζpd + ζp−1 . d

4.8 Finite Arithmetic Groups

277

But then G is a quotient of the group (Z/pd Z)∗ /{±1}, which is cyclic. Now, having proven G is cyclic, we invoke the Kronecker–Weber theorem again   to conclude that in our situation we have the inclusion K ⊂ Q ζpd + ζp−1 for d some d > 0, and to complete  the proof of Theorem 4.65 it remains to examine the case K = Q ζpd + ζp−1 . d   Lemma 4.66 Conjecture 4.60∗∗ is true for K = Q ζpd + ζp−1 . d The proof uses a general construction that may also be useful in other situations. Namely, for an arbitrary Galois extension K/Q with Galois group G, we define a Galois subextension M/Q as follows: Fix a prime p, consider an extension wp |p, and let p = O ∩ pwp be the corresponding ideal in O and r be the smallest positive integer for which p ∈ / pr(p−1) . We then let H denote the subgroup of G generated by the rth ramification groups G (r) (wp ) for all extensions wp |p, and set M = K H . The subextension M/Q obviously depends on the choice of p. However, for every prime p, it is a Galois extension, and, more importantly, for every finite G-invariant subgroup 0 of GLn (O), we have the inclusion 0 ⊂ GLn (OM ), where OM is the ring of integers of M. Indeed, for any x ∈ 0 and any σ ∈ G (r) (wp ), we have σ (x) ≡ x(mod pr ), i.e., x−1 σ (x) ∈ 0(pr ). Therefore it suffices to establish the triviality of the congruence subgroup 0(pr ). As in the proof of Minkowski’s lemma, it suffices to show that the congruence subgroup GLn (O, pr ) contains no elements of order p. This is done by a slight modification of the computations used in the proof of that lemma. More precisely, let In 6= x ∈ GLn (O, pr ) be such that xp = In . Write x = In + y where y ≡ On (mod pm ) for some m ≥ r, but y 6≡ On (mod pm+1 ). Then we have     p 2 p p In = x = In + py + y + ··· + yp−1 + yp , 2 p−1 i.e., py = −

    p 2 p y − ··· − yp−1 − yp . 2 p−1

(4.28)

Let e denote the ramification index e(wp |p), which is the exponent of h p ini the factorization of pO. Then it follows from the definition of r that r =

e p−1

+1

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Arithmetic Groups and Reduction Theory

(where, as usual, [a] is the integral part of a), and therefore (p − 1)r > e. This estimate enables us to reach a contradiction by computing the power of p that divides the left- and right-hand sides of (4.28), respectively. By our construction, for the left side of (4.28), we have py ≡ On (mod pm+e ) but py 6≡ On (mod pm+e+1 ).
Since the binomial coefficients pi for 0 < i < p are divisible by p, we have   p i y ≡ On (mod pim+e ), i and note that for 1 < i < p we have im + e ≥ m + e + 1. Lastly, yp ≡ On (mod pmp ), where mp = m + m(p − 1) ≥ m + r(p − 1) > m + e. Thus, the right-hand side of (4.26) is ≡ On (mod pm+d+1 ), contradiction. To complete the proof of the lemma, it now suffices to establish that for Kd = Q(ζpd + ζp−1 d ), the subextension M, constructed using p, is contained in Kd−1 (since K0 = Q, this will give the required result). For this, we need to recall some standard facts about the ramification of p in the cyclotomic extension Ld /Q, where Ld = Q(ζpd ) (cf. [ANT, Chapter 3]). It is well known that Ld /Q is an abelian Galois extension of degree d ϕ(pd ) = pd−1 (p−1). The ring of integers Od of Ld is Z[ζpd ] and pOd = Pϕ(p ) , where P = (1 − ζpd )Od is a maximal ideal. In other words, the p-adic valuation has a unique extension to Ld and this extension is totally ramified. In particular, for d = 1 the maximal ideal P1 ⊂ O1 lying above p is generated by 1 − ζp . On the other hand, since Ld /L1 is an extension of degree pd−1 in which P1 d−1 d−1 is totally ramified, we have P1 Od = Pp , and therefore 1 − ζp ∈ Pp , d−1 i.e., ζp ≡ 1(mod Pp ). For our purposes, we need to rewrite this congruence in a slightly different way. Namely, for each integer a coprime to p we let σa ∈ Gal(Ld /Q) denote the automorphism given by σa (ζpd ) = ζpad . Then the previous congruence means that d−1

σa (ζpd ) ≡ ζpd (mod Pp

) for a ≡ 1(mod pd−1 ).

Since ζpd generates Od , we conclude that for such a, we have d−1

σa (x) ≡ x(mod Pp

) for all x in Od .

Computing r for Kd , we find " # " # ϕ(pd )/2 pd−1 r= + 1= + 1, p−1 2

(4.29)

(4.30)

4.8 Finite Arithmetic Groups

279

since the ramification index e of the p-adic valuation in Kd /Q equals ϕ(pd )/2. Using (4.29) and (4.30), for odd p we obtain σa (x) ≡ x(mod pr ) when a ≡ 1(mod pd−1 ) for all x in Od , since the ramification index of Ld /Kd equals 2. Set G = Gal(Kd /Q) and let w be the (unique) extension of the p-adic valuation to Kd . With this notation, the preceding congruence means that the rth ramification group G (r) (w) contains the subgroup H of G consisting of the restrictions of automorphisms σa with a ≡ 1(mod pd−1 ). But KdH = Kd−1 , so the extension M defined above is contained in Kd−1 , as required. To treat the remaining case p = 2, we need to elaborate on (4.29) and show that d−1 +2

σa (x) ≡ x(mod P2

) for a ≡ 1(mod 2d−1 ) and all x in Od .

It suffices to show that d−1 +2

−1 2 σa (ζ2d + ζ2−1 d ) ≡ ζ2d + ζ2d (mod P

).

This, however, follows easily from (4.28) and the computation −1 2 −1 2 ζ2d + ζ2−1 d = ζ2d (ζ2d + 1) = ζ2d ((ζ2d − 1) + 2ζ2d ),

in view of the fact that (ζ2d − 1)2 + 2ζ2d ∈ P2 . This completes the proof of Lemma 4.66 and of Theorem 4.65. In the analysis of special cases of Conjectures 4.60–4.60∗∗ , we can impose additional conditions of two types – on the totally real extension K/Q, and on the group 0 in Conjecture 4.60∗∗ or on the quadratic form f in Conjecture 4.60∗ , respectively. Theorem 4.65 is a sample result where conditions of the first type were imposed. We will now discuss a result that involves conditions of the second type: it shows that the space of real symmetric matrices has an open subset such that all integral matrices in that subspace (or, more precisely, the corresponding quadratic forms) satisfy Conjecture 4.60∗ . Proposition 4.67 Let f be an integral positive definite quadratic form of dimension n with matrix a = (aij ). Suppose that aii ≤ 4λ for all i = 1, . . . , n, where λ is the smallest eigenvalue of a. Then On ( f )O = On ( f )Z for any totally real extension K/Q (where, as usual, O denotes the ring of integers of K).

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Arithmetic Groups and Reduction Theory

PROOF: Let x = (xij ) ∈ On ( f )O . For each j, we will show that n X i=1

x2ij ≤

ajj λ

(4.31)

in every real embedding of K. Indeed, let vj denote the jth standard basic vector (0, . . . 0, 1, 0, . . . 0). Then xvj = (x1j , . . . , xnj ) =: wj and ajj = f (vj ) = f (wj ). If we denote by g the quadratic form on Rn for which v1 , . . . , vn is an orthonormal basis, then the left-hand side of (4.31) equals g(wj ), so it suffices to show that f (w) ≥ λg(w) for every w in Rn . Since f can be diagonalized using a transformation from On (g)R , it is enough to consider the case of f diagonal, where the desired estimate is trivial. Combining (4.31) with the condition aii ≤ 4λ, we see that in our case, n X

x2ij ≤ 4 for all j = 1, . . . , n,

i=1 K . It follows and, in particular, |xij |v ≤ 2 for all i, j = 1, . . . , n and every v in V∞ that all the xij equal twice the real parts of some roots of unity. Indeed, let a ∈ K K . Then a is totally real and be such that |a|v ≤ 2 for every v ∈ V∞ s a2 b := −1 4

is purely imaginary. Furthermore, t := 2a + b satisfies t2 − at + 1 = 0, and therefore is an algebraic integer all of whose conjugates have absolute value 1. It follows easily that t is a root of unity, so a = 20 . But R>0 has no nontrivial compact subgroups, so 1GA = 1. Thus, the unimodularity of GA 0 is finite, is equivalent to the unimodularity of GA0 . Since the quotient G∞ /G∞ the same argument shows that the unimodularity of G∞ is equivalent to the 0 . Consequently, we may assume from the start that G is unimodularity of G∞ connected.

