Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves [1st ed.] 9783030443283, 9783030443290

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Progress in Mathematics 334

Jean-Benoît Bost

Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves

Progress in Mathematics Volume 334

Series Editors Antoine Chambert-Loir, Université Paris-Diderot, Paris, France Jiang-Hua Lu, The University of Hong Kong, Hong Kong SAR, China Michael Ruzhansky, Imperial College, London, UK Yuri Tschinkel, Courant Institute of Mathematical Sciences, New York, USA

More information about this series at http://www.springer.com/series/4848

Jean-Benoît Bost

Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves

Jean-Benoît Bost Département de Mathématique Université Paris-Sud Orsay, France

ISSN 0743-1643 ISSN 2296-505X (electronic) Progress in Mathematics ISBN 978-3-030-44328-3 ISBN 978-3-030-44329-0 (eBook) https://doi.org/10.1007/978-3-030-44329-0 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dilectissimae uxori necnon consilio optimae

Contents

Preface

xi

Introduction 1

1

Hermitian Vector Bundles over Arithmetic Curves 1.1 Definitions and Basic Operations . . . . . . . . . . . . . . . . . . . 1.2 Direct Images. The Canonical Hermitian Line Bundle ω OK /Z over Spec OK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Arakelov Degree and Slopes . . . . . . . . . . . . . . . . . . . . . . 1.4 Morphisms and Extensions of Hermitian Vector Bundles . . . . . .

11 11 13 14 16

2 θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves 2.1 The Poisson Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The θ-Invariants h0θ and h1θ and the Poisson–Riemann–Roch Formula 2.3 Positivity and Monotonicity . . . . . . . . . . . . . . . . . . . . . . 2.4 The Functions τ and η . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The θ-invariants of direct sums of Hermitian line bundles over Spec Z 2.6 The Theta Function θE and the First Minimum λ1 (E) . . . . . . . 2.7 Application to Hermitian Line Bundles . . . . . . . . . . . . . . . . 2.8 Subadditivity of h0θ and h1θ . . . . . . . . . . . . . . . . . . . . . . .

25 26 28 29 33 35 36 41 44

3

49 50 53 59 62 70

4

Geometry of Numbers and θ-Invariants 3.1 Comparing h0θ and h0Ar . . . . . . . . . . . . . . . . . 3.2 Banaszczyk’s Estimates and θ-Invariants . . . . . . . 3.3 Subadditive Invariants of Euclidean Lattices . . . . . ˜ 0 (E, t) . . . . . . . . . . 3.4 The Asymptotic Invariant h Ar 3.5 Some Consequences of Siegel’s Mean Value Theorem

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Countably Generated Projective Modules over Dedekind Rings 4.1 Countably Generated Projective A-Modules . . . . . . . . 4.2 Linearly Compact Tate Spaces with Countable Basis . . . 4.3 The Duality Between CPA and CTCA . . . . . . . . . . 4.4 Strict Morphisms, Exactness and Duality . . . . . . . . . 4.5 Localization and Descent Properties . . . . . . . . . . . . 4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .

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77 . 77 . 80 . 86 . 90 . 100 . 101 vii

viii 5

Contents Ind5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

and Pro-Hermitian Vector Bundles over Arithmetic Curves Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbertizable Ind- and Pro-vector Bundles . . . . . . . . . . . . . . Constructions as Inductive and Projective Limits . . . . . . . . . . Morphisms Between Ind- and Pro-Hermitian Vector Bundles over OK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Duality Between Ind- and Pro-Hermitian Vector Bundles . . . Examples – I. Formal Series and Holomorphic Functions on Disks . Examples – II. Injectivity and Surjectivity of Morphisms . . . . . . Examples – III. Subgroups of Pre-Hilbert Spaces . . . . . . . . . .

6 θ-Invariants of Infinite-Dimensional Hermitian Vector Bundles 6.1 Limits of θ-Invariants . . . . . . . . . . . . . . . . . . . . 6.2 Upper and Lower θ-Invariants . . . . . . . . . . . . . . . . 6.3 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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Summable Projective Systems of Hermitian Vector Bundles 7.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Summable Projective Systems and Associated Measures . . . . . . 0 ˆ 7.4 Proof of Theorem 7.3.4 – I. The Equality hθ (E) = limi→+∞ h0θ (E i ) 7.5 Proof of Theorem 7.3.4 – II. Convergence of Measures . . . . . . . 7.6 A Converse Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Strongly Summable and θ-Finite Pro-Hermitian Vector Bundles . . Exact Sequences of Infinite-Dim. Hermitian Vector Bundles 8.1 Short Exact Sequences of Infinite-Dimensional Hermitian Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Short Exact Sequences and θ-Invariants of Pro-Hermitian Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Strongly Summable Pro-Hermitian Vector Bundles . . . . . . . . . 8.4 A Vanishing Criterion . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 The Category proVectOK as an Exact Category . . . . . . . . . . Infinite-Dimensional Vector Bundles over Smooth Projective Curves 9.1 Pro-vector Bundles over Smooth Curves . . . . . . . . . . . . . . . b and h0 (C, E) b . . . . . . . . . . . . . . . . 9.2 The Invariants h0 (C, E) 9.3 Successive Extensions and Wild Pro-vector Bundles over Projective Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 A Vanishing Criterion . . . . . . . . . . . . . . . . . . . . . . . . .

107 108 113 114 117 121 128 130 134 137 137 139 143 147 155 155 156 159 162 164 167 170 177 178 180 188 194 200 219 220 224 229 234

Contents

ix

10 Epilogue: Formal-Analytic Arithmetic Surfaces and Algebraization 10.1 An Algebraicity Criterion for Smooth Formal Curves over Q . . 10.2 Sections of Line Bundles and Algebraization . . . . . . . . . . . 10.3 Arithmetically Ample Hermitian Line Bundles and θ-Invariants 10.4 Pointed Smooth Formal Curves . . . . . . . . . . . . . . . . . . 10.5 Green’s Functions, Capacitary Metrics and Schwarz Lemma . . 10.6 Smooth Formal-Analytic Surfaces over Spec OK . . . . . . . . . 10.7 Arithmetic Pseudo-concavity and Finiteness . . . . . . . . . . . 10.8 Arithmetic Pseudo-concavity and Algebraization . . . . . . . . 10.9 The Isogeny Theorem for Elliptic Curves over Q . . . . . . . .

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237 237 242 251 259 262 273 283 289 294

A Large Deviations and Cram´er’s Theorem A.1 Notation and Preliminaries . . . . . A.2 Lanford’s Inequalities . . . . . . . . A.3 Cram´er’s Theorem . . . . . . . . . . A.4 An Extension of Cram´er’s Theorem . A.5 Reformulation and Complements . .

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305 306 308 312 315 318

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B Continuity of Linear Forms on Prodiscrete Modules 329 B.1 Preliminary: Maximal Ideals, Discrete Valuation Rings, and Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 B.2 Continuity of Linear Forms on Prodiscrete Modules . . . . . . . . . 331 C Measures on Countable Sets and Their Projective Limits 333 C.1 Finite Measures on Countable Sets . . . . . . . . . . . . . . . . . . 333 C.2 Finite Measures on Projective Limits of Countable Sets . . . . . . 335 D Exact Categories 341 D.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . 341 D.2 The Derived Category of an Exact Category . . . . . . . . . . . . . 343 E Holomorphic Sections of Line Bundles over Compact Complex Manifolds E.1 Spaces of Sections of Analytic Line Bundles and Multiplicity Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Proof of Proposition E.1.2 . . . . . . . . . . . . . . . . . . . . . . . E.3 Proof of Proposition E.1.3 . . . . . . . . . . . . . . . . . . . . . . .

347 347 349 351

F John Ellipsoids and Finite-Dimensional Normed Spaces 355 F.1 John Ellipsoids and John Euclidean Norms . . . . . . . . . . . . . 355 F.2 Properties of the John Norm . . . . . . . . . . . . . . . . . . . . . 355 F.3 Application to Lattices in Normed Real Vector Spaces . . . . . . . 356 Bibliography

359

Index

369

Preface A. This monograph is dedicated to the construction of suitable categories of infinite-dimensional Hermitian vector bundles in the framework of Arakelov geometry and to the study of their theta invariants, which are defined in terms of theta series associated to Euclidean lattices and take values in [0, +∞]. Our constructions are developed with a view toward applications to Diophantine geometry: using infinite-dimensional vector bundles and their theta invariants, one may establish diverse results in Diophantine geometry and transcendence theory by arguments that are formally similar to classical algebraization proofs in analytic and formal geometry, as exemplified in the last chapter of this monograph. A description of our results, intended to arithmetic geometers with a specific interest in Arakelov geometry and its applications to classical Diophantine problems, is given in the general introduction that follows this preface and in the introductions of the successive chapters. Notably the introduction of the final chapter, written to be accessible directly after the general introduction, describes how the general formalism developed in this monograph leads to “concrete” Diophantine applications, concerning, for instance, the construction of isogenies between elliptic curves over Q. The general form of our results required for their applications to Diophantine geometry, notably the need to work over a base ring that can be the ring of integers OK of an arbitrary number field K (that is, an arbitrary extension field of Q of finite degree), leads to some technicalities in their formulation and may hide their basic simplicity. In this preface, we try to present them in the simplest possible terms, by sticking to the basic case in which the base ring OK is Z, and we refer the reader to the more technical introduction that follows this preface for more complete statements and for references. B. The basic object of study in this monograph is Euclidean lattices. Recall that a Euclidean lattice E is defined by the following data: • a finite-dimensional R-vector space ER ; • a lattice E in ER , namely a subgroup of the additive group M (ER , +) for which there exists some R-basis (ei )1≤i≤n of ER such that E = Zei ; 1≤i≤n

• a Euclidean norm k.k, associated to some Euclidean scalar product h., .i, on ER . Euclidean lattices traditionally appear in mathematical physics, as mathematical models for crystalline structures, and in number theory, via the so-called xi

Preface

xii

geometry of numbers (a terminology coined by Minkowski [86]). For the past decades, with the development of lattice-based cryptography, they have also played an important role in computer science. A Euclidean lattice E admits several easily defined classical invariants. The simplest of them are its rank, defined in the above notation as rk E := dimR ER = n, and its covolume covol E, defined as the Euclidean volume of a fundamental domain for E acting on ER by translation, for instance, of X ∆ := [0, 1[ ei ; 1≤i≤n

it may be expressed in terms of the Gram determinant of the basis (ei )1≤i≤n of the lattice as (covol E)2 = det(hei , ej i)1≤i,j≤n . One also classically considers, when the rank of the Euclidean lattice E is positive, its first minimum λ1 (E) := min kvk v∈E\{0}

and its covering radius ρ(E) := max min kx − vk. x∈ER v∈E

∨

To every Euclidean lattice E is associated the dual Euclidean lattice E , defined as follows. Its underlying R-vector space is the dual of the R-vector space ER∨ , defined as the space of R-linear forms ER∨ := HomR (ER , R). ∨

The lattice E ∨ in ER∨ defining E is the subgroup of linear forms that are integral valued on E: E ∨ := {ξ ∈ ER∨ | ξ(E) ⊂ Z} ; it may be identified with HomZ (E, Z) by the restriction map (ξ 7→ ξ|E ). The ∨ Euclidean norm defining E is the dual norm k.k∨ on ER∨ , defined by the equality kξk∨ :=

max x∈ER ,kxk≤1

|ξ(x)| for every ξ ∈ ER∨ .

∨

The ranks of E and E are clearly equal, and their covolumes are easily seen to be the inverses of each other: ∨

covol(E ) = (covol E)−1 . In contrast to the simplicity of these definitions, Euclidean lattices and their invariants lead to difficult problems. Their role in cryptography actually relies on

Preface

xiii

the computational “hardness” of basic questions involving Euclidean lattices when their dimension becomes large. Such difficult problems arise, for instance, when one investigates estimates relating various classical invariants of Euclidean lattices. These estimates involve constants depending on the ranks of the Euclidean lattices under study, and the control of these constants when these ranks become large is often a delicate issue. This may be illustrated by one of the oldest results in the theory of Euclidean lattices, which goes back to Hermite and Minkowski, namely the existence, for every positive integer n, of some positive constant C(n) such that the first minimum of every Euclidean lattice E of rank n satisfies the following upper bound: λ1 (E) ≤ C(n)(covol E)1/n .

(0.0.1)

Hermite first proved this result by induction on the rank n, by developing what is known today as reduction theory for Euclidean lattices of arbitrary rank (see [64]). This approach allowed him to establish the estimate (0.0.1) with C(n) = (4/3)(n−1)/4 .

(0.0.2)

Minkowski proved that it actually holds with C(n) = 2 vn−1/n ,

(0.0.3)

where vn denotes the Lebesgue measure of the unit ball in Rn . The estimate (0.0.1) with the value (0.0.3) for C(n) is the famous Minkowski’s first theorem. Its derivation by Minkowski, in his Geometrie der Zahlen ([86, pp. 73–76]), admits the following simple physical interpretation. Let us think of the Euclidean lattice as a model for a crystal in the n-dimensional Euclidean space (ER , k.k). The molecules in this crystal are represented by the points v of the ˚k.k (v, λ1 (E)/2) of radius λ1 (E)/2 centered at lattice E. Since the open balls B these points are mutually disjoint, the density of the crystal — defined as the number of its molecules per unit volume — is at most the inverse of the volume of any of these balls, which is vn (λ1 (E)/2)n . This density is nothing but the inverse of the covolume of E. Therefore, covol(E)−1 ≤ [vn (λ1 (E)/2)n ]−1 . This estimate is precisely (0.0.1) with C(n) given by (0.0.3). Since vn = π n/2 /Γ(n/2 + 1), it follows from Stirling’s formula that as n goes to +∞, the value (0.0.3) for the constant C(n) obtained from Minkowski’s argument admits the following asymptotics: p 2vn−1/n ∼ 2n/eπ. (0.0.4) It is much smaller than the value (0.0.2) originally obtained by Hermite. The square γn of the best possible (namely, the minimal) value of the constant C(n) in the estimate (0.0.1) is known as the Hermite constant in dimension n. It

Preface

xiv −2/n

turns out that Minkowski’s upper bound 4vn on γn is of the “correct order of growth” if n goes to infinity. Actually, by combining this upper bound with some further results of Minkowski and Hlawka, one shows that log γn = log n + ε(n),

(0.0.5)

where |ε(n)| = O(1)

as n −→ +∞.

However, the exact value of γn is known only for a small number of values of n, and its precise asymptotic behavior (for instance, the possible convergence of ε(n) to some limit as n goes to +∞) is still not understood. C. Another circle of questions involving optimal constants in estimates comparing invariants of Euclidean lattices are the so-called transference estimates relating ∨ the invariants of a Euclidean lattice E and those of its dual E . For instance, consider the first minimum λ1 (E) of some Euclidean lattice E ∨ of positive rank n, and the covering radius ρ(E ) of the dual Euclidean lattice E. An application of the reduction theory of Euclidean lattices establishes that the ∨ product λ1 (E)ρ(E ) is bounded from above and from below by positive constants τ1 (n) and τ2 (n) depending only on n: ∨

τ1 (n) ≤ λ1 (E)ρ(E ) ≤ τ2 (n).

(0.0.6)

One easily sees that for every n ≥ 1, the optimal value of τ1 (n) is 1/2: simply consider the lattice Zn in Rn equipped with the Euclidean norm k.kε defined by k(x1 , . . . , xn )k2ε = ε(x21 + · · · + x2n−1 ) + x2n , with ε a small positive real number. Concerning τ2 (n), important progress was obtained in the 1990s by Banaszczyk, who showed in [7] that (0.0.6) holds with τ2 (n) = n/2, while the best possible (namely the minimal) constant τ2 (n) satisfies τ2 (n) ≥ (n/2πe)(1 + o(n))

as n −→ +∞.

To establish the upper bound ∨

λ1 (E)ρ(E ) ≤ n/2, Banaszczyk introduced a new technique, originating in harmonic analysis, based on the consideration of the measures X 2 e−πβkvk δv , (0.0.7) v∈E

xv

Preface

on the real vector space ER defined for all β ∈ R∗+ and supported by the lattice E and their Fourier transforms on the dual space ER∨ . This technique has been especially influential in the development of lattice-based cryptography during the last decades. The total mass of Banaszczyk’s measure (0.0.7) is given by the classical theta series X 2 θE (β) := e−πβkvk . (0.0.8) v∈E

Such theta series have classically played a central role in the study of integral lattices, namely of Euclidean lattices whose Euclidean scalar product is Z-valued on E × E. Indeed, the theta series associated to integral lattices turn out to define modular forms, and from Jacobi to Siegel and his followers, the development of the theory of modular forms has led to spectacular applications concerning integral lattices and related integral quadratic forms. Banaszczyk’s work has demonstrated the relevance of the theta series (0.0.8) and their measure-theoretic versions (0.0.7) in the investigation of the fine properties of general Euclidean lattices. D. In the first chapters of this monograph, we investigate in some detail the properties of the invariants of Euclidean lattices defined in terms of these series, their theta invariants; the main instance of these is the nonnegative real number: X 2 h0θ (E) := log θE (1) = log e−πkvk . (0.0.9) v∈E

Besides the “technical” motivation to study these theta invariants provided by Banaszczyk’s technique, there exists an older and more “conceptual” one, which stems from the classical analogy between number fields and function fields. It is closely related to Arakelov geometry, which itself constitutes an outgrowth of this classical analogy. Recall that in the analogy between number fields and function fields, the ring of integers OK of some number field K, together with its archimedean places1 , appears as the counterpart of a smooth projective (geometrically connected) curve C over some base field k. The field k(C) of rational functions over C, traditionally known as a “function field in one variable” over k, plays the role of the number field K. More generally, one may associate “arithmetic” counterparts over number fields to “geometric” objects over C, such as vector bundles E over C, and to their invariants, such as the rank rk E and the degree degC E of E, or the dimension h0 (C, E) := dimk Γ(C, E) of its space of global regular sections. Although the special case in which the base field k of the curve C is finite plays an important role in this analogy, readers more inclined toward analytic than 1 defined

by the field embeddings of K in C, up to complex conjugation.

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xvi

algebraic geometry may focus on the situation in which the base field k is C. Then the data of the curve C (resp., of the algebraic vector bundle E) are equivalent to those of some compact connected Riemann surface C an (resp., of some C-analytic vector bundle E an over C an ); the degree degC E of E coincides with its topological degree defined by its first Chern class c1 (E an ) in H 2 (C an , Z) ' Z, and the finitedimensional C-vector space Γ(C, E) with the space Γ(C an , E an ) of global analytic sections of E an . We will concentrate on the case K = Q, and therefore OK = Z. Then in the above analogy, the counterpart of a vector bundle E over C is precisely a Euclidean lattice E. For instance, the role of the trivial line bundle OC over C is played by the Euclidean line bundle Z, defined by the R-vector space R, the lattice Z in R, and the Euclidean norm equal to the usual absolute value |.|. To every pair (E, F) of vector bundles over C are associated the vector bundle E ⊕ F and the finite-dimensional k-vector space HomOC (E, F) of morphisms from E to F. These constructions admit counterparts in the arithmetic side. Let us indeed consider two Euclidean lattices E (resp., F ), defined by the R-vector space ER (resp., FR ), the lattice E (resp., F ), and the Euclidean norm k.kE on ER (resp., k.kF on FR ). Then their direct sum E ⊕ F is the Euclidean lattice defined by the R-vector space ER ⊕ FR , its lattice E ⊕ F, and the Euclidean norm k.k on ER ⊕ FR defined by the equality kx ⊕ yk2 := kxk2E + kyk2F

for all (x, y) ∈ ER × FR .

Moreover, HomOC (E, F) is replaced by the finite set Hom(E, F ) of R-linear maps ϕ : ER −→ FR such that ϕ(E) ⊆ F and kϕ(v)kF ≤ kvkE

for all v ∈ ER .

One may also introduce the analogue of a short exact sequence p

i

0 −→ E −→ F −→ G −→ 0 of vector bundles over C. It is a so-called admissible short exact sequence of Euclidean lattices: p i 0 −→ E −→ F −→ G −→ 0, (0.0.10) defined by the data of Euclidean lattices E, F , and G, and of elements i and p in Hom(E, F ) and Hom(F , G) such that the following conditions are satisfied: • the diagram i

p

0 −→ E −→ F −→ G −→ 0 is an exact sequence of Z-modules; this is easily seen to imply that i

p

0 −→ ER −→ FR −→ GR −→ 0 is an exact sequence of R-vector spaces;

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• the map i is an isometry from the Euclidean R-vector space (ER , k.kE ) to (FR , k.kF ), and the Euclidean norm k.kG on GR that defines G is the quotient norm induced from the norm k.kF on FR by means of the surjective R-linear map p : FR → GR .2 In this dictionary between vector bundles and Euclidean lattices, the rank rk E of the vector bundle E is replaced by the rank rk E of E, and the degree degC E by the Arakelov degree of E, defined as d E := − log covol E. deg d E is R-valued. Instead of being Z-valued like degC E, the Arakelov degree deg However, it satisfies properties formally similar to those satisfied by degC E. For instance, d (E ⊕ F ) = deg d E + deg d F, deg and more generally, for every admissible short exact of Euclidean lattices (0.0.10), d G = deg d E + deg d F. deg It turns out that the invariant h0 (C, E) attached to some vector bundle E over C admits two distinct counterparts in the classical literature. The first one, already considered in substance by Weil in [117], is the nonnegative real number h0Ar (E) := log |{v ∈ E|kvk ≤ 1}|, defined in terms of the number of points of the lattice E in the unit ball of the Euclidean vector space (ER , k.k). We have a bijection ∼

Hom(Z, E) −→ {v ∈ E|kvk ≤ 1}, defined by mapping an element ϕ in Hom(Z, E) to ϕ(1), and accordingly, the definition of h0Ar (E) may also be written h0Ar (E) := log |Hom(Z, E)|.

(0.0.11)

Actually, for every vector bundle E over the curve C, we have a bijection of k-vector spaces ∼ HomOC (OC , E) −→ Γ(C, E), 2 This condition on p, k.k , and k.k may be rephrased as follows: the transpose pt : G → F R R F G of p : FR → GR , defined using the Euclidean structures on FR and GR , is an isometry from (GR , k.kG ) to (FR , k.kF ).

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xviii

also defined by mapping ϕ in HomOC to ϕ(1), and consequently, when the base field k is finite of order q, the integer h0 (C, E) admits an expression similar to (0.0.11): h0 (C, E) = dimk Γ(C, E) = dimk HomOC (OC , E) =

log |HomOC (OC , E)| . log q

The second counterpart of the invariant h0 (C, E) is the theta invariant already defined in (0.0.9): X 2 h0θ (E) := log e−πkvk . v∈E

The fact that it is an arithmetic analogue of h0 (C, E) goes back to the work of the German school of number theory, in particular to the proofs by Hecke and Schmidt of the meromorphic continuation and the functional equation of the zeta function of some global field. Hecke [61] first treated the case of a number field, and later, Schmidt [97] the case of a function field, defined as above as k(C) with k a base field of finite order q := |k|. The comparison of these proofs shows that in the function field case, 0 |Γ(C, E)| = q h (C,E) plays a role parallel to that of 0

θE (1) = ehθ (E) in the number field case. Moreover the Riemann–Roch formula on the curve C plays, in Schmidt’s proof, a role parallel to that of the Poisson formula for theta series in Hecke’s proof. The Poisson formula indeed relates the theta functions θE and θE ∨ attached ∨ to some Euclidean lattice and to its dual lattice E : θE (β) = (covol E)−1 β −rk E/2 θE ∨ (β −1 )

for every β ∈ R∗+ .

When β = 1, after taking logarithms, it reads ∨ d E. h0θ (E) − h0θ (E ) = deg

(0.0.12)

This is formally similar to the Riemann–Roch formula for a vector bundle E over a smooth projective curve C of genus g = 1, which takes the form h0 (C, E) − h0 (C, E ∨ ) = degC E. In the same vein, for all t ∈ R, we may consider the Euclidean lattice E ⊗O(t) induced from some Euclidean lattice E by scaling its Euclidean norm by e−t . Then, from the asymptotic behavior of θE (β) as β goes to 0, we obtain d E + o(1) h0θ (E ⊗ O(t)) = rk E · t + deg

as t −→ +∞.

xix

Preface This may be seen as a counterpart of the relation h0 (C, E ⊗ OC (k · O) = rk E · k + degC E

when k ∈ N is large enough,

valid for an arbitrary vector bundle E over an elliptic curve C of origin O. The two invariants h0Ar (E) and h0θ (E) are actually closely related. For instance, using Banaszczyk’s techniques, one may establish the comparison estimates −π ≤ h0θ (E) − h0Ar (E) ≤ (n/2) log n − log(1 − 1/2π),

where n = rk E. (0.0.13)

However, the properties of h0θ (E) make it a better analogue of the number h (C, E) than h0Ar (E). For instance, one easily sees that it is additive for direct sums; namely, for every two Euclidean lattices E and F , 0

h0θ (E ⊕ F ) = h0θ (E) + h0θ (F ).

(0.0.14)

Moreover, as already observed by Quillen and Groenewegen, it is subadditive for short exact sequences; namely, for every admissible short exact sequences of Euclidean lattices (0.0.10), we have h0θ (F ) ≤ h0θ (E) + h0θ (G).

(0.0.15)

This is again a consequence of (a suitable version of) Poisson’s formula. The relations (0.0.14) and (0.0.15) are easily seen not to hold when h0θ is replaced by h0Ar . Another illustration of the closer similarity of h0θ (E) to h0 (C, E) is the following observation. From the Poisson–Riemann–Roch formula (0.0.12) and the nonnegativity of h0θ (E), we immediately derive the lower bound d E, h0θ (E) ≥ deg

(0.0.16)

which is similar to the lower bound h0 (C, E) ≥ degC E, valid for every vector bundle over a smooth projective curve C of genus g = 1. In turn, combined with (0.0.13), the lower bound (0.0.16) implies d E − c(n), h0Ar (E) ≥ deg

(0.0.17)

where c(n) = (n/2) log n − log(1 − 1/2π). As n approaches infinity, this value of c(n) is equivalent to the best possible (that is, the smallest) constant c(n) in (0.0.17).3 3 Indeed, this best constant is easily seen to satisfy c(n) ≥ (n/2) log γ , where γ denotes the n n Hermite contant in dimension n, introduced in paragraph B as the square of the best constant C(n) in the Hermite–Minkowski inequality (0.0.1). From the asymptotics (0.0.5) on γn and the above value for c(n), we derive that the best constant c(n) in (0.0.17) satisfies: c(n) = (n/2) log n + O(n).

Preface

xx

A remarkable feature of the estimates (0.0.15) and (0.0.16) concerning the theta invariant hθ0 is that the ranks of the Euclidean lattices under consideration do not appear in them, while analogous relations concerning h0Ar , such as (0.0.17), would necessarily involve these ranks. E. This last observation constitutes the starting point of the main constructions in this monograph: the fact that the theta invariants of Euclidean lattices satisfy analytic properties formally independent of their rank indicates that these invariants should make sense for some infinite-dimensional avatars of Euclidean lattices. This is similar to the following familiar observation: the fact that most constructions in finite-dimensional Euclidean geometry are independent of the dimension points toward the theory of Hilbert spaces. The main result of this monograph is that it is indeed possible to define a nice class of such “infinite-dimensional Euclidean lattices” for which the theta invariant hθ0 (E) is still well defined and satisfies natural continuity properties. Moreover, such infinite-dimensional Euclidean lattices naturally appear in arithmetic geometry, and the consideration of their theta invariants leads to natural proofs in Diophantine geometry and transcendence theory. Let us describe in elementary terms the class of Euclidean lattices of infinite rank that constitute the main object of study in this monograph. In the termi≤1 nology introduced in Chapter 5, they are the objects of the category proVectZ of “pro-Hermitian vector bundles over Spec Z” that have infinite rank. They are defined by the following data: bR and a Z-submodule E b of E bR such that there • a topological R-vector space E exists an isomorphism of topological R-vector spaces ∼ bR −→ ϕ:E RN

such that b = ZN . ϕ(E) Here the space RN of R-valued sequences is equipped with the topology of simple convergence, or equivalently, with the product topology derived from the usual topology on each factor R. • a real Hilbert space (ERHilb , k.k) and a continuous injective R-linear map with dense image bR . i : ERHilb −→ E These data i b⊂E bR ←−ERHilb , k.k E

will play the role of the data E ⊂ ER , k.k

(0.0.18)

xxi

Preface

defining a Euclidean lattice. The data (0.0.18) defining a Euclidean lattice of b — or with the terminology of this monograph, an object in the infinite rank E ≤1 category proVectZ — may look rather intricate. However, this definition arises naturally, from both a practical and a conceptual point of view. Let us try to explain how. E.a. In many Diophantine problems, one encounters a combination of formal geometry over the integers and complex analytic geometry. It turns out that such a combination may often be encoded in data of the type (0.0.18). Let us give a specific example. Let Ω be an open neighborhood of 0 in C, which will be assumed to be connected, bounded, and invariant under complex conjugation. We may consider the complex Hilbert space OL2 (Ω) of square integrable holomorphic functions on Ω, equipped with the L2 -norm defined by Z kf k2L2 (Ω) := |f (x + iy)|2 dx dy. Ω

It is equipped with a natural C-antilinear “complex conjugation”, which maps a function f in OL2 (Ω) to f defined by f (z) := f (z). The fixed points of this involution define a real Hilbert space ERHilb . Its elements are the square integrable holomorphic functions on Ω whose Taylor expansions at 0 have real coefficients. Then we may consider: b := Z[[X]] and E bR := R[[X]], equipped with the topology of simple conver• E gence of coefficients; bR = R[[X]], defined as the map sending some holomorphic • i : ERHilb −→ E function f in ERHilb to its Taylor expansion at 0: X 1 f (n) (0) X n . i(f ) := n! n∈N

The map i is injective, since Ω is connected. It is continuous by Cauchy estimates, and its image is dense in R[[X]], since it contains R[X], since Ω is bounded. ≤1 Therefore, these data define an object of the category proVectZ , which will be b denoted by H(Ω). E.b. Let us consider a projective system q0

q1

qk−1

qk

qk+1

E • : E 0 ←− E 1 ←− · · · ←− E k ←− E k+1 ←− · · · .

(0.0.19)

of Euclidean lattices, defined by a sequence (E k )k∈N of Euclidean lattices and a sequence (qk )k∈N of morphisms qk in Hom(E k+1 , E k ). Let us assume that it satisfies the following admissibility conditions, for all k ∈ N:

Preface

xxii

• the morphism of Z-modules qk : Ek+1 → Ek — and therefore the R-linear map qk : Ek+1,R → Ek,R — is surjective; • the Euclidean norm k.kk on Ek,R that defines E k is the quotient norm induced from the norm k.kk+1 on Ek+1,R by means of qk : Ek+1,R −→ Ek,R . These conditions may be rephrased as follows: for all k ∈ N, the morphism qk fits into an admissible short exact sequence of Euclidean lattices, as defined in paragraph D above (see (0.0.10)): qk

i

k 0 −→ S k −→ E k+1 −→ E k −→ 0.

(0.0.20)

(Indeed, the Euclidean lattice S k may be constructed from qk by considering the lattice ker qk|Ek+1 : Ek+1 → Ek inside ker qk : Ei+k,R → Ek,R , equipped with the restriction of the Euclidean norm k.kk+1 ; the morphism ik is then the inclusion map.) We shall also assume that the nondecreasing sequence (rk Ek )k∈N is unbounded. (Otherwise, the morphisms qk are isometric isomorphisms for k large enough.) To any admissible projective system E • of Euclidean lattices as above we may associate data (0.0.18) defining some Euclidean lattice of infinite rank by the following construction. We may consider the projective limits ( ) Y b := lim Ek := (vk )k∈N ∈ E Ek | ∀k ∈ N, qk (vk+1 ) = vk ←− k

k∈N

and ( bR := lim Ek,R := E ←−

) (vk )k∈N ∈

k

Y

Ek,R | ∀k ∈ N, qk (vk+1 ) = vk

.

k∈N

They are endowed with Q a natural topology, Qdefined as the topology induced by the product topology on k∈N Ek (resp., on k∈N Ek,R ), when each space Ek (resp., Ek,R ) is endowed with the discrete topology (resp., with its natural topology of a finite-dimensional R-vector space). The surjective morphisms of Z-modules qk : Ek+1 → Ek admit Z-linear splittings, and therefore we may construct a sequence (ϕk )k∈N of isomorphisms of Z-modules ∼ ϕk : Ek −→ Zrk Ek such that the diagrams ϕk+1

Ek+1 −−−−→ Zrk Ek+1    pr qk y y k Ek

ϕk

−−−−→ Zrk Ek

(0.0.21)

Preface

xxiii

are commutative, where prk denotes the projection prk : (xi )1≤i≤rk Ek+1 7−→ (xi )1≤i≤rk Ek . The maps ϕk induce isomorphisms of finite-dimensional R-vector spaces ∼

ϕk,R : Ek,R −→ Rrk Ek and, by going to the projective limit, an isomorphism of topological R-vector spaces ∼

bR −→ RN ϕ b:E such that b = ZN . ϕ( b E) bR is defined by a sequence (vk )k∈N in Q Every element v in E k∈N Ek,R satisfying the coherence conditions: for every k ∈ N.

qk (vk+1 ) = vk ,

The morphisms qk are norm decreasing, and therefore kvk kk ≤ kvk+1 kk+1 ,

for every k ∈ N.

Consequently, the limit kvk := lim kvk kk k→+∞

exists in [0, +∞], and we may therefore define ERHilb := {v ∈ ERHilb | kvk < +∞}. Using that each Euclidean norm k.kk is the quotient norm (via qk ) of the Euclidean norm k.kk+1 , one easily sees that (ERHilb , k.k) is actually a real Hilbert space, and that the inclusion morphism bR , i : ERHilb ,−→E is continuous with dense image. Indeed, by considering the orthogonal splittings of the surjective R-linear maps qk : Ek+1,R → Ek,R , the topological R-vector space bR may be identified with the product E Y Sk,R E0,R × k∈N

and (ERHilb , k.k) with the completed infinite direct sum E0,R ⊕

M d k∈N

Sk,R

Preface

xxiv

of finite-dimensional Euclidean R-vector spaces. bR , its submodule E, b the Hilbert space The topological R-vector space E bR so constructed from the (ERHilb , k.k), and the inclusion morphism i : ERHilb ,−→E ≤1 admissible projective system E • define an object of the category proVectZ , which will be denoted by lim E • . ←− F. It turns out that this construction of Euclidean lattices of infinite rank as pro≤1 jective limits allows one to recover (up to isomorphism) any object of proVectZ . F.a. The central point behind this fact is the construction of quotient (finiteb of proVect≤1 defined by data dimensional) Euclidean lattices of an object E Z as in (0.0.18) above, associated to saturated open submodules of the topological b Z-module E. b By the definition of the topology of E b Let U be an open submodule of E. (which is assumed to be topologically isomorphic to ZN equipped with the product topology), the quotient Z-module b EU := E/U is then finitely generated. When it is torsion-free, hence isomorphic to Zr for some r in N, the submodule U is called saturated. Then the quotient map b −→ EU ' Zr q:E is continuous (when EU is equipped with the discrete topology), and is easily seen to extend uniquely to a continuous R-linear map bR −→ EU,R := EU ⊗Z R ' Rr , qR : E which is actually open and surjective. Let us consider the composite R-linear map i qR bR −→ qR ◦ i : ERHilb ,−→ E EU,R .

bR and qR is continuous and surjective, its image is dense Since i(ERHilb ) is dense in E in EU,R , hence equals EU,R , since EU,R is a finite-dimensional R-vector space. We may therefore consider the Euclidean norm k.kU of EU,R defined as the quotient norm induced from the Hilbert norm k.k on ERHilb by means of the continuous surjective map qR ◦ i. Finally, we may define a Euclidean lattice (of finite rank) E U by the lattice EU in the R-vector space EU,R equipped with the Euclidean norm k.kU : E U : EU ⊂ EU,R , k.kU . For instance, let us consider the Euclidean lattice of infinite rank lim E • , ←− constructed above from the admissible projective system E • . For every k ∈ N, we may consider the canonical projection b −→ Ek prk : E

Preface

xxv

b seen as a submodule of Q that maps an element v = (vi )i∈N of E i∈N Ei to its kth b component vk . Its kernel Uk is a saturated open submodule of E, and the quotient b k is easily seen to be canonically isomorphic to Ek , and the Euclidean EUk := E/U b to be isomorphic to E . lattice E Uk

k

b is an arbitrary object in proVect≤1 , then we may choose a In general, if E Z decreasing sequence U0 ⊃ U1 ⊃ · · · ⊃ Uk ⊃ Uk+1 ⊃ · · · b that constitutes a basis of neighborhoods of 0 of saturated open submodules of E b and the quotient b and we may consider the associated Euclidean lattices E in E, Uk morphisms qk : EUk ,R −→ EUk+1 ,R . They define a projective system of Euclidean lattices q0

qk−1

q1

qk+1

qk

E U0 ←− E U1 ←− · · · ←− E Uk ←− E Uk+1 ←− · · · that satisfies the admissibility condition introduced in E.b. Accordingly, we may ≤1 construct the object limk E Uk in proVectZ , and there exists a canonical isomor←− phism: ∼ b −→ E lim E Uk . ←− k

b := F.b. Let us apply this construction to the Euclidean lattice of infinite rank E H(Ω) introduced in E.a. Then a natural choice for the decreasing sequence (Uk )k∈N b := Z[[X]] is of saturated open submodules of E ( ) X n Uk := an X | a0 = · · · = ak−1 = 0 = X k Z[[X]] for all k ∈ N. n∈N

Then EUk = Z[[X]]/X k Z[[X]] ' Z[X]k (resp., by N≥k ) the set of nonnegative integers greater than k (resp., greater than or equal to k). By countable, we mean “of cardinality at most the cardinality of N.” If M is a module over a ring A, and if B is a commutative A-algebra, we denote by MB the “base changed” module M ⊗A B. Similarly, if ϕ : M −→ N is a morphism of A-modules, we let ϕB := ϕ ⊗A IdB : MB −→ NB , and if X is some A-scheme, we let XB := X ×Spec A Spec B. When K is a number field with ring of integers OK , and σ : K,−→C is a field embedding, these base change constructions may be applied to A := OK and B := C, considered as an A-algebra by means of σ. Then we denote MB (resp., XB ) by Mσ (resp., Xσ ). 10 An earlier version, relying on the arguments discussed after the proof of Proposition 4.4.11, had to make a countability assumption on A instead of the condition Ded3 .

Chapter 1

Hermitian Vector Bundles over Arithmetic Curves In this chapter, we gather some basic results concerning Hermitian vector bundles over arithmetic curves, namely affine schemes defined by rings of integers of number fields. These results are well known, with the exception of the content of paragraph 1.4.7, and appear, for instance, in [110], [88], [19], [13], [106], and [20]. This chapter is primarily intended as a reference chapter. It might be skipped by readers familiar with Arakelov geometry, who could refer to it only when needed. We denote by K a number field, by OK its ring of integers, and by π : Spec OK −→ Spec Z the morphism of schemes from Spec OK to Spec Z, defined by the inclusion morphism Z ,→ OK .

1.1

Definitions and Basic Operations

1.1.1 Hermitian vector bundles over arithmetic curves. A Hermitian vector bundle over Spec OK is a pair E = (E, (k.kσ )σ:K,→C ) consisting of a finitely generated projective OK -module E and a family (k.kσ )σ:K,→C of Hermitian norms k.kσ on the complex vector spaces Eσ := E ⊗OK ,σ C defined by means of the field embeddings σ : K ,→ C. The family (k.kσ )σ:K,→C is, moreover, required to be invariant under complex conjugation.1 When K = Q, a Hermitian vector bundle E = (E, k.k) over Spec OK = Spec Z may be identified with a Euclidean lattice, defined by a free Z-module of finite rank E and a Euclidean norm k.k on ER := E ⊗Z R. (Indeed, for every such 1 Namely, for every embedding σ : K ,→ C, we may consider the complex conjugate embedding ∼ σ : K ,→ C and the C-antilinear isomorphism F∞ : Eσ = E⊗OK ,σ −→ Eσ = E⊗OK ,σ defined by F∞ (e ⊗ λ) = e ⊗ λ. The Hermitian norms k.kσ and k.kσ have to satisfy kF∞ (.)kσ = k.kσ .

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0_1

11

12

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

E, the data of some Hermitian norm on EC := E ⊗Z C invariant under complex conjugation and some Euclidean norm on ER are equivalent.) The rank of a Hermitian vector bundle E as above is the rank of the OK module E, or equivalently the dimension of the complex vector spaces Eσ . A Hermitian line bundle is a Hermitian vector bundle of rank one. When confusion may arise, the family of Hermitian norms underlying some Hermitian vector bundle E over Spec OK will be denoted by (k.kE,σ )σ:K,→C . An isometric isomorphism, or simply an isomorphism, between two Hermi∼ tian vector bundles E and F over Spec OK is an isomorphism ψ : E −→ F between the underlying OK -modules that, after every base change σ : K ,→ C, defines an isometry of complex normed vector spaces between (Eσ , k.kE,σ ) and (Fσ , k.kF ,σ ). 1.1.2 Pullback. Tensor operations. Let L be a number field extension of K. The inclusion OK ,→ OL defines a morphism of arithmetic curves f : Spec OL −→ Spec OK . The pullback f ∗ E is defined as the Hermitian vector bundle f ∗ E := (f ∗ E, (k.kτ )τ :L,→C ) over Spec OL where f ∗ E := E ⊗OK OL and where, for every field embedding τ : L ,→ C, of restriction τ|K =: σ, k.kτ denotes the Hermitian norm k.kσ on (f ∗ E)τ := (E ⊗OK OL ) ⊗OL ,τ C ' E ⊗OK ,σ C =: Eσ . To every Hermitian vector bundle E over Spec OK we may associate its ∨ dual Hermitian vector bundle E and its exterior powers ∧k E, k ∈ N. They are defined by means of the (compatible) constructions of duals and exterior powers for projective OK -modules and Hermitian complex vector spaces. Similarly, for every two Hermitian vector bundles E and F over Spec OK , we may construct their direct sum E ⊕F and their tensor product E ⊗F as Hermitian vector bundles over Spec OK . These tensor operations are compatible with the pullback construction de∨ fined above. Namely, we have canonical identifications (f ∗ E)∨ ' f ∗ (E ), f ∗ (∧k E) ' ∧k (f ∗ E), . . . 1.1.3 The Hermitian line bundles O(δ). For every δ ∈ R, we may consider the Hermitian line bundle over Spec Z defined by O(δ) := (Z, k.kO(δ) ), where k1kO(δ) := e−δ . Its Arakelov degree (as defined in 1.3.1 below) is d O(δ) := − log k1k deg O(δ) = δ.

1.2. Direct Images. The Canonical Hermitian Line Bundle ω OK /Z over Spec OK 13 We may also consider its pullback by the morphism π : Spec OK −→ Spec Z: OSpec OK (δ) := π ∗ O(δ). For every Hermitian vector bundle E over Spec OK , the Hermitian vector bundle E ⊗ OSpec OK (δ) may be identified with the Hermitian vector bundle that admits the same underlying OK -module E as E, and whose Hermitian structure is defined by the Hermitian norms defining E scaled by the factor e−δ . For simplicity, we shall often write E ⊗ O(δ) instead of E ⊗ OSpec OK (δ). The “trivial” Hermitian line bundle — namely, OSpec OK (0) — will also be denoted by OSpec OK and, when no confusion may arise, simply by O.

1.2

Direct Images. The Canonical Hermitian Line Bundle ω OK /Z over Spec OK

1.2.1 To every Hermitian vector bundle E = (E, (k.kσ )σ:K,→C ) is attached its direct image 2 π∗ E over Spec Z. Observe that we have an isomorphism of C-algebras: L ∼ OK ⊗Z C −→ σ:K,→C C, α ⊗ λ 7−→ (σ(α)λ)σ:K,→C . Therefore, for every OK -module M , if π∗ M denotes the underlying Z-module, we have (π∗ M )C := π∗ M ⊗Z C ' π∗ M ⊗OK (OK ⊗Z C) M ' (M ⊗OK ,σ C). σ:K,→C

Using this observation, the direct image π∗ E may be defined as the Hermitian vector bundle of rank [K : Q] · rk E over Spec Z: π∗ E := (π∗ E, k.kπ∗ E ), L where for all v = (vσ )σ:K,→C in (π∗ E)C ' σ:K,→C Eσ , X kvk2π∗ E := kvσ k2σ . σ:K,→C

Clearly, we have rk π∗ E = [K : Q] · rk E. 2 The construction of direct images of Hermitian vector bundles introduced here actually makes sense in a considerably more general setting; see [20, Section 1.2.1]. In particular it may be extended to any morphism of arithmetic curves f : Spec OL −→ Spec OK .

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

14

1.2.2 The compatibility of the direct image operation π∗ and of the duality of Hermitian vector bundles involves the canonical Hermitian line bundle ω OK /Z := (ωOK /Z , (k.kσ )σ:K,→C ) over Spec OK . It is defined as the “canonical module” ωOK /Z := HomZ (OK , Z) — also known as the “inverse of the different” — equipped with the Hermitian norms defined by ktrK/Q kσ = 1 for every embedding σ : K ,→ C, where trK/Q denotes the trace map from K to Q (it is indeed a nonzero element in HomZ (OK , Z)). Namely, for every Hermitian vector bundle E over Spec OK , we have a canonical (!!) isometric isomorphism of Hermitian vector bundles over Spec Z, ∨

∼

π∗ (E ⊗ ω OK /Z ) −→ (π∗ E)∨ .

(1.2.1)

(See, for instance, [20, Proposition 3.2.2]. For every ξ in E ∨ := HomOK (E, OK ) and λ in ωOK /Z , the “relative duality isomorphism” (1.2.1) maps ξ ⊗ λ to λ ◦ ξ.)

1.3

Arakelov Degree and Slopes

1.3.1 Definition and basic properties. The Arakelov degree of a Hermitian line bundle L := (L, (k.kσ )σ:K,→C ) over Spec OK is defined by the following equality, valid for all s ∈ L \ {0}: X d L := log |L/OK s| − deg log kskσ (1.3.1) σ:K,→C

=

X

vp (s) log N p −

p

X

log kskσ .

(1.3.2)

σ:K,→C

In the last equality, p runs over the closed points of Spec OK — that is, over the nonzero prime ideals of OK — and vp (s) denotes the p-adic valuation of s, seen as a section of the invertible sheaf over Spec OK associated to L. Moreover, N p := |OK /p| denotes the norm of p. This definition is extended to Hermitian vector bundles of arbitrary rank over Spec OK by means of the formula d E := deg d ∧rk E E. deg If E is a Hermitian vector bundle over Spec Z, or in other words, a Euclidean lattice, its Arakelov degree may expressed in terms of its covolume3 covol(E); namely, d E = − log covol(E). deg 3 See,

for instance, paragraph 2.1.1 below.

15

1.3. Arakelov Degree and Slopes

For every two Hermitian line bundles L1 and L2 over Spec OK , the expression (1.3.2) for their Arakelov degree shows that d L1 + deg d L2 . d (L1 ⊗ L2 ) = deg deg

(1.3.3)

For every two Hermitian vector bundles E 1 and E 2 over Spec OK , there is a canonical4 isomorphism of Hermitian line bundles over Spec OK , ∼

∧rk (E1 ⊕E2 ) (E 1 ⊕ E 2 ) −→ ∧rk E1 E 1 ⊗ ∧rk E2 E 2 , and the additivity relation (1.3.3) applied to Li := ∧rk Ei E i (i = 1, 2) takes the form d (E 1 ⊕ E 2 ) = deg d E 1 + deg d E2. deg (1.3.4) Similarly, by reducing to the case of Hermitian line bundles, one shows that for every Hermitian vector bundle E over Spec OK , we have d E ∨ = − deg d E. deg 1.3.2 Arakelov degree and direct image. of a number field K shows that

The definition of the discriminant ∆K

covol(π∗ OSpec OK ) = |∆K |1/2 . In other words, d π∗ OSpec O = − 1 log |∆K |. deg K 2

(1.3.5)

More generally, for every Hermitian vector bundle E over Spec OK , we have d π∗ E = deg d E − 1 (log |∆K |) rk E. deg 2

(1.3.6)

Observe that the compatibility of the “relative duality isomorphism” (1.2.1) and the “relative Riemann–Roch formula” (1.3.6) applied to the Hermitian line bundles E = OSpec OK and E = ω OK /Z shows that d ω O /Z = log |∆K |. deg K 1.3.3 Slopes. If E is a Hermitian vector bundle over Spec OK of positive rank, its slope µ b(E) is defined as the quotient µ b(E) := 4 Up

to a sign.

dE deg . rk E

16

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

bmax (E) (resp., its minimal slope µ Its maximal slope µ bmin (E)) is the maximum (resp., the minimum) of the slopes µ b(F ), where F is the Hermitian vector bundle defined by an OK -submodule (resp., a torsion-free quotient) of positive rank F of E, equipped with the restrictions to Fσ ⊂ Eσ of the Hermitian metrics (k.kσ )σ:K,→C (resp., with the quotient metrics on the vector spaces Fσ of these metrics). One easily checks that ˇ µ bmax (E) = −b µmin (E).

(1.3.7)

It is convenient to define the maximum (resp., minimum) slope of Hermitian vector bundles of rank zero to be −∞ (resp., +∞).

1.4

Morphisms and Extensions of Hermitian Vector Bundles

1.4.1 The filtration Hom≤• OK (E, F ) on HomOK (E, F ). For every two Hermitian vector bundles E and F on Spec OK and for all λ ∈ R+ , we may introduce the subset Hom≤λ OK (E, F ) of the OK -module HomOK (E, F ) consisting of the OK -linear maps ψ : E −→ F such that for every σ : K,−→C, the induced C-linear map ψσ : Eσ −→ Fσ has an operator norm ≤ λ when Eσ and Fσ are equipped with the Hermitian norms k.kE σ and k.kF σ . Observe that if E 1 , E 2 , and E 3 are Hermitian vector bundles over Spec OK , 1 then for all (λ1 , λ2 ) ∈ R2+ , the composition of an element ψ1 in Hom≤λ OK (E 2 , E 1 ) ≤λ2 and an element ψ2 in HomOK (E 3 , E 2 ) defines an element ψ1 ◦ ψ2 in 1 λ2 Hom≤λ (E 3 , E 1 ). OK In particular, one endows the class of Hermitian vector bundles over Spec OK with the structure of a category by defining the set of morphisms from E to F as Hom≤1 OK (E, F ). The isomorphisms in this category coincide with those already introduced in 1.1.1. Moreover, for all λ ∈ R+ , transposition defines a bijection: ∼

Hom≤λ OK (E, F ) −→ ψ 7−→

∨

∨

Hom≤λ OK (F , E ), ψ ∨ := · ◦ ψ.

(1.4.1)

1.4. Morphisms and Extensions of Hermitian Vector Bundles

17

1.4.2 Injective and surjective admissible morphisms. Observe that Hom≤1 OK(E, F ) contains the injective admissible morphisms, defined as the OK -linear maps ψ : E −→ F such that ψ is injective with torsion-free cokernel and such that for all σ : K ,→ C, the C-linear map ψσ : Eσ −→ Fσ is an isometry with respect to the Hermitian norms k.kE,σ and k.kF ,σ . The set Hom≤1 OK (E, F ) also contains the surjective admissible morphisms, defined as the surjective OK -linear maps ψ : E −→ F such that for every field embedding σ : K ,→ C, the map ψσ is a “co-isometry”: the norm k.kF σ on Fσ is the quotient norm induced from the norm k.kE σ on Eσ by means of the surjective C-linear map ψσ : Eσ −→ Fσ ; equivalently, the C-linear map ψσ : Fσ∨ −→ Eσ∨ is an isometry with respect to the Hermitian norms k.kF ∨ ,σ and k.kE ∨ ,σ . The transposition map (1.4.1), with λ = 1, exchanges injective and surjective admissible morphisms. 1.4.3 Heights of morphisms. Recall that to every nonzero K-linear map ϕ : EK −→ FK is attached its height with respect to E and F , defined as X X ht(E, F , ϕ) := log kϕkp + log kϕkσ . (1.4.2) p

σ:K,→C

(See [15, Section 4.1.4], where ht(E, F , ϕ) is denoted by h(E, F , ϕ).) On the righthand side of (1.4.2), p varies over the maximal ideals of OK , and kϕkp denotes the p-adic norm of ϕ ∈ HomK (EK , FK ) ' (E ∨ ⊗OK F )K , defined by the equivalence5 kϕkp ≤ 1 ⇐⇒ ϕ ∈ (E ∨ ⊗OK F )OK,p and the normalization condition k$ϕkp = (N p)−1 kϕkp , where $ denotes a uniformizing element of OK,p , and N p := |OK /p| the norm of p. Moreover, kϕkσ denotes the operator norm of ϕσ : Eσ −→ Fσ defined from the Hermitian norms k.kE σ and k.kF σ . It is convenient to define the height ht(E, F , 0) of the zero morphism as −∞. Like its local norms kϕkp and kϕkσ , the height of a K-linear map ϕ is invariant under transposition: ∨

∨

ht(F , E , ϕ∨ ) = ht(E, F , ϕ). Clearly, for every ϕ in Hom≤1 OK (E, F ) \ {0}, all the norms kϕkp and kϕkσ belong to ]0, 1], and therefore ht(E, F , ϕ) belongs to R− . Furthermore, when E and F are Hermitian line bundles, then for every nonzero element ϕ of the K-vector space HomK (EK , FK ), definition (1.4.2) of its height shows that d (E ∨ ⊗ F ) = −ht(E, F , ϕ), deg 5 We

denote by OK,p the discrete valuation ring defined as the localization of OK at p.

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

18 or equivalently,

d E − deg d F. ht(E, F , ϕ) = deg

(1.4.3)

This equality may be extended to arbitrary Hermitian vector bundles, in the form of the following “slope inequalities”: Proposition 1.4.1. Let E and F be two Hermitian vector bundles, and let ϕK : EK −→ FK be a K-linear map. (1) If ϕ is injective, then d E ≤ rk E [b deg µmax (F ) + ht(E, F , ϕ)].

(1.4.4)

(2) If ϕ is surjective, then d F ≥ rk F [b deg µmin (E) − ht(E, F , ϕ)].

(1.4.5)

The inequality (1.4.4) is a straightforward consequence of (1.4.3) applied to the exterior powers of ϕ (see [15, Proposition 4.5]). Then (1.4.5) follows by duality. Observe that in the estimate (1.4.4) (resp., (1.4.5)), when E (resp., F ) has rank zero, we follow the usual convention 0 · (−∞) = 0 (resp., 0.(+∞) = 0) to define its right-hand side. 1.4.4 Admissible short exact sequences of Hermitian vector bundles. An admissible short exact sequence (also called an admissible extension) of Hermitian vector bundles over Spec OK is a diagram i

p

E : 0 −→ E −→ F −→ G −→ 0,

(1.4.6)

where E = (E, (k.kE,σ )σ:K,→C ), F = (F, (k.kF ,σ )σ:K,→C ), and G = (G, (k.kG,σ )σ:K,→C ) are Hermitian vector bundles over Spec OK , and where i

p

E : 0 −→ E −→ F −→ G −→ 0 is a short exact sequence of OK -modules such that for every field embedding σ : K ,→ C, the short exact sequence of complex vector spaces i

pσ

σ 0 −→ Eσ −→ Fσ −→ Gσ −→ 0

(derived from E by the base change σ) is compatible with the Hermitian norms k.kE,σ , k.kF ,σ , and k.kG,σ .6 6 Namely,

iσ is required to be an isometry for the norms k.kE,σ and k.kF ,σ , and the norm k.kG,σ is required to be the quotient norm induced from k.kF ,σ by means of pσ . Equivalently, i and p are respectively injective and surjective admissible morphisms from E to F and from F to G.

19

1.4. Morphisms and Extensions of Hermitian Vector Bundles

The additivity of the Arakelov degree for direct sums (1.3.4) extends to admissible short exact sequences. Namely, for every admissible short exact sequence (1.4.6) of Hermitian vector bundles over Spec OK , the following equality holds: d F = deg d E + deg d G. deg

(1.4.7)

This follows from the existence of a canonical7 isomorphism of Hermitian line bundles ∧rk F F ' ∧rk E E ⊗ ∧rk G G attached to the short exact sequence (1.4.6). An admissible short exact sequence of Hermitian vector bundles (1.4.6) is said to be split when there exists an admissible injective morphism of Hermitian vector bundles s : G −→ F such that p ◦ s = IdG . The morphism s is then uniquely determined. Indeed, for every embedding σ : K ,→ C, the C-linear map sσ : Gσ −→ Fσ derived from s by the extension of scalars σ : OK −→ C necessarily coincides with the orthogonal splitting s⊥ σ of the short exact sequence of complex vector spaces pσ

i

σ Eσ : 0 −→ Eσ −→ Fσ −→ Gσ −→ 0.

(1.4.8)

attached to the Hermitian metric k.kF ,σ on Fσ . (Recall that s⊥ σ is defined as ⊥ the (linear) section sσ of pσ that maps Gσ isomorphically onto the orthogonal complement iσ (Eσ )⊥ of iσ (Eσ ) in Fσ equipped with the Hermitian metric k.kF ,σ .) 1.4.5

Arithmetic extensions.

Let E and G be two finitely generated projective 1 d O (G, E) of OK -modules. In [20] is introduced the arithmetic extension group Ext K G by E. In the setting of this book, where we work over arithmetic curves of the form Spec OK , it may be defined as 1 ∼ ( d O (G, E) := HomO (G, E) ⊗Z R/Z −→ Ext K K

L

HomC (Gσ , Eσ )) HomOK (G, E)

σ:K,→C

F∞

.

(1.4.9) (See [20, Example 2.2.3]. As usual, F∞ denotes complex conjugation.) 1 d (G, E) classifies, up to isomorphism, the arithmetic extenThe group Ext OK sions of G by E. Recall that such an arithmetic extension is a pair (E, (s(σ))σ:K,→C ), where i

p

E : 0 −→ E −→ F −→ G −→ 0 7 Up

to a sign.

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

20

is an extension (of OK -modules) of G by E, and where (s(σ))σ:K,→C is a family, invariant under complex conjugation, of C-linear splittings s(σ) : Gσ −→ Fσ of the extension of complex vector spaces pσ

i

σ Fσ −→ Gσ −→ 0. Eσ : 0 −→ Eσ −→

Indeed, observe that for every arithmetic extension as above, one may choose some OK -linear splitting sint ∈ HomOK (G, F ) of E. Then, for all σ : K ,→ C, pσ ◦ (s(σ) − sσint ) = 0, and therefore there exists a uniquely determined Tσ ∈ HomC (Gσ , Eσ ) such that s(σ) − sint σ = iσ ◦ Tσ . L

F∞

. Moreover, its 1 d class [(Tσ )σ:K,→C ] modulo HomOK (G, E) in the extension group ExtOK (G, E) defined by (1.4.9) does not depend on the choice of the integral splitting sint . By definition, it is the class [(E, (s(σ))σ:K,→C )] of the arithmetic extension (E, (s(σ))σ:K,→C ). The family (Tσ )σ:K,→C belongs to (

σ:K,→C

HomC (Gσ , Eσ ))

1.4.6 Admissible short exact sequences and arithmetic extensions. sible short exact sequence i

An admis-

p

E : 0 −→ E −→ F −→ G −→ 0 of Hermitian vector bundles over Spec OK determines an arithmetic extension (E, (s⊥ σ )σ:K,→C ) of G by E, defined by means of the orthogonal splittings with respect to the Hermitian norms k.kF ,σ of the extensions of complex vector spaces Eσ . According to the discussion in 1.4.4, its class [E] := [(E, (s⊥ σ )σ:K,→C )] 1

d (G, E) if and only if the admissible short exact sequence E is vanishes in Ext OK split. Furthermore, for every two Hermitian vector bundles E and G over Spec OK , 1 d O (G, E) may be realized by the previous construction, starting every class in Ext K

from some suitable admissible short exact Lsequence E as above. F Indeed, for T = (T (σ))σ:K,→C in ( σ:K,→C HomC (Gσ , Eσ )) ∞ , we may introduce the Hermitian vector bundle T

E ⊕ G := (E ⊕ G, (k.kT (σ) )σ:K,→C )

1.4. Morphisms and Extensions of Hermitian Vector Bundles

21

over Spec OK , where for every field embedding σ : K ,→ C and (e, g) ∈ Eσ ⊕ Gσ , we have 2 k(e, g)k2T (σ) := ke − Tσ (g)kE + kgk2Gσ . σ One easily checks that the maps i : E −→ e 7−→ and

E ⊕ G, (e, 0),

p : E ⊕ G −→ (e, g) 7−→

G, g,

fit into an admissible short exact sequence T

i

p

E(T ) : 0 −→ E −→ E ⊕ G −→ G −→ 0

(1.4.10)

of Hermitian vector bundles over Spec OK and that the associated class in 1 d (G, E) is [T ]. Ext OK

1.4.7 Direct images of arithmetic and admissible extensions. Consider two finitely generated projective OK -modules E and G and the underlying Z-modules π∗ E and π∗ G. There is an obvious inclusion map between modules of OK -linear and Z-linear morphisms: HomOK (G, E),−→ HomZ (π∗ G, π∗ E). We will sometimes denote by π∗ ϕ the image in HomZ (π∗ G, π∗ E) of an element ϕ of the group HomOK (G, E). In this paragraph, we want to investigate the “derived” analogue of this map π∗ , defined between the relevant arithmetic extension groups. Consider an arithmetic extension (E, s) of E by G over Spec OK , defined by an extension p i E : 0 −→ E −→ F −→ G −→ 0 of OK -modules and a family s = (sσ )σ:K,→C , invariant under complex conjugation, of C-linear splittings sσ : Gσ −→ Fσ of the extensions of complex vector spaces i

pσ

σ Eσ : 0 −→ Eσ −→ Fσ −→ Gσ −→ 0.

We may define its direct image by π : Spec OK −→ Spec Z as the arithmetic extension (π∗ E, π∗ s) of π∗ E by π∗ G over Spec Z defined by the extension of Zmodules p i π∗ E : 0 −→ π∗ E −→ π∗ F −→ π∗ G −→ 0, equipped with the splitting L π∗ s : G ⊗Z C ' σ:K,→C Gσ (gσ )σ:K,→C

−→ 7−→

L E ⊗Z C ' σ:K,→C Eσ , (sσ (gσ ))σ:K,→C .

Chapter 1. Hermitian Vector Bundles over Arithmetic Curves

22

Proposition 1.4.2. (1) The above construction defines a morphism of Z-modules 1

d (G, E) −→ ˆ∗1 : Ext π OK [(E, s)] 7−→

1

d (π∗ G, π∗ E), Ext Z [(π∗ E, π∗ s)].

(2) Via the identifications L F∞ 1 ∼ ( σ:K,→C HomC (Gσ , Eσ )) d O (G, E) −→ Ext K HomOK (G, E) and

F∞ 1 ∼ HomC (π∗ G ⊗Z C, π∗ E ⊗Z C) d Z (π∗ G, π∗ E) −→ Ext , HomZ (π∗ G, π∗ E)

the ˆ∗1 coincides with the map defined, for all (T (σ))σ:K,→C in L map π ( σ:K,→C HomC (Gσ , Eσ ))F∞ , by π ˆ∗1 [(T (σ))σ:K,→C ] = [⊕σ T (σ)], where ⊕σ T (σ) is the element of HomC (π∗ G ⊗Z C, π∗ E)F∞ defined by L L ⊕σ T (σ) : π∗ G ⊗Z C ' σ:K,→C Gσ −→ π∗ E ⊗Z C ' σ:K,→C Eσ , (gσ )σ:K,→C 7−→ (Tσ (gσ ))σ:K,→C . (3) Every admissible short exact sequence i

p

E : 0 −→ E −→ F −→ G −→ 0 of Hermitian vector bundles over Spec OK defines, by direct image, an admissible short exact sequence i

p

π∗ E : 0 −→ π∗ E −→ π∗ F −→ π∗ G −→ 0 of Hermitian vector bundles over Spec Z. Moreover, this construction is compatible with the direct image map π ˆ∗1 between arithmetic extension groups. Namely, with the above notation, the following equality holds in 1 d Z (π∗ G, π∗ E): Ext π ˆ∗1 [E] = [π∗ E]. (4) The map π ˆ∗1 is injective. Proof. Assertions (1), (2), and (3) are left as an easy exercise for the reader. To prove the injectivity of π ˆ∗1 , we use its description in (2), and we are left to show that, with the notation of (2), if ⊕σ T (σ) belongs to HomZ (π∗ G, π∗ E), then there exists T in HomOK (G, E) such that for every σ : K ,→ C, T (σ) = Tσ .

1.4. Morphisms and Extensions of Hermitian Vector Bundles

23

In other words, we have to show that if some U in HomZ (π∗ G, π∗ E) is such that UC : π∗ G ⊗Z C −→ π∗ E ⊗Z C is block-diagonal with respect to the direct-sum decompositions M M Gσ and π∗ E ⊗Z C ' Eσ , π ∗ G ⊗Z C ' σ:K,→C

σ:K,→C

then U : G −→ E is OK -linear. This is indeed the case, since every such U is OK -linear if and L only if UC := U ⊗Z IdC is linear as a map of modules over the  ring OK ⊗Z C ' σ:K,→C C.

Chapter 2

θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves This chapter is devoted to the definitions and to some basic properties of the θinvariants of (finite-rank) Hermitian vector bundles over arithmetic curves. Their extensions to infinite-dimensional Hermitian vector bundles established in the latter chapters of this monograph, as well as their Diophantine applications discussed in its sequel, will depend on these properties. As explained in the introduction (see 0.1.2 above), the definitions of the invariants h0θ and h1θ have their origins in the classical works of Hecke and F. K. Schmidt on the functional equations of number fields and functions fields over finite fields. Formal definitions and some of their basic properties already appear in [96, Section 6.2] and, most importantly from the perspective of this monograph, in the work of van der Geer and Schoof [116] and Groenewegen [55]. In this chapter, we give a self-contained and streamlined presentation of the theory of θ-invariants, adapted to our later use of them. We especially emphasize the importance of their subadditivity properties (see Section 2.8), which will play a key role in the proof of the main results of this monograph in Chapter 7. Among the results in this part that, by their novelty, might be of special interest to readers already familiar with the content of [116] and [55], we mention: • Proposition 2.3.3 and its corollaries, which provide useful comparison estimates for the θ-invariants of Euclidean lattices with the same underlying Q-vector spaces. • The improved bounds on the invariant h0θ (L) of a Hermitian line bundle L over an arithmetic curve. These bounds turn out to be remarkably similar to the well-known bounds on the dimension h0 (C, L) of the space of sections of a line bundle L over a projective curve C (see Propositions 2.7.2 and 2.7.3). • The discussion of the subadditivity properties of h0θ and h1θ in Section 2.8, particularly the results concerning the “subadditivity measure” hθ (E) attached to an admissible short exact sequence of Euclidean lattices E that are established in Propositions 2.8.3 and 2.8.4. Let us indicate that the content of Section 10.3 below, which depends only on the results in this chapter, will illustrate how the basic formalism of θ-invariants naturally combines with the classical developments of Arakelov geometry, concerning higher-dimensional projective schemes over Spec Z. The reader interested in such applications may move directly to Section 10.3 after this chapter. © Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0_2

25

26 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves As in Chapter 1, we denote by K a number field, by OK its ring of integers, and by π the morphism of schemes form Spec OK to Spec Z.

2.1

The Poisson Formula

We first review some basic results related to the Poisson formula. This gives us the opportunity to introduce various conventions and notation that will be used in the next chapters. 2.1.1 The Poisson formula for Schwartz functions. Let V be a Hermitian vector bundle over Spec Z (or equivalently, a Euclidean lattice). We shall denote by λV the Lebesgue measure on VR . It is the unique translation-invariant Radon measure on VR that satisfies the following normalization condition: for every orthonormal basis (e1 , . . . , eN ) of (VR , k.kV ), λV

N X [0, 1[ei

! = 1.

i=1

This normalization condition may be equivalently expressed in terms of a Gaussian integral: Z 2

VR

e−πkxkV dλV (x) = 1.

(2.1.1)

Then the covolume of V may be defined as covol(V ) := λV

N X [0, 1[vi

!

i=1

for every Z-basis (v1 , . . . , vN ) of V. Every function f in the Schwartz space S(VR ) of VR admits a Fourier transform fˆ in the Schwartz space S(VR∨ ) attached to the dual vector space VR∨ , defined by Z fˆ(ξ) :=

VR

f (x) e−2πihξ,xi dλV (x), for all ξ ∈ VR∨ .

With this notation, the Poisson formula for V reads as follows: X v∈V

f (x − v) = (covol V )−1

X

∨ fˆ(v ∨ ) e2πihv ,xi , for all x ∈ VR .

(2.1.2)

v ∨ ∈V ∨

This is nothing but the Fourier series expansion of the function which is V -periodic on VR .

P

v∈V

f (. − v),

27

2.1. The Poisson Formula

It is convenient to have at one’s disposal the following more “symmetric” form of the Poisson formula: X X ∨ f (x − v) e2πihξ,vi = (covol V )−1 e2πihξ,xi fˆ(ξ − v ∨ ) e−2πihv ,xi , (2.1.3) v ∨ ∈V ∨

v∈V

VR × VR∨ . (The −2πihξ,.i

which is valid for all (x, ξ) ∈ (2.1.2) applied to the function e

identity (2.1.3) indeed follows from

f.)

2.1.2 The Poisson formula for Gaussian functions. In this monograph, a key role will be played by the Poisson formula applied to Gaussian functions. Namely, we apply the above formula to the Gaussian function f defined by 2

f (x) := e−πkxkV . Then for all ξ ∈ VR∨ , we have 2 fˆ(ξ) = e−πkξkV ∨ ,

and the identities (2.1.2) and (2.1.3) take the form X X ∨ 2 ∨ 2 e−πkx−vkV = (covol V )−1 e−πkv kV ∨ +2πihv ,xi and X

(2.1.4)

v ∨ ∈V ∨

v∈V

2

e−πkx−vkV +2πihξ,vi = (covol V )−1 e2πihξ,xi

X

e−πkξ−v

∨ 2 kV ∨ −2πihv ∨ ,xi

.

v ∨ ∈V ∨

v∈V

(2.1.5) In particular, when x = 0, the identity (2.1.4) becomes X X ∨ 2 2 e−πkvkV = (covol V )−1 e−πkv kV ∨ .

(2.1.6)

v ∨ ∈V ∨

v∈V

Observe also that (2.1.4) implies that for all x ∈ VR , X X 2 2 e−πkx−vkV ≤ e−πkvkV , v∈V

(2.1.7)

v∈V

and that equality holds in (2.1.7) if and only if x ∈ V. It also implies that for all x ∈ VR , X X X ∨ 2 2 2 e−πkx−vkV + e−πkvkV = (covol V )−1 e−πkv kV ∨ [1 + cos(2πhv ∨ , xi)] v∈V

v ∨ ∈V ∨

v∈V

X

= 2(covol V )−1

e−πkv

∨ 2 kV ∨

cos2 (πhv ∨ , xi),

v ∨ ∈V ∨

and therefore, X v∈V

2

e−πkx−vkV +

X v∈V

2

e−πkvkV ≥ 2(covol V )−1 .

(2.1.8)

28 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves

2.2

The θ-Invariants hθ0 and h1θ and the Poisson–Riemann–Roch Formula

2.2.1 The θ-invariants of a Euclidean lattice and the Poisson formula. For every Hermitian vector bundle E := (E, k.k) over Spec Z (that is, for every Euclidean lattice), we let X 2 (2.2.1) h0θ (E) := log e−πkvkE v∈E

and

∨

h1θ (E) := h0θ (E ).

(2.2.2)

The Poisson formula (2.1.6) for the Euclidean lattice E reads X X 2 2 e−πkwkE∨ = covol(E). e−πkvkE . w∈E ∨

v∈E

It may be rewritten in terms of the θ-invariants h0θ (E) and h1θ (E) and the Arakelov d E as the following relation: degree deg d E. h0θ (E) − h1θ (E) = deg

(2.2.3)

Observe the similarity of (2.2.2) and (2.2.3) to the Serre duality and the Riemann– Roch formula for vector bundles over an elliptic curve. 2.2.2 The θ-invariants of Hermitian vector bundles over a general arithmetic curve and the Poisson–Riemann–Roch formula. We extend the above definitions of h0θ and h1θ to Hermitian vector bundles over an arbitrary “arithmetic curve” Spec OK as above by defining hiθ (E) := hiθ (π∗ E) for i = 0, 1.

(2.2.4)

In other words, we have h0θ (E) = log

X

e

−πkvk2π

∗E

v∈E

and h1θ (E)

=

h0θ (E)

" # X −πkvk2 d π E ∗ − deg π∗ E = log covol(π∗ E) e . v∈E

Observe that as a consequence of the “relative duality isomorphism” (1.2.1), we have ∨ h1θ (E) = h0θ (E ⊗ ω OK /Z ). (2.2.5) This “Hecke–Serre duality formula” could have been used as an alternative definition of h1θ (E).

29

2.3. Positivity and Monotonicity

Observe finally that as a consequence of the Poisson formula (2.2.3) for Euclidean lattices and the expression (1.3.6) for the Arakelov degree of a direct image, we obtain the general version of the Poisson–Riemann–Roch formula, where we denote by ∆K the discriminant of the number field K: d E − 1 (log |∆K |) rk E. h0θ (E) − h1θ (E) = deg 2

2.3

(2.2.6)

Positivity and Monotonicity

2.3.1 Some elementary estimates. The following observation is a straightforward consequence of the definitions (2.2.1) and (2.2.2) of the θ-invariants, but it plays an important conceptual role in this monograph and its applications: Proposition 2.3.1. For every Hermitian vector bundle E over Spec OK , the real numbers h0θ (E) and h1θ (E) are nonnegative, and are positive if E has positive rank.  Together with the “Poisson–Riemann–Roch formula” (2.2.6), this nonnegativity shows that for every Hermitian vector bundle, the following avatar of the “Riemann inequality” is satisfied: d E − 1 log |∆K | · rk E. h0θ (E) ≥ deg 2

(2.3.1)

In particular, for every Euclidean lattice E, the following lower bound is satisfied: d E. h0θ (E) ≥ deg

(2.3.2)

Proposition 2.3.2. Let E and F be two Hermitian vector bundles over Spec OK , and let ϕ : E −→ F be a map in Hom≤1 OK (E, F ). Let ϕK : EK −→ FK denote the induced morphism of K-vector spaces. (1) If ϕK is injective (or equivalently, if ϕ is injective), then h0θ (E) ≤ h0θ (F ).

(2.3.3)

h1θ (E) ≥ h1θ (F ).

(2.3.4)

(2) If ϕK is surjective, then

Moreover, equality holds in either (2.3.3) or (2.3.4) if and only if ϕ is an isometric isomorphism from E onto F . Proof. (1) Let us assume that ϕK is injective.

30 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves By successively using that the maps ϕσ have operator norm at most 1 and the injectivity of ϕ, we obtain X −πkvk2 X 2 π∗ E ≤ e (2.3.5) e−πkϕ(v)kπ∗F v∈E

v∈E

≤

X

e

−πkwk2π

∗F

.

(2.3.6)

w∈F

This establishes (2.3.3). Equality holds in (2.3.5) if and only if for every field embedding σ : K ,→ C and every v ∈ E, kϕσ (v)k2F ,σ = kvk2E,σ . Since the Hermitian norms (k.kE,σ )σ:K,→C and (k.kF ,σ )σ:K,→C are invariant under complex conjugation, this holds precisely when the ϕσ are isometries. Moreover, equality holds in (2.3.6) if and only if ϕ(E) = F . This shows that equality holds in (2.3.3) if and only if ϕ is an isometric isomorphism. (2) When ϕK is surjective, we consider the morphism π∗ ϕ ∈ Hom≤1 Z (π∗ E, π∗ F ) and its transpose ∨ ∨ (π∗ ϕ)∨ ∈ Hom≤1 Z ((π∗ F ) , (π∗ E) ).

It is injective, and according to (1), h0θ ((π∗ F )∨ ) ≤ h0θ ((π∗ E)∨ ).

(2.3.7)

This establishes (2.3.4). Moreover, equality holds in (2.3.4), or equivalently in (2.3.7), if and only if (π∗ ϕ)∨ is an isometric isomorphism from (π∗ F )∨ onto (π∗ E)∨ . This is easily seen to be equivalent to the fact that ϕ itself is an isometric isomorphism.  2.3.2 Comparing the θ-invariants of generically isomorphic Hermitian vector bundles. Combined to the Poisson–Riemann–Roch formula and the basic properties of the height of morphisms (cf. 1.4.3 above, the simple estimates established in Proposition 2.3.2 may be extended to more general situations in which one deals with two Hermitian vector bundles E and F over Spec OK such that EK and FK are isomorphic. Proposition 2.3.3. Let E and F be two Hermitian vector bundles over Spec OK of the same rank n and let ϕ : E −→ F be a map in Hom≤1 OK (E, F ). If ϕ is injective, or equivalently if det ϕK : ∧n EK −→ ∧n FK

2.3. Positivity and Monotonicity

31

is not zero, then the following inequalities hold: hθ0 (E) ≤ h0θ (F ) ≤ h0θ (E) − ht(∧n E, ∧n F , det ϕK )

(2.3.8)

and h1θ (F ) ≤ h1θ (E) ≤ h1θ (F ) − ht(∧n E, ∧n F , det ϕK ). Proof. The inequalities h0θ (E) ≤ h0θ (F )

(2.3.9)

h1θ (F ) ≤ h1θ (E)

(2.3.10)

and are special cases of Proposition 2.3.2. Moreover, according to (1.4.3), d ∧n E − deg d ∧n F = deg d E − deg d F. ht(∧n E, ∧n F , det ϕK ) = deg The inequality h0θ (F ) ≤ h0θ (E) − ht(∧n E, ∧n F , det ϕK ) may therefore be written d F ≤ h0 (E) − deg d E. h0θ (F ) − deg θ Taking the Poisson–Riemann–Roch formula (2.2.6) for E and F into account, this inequality reduces to (2.3.10). Similarly, the inequality h1θ (E) ≤ h1θ (F ) − ht(∧n E, ∧n F , det ϕK ) follows from (2.3.9).



Corollary 2.3.4. For every Hermitian vector bundle F := (F, (k.kσ )σ:K,→C ) over Spec OK and every OK -submodule E of F such that EK = FK — or equivalently, such that the quotient F/E is finite — the θ-invariants of the Hermitian vector bundle E := (E, (k.kσ )σ:K,→C ) satisfy h0θ (F ) − log |F/E| ≤ h0θ (E) ≤ h0θ (F ) and h1θ (F ) ≤ h1θ (E) ≤ h1θ (F ) + log |F/E|. Proof. Indeed, if ϕ : E −→ F denotes the inclusion morphism, we have d E + deg d F = log |F/E|. −ht(∧n E, ∧n F , det ϕK ) = − deg



32 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves Recall that by definition, an Arakelov divisor X  D := ni pi , (δσ )σ:K,→C i

P over Spec OK consists of a divisor i ni pi in Spec OK and a family, invariant under complex conjugation, of [K : Q] real numbers (δσ )σ:K,→C . It is effective if the divisor D and the δσ s are nonnegative. To D is attached the Hermitian line bundle over Spec OK O(D) := (OK (D), (k.kσ )σ:K,→C ), Q where OK (D) := i pi −ni and where the metric k.kσ on OK (D)σ ' C is defined by k1kσ := e−δσ . The Arakelov degree of D is defined as X X d D := deg d O(D) = deg ni log N pi + δσ . σ

i

Corollary 2.3.5. Let E be a Hermitian vector bundle over Spec OK . (1) For every effective Arakelov divisor D over Spec OK , we have d D. 0 ≤ h0θ (E) − h0θ (E ⊗ O(−D)) ≤ rk E · deg

(2.3.11)

(2) For all δ in R+ , 0 ≤ h0θ (E) − h0θ (E ⊗ O(−δ)) ≤ rk E · [K : Q] · δ.

(2.3.12)

Proof. (1) Let us consider the Hermitian vector bundle over Spec OK 0

E := E ⊗ O(−D). Since D is effective, the module E 0 = E ⊗ OK (−D) =

Y

pni i E

i

is an OK -submodule of E, of the same rank rk E as E, and the inclusion morphism ϕ : E 0 ,→ E has Hermitian operator norms kϕσ k = e−δσ ≤ 1. Moreover, the norms of det ϕK are easily computed: k det ϕK kp = 1 if p ∈ / {pi }i = (N pi )−rk E.ni if p = pi . and for every embedding σ : K ,→ C, k det ϕK kσ = e−rk E·δσ .

33

2.4. The Functions τ and η Therefore, 0

h(∧rk E E , ∧rk E E, det ϕK ) = −rk E

X

X

ni log N pi +

i

 d D, δσ = −rk E · deg

σ:K,→C 0

and (2.3.11) follows from (2.3.8) applied to the morphism ϕ in Hom≤1 OK (E , E). (2) The Hermitian line bundle OSpec OK (δ) is of the form O(D) for some effective Arakelov divisor D of Arakelov degree d D = [K : Q] · δ, deg namely for D := (0, (δσ )σ:K,→C ), where δσ := δ for every σ : K ,→ C. Therefore, (2.3.12) follows from (2.3.11). 

2.4

The Functions τ and η

To express the θ-invariants of Euclidean lattices of rank one, it will be convenient to introduce the two functions τ : R∗+ −→ R∗+ and η : R −→ R∗+ defined as follows. For all x ∈ R∗+ , we let τ (x) := log

X

2

e−πxn .

n∈Z

This definition may also be written τ (x) = h0θ (O(−(1/2) log x)), and then the Poisson formula (2.1.6) (or equivalently the Poisson–Riemann formula (2.2.3)) applied to O(− 21 log x) becomes: τ (x) = τ (x−1 ) −

1 log x. 2

(2.4.1)

Observe also that τ (x) = 2e−πx + O(e−2πx ) as x −→ +∞.

(2.4.2)

Together with the Poisson formula (2.4.1), this implies 1 τ (x) = − log x + 2e−π/x + O(e−2π/x ) as x −→ 0+ . 2

(2.4.3)

34 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves For all t ∈ R, we also let η(t) := τ (e2|t| ), which is a continuous nonincreasing function of |t|. In particular, its maximal value is max η(t) = η(0) = τ (1). t∈R

We shall denote this number by η. In other words, η = hθ0 (O) = log ω, where ω :=

X

2

e−πn .

n∈Z

The number η is denoted by η(Q) in [116]. As explained in [116, proof of Proposition 4], the positive real number ω may be expressed as ω=

π 1/4 . Γ(3/4)

Numerically, one obtains ω = 1.0864348 . . . and η = 0.0829015 . . . . Moreover, according to (2.4.1), we have η(t) = τ (e−2t ) − t+ ,

(2.4.4)

where t+ := max(0, t), or equivalently, h0θ (O(t)) = t+ + η(t). In fact, as |t| goes to +∞, η(t) goes to zero very fast. Indeed, according to (2.4.2), we have 2|t| 2|t| η(t) = 2e−πe + O(e−2πe ) as |t| −→ +∞. (2.4.5) Observe that this implies the existence of some constant c ∈ R∗+ such that for all t ∈ R, 2|t| η(t) ≤ ce−πe , (2.4.6) or equivalently, such that for every x ∈ [1, +∞[, τ (x) ≤ ce−πx . An elementary computation shows that this holds as soon as  e−3π  c≥2 1+ = 2.0002 . . . . 1 − e−5π For instance, we may choose c = 3.

(2.4.7)

2.5. The θ-invariants of direct sums of Hermitian line bundles over Spec Z

35

2.5 Additivity. The θ-invariants of direct sums of Hermitian line bundles over Spec Z Observe that for every two Euclidean lattices E and F , the following identity holds: X X X 2 2 2 e−πkvkE · e−πk(v,w)kE⊕F = e−πkwkF . v∈E

(v,w)∈E⊕F

w∈F

This immediately implies that the invariants h0θ and h1θ satisfy the following additivity property: Proposition 2.5.1. For every two Hermitian vector bundles E and F over Spec OK , hiθ (E ⊕ F ) = hiθ (E) + hiθ (F ),

i = 0, 1.

(2.5.1) 

If n denotes a positive integer and λ := (λ1 , . . . , λn ) an element of R∗n + , we may consider the Hermitian vector bundle V λ over Spec Z attached to the lattice Zn in Rn equipped with the Euclidean norm k.kλ defined by k(x1 , . . . , xn )k2λ :=

n X

λi x2i .

i=1

Clearly, M

Vλ '

O(−(1/2) log λi ).

1≤i≤n

The Arakelov degree and the θ-invariants of Vλ are easily computed, using their additivity (1.4.7) and (2.5.1) and the Poisson–Riemann–Roch formula (2.2.3): Proposition 2.5.2. For every positive integer n and λ ∈ R∗n + , we have d V λ = −1 deg 2 h1θ (V λ ) =

n X

log λi , h0θ (V λ ) =

i=1

n X

τ (λi )

and

i=1

n   X 1 τ (λi ) + log λi . 2 i=1



These formulas may be rewritten as follows, in terms of the function η: Proposition 2.5.3. For every positive integer n and all Hermitian line bundles L1 , . . . , Ln over Spec Z, we have d deg

n M i=1

Li =

n X i=1

d Li , deg

36 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves n n M X d + Li + η( deg d Li )), h0θ ( Li ) = ( deg i=1

and

(2.5.2)

i=1

n n M X d − Li + η( deg d Li )). h1θ ( Li ) = ( deg i=1

(2.5.3)

i=1



d + Li and deg d − Li the positive and negative parts of We have denoted by deg d the real number deg Li , defined by d ± Li := 1 (| deg d Li | ± deg d Li ). deg 2 Observe the similarity of (2.5.2) and (2.5.3) with the expressions for the dimensions of the coherent cohomology groups of a direct sum of line bundles over an elliptic curve.

2.6

The Theta Function θE and the First Minimum λ1 (E)

In this section, we consider Hermitian vector bundles over Spec Z — in other words, Euclidean lattices — and we discuss some relations between their θ-invariants and the invariants classically associated to them in the geometry of numbers, such as their first minimum or their number of lattice points in the unit ball. Most of the content of this section is due to Groenewegen [55]. We shall pursue our study of the relations between θ-invariants and more classical invariants attached to Euclidean lattices in Section 3.2 below, where we shall reformulate Banaszczyk’s results [7]. In particular, in paragraph 3.2.2, we shall compare Groenewegen’s and Banaszczyk’s results relating the first minimum of a Euclidean lattice with its θ-invariants. 2.6.1 The Theta Function θE . For every Hermitian vector bundle E := (E, k.k) over Spec Z, it will be convenient to consider the associated theta function, defined for all t in R∗+ by X 2 θE (t) := e−πtkvk . v∈E

We have, by the definition of

h0θ (E),

h0θ (E) = log θE (1). Moreover, for all δ in R, if we denote by O(δ) the Hermitian line bundle d O(δ) = δ (already introduced in (Z, e−δ |.|) over Spec Z of Arakelov degree deg

2.6. The Theta Function θE and the First Minimum λ1 (E)

37

paragraph 1.1.3), then θE⊗O(δ) (t) =

X

−2δ

e−πte

kvk2

= θE (e−2δ t).

v∈E

Consequently, h0θ (E ⊗ O(δ)) = log θE (e−2δ ), and for all t ∈ R∗+ , log θE (t) = h0 (E ⊗ O(−(log t)/2)). In other words, log θE is a kind of “arithmetic Hilbert function” for the Hermitian vector bundle E over Spec Z. It turns out to be a convex function on R∗+ . Indeed, a straightforward computation shows that 00 0 θE (t)2 d2 log θE (t)/dt2 = θE (t)θE (t) − θE (t)2

!2 =

X

e

−πtkv|2

v∈E

X

2

4 −πtkv|2

π kvk e

−

v∈E

X

2 −πtkv|2

πkvk e

,

v∈E

and this is nonnegative by the Cauchy–Schwarz inequality (it is actually positive when rk E > 0). Observe also that the classical functional equation for the theta function may be written 1 θE (t) = t− 2 rk E (covol E)−1 θE ∨ (t−1 ), (2.6.1) or equivalently, 1 d E + log θ ∨ (t−1 ), log θE (t) = − rk E · log t + deg E 2

(2.6.2)

which is nothing but the Poisson–Riemann–Roch formula (2.2.3) for the Hermitian vector bundle E ⊗ O(−(log t)/2). The theta function θE may be related to the “naive” counting function NE : R+ −→ N, which controls the number of points in the Euclidean lattice E in balls centered at the origin. Indeed, if we define, for every x ∈ R+ , NE (x) := |{v ∈ E | kvk ≤ x}|, √ then the theta function θE is basically the Laplace transform of NE ( .): Proposition 2.6.1. With the above notation, for every t ∈ R∗+ , we have Z θE (t) = πt

0

+∞

√ NE ( x)e−πtx dx.

38 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves Proof. Observe that for all x ∈ R+ , X √ 1[kvk2 ,+∞[ (x) = NE ( x). v∈E

Consequently, we have θE (t) :=

X

=

X

e−πtkvk

2

v∈E +∞

Z

1[kvk2 ,+∞[ (x)e−πtx dx

πt 0

v∈E

Z

+∞

= πt 0

√ NE ( x)e−πtx dx.

(The last equality follows, for instance, from Lebesgue’s monotone convergence theorem.)  2.6.2 First minima of Euclidean lattices and θ-invariants. For every Euclidean lattice E := (E, k.k) of positive rank, we may consider the first of its successive minima, λ1 (E) := min {kvk, v ∈ E \ {0}} , and the number of its “short vectors”, ν := {v ∈ E | kvk = λ1 (E)} . These quantities are easily seen to control the asymptotic behavior of θE (t) as t goes to +∞, or equivalently, that of h0θ (E ⊗ O(δ)) as δ goes to −∞. Indeed, as t goes to +∞, 2 02 θE (t) = 1 + νe−πλ1 (E) t + O(e−πλ t ), for some λ0 > λ1 (E). Consequently, for some positive ε, 2

log θE (t) = νe−πλ1 (E) t (1 + O(e−εt )) as t goes to +∞ and h0θ (E ⊗ O(δ)) = νe−πλ1 (E)

2 −2δ

e

(1 + O(e−εe

−2δ

)) as δ goes to −∞.

(2.6.3)

It is actually possible to derive an explicit upper bound on h0θ (E) in terms of λ1 (E): Proposition 2.6.2. For every Euclidean lattice E of positive rank n and first minimum λ := λ1 (E), the following estimates hold: 0

h0θ (E) ≤ ehθ (E) − 1 ≤ C(n, λ),

(2.6.4)

2.6. The Theta Function θE and the First Minimum λ1 (E) where n

2 −n/2

39

+∞

Z

un/2 e−u du.

C(n, λ) := 3 (πλ )

(2.6.5)

πλ2

Moreover, if λ > (n/2π)1/2 , then  C(n, λ) ≤ 3n 1 −

n −1 −πλ2 e . 2πλ2

(2.6.6) 

This is a slightly improved version of Proposition 4.4 in [55], and we will follow Groenewegen’s arguments — based on the expression of the theta function √ θE as the Laplace transform of NE ( .) — with minor modifications. Observe that the estimates (2.6.4) and (2.6.6) applied to E ⊗ O(δ), compared with the asymptotic expression (2.6.3) as δ goes to −∞, show that ν ≤ 3n . Indeed, it is classically known1 that ν ≤ 2(2n − 1). ˜ by Observe also that if we define λ r n ˜ λ, λ1 (E) = 2π then the upper bounds (2.6.4)–(2.6.6) may be written  n ˜ −2 )−1 3e−λ˜ 2 /2 ˜ > 1. h0θ (E) ≤ (1 − λ when λ Lemma 2.6.3 (cf. [55, Lemma 4.2]). With the notation of Proposition 2.6.2, for all x in R+ , we have  n 2x NE (x) ≤ +1 . (2.6.7) λ Consequently, for all x in [λ, +∞[,  NE (x) ≤

3x λ

n .