5.3 Compactness and the Finiteness of the Volume of GA /GK

315

In the connected case, the Haar measure on GA can be constructed using a left-invariant rational K-defined differential form ω on G of degree n = dim G. More precisely, we know from Theorem 3.71 that, for each v ∈ V K , the differential form ω induces a left-invariant measure ωv on GKv . Let us choose numbers λv for v ∈ VfK (called convergence factors) so that the prodQ uct v λv ωv (GOv ) converges absolutely (for example, we can take λv = ωv (GOv )−1 ). Since GA is the restricted topological product of the groups GKv with respect to the distinguished subgroups GOv , we can apply the construction described in §3.5 (on pages 188–189) to obtain a Haar measure τ on GA , called the Tamagawa measure corresponding to the set of convergence factors λ = (λv ). Note that τ is actually independent of the choice of ω. Indeed, any other K-defined left-invariant rational differential form ω0 of degree n is of the form ω0 = cω for some c ∈ K ∗ . Then for any v ∈ V K , the corresponding measures ωv0 and ωv are related by the equation ωv0 = ||c||nv ωv , where || ||v is the normalized valuation introduced in §1.2.1. Thus, if we construct τ 0 by using ω0 and the same set of convergence coefficients, then Y  τ0 = ||c||nv τ = τ v

by the product formula (cf. §1.2.1). It follows from the construction of τ that the group GA is unimodular if and only if all the groups GKv are unimodular. However, according to Theorem 3.71, the group GKv is unimodular for some v if and only if the differential form ω is right-invariant, in which case the groups GKv are unimodular for all v. Thus, either both groups GA and G∞ are unimodular, or neither one is unimodular. Next, we have the following argument, which applies to the proofs of both statements (1) and (2). By Proposition 5.17, there exists a fundamental set in GA relative to GK of the form  = B × C, where B ⊂ G∞ is a closed fundamental set relative to GO and C is an open compact subset of GA(∞) . The properties of fundamental sets imply that the quotient GA /GK is compact if and only if the fundamental set  is compact, i.e., the set B is compact. Similarly, the existence of a fundamental set in GA relative to GK of finite volume is equivalent to the fact that the set B has finite volume. Since the unimodularity of G∞ is equivalent to that of GA , we have the following equivalences:



{GA /GK is compact} ⇐⇒ {G∞ /GO is compact},    GA /GK has finite G∞ /GO has finite ⇐⇒ . invariant volume invariant volume

Adeles

316

Therefore, parts (1) and (2) of Theorem 5.22 follow from the respective assertions in Theorem 4.58. The convergence factors used in the definition of the Tamagawa measure can be chosen canonically, and in fact, for G a semisimple group, they are not needed at all, i.e., one can set λv = 1 for all v (cf. Weil [1982, Appendix 2]). If the volume of GA /GK with respect to the Tamagawa measure is finite, it is called the Tamagawa number of G and denoted τ (G). Example 5.23 Let G = SL2 over Q. We will now show that τ (G) = 1. For this, let us consider the differential form ω on G of
degree three, which x y in terms of the coordinates x, y, z of X = ∈ G is given by z t ω = 1x dx ∧ dy ∧ dz. In Example 3.69, we have seen that this is a left-invariant rational form on G, and have found the volume ωp (SL2 (Zp )) with respect to the corresponding Haar measure ωp on SL2 (Qp ) to be equal to 1 − p−2 . Then the Q product p ωp (SL2 (Zp ))−1 is exactly the Euler product for the Riemann zeta Q function ζ (s) at s = 2, and therefore p ωp (SL2 (Zp )) converges absolutely to ζ (2)−1 . (Note that the convergence factors are indeed not needed in this case.) Next, let 6 be the fundamental domain in SL2 (R) relative to SL2 (Z) constructed in Example 4.55. We showed at the end of Example 3.69 that in terms of the coordinates ϕ, a and u on SL2 (R) given by the Iwasawa decomposition, we have ω = a dϕ da du, and therefore the computations in §4.6 for the corresponding Haar measure 2 ω∞ on SL2 (R) give the value ω∞ (6) = π6 . It remains to note that the proof Q of Proposition 5.13 readily shows that  = F × p SL2 (Zp ) is a fundamental domain in GA relative to GK , i.e., it satisfies conditions (1) and (20 ) of §3.5. Thus Y π2 τ (G) = ω∞ (F) × SL2 (Zp ) = ζ (2)−1 = 1, 6 p since ζ (2) =

π2 6

(cf. Serre [1973]).

In this example, we were able to give an explicit description of the fundamental domain in GA relative to GK and to compute its volume, yielding thereby the value of the Tamagawa number τ (G). While such explicit computations are not feasible in the general case, the problem of determining τ (G) in other situations is very important. Its significance became apparent after Kneser and Tamagawa independently noted that for G = SOn (f ), where f is a nondegenerate quadratic form in n variables with rational coefficients, the

5.3 Compactness and the Finiteness of the Volume of GA /GK

317

equality τ (G) = 2 is essentially equivalent to one of Siegel’s fundamental results in the analytic theory of quadratic forms (the formula for the weight of a genus – cf. Kneser’s lecture in [ANT]). According to Theorem 5.22, the Tamagawa number τ (G) is finite for any semisimple group G, so its determination was regarded at that time as a major problem. Ono (1964, 1965b, 1966) showed that, in fact, it suffices to compute the Tamagawa number for one group in every isogeny class. More precisely, we have the following elegant result. ˜ → G be a K-defined universal cover, Let G be a semisimple K-group, π : G F = ker π be the fundamental group of G, and X(F) be its character group. Then 0 ˜ · h (X(F)) , τ (G) = τ (G) i1 (X(F)) where h0 (X(F)) = |H 0 (K, X(F))| = |X(F)K | and i1 (X(F)) is the order of the Q kernel of the canonical map H 1 (K, X(F)) → H 1 (Kv , X(F)). v

Thus, it suffices to compute τ (G) for all simply connected groups. A conjecture due to Weil asserts that for G simply connected, we have τ (G) = 1. Weil (1961b, 1964) developed a method for computing Tamagawa numbers that uses induction, the residues of some analogs of the zeta function, and the Poisson summation formula. This method enables one to prove his conjecture for many classical groups and also some exceptional groups. Later, Mars (1969, 1971) computed the Tamagawa number for unitary groups of type An , and thereby completed the proof of the Weil conjecture for classical semisimple groups over number fields. A uniform proof of the Weil conjecture for Chevalley groups was given by Langlands (1966), while Lai (1976, 1980) computed τ (G) for G quasisplit. A complete proof of the Weil conjecture was obtained by Kottwitz (1988) modulo the validity of the Hasse principle for the Galois cohomology of simply connected semisimple algebraic groups. So, when Chernousov (1989b) completed the proof of the Hasse principle for groups of type E8 (cf. [AGNT, Chapter 6]), the Weil conjecture became a theorem for all semisimple simply connected algebraic groups over number fields. One of the most important recent developments in this area is the proof of the Weil conjecture over global fields of positive characteristic by Gaitsgory and Lurie (2019). The definition of the Tamagawa number requires some modification for reductive groups that are not semisimple since for many cases arising in applications (such as the 1-dimensional split torus Gm ), the volume of GA /GK is

Adeles

318

infinite. This motivates us to find some other homogeneous spaces that are also associated with adele groups, but have finite invariant volume. Since the obstruction to the finiteness of the volume of GA /GK comes from the existence of nontrivial K-defined characters, or equivalently, of a nontrivial almost direct factor that is a K-split torus, it is natural to begin our analysis precisely with the 1-dimensional K-split torus S = Gm . Here, SA is isomorphic to JK , the idele group of K. Although the quotient JK /K ∗ is obviously noncompact, a classical result from algebraic number theory (cf. Lang [1994]) asserts that a similar quotient JK1 /K ∗ of the group of special ideles JK1 (i.e., the kernel of the conQ tinuous homomorphism cK : JK → R>0 given by cK ((xv )) = v ||xv ||v , cf. §1.2.1) is already compact. In the general case, an analog of JK1 can be defined as follows: We associate with each character χ in X(G)K the continuous homoQ morphism cK (χ) : GA → R>0 given by cK (χ)((gv )) = v ||χ(gv )||v . Then we define (1)

GA =

\

ker cK (χ).

χ∈X(G)K

Clearly, this infinite intersection can be replaced by a finite one, since if T (1) χ1 , . . . , χr constitute a base of X K (G), we have GA = ri=1 ker cK (χi ). As an exercise, the reader may show that this relation also holds if χ1 , . . . , χr generate a subgroup of XK (G) of finite index. The product formula implies the (1) inclusion GA ⊃ GK . (1)

Theorem 5.24 Let G be a connected K-group. Then the topological group GA (1) is unimodular and the quotient GA /GK has finite invariant volume. The quo(1) tient GA /GK is compact if and only if the semisimple part of G is anisotropic over K. We note that the assumption in the second part of the theorem can be reformulated as follows: every unipotent element contained in GK lies in the unipotent radical of G. Moreover, if G is connected and X(G)K = 1, then (1) GA = GA ; thus, for connected groups, Theorem 5.24 is a generalization of Theorem 5.22. PROOF: It is convenient to begin by reducing to the case K = Q. Let H = RK/Q (G) be the group obtained from G by restriction of scalars. We %

shall show that the isomorphism GAK 'HAQ from Proposition 5.12 induces

5.3 Compactness and the Finiteness of the Volume of GA /GK (1)

319

(1)

an isomorphism GAK ' HAQ . To do so, note that any K-defined character χ : G → Gm induces a morphism χ˜ = RK/Q (χ) : H → RK/Q (Gm ). Composing χ˜ with the norm map N : RK/Q (Gm ) → Gm , we obtain a character κ = N ◦ χ˜ ∈ X(H)Q . Using factorization (2.4), one easily shows that the correspondence χ 7→ κ defines an isomorphism η : X(G)K → X(H)Q of the corresponding groups of characters, so that for any character χ in X(G)K , the diagram GK

%

/ HQ

χ

 K∗

η(χ)

NK/Q

 / Q∗

commutes. We leave it to the reader to verify that this diagram also extends to the corresponding groups of adeles. The formulas in §1.2.3 imply that cK (χ) = cQ (η(χ)). It follows that % induces an isomorphism of the topological groups (1) (1) (1) (1) GAK and HAK and also of the quotients GAK /GK and HAQ /HQ . Thus, we may assume from the outset that K = Q. This reduction simplifies the argument, since in this case one can give a nice description of (1)

(1)

GA(∞) = GA ∩ GA(∞) . (1)

By definition GA(∞) = GR ×GAf (∞) , and clearly GAf (∞) ⊂ GA(∞) since GAf (∞) is compact and R>0 contains no compact subgroups. Thus (1)

GA(∞) = LR × GAf (∞) , where L ⊂ G consists of those g for which χ(g) = ±1 for all χ in X(G)Q . Lemma 5.25 L is a Q-defined Zariski-closed subgroup of G, and X(L0 )Q = 1. Moreover, the semisimple parts of G and L0 are the same. PROOF: Pick a Z-basis χ1 , . . . , χr of X(G)Q , and let ϕ : G → Grm be the Q-morphism of algebraic groups given by ϕ(g) = (χ1 (g), . . . , χr (g)). Then L = ϕ −1 (D), where D ⊂ Grm is the Q-defined closed subgroup consisting of {(±1, . . . , ±1)}, which yields the first assertion of the lemma. Let

Adeles

320

G = HU be the Levi decomposition of G, and let S be the maximal central torus of H, and S = S1 S2 be its factorization as an almost direct product of Q-split and Q-anisotropic tori, respectively. Then it is easy to see that L0 = (BS2 )U, where B = [H, H] is the semisimple part of G, from which the rest of the lemma follows immediately. It follows from Lemma 5.25 and Theorem 4.47 that the group LR is uni(1) modular, and then the group GA(∞) = LR × GAf (∞) is also unimodular since (1)

the group GAf (∞) is compact. Our objective is to show that GA is unimodu-

lar, i.e., the modular function 1 = 1G(1) : GA → R>0 is actually trivial, i.e. (1)

(1)

(1)

A

1 ≡ 1. Since GA(∞) is open in GA and unimodular, the restriction of 1 to (1)

GA(∞) equals 1. (1)

We will next show that the restriction of 1 to GQ also equals 1. Since GA (1) is a normal subgroup of GA , the quotient GA /GA is a group, hence has an invariant measure, so we conclude that 1 coincides with the restriction of 1GA (1) to GA . But then one can use the Tamagawa measure to compute 1. Let ω be a Q-defined left-invariant rational differential form on G of degree n = dim G, and let %g be the right translation by an element g of G. Then, as in the proof of Theorem 4.47, there exists a character χ in X(G)Q such that %g∗ (ω) = χ(g)ω. Now, if v ∈ V Q and ωv is the corresponding Haar measure on GQv , then 1GQv (g) equals ||χ(g)||nv for g in GQv . So, by the product formula for g in GQ we have Y 1(g) = ||χ(g)||nv = 1. v

Now, it follows from Theorem 5.8 that there are only finitely many double (1) (1) cosets GA(∞) \GA /GQ , so since 1 is a group homomorphism with trivial (1)

(1)

restrictions to GA(∞) and GQ , we conclude that the image 1(GA ) must be finite. Thus, in the end 1(GA ) = 1 since R>0 has no nontrivial finite (1) subgroups, proving the unimodularity of GA . Now let us consider the finite double coset decomposition (1)

(1)

GA =

r [

(1)

GA(∞) xi GQ .

i=1 (1)

Lemma 5.16 implies that each conjugate group x−1 i GA(∞) xi is commensurable (1)

(1)

with GA(∞) , so there is a finite set y1 , . . . , yl in GA such that

5.4 Reduction Theory for S-Arithmetic Subgroups

(1) GA

=

l [

321

(1)

yj GA(∞) GQ .

j=1 (1)

Thus, it is enough to determine when GA(∞) GQ /GQ is compact or has finite (1)

volume. But, in view of the factorization GA(∞) = LR × GAf (∞) and the fact that GZ = LZ , we have (1)

(1)

(GA(∞) GQ )/GQ ' GA(∞) /GZ ' LR /LZ × GAf (∞) . (1)

Since GAf (∞) is compact, the quotient (GA(∞) GQ )/GQ is compact (resp., has finite volume) if and only if LR /LZ is compact (resp., has finite volume). But Theorems 4.42 and 4.47 imply that LR /LZ always has finite volume, and is compact if and only if the semisimple part of L0 , which in view of Lemma 5.25 is the same as the semisimple part of G, is anisotropic over Q. (1)

As in the case of GA /GK , the measure τ (1) on GA /GK can be defined (1) canonically, and then the volume of τ (1) (GA /GK ) is also called the Tamagawa number of G. Note that for G connected, the quotient GA /GK has finite (1) volume if and only if X(G)K = 1, in which case GA = GA . Thus, the new definition indeed generalizes our former one to arbitrary connected groups. Ono (1963a) computed the Tamagawa number of an algebraic K-torus T in the form |H 1 (K, X(T))| , X(T) Q where X(T) = ker (H 1 (K, T) → v H 1 (Kv , T)) is the Shafarevich–Tate group of T. Based on this result, one can construct examples of tori and semisimple groups for which τ (T) is not an integer. Moreover, one can combine the above formulas for the Tamagawa numbers of semisimple groups and of tori into a single formula (noting that the Tamagawa number of a unipotent group U is always 1), describing the Tamagawa number of an arbitrary connected K-group G in terms of the cohomology of the Picard module Pic G (cf. Sansuc [1981]). τ (T) =

5.4 Reduction Theory for S-Arithmetic Subgroups In this section, we will use the reduction theory developed for the groups of adeles in order to obtain analogous results for S-arithmetic subgroups. HenceK , and O(S) will denote the forth, S will be a finite subset of V K containing V∞

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322

ring of S-integers of K. For an algebraic K-group G ⊂ GLn , we let GO(S) denote the group of S-integral points, also called the group of S-units of G. Recall that a subgroup 0 ⊂ G is said to be S-arithmetic if it is commensurable with GO(S) . One shows that the class of S-arithmetic subgroups is invariant under K-isomorphisms, either by modifying Proposition 4.5 appropriately or by using the equality GO(S) = GA(S) ∩ GK and applying the remark made after Lemma 5.16. This equality also implies that GO(S) is a discrete subgroup of GA(S) . Since GA(S) = GS × GAS (S) and GAS (S) is compact, GO(S) is also a disQ crete subgroup of GS = v∈S GKv (which is easily seen also without using adeles). So, one can consider the problem of developing a reduction theory for GO(S) in GA(S) as well as in GS . It is natural to define fundamental sets in this context as follows, in analogy with the respective definitions for arithmetic groups and adele groups: Definition (1) A subset  of GA(S) is a fundamental set for GO(S) if (Fl)A(S) GO(S) = GA(S) , (F2)A(S) −1 ∩ GO(S) is finite. (2) A subset  of GS is a fundamental set for GO(S) if (Fl)S GO(S) = GS , (F2)S for any a, b ∈ GK , the set of x in GO(S) satisfying axb ∩  6= ∅ is finite. It is easy to see that, for S finite, the problems of constructing fundamental sets in GA(S) and in GS respectively are closely related. More precisely, if  ⊂ GS is a fundamental set as defined in (2), then  × GAS (S) ⊂ GA(S) is a fundamental set as defined in (1). For this reason, henceforth we will concern ourselves only with constructing fundamental sets in GS . Note that for S infinite, definition (2) becomes meaningless, whereas all the results for GA(S) remain valid (cf. Borel [1963, §8]). Proposition 5.26 Let B be a fundamental set for GO in G∞ . Then there is an open compact subset C of GS\V∞ K such that  = B × C ⊂ GS is a fundamental set for GO(S) . PROOF: The argument is similar to the proof of Proposition 5.17. Indeed, Theorem 5.8 implies that there is a finite decomposition GA(S) =