(2.6.8) ◦

Proof. For all P ∈ ER and r ∈ R+ , let us denote by B(P, r) the open ball of center P and radius r in the normed vector space (ER , k.k). We shall also denote by λE the normalized Lebesgue measure on ER defined by the Euclidean norm k.k (see 2.1.1). For every two points v and w of E, we have ◦

◦

v 6= w =⇒ B (v, λ/2) ∩ B (w, λ/2) = ∅. 1 Since each “short vector” of E is also a facet vector — that is, a vector v in E \ {0} such that the Voronoi domains of 0 and v have a nonempty intersection of dimension n − 1 — and, as shown by Minkowski, the number of facet vectors of a Euclidean lattice of rank n is at most 2(2n − 1).

40 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves Consequently, 



◦

X

λE (B (v, λ/2)) = λE 

v∈E,kvk≤x

◦

[

◦

B (v, λ/2) ≤ λE (B (0, x + λ/2)).

v∈E,kvk≤x

If vn denotes the volume of the n-dimensional unit ball, this estimate may be written vn NE (x)(λ/2)n ≤ vn (x + λ/2)n . This is clearly equivalent to (2.6.7).



Lemma 2.6.4. For every positive integer n and all β ∈]n/2, +∞[, we have  −1 Z +∞ n n/2 −u u e du ≤ 1 − β n/2 e−β . 2β β n Proof. The derivative 2u − 1 of n2 log u − u is bounded from above by u belongs to [β, +∞[. Therefore, for all u in this interval, n  n n log u − u ≤ log β − β + (u − β)( − 1). 2 2 2β

Consequently, we have Z +∞ Z n/2 −u u e du = β

+∞

exp

n 2

β

n

n 2β

− 1 when

 log u − u du Z

+∞

e−(1−n/2β)(u−β) du 2 β n  n −1 log β − β 1 − β . = exp 2 2

≤ exp

log β − β



Proof of Proposition 2.6.2. From the definition of h0θ (E), Proposition 2.6.1, and the fact that NE (x) = 1 if x ∈ [0, λ[, we get 0

ehθ (E) − 1 = θE (1) − 1 = π Z ≤

Z

+∞

0 +∞

πλ2

√ [NE ( x) − 1]e−πx dx

p NE ( u/π)e−u du.

(2.6.9)

Moreover, the upper bound (2.6.8) on NE over [λ, +∞[ shows that  n Z +∞ Z +∞ p 3 −u √ NE ( u/π)e du ≤ un/2 e−u du. λ π πλ2 πλ2 This establishes (2.6.4). Finally, (2.6.6) follows from the integral estimate in Lemma 2.6.4, applied with β = πλ2 . 

41

2.7. Application to Hermitian Line Bundles

By duality, the upper bound on h0θ (E) in terms of λ1 (E) in Proposition 2.6.2 ∨ may be used to derive an upper bound on h1θ (E) := h0θ (E ) in terms of the last of its successive minima, namely λrk E (E) := min{r ∈ R+ | B(0, r) ∩ E contains a basis of ER }. Indeed, for every Euclidean lattice E of positive rank, the following elementary “transference estimate” holds: ∨

λ1 (E ) · λrk E (E) ≥ 1. ∨

(Consider a “short vector” ξ of E ∨ such that kξk∨ = λ1 (E ) and some element v E in B(0, λrk E (E)) ∩ E such that hξ, vi = 6 0 and observe that since hξ, vi is an integer, ∨

1 ≤ |hξ, vi| ≤ kξkE ∨ kvkE ≤ λ1 (E ) · λrk E (E).) We therefore immediately derive from Proposition 2.6.2 the following corollary: Corollary 2.6.5. For a Euclidean lattice E of positive rank n and last minimum λn (E), the following estimate holds: 1

h1θ (E) ≤ ehθ (E) − 1 ≤ C(n, λn (E)−1 ),

(2.6.10)

where C(n, .) is defined by (2.6.5). Moreover, if r n ˜ λn := λn (E) < 1, 2π then

2.7

 n ˜ 2 )−1 3 e−λ˜ −2 n /2 C(n, λn (E)−1 ) ≤ (1 − λ . n

(2.6.11) 

Application to Hermitian Line Bundles

Let L be a Hermitian line bundle over Spec OK . 2.7.1 The first minimum of the direct image of a Hermitian line bundle. The direct image π∗ L is a Euclidean lattice of rank n := [K : Q] over Spec Z. Its first minimum λ1 (π∗ L) admits a simple lower bound in terms of the normalized Arakelov degree 1 d L := dL deg deg n [K : Q]

42 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves of L. Indeed, as shown in [55, Lemma 7.1] or [20, Proposition 3.3.1], we have λ1 (E)2 d ≥ e−2degn L . n

(2.7.1)

2 d L < 1 log(2π) =⇒ λ1 (E) > 1 . deg n 2 n 2π

(2.7.2)

In particular, we have

d L < (1/2) log(2π), we Therefore, for every Hermitian line bundle satisfying deg n may apply Proposition 2.6.2 and derive an upper bound on h0θ (L) := h0θ (π∗ L). 2.7.2 Hermitian line bundles of negative degree. from (2.7.1) we obtain λ1 (E)2 ≥ n,

d L ≤ 0, For instance, when deg

and the upper bounds (2.6.4) and (2.6.6) become 0

2

ehθ (L) − 1 ≤ 3n (1 − 1/2π)−1 e−πλ1 (E) .

(2.7.3)

Using (2.7.1) again, we finally obtain the following minor variant of [55, Proposition 7.2] (which, in a slightly less precise form, already appears as Corollary 1 to Proposition 2 in [116]): Proposition 2.7.1. For every Hermitian line bundle L over Spec OK such that d L ≤ 0, we have deg   d h0θ (L) ≤ 3[K:Q] (1 − 1/2π)−1 exp −π[K : Q]e−2degn L . (2.7.4) The right-hand side of (2.7.4) is always ≤ 1, and actually goes to zero when d L goes to +∞. Indeed, from (2.7.4), we immediately obtain [K : Q] or − deg   d h0θ (L) ≤ c[K:Q] exp −π[K : Q](e−2degn L − 1) , (2.7.5) where c := 2.7.3

3e−π = 0.154180 . . . . 1 − 1/2π

Hermitian line bundles of positive degree.

Proposition 2.7.2. There exists c ∈]0, 1[ such that for every number field K and d L ≥ 0, we have Hermitian line bundle L over Spec OK such that deg d L. h0θ (L) ≤ c[K:Q] + deg

(2.7.6)

2.7. Application to Hermitian Line Bundles

43

The proof will show that we may choose the same constant c as in (2.7.5). Clearly, Proposition 2.7.2 establishes the validity, for every Hermitian line bundle of nonnegative degree over Spec OK , of the following inequality, familiar in a geometric context: d L. h0θ (L) ≤ 1 + deg (2.7.7) Proposition 2.7.2 improves on [55, Proposition 7.3], where a similar upper bound on h0θ (L) is established, with a constant of order (1/2)[K : Q] log[K : Q] instead of c[K:Q] . d L = 0, the estimate (2.7.6) takes the form Proof of Proposition 2.7.2. When deg h0θ (L) ≤ c[K:Q] and follows from (2.7.5). To derive the general validity of (2.7.6) from this special case, we may use the inequality (2.3.12) established in Corollary 2.3.5. Indeed, if L is a Hermitian line bundle over Spec OK of nonnegative Arakelov degree and if d L, d L := [K : Q]−1 deg δ = deg n then the Hermitian line bundle OSpec OK (δ) satisfies d L. d OSpec O (δ) = deg deg K Therefore, d (L ⊗ O(−δ)) = 0 deg and, according to the special case above, h0θ (L ⊗ O(−δ)) ≤ c[K:Q] .

(2.7.8)

Furthermore, according to (2.3.12), h0θ (L) ≤ h0θ (L ⊗ O(−δ)) + [K : Q]δ. The inequality (2.7.6) follows from (2.7.8) and (2.7.9).

(2.7.9) 

2.7.4 A scholium. For later reference, we spell out the following straightforward consequences of the upper bounds on h0θ (L) established in the previous paragraphs: Proposition 2.7.3. For every Hermitian line bundle L over Spec OK and t ∈ R d L ≤ t, we have such that deg h0θ (L) ≤ 1 + t if t ≥ 0, and

  h0θ (L) ≤ exp −π[K : Q](e−2t/[K:Q] − 1) ≤ exp(2πt) if t ≤ 0.



44 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves These estimates may be seen as an arithmetic counterpart of the basic inequality (2.7.10) h0 (C, L) := dimk Γ(C, L) ≤ (1 + degC L)+ , valid for every line bundle L over a smooth, projective, geometrically connected curve C over a field k, of degree degC L := degk c1 (L) · [C].

2.8

Subadditivity of h0θ and h1θ

2.8.1

The basic subadditivity property.

Proposition 2.8.1. For every admissible short exact sequence p

i

E : 0 −→ E −→ F −→ G −→ 0

(2.8.1)

of Hermitian vector bundles over the arithmetic curve Spec OK , the following inequality holds: hθ (E) := h0θ (E) − h0θ (F ) + h0θ (G) ≥ 0. (2.8.2) Moreover, equality holds in (2.8.2) if and only if the admissible short exact sequence (2.8.1) is split. Observe that the additivity of the Arakelov degree in short exact sequences (1.4.7), together with the Poisson–Riemann–Roch formula (2.2.6) shows that h1θ (E) − h1θ (F ) + h1θ (G) = h0θ (E) − h0θ (F ) + h0θ (G).

(2.8.3)

Accordingly, Proposition 2.8.1 also holds with h0θ replaced by h1θ . 2.8.2 Proof of Proposition 2.8.1. When OK = Z, Proposition 2.8.1 appears in Quillen’s notebooks [91] in the entry of 26/04/1973, and is established as Lemma 5.3 in [55]. The inequality (2.8.2) actually follows from the following lemma, of independent interest, which may be seen as a “pointwise version” of (2.8.2) and will play a key role in deriving the main results of this monograph (in particular to establish Lemma 7.5.1, crucial to the proof of Theorem 7.1.1 below): Lemma 2.8.2. Consider an admissible short exact sequence i

p

0 −→ E −→ F −→ G −→ 0 of Hermitian vector bundles over Spec Z. Then for every g ∈ G, its preimage p−1 (g) in F satisfies X 2 2 X 2 e−πkf kF ≤ e−πkgkG e−πkekE . (2.8.4) f ∈p−1 (g)

e∈E

2.8. Subadditivity of hθ0 and h1θ

45

Indeed, by summing (2.8.4) over g ∈ G, we get the inequality X X X 2 2 2 e−πkf kF ≤ e−πkgkG · e−πkekE . f ∈F

g∈G

e∈E

Taking the logarithm of both sides establishes (2.8.2) when OK = Z. The case of a general number field K follows from this special case, applied to the admissible short exact sequence over Spec Z derived from (2.8.1) by taking its direct image on Spec Z. Proof of Lemma 2.8.2. Let us denote by s⊥ : GR −→ FR the orthogonal splitting of the surjective linear map pR : FR −→ GR . (Its image is the orthogonal complement (ker pR )⊥ of ker pR in FR , defined by means of the Euclidean structure on FR attached to k.kF .) We may choose an element f0 in p−1 (g). Then we have p−1 (g) = f0 + i(E). The element f0 − s⊥ (g) of FR belongs to ker pR = im iR , and may be written iR (δ) for some (unique) element δ of ER . Then for all e ∈ E, we have kf0 + i(e)k2F = ks⊥ (g) + i(δ + e)k2F = kgk2G + kδ + ek2E . Together with (2.1.7) (applied to V = E and x = δ), this shows that X X 2 2 2 X 2 2 e−πkf kF = e−π(kgkG +kδ+ekE ) ≤ e−πkgkG e−πkekE . f ∈p−1 (g)

e∈E



e∈E

To complete the proof of Proposition 2.8.1, we are left to show that equality holds in (2.8.2) if and only if the admissible short exact sequence (2.8.1) over Spec OK is split. Observe that according to Proposition 1.4.2, i

p

π∗ E : 0 −→ π∗ E −→ π∗ F −→ π∗ G −→ 0 is an admissible extension of Hermitian vector bundles over Spec Z, and is split (over Spec Z) if and only if E is split (over Spec OK ). Moreover, by the definition of h0θ , we have hθ (E) = h0θ (E) − h0θ (F ) + h0θ (G) = h0θ (π∗ E) − h0θ (π∗ F ) + h0θ (π∗ G) = hθ (π∗ E). Therefore, to complete the proof of Proposition 2.8.1, we may assume that Spec OK = Spec Z. In this case, the result will follow from an analysis of the equality case in the above arguments.

46 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves To expound this analysis, it is convenient to introduce the following definition. Under the assumption that OK = Z, for every T in HomC (GC , EC )F∞ ' HomZ (G, E) ⊗ R ' HomR (GR , ER ), we define

2

X

GextE,G (T ) :=

2

e−π(ke−T (g)kE +kgkG ) .

(2.8.5)

(e,g)∈E×G

The proof of Proposition 2.8.1 is now completed by the following proposition, the formulation of which uses the formalism of arithmetic and admissible extensions recalled in paragraphs 1.4.4–1.4.6: Proposition 2.8.3. Consider an admissible short exact sequence of Euclidean lattices: p i E : 0 −→ E −→ F −→ G −→ 0. 1

d Z (G, E) coincides with the class For every T ∈ HomR (GR , ER ), whose class in Ext of the admissible extension E, the following equality holds: exp(−hθ (E)) =

GextE,G (T ) GextE,G (0)

.

(2.8.6)

Moreover we have GextE,G (T ) = covol(E)−1

X

e−π(ke

∨ 2 kE ∨ +kgk2G )

e2πihe

∨

⊗g,T i

(2.8.7)

(e∨ ,g)∈E ∨ ×G

and 1 ≤ GextE,G (T ) ≤ GextE,G (0),

(2.8.8)

and the equality GextE,G (T ) = GextE,G (0) holds if and only if T belongs to HomZ (G, E). On the right side of (2.8.7), e∨ ⊗ g belongs to E ∨ ⊗ G, and T to ∨ HomR (GR , ER ) ' G∨ R ⊗R ER ' (G ⊗ E)R ,

and their pairing he∨ ⊗ g, T i is equal to the real number e∨ (T (g)). Proof. From the definition of GextE,G (T ), we get T

h0θ (E ⊕ G ) = log GextE,G (T ). In particular, h0θ (E) + h0θ (G) = log GextE,G (0). The relation (2.8.6) follows from these two equalities.

2.8. Subadditivity of hθ0 and h1θ

47

The equality (2.8.7) follows from the Poisson formula (2.1.4) applied to V = E and x = T (g). The inequality GextE,G (T ) ≥ 1 is clear, and the inequality GextE,G (T ) ≤ GextE,G (0) follows from (2.8.7). Finally, the expression (2.8.7) for GextE,G (T ) shows that for all T in HomR (GR , ER ) ' HomZ (G, E) ⊗Z R, the following conditions are equivalent: 1. GextE,G (T ) = GextE,G (0); 2. for any (e∨ , g) ∈ E ∨ × G, e2πihe ∨

∨

∨

⊗g,T i

= 1;

∨

3. for any (e , g) ∈ E × G, he ⊗ g, T i ∈ Z; 4. T belongs to G∨ ⊗ E ' HomZ (G, E).



2.8.3 The average value of exp(−hθ (E)). Proposition 2.8.3 not only makes clear the nonnegativity of hθ (E). It also allows us to compute its “geometric average” 1 d Z (G, E). over the arithmetic extension group Ext The group 1 d Z (G, E) ' HomZ (G, E) ⊗Z R/Z Ext has indeed a natural structure of a compact Lie group (it is a “compact torus” of dimension rk E · rk F ), and as such, is equipped with a canonical Haar measure, normalized by the condition Z dµ = 1. 1 d (G,E) Ext Z

Proposition 2.8.4. Let E and G be two Hermitian vector bundles over Spec Z. For 1 d Z (G, E) and let all T ∈ HomR (GR , ER ), let [T ] denote its class in Ext i

T

p

E(T ) : 0 −→ E −→ E ⊕ G −→ G −→ 0 be the associated admissible extension of class [T ] (see 1.4.6 and (1.4.10) above). Then we have Z    −h1θ (E) −h0θ (G) −hθ (E(T )) e dµ([T ]) = 1 − 1 − e 1 − e . (2.8.9) 1 d (G,E) [T ]∈Ext Z

Proof. According to (2.8.6), we have Z Z −1 −hθ (E(T )) e (0) dµ([T ]) = Gext E,G 1 d (G,E) [T ]∈Ext Z

1

GextE,G (T ) dµ([T ]).

d (G,E) [T ]∈Ext Z

(2.8.10)

48 Chapter 2. θ-Invariants of Hermitian Vector Bundles over Arithmetic Curves Moreover, from (2.8.7), we get ∨ 2 kE ∨ +kgk2G )

X

GextE,G (0) = covol(E)−1

e−π(ke

(2.8.11)

(e∨ ,g)∈E ∨ ×G 1

0

= covol(E)−1 ehθ (E) ehθ (G) ,

(2.8.12)

GextE,G (T ) dµ([T ])

(2.8.13)

and Z d 1 (G,E) [T ]∈Ext Z

X

= covol(E)−1

e−π(ke

∨ 2 kE ∨ +kgk2G )

(e∨ ,g)∈E ∨ ×G e∨⊗g=0

 = covol(E)−1 

 X

e−πke

∨ 2 kE ∨

e∨ ∈E ∨



= covol(E)−1 e

h1θ (E)

+

X

2

e−πkgkG − 1

g∈G

+e

h0θ (G)



−1 .

Formula (2.8.9) follows from (2.8.10), (2.8.11), and (2.8.14).

(2.8.14) 

Corollary 2.8.5. When E and G have positive rank, there exists an admissible b by E b such that extension E of G h   i 1 0 hθ (E) > − log 1 − 1 − e−hθ (E) 1 − e−hθ (G) . Proof. When E and G have positive rank, exp(−hθ (E(T )) defines a nonconstant 1 1 d Z (G, E). There exists some point of Ext d Z (G, E) continuous function of [T ] ∈ Ext where it assumes a value larger than its mean value given by (2.8.9).  Proposition 2.8.4 and Corollary 2.8.5 should be compared to the upper bound hθ (E) ≤ min(h0θ (G), h1θ (E)), b by E, b which follows from valid for every admissible admissible extension E of G the relations (2.3.3), (2.3.4), and (2.8.2) (see also paragraph 3.3.1 below).

Chapter 3

Geometry of Numbers and θ-Invariants In this chapter, we pursue our study of the invariants hiθ (E) attached to a Hermitian vector bundle E over an arithmetic curve Spec OK . We focus on the basic situation in which OK = Z — that is, when E is a Euclidean lattice — and we relate the θ-invariants h0θ (E) and h1θ (E) to various invariants of E classically considered in the geometry of numbers. The results of the present chapter will not be used in the next chapters of this monograph. They are intended to clarify the meaning of the θ-invariants from the perspective of the theory of Euclidean lattices, and their proofs will combine three lines of thought: (i) The methods introduced by Banaszczyk in his work [7] to establish “transference inequalities”, relating invariants of Euclidean lattices and their dual lattices, with essentially optimal constants.1 (ii) An extended version of Cram´er’s theory of large deviations, valid on a measure space of infinite total mass. This theory is presented in Appendix A, and the reader will find a self-contained summary of its results required for our study of Euclidean lattices in Section A.5, where they are stated in a form that emphasizes their relations to the formalism of statistical physics. (iii) Siegel’s mean value theorem and its use for showing the existence of Euclidean lattices with remarkable density properties (`a la Minkowski-Hlawka) by probabilistic arguments. As explained in the introduction, Banaszczyk’s techniques allow us to establish in particular that the invariants h0θ (E) and h0Ar (E) of certain Euclidean lattices E differ by some error term bounded in terms of the rank of E only (see (0.2.1) and Theorem 3.1.1). Although these techniques were developed for the investigation of classical invariants of the geometry of numbers, independently of the study of the θ-invariants, they constitute a tool of choice in the study of the latter. Conversely, the use of θ-invariants sheds some light on Banaszczyk’s arguments, and in Section 3.2, we give a self-contained presentation of some of the most important results in [7] from this perspective. 1 See [47, Section 3] for an early application of these methods to Arakelov geometry. Banasczyk’s results have played an important role in lattice-based cryptography, and we refer the reader to [84] and [114] for improvements and applications of the original results in [7] related to lattice-based cryptography, and for further references.

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0_3

49

Chapter 3. Geometry of Numbers and θ-Invariants

50

The large deviation theorems in Appendix A admit as consequences the du˜ 0 (E, t) := ality relations (0.2.3) and (0.2.4) between the asymptotic invarianth Ar ⊕n 1 0 limn→+∞ n hAr (E , nt) and the function log θE . Moreover, their thermodynamic interpretation leads us to the expression   ˜ 0 (E 1 ⊕ E 2 , t) = max h ˜ 0 (E 1 , t1 ) + h ˜ 0 (E 2 , t2 ) h Ar Ar Ar t1 ,t2 >0

t1 +t2 =t

˜ 0 (E 1 ⊕ E 2 , t) attached to the direct sum E 1 ⊕ E 2 for the asymptotic invariant h Ar of two Euclidean lattices E 1 and E 2 . This expression may actually be understood as an avatar of the second law of thermodynamics (see Proposition 3.4.6). ˜ 0 (E, t) admits the following interpretation In the same vein, we show that h Ar in terms of “information-theoretic entropy”. Let us define n o X C := p ∈ [0, 1]E | p(e) = 1 e∈E

and, for all p ∈ C, I(p) := −

X

p(e) log p(e).

e∈E

Then

( ) X 2 0 ˜ hAr (E, t) = max I(p) ; p ∈ C and p(e)kek = t e∈E

(see Proposition 3.4.7 and (3.4.14)). The probabilistic arguments based on Siegel’s mean value theorem will establish the existence, for all (n, δ) in N≥2 × R, of a Euclidean lattice E of rank n and degree δ such that h0θ (E) < log(1 + eδ ). These probabilistic arguments will also show that the constants in the comparison estimates established in this chapter are essentially optimal.

3.1

Comparing h0θ and h0Ar

3.1.1

The invariants h0Ar (E), h0Ar− (E), and h0Bl (E). Following [51], we define h0Ar (E) := log |{v ∈ E | kvk ≤ 1}|.

This definition is motivated by the classical philosophy in Arakelov geometry (see, for instance, [82, Section 2.3]) according to which the finite set {v ∈ E | kvk ≤ 1}

0 3.1. Comparing h0θ and hAr

51

should be interpreted as the “global sections” of the Hermitian vector bundle E over Spec Z “completed” by its archimedean place. It turns out that the invariant h0Ar (E), defined as above as the logarithm of the number of points in the unit ball of the Euclidean lattice E and the θ-invariant h0θ (E) coincide up to an error term bounded by a function of rk E only: Theorem 3.1.1. For every Euclidean lattice of positive rank E, we have 1 h0θ (E) − rk E · log rk E + log(1 − 1/2π) ≤ h0Ar (E) ≤ h0θ (E) + π. (3.1.1) 2 We shall actually establish a slightly stronger version of the first inequality in (3.1.1), namely2 1 h0Ar− (E) := log |{v ∈ E | kvk < 1}| ≥ h0θ (E) + log(1 − 1/2π) − rk E · log rk E. 2 (3.1.2) Together with the “Riemann inequality” over Spec Z (2.3.2), inequality (3.1.2) entails the following strengthened variant of Minkowski’s first theorem: 1 d E. h0Ar− (E) ≥ log(1 − 1/2π) − rk E · log rk E + deg 2 Corollary 3.1.2. For every Euclidean lattice of positive rank E, 1 λ1 (E) ≥ 1 =⇒ h0θ (E) ≤ − log(1 − 1/2π) + rk E · log rk E. 2 Proof. The condition λ1 (E) ≥ 1 is equivalent to the equality h0Ar− (E) = 0.8



Observe that from Proposition 2.6.2, we also obtain a lower bound on h0θ (E) for every Euclidean lattice E with n := rk E > 0 and λ1 (E) ≥ 1, namely   Z √ n +∞ n/2 −u 0 hθ (E) ≤ log(1 + C(n, 1)) = log 1 + (3/ π) u e du . π

This estimate is slightly weaker than the one in Corollary 3.1.2. Indeed, as n goes to +∞, its right-hand side is of the form 12 n · log n + α · n + o(n) with α > 0. We may also introduce a variant `a la Blichfeldt of h0Ar (E), namely h0Bl (E) := log max |{v ∈ E | kv − xk ≤ 1}| x∈ER

(see paragraph 3.3.2 below). It is straightforward that h0Ar− (E) ≤ h0Ar (E) ≤ h0Bl (E), and we shall establish the following strengthening of the second inequality in (3.1.1): Proposition 3.1.3. For every Euclidean lattice of positive rank E, we have h0Bl (E) ≤ h0θ (E) + π. 2 Since

h0Ar− (E)

h0Ar (E

= ⊗ O(−ε)) for every small enough ε in from (3.1.1) applied to E ⊗ O(−ε), by taking the limit ε → 0+ .

(3.1.3) R∗+ ,

(3.1.2) actually follows

Chapter 3. Geometry of Numbers and θ-Invariants

52

3.1.2 Proof of Proposition 3.1.3 and Theorem 3.1.1. Let E be a Euclidean lattice of positive rank. To prove Proposition 3.1.3, namely that h0Bl (E) ≤ h0θ (E) + π, we simply observe that from the definition of h0θ (E) and the inequality (2.1.7), we have for all x ∈ ER , X X X 2 2 2 h0θ (E) = log e−πkvk ≥ log e−πkv−xk ≥ log e−πkv−xk , v∈E

v∈E

v∈E,kv−xk≤1

and that the last sum is at least e−π · |{v ∈ E | kv − xk ≤ 1}|. The second inequality in (3.1.1), namely h0Ar (E) ≤ h0θ (E) + π, follows from the special case x = 0 (which does not require the use of (2.1.7)) of the previous argument. We split the proof of (3.1.2) into a succession of auxiliary statements, of independent interest. The following assertions are variants of results in [7, Section 1]. Lemma 3.1.4. (1) The expression log θE (t) defines a decreasing function of t in R∗+ , and the expression 1 log θE (t) + rk E · log t 2

(3.1.4)

an increasing function of t in R∗+ . (2) We have X

2

kvk2 e−πtkvk ≤

v∈E

rk E X −πtkvk2 e . 2πt

(3.1.5)

v∈E

(3) For all t and r in R∗+ , we have X v∈E,kvk (n/2π)1/2 , obtained by means of Banaszczyk’s methods, with the upper bound derived in Proposition 2.6.2 using Groenewegen’s argument, namely  −1 2 n n h0θ (E) e −1≤3 1− e−πλ1 (E) . (3.2.7) 2 2πλ1 (E)

Chapter 3. Geometry of Numbers and θ-Invariants

56

To achieve this comparison, observe that  −n/2 2 ˜ −(1/2)(λ˜ 2 −1) ]n = n [λe λ1 (E)n e−πλ1 (E) . 2πe

(3.2.8)

This shows that as λ1 (E) goes to +∞ and n is fixed, Groenewegen’s bound (3.2.7) is better than (3.2.6) by a factor  n  n n/2 3 n −n ˜ −n . 3 λ1 (E) = √ λ 2πe e Moreover, for every fixed value of n, Groenewegen’s bound is also p better than Banaszczyk’s as λ1 (E) goes to (n/2π)1/2 . However, when λ1 (E) = n/2 or ˜ = √π, Banaszczyk’s bound improves on Groenewegen’s by a factor equivalently λ (1 − π −1 )

 πe n/2 9

as n goes to infinity. (Observe that πe/9 = 0.9488 . . . < 1.) 3.2.3 Covering radius and Banaszczyk’s transference estimate. At this stage, we can easily recover Banaszczyk’s transference estimate relating the first minimum and the covering radius of a Euclidean lattice and of its dual. Recall that the covering radius of a Euclidean lattice E of positive rank is defined as the positive real number ( ) [ ◦ + ρ(E) := max min kv − xk = inf r ∈ R | B(v, r) = ER x∈ER v∈E

v∈E

and that Banaszczyk’s transference estimate is the second estimate in the following proposition: Proposition 3.2.4 ([7, Theorem 2.2]). For every Euclidean lattice E of positive rank n, we have ∨ 1/2 ≤ ρ(E) λ1 (E ) ≤ n/2. ∨

The first inequality 1/2 ≤ ρ(E).λ1 (E ) is elementary4 . To establish the second one, we first derive another corollary of Proposition 3.2.2: Corollary 3.2.5.pLet E be a Euclidean lattice of positive rank n and of covering radius ρ(E) ≥ n/2π. If we define ρ˜ in [1, +∞[ by the relation r n ρ(E) = ρ˜, 2π 4 Simply

∨

consider a “short vector” ξ of E , an element x ∈ ER such that ξ(x) = 1/2, and v ∈ E ∨ such that kv − xk ≤ ρ(E), and observe that ρ(E) · λ1 (E ) ≥ kξk.kv − xk ≥ |ξ(v) − ξ(x)| ≥ 1/2.

57

3.2. Banaszczyk’s Estimates and θ-Invariants then there exists x in ER such that P −πkv−xk2 2 v∈E e P ≤ [˜ ρe−(ρ˜ −1)/2 ]n . 2 −πkvk v∈E e

(3.2.9)

Proof. By the definition of ρ(E), there exists x in ER such that kv − xk ≥ ρ(E) for every v in E. For this choice of x, (3.2.9) follows from (3.2.4) applied with  r = ρ(E). The following lemma is a straightforward reformulation of already established properties of h0θ . Lemma 3.2.6. Let E be a Euclidean lattice of positive rank n. (1) For every x in ER , we have P −πkv−xk2 ∨ 0 v∈E e P ≥ 2e−hθ (E ) − 1. −πkvk2 e v∈E (2) Assume that the first minimum of the dual lattice E p ˜ ∨ in [1, +∞[ by the relation n/2π. If we define λ r n ˜∨ ∨ λ1 (E ) = λ , 2π then we have

0

2e−hθ (E

∨

)

∨

(3.2.10) ∨

satisfies λ1 (E ) ≥

˜ ∨ e−(λ˜ ∨2 −1)/2 ]n . − 1 ≥ 1 − 2[λ

(3.2.11)

Proof. (1) According to (2.1.8), P −πkv−xk2 2 v∈E e P ≥ − 1. P −πkvk2 −πkvk2 e (covol E) v∈E v∈E e We conclude the proof of (3.2.10) using the Poisson formula (2.1.6) and the defi∨ nition of h0θ (E ). (2) To prove (3.2.11), we just apply the bound (3.2.5) to the dual lattice ∨ E .  From Corollary 3.2.5 and Lemma 3.2.6, we easily derive Banaszczyk’s transference estimate in the following more precise form: ∼

Proposition 3.2.7. Let ψ : [1, +∞[−→]0, 1] be the decreasing homeorphism defined 2 by ψ(t) := te−(t −1)/2 , and for every positive integer n, let tn := ψ −1 (3−1/n ). Then we have p (3.2.12) tn = 1 + log 3/n + O(1/n) as n −→ +∞. Moreover, for every Euclidean lattice E of positive rank n, we have ∨

ρ(E) λ1 (E ) ≤

t2n n . 2π

(3.2.13)

Chapter 3. Geometry of Numbers and θ-Invariants

58

Observe that √ √ tn ≤ π ⇐⇒ ψ(tn ) ≥ ψ( π) √ ⇐⇒ 3−1/n ≥ π exp(−(π − 1)/2) ⇐⇒ −(log 3)/n ≥ −(π − 1)/2 + (log π)/2.

(3.2.14)

Since log 3 = 1.0986 . . .

and (π − 1)/2 − (log π)/2 = 0.4984 . . . ,

the above inequalities hold for every integer n ≥ 3. ∨ Consequently, Proposition 3.2.7 implies Banaszczyk’s inequality ρ(E) λ1 (E ) ≤ n/2 when n ≥ 3. This inequality is trivial when n = 1. When n = 2, it follows ∨ from elementary considerations involving reduced bases of E and E . Proof of Proposition 3.2.7. As x goes to zero, we have ψ(1 + x) = (1 + x) exp(−(x + 1)2 /2 + 1/2) = exp(log(1 + x) − x − x2 /2) 2

(3.2.15)

3

= 1 − x + O(x ). Therefore, as y ∈ R+ goes to zero, ψ −1 (1 − y) = 1 +

√

y + O(y).

(3.2.16)

Moreover, as n goes to +∞, 3−1/n = exp(−(log 3)/n) = 1 − (log 3)/n + O(n−2 ).

(3.2.17)

The asymptotic (3.2.12) for tn follows from (3.2.16) and (3.2.17). Let us consider a Euclidean lattice E of positive rank n and let us define ρ˜ ˜ ∨ as in Corollary 3.2.5 and Lemma 3.2.6. From the estimates (3.2.9), (3.2.10), and λ and (3.2.11), it follows that ˜ ∨ are ≥ 1, then ψ(˜ ˜ ∨ )n ≤ 1. if ρ˜ and λ ρ)n + 2ψ(λ ˜ ∨ = t, then 3ψ(t)n ≤ 1 if Consequently, if for some t ∈ R∗+ , we have ρ˜ = λ t ≥ 1, and therefore t ≤ tn and p ∨ ρ(E) = λ1 (E ) ≤ tn n/2π. ∨

This establishes (3.2.13) when ρ(E) = λ1 (E ). To derive the general validity of (3.2.13) from this special case, simply observe that replacing the Euclidean lattice E by E ⊗ O(δ) for some δ ∈ R (that is, scaling ∨ the metric of E by a positive factor e−δ ) does not change the product ρ(E) λ1 (E ) and that by a suitable choice of δ, the condition ρ(E ⊗ O(δ)) = λ1 ((E ⊗ O(δ))∨ )

59

3.3. Subadditive Invariants of Euclidean Lattices

may be achieved. Indeed, from the definitions of the covering radius and the first minimum, we obtain ρ(E ⊗ O(δ)) = e−δ ρ(E) and ∨

λ1 ((E ⊗ O(δ))∨ ) = eδ λ1 (E ).

3.3



Subadditive Invariants of Euclidean Lattices

3.3.1 Alternating inequalities. With the notation of Proposition 2.8.1, one has the following series of alternating inequalities: h0θ (E) ≥ 0, h0θ (E) h0θ (E) h0θ (E) h0θ (E) h0θ (E)

− − − − −

(3.3.1)

h0θ (F ) h0θ (F ) h0θ (F ) h0θ (F ) h0θ (F )

≤ 0, + + + +

(3.3.2)

h0θ (G) h0θ (G) h0θ (G) h0θ (G)

≥ 0, − − −

(3.3.3)

h1θ (E) h1θ (E) h1θ (E)

≤ 0, + +

h1θ (F ) h1θ (F )

(3.3.4) ≥ 0, −

h1θ (G)

(3.3.5) = 0.

(3.3.6)

These are precisely the inequalities we should obtain if the hkθ s were dimensions of the spaces in a long exact cohomology sequence derived from the admissible short exact sequence i

p

0 −→ E −→ F −→ G −→ 0, which would vanish in cohomological degree k > 1. Indeed, the inequalities (3.3.1)–(3.3.3) have already been established. As already observed (see (2.8.3) above), thanks to the Poisson–Riemann–Roch formula (2.2.6), the equality (3.3.6) may be written d E − deg d F + deg d G = 0, deg which precisely expresses the additivity (1.4.7) of the Arakelov degree in admissible short exact sequences. Taking (3.3.6) into account, inequalities (3.3.4) and (3.3.5) are equivalent to the relations −h1θ (F ) + h1θ (G) ≤ 0 and h1θ (G) ≥ 0, which follow from Proposition 2.3.2 applied to ϕ = p and from Proposition 2.3.1. Observe that using again the Poisson–Riemann–Roch formula (2.2.6), the inequality (3.3.4) may be also written as d E + 1 log |∆K | · rk E. h0θ (G) ≤ h0θ (F ) − deg 2

(3.3.7)

60

Chapter 3. Geometry of Numbers and θ-Invariants

3.3.2 Blichfeldt pairs. Let us indicate that that the occurrence in geometry of numbers of “alternating inequalities”, similar to those satisfied by dimensions of cohomology groups has been observed by Gillet, Mazur, and Soul´e [50] in the context of the classical theorem of Blichfeldt. In that work, instead of Euclidean lattices E := (E, k.k) defined by a finitely generated free Z-module E and a Euclidean norm k.k on ER , the authors deal with so-called Blichfeldt pairs E := (E, B), defined by a Z-module E as above and a bounded Lebesgue measurable subset B in ER . They define h0 (E) := log max |E ∩ (v + B)| v∈ER

and h1 (E) := h0 (E) − log µ(B), where µ is the Haar measure on ER that gives a fundamental domain for E a measure equal to one. They define an exact sequence of Blichfeldt pairs p

i

0 −→ (E, B) −→ (F, C) −→ (G, D) −→ 0 as an exact sequence of Z-modules i

p

0 −→ E −→ F −→ G → 0 such that pR (C) and D (resp., p−1 R (pR (x)) ∩ C and iR (B) for all x ∈ C) coincide up to translation in GR (resp., in FR ). Then they establish the validity of (3.3.1)– (3.3.6), with hk instead of hkθ , for every short exact sequence of Blichfeldt pairs as above. It is also possible to define the direct sum of two Blichfeldt pairs E1 := (E1 , B1 ) and E2 := (E2 , B2 ) as E1 ⊕ E2 := (E1 ⊕ E2 , B1 × B2 ). Then the following additivity relations are easily established: hk (E1 ⊕ E2 ) = hk (E1 ) + hk (E2 ), for k = 0, 1. To every Euclidean lattice E = (E, k.k), we may attach a natural Blichfeldt pair, namely E := (E, B(E R )), where B(E R ) := {v ∈ ER | kvk ≤ 1}. Observe that, with the notation of Section 3.1, we have h0 (E) = h0Bl (E).

3.3. Subadditive Invariants of Euclidean Lattices

61

However, this construction of Blichfeldt pairs from Euclidean lattices is not compatible with short exact sequences or products of Euclidean lattices and Blichfeldt pairs. In fact, for every two Euclidean lattices E 1 and E 2 of positive ranks the Blichfeldt pair associated to their direct sum E 1 ⊕ E 2 cannot be expressed as the direct sum of two Blichfeldt pairs. This lack of compatibility prevents the association of invariants h0 and h1 to Euclidean lattices that would satisfy the alternating inequalities (3.3.1)–(3.3.6), reducing to the construction in [50]. 3.3.3 Concerning the subadditivity of h0Ar . At this point, it may be worth emphasizing that the subadditivity property (2.8.2) (or equivalently (3.3.3)) satisfied by h0θ does not hold when h0θ is replaced by h0Ar (contrary to what is claimed in [51, p. 356, Proposition 7, (i)]; see [52]). As shown by the following proposition, counterexamples may be obtained with E a small perturbation of the hexagonal rank-two lattice5 A2 : Proposition 3.3.1. For all λ ∈]0, 4[, let E λ be the Euclidean lattice defined by Eλ = Z2 inside Eλ,R = R2 equipped with the Euclidean norm k.kλ such that k(x, y)k2λ := λ(x2 − xy) + y 2 ,

(3.3.8)

and let F λ be the sub-Euclidean lattice of E λ defined by the Z-submodule Fλ := Z × {0} of Z2 . Then for all λ ∈]0, 4[, we have h0Ar (F λ ) = 0 ⇐⇒ λ > 1,

(3.3.9)

h0Ar (E λ /F λ ) ≤ log 3 ⇐⇒ λ < 3,

(3.3.10)

h0Ar (E λ ) ≥ log 5.

(3.3.11)

and Indeed, the Euclidean lattice E 1 is nothing but the hexagonal lattice A2 , and (3.3.9)–(3.3.11) show that for all λ ∈]1, 3[, h0Ar (E λ ) > h0Ar (F λ ) + h0Ar (E λ /F λ ). Proof of Proposition 3.3.1. Observe that the definition (3.3.8) of k.kλ may also be written k(x, y)k2λ = λ(x − y/2)2 + (1 − λ/4)y 2 . (3.3.12) The equivalence (3.3.9) follows from the fact that F λ may be identified with the lattice Z inside R equipped with the norm k.k such that k1k2 = λ. 5 Recall that A is defined as the lattice Z + Ze2πi/3 inside C (equipped with its usual absolute 2 value), or equivalently, as the lattice Z2 inside R2 equipped with the Euclidean norm k.kA2 defined by k(x, y)k2A2 := x2 − xy + y 2 .

Chapter 3. Geometry of Numbers and θ-Invariants

62

To establish (3.3.10), observe that as shown by (3.3.12), the orthogonal complement of Fλ,R in the Euclidean vector space (Eλ,R , k.kλ ) is the real line R(1/2, 1). Consequently, the norm in E λ /F λ of the class [(0, 1)] of (0, 1) is given by k[(0, 1)]k2E

λ /F λ

= k(1/2, 1)k2E = 1 − λ/4, λ

and accordingly, p k[(0, 2)]kE λ /F λ = 2 1 − λ/4. p Therefore, h0Ar (E λ /F λ ) ≤ log 3 if and only if 1 − λ/4 > 1/2, that is, if and only if λ < 3. Finally, the unit ball of E λ always contains the five lattice points (0, 0), (0, 1), (0, −1), (1, 1), and (−1, −1). This proves (3.3.11). 

3.4

˜ 0 (E, t) The Asymptotic Invariant h Ar

In a vein related to the discussion of subadditive invariants of Euclidean lattices in the previous section — in particular to the discussion of Blichfeldt pairs in 3.3.2 — it may be worth mentioning that h0Ar satisfies a superadditivity property that turns out to lead to another interpretation of the θ-invariants of Euclidean lattices and to relate h0Ar and h0θ to the thermodynamic formalism. ˜ 0 (E, t). To formulate the superadditiv3.4.1 The invariants h0Ar (E, t) and h Ar 0 ity of hAr , it is convenient to introduce a simple generalization of this invariant. Namely, for every Euclidean lattice E = (E, k.k) and positive real number t, we let  h0Ar (E, t) := log v ∈ E | kvk2 ≤ t (3.4.1) = h0Ar (E ⊗ O((log t)/2)). The following observation is straightforward: Lemma 3.4.1. For every two Euclidean lattices E 1 = (E1 , k.k1 ) and E 2 = (E2 , k.k2 ), and every two positive real numbers t1 and t2 , we have h0Ar (E 1 , t1 ) + h0Ar (E 2 , t2 ) ≤ h0Ar (E 1 ⊕ E 2 , t1 + t2 ).

(3.4.2)

Proof. This follows from the inclusion   v1 ∈ E1 | kv1 k21 ≤ t1 × v2 ∈ E2 | kv2 k22 ≤ t2 o n ⊂ v ∈ E1 ⊕ E2 | kvk2E 1 ⊕E 2 ≤ t1 + t2 .



˜ 0 (E, t) 3.4. The Asymptotic Invariant h Ar

63

In particular, for every Euclidean lattice E, the sequence (h0Ar (E is superadditive; namely, it satisfies, for every (n1 , n2 ) ∈ N2≥1 , h0Ar (E

⊕n1

, n1 t) + h0Ar (E

⊕n2

, n2 t) ≤ h0Ar (E

⊕(n1 +n2 )

⊕n

, (n1 + n2 )t).

, nt))n≥1

(3.4.3)

Moreover, this sequence grows at most linearly with n: Lemma 3.4.2. For every Euclidean lattice E, as n goes to +∞, h0Ar (E

⊕n

, nt) = O(n).

(3.4.4)

Proof. For every Euclidean lattice V := (V, k.k) and for all (P, r) ∈ VR × R+ , we ◦

shall denote by B V (P, r) the open ball of center P and radius r in the normed vector space (VR , k.k). We shall also denote by vn the volume of the n-dimensional unit ball. Recall that vn = π n/2 /Γ((n/2) + 1) and that consequently, as n goes to +∞, log vn = −(n/2) log n + O(n).

(3.4.5)

Let λ be the first minimum of E. Observe that for every two points v and w of E ⊕n ,     ◦ ◦ n n ∩ w + B E (0, λ/2) = ∅. v 6= w =⇒ v + B E (0, λ/2) (Compare with the proof of Lemma 2.6.3.) n o Furthermore, if E(n, t) := v ∈ E ⊕n | kvk2 ⊕n ≤ nt , then we have E



[

◦

n

v + B E (0, λ/2)

v∈E(n,t)

 ⊂

[

◦ ◦ √ √ √ B E ⊕n (v, λ n/2) ⊂ B E ⊕n (0, n( t+λ/2)).

v∈E(n,t)

(3.4.6) If e := rk E, we finally obtain, by considering the Lebesgue measures of the first and last sets in (3.4.6), √ √ ven (λ/2)n |E(n, t)| ≤ vne [ n( t + λ/2)]ne . (3.4.7) Finally, as n goes to +∞, from (3.4.7) and (3.4.5), we obtain h0Ar (E

⊕n

, nt) = log |E(n, t)| ≤ ne log

√

n + log vne + O(n) = O(n).