r [ i=1

GA(∞) xi (GK ∩ GA(S) ) =

r [ i=1

GA(∞) xi GO(S) ,

5.4 Reduction Theory for S-Arithmetic Subgroups

323

with xi ∈ GS\V∞ K , which leads to the decomposition GS =

r [

Dxi GO(S) ,

i=1

Q Sr where D = G∞ × U and U = v∈S\V∞ K GOv . Set C = i=1 Uxi U. Then one can easily verify that for  = B × C, the equality GS = GOS holds, so it remains to check that 6 := {x ∈ GO(S) :  ∩ axb 6= ∅} is finite for any a, b in GK . If x ∈ 6, then, passing to the projection to the non-Archimedean part of GS , we obtain x ∈ a−1 C −1 Cb−1 , and then x−1 ∈ bC −1 Ca. As we have seen earlier, the compactness of C −1 C implies that there exists r in O such that 6 ⊂ Gr := {x ∈ GK : rx , rx−1 ∈ Mn (O)} (assuming that G ⊂ GLn ). Therefore, the finiteness of 6 follows immediately from the obvious generalization of Lemma 4.27. Proposition 5.26 easily yields Theorem 5.27 (1) GS /GO(S) has finite invariant volume if and only if X(G0 )K = 1; (2) GS /GO(S) is compact if and only if the reductive part of the connected component of G is anisotropic over K. PROOF: We have seen in the proof of Theorem 5.22 that either all of the groups GKv for v ∈ V K are unimodular or none of them is unimodular. So, the unimodularity of GS is equivalent to the unimodularity of G∞ . We choose a closed fundamental set for GO in G∞ in the sense of §3.5, and then use it as the set B in Proposition 5.26 to obtain a fundamental set for GO(S) in GS of the form B × C, where C ⊂ GS\V∞ K is compact. The existence of such a fundamental set implies that GS /GO(S) has finite invariant volume (resp., is compact) if and only if the respective property holds for G∞ /GO . Therefore our assertions follow from the corresponding parts of Theorem 4.58. Applying reduction theory, we previously obtained theorems on finiteness of orbits both for arithmetic groups and adele groups (cf. Theorems 5.18 and 4.26). A similar theorem also holds for S-arithmetic subgroups. To formulate the result, we define an S-lattice in K n as a finitely generated O(S)-submodule of K n containing a basis.

324

Adeles

Theorem 5.28 Let G be a reductive algebraic K-group, and let % : G → GLm be a K-defined representation of G. For w in K m , if the orbit X = w%(G) is Zariski-closed, then for any S-lattice L ⊂ K m invariant under GO(S) , the intersection XS ∩ L is a union of a finite number of orbits of GO(S) . The proof follows directly from the following two propositions. Proposition 5.29 XS consists of a finite number of orbits of GS . Proposition 5.30 w%(GS ) ∩ L is a union of a finite number of orbits of GO(S) . PROOF OF PROPOSITION 5.29: The argument reduces immediately to the case where S consists of a single valuation v (recall that S is always assumed to be finite). If v is complex, then GKv acts transitively on XKv i.e., there is only one orbit. For v real, the desired finiteness is established in Corollary 3.18. For v non-Archimedean, the only known way to obtain the finiteness of the number of orbits relies on the finiteness theorem for Galois cohomology over locally compact fields – see [AGNT, §6.4]. PROOF OF PROPOSITION 5.30: The argument relies on the construction of fundamental sets described in Proposition 5.26. More precisely, in §4.7, we showed how to construct a fundamental set B ⊂ G∞ relative to GO by using restriction of scalars and the construction of fundamental sets in GR relative to GZ (cf. §4.3). Then it follows from the proof of Theorem 5.31 below that the set B obtained this way has the following property: if % : G → GLm is a K-defined representation, then the intersection w%(B) ∩ Om is finite for any w in K m such that the orbit X = w%(G) is Zariski-closed. Starting with B ⊂ G∞ satisfying this property, we then use Proposition 5.26 to find a compact set C ⊂ GS\V∞ K such that  = B × C is a fundamental set for GO (S) in GS . Since L is invariant under GO(S) , it suffices to show that F := w%() ∩ L is finite. The compactness of C implies the existence of r in O such that rF ⊂ Om . Then, projecting onto the Archimedean component, we see that rF is contained in the intersection (rw)%(B) ∩ Om , which is finite by construction. We will now derive from Theorem 5.28 that the class of S-arithmetic subgroups is invariant under arbitrary epimorphisms, and that GO(S) has only finitely many conjugacy classes of finite subgroups. Theorem 5.31 Let f : G → H be a K-defined epimorphism of algebraic groups. Then for any S-arithmetic subgroup 0 of G, the image of f (0) is an S-arithmetic subgroup of H.

5.4 Reduction Theory for S-Arithmetic Subgroups

325

PROOF: The argument is essentially the same as in the proof of Theorem 4.7. We may assume that G and H are connected. Clearly, it suffices to establish that f (GO(S) ) is S-arithmetic, and we may also assume that f (GO(S) ) ⊂ HO(S) . First, we reduce the general case to the cases where G is either reductive or unipotent. Lemma 5.32 Let G = FU be the Levi decomposition. If B and D are subgroups of finite index of FO(S) and UO(S) , respectively, such that B normalizes D, then the product BD is a subgroup of GO(S) of finite index. PROOF: Repeating verbatim the proof of Corollary 4.4, we see that the index [GO(S) : FO(S) UO(S) ] is finite. Next, the fact that the product BD is a subgroup is immediate. Now, let FO(S) =

r [

xi B and UO(S) =

i=1

t [

yj D.

j=1

Then FO(S) UO(S) =

r [ t [

xi yj BD.

i=1 j=1

Indeed, if x ∈ FO(S) , y ∈ UO(S) , then writing x = xi b and byb−1 = yj d for b in B and d in D, we obtain xy = xi by = xi yj db = xi yj b(b−1 bd) ∈ xi yj BD. It follows that BD is of finite index in FO(S) UO(S) , and consequently also in GO(S) . Let G = FU be the Levi decomposition of G. Then H = f (F)f (U) is the Levi decomposition of H. If we show that both indices [f (F)O(S) : f ( FO(S) )] and [f (U)O(S) : f (UO(S) )] are finite, then Lemma 5.32 will imply that the index [HO(S) : f ( FO(S) UO(S) )], and hence also the index [HO(S) : f (GO(S)) ], is finite. This gives the reduction of the proof of Theorem 5.31 to the cases where G is either unipotent or reductive. Let us first consider the case where G is unipotent. Then by Theorem 5.27, the quotient GS /GO(S) is compact. Letting U = ker f and observing that H 1 (Kv , U) = 1 for any v in V K (Lemma 2.25), we deduce from the cohomology sequence associated with the exact sequence 1 → U → G → H → 1 that f (GKv ) = HKv . This implies that fS (GS ) = HS , so the quotient HS /f (GO(S) ) is compact. It follows that the quotient HO(S) /f (GO(S) ) is both compact and discrete, and therefore the index [HO(S) : f (GO(S) )] must be finite.

326

Adeles

Now let G be reductive. If H ⊂ GLn , then using a well-known trick (cf. p. 54), we may assume without loss of generality that H is closed in Mn . Then H can be viewed as the (closed) orbit of the identity matrix In under the action of G on Mn given by A · g = Af (g), with usual matrix multiplication in the right-hand side. It remains to note that L = Mn (O(S)) is invariant under GO(S) and that HO(S) = H ∩ L, while the orbits of GO(S) on HO(S) are precisely the cosets modulo the subgroup f (GO(S) ). Thus the finiteness of the number of orbits established in Theorem 5.28 is equivalent in this case to the finiteness of the index [HO(S) : f (GO(S) )]. Theorem 5.33 Assume that G is connected. Then the group GO(S) has finitely many conjugacy classes of finite subgroups. PROOF: Let us pick v 6∈ S and embed GO(S) into GOv as a subgroup. Then Proposition 3.13 implies the finiteness of the number of isomorphism classes of finite subgroups of GO(S) . So, it suffices to show that for a given finite group 0, there are only finitely many conjugacy classes of subgroups of GO(S) that are isomorphic to 0. We first consider the case where G is reductive. The argument in this case essentially repeats the proof of Proposition 3.13. Let R(0, G) be the variety of representations of 0 in G. Then G naturally acts on R(0, G) by conjugation, and it is enough to show the finiteness of the number of orbits of GO(S) on R(0, G)O(S) . By Theorem 2.52, the group G has a finite number of orbits on R(0, G), and all these orbits are Zariski-closed. Let X be one of the orbits for which XO(S) 6= ∅. If G ⊂ GLn and |0| = d, then X can be realized as a closed subset of V = Mn × · · · × Mn (d factors), and the action of G extends naturally to V . Therefore, applying Theorem 5.28 to the S-lattice L = Mn (O(S)) × · · · × Mn (O(S)), we conclude that there are only finitely many orbits of GO(S) on XO(S) , as desired. In the general case, let us consider the Levi decomposition G = HU, where U is the unipotent radical of G and H is reductive. Let π : G → G/U be the canonical morphism. We can find a matrix realization of G/U in such a way that π(GO(S) ) ⊂ (G/U)O(S) . Since the index [(G/U)O(S) : π(GO(S) )] is finite and Theorem 5.33 has already been established in the reductive case, we see that the group π(GO(S) ) has finitely many conjugacy classes of finite subgroups. Let us consider an arbitrary finite subgroup 0 = {γ1 , . . . , γd } of GO(S) , and define a closed subset A(0) of Gd as follows: A(0) = R(0, G) ∩ {(δ1 , . . . , δd ) : π(δi ) = π(γi )}. Then U naturally acts on A(0) by conjugation, and to complete the proof of the theorem it suffices to show that A(0)O(S) consists of a finite number of orbits

5.4 Reduction Theory for S-Arithmetic Subgroups

327

of UO(S) . Consider the morphism ϕ : U → A(0) given by ϕ(g) = g−1 γ g = (g−1 γ1 g, . . . , g−1 γd g), where γ = (γ1 , . . . , γd ). Furthermore, let U1 denote the centralizer of 0 in U, and choose a K-subvariety U2 ⊂ U so that the product morphism U1 × U2 → U is a K-isomorphism of varieties (cf. Lemma 2.5). (Note that, in general, it is not possible to choose a subgroup U2 of U satisfying this property.) Lemma 5.34 The restriction ϕ|U2 : U2 → A(0) is a K-isomorphism of varieties. PROOF: First, we show that the action of U on A(0) is transitive. Let δ = (δ1 , . . . , δd ) ∈ A(0) and 1 = {δ1 , . . . , δd }. Then 0 and 1 are reductive subgroups of G, so by Theorem 2.7 there exist x, y ∈ U such that both x−1 0x and y−1 1y are contained in H. Since for any i = 1, . . . , d we have π(x−1 γi x) = π(γi ) = π(δi ) = π(y−1 δi y), the injectivity of the restriction π|H yields that x−1 γi x = y−1 δi y,

i.e., δ = g−1 γ g = ϕ(g) for g = xy−1 .

Since the stabilizer of γ is precisely U1 , the restriction ϕ|U2 : U2 → A(0) is one-to-one. It remains to observe that being a homogeneous variety, A(0) is smooth, so the fact that ϕ is an isomorphism follows from the remark made after Theorem 2.47. Now, let us first assume that U is an abelian group. Then for U2 one can take a suitable K-subgroup of U (Lemma 2.5). It follows from Lemma 5.34 that the preimage (ϕ|U2 )−1 (A(0)AS (S) ) is a compact subset of U2AS and therefore is contained in the union of a finite number of (left) cosets modulo the subgroup U2AS (S) . But then ϕ −1 (A(0)O(S) ) ⊂ ϕ −1 (A(0)AS (S) ) ∩ U2K is contained in the union of a finite number of cosets modulo U2O(S) = U2AS (S) ∩ U2K , which implies the desired conclusion. In the general case, we use induction on dim U. Let Z(U) be the center of 0 U, and consider G0 = G/Z(U). By induction, finite subgroups of GO (S) partition into a finite number of conjugacy classes. According to Theorem 5.31, the image of GO(S) under the canonical morphism G → G0 is an S-arithmetic subgroup, so GO(S) has a finite number of conjugacy classes of subgroups of the form 0Z(U)O(S) , where 0 is a finite subgroup of GO(S) . To complete the proof of Theorem 5.33, it suffices to show that finite subgroups of 0Z(U)O(S) partition into a finite number of conjugacy classes in GO(S) . But 0 is contained in a suitable maximal reductive K-defined subgroup F of G. The unipotent radical

Adeles

328

of the semi-direct product D = FZ(U) is abelian, and DO(S) ⊂ GO(S) . Therefore the finite subgroups DO(S) partition into a finite number of conjugacy classes in GO(S) . Remarks. 1. Using the Levi decomposition for arbitrary algebraic groups over fields of characteristic zero (see Mostow [1956]), one can remove the assumption in Theorem 5.33 that G be connected. 2. For G reductive, Theorem 5.33 can be proved in the same way as Theorem 4.9 was proved for arithmetic subgroups, using Bruhat-Tits buildings for groups over non-Archimedean local fields (cf. §3.4). We will now establish the following version of the Borel Density Theorem 4.34 for S-arithmetic subgroups, which is needed in the proof of the Strong Approximation Theorem (cf. [AGNT, §7.4]). Proposition 5.35 Let G be an almost K-simple algebraic group, and S be a K . Assume that the group G is noncompact. finite subset of V K containing V∞ S Then (1) 0 = GO(S) is Zariski-dense in G; (2) if H = RK/Q (G) and 1 is the image of 0 under the canonical isomorphism GK ' HQ , then 1 is Zariski-dense in H. (Conversely, if GS is compact, then GO(S) , being a discrete subgroup thereof, is finite, hence, of course, not Zariski-dense in G.) PROOF: It follows from the construction of restriction of scalars (cf. §2.1.2) that there exists a K-defined morphism H → G that takes 1 back to 0. This shows that part (2) of the proposition implies part (1). So, we will only prove part (2), although almost the same argument yields a direct proof of part (1). Since the quotient GS /GO(S) has finite invariant measure (cf. Theorem 5.27(1)), while the group GS is noncompact, hence has infinite Haar measure, the group 0 = GO(S) must be infinite. Being the image of 0 under the natural identification ϕ : GK → HQ , the group 1 is also infinite. Moreover, since the groups 0 and g0g−1 are commensurable for any g ∈ GK , we see that the groups 1 and h1h−1 are commensurable for any h ∈ HQ . Then, letting the bar denote the Zariski-closure, we have the following equalities for the connected components 0

0

0

1 = 1 ∩ h1h−1 = h1h−1 , 0

0

0

i.e., 1 = h1 h−1 for any h ∈ HQ . Thus, 1 is normalized by HQ , and 0 since HQ is Zariski-dense in H (cf. Theorem 2.6), we see that 1 is a Qdefined normal subgroup of H, which in fact has positive dimension (since 1 is infinite). However, H is Q-simple. (Indeed, since G is K-simple, we may assume (up to isogeny) that G = RL/K (G0 ) for some absolutely almost simple

5.4 Reduction Theory for S-Arithmetic Subgroups

329

group G0 defined over a finite extension L/K. Then H = RL/Q (G0 ), hence 0

Q-simple.) Thus, 1 = H, as required. Our next result concerns the finite presentability of S-arithmetic groups. Theorem 5.36 Any S-arithmetic subgroup of a reductive group G is a group with a finite number of generators and a finite number of defining relations. PROOF: The argument is based on Reidemeister-Schreier’s method from combinatorial group theory (cf. Lyndon and Schupp [2001, Chapter 2, §4]), which we apply to GO(S) regarded as a subgroup of 0 = GS\V∞ K . We showed in Theorem 3.57 that the group GKv is compactly presented for any v ∈ VfK . This easily Q implies that 0 = v∈S\V∞ K GKv is also compactly presented. (Recall that 0 is compactly presented if there exists a compact subset D of 0 that generates 0 and such that the relations of the form ab = c for a, b, c ∈ D constitute a defining set of relations for 0. Replacing D by D ∪ D−1 ∪ {e}, where e is the identity element of 0, we may assume that e ∈ D and D = D−1 .) For a set X , we let F(X ) denote the corresponding free group. Now, let D∗ be the set whose elements are in bijection with the elements of D under the map d ∗ 7→ d. Then the homomorphism % : F(D∗ ) → 0 defined by this bijection is surjective, and N := ker % is generated as a normal subgroup by elements of the form a∗ b∗ c∗−1 for a∗ , b∗ , c∗ ∈ D∗ satisfying %(a∗ b∗ ) = %(c∗ ), or equivalently, ab = c. A system of representatives of the cosets F(D∗ )/H (where H = %−1 (GO(S) )) needed to apply Reidemeister-Schreier’s method can be chosen as follows. According to Proposition 5.26, there exists a compact subset C of 0 such that 0 = GO(S) C, and without loss of generality we may assume that e ∈ C. We begin by choosing a system of representatives T of (right) cosets modulo GO(S) that consists of elements of C and contains e. We then let T ∗ denote a system of representatives of (right) cosets of F(D∗ ) modulo H containing the identity element e∗ of F(D∗ ) and such that % gives a one-to-one correspondence between T ∗ and T. Next, we introduce a map F(D∗ ) → T ∗ , denoted by x 7→ x¯ , that sends x to the representative x¯ of Hx lying in T ∗ ; in other words, x¯ in T ∗ is such that Hx = H x¯ . Now take any x in H and write it as x = d1 · · · dm with di ∈ D∗ ∪ D∗−1 . Following the Reidemeister–Schreier method, we write x = (d1 d¯ 1−1 )(d¯ 1 d2 (d1 d2 )−1 ) · · · (d1 · · · dm−1 dm (d1 · · · dm )−1 ), noting that d1 · · · dm = x¯ = e∗ . The factors d1 · · · di−1 di (d1 · · · di )−1 , provided by the method, have x as their product and lie in X := (T ∗ (D∗ ∪ D∗−1 )T ∗−1 ) ∩H, which thereby is a system of generators for H. Furthermore,