Recall that according to a well-known observation that goes back to Fekete [46], superadditive sequences of real numbers have a simple asymptotic behavior:

Chapter 3. Geometry of Numbers and θ-Invariants

64

Lemma 3.4.3. Let (an )n∈N≥1 be a sequence of real numbers that is superadditive 2 ). (namely, that satisfies an1 +n2 ≥ an1 + an2 for all (n1 , n2 ) ∈ N≥1 Then the sequence (an /n)n∈N≥1 admits a limit in ] − ∞, +∞]. Moreover, lim an /n = sup an /n.

n→+∞



n∈N≥1

Combined with the superadditivity property (3.4.3) and with the bound (3.4.4), Fekete’s lemma establishes the following result: Proposition 3.4.4. For every Euclidean lattice E and all t ∈ R∗+ , the limit ˜ 0 (E, t) := lim 1 h0 (E ⊕n , nt) h Ar Ar n→+∞ n exists in R+ . Moreover, we have ˜ 0 (E, t) := sup 1 h0 (E ⊕n , nt). h Ar Ar n∈N≥1 n



˜ 0 (E, t) defined in Proposition 3.4.4 is an “asymptotic verThe invariant h Ar sion” of the more naive invariant h0Ar (E, t). By its definition, its satisfies, for every positive integer k, ˜ 0 (E ⊕k , kt) = k h ˜ 0 (E, t), h Ar Ar and inherits the superaddivity property of h0Ar stated in Lemma 3.4.1; namely, with the notation of Lemma 3.4.1, we have ˜ 0 (E 1 , t1 ) + h ˜ 0 (E 2 , t2 ) ≤ h ˜ 0 (E 1 ⊕ E 2 , t1 + t2 ). h Ar Ar Ar

(3.4.8)

˜ 0 (E, t) and the Legendre transform of log θ . The invari3.4.2 The invariant h Ar E ˜ 0 (E, t) attached to a Euclidean lattice E, turns out as a function of t ∈ R∗ , ant h + Ar to be simply related to the theta function θE of E, and consequently to enjoy various properties — in particular, it is a real analytic function — that are not obvious from its original definition. To express this relation, recall that provided E has positive rank, the function log θE : R∗+ −→ R∗+ is a decreasing real analytic diffeomorphism that moreover is convex (see 2.6.1 above). In fact, the function UE := −(log θE )0 satisfies, for every β ∈ R∗+ , P UE (β) =

2

πkvk2 e−βπkvk , −βπkvk2 v∈E e

v∈E

P

˜ 0 (E, t) 3.4. The Asymptotic Invariant h Ar

65

and is easily seen to establish a decreasing real analytic diffeomorphism ∼

UE : R∗+ −→ R∗+ . Theorem 3.4.5. For every Euclidean lattice E of positive rank, the function ˜ 0 (E, .) is real analytic, increasing, concave, and surjective from R∗ to R∗ . h + + Ar Moreover, if we let ˜ 0 (E, x/π) (x ∈ R∗ ), SE (x) := h Ar +

(3.4.9)

then the functions −SE (−.) and log θE are Legendre transforms of each other. Namely, for every x ∈ R∗+ , ˜ 0 (E, x) = inf (πβx + log θ (β)), h Ar E

(3.4.10)

˜ 0 (E, x) − πβx). log θE (β) = sup(h Ar

(3.4.11)

β>0

and for every β ∈ R∗+ , x>0

0 Moreover, the derivative SE establishes a real analytic decreasing diffeomorphism ∼ 0 SE : R∗+ −→ R∗+ ,

the inverse of UE , and for every x ∈ R∗+ , the infimum on the right-hand side of (3.4.10) is attained for a unique value β, namely for 0 ˜ 0 (E, .)0 (x). β = SE (πx) = π −1 h Ar

Dually, for every β ∈ R∗+ , the supremum on the right-hand side of (3.4.11) is attained for a unique value of x, namely for x = π −1 UE (β). When E is the “trivial” Euclidean lattice of rank one O(0) := (Z, |.|), Theorem 3.4.5 may be deduced from results of Mazo and Odlyzko [83, Theorem 1]. In its general formulation above, Theorem 3.4.5 is a consequence of the extension of Cram´er’s theory of large deviations presented in Appendix A, concerning an arbitrary measure space (E, T , µ) and a nonnegative measurable function H on E. Theorem 3.4.5 gathers the results of paragraph A.5.1 (in particular Theorem A.5.1 and Corollary A.5.2) in the situation when X (E, T , µ) = (E, P(E), δv ) v∈E

— that is, the set E underlying the Euclidean lattice E equipped with the counting measure — and when H = πk.k2 .

66

Chapter 3. Geometry of Numbers and θ-Invariants

Indeed, in this situation, the function Ψ introduced in paragraph A.5.1 is nothing else than log θE , and the function S is simply the function SE defined by (3.4.9). One may also observe that specialized to this situation, the arguments in ˜ 0 (E, t) that Appendix A actually provide an alternative proof of the finiteness of h Ar relies on the properties of the theta functions of lattices and avoids the estimates in the proof of Lemma 3.4.2. 3.4.3 Euclidean lattices and thermodynamic formalism. The above construction, of data (E, T , µ) and H of the type considered in Appendix A from Euclidean lattices, isLclearly compatible with finite products: the data associated to the direct sum i∈I E i of a finite family (E i )i∈I of Euclidean lattices may be identified with the “product”, in the sense of paragraph A.5.2, of the data associated to each of the E i . This observation allows us to apply Proposition A.5.3 to analyze the invari˜ 0 of a direct sum of Euclidean lattices. In particular, we immediately obtain ants h Ar ˜0 : the following more precise form of the superadditivity (3.4.8) of h Ar Proposition 3.4.6. For every two Euclidean lattices of positive rank E 1 and E 2 and all t ∈ R∗+ ,   ˜ 0 (E 1 ⊕ E 2 , t) = max h ˜ 0 (E 1 , t1 ) + h ˜ 0 (E 2 , t2 ) . h (3.4.12) Ar Ar Ar t1 ,t2 >0

t1 +t2 =t

Moreover, the maximum is attained for a unique pair (t1 , t2 ), namely for 0 (πt).  (π −1 UE 1 (β), π −1 UE 2 (β)), where β := SE ⊕E 1

2

This proposition can be understood as an expression of the second law of thermodynamics in the context of Euclidean lattices. As a last element pleading for an interpretation of the θ-invariants of Euclidean lattices in terms of statistical thermodynamics, let us briefly translate to the framework of this section the results in the paragraph A.5.4 of Appendix A. Let us consider a Euclidean lattice of positive rank E := (E, k.k) and let us denote by n o X C := p ∈ [0, 1]E | p(e) = 1 e∈E

the space of probability measures on E. We may consider the functions ε (“energy”) and I (“information-theoretic entropy”) from C to [0, +∞] defined as follows: X ε(p) := p(e)πkek2 e∈E

and I(p) := −

X e∈E

p(e) log p(e).

˜ 0 (E, t) 3.4. The Asymptotic Invariant h Ar

67

Then Proposition A.5.5, applied to (E, T , µ) = (E, P(E), πk.k2 , becomes the following statement:

P

v∈E δv )

and H =

Proposition 3.4.7. Let u and β be two positive real numbers such that u = UE (β). For all p in C such that ε(p) = u, we have ˜ 0 (E, u/π). I(p) ≤ h Ar

(3.4.13)

Moreover, the equality is achieved in (3.4.13) for a unique p in ε−1 (u), namely for the measure pβ defined by 2

pβ (e) := θE (β)−1 e−πβkek .



˜ 0 (E, t), for all t ∈ R∗ : Proposition 3.4.7 provides the following expression of h + Ar n o X ˜ 0 (E, t) = max I(p) ; p ∈ C and h p(e)kek2 = t . (3.4.14) Ar e∈E

3.4.4 Duality and further comparison estimates. In this paragraph, we denote by E a Euclidean lattice of positive rank, and we derive additional estimates ˜ 0 (E, .), and h0 (E). comparing its invariants h0Ar (E, .), h Ar θ Ultimately, these estimates will appear as consequences of (i) the “thermodynamic” formalism of the previous paragraphs and (ii) the Poisson formula (2.6.2), which relates the theta functions θE and θE ∨ of E and its dual Euclidean lattice ∨ E . Indeed, from (2.6.2), we immediately get the following statement: Proposition 3.4.8. For all β ∈ R∗+ , we have d E, log θE (β) − log θE ∨ (β −1 ) = −(rk E/2) log β + deg

(3.4.15)

β UE (β) + β −1 UE ∨ (β −1 ) = rk E/2,

(3.4.16)

0 ≤ UE (β) ≤ rk E/(2β).

(3.4.17)

and  The expression rk E/(2β) that appears in the upper bound on UE (β) in (3.4.17) coincides with the function U (β) in the “Maxwellian” situation discussed in paragraph A.5.3. This upper bound was also the key point behind the estimates a la Banaszczyk derived in Lemma 3.1.4. ` For every integer n ≥ 1, we let C(n) := − sup [log(1 − t−1 ) − (n/2) log t]. t>1

Chapter 3. Geometry of Numbers and θ-Invariants

68 One easily shows that

C(n) = log(n/2) + (1 + n/2) log(1 + 2/n) and that 1 ≤ C(n) − log(n/2) ≤ (3/2) log 3. ∗ , we have Theorem 3.4.9. For all x ∈ R+

log θE (rk E/(2πx)) ≤ h0Ar (E, x) + C(rk E),

(3.4.18)

˜ 0 (E, x), log θE (rk E/(2πx)) ≤ h Ar

(3.4.19)

˜ 0 (E, x) ≤ log θ (rk E/(2πx)) + rk E/2. h Ar E

(3.4.20)

and The following consequence of Theorem 3.4.9, which involves only the “ele˜ 0 (E, t), seems worth mentioning: mentary” invariants h0Ar (E, t) and h Ar Corollary 3.4.10. For all x ∈ R∗+ , ˜ 0 (E, x) − h0 (E, x) ≤ C(rk E) + rk E/2. 0≤h Ar Ar

(3.4.21) 

Proof of Theorem 3.4.9. Let us begin with a straightforward consequence of Lemma 3.1.4 (after the change of notation x = r2 and β = t): Lemma 3.4.11. For all (x, β) ∈ R∗2 + such that βx > rk E/(2π), h0Ar (E, x) ≥ log(1 − rk E/(2πβx)) + log θE (β).

(3.4.22) 

From Lemma 3.4.11, we easily deduce the following: 0 Lemma 3.4.12. For all (x, β, β 0 ) ∈ R∗3 + such that βx ≥ rk E/(2π) and β > β,

h0Ar (E, x) ≥ log(1 − rk E/(2πβ 0 x)) − (rk E/2) log(β 0 /β) + log θE (β).

(3.4.23)

Proof. Lemma 3.4.11 implies that h0Ar (E, x) ≥ log(1 − rk E/(2πβ 0 x)) + log θE (β 0 ). Furthermore, we have log θE (β 0 ) ≥ log θE (β) − (rk E/2) log(β 0 /β).

(3.4.24)

This follows, for instance, from the upper bound in (3.4.17) on UE (β) := −(log θE )0 (β). 

˜ 0 (E, t) 3.4. The Asymptotic Invariant h Ar

69

To prove the inequality (3.4.18), we apply Lemma 3.4.12 with β := rk E/(2πx) and we observe that sup

[log(1 − rk E/(2πβ 0 x)) − (rk E/2) log(β 0 /β)]

β 0 ∈]β,+∞[

=

sup

[log(1 − (β 0 /β)−1 ) − (rk E/2) log(β 0 /β)] = −C(rk E).

β 0 ∈]β,+∞[

For every positive integer n, from (3.4.18) applied to E ⊕n and to nx instead of E and x, we get n log θE (rk E/(2πx)) ≤ h0Ar (E

⊕n

, nx) + C(n rk E).

Multiplying by 1/n and letting n go to +∞, we obtain (3.4.19), since C(n rk E) = o(n). An alternative proof of (3.4.19) consists in observing that as an easy consequence of (3.4.24) (which holds when β 0 ≥ β), the following inequality holds for all (β, β 0 ) ∈ R∗2 +: (rk E/2)(β 0 /β) + log θE (β 0 ) ≥ log θE (β). ˜ 0 (E, .) in terms of the Then (3.4.19) follows from the expression (3.4.10) of h Ar Legendre transform of log θE . Finally, (3.4.20) follows from (3.4.10) or (3.4.11), by chosing x and β related by βx = rk E/(2π).  The estimates in Theorem 3.4.9 show that the expressions h0Ar (E, rk E/2π) ˜ and h0Ar (E, rk E/2π) satisfy −C(rk E) ≤ h0Ar (E, rk E/2π) − h0θ (E) ≤ rk E/2

(3.4.25)

˜ 0 (E, rk E/2π) − h0 (E) ≤ rk E/2. 0≤h Ar θ

(3.4.26)

and These comparison estimates, relating Arakelov and θ-invariants of Euclidean lattices, should be compared with the comparison estimate (3.1.1) in Section 3.1. The error term, of order (1/2)rk E · log rk E in (3.1.1), is replaced by rk E/2 in (3.4.25) and (3.4.26). (These error terms will be shown to be basically optimal as rk E goes to +∞ in the next section; see paragraph 3.5.3 below.) These remarks plead for considering the positive real numbers ˜ 0 (E, rk E/2π) h0Ar (E, rk E/2π) and h Ar attached to a Euclidean lattice of positive rank E as variants of h0Ar (E) = h0Ar (E, 1) that are “better behaved” than h0Ar (E) itself.

70

Chapter 3. Geometry of Numbers and θ-Invariants

3.5 Some Consequences of Siegel’s Mean Value Theorem 3.5.1 Siegel’s mean value theorem over SLn (R)/ SLn (Z) and over L(n, δ). In this paragraph, we denote by n an integer ≥ 2. The Lie group SLn (R) is unimodular, and its discrete subgroup SLn (Z) has finite covolume. We shall denote by µn the Haar measure on SLn (R) that satisfies the following normalization condition: the measure induced by µn on the quotient SLn (R)/ SLn (Z) — which we shall still denote by µn — is a probability measure. In other words, Z dµn = 1. SLn (R)/ SLn (Z)

For every Borel function ϕ : Rn −→ [0, +∞] and g ∈ SLn (R), we may consider the sum X Σ(ϕ)(g) := ϕ(g.v). v∈Zn \{0}

Clearly, for all γ ∈ SLn (Z), we have Σ(ϕ)(g.γ) = Σ(ϕ)(g) and the function Σ(ϕ) : SLn (R)/ SLn (Z) −→ [0, +∞] so defined is a Borel function. In its most basic form, Siegel’s mean value theorem is the following statement ([103]; see also [118] and [81] for other derivations, and [119, Chapter III] for a “modern” presentation). Theorem 3.5.1. For every Borel function ϕ : Rn −→ [0, +∞] as above, the following equality holds: Z Z Σ(ϕ)(g) dµn (g) = ϕ(v) dλn (v), (3.5.1) Rn

SLn (R)/ SLn (Z)

where we denote by λn the Lebesgue measure on Rn . −δ/n

 n

n

For every g ∈ SLn (R) and δ ∈ R, the lattice e g(Z ) in R equipped with the standard Euclidean norm k.kn (defined by k(x1 , . . . , xn )k2n = x21 + · · · + x2n ) becomes a Euclidean lattice (e−δ/n g(Zn ), k.kn ) of covolume e−δ , or equivalently, of Arakelov degree δ. We shall denote by L(n, δ) the set of isomorphism classes of Euclidean lattices of rank n and Arakelov degree δ. This set may be endowed with a natural locally compact topology (in fact, with the structure of an “orbifold”) by means of the identification of the set a L(n, δ) L(n) := δ∈R

3.5. Some Consequences of Siegel’s Mean Value Theorem

71

of isomorphism classes of Euclidean lattices of rank n with the double coset space On (R) \ GLn (R)/GLn (Z). In concrete terms, the natural topology and Borel structures on L(n, δ) are the quotients of those of SLn (R)/ SLn (Z) by the surjective map SLn (R)/ SLn (Z) −→ [g] 7−→

πn,δ :

Ln,δ , [(e−δ/n g(Zn ), k.kn )].

We shall denote by µn,δ := πn,δ∗ µn the Borel measure on Ln,δ derived from the measure µn on SLn (R)/ SLn (Z) by the parametrization πn,δ of Ln,δ . Like µn , it is a probability measure: Z dµn,δ = 1. L(n,δ)

Applied to a radial function ϕ : Rn −→ [0, +∞], Siegel’s mean value formula (3.5.1) “descends” through πn,δ . Namely, to every Borel function ∗ ρ : R+ −→ [0, +∞]

we may attach the Borel function Σn (ρ) : L(n) −→ [0, +∞] that maps the isomorphism class of a Euclidean lattice E := (E, k.k) of rank n to X Σn (ρ)(E) := ρ(kvk). v∈E\{0}

For all δ ∈ R and g ∈ SLn (R), we have X Σn (ρ)(πn,δ ([g])) = ρ(e−δ/n kg.wk) = Σ(ϕδ )([g]) w∈Zk \{0}

where ϕδ is the function from Rn to [0, +∞] defined by ϕδ (v) := ρ(e−δ/n kvk). Clearly, we have Z Rn

ϕδ (v) dλn (v) = eδ

Z ρ(kvk) dλn (v), Rn

and Siegel’s mean value formula (3.5.1) applied to ϕδ becomes the following:

Chapter 3. Geometry of Numbers and θ-Invariants

72

Theorem 3.5.2. For al (n, δ) ∈ N≥2 × R and for every Borel function ρ : R+ −→ [0, +∞], the following equality holds: Z Z δ ρ(kvkn ) dλn (v). (3.5.2) Σn (ρ) dµn,δ = e L(n,δ)

Rn

 The last integral may also be written Z Z ρ(kvkn ) dλn (v) = nvn Rn

+∞

ρ(r)rn−1 dr,

0

where as previously in this monograph, vn denotes the volume of the n-dimensional ball: π n/2 vn := λn ({v ∈ Rn | kvkn < 1}) = . Γ(1 + n/2) Theorems 3.5.1 and 3.5.2 are classically used to establish the existence of Euclidean lattices satisfying suitable conditions — for instance, lattices of large enough density — by “probabilistic arguments”, based on the observation that a positive measurable function on some probability space assumes values greater than or equal to its mean value on some subset of positive measure. We refer the reader to [103] for a concise discussion of the existence of “dense lattices” as a consequence of Theorem 3.5.1 and for references to related earlier work of Minkowski and Hlawka. For later reference, we state a formal version of the above observation as the following lemma: Lemma 3.5.3. Let ϕ and ψ be two Borel functions from L(n, δ) to [0, +∞]. (1) If the integrals Z

Z

Iϕ :=

ϕ(x) dµn,δ (x) and Iψ := L(n,δ)

ψ(x) dµn,δ (x) L(n,δ)

are finite and positive, then the Borel subsets E≤ := {x ∈ L(n, δ) | ϕ(x)/Iϕ ≤ ψ(x)/Iψ } and E≥ := {x ∈ L(n, δ) | ϕ(x)/Iϕ ≥ ψ(x)/Iψ } have positive µn,δ -measures, and therefore are nonempty. (2) In particular, if the integral Iϕ is finite, then there exists x in L(n, δ) such that ϕ(x) ≤ Iϕ . If, moreover, ϕ is continuous and nonconstant, then Iϕ is positive, and the image ϕ(L(n, δ)) contains an open neighborhood of Iϕ in R∗+ . 

73

3.5. Some Consequences of Siegel’s Mean Value Theorem

0 3.5.2 Applications to hAr and h0θ . Let us begin by recovering a simple version of the classical results of Minkowski–Hlawka–Siegel alluded to above. We will express 0 (., t), in a form convenient for later reference and it in terms of the invariant hAr for comparison with similar results concerning the invariant h0θ . According to the definition of h0Ar (., t), we have

0 ehAr (E,t) − 1 = {v ∈ E \ {0} | kvk2 ≤ t} =

X

1[0,t1/2 ] (kvk)

v∈E\{0}

= Σn (1[0,t1/2 ] )([E]). For ρ = 1[0,t1/2 ] , the computation of the integral on the right-hand side of Siegel’s mean value formula (3.5.2) is straightforward — indeed, Z 1[0,t1/2 ] (kvk)dλn (v) = vn · tn/2 Rn

— and formula (3.5.2) takes the following form: Proposition 3.5.4. For all (n, δ) in N≥2 × R and t ∈ R∗+ , the following relation holds: Z 0 ehAr (E,t) dµn,δ ([E]) = 1 + vn tn/2 eδ . (3.5.3) [E]∈L(n,δ)

 In particular, when t = 1, we obtain Z 0 ehAr (E) dµn,δ ([E]) = 1 + vn eδ . [E]∈L(n,δ) 0

Observe that ehAr (.,t) −1 takes its values in N and therefore vanishes where it is 0 < 1. Therefore, if we apply Lemma 3.5.3, part (2), to the function ϕ := ehAr (.,t) −1, ∗ we obtain that for all (n, δ) in N≥2 × R and t ∈ R+ such that vn tn/2 eδ < 1, there exists a Euclidean lattice E of rank n and Arakelov degree δ such that h0Ar (E, t) = 0, or equivalently, such that λ1 (E) > t1/2 . In other words, we have established the following variant of a classical result of Minkowski: Corollary 3.5.5. For all (n, δ) in N≥2 × R, we have sup

λ1 (E) ≥ e−δ/n vn−1/n .

(3.5.4)

[E]∈L(n,δ)



74

Chapter 3. Geometry of Numbers and θ-Invariants This estimate has to be compared with the upper bound λ1 (E) ≤ 2e−δ/n vn−1/n

sup

(3.5.5)

[E]∈L(n,δ)

that follows from the so-called “Minkowski first theorem”. Let us also recall that as n goes to infinity, p vn−1/n ∼ n/(2πe), and that the positive real number γn :=

sup

λ1 (E)2

[E]∈L(n,0)

is classically known as the Hermite constant in dimension n. Thus the lower bound (3.5.4), when expressed in terms of Hermite constants, takes the following asymptotic form: lim inf γn /n ≥ 1/(2πe), n→+∞

well known in the study of sphere packings (see [35], especially Chapter 1, for additional informations and references). To compute the average value on L(n, δ) of the θ-invariants, we apply Siegel’s mean value formula (3.5.2) to the Gaussian function 2

ρ(x) := e−πx . For this choice of ρ, the integral on the right-hand side of (3.5.2) is simply Z 2 e−πkvk dλn (v) = 1, Rn

and Siegel’s mean value formula takes the following form: Proposition 3.5.6. For all (n, δ) in N≥2 × R, Z 0 ehθ (E) dµn,δ ([E]) = 1 + eδ .

(3.5.6)

[E]∈L(n,δ)

 h0θ (E)

This expression for the mean value of e lower bound h0θ (E) ≥ δ,

has to be compared with the

valid over L(n, δ) (see (2.3.2)). The function on L(n, δ) defined by h0θ is clearly continuous. Moreover, it is nonconstant (this follows, for instance, from its expression for direct sums of rank-one Euclidean lattices in Proposition 2.5.2). Therefore, we may apply the 0 last assertion of Lemma 3.5.3 to the function ehθ , and we obtain, from the value of its integral computed in (3.5.6), the following corollary:

3.5. Some Consequences of Siegel’s Mean Value Theorem

75

Corollary 3.5.7. For all (n, δ) in N≥2 × R, there exists a Euclidean lattice E of rank n and degree δ such that h0θ (E) < log(1 + eδ ).

(3.5.7) 

According to the Poisson–Riemann–Roch formula, for all [E] in L(n, δ), we have h0θ (E) − h1θ (E) = δ. Therefore, the equality (3.5.6) may be also written Z 1 ehθ (E) dµn,δ ([E]) = 1 + e−δ ,

(3.5.8)

[E]∈L(n,δ)

and the condition (3.5.7) is equivalent to h1θ (E) < log(1 + e−δ ). To put the conclusion of Corollary 3.5.7 in perspective, we may consider the “obvious” Euclidean lattice of rank n and Arakelov degree δ, namely O(δ/n)⊕n , for all (n, δ) ∈ N≥1 × R. If we define, for every t ∈ R∗+ , X 2 θ(t) := θO(0) (t) = e−πk t , k∈Z

then its θ-invariant is h0θ (O(δ/n)⊕n ) = n log θ(e−2δ/n ). When δ is fixed and n goes to infinity, this expression is equivalent to n log θ(1) = nh0θ (O) = nη. This demonstrates that the existence of a (class of) Euclidean lattice in L(n, δ) satisfying (3.5.7) is not “obvious” when n is large. 0 We may also apply the first part of Lemma 3.5.3 to the functions ehAr (.t) and h0θ e . Taking into account the expressions (3.5.3) and (3.5.6) for their integrals, we obtain the following: Corollary 3.5.8. For all (n, δ, t) ∈ N≥2 × R × R∗+ , there exist Euclidean lattices E + and E − , of rank n and Arakelov degree δ, such that h0Ar (E + , t) − h0θ (E + ) ≥ log

1 + vn tn/2 eδ 1 + eδ

h0Ar (E − , t) − h0θ (E − ) ≤ log

1 + vn tn/2 eδ . 1 + eδ

and



Chapter 3. Geometry of Numbers and θ-Invariants

76

3.5.3 Concerning the constants in the comparison estimates. From Corollary 3.5.8, one easily derives that the additive constants in various estimates relating 0 the invariants hAr (E, t) and h0θ (E) established in the previous sections are “of the correct order of growth” as the rank of the Euclidean lattice E goes to +∞. For instance, consider the first inequality in (3.1.1). It asserts that for every Euclidean lattice E of rank n ≥ 1, h0Ar (E) − h0θ (E) ≥ −(n/2) · log n + log(1 − 1/2π).

(3.5.9)

According to Corollary 3.5.8 applied with t = 1 and δ = − log vn , for every n ∈ N≥2 there exists a Euclidean lattice E − of rank n such that covol(E − ) = vn and h0Ar (E − ) − h0θ (E − ) ≤ log

(3.5.10) 2 . 1 + vn−1

Moreover, as n goes to +∞, log

2 = −(n/2) · log n + O(n). 1 + vn−1

This shows that the “best constant” on the right-hand side of (3.5.9) is equivalent to −(n/2) · log n as n goes to +∞, even if one considers only Euclidean lattices that satisfy (3.5.10). By applying Corollary 3.5.8 with t = 1 and δ very large, one actually obtains that the best constant in (3.5.9) is at most log vn . Consider now the estimates, valid for every Euclidean lattice E of rank n ≥ 1, ˜ 0 (E, n/2π) − h0 (E) ≤ n/2, h0Ar (E, n/2π) − h0θ (E) ≤ h Ar θ

(3.5.11)

already considered in (3.4.25) and (3.4.26). (These estimates follow from the defi˜ 0 and from (3.4.20).) nition of h Ar According to Corollary 3.5.8 applied with t = n/2π, for all n ∈ N≥2 and δ ∈ R, there exists a Euclidean lattice E + of rank n and Arakelov degree δ such that 1 + vn (n/2π)n/2 eδ h0Ar (E + , n/2π) − h0θ (E + ) ≥ log . 1 + eδ Moreover, 1 + vn (n/2π)n/2 eδ lim log = log[vn (n/2π)n/2 ], δ→+∞ 1 + eδ and, as n goes to +∞, log[vn (n/2π)n/2 ] = n/2 + O(log n). This shows in particular that the “best constant” on the right-hand side of (3.5.11) is equivalent to n/2 as n goes to +∞.

Chapter 4

Countably Generated Projective Modules and Linearly Compact Tate Spaces over Dedekind Rings In this chapter, we denote by A a Dedekind ring (in the sense of Bourbaki, [21, Vii.2.1]; in other words, A is either a field or a Noetherian integrally closed domain of dimension 1), and we introduce the categories of (topological) modules CPA and CTCA attached to A. When the Dedekind ring A is the ring OK of integers in some number field K, the modules in these categories will occur in the following chapters as the OK -modules underlying the “infinite-dimensional Hermitian vector bundles” over Spec OK investigated in this monograph. The objects in the dual categories CPA and CTCA are easily described. Namely, an object of CPA is an A-module that is either finitely generated and projective or isomorphic to A(N) . An object of CTCA is a topological A-module1 that is either a finitely generated projective A-module equipped with the discrete topology or isomorphic to AN equipped with the product of the discrete topology on every factor A. Handling the morphisms in these categories requires more care. The strict morphisms in CTCA play an especially important role, as shown in Section 4.4, and a number of “pathologies” concerning the morphisms in CPA and CTCA occur naturally, as demonstrated by the examples in Section 4.6.

4.1

Countably Generated Projective A-Modules

4.1.1 The category CPA . The following proposition is a simple consequence of the fact that over a Dedekind ring, a finitely generated module is projective when it is torsion-free. Proposition 4.1.1. For every A-module M , the following conditions are equivalent: (1) The A-module M is countably generated and projective. (2) The A-module M is isomorphic to a direct summand of A(N) . (3) The A-module M is isomorphic to some A-submodule of A(N) . 1 By a topological A-module, we mean a topological A-module over the ring A equipped with the discrete topology.

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0_4

77

78

Chapter 4. Countably Generated Projective Modules over Dedekind Rings

(4) There exists a family (Mi )i∈N of A-submodules of M such that: (i) for all i ∈ N, Mi is a finitely generated torsion-free A-module; (ii) for all i ∈ N, Mi is a saturated A-submodule of Mi+1 ; S (iii) M = i∈N Mi . (5) The A-module M is a countable direct sum of finitely generated projective A-modules. Proof of Proposition 4.1.1. The implications (5) ⇒ (1) ⇒ (2) ⇒ (3) are clear. When (3) holds, we may consider the filtration (Ni )i∈N of N := A(N) defined by Ni := {(ak )k∈N ∈ A(N) | ∀k ∈ N≥i , ak = 0}. Then the filtration (Mi )i∈N of M defined by Mi := M ∩ Ni satisfies (4). When (4) holds, for every i ∈ N, the quotient Mi+1 /Mi is a finitely generated A-module, which is torsion-free, hence projective. Therefore, the short exact sequence of A-modules 0 −→ Mi −→ Mi+1 −→ Mi+1 /Mi −→ 0 is split, and there exists a (necessarily finitely generated and projective) A-submodule Pi of Mi+1 such that Mi+1 = Mi ⊕ Pi . Then we obtain the following decomposition of M : M M = M0 ⊕ Pi . i∈N

This displays M as a countable direct sum of finitely generated projective A modules. We define the A-linear category CPA of countably generated projective Amodules as the category whose objects are A-modules satisfying the equivalent conditions in Proposition 4.1.1 and whose morphisms are A-linear maps. For every object M of CPA , we denote by F(M ) the family of finitely generated A-submodules of M , and by FS(M ) the family of saturated finitely generated A-submodules of M . It is straightforward that for every submodule N in F(M ), the saturation N sat := {n ∈ M | ∃a ∈ A \ {0}, an ∈ N } of N in M belongs to M . In particular, FS(M ) is cofinal in F(M ) ordered by inclusion. We shall also denote by coF(M ) (resp., by coFS(M )) the family of Asubmodules M 0 of M such that M/M 0 is a finitely generated A-module (resp., a finitely generated torsion-free A-module). Observe that according to the equivalences of conditions (1) and (3) in Proposition 4.1.1, every A-submodule M 0 of an object M of CPA is again an object of CPA . However, the quotient A-module M/M 0 — even if assumed torsion-free —

4.1. Countably Generated Projective A-Modules

79

is not always an object in CPA (see, for instance, when A = Z, the constructions in Proposition 4.6.1 below, notably the short exact sequence (4.6.2)). The A-linear category CPA admits obvious finite direct sums that are also finite direct products, and it is actually an additive category. It also admits countable direct sums. If B denotes a Dedekind ring that is an A-algebra, the tensor product defines an additive functor . ·A B : CPA −→ CPB . 4.1.2 A theorem of Kaplansky. Since every finitely generated projective Amodule is a (finite) direct sum of invertible A-modules, condition (5) is equivalent to M being a countable direct sum of invertible A-modules. Indeed, when M has infinite rank, this observation admits the following strengthening, proved by Kaplansky [69, Theorem 2] in a more general setting: Proposition 4.1.2. If an A-module M satisfies the conditions in Proposition 4.1.1 and has infinite rank (or equivalently, is not finitely generated), then it is free, hence isomorphic to A(N) . When A is principal (e.g., when A = Z, a case of special interest in this monograph), this is straightforward. The part of Kaplansky’s argument in [69] relevant to the derivation of Proposition 4.1.2 for a general Dedekind ring A may be summarized as follows. Firstly, one shows that, for every element m of a projective countably generated A-module M of finite rank, there exists a direct summand P in M , free and of finite rank, that contains m. To achieve this, observe that M may written as an infinite countable direct L sum i∈N Ii of invertible submodules Ii of M , and recall that for every two in2 vertible A-modules I and J, the A-modules I ⊕ J and A ⊕ (I N⊗ J) are isomorphic . This lastL fact implies that, for all n ∈ N, if we define Jn := 0≤i≤n Ii , then the Amodule 0≤i≤n Ii ⊕ Jn∨ is free of rank n + 1, and that there exists an isomorphism of A-modules: ∼

ϕ : In+1 ⊕ In+2 −→ Jn∨ ⊕ (Jn ⊗ In+1 ⊗ In+2 ). L Therefore, if n is chosen so large that 0≤i≤n Ii contains m, then the submodule L P := 0≤i≤n Ii ⊕ ϕ−1 (Jn∨ ⊕ {0}) of M is a free direct summand, of rank n + 1, and contains m. Secondly, one considers a countable family of generators (mi )i∈N>0 of M , and by means of the above fact, one constructs inductively projective A-submodules (Pi )i∈N>0 and (M i )i∈N of M such that the Pi are free of finite rank and the M i are countably generated, and such that the following conditions are satisfied: 2 See, for instance, [85, Lemma 1.7]. More generally, every finitely generated projective Amodule E of positive rank is isomorphic to Ark E−1 ⊕ ∧rk E E; see [85, Theorem 1.6].

80

Chapter 4. Countably Generated Projective Modules over Dedekind Rings

(1) M 0 = M ; L (2) for all i ∈ N>0 , one has M i−1 = Pi ⊕ M i , and mi ∈ 1≤j≤i Pj . L Thus we obtain a decomposition M = i∈N>0 Pi , which shows that M is a countable direct sum of free modules of finite ranks which completes the proof.

Linearly Compact Tate Spaces with Countable Basis

4.2

4.2.1 Basic definitions. We define the A-linear category CTCA of linearly compact Tate spaces with countable basis over A as follows.3 An object N of CTCA is a topological module over the ring A equipped with the discrete topology that satisfies the following two conditions: CTC1 1 : The topology of N is Hausdorff and complete. CTC2 1 : Their exists a countable basis of neighborhoods U of 0 in N consisting of A-submodules of N such that N/U is a finitely generated projective A-module. Morphisms in the category CTCA are A-linear continuous maps: for every two objects N1 and N2 in CTCA , we let cont HomCTCA (N1 , N2 ) := HomA (N1 , N2 ).

For every subset U of an A-module N , we may consider the condition appearing in CTC2 : CU : U is an A-submodule of N , and N/U is a finitely generated projective Amodule. Then we have the following lemma: Lemma 4.2.1. Let N be an object of CTCA . A subset U of N is a neighborhood of 0 and satisfies condition CU if and only if U is an open saturated submodule of N. Proof. The necessity is clear. Conversely, if U is an open saturated submodule of N , then it contains a neighborhood U0 of 0 that satisfies CU0 . Then U/U0 is a saturated submodule of the finitely generated projective A-module N/U0 , and consequently, N/U ' (N/U0 )/(U/U0 ) also is a finitely generated projective A-module. 3 The

terminology of Tate space is borrowed from Drinfeld [39].



81

4.2. Linearly Compact Tate Spaces with Countable Basis

For an object N of CTCA , we shall denote the family of open saturated submodules of N by U(N ). It is stable under finite intersection. Every finitely generated projective A-module equipped with the discrete topology becomes an object of CTCA . In this way, the category of finitely generated projective A-modules and A-linear maps appears as a full subcategory of CTCA . According to the countability assumption in CTC2 , for every object N of CTCA , there exists a “nonincreasing” sequence U0 ←- U1 ←- U2 ←- · · · of submodules in U(N ) that constitute a basis of neighborhoods of 0 in N . We b N a filtration defining the topology of shall call any such sequence (Ui )i∈N in U(E) b or for brevity a defining filtration in U(E) b N. E, b N we may construct a countable From any defining filtration (Ui )i∈N in U(E) projective system of finitely generated projective A-modules b 1 ←− E/U b 0 ←− E/U b 2 ←− · · · , E/U b −→ lim E/U b i , defined by the and we may consider the canonical morphism E ←−i b b quotient maps E −→ E/Ui . According to condition CTC1 , this morphism is bijective, and it becomes an isomorphism of topological A-modules b i b ' lim E/U E ←−

(4.2.1)

i

b i is equipped with the projective limit topology derived from the when limi E/U ←− b i. discrete topology on the finitely generated projective modules E/U Conversely, for every projective system q2

q1

q0

E0 ←− E1 ←− E2 ←− · · ·

(4.2.2)

of surjective morphisms between finitely generated projective A-modules, the projective limit b := lim Ei , E ←− i

equipped with its natural prodiscrete topology, defines an object of CTCA . b := lim Ei and Fb := lim Fj are two objects of CTCA , realized Moreover, if E ←−j ←−i as limits of projective systems of finitely generated projective A-modules as above, we then have a canonical identification b Fb) := Homcont (lim Ei , lim Fj ) ' lim lim HomA (Ei , Fj ). HomCTCA (E, A ←− ←− −→ ←− i

j

j

i

If B denotes a Dedekind ring that is an A-algebra, the completed tensor product defines an additive functor b A B : CTCA −→ CTCB . ·. ⊗

82

Chapter 4. Countably Generated Projective Modules over Dedekind Rings

Observe that for every projective system (4.2.2) of surjective morphisms of finitely projective A-modules, we get, by extending the scalars from A to B, a projective system of surjective morphisms of finitely projective B-modules q0,B

q2,B

q1,B

E0,B ←− E1,B ←− E2,B ←− · · ·

(4.2.3)

b⊗ b A B of CTCB derived from the projective Its projective limit “is” the object E b limit E of (4.2.2) by the completed tensor product functor. Indeed, we have a canonical isomorphism of prodiscrete B-modules: ∼

ˆ A B −→ lim Ei,B . (lim Ei )⊗ ←− ←− i

i

The following proposition is included for later reference in Section 9.1. Proposition 4.2.2. With the above notation, the completed tensor product defines a map b⊗ b −→ U(E b A B), b A B : U(E) (4.2.4) ·⊗ which is injective when B is flat over A (or equivalently when the morphism A → B is injective), and bijective when the morphism Spec B → Spec A is an open immersion. b is a finitely generated Proof. This easily follows from the special case in which E b b b A B is the tensor product M ⊗A B, and U(E) projective A-module M . Then E ⊗ b⊗ b A B)) is the set of saturated A-submodules of M (resp., of saturated (resp., U(E B-submodules of M ⊗A B). We leave the details to the interested reader. 

4.2.2

Subobjects and countable products.

Proposition 4.2.3. Let N be an object of CTCA . Every closed A-submodule N 0 of N equipped with the induced topology is an object of CTCA . Proof. Equipped with the induced topology, N 0 is clearly Hausdorff and complete. Therefore, the topological A-module N 0 satisfies CTC1 . Moreover, if some neighborhood U of 0 in N satisfies CU , then U 0 := U ∩ N 0 is a neighborhood of 0 in N 0 that is clearly an A-submodule of N 0 . Moreover, the inclusion N 0 ,→ N defines an injective morphism of A-modules N 0 /U 0 → N/U ; therefore, N 0 /U 0 — like every submodule of a finitely generated projective module over a Dedekind ring — is also a finitely generated projective A-module. In other words, U 0 satisfies CU 0 . This immediately implies that condition CTC2 also is inherited by N 0 .  Observe that with the notation of Proposition 4.2.3, the topological A-module N/N 0 , even if assumed torsion-free, may not be an object of CTCA .

4.2. Linearly Compact Tate Spaces with Countable Basis

83

For instance, when A = Z, the short exact sequences4 (4.6.5) and (4.6.13) in Proposition 4.6.1 below and its proof display the ring of p-adic integers Zp , equipped with its p-adic topology, as a quotient of ZN by a closed submodule. In fact, one may easily show that the topological Z-modules that may be realized as quotients of an object of CTCZ by a closed subobject are precisely the commutative Polish topological groups G admitting a basis of neighborhoods of 0 that are open subgroups U such that G/U is finitely generated. The A-linear category CTCA admits obvious finite direct sums, which are also finite direct products, and is actually an additive category. It also admits countable products. Indeed, if (Ni )i∈I is a countable family of objects in CTCA , then the A-module Y Ni , N := i∈I

equipped with the product topology, is easily seen to define an object in CTCA . The projection maps pri : N −→ Ni are morphisms in CTCA , and (N, (pi )i∈I ) is a product of the Ni s in the category CTCA . More generally, every projective system M0  M1  M2  M3  · · · of surjective open morphisms in CTCA admits a (projective) limit limi Mi in ←− CTCA , defined by the A-module projective limit of the Mi s, equipped with the projective limit of their given topology. The following proposition shows that up to isomorphism, every object of CTCA is a product of finitely generated projective A-modules (equipped with the discrete topology). Its easy proof is left to the reader. Proposition 4.2.4. Consider a projective system of surjective morphisms of finitely generated projective A-modules: q0

q1

q2

E0 ←− E1 ←− E2 ←− · · · .

(4.2.5)

For every i ∈ N, there exists an A-linear section σi : Ei −→ Ei+1 of qi , and if we define ( ker qi−1 if i ≥ 1, Si := E0 if i = 0, then the A-modules Si are finitely generated and projective, and for every i ∈ N>0 , we have Ei = Si ⊕ σi−1 (Ei−1 ). (4.2.6) The direct sum decompositions (4.2.6) determine a family of isomorphisms of A-modules M ∼ ιn : En −→ Si 0≤i≤n 4 Observe

that these are actually strict short exact sequences of topological abelian groups.

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

such that the map ιn ◦ qn ◦ ι−1 n+1 :

M

Si −→

M

Si

0≤i≤n

0≤i≤n+1

is the projection on the first n + 1th factors, and consequently an isomorphism in CTCA : Y ∼ ι : lim En −→  Si . ←− n

i∈N

4.2.3 Continuity of morphisms of A-modules between objects of CTCA . In this paragraph, we want to indicate that for a large class of Dedekind rings A, every morphism of A-modules between two objects N1 and N2 in CTCA is automatically continuous. In other words, for these rings, the forgetful functor from the category CTCA to the category of A-modules is fully faithful. This will follow from the variant of results of Specker [107] and Enochs [44] discussed in Appendix B. Observe that for every Dedekind ring A, precisely one of the following three conditions is satisfied: Ded1 : A is a field; Ded2 : A is a complete discrete valuation ring; Ded3 : there exists a nonzero prime ideal p of A such that A is not p-adically complete.5 For instance, every countable Dedekind ring A that is not a field satisfies Ded3 , since the cardinality of a complete discrete valuation ring is at least the cardinality of the continuum. When the Dedekind ring A satisfies Ded1 or Ded2 , there exist many “linear forms” in HomA (AN , A) that are not continuous when AN (resp., A) is equipped with its natural prodiscrete (resp., discrete) topology, or equivalently, when there exists an A-linear map ξ : AN −→ A that is not of the form X ξ((xn )n∈N ) = ξn xn (4.2.7) n∈N

for some (ξn )n∈N in A(N) . Indeed, when A is a field, every nonzero linear form on the vector space AN that vanishes on its subspace A(N) is such a map. When A is a complete discrete valuation ring of maximal ideal m, then for every sequence (ξn )n∈N in AN \A(N) that converges to zero in the m-adic topology, the formula (4.2.7) defines an element ξ ∈ HomA (AN , A) of the required type. In contrast, for Dedekind rings satisfying Ded3 , the following continuity results holds: 5 It is straightforward that when Ded holds, the ring A is not p-adically complete for every 3 nonzero prime ideal p of A.

4.2. Linearly Compact Tate Spaces with Countable Basis

85

Proposition 4.2.5. If the Dedekind ring A satisfies Ded3 , then for every two objects N and N 0 in CTCA , we have cont HomA (N, N 0 ) = HomA (N, N 0 ).

(4.2.8)

Proof. When N 0 = A, this follows from Corollary B.2.2 applied to R := A and M := N . (Indeed, we may choose as m any nonzero prime ideal p of A: the local ring A(p) is not complete, since A satisfies Ded3 .) The validity of (4.2.8) when N 0 = A implies its validity when N 0 = A⊕n for some n ∈ N, hence its validity for every finitely generated projective A-module N 0 equipped with the discrete topology. (Indeed, such a module may be realized as a submodule — in fact, as a direct summand — of a free A-module A⊕n .) To complete the proof of Proposition 4.2.5, observe that every object N 0 of CTCA may be realized as the projective limit N 0 = limi Ni0 of a projective system ←− of finitely generated projective A-modules N00 ← N10 ← N20 ← · · · , and that we have natural identifications HomA (N, N 0 ) ' lim HomA (N, Ni0 ) ←− i

and cont 0 lim HomA (N, Ni0 ). Homcont A (N, N ) ' ← − i

The validity of (4.2.8) consequently follows from the already established equalities cont (N, Ni0 ). HomA (N, Ni0 ) = HomA



4.2.4 The topology on objects in CTCA when A is a topological ring. Let us assume that besides its discrete topology, the ring A is equipped with a topology that makes A a topological ring. We shall denote by Aan the topological ring defined by A equipped with this coarser topology. For instance, A may be a discrete valuation ring and Aan the ring A equipped with the topology associated to its discrete valuation, or a local field and Aan the field A equipped with its “usual” locally compact topology. In this situation, besides its topology of prodiscrete A-module, any object N of CTCA is canonically endowed with a coarser topology, which makes it a topological Aan -module N an . Indeed, any finitely generated projective A-module P is equipped with a canonical topology of Aan -module: if P is embedded as a direct summand in the direct sum A⊕n of a finite number of copies of A, this topology is the one induced by the product topology on (Aan )n . We shall denote by P an the so-defined topological Aan -module. Observe that any A-linear morphism ϕ : P1 −→ P2 between finitely generated projective A-modules defines a continuous morphism ϕ : P1an −→ P2an of topological Aan -modules.