Adeles

330

by assumption, N is generated as a normal subgroup of F(D∗ ) by elements of the form abc−1 for a, b, c ∈ D∗ satisfying %(ab) = %(c). This means that any n in N can be written as n=

e Y

gi (abc−1 )εi gi−1 ,

i=1

with gi ∈ and εi = ±1. Writing gi = hi ti with hi ∈ H and ti ∈ T ∗ , we see that N is generated as a normal subgroup of H by elements of the form xyz−1 where x, y, z ∈ T ∗ D∗ T ∗−1 and %(xy) = %(z). Since xyz−1 = e∗ , we have F(D∗ )

xyz−1 = (x¯x−1 )(¯xy(xy)−1 )((xyz−1 )z(xy)−1 )−1 , implying that N is generated as a normal subgroup of H by elements of the form xyz−1 , where x, y, z ∈ T ∗2 D∗ T ∗−2 ∩H satisfy %(xy) = %(z). This suggests that it may be helpful to extend the system of generators X to the following set: Y Y = %−1 (BC 2 DC −2 B ∩ GO(S) ) where B = GOv . K v∈S\V∞

(We note that the definition of X clearly implies the inclusion X ⊂ Y .) Then N is generated as a normal subgroup of H by elements of the form xyz−1 , where x, y, z ∈ Y and %(xy) = %(z). We will now show how to obtain from Y a finite set of generators for GO(S) . For this, we first choose a subset Z of Y for which % induces a bijection Z → %(Y ), and then define a map π : Y → Z by the condition %(x) = %(π(x)) for α1 all x in Y . Consider the mutually inverse homomorphisms F(Z) −→ ←− F(Y ) of the α2

corresponding free groups induced by the inclusion Z ⊂ Y and the map π. We then have the commutative diagram α1

F(Z) o

/ F(Y )

α2 %2

{ # GO(S),

(5.10)

%1

where %2 = %1 ◦ α1 and %1 is obtained as the compositions of % and the homomorphism τ : F(Y ) → H, which yields the equality ker %2 = α2 (ker %1 ). But ker %1 is generated by ker τ and elements of the form xyz−1 , for x, y, z ∈ Y such that %(xy) = %(z). It follows that ker %2 is generated by ker (τ ◦α1 ) and elements of the form xyz−1 , where x, y, z ∈ Z and %(xy) = %(z). By assumption, % yields a bijection between Y and E ∩ GO(S) , where E = BC 2 DC −2 B. But E is a compact subset of 0 and therefore can be covered by a finite number of

5.4 Reduction Theory for S-Arithmetic Subgroups

331

translations of the open subgroup B. Thus, there exists a finite set of elements y1 , . . . , yr in E ∩ GO(S) such that E ∩ GO(S) =

r [

yi (GO(S) ∩ B) =

i=1

r [

yi GO .

i=1

(We note that the equalities E = BE = EB imply that E ∩ GO(S) = GO (E ∩ GO(S) ) = (E ∩ GO(S) )GO .) By Theorem 4.8, we can choose a finite set z1 , . . . , zt of generators of GO . Set [ U = {z1 , . . . , zt } and W = {y1 , . . . , yr } U, and using the bijection between Z and E ∩ GO(S) , identify these sets with the corresponding subsets of Z. By assumption, we have an epimorphism ϕ : F(U) → GO . Taking a section ψ : GO → F(U) for ϕ that is the idenσ tity map on U, we can define a map Z →F(W ) that sends yi g to yi ψ(g) for g in GO . This map induces a homomorphism β2 : F(Z) → F(W ), which is the inverse of the homomorphism β1 : F(W ) → F(Z) given by the inclusion W ⊂ Z. Moreover, we have the commutative diagram β1

F(W ) o

/ F(Z)

β2 %3

{ # GO(S)

(5.11)

%2

in which %3 = %2 ◦ β1 and %2 is taken from (5.10). Diagram (5.11) yields the equality ker %3 = β2 (ker %2 ), implying that ker %3 is generated by ker (τ ◦ α2 ◦ β2 ) and elements of the form xyz−1 , where x, y, z ∈ σ (Z) and %3 (xy) = %3 (z). We will now show that all the relations xy = z for such x, y, z, can in fact be reduced to a finite number of them. First, let us consider the relations where x ∈ ψ(GO ). Since GO (E∩GO(S) ) = E ∩ GO(S) , we have relations zi yj = yk wij for all i = 1, . . . , t and j = 1, . . . , r with suitable k ∈ {1, . . . , r} and wij ∈ GO . We then introduce the relations zi yj = yk ψ(wij ),

for i = 1, . . . , t

and j = 1, . . . , r.

(5.12)

Adding (5.12) to a finite system of relations in terms of the zi that define GO (cf. Theorem 4.58(2)), we obtain a finite system of relations from which all the relations of the form xy = z with x ∈ ψ(GO ) can be derived. Indeed, any x in ψ(GO ) is a word in zi ’s. Therefore, writing y = yj b and z = yk c with

332

Adeles

b, c ∈ ψ(GO ), and using (5.12) repeatedly, we reduce xy = z to a relation of the form yl ab = ykc with a ∈ ψ(GO ). The equality %3 (xy) = %3 (z) implies that k = l, so canceling yk , we obtain a relation ab = c between the words in the zi ’s. However, by assumption, all such relations have already been incorporated. Now, let us take any x in σ (Z) and write it as x = yi a with a ∈ ψ(GO ). By what we have already proved, any relation of the form xy = z can be reduced to a relation yi yj = yk ψ(rij ), (5.13) where rij ∈ GO is the element determined by the relation yi yj = yk rij in GO(S) . Thus, all relations of the form xy = z, where x, y, z ∈ σ (Z) and %3 (xy) = %3 (z), can be derived from a finite number of relations. Equivalently, the normal subgroup of F(W ) generated by all the elements xyz−1 for such x, y, z, is actually generated by a finite number of these elements. Therefore, if we let 8 = α2 β2 (F(W )), then our computation of ker %3 implies that ker %|8 = 8 ∩ W is generated as a normal subgroup of 8 by a finite number of elements. It remains to note that since 8 is a finitely generated subgroup of F(D∗ ), it is a free group of finite rank (by the Nielsen–Schreier Theorem). Therefore, the isomorphism GO(S) ' 8/(8 ∩ N) yields the desired finite presentation of GO(S) . Finally, an arbitrary S-arithmetic subgroup of G, being commensurable with GO(S) , is also finitely presented. Our proof of Theorem 5.36 is a formalized version of the original argument due to Kneser (1964). The reader familiar with combinatorial group theory may find such formalization superfluous and prefer to argue directly in the group 0 without introducing the free groups F(Y ), F(Z), etc. (such “simplifications” are indeed possible and in fact used by Kneser). In our exposition, we carefully wrote down all formal details, which will probably be helpful to the reader with less experience in combinatorial group theory. At the same time, this formalization does not alter the main idea of the argument, which is to K , where G reduce the general case to the case S = V∞ O(S) = GO whose finite presentability has already been established in Theorem 4.58. This reduction K | by regarding G can also be carried out by induction on |S\V∞ O(S) as a subK group of GKv for some v ∈ S\V∞ . In certain cases, a suitable modification of this induction argument can make it possible to determine explicit generators and relations for GO(S) once one has an explicit presentation for GO (see Serre [2003] for G = SL2 ). K is There is also another proof of Theorem 5.36 in which the case S = V∞ no different from the other cases (cf. Borel and Serre [1973]). This proof is similar to the proof of Theorem 4.8 and uses the discrete action of GO(S) on

5.4 Reduction Theory for S-Arithmetic Subgroups

333

a suitable simply connected space, which is the product of the quotient space of G∞ by a maximal compact subgroup and the Bruhat–Tits buildings for the K . As we have already noted, this approach also yields groups GKv for v ∈ S\V∞ another proof of Theorem 5.33 for reductive groups. It should be noted that in Theorem 4.8, we did not require that G be reductive. Theorem 5.36, however, is not valid for nonreductive groups in the general K , the additive group of O(S) is not finitely case. First of all, for S 6= V∞ generated, and consequently any S-arithmetic subgroup of the 1-dimensional unipotent n group Ga is not ofinitely generated either. On the other hand, for α β B= : α 6= 0 , a Borel subgroup of SL2 , any S-arithmetic sub0 α −1 group is finitely generated. In this regard, we point out a criterion (proved by Kneser [1964]) for an S-arithmetic group to be finitely generated (resp., finitely presented): the S-arithmetic subgroups of G are finitely generated (resp., finitely presented) if and only if the groups GKv are compactly genK . Since a reductive group erated (resp., compactly presented) for all v in S\V∞ is always compactly presented (Theorem 3.57), a criterion for compact generation in the general case can be formulated in terms of the action of G on the unipotent radical Ru (G). We recommend this as an exercise for the reader. See Abels (1987) regarding compact presentability of solvable groups. We conclude this chapter with a description of S-arithmetic subgroups of tori. Theorem 5.37 contains as special cases both Dirichlet’s classical theorem on S-units in algebraic number fields, as well as the description of usual arithmetic subgroups of tori given in §4.5 (so, this theorem can be called the generalized Dirichlet theorem). Its proof, published by Shyr (1977b), is practically identical to the proof of Theorem 9 in Chapter 4 of Weil (1995). Theorem 5.37 Let T be a torus defined over a number field K, and let S K . Then the group of S-units T be a finite subset of V K containing V∞ O(S) is isomorphic to the product of a finite group and a free abelian group of rank P s = v∈S rankKv T − rankK T. PROOF: Clearly TO(S) = TA(S) ∩ TK is a discrete subgroup of TA(S) . However, this does not immediately yield results about TO(S) since the quotient TA(S) /TO(S) , in general, is noncompact. To obtain a compact quotient space, (1) (1) (1) one needs to pass from TA(S) to TA(S) = TA(S) ∩ TA , with TA defined as in (1)

(1)

§5.3. Then, since TK ⊂ TA , we have TO(S) = TA(S) ∩ TK , while the quo(1)

(1)

tient TA(S) /TO(S) can be identified with a clopen subspace of TA /TK , which is compact by Theorem 5.24.

Adeles

334

(1)

We will next describe precisely the structure of TA(S) . By definition, Y Y TA(S) = TKv × TOv . v∈S

We already know the structure of TKv for v

v6∈S K in V∞

(cf. proof of Corollary 4.39):

TKv ' Rrv × B, where rv = rankKv T and B is compact. K . Consider the decomposition T = T T as an almost direct Now let v ∈ S\V∞ 1 2 product of a maximal Kv -split torus T1 and a maximal Kv -anisotropic torus T2 (cf. §2.1.7), and let B be the (unique) maximal compact subgroup of TKv . Then T2Kv ⊂ B, since T2Kv is compact (Theorem 3.1). So, if ϕ : T → T3 = T/T2 is the corresponding quotient morphism, then TKv /B ' ϕ(TKv )/ϕ(B). But T3 is Kv -split, so (T3 )Kv ' (Kv∗ )rv ' Zrv × U,

where rv = dim T3 = rankKv T and U is compact. Since B is a maximal compact subgroup containing T2Kv , we conclude that ϕ(B) = U ∩ ϕ(TKv ). It follows that ϕ(TKv )/ϕ(B) ⊂ Zrv , implying that ϕ(TKv )/ϕ(B) ' Zt for some t ≤ rv . But T1 is also a Kv -split subtorus of rank rv , whence T1Kv ' Zrv × U1 for some compact subgroup U1 . Furthermore, the subgroup of T1Kv isomorphic to Zrv is discrete and therefore does not intersect B. Thus, TKv /B contains a free abelian group of rank rv . Finally, TKv /B ' Zrv , implying easily that TKv ' Zrv × B. Combining our results on the Archimedean and non-Archimedean components, we obtain the following factorization of TA(S) : TA(S) ' Rα × Zβ × W , P rankKv T and β = v∈S\V∞ K

where α = v∈V∞ rankKv T, while W is K a compact subgroup. (1) It is now easy to describe the structure of TA(S) . First, we recall that the P

(1)

subgroup TA was defined as the intersection of the kernels of all continuous homomorphisms cK (χ) : TA → R>0 for χ in X(T)K , where Y cK (χ)((gv )) = ||χ(gv )||v . v

Let χ1 , . . . , χr , where r = rankK T, be a Z-basis of X(T)K . We then have a continuous homomorphism θ : TA → (R>0 )r given by θ

g 7→ (cK (χ1 )(g), . . . , cK (χr )(g)), K . Indeed, and TA = ker θ. We claim that θ(TA(S) ) = (R>0 )r for any S ⊃ V∞ since χ1 , . . . , χr are linearly independent, the morphism ϕ : T → Grm given (1)

5.4 Reduction Theory for S-Arithmetic Subgroups

335

ϕ

by g 7→ (χ1 (g), . . . , χr (g)) is surjective. Therefore, applying Corollary 3.7, we K , the image ϕ(T ) is open and therefore contains the see that, for any v ∈ V∞ Kv connected component of the identity in (Kv∗ )r , which for v real is (R>0 )r and for v complex is C∗r . It remains to observe that the restriction of θ to TKv (naturally embedded in TA ) is the composition of ϕ and of the rth Cartesian power of the normalized valuation map || ||v : Kv∗ → R>0 , implying that θ(TKv ) is (R>0 )r K . Since the groups R and R>0 are topologically isomorphic, we for any v in V∞ are in a position to apply Lemma 5.38 Let 0 = Rα × Zβ × W , where W is a compact group. Given a continuous surjective homomorphism θ : 0 → Rγ , we have γ ≤ α and ker θ ' Rα−γ × Zβ × W . PROOF: Left to the reader as an exercise. Thus, we eventually conclude that there is an isomorphism TA(S) ' Rα−γ × Zβ × W . (1)

(1)

We showed earlier that TO(S) is a discrete subgroup of TA(S) with compact (1)

quotient TA(S) /TO(S) . Now, Theorem 5.37 follows from Lemma 4.40 and the observation that (α − r) + β is precisely the number s in the statement of the theorem. Corollary 5.39 (DIRICHLET S-UNIT THEOREM) Let K be a number field, and K . Then the group of S-units let S be a finite subset of V K containing V∞ E(S) = {x ∈ K ∗ : |x|v = 1 for all v 6∈ S} is isomorphic to the product of the group E of the roots of unity in K and a free abelian group of rank |S| − 1. This immediately follows from Theorem 5.37 applied to the 1-dimensional split K-torus Gm . Bibliographical Note In our description of the construction of the space of adeles XA associated to an arbitrary K-defined algebraic variety X , we followed Weil (1982, Chapter. 1). We refer to Conrad (2012a) for a detailed explanation of the fact that XA coincides with the set of adelic points in the sense of Grothendieck. The basic results of reduction theory for adele groups and S-arithmetic subgroups in the number-theoretic case can be found in Borel (1963). In his exposition, Borel uses in an essential way the reduction theory

336

Adeles

for arithmetic groups that he had developed with Harish-Chandra. Later, Godement (1964) showed how the same theorems can be obtained along different lines, independent of the reduction theorems for arithmetic groups. By elaborating on Godement’s method, Harder (1969) was able to develop a reduction theory for adele groups over global fields of positive characteristic (cf. also Springer [1994]). The basic theorems here are the same as in the number field case, except for the fact that the analog of Theorem 5.8 looks as follows: If S is a nonempty subset of V K , then there are only finitely many double cosets GA(S) \GA /GK . However, many fundamental properties of S-arithmetic groups over global function fields are substantially different from those of their counterparts over number fields. These differences are best seen in the context of higher finiteness properties. We recall that a group 0 is of type Fn (n ≥ 1) if it admits a classifying space with finite n-skeleton, or equivalently, there is a free action of 0 on a contractible CW -complex whose n-skeleton consists of finitely many orbits of cells. It is known that a group is of type F1 if and only if it is finitely generated, and it is of type F2 if and only if it is finitely presented. So, in this language, our Theorem 5.36 means that every S-arithmetic subgroup 0 of a reductive algebraic group over a number field K is of type F2 for any finite set S of places of K containing all Archimedean ones. It is remarkable that in fact such a group 0 is of type Fn for all n ≥ 1. This result was proved by Raghunathan (1967/68) for arithmetic groups and by Borel and Serre (1976) for S-arithmetic ones. Let now K be a global function field, and let G be a reductive K-group such that the semisimple part [G, G] is K-anisotropic. In this case, the above result of Borel and Serre remains valid for any finite set of places S. However, for semisimple K-isotropic groups the situation is substantially different. First, S-arithmetic subgroups in this case do not even need to be finitely generated, e.g., the group SL2 (k[t]) for any finite field k is not finitely generated (cf. Serre [2003 Chapter II, §1]). Behr (1969) developed a version of reduction theory that enabled him to essentially settle the question of finite generation for (most) S-arithmetic groups in positive characteristic. Furthermore, Behr (1979) showed that the group SL3 (k[t]) is finitely generated but is not finitely presented; in other words, it is of type F1 but not of type F2 . Stuhler (1976, 1980) obtained the first results involving finiteness properties beyond finite presentability; in particular, he showed that the group SL2 (OS ) over the ring OS of S-integers of a global function field for a finite set S is of type F|S|−1 but not of type F|S| . Abels (1991) and Abramenko (1987) independently showed that SLd+1 (k[t]) is of type Fd−1 but not of type Fd provided that the finite field k is sufficiently large. Both used the action of SLd+1 (k[t]) on the associated Bruhat–Tits building, and their (slightly different) lower bounds on the size

5.4 Reduction Theory for S-Arithmetic Subgroups

337

of k arise from the fact that the building needs to be sufficiently thick. Later, Abramenko (1994) generalized his results to other classical groups, and then in (1996) used twin buildings to investigate higher finiteness properties of classical groups over the rings of Laurent polynomials. These results have led to the following Rank Conjecture: Let G be an absolutely almost simple isotropic algebraic group over a global function field K, let S be a finite set of places P of K, and let d = v∈S rkKv G denote the S-rank of G. Then the S-arithmetic group G(OS ) is of type Fd−1 but not of type Fd (cf. Bux and Wortman [2007, 2011] for the details, including some history). We note that d is the dimenQ sion of the product X := v∈S Xv of the Bruhat–Tits buildings associated to G over Kv for v ∈ S on which the group G(OS ) naturally acts. The negative part of the Rank Conjecture was proved in full by Bux and Wortman (2007); see Gandini (2012) for a different argument. Furthermore, Bux and Wortman (2011) proved the Rank Conjecture for groups G of K-rank one. The proof of the Rank Conjecture in full generality was obtained by Bux, Köhl, and Witzel (2013). This result, which is now called the Rank Theorem, completely settles questions about higher finiteness properties of S-arithmetic subgroups of reductive groups over global function fields. Returning to the comparison between the number field and the function field situations, we now see concretely how much different the higher finiteness properties are in these cases. Interestingly, it appears that these differences can be explained solely by the fact that in the number field situation, we consider the finite sets S that contain all Archimedean places. More precisely, let K be a number field and let S be a finite set of places of K that contains no Archimedean places. Given an absolutely almost simple algebraic K-group G, one can define an approximate S-arithmetic lattice G(OS ) and extend the notion of finiteness properties to approximate groups. Hartnick and Witzel (2022) showed that the Rank Theorem extends verbatim to this setting. Thus, if S contains no Archimedean places of a number field K, approximate S-arithmetic groups behave like S-arithmetic groups over global function fields. The above survey of developments related to the Rank Conjecture and its resolution was kindly provided by Kai-Uwe Bux. We would like to thank him for his detailed write-up, and Peter Abramenko for useful discussions of the topic.