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

By definition, if N denotes an object of CTCA , we have a canonical isomorphism of topological A-modules ∼

N −→

lim ←−

N/U,

(4.2.9)

U ∈U (N )

where limU ∈U (N ) N/U is equipped with the prodiscrete topology. We make N a ←− topological Aan -module N an by declaring (4.2.9) to be an isomorphism of topological Aan -modules ∼ (4.2.10) N an −→ lim (N/U )an , ←− U ∈U (N )

an

where limU ∈U (N ) (N/U ) is equipped with the projective limit topology derived ←− from the canonical topology of an Aan -module on each finitely generated projective A-module N/U . Every U ∈ U(N ) is an object of CTCA and as such is equipped with the “analytic” topology U an . One easily sees that U is actually closed in N an and that the topology of U an is the topology induced by the topology of N an . Every morphism f : N 0 −→ N in CTCA defines a continuous morphism of topological Aan -modules f : N 0an −→ N an . However, the so defined injective map cont 0an 0 Homcont , N an ) A (N , N ),−→ HomAan (N

is not surjective in general. (For instance, when A is a complete discrete valuation ring with maximal ideal m, for every sequence (ξn )n∈N in (A\{0})N that converges to 0 in the m-adic topology, the linear form ξ : AN −→ A defined by (4.2.7) defines cont N an an N an element of Homcont Aan ((A ) , A ) that does not belong to HomA (A , A).) an When A is the field R (resp., C) equipped with its usual “analytic” topology, the topological Aan -module N an is a topological vector space over R (resp., over C), isomorphic to Rn (resp., to Cn ) if n := dimAan N is finite, and to RN (resp., to CN ) if n is infinite. In particular, it is a Fr´echet locally convex vector space over R (resp., over C).

4.3 The Duality Between CPA and CTCA In this section, we discuss the (anti)equivalence of A-linear categories between CPA and CTCA defined by duality, and some of its consequences. 4.3.1 The duality functors. (i) To every A-module M we may attach its dual topological A-module, namely the A-module M ∨ := HomA (M, A)

(4.3.1)

equipped with the topology of pointwise convergence. If α : M2 −→ M1 is a morphism of A-modules, then its transpose α∨ := . ◦ α : M1∨ −→ M2∨

(4.3.2)

4.3. The Duality Between CPA and CTCA

87

is clearly A-linear and continuous. For every family (Mi )i∈I of A-modules, there is a canonical identification of topological A-modules: M Y ∼ Mi∨ . Mi )∨ −→ ( i∈I

i∈I

Moreover, if M is a finitely generated projective A-module, then its dual M ∨ also is finitely generated and projective, and the topology of M ∨ is discrete. (Indeed, this holds for every finitely generated free A-module A⊕n , and consequently for every direct summand of such an A-module.) Since every object in CPA is a countable direct sum of finitely generated projective A-modules, these observations show that the constructions (4.3.1) and (4.3.2) define a contravariant A-linear duality functor .∨ : CPA −→ CTCA .

(4.3.3)

(ii) Conversely, to any topological A-module N we may attach its topological dual, namely the A-module cont (N, A) N ∨ := HomA

(4.3.4)

consisting of continuous A-linear maps from N to A. Then every continuous Alinear morphism β : N1 −→ N2 of topological A-modules defines, by transposition, a morphism of A-modules β ∨ := . ◦ β : N2∨ −→ N1∨ .

(4.3.5)

For every projective system of topological A-modules p0

p2

p1

N0 ←− N1 ←− N2 ←− · · ·

(4.3.6)

we may form the projective limit limi Ni , as a topological A-module, and we may ←− also consider the dual inductive system of A-modules p∨

p∨

p∨

0 1 2 ··· . N0∨ −→ N2∨ −→ N1∨ −→

Then there is a canonical identification of A-modules ( lim Ni )∨ ' lim Ni∨ . ←− −→ i

i

Furthermore, if N is a finitely generated projective A-module, equipped with the discrete topology, then N ∨ = Homcont A (N, A) = HomA (N, A) is also a finitely generated projective A-module. Moreover, if q : N 0 → N is a surjective morphism of finitely generated projective A-modules, then p∨ : N ∨ → N 0∨ is injective, and its image is a direct summand in N 0∨ .

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

Since every object in CTCA is (up to isomorphism) the projective limit of some countable projective system E0  E1  E2  · · · of surjective morphisms between finitely generated projective A-modules, these observations show that the constructions (4.3.4) and (4.3.5) define a contravariant A-linear duality functor (4.3.7) ·∨ : CTCA −→ CPA . 4.3.2 Duality as an adjoint equivalence. Observe that for every object M of cont CPA and every object N of CTCA , the A-modules HomA (M, N ∨ ) and ∨ HomA (N, M ) may both be identified with the module of A-bilinear maps M × N −→ A that are continuous in the first variable. In this way, we obtain a system of bijections ∨ ∨ ∨ op HomCTCA (M, N ∨ ) := Homcont A (M, N ) ' HomA (N, M ) =: HomCPA (M , N ). (4.3.8)

Proposition 4.3.1. The bijections (4.3.8) defines an adjunction of functors op ·∨ : CTCA  CPA : ·∨ .

It is actually an adjoint equivalence, whose unit and counit are the natural isomorphism η : IdCTCA ' ·∨∨ and ε : ·∨∨ ' IdCPop defined by the biduality A isomorphisms ∼

cont HomA (HomA (M, A), A), (ξ 7−→ ξ(m)),

∼

HomA (Homcont A (N, A), A), (ζ 7−→ ζ(n)),

ε−1 M : M m

−→ 7−→

N n

−→ 7−→

and ηN :

associated to any object M of CPA and to any object N of CTCA . Proof. We only sketch the proof and leave the details to the reader. The naturality with respect to M (in CTCA ) and to N (in CPop A ) of the and as well. bijections (4.3.8) is straightforward, and the expressions for ε−1 η N M To complete the proof, we are left to establish that for all M (resp., N ) −1 (resp., ηN ) is an isomorphism in CPA (resp., in in CPA (resp., in CTCA ), εM CTCA ). The compatibility of the duality functors with countable direct sums (in CPA ) and countable products (in CTCA ), and the fact that every object in CPA (resp., in CTCA ) is a countable direct sum (resp., a countable product) of finitely generated projective A-modules, allow us to reduce the proof to the special case in which M (resp., N ) is a finitely generated projective A-module. Then it is  straightforward.

4.3. The Duality Between CPA and CTCA

89

Corollary 4.3.2. The duality functors (4.3.3) and (4.3.7) are (anti)equivalences of categories.  Let B be a Dedekind ring that is an A-algebra. The reader may easily establish the compatibility between the duality functors between CPA and CTCA and between CPB and CTCB , and of the “base change” functors · ⊗A B : CPA −→ ˆ A B : CTCA −→ CTCB . CPB and · ⊗ 4.3.3 Applications. Various results about the category CPA may be transferred by duality to the category CTCA . (i) For instance, the morphisms ϕ in HomA (A(N) , A(N) ) are in bijection with “infinite matrices” (ϕij )(i,j)∈N×N in AN×N that admit only a finite number of nonzero entries in each column, by the usual formula, valid for all x = (xi )i∈N and y = (yj )j∈N in A(N) : y = ϕ(x) ⇐⇒ yi =

X

ϕij xj .

(4.3.9)

j∈N N N By duality, this implies that the morphisms ϕ in Homcont A (A , A ) are in N×N bijection with the matrices (ϕij )(i,j)∈N×N in A that contain only a finite number of nonzero entries in each row, still by means of formula (4.3.9).

(ii) From Kaplansky’s result concerning the freeness of modules of infinite rank in CPA (Proposition 4.1.2), we get the following refinement of the description of objects of CPA as countable products of finitely generated projective A-modules in Proposition 4.2.4: Proposition 4.3.3. Let N be an object in CTCA . Either the topology of N is discrete and the A-module N is finitely generated and projective, or N is isomorphic to AN as a topological A-module.  (iii) When A is a field k, the objects of CTCA are precisely the linearly compact k-vector spaces, introduced by Lefschetz and Chevalley [78, Chapter II, §6] that admit a countable basis of neighborhoods of zero. In this case, the category CPA is the category of k-vector spaces of countable dimension. A number of results from linear algebra over a field k may be transported, by duality, to results concerning CTCk . In this way, from the basic facts about k-vector spaces of countable dimension and their k-linear maps, we derive the following results (which go back to Toeplitz and K¨ othe; see notably [115] and [73]; see also [38]): Proposition 4.3.4. Let k be a field. (1) In the category CTCk , every object is isomorphic either to k n for some nonnegative integer n or to k N .

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

(2) For every morphism ϕ : N1 −→ N2 in CTCk , there exist objects K, N, and C and isomorphisms ∼

∼

u : N1 −→ K ⊕ N and v : N2 −→ C ⊕ N in CTCk such that the morphism ϕ˜ := vϕu−1 : K ⊕ N −→ C ⊕ N is the “block diagonal” morphism 0 ⊕ IdN , which sends (k, n) ∈ K ⊕ N to (0, n) in  C ⊕ N.

4.4 Strict Morphisms, Exactness and Duality 4.4.1

Strict morphisms in CTCA . We shall say that a morphism ϕ : N1 −→ N2

(4.4.1)

in CTCA is strict if ϕ is a strict morphism of topological groups, namely if the induced map ϕ˜ : N1 / ker ϕ −→ im ϕ, [x] 7−→ ϕ(x), is a homeomorphism when N1 / ker ϕ (resp., im ϕ) is equipped with the topology quotient of the topology of N1 (resp., with the topology induced by the topology of N2 ). Since N1 and therefore its quotient N1 / ker ϕ are complete, if the morphism (4.4.1) is strict, its image im ϕ also is complete, hence closed in N2 , and therefore defines an object of CTCA by Proposition 4.2.3. cont (N1 , N2 ) are precisely the maps Accordingly, the strict morphisms in HomA ϕ : N1 −→ N2 for which there exist an object I in CTCA , defined by some closed A-submodule of N2 , and an open surjective morphism ϕ0 in Homcont A (N1 , I) such that ϕ admits the factorization ϕ0

ϕ : N1  I ,→ N2 . 4.4.2 Finite-rank morphisms. We want to show that every morphism in CTCA whose image is finitely generated is strict. This will follow from the following result, of independent interest: Proposition 4.4.1. Let P be a finitely generated projective A-module, and let N be an object of CTCA . Let K denote the field of fractions of A. For every morphism of A-modules ϕ : P −→ N, the following conditions are equivalent: (1) ϕ is injective; ˆ A K is injective; (2) the K-linear map ϕK : PK := P ⊗A K −→ NK := N ⊗

4.4. Strict Morphisms, Exactness and Duality

91

(3) there exists U in U(N ) such that the composite morphism of A-modules ϕ

P −→ M  N/U is injective. Proof. The implications (3) ⇒ (2) ⇒ (1) are straightforward. To prove the implication (1) ⇒ (3), observe that the inverse images ϕ−1 (V ) of the submodules V of N in U(N ) constitute a family of saturated submodules of E stable under finite intersection. Since P has finite rank, we may consider U in U(N ) such that ϕ−1 (U ) has minimal rank. Then for every V ∈ U(N ), the intersection ϕ−1 (U ) ∩ ϕ−1 (V ) is saturated in N , of rank at most the rank of ϕ−1 (U ), and therefore coincides with ϕ−1 (U ). This shows that \ ϕ−1 (U ) = ϕ−1 (V ). (4.4.2) V ∈U (N )

When (1) is satisfied, we have \ \ ϕ−1 (V ) = ϕ−1 ( V ∈U (N )

V ) = ϕ−1 (0) = {0},

V ∈U (N )

and therefore ϕ−1 (U ) = {0}, and (3) is satisfied.



Corollary 4.4.2. If N is an object of CTCA and if P is a finitely generated Asubmodule of N , then P is a finitely generated projective A-module, and the topology of N induces the discrete topology on P . Moreover, the saturation P sat := {n ∈ N | ∃α ∈ A \ {0}, αn ∈ P } of P in N is also a finitely generated A-module. Proof. The projectivity of P is clear. Moreover, according to Proposition 4.4.1, we may choose an element U of U(N ) such that U ∩ P = 0. Since U is open in N , this shows that the topology on P induced by that of M is the discrete topology. To prove that P sat is finitely generated, observe that the composite map U ,−→N −→ N/P is injective and fits into an exact sequence of A-modules: q

0 −→ U −→ N/P −→ N/(U + P ) −→ 0. Since U is torsion-free, the map q defines by restriction an injection of torsion submodules: q : (N/P )tor ,−→(N/(U + P ))tor .

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

Since N/U hence N/(U + P ) is a finitely generated A-module, this proves that (N/P )tor also is finitely generated. Finally, the short exact sequence 0 −→ P −→ P sat −→ (N/P )tor −→ 0 shows that P sat is finitely generated.



Corollary 4.4.3. If the image of some morphism in CTCA is a finitely generated  A-module, then that morphism is strict. The following proposition completes Proposition 4.4.1 when ϕ has a saturated image. Proposition 4.4.4. Let P be a finitely generated projective A-module, and let N be an object of CTCA . For every morphism of A-modules ϕ : P −→ N, the following conditions are equivalent: (1) The morphism ϕ is injective and its image ϕ(P ) is a saturated A-submodule of N . (2) There exists U in U(N ) such that the composite morphism of A-modules ϕ

P −→ M  N/U is injective and its image is a saturated A-submodule of N/U . Proof. The implication (2) ⇒ (1) is straightforward. To establish the converse implication, let us assume that ϕ is injective with saturated image, and consider a defining sequence (Ui )i∈N in U(N ). We define Ni := M/Ui , and we denote by pi : N −→ Ni the canonical quotient map and by ϕi := pi ◦ ϕ : P −→ Ni its composition with ϕ. Let p be a nonzero prime ideal of A, and let Fp := A/p be its residue field. The Fp -vector space NFp := N/pN may be identified with the tensor product N ⊗A Fp . Equipped with the topology quotient of the topology of N , it coincides ˆ A Fp , and with the object of CTCFp defined as the completed tensor product N ⊗ also with the projective limit limi Ni,Fp of the finite-dimensional Fp -vector spaces ←− Ni,Fp equipped with the discrete topology. The Fp -linear map ϕFp : PFp −→ NFp

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4.4. Strict Morphisms, Exactness and Duality

is injective, since ϕ is injective and its image is saturated in N . The map ϕFp is also defined by the compatible system of Fp -linear maps ϕi,Fp : PFp −→ Ni,Fp . Therefore, \

ker ϕi,Fp = ker ϕFp = {0}.

i∈N

This shows the existence of some ι(p) in N such that ker ϕi,Fp = {0} for every i ∈ N≥ι(p) . Moreover, according to Proposition 4.4.1, there exists ι0 in N such that ϕι0 (and consequently ϕi for all i ∈ N≥ι0 ) is injective. The set T of nonzero prime ideals in A such that ϕι0 ,Fp : PFp −→ Nι0 ,Fp is noninjective is finite. It is indeed defined by the vanishing of the nonzero section ∧rk P ϕι0 of (∧rk P P )∨ ⊗ ∧rk P Nι0 over Spec A. Finally, if we let ι1 := max(ι0 , max ι(p)), p∈T

then for all i ∈ N≥ι1 and every nonzero prime ideal p of A, ϕi,Fp is injective, and therefore ϕi is injective with a saturated image.  When condition (2) in Proposition 4.4.4 is satisfied by some U in U(N ), it is clearly also satisfied by every element U 0 in U(N ) contained in U . This implies the following corollary: Corollary 4.4.5. Let N be an object of CTCA and let P be a finitely generated A-submodule of N . The quotient topological A-module N/P is an object of CTCA if (and only if ) P is saturated in N .  4.4.3 Strict short exact sequences and duality. f

g

We shall say that a diagram

0 −→ N1 −→ N2 −→ N3 −→ 0

(4.4.3)

of object and morphisms in CTCA is a strict short exact sequence if it is a short exact sequence of A-modules and if the morphisms f and g are strict — in other words, if f establishes an isomorphism of topological A-modules from N1 onto ker g and if g is a surjective open map. This last condition on g is satisfied in particular if g admits a section h (that cont (N3 , N1 ). When this holds, the strict short exact is, a right inverse) in HomA sequence (4.4.3) is said to be split, and h is called a splitting of (4.4.3). We shall say that a closed A-submodule N 0 of some object N in CTCA is supplemented in CTCA when there exists a closed A-submodule N 00 of N such that the sum map N 0 ⊕ N 00 −→ N, (n0 , n00 ) 7−→ n0 + n00 ,

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

is an isomorphism of topological A-modules (necessarily in CTCA , by Proposition 4.2.3). Observe that the strict short exact sequence (4.4.3) is split precisely when the closed submodule f (N1 ) of N2 is supplemented in CTCA . Indeed, the splittings h in of (4.4.3) are in bijection with the “topological supplements” N 00 in CTCA of f (N1 ) by the map that sends h to its image h(N3 ). Let M1 , M2 , and M3 be objects in CPA , and let N1 := M1∨ , N2 := M2∨ , and N3 := M3∨ be the dual objects in CTCA . Let S (resp., T ) be the subset of HomA (M3 , M2 ) × HomA (M2 , M1 ) (resp., cont of Homcont A (N1 , N2 ) × HomA (N2 , N3 )) consisting of the pairs of morphisms (i, p) (resp., (j, q)) such that the diagram p

i

0 −→ M3 −→ M2 −→ M1 −→ 0

(4.4.4)

is an exact sequence of A-modules (resp., such that j

q

0 −→ N1 −→ N2 −→ N3 −→ 0

(4.4.5)

is a strict short exact sequence in CTCA ). For each i ∈ {1, 2, 3}, the topological dual Ni∨ of Ni will be identified with Mi by the biduality isomorphisms εMi of Proposition 4.3.1. ∼

Proposition 4.4.6. With the above notation, one defines a bijection δ : S −→ T by the formula (4.4.6) δ(i, p) := (p∨ , i∨ ). The inverse bijection δ −1 is given by δ −1 (j, q) = (q ∨ , j ∨ ).

(4.4.7)

Moreover, for all (i, p) in S (resp., for all (j, q) in T ), the short exact sequence of A-modules (4.4.4) (resp., the strict short exact sequence in CTCA (4.4.5)) is split. Proof. (1) For all (i, p) in S, the associated short exact sequence of A-modules (4.4.4) is split, since M1 is projective. This implies that the diagram p∨

i∨

0 −→ M1∨ −→ M2∨ −→ M3∨ −→ 0 derived from (4.4.4) by duality is a split strict short exact sequence in CTCA . This shows in particular that (p∨ , i∨ ) belongs to T . (2) Consider an element (j, q) of T . cont (., A) to the strict short exact sequence If we apply the functor HomA (4.4.5), we get the following exact sequence of A-modules: q∨

j∨

0 −→ N3∨ −→ N2∨ −→ N1∨ .

95

4.4. Strict Morphisms, Exactness and Duality

Indeed, the injectivity of q ∨ follows from the surjectivity of Q, and the equality im q ∨ = ker j ∨ means that the continuous A-linear maps from N2 to A that vanish on j(N1 ) are in bijection — by means of factorization through q — with the A-linear forms from N3 to A. This follows from the fact that q is continuous and open, with kernel j(N1 ). The morphism j ∨ : N2∨ −→ N1∨ may be factorized as [j ∨ ]

ι

j ∨ = ι ◦ [j ∨ ] : N2∨ −→ im j ∨ ,−→ N1∨ ,

(4.4.8)

where ι denotes the inclusion morphism. Moreover, the A-module im j ∨ , as every submodule of N1∨ , is an object of CPA . Finally, the diagram q∨

[j ∨ ]

0 −→ N3∨ −→ N2∨ −→ im j ∨ −→ 0

(4.4.9)

is a short exact sequence in CPA . According to part (1) of the proof, the short exact sequence (4.4.9) is split and determines by duality a split short exact sequence in CTCA . Moreover, the factorization (4.4.8) shows that j = [j ∨ ]∨ ◦ ι∨ . Therefore, the following diagram in CTCA is commutative, and its rows are short strict exact sequences: [j ∨ ]∨

0 −−−−→ (im j ∨ )∨ −−−−→ x  ι∨ 0 −−−−→

N1

q

N2 −−−−→



N3 −−−−→ 0



(4.4.10)

q

j

−−−−→ N2 −−−−→ N3 −−−−→ 0.

This implies that ι∨ is an isomorphism in CTCA . In particular, the second line in (4.4.10), like the first one, is a split short exact sequence in CTCA . Moreover, since the duality functor ·∨ is an equivalence of categories from CPA to CTCA , this also proves that ι is an isomorphism in CPA . This establishes the equality im j ∨ = N1∨ and the exactness of q∨

j∨

0 −→ N3∨ −→ N2∨ −→ N1∨ −→ 0. This shows that (q ∨ , j ∨ ) belongs to S. The fact that the maps between S and T defined by (4.4.6) and 4.4.7) are inverse to each other is a straightforward consequence of the “biduality” established  in Proposition 4.3.1. For later reference, we spell out some consequences of the results on short exact sequences in CPA and CTCA established in the previous proposition: Proposition 4.4.7. Let M and M 0 be two objects in CPA and let N := M ∨ and N 0 := M 0∨ be their duals in CTCA . Let α : M −→ M 0 be a morphism of A-modules and let β := α∨ : N 0 −→ N 0 denote the dual morphism, in Homcont A (N , N ).

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Chapter 4. Countably Generated Projective Modules over Dedekind Rings

(1) The following two conditions are equivalent: E1 : The morphism β is surjective and strict. E2 : The morphism α is injective and its cokernel is a projective A-module. When these conditions are realized, im α is a direct summand of M 0 and ker β is a closed submodule of N supplemented in CTCA . (2) The following two conditions are equivalent: F1 : The morphism β is injective and strict, and the topological A-module N/im β is an object of CTCA . F2 : The morphism α is surjective. When these conditions are realized, ker α is a direct summand of M and im β  is a closed submodule of N supplemented in CTCA . Corollary 4.4.8. A closed A-submodule N 0 of an object N in CTCA is supplemented in CTCA if and only if the quotient topological A-module N/N 0 is an object of CTCA .  Corollary 4.4.9. Let M be an object of CPA and let N := M ∨ be the dual object in CTCA . Every submodule of M (resp., of N ) in FS(M ) (resp., in U(N )) is supplemented in M (resp., in N ). Moreover, there is an inclusion-reversing bijection ∼

·⊥ : FS(M ) −→ U(N ). It sends a module M 0 in FS(M ) to M 0⊥ := {ξ ∈ N | ∀m ∈ M 0 , ξ(m) = 0}. The inverse bijection sends an element U in U(N ) to U ⊥ := {m ∈ M | ∀ξ ∈ U, ξ(m) = 0}. Moreover, when U = M 0⊥ , there exists a unique isomorphism of A-modules ∼ I : N/U −→ M 0∨ such that if we denote by iM 0 : M 0 ,→ M the inclusion morphism, the following diagram is commutative: N   pU y

=

−−−−→ M ∨  i∨ y M0

(4.4.11)

I

∼

N/U −−−−→ M 0∨ . 

4.4. Strict Morphisms, Exactness and Duality

97

4.4.4 Strict morphisms and conditions Ded1 , Ded2 , Ded3 . The significance of being strict for a morphism in CTCA turns out to depend on which of the conditions Ded1 , Ded2 , Ded3 the base ring A satisfies. In this paragraph, we present several results that illustrate this point. We will return to constructions of nonstrict morphisms in the next section, devoted to examples. Proposition 4.4.10. If the Dedekind ring A satisfies Ded1 — that is, if A is a field — then every morphism in CTCA is strict. Proof. This follows from the description of morphisms in CTCk in Proposition 4.3.4. Indeed, with the notation of this proposition, ϕ˜ = 0 ⊕ IdN is clearly a strict morphism (with kernel K ⊕ {0} and image {0} ⊕ N ), and consequently ϕ = v −1 ϕu ˜ is also strict.  In paragraph 4.6.2 below, we shall see that if A satisfies Ded2 — that is, if A is a complete discrete valuation ring — then there exist bijective morphisms in CTCA that are not strict. Proposition 4.4.11. When the Dedekind ring A satisfies Ded3 , a morphism ϕ : N1 −→ N2 in CTCA is strict if and only if its image im ϕ is closed in N2 . Moreover, for every two objects N1 and N2 of CTCA , an A-linear map ϕ : N1 −→ N2 is continuous (hence defines a morphism in CTCA ) if and only if its graph is closed in N1 ⊕ N2 . Proof. (1) We have already observed that the image of every strict morphism is closed (for an arbitrary Dedekind ring A). Conversely, if a morphism ϕ : N1 −→ N2 in CTCA has its image im ϕ closed in N2 , then we may form the following diagram in CTCA , which is an exact sequence of A-modules: j

ϕ

0 −→ ker ϕ −→ N1 −→ im ϕ −→ 0,

(4.4.12)

where j denotes the inclusion morphism. By applying the functor HomA (., A) — cont (., A) on CTCA when A satisfies Ded3 , as shown in which coincides with HomA Proposition 4.2.5 — to the exact sequence (4.4.12), we obtain an exact sequence of A-modules: ϕ∨ j∨ 0 −→ (im ϕ)∨ −→ N1∨ −→ (ker ϕ)∨ . Since every A-submodule of some object in CPA is again an object in CPA , we finally obtain the following exact sequence in CPA : ϕ∨

j∨

0 −→ (im ϕ)∨ −→ N1∨ −→ im j ∨ −→ 0. The equivalence of conditions E1 and E2 in Proposition 4.4.7 finally shows that the morphism ϕ : N1 −→ im ϕ, already known to be surjective, is also strict. This shows that ϕ : N1 −→ N2 is strict.

98

Chapter 4. Countably Generated Projective Modules over Dedekind Rings (2) For every map ϕ : N1 −→ N2 , its graph Gr ϕ := {(n, ϕ(n)), n ∈ N}

is also the inverse image of the diagonal ∆N2 in N2 × N2 by the map (ϕ, IdN2 ) : N1 × N2 −→ N2 × N2 . When ϕ is continuous, (ϕ, IdN2 ) also is continuous, and Gr ϕ = (ϕ, IdN2 )−1 (∆N2 ) is closed, since the topology of N2 is Hausdorff and accordingly ∆N2 is closed in N2 × N2 . Conversely, if ϕ is a continuous A-linear map and if its graph Gr ϕ is closed in N1 × N2 , then Gr ϕ defines an object of CTCA by Proposition 4.2.3, and the two projections pri : Gr ϕ −→ Ni , i = 1, 2, are morphisms in CTCA . By construction, the map pr1 : Gr ϕ −→ N1 is bijective. According to part (1) of the proof, it is therefore an isomorphism in CTCA , and therefore ϕ = pr2 ◦ pr−1 1  is finally a morphism in CTCA . When the ring A is countable — notably when A is the ring of integers of some number field — Proposition 4.4.11 admits an alternative proof that does not rely on the “automatic continuity” of morphisms of A-modules established for general Dedekind rings satisfying Ded3 . Indeed, for every object M of CTCA , the topology of M may be defined by a complete metric (this follows from the countability assumption in condition CTC2 ). If, moreover, A is countable, the topological space M is separable6 and therefore the additive group (M, +) is a Polish topological group. Remarkably, the open mapping and the closed graph theorems are valid for continuous morphisms of Polish topological groups, by a classical theorem of Banach,7 and these theorems immediately yield the conclusions of Proposition 4.4.11 if A is countable. Corollary 4.4.12. When the Dedekind ring A satisfies Ded3 , every morphism ϕ : N −→ N 0 in CTCA whose cokernel coker ϕ := N 0 /ϕ(N ) is a finitely generated A-module is an open map, and therefore a strict morphism. Proof. Let (n01 , . . . , n0k ) be a finite family of elements of N2 the classes of which generate the A-module coker ϕ, and let ϕ˜ : N ⊕ A⊕k −→ N 0 6 This follows from Propositions 4.2.4 and 4.3.3, which also show that, conversely, if M 6= 0 and the topological space M is separable, then A is necessarily countable. 7 See [5]. This theorem now appears as a special case of the “th´ eor` eme du graphe souslinien”, concerning morphisms of Polish topological groups, presented, for instance, in [23, Chapitre IX, §6, no. 8, Th´ eor` eme 4].

4.4. Strict Morphisms, Exactness and Duality

99

be the continuous morphism of A-modules defined by the formula ϕ(n, ˜ a1 , . . . , ak ) := ϕ(n) + a1 n01 + · · · + ak n0k for all n ∈ N and all (a1 , . . . , ak ) ∈ Ak . By construction, the morphism ϕ˜ is surjective, and therefore, according to Proposition 4.4.11, it is strict. In particular, it is an open map. Consequently, the map ϕ : N −→ N 0 — which is the composition of ϕ˜ and the inclusion map N ,−→N ⊕ A⊕k , which is clearly open — is also open.  Let us finally indicate that for any Dedekind ring A that satisfies Ded3 , there exist continuous endomorphisms of the A-module AN that are injective, with dense image, but are not strict (see, for instance, Proposition 4.6.4 below).

4.4.5 Strict injective morphisms and extensions of scalars. The following proposition is stated for further reference. Its proof is straightforward and left to the reader. Proposition 4.4.13. Let f : N 0 −→ N be a strict injective morphism in CTCA , and let (Ui )i∈N be a defining sequence in U(N ). (1) The sequence (Ui0 )i∈N := (f −1 (Ui ))i∈N is a defining sequence in U(N 0 ). Moreover, for every i ∈ N, the morphism of (finitely generated projective) Amodules fi : Ni0 := N 0 /Ui0 −→ Ni := N/Ui , defined by fi (x0 + Ui0 ) = f (x0 ) + Ui for all x0 ∈ M 0 , is injective. When the topological A-modules N and N 0 are identified with the projective limits limi Ni and limi Ni0 , the morphism f : N 0 −→ N is identified ←− ←− with the morphism limi fi . ←− (2) For every field extension L of the field of fractions K of A, the topological 0 L-modules ML0 and ML may be identified with limi Ni,L and limi Ni,L , and ←− ←− 0 the morphism fL : NL −→ Nl with the morphism limi fi,L . ←− In particular, like the morphisms fi,L , the morphism fL is injective.



Let us emphasize that with the notation of Proposition 4.4.13, if the injective morphism f : N 0 −→ N is not assumed to be strict, its base change fK may be noninjective. (For instance, the morphism β1 : ZN −→ ZN in CTCZ considered in Proposition 4.6.1, 1) below is injective, but β1Q : QN −→ QN admits the line Q.(p−k )k∈N as kernel.)

100 Chapter 4. Countably Generated Projective Modules over Dedekind Rings

4.5 Localization and Descent Properties In this monograph, we shall not investigate systematically the descent 4.5.1 properties satisfied by the categories CPA and CTCA or by the diverse “infinitedimensional vector bundles” defined in terms of them, and we shall content ourselves with a few observations in this paragraph and in paragraph 9.1.3 below. As already indicated, the duality between the categories CP. and CTC. is easily seen to be compatible with the base change functors · ⊗A B : CPA −→ CPB

and

b A B : CTCA −→ CTCB ·⊗

associated to some ring morphism A −→ B between Dedekind rings. This allows one, for proving localization or descent properties of these categories, to concentrate on the categories CP. of countably generated projective modules over Dedekind rings. The descent properties of projectivity have been investigated in a general setting by Gruson and Raynaud in [94]. They show8 in particular that for every commutative ring R and every faithfully flat commutative R-algebra R0 , an Rmodule M is projective if (and only if) the R0 -module M ⊗R R0 is projective. In the next paragraphs, we simply spell out two simple consequences of this projectivity criterion, combined with basic faithfully flat descent (see, for instance, [12, Section 6.2, Examples A and D]). Since this monograph concentrates on “provector bundles” rather than on “ind-vector bundles”, we formulate them in terms of the categories CTC. . 4.5.2

Let U and V be two open nonempty subsets of Spec A such that Spec A = U ∪ V.

In other words, U and V are the complements U := Spec A \ F

and V := Spec A \ G

of two disjoint finite subsets F and G of the set (Spec A)0 of nonzero prime ideals of A. In fact, U, V , and U ∩ V define affine open subschemes of Spec A. If K denotes the fraction field of A, and for all p ∈ (Spec A)0 , vp denotes the p-adic valuation of K, we have AU := Γ(U, OSpec A ) = {x ∈ K | ∀p ∈ (Spec A)0 \ F, vp (x) ≥ 0}, and similar descriptions of AV := Γ(V, OSpec A ) and AU ∩V := Γ(U ∩ V, OSpec A ), with F replaced by G and by F ∪ G, respectively. 8 See [94, Seconde partie, §3.1]. For a proof of this faithfully flat descent property, the reader may refer to [108, Sections 10.79–10.94]. The importance of the criteria of projectivity in [94] for the study of “infinite-dimensional vector bundles in algebraic geometry” was emphasized by Drinfeld in [39].

4.6. Examples

101

With this notation, the datum of an object N in CTCA is equivalent to the data of some objects NU of CTCAU and NV of CTCAV and of some “gluing isomorphism” in CTCAU ∩V : ∼

b AV AU ∩V . b AU AU ∩V −→ NV ⊗ NU ⊗ b A AU and Indeed, the equivalence maps N to the topological modules NU := N ⊗ b A AV and to the gluing isomorphism derived from the canonical isoNV := N ⊗ morphisms of A-algebras AU ⊗AU AU ∩V ' AU ∩V ' AV ⊗AV AU ∩V . 4.5.3 Let U := Spec A \ F be a nonempty open subscheme of Spec A. For all p in F, we may consider the complete discrete valuation ring Aˆp defined as the ˆ p is the completion of K with respect p-adic completion of A; its fraction field K to the p-adic valuation vp . Then the datum of an object N in CTCA is equivalent to the data of some object NU of CTCAU , and for every p ∈ F, of an object Np in CTCAˆp and some gluing isomorphism in CTCKˆ p : ∼

ˆ p. ˆ p −→ Np ⊗ b Aˆ K b AU K NU ⊗ p b A AU and Np := The equivalence maps N to the topological modules NU := N ⊗ ˆ b N ⊗A Ap and to the gluing morphisms deduced from the canonical isomorphism of A-algebras ˆ p ' Aˆp ⊗ ˆ K ˆp ' K ˆ p. AU ⊗AU K Ap

4.6 Examples In this section, we discuss some simple examples of objects and morphisms in the categories CPA and CTCA when A is the ring Zp or Z. These examples should make clear that nonstrict morphisms in the categories CTCZp and CTCZ are not “pathologies” but occur “naturally”, and that exact sequences in the categories CPZ and CTCZ and their duality properties must be handled with some care. We denote by p a prime number and by |.|p the p-adic norm on the ring Zp of p-adic integers, defined by |pn u|p := p−n for every n ∈ N and every u ∈ Zp× . 4.6.1 Subobjects and duality in CPZ and CTCZ . Besides condition E2 (introduced in Proposition 4.4.7 above), we may consider the following condition: E02 : The morphism α is injective and its image is saturated in M 0 . Similarly, besides condition F1 , we may consider the following:

102 Chapter 4. Countably Generated Projective Modules over Dedekind Rings F10 : The morphism β is injective and its image is closed and saturated in N2 . Clearly, condition E2 implies condition E02 , and condition F1 implies condition F10 . However, the converse implications do not hold in general. This is demonstrated, when A = Z, by Proposition 4.6.1 below. Let (εn )n∈N denote the canonical basis of Z(N) , defined by εn (k) := δnk for every (n, k) ∈ N2 . We shall identify the dual in CTCZ of the object M := Z(N) of CPZ with the Z-module N := ZN equipped with the product topology of the discrete topology on Z, by means of the map ∼

HomZ (Z(N) , Z) −→ ξ 7−→

ZN , (ξ(εn ))n∈N .

(4.6.1)

Proposition 4.6.1. (1) If we define two morphisms of Z-modules α1 : Z(N) −→ Z(N) and π1 : Z(N) −→ Z[1/p] by α1 (εn ) := εn − pεn+1 and π1 (εn ) :=

1 pn

for every n ∈ N, then α1 and π1 fit into the following short exact sequence: α

π

1 1 0 −→ Z(N) −→ Z(N) −→ Z[1/p] −→ 0.

(4.6.2)

Moreover, the dual morphism β1 := α1∨ ∈ Homcont (ZN , ZN ) is injective, Z N and if we define a Z-linear map σ1 : Z −→ Zp /Z by " # X k σ1 ((yk )k∈N ) := p yk , k∈N

then the following diagram is a short exact sequence: β1

σ

1 0 −→ ZN −→ ZN −→ Zp /Z −→ 0.

(4.6.3)

(2) If we define two morphisms of Z-modules α2 : Z(N) −→ Z(N) and π2 : Z(N) −→ Z[1/p]/Z by ( −pε0 when n = 0, α2 (εn ) := εn−1 − pεn when n ≥ 1, and by  π2 (εn ) :=

1 pn+1

 for every n ∈ N,

then α2 and π2 fit into the following short exact sequence: α

π

2 2 0 −→ Z(N) −→ Z(N) −→ Z[1/p]/Z −→ 0.

(4.6.4)

4.6. Examples

103

Moreover, the dual morphism β2 := α2∨ in Homcont (ZN , ZN ) is injective, Z and if we define a continuous Z-linear map σ2 : ZN −→ Zp by X σ2 ((yk )k∈N ) := pk yk , k∈N

then the following diagram is a short exact sequence: β2

σ

2 0 −→ ZN −→ ZN −→ Zp −→ 0.

(4.6.5)

Observe that α1 satisfies E20 and not E2 , since β1 does not satisfy E1 , and that β2 satisfies F01 and not F1 , since α2 does not satisfy F2 . As already mentioned, Proposition 4.6.1 also demonstrates how the categories CPZ and CTCZ are “badly behaved” with respect to quotients. Proof. (1) We use the identification of Z-modules Z(N) εn

∼

−→ 7−→

Z[X], X n,

and

ZN (yk )k∈N

∼

−→ 7−→

Z[[X]], P k k∈N yk X .

The isomorphism (4.6.1) becomes the isomorphism ∼

HomZ (Z[X], Z) −→ Z[[X]] induced by the residue pairing Z[[X]] × Z[X] −→ (f, P ) 7−→

Z, ResX=0 [f (X)P (1/X)dX/X].

(4.6.6)

The diagram (4.6.2) may be written α

π

1 1 0 −→ Z[X] −→ Z[1/p] −→ 0, Z[X] −→

(4.6.7)

α1 (P ) := (1 − pX)P

(4.6.8)

where and π1 (P ) = P (1/p). The exactness of (4.6.7), hence of (4.6.2), follows from the basic properties of polynomials with integer coefficients. Moreover, the expressions (4.6.8) for α1 and (4.6.6) for the duality pairing between Z[[X]] and Z[X] show that the dual morphism β1 := α1∨ : Z[[X]] −→ Z[[X]] is given by β1 (f ) = (1 − p/X)f + pf (0)/X.

(4.6.9)

104 Chapter 4. Countably Generated Projective Modules over Dedekind Rings The vanishing of β1 (f ) therefore implies the following equality in Q[[X]]: X f (X) = f (0)(1 − p−1 X)−1 = f (0) p−k X k . k∈N

Clearly, the only f in Z[[X]] satisfying this condition is f = 0, and accordingly β1 is injective. To complete the proof of the exactness of (4.6.3), we shall use the following lemma: Lemma 4.6.2. The evaluation morphism η : Z[[X]] −→ f 7−→

Zp , f (p),

(4.6.10)

is surjective. Its kernel is (X − p) Z[[X]]. Proof of Lemma 4.6.2. The surjectivity of η (X − p)Z[[X]] ⊂ ker η also. To establish the converse inclusion, observe map ηZp : Zp [[X]] −→ f 7−→

is clear, and the inclusion that the kernel of the evaluation Zp , f (p),

is (X − p)Zp [[X]], say by the Weierstrass preparation theorem. Therefore, every element f of Z[[X]] in the kernel of η may be written f = (X − p)g

(4.6.11)

for some g in Zp [[X]]. The polynomial (X − p) = −p(1 − X/p) is a unit in Z[1/p][[X]], and equation (4.6.11) shows that the series g, seen as an element of Qp [[X]], actually belongs to Z[1/p][[X]]. Consequently, g belongs to Zp [[X]] ∩ Z[1/p][[X]] = Z[[X]].  The morphism σ1 : Z[[X]] −→ Zp /Z maps g ∈ Z[[X]] to the class of g(p) in Zp /Z. It is clearly surjective, and to establish the exactness of (4.6.3), we are indeed left to show that for all g ∈ Z[[X]], the following two conditions are equivalent: (i) there exists f ∈ Z[[X]] such that g = (1 − p/X)f + pf (0)/X; (ii) the element g(p) of Zp belongs to Z. When (i) holds, then g(p) = f (0) and therefore (ii) also holds. Conversely, when (ii) holds, then g − g(p) is an element of Z[[X]] in the kernel of the evaluation morphism (4.6.10) and, for some h ∈ Z[[X]], we have g − g(p) = (X − p)h.

4.6. Examples

105

Therefore, condition (i) is satisfied by f := Xh + g(p). (2) Similarly, the exact sequences (4.6.4) and (4.6.5) may be written as π

α

2 2 0 −→ Z[X] −→ Z[X] −→ Z[1/p]/Z −→ 0,

where α2 (P ) :=

(4.6.12)

1 − pX P (0) P − P (0) P− = − pP X X X

and π2 (P ) = (1/p) · P (1/p)

mod Z,

and as β2

σ

2 0 −→ Z[[X]] −→ Z[[X]] −→ Zp −→ 0,

(4.6.13)

where β2 (f ) = α2 (f ) := (X − p) f and σ2 (f ) := f (p). We leave the proof of the exactness of (4.6.12) as an elementary exercise on polynomials with integer coefficients. The exactness of (4.6.13) is basically the content of Lemma 4.6.2.  4.6.2 A non-strict bijective morphism in CTCZp . We want to point out that when the base ring A satisfies Ded2 — that is, when A a complete discrete valuation ring — the open mapping and closed graph theorems (as stated in Proposition 4.4.11 for a Dedekind ring A satisfying Ded3 ) do not hold. For definiteness, let us assume that A = Zp , and let N be the object of CTCZp defined as N := ZN p equipped with the product of the discrete topology on each factor Zp . Lemma 4.6.3. Let (ξn )n∈N be an element of ZN such that limn→+∞ |ξn |p = 0. For Pp all x := (xn ) ∈ N = ZN , the series ξ(x) := p n∈N ξn xn converges in Zp equipped with |.|p and defines a Zp -linear map ξ : N −→ Zp . The map ξ belongs to HomCTCZp (N, Zp ) if and only if (ξn )n∈N belongs to (N)

Zp . The graph Gr ξ of ξ is closed in the topological module N ⊕ Zp of CTCZp . Proof. All the assertions are immediate except but possibly the last one. To prove that Gr ξ of ξ is closed in N ⊕ Zp = ZN p ⊕ Zp equipped with the product of the discrete topology on each factor Zp , consider x in N and a in Zp such that (x, a) ∈ / Gr ξ, or equivalently, such that ξ(x) 6= a. There exists a positive integer n0 such that for every n ∈ N>n0 , |ξn |p < |ξ(x) − a|p . N

Then U := {0}{0,...,n0 } × Zp >n0 is an open neighborhood of 0 in N = ZN p such that for all u ∈ U, |ξ(u)|p < |ξ(x) − a|p .

106 Chapter 4. Countably Generated Projective Modules over Dedekind Rings Consequently, for all x ˜ ∈ x + U, |ξ(˜ x) − a|p = |ξ(x) − a|p 6= 0, and (x, a) + U ⊕ {0} is an open neighborhood of (x, a) in N ⊕ Zp disjoint from  Gr ξ. Let us keep the notation of Lemma 4.6.3. According to Proposition 4.2.3, the A-module Gr ξ, equipped with the topology induced by that of N ⊕ Zp , becomes an object of CTCZp . Clearly, the first projection pr1|Gr ξ : Gr ξ −→ N defines a continuous bijective morphism of topological A-modules. By construction, it is a homeomorphism — or equivalently, a strict morphism — precisely when ξ : N −→ Zp is continuous. (N) such that limn→+∞ |ξn |p = 0, This shows that for all (ξn )n∈N in ZN p \ Zp the map pr1|Gr ξ is a nonstrict bijective morphism in HomCTCZp (Gr ξ, N ). 4.6.3 Nonstrict injective morphisms in CTCA when A satisfies Ded3 . The construction in Proposition 4.6.1, (2), may easily be extended to a more general setting, and it provides, for every Dedekind ring A that satisfies Ded3 , examples of injective morphisms in CTCA that are not strict. The ring A[[X]] of formal series with coefficients in A will be equipped with its natural prodiscrete topology (say, defined by its identification with limn A[X]/(X n )). Then it becomes an object of CTCA . ←− By mimicking the arguments in the proof of Proposition 4.6.1, (1), one easily establishes the following proposition. (To prove its assertion (3), choose some nonzero prime ideal p dividing a, and consider the evaluation map g 7→ g(a) from A to the p-adic completion Aˆp of A. On the image ϕa (A[[X]]) of ϕa , this evaluation map takes values in A.) We leave the details of its proof to the reader. Proposition 4.6.4. For all a ∈ A, we define a continuous A-linear map ϕa : A[[X]] −→ A[[X]] by letting ϕa (f ) := (1 − a/X)f + af (0)/X = f − a (f − f (0))/X. (1) The map ϕa is injective if and only if a ∈ / A× . (2) For all a in A, the map ϕa sends A[X] bijectively onto A[X]. In particular, its image ϕa (A[[X]]) is dense in A[[X]]. (3) If A satisfies Ded3 , then for all a ∈ A \ (A× ∪ {0}), the map ϕa is not  surjective.

Chapter 5

Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves In this chapter, we introduce several categories of infinite-dimensional Hermitian vector bundles over an arithmetic curve Spec OK defined by the ring of integers OK of a number field K. These categories are constructed from the categories CPA and CTCA investigated in the previous chapters, specialized to the case of the Dedekind ring A = OK (of type Ded3 ), by enriching the objects and the morphisms by “Hermitian data”. In the applications to Diophantine geometry, we will be mainly interested in pro-Hermitian vector bundles: their underlying “algebraic” objects will be the topological OK -modules in CTCOK . In the basic case OK = Z, these are precisely the pro-Euclidean lattices described in the introduction (see 0.1.1). As already indicated there, they admit alternative descriptions, either (i) in terms of objects b of CTCO and Hilbert spaces EσHilb densely embedded in the completed tensor E K bσ ' E b⊗ ˆ σ C associated to the diverse complex embeddings σ : K ,→ C, products E or (ii) in terms of projective systems q0

q1

qi−1

qi

qi+1

E • : E 0 ←− E 1 ←− · · · ←− E i ←− E i+1 ←− · · · of surjective admissible morphisms of Hermitian vector bundles over Spec OK . The equivalence of these descriptions is elementary, but quite useful in practice. Other constructions described in this chapter are mostly formal, and their details could be skipped at first reading. The last sections of this chapter are devoted to examples that are quite elementary but should convey some feeling of the “concrete significance” of proHermitian vector bundles and the technical subtleties one may encounter when handling them. In particular, in Section 5.6, we introduce the “arithmetic Hardy spaces” b b H(R) and the “arithmetic Bergman spaces” B(R). These pro-Euclidean lattices constitute the archetypes of the pro-Hermitian vector bundles over arithmetic curves that we shall investigate in the sequel, for instance in Chapter 10, when we apply the formalism developed here to study the interaction of complex analytic geometry and formal geometry (over the integers) in diverse Diophantine settings. Furthermore, in Section 5.7 we show, by explicit examples, that the injectivity or surjectivity properties of the morphisms of topological OK -modules and complex Fr´echet and Hilbert spaces underlying a morphism of pro-Hermitian vector bundles are in general rather subtly related. © Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0_5

107

108 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves We denote by K a number field, by OK its ring of integers, and by π : Spec OK −→ Spec Z the morphism of schemes from Spec OK to Spec Z.

5.1

Definitions

5.1.1 Ind-Hermitian vector bundles. We define an ind-Hermitian vector bundle over the arithmetic curve Spec OK as a pair F := (F, (k.kσ )σ:K,→C ), where F is an object of CPOK — namely a countably generated projective OK module — and (k.kσ )σ:K,→C is a family of pre-Hilbert norms on the complex vector spaces Fσ := F ⊗σ C derived from the OK -module E by the base changes σ : OK −→ C. Moreover, the family (k.kσ )σ:K,→C is required to be invariant under complex conjugation.1 0 An isometric isomorphism ϕ : F −→ F between two ind-Hermitian vector 0 bundles F := (F, (k.kσ )σ:K,→C ) and F := (F 0 , (k.k0σ )σ:K,→C ) over Spec OK is ∼ an isomorphism of OK -modules ϕ : F −→ F 0 such that for every embedding ∼ σ : K ,→ C, the C-linear isomorphism ϕσ : Fσ −→ Fσ0 is isometric with respect to 0 the norms k.kσ and k.kσ . 5.1.2 Pro-Hermitian vector bundles. dle over Spec OK is the data

By definition, a pro-Hermitian vector bun-

b := (E, b (E U ) E b ) U ∈U (E) b of CTCO , and for every U in U(E) b (that is, for every open of an object E K b of a structure of a Hermitian vector bundle saturated OK -submodule U of E), E U := (EU , (k.kU,σ )σ:K,→C ) b over Spec OK on the finitely generated projective OK -module EU := E/U. b such Moreover, for every two open saturated OK -submodules U and U 0 of E 0 that U ⊂ U , the surjective morphism of OK -modules pU 0 U : EU −→ EU 0 is required to define a surjective admissible morphism of Hermitian vector bundles from E U onto E U 0 (see (1.4.2)). 1 Namely, for every field embedding σ : K ,→ C, the norm k.k on F and the norm k.k on F σ σ σ σ attached to the complex conjugate embedding σ coincide through the C-antilinear isomorphism

F ⊗σ C e⊗λ

' 7−→

In other words, kvkσ = kvkσ for every v ∈ Fσ .

F ⊗σ C, e ⊗ λ := e ⊗ λ.

5.1. Definitions

109

b −→ E b 0 between two pro-Hermitian vector An isometric isomorphism ψ : E bundles b = (E, b (E U ) E b ) U ∈U (E) and

b 0 := (E b 0 , (E 0U 0 ) 0 E b0 ) ) U ∈U (E

∼ b0 b −→ is an isomorphism ψ : E E of topological2 OK -modules such that for every U 0 b with image U := ψ(U ) (necessarily in U(E b 0 )), the induced isomorphism in U(E), of OK -modules b b 0 /U 0 ψU : EU := E/U −→ EU0 0 := E 0

defines an isometric isomorphism of Hermitian vector bundles from E U onto E U 0 . 5.1.3 The complex topological vector spaces associated to a pro-Hermitian vecb := (E, b (E U ) tor bundle. Let E b ) be a pro-Hermitian vector bundle over U ∈U (E) Spec OK , as above, and let σ : K,−→C be a field embedding. 5.1.3.1. We may apply the functor ˆ OK ,σ : CTCOK −→ CTCC .⊗ b We thus define the completed tensor product to the topological OK -module E. bσ := E b⊗ ˆ OK ,σ C. E bσ may be identified with an inverse limit of finite-dimensional By its definition, E complex vector spaces equipped with the discrete topology: bσ ' E

lim EU,σ . ←− b U ∈U (E)

Besides this “prodiscrete” topology, which makes it an object of CTCC , the bσ also admits a canonical separated and locally convex complex vector space E topology (see 4.2.4): it is defined by taking the projective limit limU ∈U (E) b EU,σ ←− bσ in the category of locally convex complex vector spaces when each that defines E finite-dimensional complex vector space EU,σ is equipped with its usual (separated and locally convex) topology. bσ is a nuclear Fr´echet space. In fact, in the Equipped with this topology, E ∼ bσ −→ category CTCC , there exists an isomorphism ϕ : E CI for a countable set I I (where C is equipped with the product of the discrete topology on each factor C), and every such isomorphism is an isomorphism of complex locally convex vector ∼ b0 b −→ to Proposition 4.2.5, every isomorphism ψ : E E of OK -modules is actually an isomorphism of topological OK -modules. 2 According

110 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves bσ (resp., CI ) is equipped with its natural locally convex topology spaces when E (resp., with the product of the usual topology on each factor C). bσ associated to some proFrom now on, the completed tensor products E b b in CTCO ) will Hermitian vector bundle E (or more generally to some object E K be always be endowed with their canonical topologies of a complex Fr´echet space. We shall denote by b −→ E/U b pU : E =: EU the quotient map, and by bσ −→ EU,σ pU,σ : E its “completed complexification”. 5.1.3.2. Observe that the Hermitian vector spaces E U,σ and the “admissible” Clinear maps pU 0 U,σ : EU,σ −→ EU 0 ,σ also constitute a projective system in the category whose objects are the complex normed spaces and the morphisms, the continuous linear maps of operator norm ≤ 1. This projective system admits a limit in this category, which may be described as follows. Its underlying C-vector space is the subspace of bσ := E

lim EU,σ ←− b U ∈U (E)

n = (xU )U ∈U (E) b ∈

Y

b 2, EU,σ | for all (U, U 0 ) ∈ U(E)

b U ∈U (E)

U ⊂ U 0 =⇒ pU 0 U (xU ) = xU 0 defined by “uniformly bounded” elements, namely n EσHilb := limHilb EU,σ = (xU )U ∈U (E) b ∈ lim EU,σ | ←− ←− U

U

o

o sup kxU kE U,σ < +∞ . b U ∈U (E)

Its norm is the norm k.kEσHilb defined by the equality kxkEσHilb :=

sup kxU kE U,σ

(5.1.1)

U ∈U (E)

Hilb for every element x = (xU )U ∈U (E) . In fact, kxU kE U,σ is a nondecreasing b in Eσ b function of U ∈ U(E), and therefore we also have

kxkEσHilb =

lim kxU kE U,σ .

U ∈U (E)

The following proposition is a straightforward consequence of the definitions, and its proof is left to the reader:

5.1. Definitions

111

bσHilb Proposition 5.1.1. Equipped with the norm k.kEσHilb , the complex vector space E b becomes a separable Hilbert space. For all U ∈ U(E), the map pU,σ|EσHilb : EσHilb −→ EU,σ is a “co-isometry”. In other words, the Hermitian norm k.kE U ,σ coincides with the quotient norm derived from the Hilbert norm k.kEσHilb by means of the surjective C-linear map pU,σ|EσHilb . bσHilb is equipped with its topology of a Hilbert space, and E bσ Moreover, when E with its canonical topology of a separated locally convex complex vector space, the inclusion morphism bσ iσ : EσHilb −→ E is continuous with dense image.



Finally, observe that the constructions in 5.1.3.1 and 5.1.3.2 are clearly compatible with isometric isomorphisms of pro-Hermitian vector bundles. 5.1.4 An alternative description of pro-Hermitian vector bundles. 5.1.4.1. The previous constructions lead us to the following alternative definition of pro-Hermitian vector bundles over Spec OK , which turns out to be more flexible than their initial definition in terms of projective systems of (finite-dimensional) Hermitian vector bundles. We may define a pro-Hermitian vector bundle over Spec OK as the data
b := (E, b EσHilb , k.kσ , iσ E ), σ:K,→C

(5.1.2)

b is an object of CTCO and where for every field embedding σ : K ,→ C, where E K Hilb bσ a Eσ denotes a complex Hilbert space, k.kσ its norm, and iσ : EσHilb −→ E continuous injective C-linear map with dense image. The data
EσHilb , k.kσ , iσ σ:K,→C are required to be compatible with complex conjugation. Namely, one requires the ∼ existence, for every σ : K ,→ C, of a C-antilinear bijective isometry γσ : EσHilb −→ Hilb Eσ such that the following relation holds, where · denotes the C-antilinear isobσ := E b⊗ bσ := E b⊗ ˆ σ C onto E ˆ σ C derived from the complex conjumorphism from E gation on C: (5.1.3) iσ ◦ γσ = · ◦ iσ . When they exist, the maps γσ are uniquely determined by the relations (5.1.3). Moreover, they are easily seen to exist when one is given a pro-Hermitian b := (E, b (E ) vector bundle E ) in the sense of paragraph 5.1.2, and when U U ∈U (E) b

EσHilb , k.kσ := k.kEσHilb and iσ are defined as in paragraph 5.1.3.2.

112 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves b one defines Conversely, starting from the data (5.1.2), for every U ∈ U(E), the structure of a Hermitian vector bundle E U := (EU , (k.kU,σ )σ:K,→C ) b on the finitely generated projective OK -module EU := E/U by defining the norm k.kU,σ as the quotient norm induced from the Hilbert norm k.kσ on EσHilb by requiring the C-linear maps pU,σ ◦ iσ : EσHilb −→ EU,σ bσ means precisely to be co-isometries. (Observe that the density of iσ (EσHilb ) in E b that this map is surjective for every U ∈ U(E).) In this way, one constructs a pro-Hermitian vector bundle b := (E, b (E U ) E b ) U ∈U (E) in the sense of paragraph 5.1.2 from the data (5.1.2). The reader will easily check that these two constructions are inverses of each other. 5.1.4.2. When dealing with pro-Hermitian vector bundles defined by data of type (5.1.2), we shall occasionally write iE σ instead of iσ to make the dependence on b E explicit, notably when discussing “concrete” examples of pro-Hermitian vector b

bundles occurring in Diophantine geometry. Conversely, when investigating the general properties of pro-Hermitian vector bundles, we shall also sometimes avoid naming explicitly the morphisms iσ and γσ , and identify EσHilb with its image by iσ . Accordingly, a pro-Hermitian vector bundle will be often denoted by b := (E, b (EσHilb , k.kσ )σ:K,→C ). E bσ := E b⊗ ˆ σ C, we may also extend Having identified EσHilb with a subspace of E Hilb the Hilbert norm k.kσ on Eσ to a function bσ −→ [0, +∞] k.kσ : E by letting bσ \ EσHilb . kvkσ := +∞ for any v ∈ E Then the relations kvkσ =

sup kpU,σ (v)kE U ,σ = b U ∈U (E)

lim kpU,σ (v)kE U ,σ

b U ∈U (E)

bσ . hold for all v ∈ E Observe that for every R ∈ R+ , the ball {v ∈ EσHilb | kvk ≤ R} is closed in bσ . (Indeed, it is convex and compact in the the locally convex C-vector space E Hilb bσ .) In other words, k.kσ weak topology of Eσ , hence in the weak topology of E b is lower semicontinuous on Eσ .

5.2. Hilbertizable Ind- and Pro-vector Bundles

113

5.1.5 Direct images. Ind- and pro-Euclidean lattices. The construction of the direct image of a Hermitian vector bundle over Spec OK by the morphism π : Spec OK −→ Spec Z discussed in Section 1.2 above extends to ind- and proHermitian vector bundles. b over Spec O , we may For instance, for every pro-Hermitian vector bundle E K define its direct image by π as the pro-Hermitian vector bundle over Spec Z b := (π E, b (ECHilb , k.kC )), π∗ E ∗ b is nothing but E b considered as a topological Z-module, and where where π∗ E ECHilb denotes the complex Hilbert space defined as M ECHilb := EσHilb , σ:K,→C

equipped with the norm k.kC such that for every (xσ )σ:K,→C ∈ ECHilb , X k(xσ )σ:K,→C k2C := kxσ k2σ . σ:K,→C

L b⊗ bσ , the Hilbert space E Hilb is indeed ˆ Z C ' σ:K,→C E Observe that since E C b C . Moreover, it is endowed with a canonical C-antilinear a dense subspace of (π∗ E) involution. This construction of direct images reduces many questions concerning indand pro-Hermitian vector bundles over Spec OK to questions concerning ind- and pro-Hermitian vector bundles over the “final” arithmetic curve Spec Z. We will often say ind-Euclidean lattice (resp., pro-Euclidean lattice) instead of ind-Hermitian (resp., pro-Hermitian) vector bundle over Spec Z. b = (E, b (E Hilb , k.kC )) on Observe also that a pro-Hermitian vector bundle E C b (E Hilb , k.kR )), where (E Hilb , k.kR ) Spec Z may be equivalently defined as a pair (E, R R denotes a real Hilbert space equipped with a continuous R-linear injection with dense image bR := E b⊗ ˆ Z R. ERHilb ,−→E Indeed, the real Hilbert space (ERHilb , k.kR ) is derived from (ECHilb , k.kC ) by taking its fixed point under complex conjugation. Conversely, we recover (EC , k.kC ) from (ER , k.kR ) by extending the scalars from R to C. Similar remarks concerning ind-Hermitian vector bundles may be developed and are left to the reader. We only observe that an ind-Hermitian vector bundle F over Spec Z may be defined as a pair (F, k.k), where F is a countable free Z-module and k.k is a pre-Hilbert norm on the real vector space FR := F ⊗Z R.

5.2

Hilbertizable Ind- and Pro-vector Bundles

In applications, it is convenient to have at one’s disposal weakened variants of the notions of ind- and pro-Hermitian vector bundles, where the (pre-)Hilbert

114 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves norms that enter into their definitions in paragraphs 5.1.1 and 5.1.4 are replaced by equivalence classes of (pre-)Hilbert norms. Thus we shall define a Hilbertizable ind-vector bundle over Spec OK as the data ... top F := (F, (Fσ )σ:K,→C ) of an object F of CPOK and of structures Fσtop of complex topological vector spaces on the C-vector spaces Fσ := F ⊗σ:K,→C that may be defined by pre-Hilbert ∼ norms. The conjugation maps Fσ −→ Fσ are required to be homeomorphisms. To every ind-Hermitian vector bundle F...:= (F, (k.kσ )σ:K,→C ) is attached its underlying Hilbertizable ind-vector bundle F := (F, (Fσtop )σ:K,→C ), where Fσtop denotes Fσ equipped with the norm topology defined by k.kσ . Similarly, we shall define a Hilbertizable pro-vector bundle over Spec OK as a pair ˜ := (E, b (E Hilb , iσ )σ:K,→C ), E σ b is an object of CTCO and where for every field embedding σ : K ,→ C, where E K EσHilb denotes a complex Hilbertizable vector space (that is, a topological complex vector space whose topology may be defined by a Hilbert space structure) and bσ a continuous injective C-linear map with dense image. These iσ : EσHilb −→ E data are required to be compatible with complex conjugation (namely, one requires ∼ the existence of C-antilinear isomorphisms γσ : EσHilb −→ EσHilb that satisfy the relations (5.1.3)). Observe that by the closed graph theorem, the “Hilbertizable” topology on EσHilb is the unique topology of a Fr´echet space on the complex vector subspace bσ that makes continuous the injection from EσHilb into E bσ . EσHilb of E To every pro-Hermitian vector bundle
b := (E, b E Hilb , k.kσ , iσ E ) σ σ:K,→C is attached its underlying Hilbertizable pro-vector bundle ˜ := (E, b (EσHilb , iσ )σ:K,→C ). E

5.3

Constructions as Inductive and Projective Limits

5.3.1 Construction of ind-Hermitian vector bundles as inductive limits. sider an inductive system j0

j1

ji−1

ji

Con-

ji+1

F • : F 0 −→ F 1 −→ · · · −→ F i −→ F i+1 −→ · · · of injective admissible morphisms of Hermitian vector bundles over Spec OK (as defined in 1.4.2).

5.3. Constructions as Inductive and Projective Limits

115

To F • , we may attach an ind-Hermitian vector bundle lim F i := (F, (k.kσ )σ:K,→C ) −→ i

defined by the following simple construction. Its underlying OK -module is the inductive limit F := lim Fi . −→ i

By construction, it satisfies condition (4) in Proposition 4.1.1, and therefore is indeed an object of CPOK . Moreover, for every field embedding σ : K ,→ C, the maps ji,σ : Fi,σ −→ Fi+1,σ are isometric with respect to the Hermitian norms k.kF i,σ and k.kF i+1,σ . Therefore, there is a unique norm k.kσ on Fσ := limi Fi,σ such that the canonical −→ maps Fi,σ ,−→Fσ are isometric with respect to the norms k.kF i,σ and k.kσ . The norm k.kσ thus defined, like the norms k.kF i,σ , is clearly a pre-Hilbert norm. Observe that up to isometric isomorphism, every ind-Hermitian vector bundle F := (F, (k.kσ )σ:K,→C ) over Spec OK is the limit of an inductive system F • of Hermitian vector bundles as above. Indeed, we may consider a sequence (Fi )i∈N of OK -submodules of F satisfying condition (4) in Proposition 4.1.1 (with A = OK ) and endow each Fi with the Hermitian norm restrictions of the given norms (k.kσ )σ:K,→C : the Hermitian vector bundles F i thus defined, define an inductive system F • the whose limit limi F i is canonically isomorphic to F . −→ 5.3.2 Construction of pro-Hermitian vector bundles as projective limits. Consider a projective system q0

q1

qi−1

qi

qi+1

E • : E 0 ←− E 1 ←− · · · ←− E i ←− E i+1 ←− · · · of surjective admissible morphisms of Hermitian vector bundles over Spec OK . To E • , we may associate a pro-Hermitian vector bundle b (E U ) lim E i = (E, b ) U ∈U (E) ←− i

over Spec OK defined as follows. Its underlying topological OK -module is the object of CTCOK defined as the projective limit b := lim Ei . E ←− i

b −→ Ei . Let us consider the kernels Ui := ker pi of the canonical projections pi : E The sequence (Ui )i∈N is nonincreasing and constitutes a basis of a neighborhood b of 0 in U(E).

116 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves b there exists i ∈ N such that U contains Ui . Then the For every U in U(E), quotient map b i −→ EU := E/U b i pU Ui : Ei := E/U is surjective. Consequently, for every embedding σ : K ,→ C, the C-linear map pU Ui ,σ : Ei,σ −→ EU,σ also is surjective, and EU,σ may be endowed with the Hermitian norm k.kEU,σ defined as the quotient norm, defined by means of pU Ui ,σ , of the Hermitian norm k.kE i ,σ on Ei,σ . By construction, the Hermitian vector bundle E U := (EU , (k.kEU,σ )σ:K,→C ) over Spec OK is such that pU Ui : Ei −→ EU becomes a surjective admissible morphism from E i to E U . Using the fact that the morphisms qi : E i+1 −→ E i , and therefore their compositions pUi Ui0 = qi ◦ · · · ◦ qi0 −1 : E i0 −→ E i0 −1 −→ · · · −→ E i , are surjective admissible, one easily checks that the construction of the Hermitian structure on E U does not depend on the choice of the open saturated submodule Ui contained in U . b and every i ∈ N such that Ui ⊂ U ⊂ U 0 , Finally, for every U and U 0 in U(E) the commutativity of the diagram Ei

pU U i

pU 0 Ui

/ EU pU 0 U

!  EU 0

and the fact that pU Ui (resp., pU 0 Ui ) is a surjective admissible surjective morphism from E i to E U (resp., from E i to E U 0 ) implies that pU 0 U is a surjective admissible morphism from E U to E U 0 . Observe that up to isometric isomorphism, every pro-Hermitian vector bundle b := (E, b (E U ) E b ) U ∈U (E) over Spec OK is the limit of a projective system of Hermitian vector bundles E • b that as above. Indeed, we may choose a decreasing sequence (Ui )i∈N in U(E)) b constitutes a basis of neighborhoods of 0 in E — that is, a defining filtration in the sense of paragraph 4.2.1 — and consider the projective system defined by the Hermitian vector bundles E i := E Ui .

5.4. Morphisms Between Ind- and Pro-Hermitian Vector Bundles over OK

117

5.4 Morphisms Between Ind- and Pro-Hermitian Vector Bundles over OK For every two normed complex vector spaces (V, k.k) and (V 0 , k.k0 ) and for every λ in R+ , we may consider the set of C-linear maps of operator norm at most λ from (V, k.k) to (V 0 , k.k0 ): ≤λ HomC ((V, k.k)), (V 0 , k.k0 )) := {T ∈ HomC (V, V 0 ) | kT k :=

sup

kT vk0 ≤ λ}.

v∈V,kvk≤1

Their union cont HomC ((V, k.k)), (V 0 , k.k0 )) :=

[

≤λ ((V, k.k)), (V 0 , k.k0 )) HomC

λ∈R+

is the C-vector space of continuous linear maps from (V, k.k) to (V 0 , k.k0 ). 5.4.1 Categories of ind-Hermitian vector bundles. Let F 1 and F 2 be two indHermitian vector bundles over OK . For every λ in R+ , we define Hom≤λ OK (F 2 , F 1 ) as the subset of HomOK (F2 , F1 ) consisting of the OK -linear maps ψ : F2 −→ F1 such that for every embedding σ : K,−→C, the induced C-linear map ψσ : F2,σ −→ F1,σ is continuous, with operator norm ≤ λ, when F2,σ and F1,σ are equipped with the pre-Hilbert norms k.kF 2,σ and k.kF 1,σ . Clearly, if F 1 , F 2 , and F 3 are ind-Hermitian vector bundles over OK , and if λ and µ are two elements of R+ , the composition of an element ψ in Hom≤λ OK (F 2 , F 1 ) ≤µ ≤λµ 0 0 and an element ψ in HomOK (F 3 , F 2 ) defines an element ψ◦ψ in HomOK (F 3 , F 1 ). Consequently, it is possible to define a category whose objects are the indHermitian vector bundles over OK by either of the following constructions: (a) by defining the morphisms from F 2 to F 1 to be Homcont OK (F 2 , F 1 ) :=

[

Hom≤λ OK (F 2 , F 1 )

λ∈R+

= {ψ ∈ HomOK (F2 , F1 ) | for every σ : K ,→ C, o ψσ ∈ Homcont ((F , k.k ), (F , k.k )) . 2,σ 1,σ C F 2 ,σ F 1 ,σ The category indVectOK thus defined is clearly OK -linear.

118 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves (b) by defining the morphisms from F 2 to F 1 to be Hom≤1 OK (F 2 , F 1 ). The ≤1

category thus defined will be denoted by indVectOK . ∼

≤1

Observe that an isomorphism ψ : F 2 −→ F 1 in indVectOK (resp., in ∼ indVectOK ) is an isomorphism of OK -modules ψ : F2 −→ F2 such that the ∼ C-linear isomorphism ψσ : F2,σ −→ F1,σ is an isometry (resp., a homeomorphism) between the normed vector spaces (F2 , k.kF 2,σ ) and (F1 , k.kF 1,σ ). In particular, isometric isomorphisms of ind-Hermitian vector bundles (as defined in paragraph ≤1 5.1.1) are exactly the isomorphisms in indVectOK . The inductive limit limi F i of an inductive system F • of injective admis−→ sible morphisms of Hermitian vector bundles, as considered in paragraph 5.3.1, together with the obvious inclusion maps F k −→ limi F i , is easily checked to be −→ ≤1 an inductive limit of F • in the category indVectOK . ] OK whose objects are We may also introduce an OK -linear category indVect ] OK , the Hilbertizable ind-vector bundles over Spec OK : in the category indVect the set of morphisms between two Hilbertizable ind-vector bundles ... ... top top F 2 := (F2 , (F2,σ )σ:,→K ) and F 1 := (F1 , (F1,σ )σ:,→K ) over Spec OK is defined as the OK -module ... ... Homcont OK ( F 2 , F 1 ) = {ψ ∈ HomOK (F2 , F1 ) | top top for every σ : K ,→ C, ψσ : F2,σ −→ F1,σ is continuous . ] OK , Finally, there is a natural forgetful functor from indVectOK to indVect F to the associated Hilbertizable which maps an ind-Hermitian vector bundle ... ind-vector bundle F , and is the identity on morphisms. It is easily seen to be an equivalence of categories. 5.4.2 Categories of pro-Hermitian vector bundles. For every two pro-Hermitian b and E b over Spec O and for every λ in R , we define vector bundles E 1 2 K + b b Hom≤λ lim lim Hom≤λ OK (E 1 , E 2 ) := ← OK (E 1,U1 , E 2,U2 ). − −→

(5.4.1)

U2 U1

In the inductive (resp., projective) limit, U1 (resp., U2 ) varies in the filtered set b1 ) (resp., U(E b2 )), ordered by ⊇. U(E b as the b and E We also define the set of continuous OK -morphisms from E 1 2 OK -module [ b b b b Hom≤λ (5.4.2) Homcont OK (E 1 , E 2 ) := OK (E 1 , E 2 ). λ∈R+

5.4. Morphisms Between Ind- and Pro-Hermitian Vector Bundles over OK

119

b ) is uniquely determined by its cont b (E 1 , E Observe that an element ϕ˜ in HomO 2 K image ϕˆ in the OK -module b b lim lim HomOK (E1,U1 , E2,U2 ) ' Homcont OK (E1 , E2 ) ←− −→ U2 U1

b1 to E b2 . of OK -linear continuous maps from E The following proposition is a direct consequence of the construction of the Hilbert spaces associated to a pro-Hermitian vector bundle: b b Proposition 5.4.1. With the above notation, an element ϕˆ in Homcont OK (E1 , E2 ) may b ,E b ) if and only if for every embedding be lifted to an element ϕ˜ in Hom≤λ (E OK

1

2

σ : K ,→ C, there exists a continuous C-linear map Hilb Hilb ϕσ : E1,σ −→ E2,σ Hilb

Hilb

with operator norm ≤ λ between the Hilbert spaces E 1,σ and E 2,σ such that the following diagram is commutative: ϕσ

Hilb Hilb E1,σ −−−−→ E2,σ    Eb b  1y yiσ 2 iE σ

(5.4.3)

ϕ ˆσ

b1,σ −−−−→ E b2,σ . E When this holds, these morphisms ϕσ are unique.



b b According to Proposition 5.4.1, an element of Homcont OK (E 1 , E 2 ) (resp., b )) may be described as a pair b ,E Hom≤λ (E OK

1

2

ϕ˜ := (ϕ, ˆ (ϕσ )σ:K,→C ),

(5.4.4)

consisting of the following data: 1. a continuous morphism of topological OK -modules b1 −→ E b2 ; ϕˆ : E 2. for every embedding σ : K ,→ C, a continuous C-linear map (resp., a C-linear map of operator norm ≤ λ) Hilb Hilb ϕσ : E1,σ −→ E2,σ Hilb

Hilb

between the Hilbert spaces E 1,σ and E 2,σ that is compatible with ϕ, ˆ in the sense that the diagram (5.4.3) is commutative for every embedding σ : K ,→ C.

120 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves In the following sections, and in the sequel of this monograph, we shall freely b ). cont b (E 1 , E use this alternative description of HomO 2 K As a special case of this description, observe that for every Hermitian vecb over tor bundle E over Spec OK and for every pro-Hermitian vector bundle F Spec OK , we have a natural identification b ∼ Hilb b Homcont ). OK (E, F ) −→ HomOK (E, F ∩ FC b , and E b over Spec O , b ,E For every three pro-Hermitian vector bundles E 1 2 3 K there is a natural OK -bilinear composition map b b b b cont b cont b · ◦ · : Homcont OK (E 2 , E 3 ) × HomOK (E 1 , E 2 ) −→ HomOK (E 1 , E 3 ),

(5.4.5)

µ b b b b that for every (λ, ≤ µ) ∈ R2+ , maps Hom≤λ OK (E 2 , E 3 ) × HomOK (E 1 , E 2 ) to b b Hom≤λµ OK (E 1 , E 3 ). It is derived from the composition of morphisms ≤µ b ≤λµ b b b b b Hom≤λ OK (E 2,U2 , E 3,U3 ) × HomOK (E 1,U1 , E 2,U2 ) −→ HomOK (E 1,U1 , E 3,U3 )

by passage to the projective and inductive limits involved in the definitions (5.4.1) and (5.4.2) of continuous OK -morphisms of pro-Hermitian vector bundles. In terms of the description of the morphisms of pro-Hermitian vector bundles as pairs of the form (5.4.4), their composition law (5.4.5) may be described as ˆ (ψσ )σ:K,→C )) is an element of follows: if ϕ˜ := (ϕ, ˆ (ϕσ )σ:K,→C ) (resp., ψ˜ := (ψ, b b cont b cont b ˜ ϕ) Hom (E 1 , E 2 ) (resp., of Hom (E 2 , E 3 )), then it maps (ψ, ˜ to OK

OK

ψ˜ ◦ ϕ˜ := (ψˆ ◦ ϕ, ˆ (ψσ ◦ ϕσ )σ:K,→C ). b to be Homcont (E b ,E b ) and the b to E By defining the morphisms from E 1 2 1 2 OK composition of morphisms as above, the class of pro-Hermitian vector bundles over Spec OK becomes an OK -linear category. We shall denote it by proVectOK . We may define another category, the objects of which are again the prob to Hermitian vector bundles over Spec OK , by defining the morphisms from E 1 ≤1 b b b E 2 to be HomOK (E 1 , E 2 ), and by defining the composition as above. We shall ≤1

denote this category by proVectOK . Finally, we may formulate some observations concerning these categories of pro-Hermitian vector bundles similar those concerning ind-Hermitian vector bundles at the end of paragraph 5.4.1: ∼ b b −→ (i) An isomorphism ϕ˜ : E E 2 in proVectOK is the data of an isomor1 ∼ b b phism ϕˆ : E1 −→ E2 of topological OK -modules and of C-linear homeomorphisms Hilb ∼ Hilb ϕσ : E1,σ −→ E2,σ compatible with ϕ. ˆ ≤1 ∼ b b An isomorphism ϕ˜ : E −→ E in proVect is precisely an isometric 1

2

isomorphism of pro-Hermitian vector bundles.

OK

5.5. The Duality Between Ind- and Pro-Hermitian Vector Bundles

121

(ii) The projective limit limi E i of a projective system E • of surjective admis←− sible morphisms of Hermitian vector bundles (as considered in paragraph 5.3.2), equipped with the projection maps limi E i −→ E k , is easily checked to be a pro←− ≤1 jective limit of E • in proVectOK . ] O , whose objects (iii) We may introduce the OK -linear category proVect K are the Hilbertizable pro-vector bundles over Spec OK , and a forgetful functor ]O proVectOK −→ proVect K b of proVect ˜ ] — it sends an object E OK to the underlying object E of proVectOK — that is actually an equivalence of category.

5.5

The Duality Between Ind- and Pro-Hermitian Vector Bundles

In this section, we construct a duality between the categories proVectOK and ≤1

≤1

indVectOK and between the categories proVectOK and indVectOK by combining the duality between CPOK and CTCOK described in Section 4.3 and the classical duality theory of (pre-)Hilbert spaces. 5.5.1 The duality functors. 5.5.1.1. Let F := (F, (k.kσ )σ:K,→C ) be an ind-Hermitian vector bundle over Spec OK . To F , we may attach a dual pro-Hermitian vector bundle F

∨

:= (F ∨ , (Fσ∨Hilb , k.k∨ σ )σ:K,→C )

(5.5.1)

over Spec OK , defined as follows. Its underlying topological OK -module is the OK -module F ∨ := HomOK (F, OK ) equipped with the topology of pointwise convergence. In other words, it is the dual in CTCOK of the object F of CPOK . For every embedding σ : K ,→ C, we get canonical isomorphisms of complex vector spaces ˆ OK ,σ C ' HomOK ,σ (F, C) ' HomC (Fσ , C). Fσ∨ := F ∨ ⊗

(5.5.2)

Indeed, the Fr´echet topology on Fσ∨ (derived from the structure of a topological OK -module on F ∨ , as explained in 5.1.3.1) coincides with the locally convex topology on HomC (Fσ , C) defined by pointwise convergence on Fσ . By means of the identifications (5.5.2), we may introduce the vector subspace Fσ∨Hilb := Homcont C ((Fσ , k.kσ ), C)

122 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves of Fσ∨ consisting of the linear forms on Fσ continuous with respect to the Hermitian norm k.kσ . Equipped with its operator norm, defined by kξk∨ σ :=

|ξ(f )|,

sup f ∈Fσ ,kf kσ ≤1

this vector space becomes a separable Hilbert space (Fσ∨Hilb , k.kσ∨ ), and the inclusion Fσ∨Hilb ,−→Fσ∨ is easily seen to be continuous and to have a dense image.3 This construction is compatible with complex conjugation, and the righthand side of (5.5.1) actually defines a pro-Hermitian vector bundle over Spec OK , in terms of the alternative approach to pro-Hermitian vector bundles presented in paragraph 5.1.4. Let λ be a positive real number, and let F 1 := (F1 , (k.k1,σ )σ:K,→C ) and F 2 := (F2 , (k.k2,σ )σ:K,→C ) be two ind-Hermitian vector bundles over Spec OK . Consider a morphism α in Hom≤λ OK (F 1 , F 2 ). By definition, α is an element of HomOK (F1 , F2 ) such that, for any σ : K ,→ C, the C-linear map ασ : F1,σ −→ F2,σ is continuous of operator norm ≤ λ from (F1σ , k.k1,σ ) to (F2σ , k.k2,σ ). The dual (or adjoint) morphism α∨ := · ◦ α : F2∨ := HomOK (F2 , OK ) −→ F1∨ := HomOK (F1 , OK ) becomes, after a “completed base change” by the embedding σ : OK ,→ C, the C-linear map ∨ ∨ ασ∨ := . ◦ ασ : F2,σ ' HomC (F2,σ , C) −→ F1,σ ' HomC (F1,σ , C). ∨ ∨ It is a continuous C-linear map between the Fr´echet spaces F2,σ and F1,σ . cont ∨Hilb ∨Hilb Moreover, it sends F2,σ := HomC ((F2,σ , k.k2,σ ), C) to F1,σ := Homcont ((F , k.k ), C), and defines a continuous C-linear map with operator 1,σ 1,σ C ∨Hilb ∨Hilb norm ≤ λ from (F2,σ , k.k∨ , k.k∨ 2,σ ) to (F1,σ 1,σ ). ∨

∨

In conclusion, the dual map α∨ defines a morphism in Hom≤λ OK (F 2 , F 1 ). Moreover, the maps Hom≤λ OK (F 1 , F 2 ) −→ α 7−→

∨

∨

∨

∨

Hom≤λ OK (F 2 , F 1 ) α∨ ·

thus defined define an OK -linear map cont ·∨ : Homcont OK (F 1 , F 2 ) −→ HomOK (F 2 , F 1 ). 3 Indeed, from any countable C-basis of F , by orthonormalization we get a countable orσ thonormal basis of Fσ . By means of such an orthonormal basis, we get compatible isomorphisms ∼ ∼ ∼ Fσ −→ C(N) , Fσ∨ −→ CN , and Fσ∨Hilb −→ `2 (N).

5.5. The Duality Between Ind- and Pro-Hermitian Vector Bundles

123

This construction is clearly functorial and defines two contravariant duality functors, ≤1 ≤1 ·∨ : indVectOK −→ proVectOK and ·∨ : indVectOK −→ proVectOK , the second of which is OK -linear. Observe that every inductive system j0

ji−1

j1

ji+1

ji

F • : F 0 −→ F 1 −→ · · · −→ F i −→ F i+1 −→ · · · of injective admissible morphisms of Hermitian vector bundles over Spec OK determines by duality a projective system ∨

∨ j∨

∨ ji−1

∨ j∨

∨ j∨

∨

∨ ji+1

0 1 i F • : F 0 ←− F 1 ←− · · · ←− F i ←− F i+1 ←− · · ·

of surjective admissible morphisms of Hermitian vector bundles over Spec OK . The injection morphisms jk : F k −→ limi F i define, by duality, morphisms in −→ proVectOK , ∨

jk∨ : (lim F i )∨ −→ F k , −→ i

∨

which in turn define a morphism from (limi F i )∨ to the projective limit limi F i of −→ ←− ∨ F • , which is easily seen to be an isometric isomorphism of pro-Hermitian vector bundles over Spec OK : ∨ ∼ (lim F i )∨ −→ lim F i . (5.5.3) −→ ←− i

i

5.5.1.2. Conversely, to any pro-Hermitian vector bundle over Spec OK , b := (E, b (E Hilb , k.kσ )σ:K,→C ), E σ we may attach a dual ind-Hermitian vector bundle b ∨ := (E b ∨ , (k.k∨ E σ )σ:K,→C )

(5.5.4)

defined as follows. Its underlying projective OK -module is the dual b ∨ := Homcont b E OK (E, OK ) b of CTCO . In fact, as a consequence of Proposition in CPOK of the object E K ∨ b b OK ) of the OK -module E. b 4.2.5, E coincides with the algebraic dual HomOK (E,

124 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves For every embedding σ : K ,→ C, we have canonical isomorphisms b ∨ ⊗O ,σ C ' ( lim EU∨ ) ⊗O ,σ C ' b ∨ )σ := E (E K K −→ b U ∈U (E)

cont b ∨ lim EU,σ (Eσ , C). ' HomC −→ b U ∈U (E)

bσ is continuous with dense image, its transpose Since the injection iσ : EσHilb ,−→E defines an injective map (also with dense image) b ∨ )σ ,−→(EσHilb )∨ iσ∨ : (E b ∨ )σ to the dual Hilbert space (E Hilb )∨ , and we define the norm k.k∨ as from (E σ σ b ∨ )σ , the dual of the norm k.kσ b ∨ )σ of the Hilbert norm on (E the restriction to (E cont b on EσHilb . (In other words, the norm kξk∨ σ of a linear form ξ in HomC (Eσ , C) is Hilb the operator norm, with respect to k.kσ , of its restriction to Eσ .) It is straightforward that the construction of the norms (k.k∨ σ )σ:K,→C on the ∨ b complex vector spaces ((E )σ )σ:K,→C is compatible with complex conjugation. Consequently, the right-hand side of (5.5.4) indeed defines an ind-Hermitian vector bundle over Spec OK . Let λ be a positive real number, and let b := (E b1 , (E Hilb , k.k1,σ )σ:K,→C ) E 1 1,σ and b := (E Hilb b2 , (E2,σ E , k.k2,σ )σ:K,→C ) 2 be two pro-Hermitian vector bundles over Spec OK . Consider a morphism β in b b cont b b Hom≤λ OK (E 1 , E 2 ). By definition, β is an element in HomOK (E1 , E2 ) such that for every embedding σ : K ,→ C, the continuous linear map of Fr´echet spaces b1,σ −→ E b2,σ βσ : E

(5.5.5)

Hilb Hilb maps E1,σ to E2,σ , with an operator norm (with respect to the Hilbert norms k.k1,σ and k.k2,σ ) at most λ. The dual morphism cont b b2∨ := Homcont b b∨ β ∨ := · ◦ β : E OK (E2 , OK ) −→ E1 := HomOK (E1 , OK ),

after the base change σ : OK ,−→C, becomes the transpose of the map (5.5.5), namely cont b b2∨ )σ ' Homcont b b∨ βσ∨ = · ◦ βσ : (E C (E2,σ , C) −→ (E1 )σ ' HomC (E1,σ , C),

b ∨ )σ , and therefore satisfies, for every ξ ∈ (E 2 ∨ ∨ kβσ∨ (ξ)k∨ 1,σ = kξ ◦ βσ k1,σ ≤ λkξk2,σ .

5.5. The Duality Between Ind- and Pro-Hermitian Vector Bundles

125

b∨ b∨ This shows that β ∨ belongs to Hom≤λ OK (E 2 , E 1 ). The maps b∨ b∨ Hom≤λ OK ( E 2 , E 1 ) β∨·

b b Hom≤λ OK (E 1 , E 2 ) −→ β 7−→ thus defined define an OK -linear map

∨ b b b∨ cont b ·∨ : Homcont OK (E 1 , E 2 ) −→ HomOK (E 2 , E 1 ).

This construction is clearly functorial and defines two contravariant duality functors, ≤1 ≤1 ·∨ : proVectOK −→ indVectOK (5.5.6) and ·∨ : proVectOK −→ indVectOK , the second of which is OK -linear. The duality functor (5.5.6) is compatible with the construction of pro-Hermitian vector bundles as limits of projective systems of surjective admissible maps of Hermitian vector bundles over Spec OK discussed in paragraph 5.3.2. Namely, if qi−1 qi+1 q0 q1 qi E • : E 0 ←− E 1 ←− · · · ←− E i ←− E i+1 ←− · · · is such a projective system, we may form the dual inductive system ∨

∨ q∨

∨ qi−1

∨ q∨

∨ q∨

∨

∨ qi+1

0 1 i E • : E 0 −→ E 1 −→ · · · −→ E i −→ E i+1 −→ · · ·

of injective admissible morphisms of Hermitian vector bundles over Spec OK . The ≤1 projections pk : limi E i −→ E k define, by duality, morphisms in indVect (OK ), ←− ∨

p∨ E )∨ , k : E k −→ (lim ←− i i

∨

which in turn define a morphism from the inductive limit of E • to (limi E i )∨ . This ←− morphism is easily seen to be an isometric isomorphism of ind-Hermitian vector bundles over Spec OK : ∨ ∼ lim E i −→ (lim E i )∨ . (5.5.7) −→ ←− i

i

b := (E, b (EσHilb , k.kE,σ )σ:K,→C ) be an 5.5.2 Duality as adjoint equivalences. Let E object of the category proVectOK and let F := (F, (k.kF,σ )σ:K,→C be an object of b ∨ = (E b ∨ , (k.k∨ ) ) and indVect . We may consider their dual objects E F

∨

= (F

OK ∨

, (Fσ∨Hilb , k.k∨ F,σ )σ:K,→C ),

E,σ σ:K,→C

in indVectOK and proVectOK respectively.

126 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves As discussed in 4.3.2, we have natural isomorphisms that define the duality between the OK -linear categories CTCOK and CPOK : b ∨ ∼ b∨ Homcont OK (E, F ) −→ HomOK (F, E ).

(5.5.8)

cont b b ∨ ) may be identified with the OK Indeed, both HomO (E, F ∨ ) and HomOK (F, E K module consisting of the OK -bilinear maps

b × F −→ OK b:E that are continuous in the first variable, or equivalently with the OK -module lim EU∨ ⊗OK F ∨ . −→ b U ∈U (E)

Similarly, for every field embedding σ : K ,→ C, we have a duality isomorphism of complex vector spaces ∼ cont b bσ∨ ). HomC (Eσ , Fσ∨ ) −→ HomC (Fσ , E

(5.5.9)

cont b bσ∨ ) may be identified with the comIndeed, both HomC (Eσ , Fσ∨ ) and HomC (Fσ , E plex vector space consisting of the C-bilinear maps

˜b : E bσ × Fσ −→ C that are continuous in the first variable, or equivalently with the complex vector space ∨ lim EU,σ ⊗C Fσ∨ . −→ b U ∈U (E)

Moreover, for every λ ∈ R+ , the bijection (5.5.9) defines by restriction a bijection ≤λ ∨ ∨ Hilb b )) Homcont , k.kE,σ ), (Fσ∨Hilb , k.kF,σ C (Eσ , Fσ ) ∩ HomC ((Eσ ∼

∨ b∨ −→ Hom≤λ C ((Fσ , k.kF,σ ), (Eσ , k.kE,σ )). (5.5.10) cont b bσ∨ ) defined by both sides (Eσ , Fσ∨ ) and HomC (Fσ , E Indeed, the subsets of HomC of (5.5.10) may be identified with the spaces of C-bilinear maps ˜b as above such that when considered as bilinear maps on EσHilb × Fσ , their ε-norms

n o k˜bkε := sup |˜b(x, y)|; (x, y) ∈ EσHilb × Fσ , kxkE,σ ≤ 1, kykF,σ ≤ 1 are at most λ.

5.5. The Duality Between Ind- and Pro-Hermitian Vector Bundles

127

Furthermore, the identifications (5.5.8) and (5.5.9) are compatible with “extension of scalars by σ : K ,→ C.” In other words, (5.5.8) and (5.5.9) fit into a commutative diagram: ∼ cont b b∨ ) HomO (E, F ∨ ) −−−−→ HomOK (F, E K   ·⊗ 1 ·⊗ 1 y σ C y σ C ∼ ∨ b b∨ Homcont C (Eσ , Fσ ) −−−−→ HomC (Fσ , Eσ ).

Together with the bijections (5.5.10), this shows that by restriction, the bijection (5.5.8) defines a bijection ≤λ b ∨ ∼ b∨ Hom≤λ OK (E, F ) −→ HomOK (F , E )

(5.5.11)

for every λ ∈ R+ , and consequently some OK -linear isomorphism b ∨ ∼ b∨ cont Homcont OK (E, F ) −→ HomOK (F , E ).

(5.5.12)

We may finally formulate the duality between the categories of pro- and indHermitian vector bundles over Spec OK as the following statement, to be compared with the duality between CTCOK and CPOK that constitutes the special case of Proposition 4.3.1 in which A = OK : Proposition 5.5.1. The bijections (5.5.11) with λ = 1 and (5.5.12) define adjunctions of functors op ≤1 ·∨ : proVectOK  indVectOK : ·∨ and

≤1 op

·∨ : proVectOK  proVectOK

: ·∨ .