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Index

actions of algebraic groups on varieties, 114 adele topology, 11, 12 adeles, 11, 293, see also group of adeles; ring of adeles adelic space, 294 adelization of a regular map, 295–297 adjoint representation, 58 algebraic group, 54 absolutely almost simple, 71 adjoint, 71 defined over K, 55 diagonalizable, 59 K-anisotropic, 75 K-isotropic, 75 of compact/noncompact/mixed type, 243 pseudo-reductive, 66 quasisplit, 66 reductive, 66 semisimple, 66 simply connected, 71 split, 66 unipotent, 64 algebraic number field, 1 algebraic torus, 60, see also torus analytic structure, 128 anisotropic kernel, 77 approximation strong, 15, 301 weak, 15 arithmetic group, 198 birational isomorphism, 113 BN-pair, 174

Borel Density Theorem, 243 Borel subgroup, 65 Brauer group, 31 Bruhat decomposition, 68 relative, 76 building, 173 Cartan decomposition, 175 central isogeny, 71 character of an algebraic group, 60 Chebotarev Density Theorem, 10 Chevalley basis, 73 Chevalley group, 74 Chevalley’s theorem, 115 stronger version, 115 class number, 2, 302 clopen set, 138 coboundary map, 25, 84 cohomology, 17 continuous, 22 Galois, 23 non-abelian, 23 of algebraic groups, 83 commensurability subgroup, 244 compactness criterion for GA /GK , 314 for GR /GZ , 249 completion, 4 congruence subgroup, 155, 200 connected component, 58 convergence factors, 315 corestriction map, 22

358

Index

decomposition group, 7 Dieudonné determinant, 44 differential form, 193 associated measure, 196 invariant, 193 Dirichlet Unit Theorem, 248, 335 generalized, 333 distinguished vertex, 76 dominant morphism, 111 Eisenstein polynomial, 10 exponential map, 135 truncated, 64 field local, 5 non-Archimedean, 5 of definition, 111 field extension totally ramified, 6 totally real, 271 unramified, 6 finite presentability of arithmetic groups, 228 of S-arithmetic groups, 329 finiteness of the volume of GA /GK , 314 of GR /GZ , 253 finiteness theorem for the orbits of arithmetic groups, 225 of S-arithmetic groups, 324 forms of absolutely almost simple K-groups, 89 inner, 76, 90 outer, 76 of algebraic K-tori, 87 of K-objects, 77 classification in terms of Galois cohomology, 78 fractional ideal, 2 Frobenius automorphism, 7 fundamental domain, 190 fundamental group (of a semisimple algebraic group), 72 fundamental set, 191, 219, 225, 305, 322 Galois cohomology of algebraic groups, 83 group algebraic, 54, see algebraic group compactly presented, 177 of characters, 60

359

of cocharacters, 61 of S-units, 322 of units, 199 profinite, 159 pro-p, 160 self-adjoint, 144 unimodular, 187 with bounded generation, 237 group of adeles, 300 finite, 301 integral, 300 principal, 300 S-integral, 300 group of S-adeles, 300 Haar measure, 186 Hasse norm principle, 17 Hasse–Minkowski Theorem, 16 higher finiteness properties, 336 Hilbert’s Theorem 90, 81 Hochschild–Serre spectral sequence, 22 for non-abelian cohomology, 27 ideal class group, 2 ideles (group of), 12 integral, 13 principal, 13 S-integral, 13 special, 14 idele topology, 13 index of a central simple/division algebra, 29 inertia group, 11 inflation map, 21 invariant of a central simple/division algebra, 32 Inverse Function Theorem, 128 inverse/projective limit, 159 inverse/projective system, 159 involution (of an algebra), 96 of the first kind, 96 of orthogonal type, 98 of symplectic type, 98 of the second kind, 96 isogeny, 71 Iwahori subgroup, 173 Iwasawa decomposition for a reductive group, 152 for GLn (R), 149 Jordan decomposition, 59

360

Index

Killing form, 58 lattice in a locally compact topological group, 189, 262 in a vector space, 47 lemma Hensel’s, 165 Minkowski’s, 274 on closed orbits, 115 Shapiro’s, 22, 28 Levi decomposition, 66 Lie algebra of an algebraic group, 58 of an analytic group, 135 Lie subgroup, 135 local-global (Hasse) principle, 16 logarithmic map, 135 truncated, 64 Mahler’s compactness criterion, 250 measure Borel, 186 Haar, 186 Tamagawa, 315 modular function, 187 modulus of an automorphism, 186 morphism dominant, 111 K-defined, 55 of algebraic groups, 54 multidimensional conjugacy classes, 116 multinorm torus, 62 non-abelian cohomology, 23 nonsingular point, see simple point norm torus, 62 order (in a finite-dimensional algebra), 48 maximal, 48 orthogonal group, 93 special, 93 Ostrowski’s theorem, 3 parabolic subgroup, 66 standard, 71 parahoric subgroup, 173 polar decomposition for a reductive group, 146 for GLn (R) and GLn (C), 142, 144

Prasad volume formula, 263 applications, 263 product formula, 13 profinite group, 159 projective/inverse limit, 159 projective/inverse system, 159 pro-p group, 160 pseudobasis, 47 quotient varieties (of algebraic groups), 59 radical, 66 unipotent, 66 ramification groups, 11 ramification index, 6 rank of an algebraic group absolute, 65 over K, 75 Rank Theorem, 337 reduced norm, 30 reduction modulo a maximal ideal, 164 smooth, 165 reduction theory for arithmetic groups, 204, 220 in GLn (R), 203 for S-arithmetic groups, 321 representation variety, 119 residual degree, 6 residue algebra, 31 residue field, 6 restricted topological product, 188 restriction map, 21 restriction of scalars, 56 ring of adeles, 11, 12 integral, 12 principal, 12 S-integral, 12 root system, 67 relative, 75 S-arithmetic group, 203, 322 Schur multiplier, 19 self-adjoint group, 144 Shapiro’s Lemma, 22 noncommutative version, 28 Siegel set, 205, 255 generalized, 266 simple point, 112 singular point, 112 special cover, 88 special linear group, 90

Index

spinor group, 95 splitting field of a central simple algebra, 30 of a diagonalizable group, 60 stabilizer of a lattice, 51 strong approximation, 301 absolute, 301 Sylow pro-p-subgroup, 161 symplectic group, 93 system of simple roots, 68 Tamagawa measure, 315 Tamagawa number, 316, 321 tangent space, 112 Tannaka–Artin problem, 30 theorem of Albert–Hasse–Brauer–Noether, 42 Bartels–Kitaoka, 276 Behr, 177 Borel–Harish-Chandra, 224 Eichler, 43 Harish-Chandra, 213 Hermite, 10 Matsumoto, 161 Mostow, 144, 148 Noether, 166 Platonov–Yanchevski˘ı, 34 Prasad–Raghunathan, 40 Riehm, 133 Skolem–Noether, 29 Wang, 44 Whitney, 138 theorem on dimension of fibers, 111 Tits index, 77 Tits system, see BN-pair topology adele, 11, 12 associated with a valuation, 125, see also v-adic topology idele, 13

torus anisotropic, 60 multinorm, 62 norm, 62 quasisplit, 62 split, 60 transgression map, 22 twisting, 26, 78, 79 uniformizer, 6, 32 unimodular group, 187 unirational variety, 114 unitary group, 97 special, 97 universal cover, 72 K-defined, 88 v-adic topology, 125 valuation, 2 Archimedean, 3 complex, 5 extension of, 3 logarithmic, 3 non-Archimedean, 3 normalized, 13 p-adic, 3 real, 5 valuation ideal, 5 valuation ring, 5 variety K-defined, 111 of Borel subgroups, 123 of maximal tori, 121 rational, 113 smooth, 113 unirational, 114 Weyl group, 68, 75, 174 Whitehead group, reduced, 30 Witt index, 95

361