These are actually adjoint equivalences. Their unit and counit are the natural isomorphisms η and ε defined by the isometric isomorphisms ∼

ε−1 : F −→ F F

∨∨

∨∨ ∼ b b −→ and η b : E E , E

b in proVect , whose associated to any object F in indVectOK and any object E OK b to E b ∨∨ are the biduality isounderlying morphisms from F to F ∨∨ and from E morphisms: ∼

ε−1 F : F f

−→ 7−→

b ηEb : E e

−→ 7−→

and

∼

Homcont OK (HomOK (F, OK ), OK ), (ξ 7−→ ξ(f )), b HomOK (Homcont OK (E, OK ), OK ), (ζ 7−→ ζ(e)).

128 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves Proof. This may be deduced from the duality between CTCOK and CPOK established in Proposition 4.3.1, combined with basic results concerning the duality of (pre-)Hilbert spaces. At this stage, we may also argue directly as follows. The naturality (in each b and F of the bijections (5.5.11) of the relevant categories) with respect to E with λ = 1 and (5.5.12) is straightforward. We are left to show that the unit εF and counit η b of these adjunctions are isometric isomorphisms with underlying E isomorphisms the biduality isomorphisms εF and ηEb . b This directly follows from (i) the validity of these properties when F and E are Hermitian vector bundles, (ii) the compatibility of the duality functors with inductive and projective limits (see (5.5.3) and (5.5.7)), and (iii) the fact that every ind- (resp., pro-)Hermitian vector bundle over Spec OK may be realized as an inductive (resp., projective) limit of an admissible system of Hermitian vector bundles (cf. Section 5.3). 

5.6

Examples – I. Formal Series and Holomorphic Functions on Disks

Let R be a positive real number. Let us consider the open disk in C of radius R, D(R) := {z ∈ C | |z| < R}, and the space Oan (D(R)) of holomorphic functions on D(R). Equipped with the topology of uniform convergence on compact subsets of D(R), it is a Fr´echet space. Moreover, Taylor expansion at 0 defines an inclusion iR : Oan (D(R)) ,−→ f 7−→

C[[X]], P (n) (0) X n . n∈N (1/n!) f

When C[[X]] ' CN is equipped with its natural Fr´echet topology, defined by the simple convergence of coefficients, the map iR is continuous with dense image. Let f be an element of Oan (D(R)) and let f (z) =

X

an z n

n∈N

be its expansion at the origin. For every r ∈ [0, R[, we have Z 0

1

|f (re2πit )|2 dt =

X n∈N

r2n |an |2 .

(5.6.1)

5.6. Examples – I. Formal Series and Holomorphic Functions on Disks

129

This relation shows that for every r ∈]0, R[, one may define a Hermitian norm k.kr on Oan (D(R)) by letting kf k2r

1

Z

|f (re2πit )|2 dt,

= 0

and that the Hardy space ( 2

H (R) :=

) an

f ∈ O (D(R)) | sup

kf kr2

< +∞

r∈[0,R[

becomes a Hilbert space when equipped with the norm k.kR defined by X 2 := sup kf kr2 = R2n |an |2 . kf kR r∈[0,R[

n∈N

From (5.6.1), we also derive Z

|f (x + iy)|2 dx dy

kf kL2 (D(R)) :=

(5.6.2)

x+iy∈D(R)

Z

R

Z

= 0

=

X

2π

|f (reiθ )|2 rdr dθ

(5.6.3)

0

π(n + 1)−1 R2(n+1) |an |2 ,

(5.6.4)

n∈N

and equipped with the norm k.kL2 (D(R)) , the Bergman space B(R) := Oan (D(R)) ∩ L2 (D(R)) is also a Hilbert space. Observe that the expressions (5.6.1) and (5.6.2) for the norms k.kR and k.kL2 (D(R)) show that the vector space C[T ] is dense both in H 2 (R) and in B(R). This shows that the composite injections an 2 iH R : H (R),−→O (D(R)),−→C[[X]]

and an iB R : B(R),−→O (D(R)),−→C[[X]],

which are clearly continuous, have dense images. Moreover, H 2 (R) and B(R) are subspaces of Oan (D(R)) invariant under the operation of complex conjugation ·, defined by X f (z) := f (z) = an z n , n∈N

130 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves and the norms k.kR and k.kL2 (D(R)) also are invariant under complex conjugation. Finally, we may define the following pro-Hermitian vector bundles over Spec Z: b H(R) := (Z[[X]], (H 2 (R), k.kR ), iH (5.6.5) R) and b B(R) := (Z[[X]], (B(R), k.kL2 (D(R)) , iB R )), which may be seen as arithmetic avatars of the classical Hardy and Bergman spaces. Let us emphasize that the isomorphism class in proVectOK (and a fortiori ≤1 b b in proVect ) of H(R), or of B(R), varies with R ∈ R∗ . (This may be shown OK

+

b be considering their θ-invariants hiθ (H(R) ⊗ O(δ)); see Proposition 6.4.3 below.)

5.7

Examples – II. Injectivity and Surjectivity of Morphisms of Pro-Hermitian Vector Bundles

In this paragraph, we gather some observations and examples that demonstrate that if b −→ F b f :E is a morphism of pro-Hermitian vector bundles over Spec OK , then the injectivity b −→ Fb, fˆσ : (resp., surjectivity) properties of the underlying morphisms fˆ : E Hilb Hilb b b Eσ −→ Fσ , and fσ : Eσ −→ Fσ (of topological OK -modules and of complex Fr´echet and Hilbert spaces) are in general loosely related. 5.7.1 Concerning their injectivity, on the positive side, let us observe that each of the following assertions implies the following one: b −→ Fb of topological OK -modules is injective and strict; (i) the morphism fˆ : E bσ −→ Fbσ is injective; (ii) the C-linear map fˆσ : E (iii) the C-linear map fσ : EσHilb −→ FσHilb is injective. Indeed, the implication (i)=⇒(ii) follows from Proposition 4.4.13, and the implication (ii)=⇒ (iii) is clear. Moreover, the injectivity (ii) of fˆσ immediately implies the injectivity of fˆ. 5.7.2 Simple examples of morphisms of pro-Hermitian vector bundles that demonstrate that fˆ or fσ may be injective, while fˆσ is not, are easily obtained by the constructions in Proposition 4.6.1, (2), and Proposition 4.6.4. Indeed, for all a ∈ Z, the map ϕa considered in Proposition 4.6.4 with A = Z — namely the map from Z[[X]] to itself defined by the formula ϕa (f ) := (1 − a/X)f + af (0)/X = f − a (f − f (0))/X

(5.7.1)

5.7. Examples – II. Injectivity and Surjectivity of Morphisms

131

— defines a morphism between “arithmetic Hardy spaces” (as defined in (5.6.5)) b b 0) Φa : H(R) −→ H(R for any two positive real numbers R and R0 such that R0 ≤ R. By definition, ca := ϕa , and the morphisms Φ caC : C[[X]] −→ C[[X]] ϕa,C := Φ and ϕHilb := Φa,C : H 2 (R) −→ H 2 (R0 ) a are still defined by formula (5.7.1), and they make sense for all a ∈ C. According to Proposition 4.6.4, the map ca := ϕa : Z[[X]] −→ Z[[X]] Φ is injective if and only if a ∈ / {1, −1}, is surjective if and only if a ∈ {−1, 0, 1}, but satisfies ϕa (Z[X]) = Z[X]. Moreover, ϕa,C satisfies ϕa,C (C[X]) = C[X] and is injective if and only if a = 0. These properties are complemented by the following proposition, which we leave as an easy exercise: Proposition 5.7.1. For all a ∈ C∗ , the map ϕa,C : C[[X]] −→ C[[X]] P is surjective, and its kernel is the line C. k∈N a−k X k . For every a ∈ C and two positive real numbers R and R0 such that R0 ≤ R, the continuous linear map ϕHilb : H 2 (R) −→ H 2 (R0 ) a is injective if and only if |a| ≤ R. When |a| > R, its kernel is the line C·(a−X)−1 . Moreover, the following conditions are equivalent: (i) ϕHilb is onto; a (ii) ϕHilb is a strict morphism of complex topological vector spaces; a (iii) R = R0 and |a| = 6 R.



In particular, we obtain the following: Scholium 5.7.2. For every a ∈ Z \ {1, 0, −1} and R ∈ R∗+ , the morphism of proHermitian vector bundles over Spec Z, b b −→ H(R), Φa : H(R) ca : Z[[X]] −→ Z[[X]] is injective and not strict, and Φ caC : C[[X]] −→ is such that Φ 2 C[[X]] is strict and not injective. Moreover, Φa,C : H (R) −→ H 2 (R) is an isomorphism (resp., injective with dense image, but not strict; resp., surjective but not injective) if |a| < R (resp., if |a| = R; resp., if |a| > R). 

132 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves 5.7.3 Concerning the surjectivity properties of the underlying morphisms atb −→ Fb over an tached to a morphism of pro-Hermitian vector bundles fˆ : E arithmetic curve Spec OK , we may consider the following conditions, where we denote by σ a field embedding of K into C: b −→ Fb is dense in Fb; b of the morphism fˆ : E (i) the image fˆ(E) bK −→ FbK is surjective; (ii) the K-linear map fˆK : E bσ −→ Fbσ is surjective; (iii) the C-linear map fˆσ : E (iv) the image fσ (EσHilb ) of the continuous C-linear map fσ : EσHilb −→ FσHilb is dense in FσHilb . The properties of the morphisms in CTCk when k is a field presented in paragraphs 4.3.3 and 4.4.4 (see especially Propositions 4.3.4 and 4.4.10) show that the morphisms fˆK and fˆσ in CTCK and CTCC are necessarily strict, and are therefore surjective if and only if they have a dense image. Proposition 4.3.4 also shows that the surjectivity of fˆK and that of fˆσ are equivalent. From these observations, one immediately derives the validity of the following implications: (i) =⇒ (ii) ⇐⇒ (iii) ⇐= (iv). The converse implications (ii)=⇒(i) and (iii)=⇒(iv) do not hold in general. This will be demonstrated by the examples in 5.7.4 and 5.7.5 below. 5.7.4

Let R be a positive real number and let p be a prime number. Let us consider the morphism b b f : H(R) −→ H(R)

defined by the morphism of multiplication by X − p. Namely, fˆ := (X − p)· : Z[[X]] −→ Z[[X]] is the morphism β2 considered in Proposition 4.6.1, (2), and the continuous Clinear map fC := H 2 (R) −→ H 2 (R) is multiplication by the function (z 7→ z − p). Then, according to Proposition 4.6.1, the cokernel of fˆ may be identified, as a topological Z-module, with the p-adic integers Zp . Moreover, the maps fˆQ : Q[[X]] −→ Q[[X]] and fˆC : C[[X]] −→ C[[X]] are still defined by the multiplication by X − p, and are therefore isomorphisms. Finally, the injective map fC is an isomorphism (resp., has a dense image, but is not surjective; resp., has a closed image of codimension 1) if R < p (resp., if R = p; resp., if R > p).

5.7. Examples – II. Injectivity and Surjectivity of Morphisms

133

b −→ F b of proWe finally construct an example of a morphism f : E 5.7.5 ˆ ˆ Hermitian vector bundles over Spec OK such that f (and consequently fK and fˆσ for every embedding σ : K ,→ C) is an isomorphism, and fσ is an isometry with infinite-dimensional cokernel. For simplicity, we shall assume that K = Q. Let I = [a, b] be a closed bounded interval in R, of length b − a > 0. To I, we may attach the ind-Hermitian vector bundle V I := (VI , k.kI ) over Spec Z defined by the Z-module VI := Z[X] equipped with the L2 -norm k.kI on VI,C = C[X] defined by Z kP k2I :=

|P (t)|2 dt.

I

The completion of the complex normed space (VI,C , k.kI ) is the Hilbert space L2 (I). Consequently, if we consider the pro-Hermitian vector bundle b := V ∨ F I I Hilb dual to V I , its underlying Hilbert space FIC may also be identified4 with L2 (I). 0 Let I be another closed bounded interval of positive length, and let V I 0 and

b 0 := V ∨0 F I I be the associated ind- and pro-Hermitian vector bundles over Spec Z. Let us assume, moreover, that I 0 ⊂ I. The identity map IdZ[X] defines a morphism ρI 0 I in Hom≤1 Z (V I , V I 0 ). Its adjoint map ≤1 b b ηII 0 := ρ∨ I 0 I ∈ HomZ (F I 0 , F I ) is easily seen to be the morphism of pro-Hermitian vector bundles over Spec Z defined by the morphism ηˆII 0 := IdZ[X]∨ of topological Z-modules and the continuous C-linear map “extension by zero” ηII 0 ,C : L2 (I 0 ) −→ L2 (I) that sends a function ψ in L2 (I 0 ) to the function ηII 0 ,C (ψ) such that ( ψ(x) if x ∈ I 0 , ηII 0 ,C (ψ)(x) := 0 if x ∈ I \ I 0 . This map ηII 0 ,C is an isometry, and its cokernel may be identified with L2 (I \ I 0 ), which is infinite-dimensional if I 0 6= I. 4 The

Hilb is characterized by the fact that the composite natural isomorphism L2 (I) ' FIC Hilb ˆ map R ' FIC ,−→FI,C := HomC (VI,C , C) sends a function ϕ ∈ L2 (I) to the linear form (P 7→ I ϕ(t)P (t) dt) on VI,C = C[X].

L2 (I)

134 Chapter 5. Ind- and Pro-Hermitian Vector Bundles over Arithmetic Curves

5.8 Examples – III. Subgroups of Pre-Hilbert Spaces and Ind-Euclidean Lattices Let (H, k.k) be a real pre-Hilbert space, and let Γ be a subgroup of (H, +). The inclusion morphism i : Γ −→ H uniquely extends to an R-linear map iR : ΓR := Γ ⊗R R −→ H. Its image is X R · f. im iR = f ∈Γ

Proposition 5.8.1. The following two conditions are equivalent: (i) The map iR is injective and Γ is an object of CPZ . (ii) The group Γ is countable, and the following condition is satisfied: P (F) For every finite P subset F of Γ, the abelian group f ∈F Z · f has finite index in Γ ∩ f ∈F R · f. Recall that the objects of CPZ are precisely the countable free Z-modules. Observe also that iR is injective if and only if kiR (.)k is a pre-Hilbert norm on ΓR . Consequently, condition (i) may be rephrased as (i0 ) The pair (Γ, kiR (.)k) defines an ind-Euclidean lattice. Proof. The direct implication (i) ⇒ (ii) is straightforward, since condition (F) is satisfied by the subgroup Γ = Z(I) of the real vector space ΓR ' R(I) , for every (countable) set I. To establish the converse implication (ii) ⇒ (i), let us assume that (ii) is satisfied, and consider a sequence (fn )n∈N ∈ ΓN in which every element of Γ occurs. For every i ∈ N, let us consider X Γi := Γ ∩ R · fi . 0≤n≤i

P It is a torsion-free Z-module, which contains 0≤n≤i Z.fi as a submodule of finite index. Consequently Γi is a finitely generated torsion-free Z-module. Moreover, for all i ∈ N, Γi is clearly a saturated Z-submodule of Γi+1 , and Γ = ∪i∈N Γi . Therefore, the Z-module Γ is countably generated and projective, hence isomorphic to Z(I) , with I at most countable infinite (this follows from the implications (4) ⇒ (1) ⇒ (5) in Proposition 4.1.1). We may assume that I = {n ∈ N | n < N } with N := |I| ∈ N ∪ {+∞}. Let (ei )i∈I be an Z-basis of Γ. It is also a R-basis of ΓR , and to prove the injectivity of iR , we are left to show that the family (ei )i∈I , considered as a family of vectors in H, is free over R.

5.8. Examples – III. Subgroups of Pre-Hilbert Spaces

135

P To achieve this, observe that for every n ∈ I, the Z-module P0≤ik (1) a sequence (H 0 0 of morphisms b k−1 −→ H bk pk : H of topological OC -modules; (2) a sequence (Fk )k>k0 of (finite-rank) vector bundles over C and a sequence (δk )k>k0 of maps of OC -modules b k−1 ; δk : Fk −→ H and let us assume that for every k ∈ N>k0 , the diagram pk

δ

k b k−1 −→ H b k −→ 0 Fk −→ H

is an exact sequence of OC -modules. Proposition 9.4.1. With the above notation, if the following two conditions are satisfied, b k ) < +∞, Fin : for every k ∈ N≥k0 , h0 (C, H and Amp :

lim inf

k→+∞

1 µmin (Fk ) > 0, k

then there exists k1 ∈ N>k0 such that the map δk vanishes for all k ∈ N≥k1 , or b k−1 −→ H b k is an isomorphism in equivalently, such that for all k ∈ N≥k1 , pk : H proVectC . One easily sees that ker δk is a saturated coherent subsheaf of Fk , and therefore that ∼ im δk −→ Fk / ker δk

9.4. A Vanishing Criterion

235

is a vector bundle over C. Since the cohomology groups of im δk are finite-dimensional k-vector spaces, the long exact sequence of cohomology b k−1 ) −→ Γ(C, H b k ) −→ H 1 (C, im δk ) Γ(C, im δk ) −→ Γ(C, H associated to the short exact sequence of OC -modules pk

b k−1 −→ H b k −→ 0 0 −→ ker pk = im δk ,−→H b k ) < +∞ for some k ∈ shows that condition Fin is satisfied as soon as h0 (C, H N≥k0 . In condition Amp, µmin (Fk ) denotes the minimal slope of Fk , namely the infimum of the slopes µ(V ) := degC V /rk V of the vector bundles V of positive rank that arise as quotients of Fk . (It is +∞ if rk Fk = 0.) Proof. For all k ∈ N≥k0 , we may consider the surjective morphism of topological OC -Modules qk : Hk0 −→ Hk defined as the composition of the morphisms (pi )k0 k0 , we have an exact sequence of OC -modules qk−1

0 −→ ker qk−1 ,−→ ker qk −→ ker pk = im δk −→ 0. Since ker qk0 = 0, this implies inductively that the OC -Modules ker qk are vector bundles over C and that if for every k ∈ N≥k0 we let nk := rk im δk , we have rk ker qk − rk ker qk−1 = nk

(9.4.1)

degC ker qk − degC ker qk−1 ≥ nk µmin (Fk ).

(9.4.2)

and (If Fk = 0, then nk = 0 and nk µmin (Fk ) = 0.) The validity of Amp may be rephrased as the existence of η in R∗+ and of c in R+ such that for all k ∈ N≥k0 , µmin (Fk ) ≥ ηk − c. From this lower bound on µmin (Fk ) combined with the relations (9.4.1) and (9.4.2), we derive that for every k ∈ N≥k0 : X ni rk ker qk = k0 0 when D is large enough. Theorem 10.3.2. If the Hermitian line bundle L is arithmetically ample on X , then there exists κ ∈ R∗+ such that ⊗D

h0θ (π∗ LJ ) =

1 h (X )Dd + o(Dd ) d! L

as D −→ +∞,

(10.3.3)

and ⊗D

h1θ (πZ∗ LJ ) ≤ e−e

κD

for every large enough positive integer D.

(10.3.4)

Proof. Observe that dim XK n(D) := rk Z πZ∗ L⊗D = [K : Q] dimK Γ(Xk , L⊗D ) = O(Dd−1 ). K ) = O(D (10.3.5) Let D0 and η be as in the second part of Theorem 10.3.1. For all D ≥ D0 , πZ∗ L⊗D admits a Z-basis consisting of elements s whose John norms satisfy

kskJ ≤ (2n(D))1/2 kskL∞ ≤ (2n(D))1/2 e−ηD .

10.3. Arithmetically Ample Hermitian Line Bundles and θ-Invariants

255 ⊗D

Therefore, the last of the successive minima of the Euclidean lattice πZ∗ LJ satisfies ⊗D (10.3.6) λn(D) (πZ∗ LJ ) ≤ (2n(D))1/2 e−ηD . The existence of κ > 0 such that (10.3.4) holds follows from (10.3.5), (10.3.6), and from the upper bound on h1θ in terms of the last of the successive minima established in Corollary 2.6.5. ⊗D According to (10.3.1) applied to E := L and to (10.3.5), we also have ⊗D

d πZ∗ L | deg J


− χk.kL∞ πZ∗ L⊗D | = O(n(D) log n(D)) = O(Dd−1 log D).

Together with the first assertion in Theorem 10.3.1, this shows that d πZ∗ L⊗D = 1 h (X )Dd + o(Dd ) deg J d! L

as D −→ +∞.

(10.3.7)

Finally, the Poisson–Riemann–Roch formula ⊗D ⊗D d πZ∗ L⊗D h0θ (πZ∗ LJ ) − h1θ (πZ∗ LJ ) = deg J

and the asymptotic expressions (10.3.7) and (10.3.4) imply (10.3.3).



10.3.3 Lower bounds on the invariant h0θ of Hermitian vector bundles of sections of O(D). For the proof of the Diophantine algebraization criteria (Theorems 10.1.2 and 10.8.1), we will rely on the following weaker variant of Theorem 10.3.2: Corollary 10.3.3. Let N be a positive integer and let X be a closed integral subscheme of PN OK , flat over Spec OK . Let π : X → Spec OK denote its structural morphism. There exists a Hermitian line bundle L over X , a sequence (E D )D∈N of Hermitian vector bundles over Spec OK , and a sequence (ιD )D∈N of injective morphisms of OK -modules ιD : ED ,−→π∗ L⊗D such that for some D0 ∈ N and some c ∈ R∗+ , the following conditions are satisfied for every integer D ≥ D0 and every field embedding σ : K ,→ C: kskL∞ ,σ ≤ kskE D ,σ

for every section s ∈ (π∗ L⊗D )σ ' Γ(Xσ , L⊗D σ );

(10.3.8)

and h0θ (E D ) ≥ c · Ddim X .

(10.3.9)

This statement may be seen as a Diophantine counterpart of the geometric lower bounds in Proposition 10.2.2, concerning the dimension of spaces of sections of powers of (ample) line bundles on projective varieties over a field. It is a

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

256

straightforward consequence of Theorem 10.3.2, which actually shows that Corollary 10.3.3 holds for every arithmetically ample Hermitian line bundle L over X , by letting ⊗D E D := π∗ LJ for every positive integer D. However, like its geometric counterpart stated in Proposition 10.2.2, Corollary 10.3.3 also admits a direct proof, using Noether’s normalization and some basic results concerning sections of the line bundles O(D) on projective spaces (compare to paragraph 10.2.2). This direct proof will use some elementary properties of natural Hermitian norms on symmetric powers of Hermitian vector spaces and on spaces of homogeneous polynomials, which we now recall. Let k be a natural integer. For every δ ∈ R, let us equip M (Ck+1 )∨ := CXi 0≤i≤k

with the Hermitian norm k.kδ such that (Xi )0≤i≤k becomes an orthogonal basis and kX0 kδ = · · · = kXk kδ = e−δ . Then for all D ∈ N, we may equip C[X0 , . . . , Xk ]D ' S D (Ck+1∨ ) with the Hermitian norm k.kD,δ induced from k.kδ by taking its Dth symmetric power; by definition, it is the quotient norm of the Hermitian norm on (Ck+1∨ )⊗D induced from k.kδ by the tensor product. Let MD := {I := (i0 , . . . , ik ) ∈ Nk+1 | |I| := i0 + · · · + ik = D}, and as usual, let X I := X0i0 · · · Xkik

and I! := i0 ! · · · ik ! for all I = (i0 , . . . , ik ) ∈ Nk+1 .

A straightforward computation shows that (X I )I∈MD is an orthogonal basis of C[X0 , . . . , Xk ]D equipped with k.kD,δ and that kX I k2D,δ =

I! −2δD e ≤ e−2δD . D!

(10.3.10)

Moreover, we may equip the canonical quotient bundle O(1) on PkC := P(C ) with the Hermitian norm induced from the norm k.kδ on Ck+1∨ . For all D in N, this Hermitian norm induces, by tensor product, a Hermitian norm k.kO(D),δ on O(D) over PkC . k+1∨

10.3. Arithmetically Ample Hermitian Line Bundles and θ-Invariants

257

k For all P in C[X0 , . . . , Xk ]D , identified with some section in Γ(PC , O(D)), and every point x := (x0 : · · · : xk )

in the projective space Pk (C), the norm of the value P (x) ∈ O(D)x of P at x satisfies, as a straightforward consequence of the definitions, |P (x0 , . . . , xk )| kP (x)kO(D),δ = e−δD Pk . ( i=0 |xi |2 )D/2 Observe that if P =

X

aI X I ,

I∈MD

then according to (10.3.10), kP k2D,δ =

X

|aI |2 kX I k2D,δ =

I∈MD

X

|aI |2

I∈MD

I! −2δD e . D!

Therefore, by the Cauchy–Schwarz inequality, 2 X X D! 2 I |P (x0 , . . . , xk )| = aI x ≤ kP k2D,δ |x2I | e2δD I! I∈MD

I∈MD

n X = kP k2D,δ e2δD ( |xi |2 )D . i=0

This shows that kP (x)kO(D),δ ≤ kP kD,δ .

(10.3.11)

Direct proof of Corollary 10.3.3. Let d := dim X . It is a positive integer and dim XK = d − 1. After possibly composing the embedding X ,→ PN OK with the automorphism of PN defined by some element g ∈ SL (O ), we may assume that XK is disjoint k+1 K OK from the linear subspace d−1 \ −d PN = div Xi K i=0

defined by the vanishing of the first d homogeneous coordinates X0 , . . . , Xd−1 9 . Then the linear projection p:

N −d PN −→ OK \ P OK (x0 : . . . : xN ) 7−→

Pd−1 OK , (x0 : . . . xd−1 ),

−d by the properness of PN , the condition gX (K) ∩ PN −d (K) = ∅ on g ∈ SLN +1 (K) K defines an open subscheme in SLN +1,K . By Noether’s normalization, this open subscheme is not empty, and therefore meets SLN +1 (OK ), which is Zariski dense in SLN +1,K . 9 Indeed,

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

has a well-defined restriction d−1 p|XK : XK −→ PK ,

which is a finite surjective morphism. For all D ∈ N, we define d−1 ED := Γ(PO , O(D)) ' OK [X0 , . . . , Xd−1 ]D . K

The pullback p∗ O(1) may be identified with the restriction O(1)|PN

N −d OK \POK

, and

the pullback map N −d N −d ∗ N p∗ : ED −→ Γ(PO \ PO , p O(D)) ' Γ(PN OK \ POK , O(D)) K K

has its image contained in Γ(PN OK , O(D)) and may actually be identified with the inclusion morphism OK [X0 , . . . , Xd−1 ]D ,−→OK [X0 , . . . , XN ]D . Furthermore, let us define L := O(1)|X . By composing p∗ and the map “restriction to X ”, we define a morphism of OK modules p∗

N , O(D)) iD : ED −→ Γ(PO K

restriction

−→

Γ(X , O(D)) ' Γ(X , L⊗D ).

∗ The pullback p|X O(1) coincides with O(1)XK ' LK , and therefore the map K d−1 iD,K : ED,K = Γ(PK , O(D)) −→ Γ(XK , L⊗D K )

may be identified with the pullback by the finite surjective morphism p|XK , and is therefore injective. To complete the proof of Corollary 10.3.3, we are left to show that there exist Hermitian metrics on L and on the ED s such that the conditions (10.3.8) and (10.3.9) are satisfied. ∗ To achieve this, let us choose δ ∈ R+ , and let us use the above construction of k the metrics k.kO(1),δ (on the line bundle O(1) over PC ) and k.kD,δ (on Γ(PkC , O(D))) in the special case k = d − 1. For every embedding σ : K ,→ C, we have an identification Lσ ' p∗σ|Xσ O(1) of line bundles over Xσ . We define L as L equipped with the metric defined on Xσ as the pullback of k.kO(1),δ by pσ . Furthermore, we define E D as ED equipped with the metric k.kE D ,σ := k.kD,δ for every embedding σ : K ,→ C. With these choices of Hermitian structures, the validity of (10.3.8) follows from the norm estimates (10.3.11).

10.4. Pointed Smooth Formal Curves

259

To establish (10.3.9), let us observe that E D may be written as the direct sum of Hermitian line bundles M ED = OK X I I∈MD

and that according to (10.3.10), d OK X I = −[K : Q](1/2) log deg

 I! D!

 e−2δD ≥ [K : Q]δD.

Therefore, d E D ≥ [K : Q]δD rk ED , deg and by the “Riemann inequality” (2.3.1), d E D − 1 log |∆K | · rk ED h0θ (E D ) ≥ deg 2 1 ≥ ([K : Q] δD − log |∆K ) rk ED . 2 Since  rk ED =

 D+d−1 1 D d−1 , ≥ (d − 1)! d−1

this proves that for every given c < [K : Q] δ/(d − 1)!, we have h0θ (E D ) ≥ c.Dd , when D is large enough.

10.4



Pointed Smooth Formal Curves

Let us consider a Noetherian affine base scheme S := Spec A. 10.4.1 Definitions. We define a pointed smooth formal curve over S as the data b π, P) of a Noetherian affine formal scheme10 Cb = Spf B, of a morphism of (C, (formal) schemes π : Cb −→ S, and of a section P : S −→ Cb of π such that the following conditions are satisfied: 10 See

EGA I [57, Section 1.10].

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PSFC1 : The ideal I of B defined as the kernel of the morphism of rings b O b) −→ A = Γ(S, OS ) P ∗ : B = Γ(C, C is an ideal of definition of the (Noetherian adic) ring B. PSFC2 : The morphism of topological rings b O b) π ∗ : A = Γ(S, OS ) −→ B = Γ(C, C makes the (Noetherian adic) ring B a formally smooth11 algebra over A (equipped with the discrete topology). PSFC3 : The fiber of π over any point of S with value in a field is one-dimensional. When PSFC1 is satisfied, we may introduce I := I ∆ , the Ideal of definition b of C corresponding to I. For every n ∈ N, we may consider the affine S-schemes of finite type b O b/I n+1 ) ' Spec B/I n+1 , Cn := (|C|, C and Cb may be identified with their direct limit in the category of formal schemes: Cb ' lim Cn . −→ n

Moreover, the morphisms π|C0 and P define isomorphisms, inverse to each other, between C0 and S. Still assuming that PSFC1 holds, conditions PSFC2 and PSFC3 are equivalent to the following one: PSFC02−3 : The coherent sheaf I/I 2 over C0 (' P) is a line bundle, and for every positive integer n, the canonical morphism of OC0 -Modules12 ϕn : (I/I 2 )⊗n −→ I n /I n+1 is an isomorphism. This follows from EGA IV1 [56, Chapitre 0, 19.5.4, equivalence of a) and b)]. When conditions PSFC1−3 hold, the line bundle I/I 2 (or its pullback b We will also consider P ∗ I/I 2 over S) may be called the conormal bundle of P in C. its dual, NP Cb := P ∗ (I/I 2 )∨ , b the normal bundle of P in C. 11 See

EGA IV1 [56, Chapitre 0, 19.3.1.] every local section s of I of class [s] in I/I 2 , ϕn maps [s]⊗n to the class of sn in I n /I n+1 . 12 For

10.4. Pointed Smooth Formal Curves

261

10.4.2 Examples and remarks. (i) Observe that if f : C −→ S is a smooth relative curve over S (namely, a smooth morphism of schemes of relative dimension 1), then every section P : S −→ C of f becomes, after possibly replacing C by an open subscheme13 , a closed immersion. We may therefore consider the formal completion Cb of C along the image of P . By completion, from f and P, one deduces morphisms of formal schemes π : Cb −→ S

b and P : S −→ C,

b π, P) defines a pointed smooth formal curve over and it is straightforward that (C, S. If Tf denotes the relative tangent bundle of f , we have a canonical isomorphism ∼

NP Cb −→ P ∗ Tf . (ii) For every line bundle L over S, we may consider the affine S-scheme VS (L∨ ) := SpecS SymOS L∨ . (It is defined as the spectrum of the quasi-coherent OS -algebra M L∨⊗n , SymOS L∨ := n∈N

and may be thought of as the “total space” of the line bundle L.) It is smooth over S, of relative dimension 1, and equipped with the “zero-section” ε, defined by the augmentation ideal M L∨⊗n = L∨ · SymOS L∨ Iε := 0 ⊕ n∈N>0

of the symmetric algebra SymOS L∨ . Applied to C := VS (L∨ ) and P := ε, the previous construction defines a b S (L∨ ), π, ε). By construction, we have a canonical pointed smooth formal curve (V isomorphism ∼ b S (L∨ ) −→ L. Nε V b S (L∨ ) may be described as follows In concrete terms, the S-formal scheme V in terms of the projective A-module of rank one L∨ := Γ(S, L∨ ): it is the affine formal scheme attached to the topological A-algebra d A L∨ , B := Sym defined as the completion of the symmetric algebra M L∨⊗A n SymA L∨ := n∈N 13 This

is not needed if f , or equivalently C, is separated.

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

with respect to the adic topology defined by the augmentation ideal M I := L∨ · SymA L∨ = 0 ⊕ L∨⊗A n . n∈N>0

As a topological A-module, ∼

B −→

Y

L∨⊗A n ,

n∈N

where the right-hand side is equipped with product topology of the discrete topology on each of the factors L∨⊗A n . b π, P) over S is isoIt turns out that every pointed smooth formal curve (C, ∨ b S (L ), π, ε), where L := NP Cb (see EGA morphic (noncanonically in general) to (V IV1 [56, Chapitre 0, 19.5.4, equivalence of a) and c)]). When S is the spectrum Spec k of a field k, a pointed smooth formal curve over S is often called a smooth germ of formal curve over k, and the previous classification result boils down to the classical fact that any smooth germ of formal curve over k is isomorphic to Spf k[[T ]]. (iii) Let us finally observe that for every morphism of affine Noetherian schemes S 0 −→ S, we may “base change” any pointed smooth formal curve b π, P) over S to S 0 , and define a pointed smooth formal curve (CbS 0 , πS 0 , PS 0 ) (C, over S 0 .

10.5 Green’s Functions, Capacitary Metrics and Schwarz Lemma on Compact Riemann Surfaces with Boundary 10.5.1 Compact Riemann surfaces with boundary. A Riemann surface with boundary is the data (V, V + ) of a Riemann surface V + (without boundary) and a closed C ∞ submanifold with boundary (of codimension 0) V of V + . ˚ of V in V + is then a Riemann surface (without boundary), The interior V its boundary ˚ ∂V := V \ V is a closed 1-dimensional real C ∞ submanifold of V + , and we may consider the open and closed immersions jV iV ˚ ,−→ V ,−→ V + . V

We shall consider that shrinking the “ambient” Riemann surface V + to a 0 smaller open neighborhood V + of V does not change the surface with boundary (V, V + ). In other words, V + should be understood as a germ of a Riemann surface around V .

10.5. Green’s Functions, Capacitary Metrics and Schwarz Lemma

263

An alternative definition, formally more satisfactory, would be to define the Riemann surface with boundary associated to (V, V + ) as the ringed space (V, OVan ), where OVan denotes the inverse image jV−1 OVan+ of the sheaf of C-analytic functions OVan+ on V + . An analytic vector bundle over the Riemann surface with boundary V is, by definition, a germ of an analytic vector bundle along V in V + . In other words, by definition, an analytic vector bundle over V extends to an analytic vector bundle on any small enough open neighborhood of V in V + . (Such analytic vector bundles precisely correspond to locally free sheaves of finite rank over the ringed space (V, OVan ).) A C ∞ Hermitian metric k.k on an analytic vector bundle E on V is, by definition, a C ∞ Hermitian metric on the C ∞ vector bundle E|V on the C ∞ manifold (with boundary) V . The pair (E, k.k) will the be called a Hermitian analytic vector bundle over V . Clearly, the usual tensor operations (such as direct sums and tensor products) make sense for analytic vector bundles and Hermitian analytic vector bundles over Riemann surfaces with boundary. A volume form ν on the Riemann surface with boundary V is a C ∞ 2-form on the oriented C ∞ manifold (with boundary) V that is everywhere positive. 10.5.2 Green’s functions and capacitary metrics. In this paragraph, we recall some well-known facts concerning Green’s functions on compact Riemann surfaces with boundary. For a detailed discussion and references on this topic, we refer the reader to [14, Section 3.1 and Appendix]. We shall actually not need the “refined” theory (which allows domains with rough boundaries) presented in that work, and the results below may also be obtained as special cases of some basic results concerning elliptic boundary problems (see, for instance, [112, Sections 5.1-2]). Let V be a connected compact Riemann surface with boundary V such that ∂V is non-empty. ˚, one defines the Green’s function gV,O of O in V as For every point O in V the unique function gV,O : V \ {O} −→ R that satisfies the following conditions: Gr1 1 : gV,O is continuous on V \ {O} and gV,O|∂V = 0. ˚ \ {O}. Gr2 1 : gV,O is harmonic on V Gr3 1 : gV,O admits a logarithmic singularity at O; namely, if ∼

z : U −→ z(U ) denotes an analytic chart with domain an open neighborhood U of O in ˚ and range an open subset z(U ) in C, we have: V gV,O (P ) = log |z(P ) − z(O)|−1 + O(1)

as P → O.

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

264

When conditions Gr1 and Gr2 are satisfied, the function of P gV,O (P ) − log |z(P ) − z(O)|−1 is harmonic on U \ {O} and remains bounded as P goes to O, hence extends to some harmonic function h on U . By construction, we have, for every P ∈ U \ {O}, gV,O (P ) = k∂/∂zkcap V,O + h(P ). ˚. ˚, hence a current on V In particular, gV,O defines a locally L1 function on V Together with the Poincar´e–Lelong equation on C, (i/2π) ∂ ∂ log |.|2 = δ0 , these observations show that conditions Gr2 and Gr3 imply ˚ that satisfies the equation of currents Gr2−3 : gV,O defines a current on V i ∂∂gV,O = −π δO .

(10.5.1)

Conversely, using that every current in the kernel of ∂∂ is actually a C ∞ (harmonic) function, one easily see that Gr2−3 implies Gr2 and Gr3 . With the above notation, the capacitary metric k.kcap V,O on the tangent space TO V of V at O is the Hermitian metric on this complex line defined by −h(P ) k∂/∂zkcap . V,O := e

(10.5.2)

In other words,   −1 log k∂/∂zkcap − gV,O (P ) . V,O = lim log |z(P ) − z(O)| P →O

(10.5.3)

The relation (10.5.3) makes clear that the definition (10.5.2) of the capacitary metric is actually independent of the choice of the local coordinate z. ˚ \ {O} (this is a straightforLet us finally recall that gV,O is positive on V ward consequence of the maximum principle for harmonic functions on Riemann surfaces). Moreover, gV,O is C ∞ up to the boundary — that is, C ∞ on the surface with boundary V \ {O} — and its differential dgV,O does not vanish on ∂V (see, for instance, [112, 5.1-2]). 10.5.3

Examples. (i) For all R ∈ R+ , we let ˚ R) := {z ∈ C | |z| < R} D(0,

and D(0, R) := {z ∈ C | |z| ≤ R}.

10.5. Green’s Functions, Capacitary Metrics and Schwarz Lemma

265

When R > 0, D(0, R) is a compact connected Riemann surface with nonempty boundary. It is straightforward that for every z ∈ D(0, R) \ {0}, gD(0,R),0 (z) = log

R . |z|

Consequently, we have cap k∂/∂zkD(0,R),0 = 1/R.

(ii) Let V ,→ P1 (C) be a closed submanifold with boundary (of codimension 0) of the complex projective line. Let us assume that V is connected and contains the point ∞ := (0 : 1). On P1C := Proj(CX0 ⊕ CX1 ), we may consider the rational function z := ∼ X1 /X0 — it defines the usual identification P1C \ {∞} −→ A1C — and its inverse t := X0 /X1 , which defines a local coordinate on P1 (C) \ {0} and vanishes at ∞. By the definitions of the Green’s function and the capacitary metric attached ˚, we have, as P ∈ A1 (C) goes to ∞, to the point ∞ of V gV,∞ (P ) = log |t(P )|−1 − log k∂/∂tkcap V,∞ + o(1) = log |z(P )| − log k∂/∂tkcap V,∞ + o(1). This shows that gV,∞ coincides with the classical Green’s function of the ˚, and that − log k∂/∂tkcap is the so-called Robin compact set K := A1 (C) \ V V,∞ constant of K and k∂/∂tkcap its two-dimensional capacity c(K) (see, for instance, V,∞ [93, Chapter 5]). ˚ R) \ {0} and let (iii) Let F := {a1 , . . . , an } be a finite subset of D(0, ε := (ε1 , . . . , εn ) be an element of R∗n + such that εi < R − |ai | for all i ∈ {1, . . . , n} and εi + εj < |ai − aj | for all (i, j) ∈ {1, . . . , n}2 such that i 6= j. Then the disks D(ai , εi ) are contained in [ Vε := D(0, R) \

˚ R) and pairwise disjoint, and D(0, ˚ i , εi ) D(a

1≤i≤n

is a compact Riemann surface with boundary. As ε goes to (0, . . . , 0) in Rn , the surface Vε shrinks to D(0, R)\F . Moreover, the fact that the finite subset F is polar in C implies that for all z ∈ D(0, R) \ {0}, lim ε→(0,...,0)

gVε ,0 (z) = gD(0,R),0 (z) = log

R |z|

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

and lim ε→(0,...,0)

cap = k∂/∂zkD(0,R),0 k∂/∂zkVcap = 1/R. ε ,0

(10.5.4)

(See [93], notably the properties of the two-dimensional capacity discussed in Chapter 5.) 10.5.4 Hilbert spaces of analytic sections and Fr´echet spaces of formal sections. Let V be a compact Riemann surface with boundary, ν a volume form on V , and ˚, namely an analytic vector (E, k.k) a Hermitian analytic vector bundle over V ˚ ˚. bundle E over V equipped with a Hermitian metric k.k, C ∞ on V ˚ The space of analytic sections s of E over V such that Z 2 kskL := ksk2 ν < +∞, ˚ V

when equipped with the L2 -norm k.kL2 , defines a Hilbert space that we shall denote by ΓL2 (V, ν; E, k.k). ˚. We may consider its successive infinitesimal Let us choose a point O in V 14 neighborhoods On in V , and the formal completion VbO := lim On −→ n of V at O. The space of sections of E over VbO , ∼

Γ(VbO , E) −→ lim E|On , ← − n is equipped with a natural Fr´echet space topology, as the projective limit of the finite-dimensional complex vector spaces E|On endowed with their natural topology of (separated) topological vector spaces. We may finally introduce the restriction map ηˆ :

ΓL2 (V, ν; E, k.k) −→ s 7−→

Γ(VbO , E), s|VbO ,

˚ to its “jet” s b at the point which maps a L2 holomorphic section s of E over V |VO O. Proposition 10.5.1. The linear map ηˆ is continuous when ΓL2 (V, ν; E, k.k) is equipped with its Hilbert space topology and Γ(VbO , E) with is natural Fr´echet space topology. 14 By definition, for all n ∈ N, O is defined by the sheaf of ideals I n+1 in O an , where I n O V O denotes the ideal sheaf of O. It is convenient to extend this definition to n = −1, so that O−1 = ∅.

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267

When V is connected, ηˆ is injective. If, moreover, the boundary ∂V is nonempty and if (E, k.k) extends to a Hermitian analytic vector bundle over V 15 , then the image of ηˆ is dense in Γ(VbO , E). ˚ by analytic conProof. The injectivity of ηˆ follows from the connectedness of V tinuation. The continuity of ηˆ is equivalent to the continuity of the maps ΓL2 (V, ν; E, k.k) −→ s 7−→

ηn :

E|On , s|On ,

when n ∈ N. This directly follows for Cauchy estimates. The density in Γ(VbO , E) of the image of ηˆ is equivalent to the surjectivity of the maps ηn . This surjectivity follows from the surjectivity of the composite maps ηn

˜ Γ(V + , E),−→Γ L2 (V, ν; E, k.k) −→ EOn , which in turn is a consequence of the fact that V is a Stein compact subset (that is, admits a basis of open Stein neighborhoods) in V + , since every connected noncompact Riemann surface is a Stein manifold (theorem of Behnke–Stein; see [58, Chapitre V, Th´eor`eme 1] for a simple proof).  Let us observe that for all n ∈ N, if s denotes an analytic section of E on an ˚ that vanishes at order at least n at O — in other open neighborhood of O in V words, that satisfies ηn−1 (s) := s|On−1 = 0 — then its jet of order n at O, ηn (s) := s|On , may be identified with some element of TO∨⊗n ⊗ EO . Let us also recall that if L := (L, k.k) denotes a Hermitian analytic line ˚, then its first Chern form c1 (L) is the real (1, 1)-form on V ˚ bundle16 over V defined by the equality 1 ∂∂ log ksk2 c1 (L) = 2πi for every local nonvanishing analytic section of L. More generally, if s denotes a ˚ and div s nonzero meromorphic section of L over a connected open subset U of V its divisor, then the function log ksk is locally L1 on U and satisfies the following equality of currents on U : i ∂∂ log ksk = δdiv s − c1 (L) π

(10.5.5)

(Poincar´e–Lelong equation). 15 Namely, if the vector bundle E is the restriction to V ˚ of an analytic vector bundle E ˜ on V + ˜ over V . and the Hermitian metric k.k extends to a C ∞ metric on E 16 That is, a Hermitian analytic vector bundle of rank one

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

268

10.5.5 Poisson–Jensen formula and Schwarz lemma over a compact Riemann surface with boundary. Let V be a connected compact Riemann surface with ˚. nonempty boundary, and let O be a point of V Let η be a positive real number and let ρ : R −→ R+ be a C ∞ function with compact support such that Z supp ρ ⊂]0, η[ and ρ(t) = 1. (10.5.6) R

Let us denote by χ the “second primitive” of ρ, namely the C ∞ function from R to R defined by χ(0) = χ0 (0) = 0 and χ00 = ρ. It is a nondecreasing convex function with values in R+ . Moreover, there exists c ∈ R such that17 for every t ∈ [η, +∞[, χ(t) = t + c. (10.5.7) Like gV,O , the function χ ◦ gV,O has a logarithmic singularity at O and is C ∞ ˚ ˚ \ {O}. It defines a current (of degree 0) on V + with compact support in V on V and with singular support {O}. ˚: Proposition 10.5.2. The following equality of currents holds on V i ∂∂(χ ◦ gV,O ) = −δO + µχ , π where

is a C ∞

(10.5.8)

i µχ := (χ00 ◦ gV,O ) ∂gV,O ∧ ∂gV,O (10.5.9) π ˚ \ {O} and satisfies positive (1, 1)-form with compact support in V Z µχ = 1. (10.5.10) ˚ V

−1 ˚, we have Proof. On the open neighborhood U := gV,O (]η, +∞]) of O in V

χ ◦ gV,O = gV,O + c, and therefore

i ∂∂(χ ◦ gV,O ) = −δO . π

˚ \ {O}, the Green’s function gV,O is C ∞ and ∂∂gV,O = 0, and Moreover, on V therefore ∂(χ ◦ gV,O ) = (χ0 ◦ gV,O ) ∂gV,O 17 A

straightforward computation shows that c = −

R +∞ 0

tρ(t)dt ∈] − η, 0[.

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269

and ∂ ∂(χ ◦ gV,O ) = ∂(χ0 ◦ gV,O ) ∧ ∂gV,O + (χ0 ◦ gV,O ) ∂∂gV,O = (χ00 ◦ gV,O ) ∂gV,O ∧ ∂gV,O . To complete the proof of the proposition, we are left to establish the equality (10.5.10). It follows from (10.5.8) and from the fact that since χ◦gV,O has compact ˚, we have support in V Z Z ∂∂(χ ◦ gV,O ) = d ∂(χ ◦ gV,O ) = 0 ˚ V

˚ V

by Stokes’s formula.



˚, Theorem 10.5.3. Let L := (L, k.k) be a Hermitian analytic line bundle L over V ˚ and let s be a nonzero analytic section of L on V and n := vO (s) its order of vanishing at the point O. Let us denote by ηn (s) := s|On the jet of order n of s at O — a nonzero element of the complex line TO∨⊗n ⊗ LO — and by k.kcap the Hermitian metric M,O,L on TO∨⊗n ⊗ LO derived from the metrics k.kcap M,O on TO M and k.k on LO by duality and tensor product. Then we have log kηn (s)kcap M,O,L Z Z Z Z = log ksk µχ − (χ ◦ gV,O ) δdiv s−nO + n gV,O µχ + (χ ◦ gV,O ) c1 (L). ˚ V

˚ V

˚ V

˚ V

(10.5.11) Let 1O(O) denote the tautological section, with divisor O, of the analytic line bundle O(O) over V . Applied to the line bundle L := (O(O), k1O(O) k := e−χ◦gV,O ) and its section s := 1O(O) , Theorem 10.5.3 shows that the penultimate integral on the right-hand side of (10.5.11) takes the value Z ˚ V

Z gV,O µχ = −c =

+∞

tρ(t) dt.

(10.5.12)

0

The equality (10.5.11) may be seen as an avatar of the classical formulas of Poisson and Jensen. To clarify the relation between Theorem 10.5.3 and these formulas, let us observe that using the Green’s function gV,O , we may construct a family of Riemann surfaces with boundary Vε as follows. Let us recall that the Green’s function gV,O is C ∞ on V \ {O}, positive on ˚ V \ {O}, and vanishes on ∂V and that its differential dgV,O does not vanish on ∂V . Moreover, the map g : V \ {O} → [0, +∞[ is proper. This implies the existence of

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

∗ ε0 in R+ such that every element of [0, ε0 [ is a regular value of gV,O . Then for all ε ∈ [0, ε0 [, Vε := g −1 ([ε, +∞]) ˚ε contains is a connected Riemann surface with nonempty boundary; moreover, V O, and admits ∂Vε := g −1 (ε)

as boundary. Finally, it is straighforward that gVε ,O = gV,O − ε on Vε and = eε k.kcap k.kVcap V,O . ε ,O

(10.5.13)

For all ε ∈]0, ε0 [, the “limit case” of (10.5.11) when ρ is replaced by δε — and accordingly χ(t) by (t − ε)+ — becomes the following formula, cap log kηn (s)kV,O, L Z Z Z = log ksk dc gV,O − (gV,O − ε) δdiv s−nO + nε + (gV,O − ε) c1 (L), ∂Vε

Vε

Vε

(10.5.14) where dc := (i/2π)(∂ − ∂). This relation may also be written as the following Poisson–Jensen formula on the Riemann surface with boundary Vε : Z Z Z c log kηn (s)kcap = log ksk d g − g δ + gVε ,O c1 (L). Vε ,O Vε ,O div s−nO V ,O,L ε

∂Vε

Vε

Vε

(10.5.15) We leave the details of the derivation of (10.5.14) and (10.5.15) to the reader. We simply observe that conversely, when supp ρ ⊂ ]0, ε0 [, we recover (10.5.11) from (10.5.14) by mutilplying both sides by ρ(ε) and integrating. ˚ \{O}. In Proof of Theorem 10.5.3. The current log ksk+ngV,O is clearly C ∞ on V fact, the logarithmic singularities of log ksk and ngV,O at O cancel, and log ksk + ngV,O defines a C ∞ function on an open neighborhood of O. Moreover, by the definition (10.5.3) of the capacitary metric k.kcap M,O , the value at O of this function cap is log kηn (s)kM,O,L . In other words, Z (log ksk + ngV,O ) δO = log kηn (s)kcap . (10.5.16) M,O,L ˚ V

˚, and its singular support and The current χ ◦ gV,O has compact support in V that of log ksk + ngV,O are disjoint, so that Z Z i i (χ ◦ gV,O ) ∂∂(log ksk + ngV,O ) = (log ksk + ngV,O ) ∂∂(χ ◦ gV,O ) (10.5.17) π π ˚ ˚ V V

10.5. Green’s Functions, Capacitary Metrics and Schwarz Lemma

271

(Green–Stokes formula). Furthermore, by the defining property (10.5.1) of the Green’s function gV,O and the Poincar´e–Lelong equation (10.5.5), the following equality of currents on ˚ holds: V i ∂∂(log ksk + ngV,O ) = δdiv s−nO − c1 (L). (10.5.18) π By means of this relation and the expression (10.5.8) for πi ∂∂(χ ◦ g), the Green– Stokes formula (10.5.17) becomes Z Z (log ksk + ngV,O )(−δO + µχ ). (χ ◦ gV,O ).(δdiv s−nO − c1 (L)) = ˚ V

˚ V

Taking (10.5.16) into account, this is precisely the equality (10.5.11) to be proved.  From the Poisson–Jensen formula (10.5.11), one easily derives the following version of the Schwarz lemma over a compact Riemann surface with boundary. Corollary 10.5.4. Let L be as in Theorem 10.5.3. For all n ∈ N and every analytic ˚ of vanishing order vO (s) at O at least n, we have section s of L over V kηn (s)kcap ≤ eα(L)+nη kskL2 (µχ ,L) , M,O,L

(10.5.19)

where the L2 -norm kskL2 (µχ ,L) is defined by Z ksk2L2 (µχ ,L) := ksk2 µχ ˚ V

and

Z α(L) := ˚ V

(χ ◦ gV,O ) c1 (L).

Proof. When vO (s) > n, the jet ηn s vanishes and (10.5.19) is clear. When vO (s) = n, it follows from the Poisson–Jensen formula (10.5.11), together with the following elementary estimates: Z log ksk µχ ≤ log kskL2 (µχ ,L) , (10.5.20) ˚ V

Z ˚ V

and

(χ ◦ gV,O ) δdiv s−nO ≥ 0,

(10.5.21)

Z ˚ V

gV,O µχ ≤ η.

(10.5.22)

Indeed, (10.5.20) follows from Jensen’s inequality applied to the concave funcR tion log, since V˚ µχ = 1. The lower bound (10.5.21) follows from the nonnegativity of χ◦g. Finally, since the support of χ00 is contained in ]0, η[, the Rexpression (10.5.9) −1 for µχ shows that its support is contained in gV,O ([0, η]). Since V˚ µχ = 1, this implies (10.5.22). The estimate (10.5.22) also follows from the relation (10.5.12). 

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

272

For later reference, let us state a straightforward consequence of Corollary 10.5.4. Scholium 10.5.5. Let V be a compact connected Riemann surface with (nonempty) ˚, and ν a volume form on V . boundary, O a point in V (1) Let E := (E, k.k) be a Hermitian analytic vector bundle over V . For all η > 0, there exists Cη in R+ such that the following condition holds: for all n ∈ N ˚ of vanishing order at O at least and every analytic section s of E over V n, the capacitary norm of its jet ηn (s) of order n at O satisfies18 the upper bound kηn (s)kcap ≤ Cη enη kskL2 (ν,E) , (10.5.23) M,O,E where ksk2L2 (ν,E) :=

Z

ksk2 ν (∈ [0, +∞]).

˚ V

(2) Let L := (L, k.k) be a Hermitian analytic vector bundle over V . For all η > 0, there exists Cη in R+ such that the following condition holds: for all ˚ of vanishing order (D, n) ∈ N2 and every analytic section s of L⊗D over V at O at least n, the capacitary norm of its jet ηn (s) of order n at O satisfies the upper bound kηn (s)kcap

⊗D

M,O,L

≤ CηD+1 enη kskL2 (ν,L⊗D ) .

(10.5.24)

Proof. Let us first establish (2). For all η > 0, we may choose ρ such that condition (10.5.6) is satisfied. Then ⊗D Corollary 10.5.4, applied to L instead of L, shows that with the notation of (2), we have kηn (s)kcap ⊗D ≤ CηD enη kskL2 (µχ ,L⊗D ) , M,O,L

where Cη := eα(L) . ⊗D

⊗D

(Indeed, c1 (L ) = Dc1 (L), and therefore α(L ) = Dα(L).) After possibly increasing Cη , we may also assume that µχ ≤ Cη2 ν. Then kskL2 (µχ ,L⊗D ) ≤ Cη kskL2 (ν,L⊗D ) , and (10.5.24) follows. Assertion (1) when rk E = 1 follows from (2) with D = 1. One reduces to this situation, thanks to the following observations: 18 As observed at the end of 10.5.4, η (s) may be seen as an element T ∨⊗n ⊗E . The capacitary n O O is the norm deduced from the norms k.kcap norm k.kcap M,O on TO M and k.k on EO by duality M,O,E and tensor product.

10.6. Smooth Formal-Analytic Surfaces over Spec OK

273

(a) the validity of (1) for Hermitian analytic vector bundles E 1 , . . . , E N over V is equivalent to its validity for the direct sum E 1 ⊕ · · · ⊕ E N ; (b) for every analytic vector bundle E over V and every two C ∞ Hermitian metrics k.k1 and k.k2 on E, the validity of (1) for (E, k.k1 ) and (E, k.k2 ) are equivalent; (c) every analytic vector bundle E over V may be trivialized, and is therefore isomorphic to a direct sum L1 ⊕ · · · ⊕ Lrk E of analytic line bundles over V . (Assertions (a) and (b) are straightforward; (c) is a consequence of the theorem of Behnke–Stein; see, for instance, [58, Chapitre V, Th´eor`eme 3].) 

10.6

Smooth Formal-Analytic Surfaces over Spec OK

10.6.1 Basic definitions. We shall define a smooth formal-analytic surface over Spec OK as a pair b (Vσ , Pσ , iσ )σ:K,→C ), V˜ := (V, where: • V˜ is a pointed smooth formal curve over Spec OK ; we shall denote by π : V˜ −→ Spec OK its structural morphism, and by P the canonical section19 of π; • for every field embedding σ : K ,→ C, Vσ is a compact connected Riemann ˚σ , and iσ is surface with nonempty boundary, Pσ is a point in its interior V an isomorphism ∼ c bσ −→ iσ : V Vσ ,Pσ between the smooth germs of formal complex curves defined by the base change bσ := V b ⊗O ,σ C V K b from OK to C through the embedding σ : OK ,→ C and the formal of V cσ ,P of Vσ at the point Pσ . completion V σ These data are, moreover, assumed to be compatible with complex conjugation. Namely, we are given a family (jσ )σ:K,→C of antiholomorphic isomorphisms of Riemann surfaces with boundary ∼

jσ : Vσ −→ Vσ such that jσ = jσ−1 , 19 If V bred denotes |V| b equipped with its structure of a reduced scheme (or equivalently the b defined by its largest ideal of definition), the map π defines an isomorphism subscheme of V ∼ bred −→ π|Vb :V Spec OK , and P coincides with the inverse of πVb . red

red

274

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization jσ (Pσ ) = Pσ ,

and such that the following diagram is commutative: iσ bσ := V b ⊗O ,σ C −−− cσ ,P V −→ V K σ    c id⊗.y yjσ ,Pσ

(10.6.1)

iσ bσ := V b ⊗O ,σ C −−− cσ ,P . V −→ V K σ

It is convenient to see the isomorphisms jσ as identifications Vσ ' Vσc.c. of the Riemann surfaces Vσ with the complex conjugates of the Riemann surfaces Vσ , and to think of them as defining a real structure on a VC := Vσ . σ:K,→C

We shall define a vector bundle E˜ over the smooth formal-analytic surface V˜ as a pair b (Eσ , ϕσ )σ:K,→C ), E˜ := (E, where: b • Eb is a vector bundle (that is, a locally free coherent sheaf) over V; • for every field embedding σ : K ,→ C, Eσ is an analytic vector bundle over bσ : Vσ and ϕσ is an isomorphism of formal vector bundles over V ∼ ˆ OK ,σ C −→ ϕσ : Ebσ := Eb⊗ i∗σ Eσ |Vcσ

,Pσ

.

iσ

∼ c bσ −→ (Recall that we are given an isomorphism V Vσ ,Pσ .) The family (Eσ , ϕσ )σ:K,→C is, moreover, assumed to be compatible with complex conjugation. Namely, we are given a family (Jσ )σ:K,→C of isomorphisms ∼

Jσ : Eσ −→ Eσc.c. over Vσ ' Vσc.c. that satisfy the relations Jσ = Jσc.c. −1 and make the diagram ϕσ Ebσ := Eb ⊗OK ,σ C −−−−→ i∗σ Eσ |Vcσ   ,Pσ  c id⊗.y yJσ ϕσ

Ebσ := Eb ⊗OK ,σ C −−−−→ iσ∗ Eσ |Vcσ

,Pσ

(10.6.2)

10.6. Smooth Formal-Analytic Surfaces over Spec OK

275

commutative. The family (Jσ ) may be understood as defining a “real structure” on the C-analytic vector bundle over VC , whose restriction to Vσ is Eσ for every field embedding σ : K ,→ C. In practice, the definition of the isomorphisms Jσ is often clear from the context and accordingly not explicitly given. Finally, a Hermitian vector bundle E˜ over V˜ is defined as a pair b (Eσ , ϕσ , k.kσ )σ:K,→C ), E˜ := (E, where: b (Eσ , ϕσ )σ:K,→C ) is a vector bundle over V, e as above; • E˜ := (E, • (k.kσ )σ:K,→C is a family of C ∞ Hermitian metrics on the C ∞ vector bundles Eσ|Vσ (on the C ∞ manifolds with boundary Vσ ) that is invariant under complex conjugation20 . Then for every σ : K ,→ C, the pair (Eσ , k.kσ ) is a Hermitian analytic vector bundle over Vσ in the sense of 10.5.1. ˜ ˜ ν; E). 10.6.2 The pro-Hermitian vector bundle ΓL2 (V, Let us consider a smooth formal-analytic surface over Spec OK b (Vσ , Pσ , iσ )σ:K,→C ) V˜ := (V, e and a Hermitian vector bundle over V, b (Eσ , ϕσ , k.kσ )σ:K,→C ). E˜ := (E, Let us, moreover, assume that for every embedding σ : K ,→ C, the Riemann surface Vσ is connected and its boundary ∂Vσ is nonempty. Finally, let ν be a C ∞ volume form on VC , invariant under complex conjugation. For every embedding σ : K ,→ C, we shall denote its restriction to Vσ by νσ := ν|Vσ . (The family (νσ )σ:K,→C is invariant under complex conjugation; namely, for every field embedding σ : K ,→ C, we have jσ∗ νσ = νσ .

(10.6.3)

Conversely, every family (νσ )σ:K,→C satisfying the conditions (10.6.3) defines a volume form ν invariant under conjugation on VC .) 20 Namely,

Eσc.c.

the isomorphisms Jσ are isometries when Eσ|Vσ and Eσ|V (which coincides with σ as C ∞ vector bundle) are equipped with the C ∞ metrics k.kσ and k.kσ .

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b of global sections of the locally free b E) We may consider the OK -module Γ(V, b b coherent sheaf E on the formal scheme V over OK . By construction, it is endowed with the structure of a topological module over OK (equipped with the discrete topology). b as a pointed smooth formal curve over Spec OK , Indeed, by the definition of V we have b = Spf B, V where B is a Noetherian adic topological OK -algebra. Indeed, as recalled in 10.4.2 (ii), if we denote by b N := Γ(Spec OK , NP V) b there exists the invertible OK -module that defines the normal bundle of P in V, an isomorphism of topological algebras Y ∼ d ∨ N ∨⊗A n , B −→ Sym (10.6.4) OK N ' n∈N

d O N ∨ is equipped with the I-adic topology associated to the ideal where Sym K ∨ b may be written (canonically) as I := N · B. Moreover, the vector bundle E over V ∆ the coherent sheaf M over Spf B associated to a finitely generated projective Bb b E) module M equipped with the I-adic topology. The OK -module of sections Γ(V, may be identified with M , and as such, is a topological OK -module. Observe that M is a direct summand of the B-module B ⊕N for some N ∈ N. A fortiori, M is a direct summand of B ⊕N as a topological OK -module. The isomorphism (10.6.4) shows that B is an object of CTCOK (see (4.2)). Consequently, the topological OK -module B ⊕N and all its closed submodules also are objects of CTCOK (cf. Proposition 4.2.3). This establishes the following result: e E) ˜ is an object of CTCO . Lemma 10.6.1. The topological OK -module Γ(V, K



For every field embedding σ : K ,→ C, we may form the completed tensor product ˜ σ := Γ(V, e E) ˜⊗ e E) ˆ K,σ C. Γ(V, bσ , Ebσ ) of sections of the vector bundle Ebσ It may be identified with the space Γ(V bσ derived from Eb and V b by the over the smooth germ of complex formal curve V base change σ. Moreover, the isomorphisms ∼

bσ −→ Vbσ,O iσ : V σ

and

∼

ϕσ : Ebσ −→ iσ∗ Eσ |Vcσ

,Oσ

provide an identification cσ ,P , Eσ ) bσ , Ebσ ) ' Γ(V Γ(V σ

(10.6.5)

10.6. Smooth Formal-Analytic Surfaces over Spec OK

277

compatible with the canonical topology of a Fr´echet space on these two complex vector spaces (cf. 5.1.3 and 10.5.4 above). Finally, we may consider the Hilbert space of analytic sections ΓL2 (Vσ , νσ ; Eσ , k.kσ ) ˚σ of the Hermitian analytic vector bundle (Eσ , k.kσ ) on the Riemann surface V equipped with the volume form νσ , and the evaluation maps ηˆσ :

ΓL2 (Vσ , νσ ; Eσ , k.kσ ) −→ s 7−→

cσ ,P , Eσ ), Γ(V σ s|Vcσ . ,Pσ

According to Proposition 10.5.1, these maps are injective, and continuous with dense image. Moreover, the construction of the maps ηˆσ is clearly compatible ∼ with the complex conjugation isomorphisms jσ : Vσ −→ Vσc.c. . Together with Lemma 10.6.1, this proves the following: Proposition 10.6.2. The pair b := (Γ(V, ˜ ν; E) e E), ˜ (ΓL2 (Vσ , νσ ; Eσ , k.kσ ), ηˆσ )σ:K,→C ) ΓL2 (V, defines a pro-Hermitian vector bundle over Spec OK .



Since the Riemann surface with boundary Vσ is compact, every two volume forms (resp., every two Hermitian metrics over Eσ ) over Vσ are comparable. This implies that the Hilbertizable pro-vector bundle over Spec OK defined by b is independent of the choice of the volume form ν and the Hermitian ˜ ν; E) ΓL2 (V, metrics (k.kσ )σ:K,→C on the complex vector bundles (Eσ )σ:K,→C . We will denote it ˜ E). ˜ by ΓL2 (V; With the above notation, the ideal I is the largest ideal of definition of the adic algebra B, since B/I ' OK is reduced. The corresponding coherent Ideal I := I ∆ of OVb is its largest ideal of definition, and the corresponding reduced bred is the image of the closed immersion P. We shall often denote it subscheme V by P. b — namely the OK If Pi denotes the ith infinitesimal neighborhood of P in V i+1 b b as the inductive subscheme of V defined by the Ideal I — we may describe V limit (in the category of formal schemes) b = lim Pi V −→

(10.6.6)

i

of the inductive system of schemes finite and flat over Spec OK : P0 := P,−→P1 ,−→ · · · ,−→Pi ,−→Pi+1 ,−→ · · ·

.

d O N ∨ , the OK -schemes Pi become In fact, when we identify B with Sym K naturally isomorphic to d O )N ∨ /I i+1 = Spec(SymO N ∨ )/I i+1 , Spec(Sym K K

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

and we have an isomorphism of OK -modules (SymOK N ∨ )/I i+1 '

i M

N ∨⊗k .

k=0

b over Spec O admits a natu˜ ν; E) The pro-Hermitian vector bundle ΓL2 (V, K b• of (finite-rank) ral description as the projective limit of a projective system E b as the inductive limit Hermitian vector bundles that reflects the description of V (10.6.6). Indeed, for every i ∈ N, we have an exact sequence of OK -modules ηi b I i+1 ),−→Γ(V, b E) b −→ 0 −→ Ui := Γ(V, Γ(Vi , Eb|Vi ) −→ 0.

b E), b and the sequence (Ui )i∈N is nondecreasing and Each module Ui is open in Γ(V, b E). b Therefore, constitutes a fundamental system of neighborhoods of zero in Γ(V, we may apply the general construction of a pro-Hermitian vector bundle as a projective limit of Hermitian vector bundles discussed in paragraph 5.3.2. b := Γ 2 (V, b may be identified with the projective limit ˜ ν; E) We obtain that E L limi E i , where the projective system ←− q0

q1

qi−1

qi

qi+1

E • : E 0 ←− E 1 ←− · · · ←− E i ←− E i+1 ←− · · · is defined by the Hermitian vector bundles E i := E Ui over Spec OK and the surjective admissible quotient morphisms qi := E Ui+1 −→ E Ui . In concrete terms, we have canonical isomorphisms ∼ b i = Γ(V, b E)/Γ( b b I i+1 ) −→ Ei := E/U V, Γ(Vi , Eb|Vi ),

and for every embedding σ : K ,→ C, ∼ ∼ ∼ Ei,σ −→ Γ(Vi , Eb|Vi )σ −→ Γ(Vi,σ , Eb|Vi ,σ ) −→ Eσ|Pσi .

(We denote by Pσi the ith infinitesimal neigborhood of Pσ in Vσ . The last isomorphism is induced by the isomorphisms iσ and ϕσ .) Accordingly, E i may be identified with (Γ(Vi , Eb|Vi ), (k.ki,σ )σ:K,→C ), where k.ki,σ is the Hermitian norm on Γ(Vσ,Pσ , Eσ ) that makes the jet map ˚σ , νσ ; Eσ , k.kσ ) −→ Eσ|P ηi,σ : ΓL2 (V σi a co-isometry21 when ΓL2 (Vσ , νσ ; Eσ , k.kσ ) is equipped with its L2 -norm k.kL2 (Vσ ,νσ ;Eσ ,k.kσ ) . 21 In

other words, k.ki,σ is the quotient norm of k.kL2 (Vσ ,νσ ;Eσ ,k.kσ ) via the continuous surjective map ηi,σ .

10.6. Smooth Formal-Analytic Surfaces over Spec OK

279

10.6.3 Morphisms from a smooth formal-analytic surface to an OK -scheme. Let X be a separated scheme of finite type over Spec OK . b (Vσ , Pσ , iσ )σ:K,→C ) as above, we define an OK -morphism For every V˜ := (V, from V˜ to X f : V˜ −→ X as a pair f := (fˆ, (fσ )σ:K,→C ), where b −→ X fˆ : V is a morphism of formal schemes over OK , and where for every embedding σ : K ,→ C, fσ : Vσ+ −→ Xσ (C) is a C-analytic morphism, or equivalently, a map of C-locally ringed spaces from an ) to the C-scheme Xσ . (Vσ , O|V σ These maps are, moreover, assumed to satisfy the following compatibility conditions: for every embedding σ : K ,→ C, the diagram of ringed spaces fˆσ

bσ V   iσ y'

−−−−→ Xσ   y=

(10.6.7)

κσ fσ cσ ,P ,−→ V Vσ −−−−→ Xσ σ

is commutative22 . This implies that the family (fσ )σ:K,→C is compatible with complex conjugation. Observe that for every vector bundle E over X , we may define the inverse e image of E by f as the following vector bundle over V: f ∗ E := (fˆ∗ E, (fσ∗ EXσ , ϕσ )σ:K,→C ), where the isomorphism ϕσ is derived from the base change isomorphism23 ∗

∼ fˆ∗ E ⊗OK ,σ C −→ fc σ Eσ

and the equality of morphisms cσ ,P −→ Xσ . fˆσ = fσ ◦ κσ ◦ iσ : V σ Namely, it is defined as the composition ∗

∗ ∗ ∗ ∗ ∗ ϕσ : (fˆ∗ E)σ = fˆ∗ E ⊗OK ,σ C ' fc σ Eσ ' iσ κσ fσ Eσ = iσ (fσ Eσ )|V c σ

,Pσ

.

22 In this diagram, κ denotes the “inclusion” morphism from V c σ σ ,Pσ to Vσ , defined by the an canonical morphism from the local C-algebra OV to its completion. ,P σ σ 23 We denote by fˆ : V bσ −→ Xσ the morphism derived from fˆ : V b −→ X by the base change σ σ : OK ,−→C.

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

Observe also that if XK is reduced and E := (E, k.k) is a Hermitian vector bundle over X , defined by a vector bundle E on X and a C ∞ metric k.k on ` ECan over X (C) = σ:K,→C Xσ (C) (invariant under complex conjugation), then for every embedding σ : K ,→ C, we may equip the analytic vector bundle fσ∗ Eσ over Vσ with the pullback of the Hermitian metric k.k on Xσ (C) by fσ . The vector bundle f ∗ E, endowed with these Hermitian metrics, defines a Hermitian vector e the inverse image of E by f . bundle f ∗ E over V, 10.6.4 Schematic image and algebraicity. In paragraph 10.2.4, we considered b −→ X from a formal the Zariski closure im f of the image of a morphism f : V b surface V over a field k to a quasi-projective k-scheme X. The dimension bound dim im f ≤ 2 provided a formal definition of the algebraicity of the image of f . It is possible to formulate a similar algebraicity property concerning a morphism e −→ X f :V e to an OK -scheme as above. from a formal-analytic surface V Indeed, with the notation of paragraph 10.6.3, to each of the morphisms b −→ X , fˆK : V bK −→ XK , fˆσ : V bσ −→ Xσ , and of locally ringed spaces fˆ : V fσ : Vσ −→ Xσ , we may attach the Zariski closure of its image im fˆ, im fˆK , im fˆσ , and im fσ , namely the smallest closed subscheme of its range such that the morphism factorizes through this closed subscheme. The formation of the Zariski closure of the image of these morphisms is compatible with the replacement of their range scheme by an open subscheme through which the morphism factorizes (as in 10.2.4, property (i) on page 249 and (10.2.14)). Moreover, each of these Zariski closures is an integral scheme. For instance, we have the following lemma: Lemma 10.6.3. If U is an open subscheme of X such that fˆ factorizes through the inclusion U ,→ X — in other words, if the image of the continuous map of b = P to |X | defined by fˆ is contained in |U | — and if topological spaces from |V| U b to U , then we have fˆ denotes fˆ seen as a morphism from V im fˆU = im fˆ ∩ U.



In fact, im fˆ contains the OK -point fˆ ◦ P of X and im fˆσ the associated K-rational point of XK . This implies that the integral scheme im fˆ is flat over Spec OK and that im fˆK is geometrically integral over K. Proposition 10.6.4. With the above notation, we have   im fˆK = im fˆ , K

(10.6.8)

10.6. Smooth Formal-Analytic Surfaces over Spec OK

281

  im fˆσ = im fˆK ,

(10.6.9)

im fσ = im fˆσ .

(10.6.10)

σ

and Proof. By the compatibility of the formation of the Zariski closure of the image of a morphism with the “shrinking” of the range scheme alluded to above, we may assume that X is a closed subscheme of PN OK . Then the graded ideal in M OK [X0 , . . . , XN ] ' Γ(PN OK , O(D)) D∈N

that defines im fˆ is the direct sum D∈N ker ηˆD , where ηˆD denotes the evaluation map b ˆ∗ ηˆD : Γ(PN OK , O(D)) −→ Γ(V, f O(D)), s 7−→ fˆ∗ s. L Similarly, im fˆK is defined by the graded ideal D∈N ker ηˆD,K in K[X0 , . . . , XN ] defined by the evaluation maps L

ηˆD,K :

Γ(PN K , O(D)) −→ s 7−→

bK , fˆ∗ O(D)), Γ(V ∗ fˆK s.

b is Since ηˆD,K is derived from ηˆD by the base change Spec K ,→ Spec OK and V flat over Spec OK , we have ker ηˆD = ηˆD,K ∩ Γ(PN OK , O(D))

for every D ∈ N.

This establishes (10.6.8). The Zariski closure im fˆσ is defined by the graded ideal C[X0 , . . . , XN ] defined by the evaluation maps ηˆD,σ :

Γ(PN C , O(D)) −→ s 7−→

L

D∈N

ker ηˆD,σ in

bσ , fˆσ∗ O(D)), Γ(V fˆσ∗ s.

The morphism ηˆD,σ may be identified with the morphism induced from ηˆD,K by the field extension σ : K,−→C, and (10.6.9) follows. L an Finally, im fσ is defined by the graded ideal D∈N ker ηˆD,σ in C[X0 , . . . , XN ] defined by the evaluation maps an ηˆD,σ :

Γ(PN C , O(D)) −→ s 7−→

Γ(Vσ , fσ∗ O(D)), fσ∗ s.

Since the Riemann surface Vσ is connected, by analytic continuation, we have an ker ηˆD,σ = ker ηˆD,σ

This establishes (10.6.10).

for every D ∈ N. 

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Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

Corollary 10.6.5. The following four conditions are equivalent: dim im fˆ ≤ 2,

(10.6.11)

dim im fˆK ≤ 1,

(10.6.12)

dim im fˆσ ≤ 1,

(10.6.13)

dim im fσ ≤ 1.

(10.6.14)

and  When conditions (10.6.11)–(10.6.14) are satisfied, we shall say that the image of f is algebraic. b −→ X is constant (that Indeed, when this holds, either the morphism fˆ : V b is, factorizes through the structural morphism π : V −→ Spec OK ) and the morphisms fσ also for every embedding σ : K ,→ C, or equality holds in the relations (10.6.11)–(10.6.14). When the latter holds, we may consider the normalization ν : C −→ im fˆK of the projective curve im fˆK . Then, by the compatibility of normalization and completion, the morphism fˆK factorizes as fˆK = ν ◦ gˆ for a uniquely determined bK −→ C. The image Q := gˆ(PK ) of the K-point PK = (V bK )red K-morphism gˆ : V defines a K-rational point of the smooth integral curve C, and gˆ defines a finite bK −→ C bQ . morphism of smooth pointed curves over K: gˆ : V ˆ In brief, fK may be written as the composition of the (nonconstant) morphism of smooth germs of formal curves over K, bK −→ C bQ , gˆ : V and the finite morphism of K-schemes ν : C −→ XK . On may actually show that for every embedding σ : K ,→ C, the morphism of complex formal germs bσ −→ C bQ,σ gˆσ : V “extends” to an analytic map gσ : Vσ+ −→ Xσ (C) and that fσ = νσ ◦ gσ . We leave this to the interested reader.

10.7. Arithmetic Pseudo-concavity and Finiteness

10.7

283

Arithmetic Pseudo-concavity and Finiteness

e := (V, b (Vσ , Pσ , iσ )σ:K,→C ) be a smooth formal-analytic surface 10.7.1 Let V b and over Spec OK . As before, we shall denote by π the structural morphism of V, by P the canonical section of π. b of P in the smooth formal surface V b is a line bundle The normal bundle NP V b σ may over Spec OK . For every field embedding σ : K ,→ C, the complex line (NP V) be identified with the tangent line TOσ at the point Oσ of the Riemann surface Vσ , and may therefore be equipped with the capacitary metric k.kcap Vσ ,Pσ . This construction defines a Hermitian line bundle over Spec OK : b (k.kcap )σ:K,→C ). N P V˜ := (NP V, Vσ ,Pσ Theorem 10.7.1. Let us keep the above notation, and let us assume that d N P V˜ > 0. deg

(10.7.1)

e the Hilbertizable pro-vector bundle Then for every vector bundle E˜ over V, ˜ ˜ ΓL2 (V; E) is θ-finite. b over V e and every conjugationMoreover, for every Hermitian line bundle L invariant volume form on VC , as the integer D goes to infinity, we have b ⊗D )) = O(D2 ). ˜ ν; L h0θ (ΓL2 (V,

(10.7.2)

Condition (10.7.1) is an arithmetic avatar of the pseudo-concavity condition (10.2.8) considered in the geometric framework of smooth formal surfaces over a field in paragraph 10.2.4, and Theorem 10.7.1 is an arithmetic counterpart of Proposition 10.2.7. The basic properties of the θ-invariant h0θ developed in Chapters 6 and 7, together with the Schwarz lemma on compact Riemann surfaces with boundary established in paragraph 10.5.5, will allow us to give a proof of this theorem formally similar to the proof of Proposition 10.2.7. 10.7.2 We shall also rely on some elementary properties of the θ-invariants of (finite-rank) Hermitian vector bundles, which we now recall in a form suitable for the proof of Theorem 10.7.1. From Proposition 2.7.3, it follows that the θ-invariant h0θ (L) of a Hermitian line bundle L over Spec OK admits the following upper bound in terms of its Arakelov degree: d L), (10.7.3) h0θ (L) ≤ χ( deg where ( χ(x) :=

1+x exp x

if x ≥ 0, if x ≤ 0.

284

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

d L < 0. A technical In fact, h0θ (L) satisfies much stronger estimates when deg advantage of the above choice for χ is that it is a convex function. As a consequence, for all x ∈ R and a ∈ R∗+ , we have χ(x) ≤

1 a

Z

x+a/2

χ(t) dt.

(10.7.4)

x−a/2

Let E be a Hermitian vector bundle over Spec OK , of rank N , and let ξ := ∨ (ξ1 , . . . , ξN ) be an N -tuple of elements of E ∨ that constitutes a K-basis of EK . The morphism ξ : E −→ OK ⊕N ⊕N

is injective and defines an element of Hom≤λ OK (E, O Spec OK ), provided λ ≥ M := max

σ:K,→C

n X

!1/2 kξi k2E ∨ ,σ

.

i=1

Therefore, for every Hermitian line bundle L over Spec OK , the tensor product ⊕N IdL ⊗ ξ defines an injective morphism in Hom≤1 ). OK (L ⊗ E, L ⊗ O Spec OK (log M ) 0 Together with the monotonicity and the additivity of hθ and the estimate (10.7.3), this implies the following upper bound: d L + log M ). (10.7.5) h0θ (L ⊗ E) ≤ rk E · h0θ (L ⊗ OSpec OK (log M )) ≤ rk E · χ( deg 10.7.3 Proof of Theorem 10.7.1-I. Let us consider a Hermitian vector bundle e over V, b (Eσ , ϕσ , k.kσ )σ:K,→C ), E˜ := (E, and let ν be a volume form on VC invariant under complex conjugation. As explained at the end of Section 10.6.2, we may describe the pro-Hermitian vector bundle b := Γ 2 (V, b ˜ ν; E) E L as the projective limit limi E i of the system of admissible surjective morphisms ←− q0

q1

qi−1

qi

qi+1

E • : E 0 ←− E 1 ←− · · · ←− E i ←− E i+1 ←− · · · where E i := (Γ(Pi , Eb|Pi ), (k.ki,σ )σ:K,→C ), and where the Hermitian norm k.ki,σ is defined as the quotient norm induced from the L2 -norm on ΓL2 (Vσ , νσ ; Eσ , k.kσ ) by the ith jet map ηi,σ : ΓL2 (Vσ , νσ ; Eσ , k.kσ ) −→ Eσ|Pσi .

10.7. Arithmetic Pseudo-concavity and Finiteness

285

The morphisms qi are the obvious restriction morphisms, induced by the inclusion Pi ,−→Pi+1 . As in Section 7.1, for every i ∈ N, we may consider the Hermitian vector bundle over Spec OK ker qi := (ker qi , (k.ki,σ )σ:K,→C ). Recall that if I denotes the ideal of P in OVb , there exist canonical isomorphisms of OVb -modules ∼

OPi −→ OVb /I i+1 and

for every i ∈ N

∼ b ∨⊗i . I i /I i+1 −→ (NP V)

(10.7.6)

(10.7.7)

(Indeed, (10.7.6) holds by the definition of Pi ; (10.7.7) holds because of the smoothb which shows that the quotient I i /I i+1 is supported by P.) ness assumption on V, From these isomorphisms, we deduce the short exact sequences of OVb -modules 0 −→ I i /I i+1 −→ OPi −→ OPi−1 −→ 0, and after taking tensor products with Eb and spaces of global sections, the short exact sequences of OK -modules qi−1 b ∨⊗i ⊗ P ∗ E),−→Γ(P b b b 0 −→ Γ(Spec OK , (NP V) i , E|Pi ) −→ Γ(Pi , E|Pi−1 ) −→ 0.

This shows that for every i ≥ 1, the OK -module ker qi−1 may be identified with b ∨ )⊗i ⊗ P ∗ E. b (the global sections of) (NP V Observe that for every σ : K ,→ C, the complex vector space (P ∗ E)σ ' Eσ,Pσ is endowed with the Hermitian norm k.kσ,Pσ , the restriction over Pσ of the Hermitian metric k.kσ over Eσ . We shall denote by P ∗ E the Hermitian vector bundle (P ∗ E, (k.kσ,Pσ )σ:K,→C ) over Spec OK . b ∨⊗i ⊗ P ∗ Eb appears as the underlying OK -module of the HermiThen (NP V) b ∨⊗i ⊗ P ∗ E. We shall denote by k.kcap the corresponding tian vector bundle (N P V) i,σ metric on b ∨⊗i ⊗ P ∗ E] b σ ' (TP Vσ )∨⊗i ⊗ Eσ,P . [(NP V) σ σ By construction, it is the metric derived from the metrics k.kcap Vσ ,Pσ on TPσ Vσ and k.kσ on Eσ by duality and tensor product. Under the identification b ∨ )⊗i ⊗ P ∗ E, b ker qi−1 ' (NP V

286

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

b ∨⊗i ⊗ P ∗ E do not coincide in the Hermitian structures on ker qi−1 and on (N P V) general. Indeed, the Hermitian norms k.ki,σ on ker qi−1 are defined as quotients of b ∨⊗i ⊗ P ∗ E are defined in L2 -norms, while the Hermitian norms k.kcap i,σ on (N P V) terms of capacitary metrics. However, in paragraph 10.5.5, we have established a version of Schwarz’s lemma on compact Riemann surfaces with boundary that allows one to compare these norms. Indeed, as a reformulation of the estimates (10.5.23) in Scholium 10.5.5 applied to the compact connected Riemann surfaces with boundary (Vσ )σ:K,→C , we obtain the following comparison estimates: Lemma 10.7.2. For all η > 0, there exists Cη ∈ R+ such that for every field embedding σ : K ,→ C and all i ∈ N, ηi k.kcap i,σ ≤ Cη e k.ki,σ .

(10.7.8) 

From the estimates (10.7.8) and the monotonicity of (1)), we derive that for every integer i ≥ 1,

h0θ

(Proposition 2.3.2,

b ∨⊗i ⊗ P ∗ E ⊗ O(η i + log Cη )). h0θ (ker qi−1 ) ≤ h0θ ((N P V)

(10.7.9)

Moreover, the upper bound (10.7.5) on the θ-invariant of the tensor product of a fixed Hermitian vector bundle and a Hermitian line bundle shows the existence of M > 0, depending only on P ∗ E, such that b ∨⊗i ⊗ P ∗ E ⊗ O(ηi + log Cη )) h0θ ((N P V) d NPV b + [K : Q](η i + log Cη + log M )). ≤ rk E.χ(−i deg When the arithmetic ampleness condition (10.7.1) is satisfied, we may choose d N P V[. b Then η in the interval ]0, [K : Q]−1 deg d NPV b − [K : Q]η a := deg is positive, and if we let b := [K : Q](log Cη + log M ), the previous estimates imply that for every integer i ≥ 1, h0θ (ker qi−1 ) ≤ rk E χ(−ai + b). Finally, we obtain +∞ X n=0

h0θ (ker qn ) ≤ rk E

∞ X i=1

χ(−ai + b) < +∞.

10.7. Arithmetic Pseudo-concavity and Finiteness

287

This shows that the projective system E • is summable. More generally, the estimates (10.7.8) show that for all δ ∈ R, b ∨⊗i ⊗ P ∗ E ⊗ O(η i + log Cη + δ)). h0θ (ker qi−1 ⊗ O(δ)) ≤ h0θ ((N P V)

(10.7.10)

Replacing (10.7.9) by this “twisted version”, we now obtain that h0θ (ker qi−1 ⊗ O(δ)) ≤ rk E χ(−ai + b + [K : Q]δ), and consequently, +∞ X

h0θ (ker qn ⊗ O(δ)) < +∞.

n=0

This establishes that E • ⊗ O(δ) is summable for every δ ∈ R, and finally that b ' lim E is θ-summable. ˜ ν; E) ΓL2 (V, ←−i i 10.7.4 Proof of Theorem 10.7.1-II. Let us now consider a Hermitian line bundle ˜ over V. ˜ ⊗D . In e For all D ∈ N, we may apply the previous construction to E˜ := L L this way, we get the upper bound +∞

b ≤ h0 (E ) + X h0 (ker q ) ˜ ν; E)) h0θ (ΓL2 (V, 0 n θ θ n=0

=

+∞ X

b ∨ )⊗i ⊗ P ∗ Lb⊗D , (k.kD,i,σ )σ:K,→C ), (10.7.11) h0θ ((NP V

i=0

where k.kD,i,σ denotes the metric on b ∨ )⊗i ⊗ P ∗ Lb⊗D ]σ ' (TP Vσ )∨⊗i ⊗ L⊗D [(NP V σ σ,Pσ derived (by “subquotient” as above) from the L2 -norm on ΓL2 (Vσ , νσ ; L⊗D σ , k.kσ ). Using the estimates (10.5.24) in Scholium 10.5.5, we now obtain the following variant of Lemma 10.7.2: Lemma 10.7.3. For all η > 0, there exists Cη ∈ R+ such that for every field embedding σ : K ,→ C and (D, i) ∈ N2 , D+1 ηi k.kcap e k.ki,σ . i,σ ≤ Cη

(10.7.12) 

The estimates (10.7.12) and the monotonicity of h0θ now imply that for all i ∈ N, b ∨ )⊗i ⊗ P ∗ Lb⊗D , (k.kD,i,σ )σ:K,→C ) h0θ ((NP V b ∨⊗i ⊗ P ∗ L⊗D ⊗ O(η i + (D + 1) log Cη )). (10.7.13) ≤ h0θ ((N P V)

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

288

Moreover, the upper bound (10.7.3) on the θ-invariant of a Hermitian line bundle in terms of its Arakelov degree shows that b ∨⊗i ⊗ P ∗ L⊗D ⊗ O(η i + (D + 1) log Cη )) h0θ ((N P V) d P ∗ L + [K : Q](η i + (D + 1) log Cη )). (10.7.14) d NPV b + D deg ≤ χ(−i deg When the arithmetic ampleness condition (10.7.1) is satisfied and η belongs d N P V[, b we obtain from (10.7.13) and (10.7.14) to the interval ]0, [K : Q]−1 deg b ∨ )⊗i ⊗ P ∗ Lb⊗D , (k.kD,i,σ )σ:K,→C ) ≤ χ(−ai + a0 D + b), h0θ ((NP V

(10.7.15)

where d NPV b − [K : Q]η > 0, a := deg d P ∗ L + [K : Q] log Cη , a0 := deg and b := [K : Q] log Cη . The asymptotic bound (10.7.2), b ⊗D )) = O(D2 ), ˜ ν; L h0θ (ΓL2 (V, now follows from (10.7.11) and (10.7.15), combined with the following elementary lemma: P Lemma 10.7.4. For all (a, a0 , b) ∈ R∗+ × R2 and D ∈ N, the series n∈N χ(−an + a0 D + b) is convergent. Moreover, as D goes to +∞, X

χ(−an + a0 D + b) = O(D2 ).

n∈N

Proof. According to the convexity estimate (10.7.4), we have X n∈N

1 χ(−an + a D + b) ≤ a 0

Z

a/2+a0 D+b

χ(t) dt.

(10.7.16)

−∞

This establishes the convergence of the series. Moreover, when a0 ≤ 0, the righthand side of (10.7.16) remains bounded as D goes to +∞. When a0 > 0, it is bounded from above by 1 1 + a a

Z

a/2+a0 D+b

(1 + t) dt = 0

a02 2 D + O(D). 2a



10.8. Arithmetic Pseudo-concavity and Algebraization

10.8

289

Arithmetic Pseudo-concavity and Algebraization

10.8.1 A Diophantine algebraization theorem. We may now establish the Diophantine algebraization theorem that constitutes the main result of this chapter. It may be seen as a Diophantine counterpart of the algebraization theorem concerning “pseudo-concave” formal surfaces discussed in paragraph 10.2.4 (cf. Theorem 10.2.8). Indeed, as explained in paragraph 10.1.3, its proof will be similar to that of Theorem 10.2.8: it will combine the upper bound for the θ-invariants of proHermitian vector bundles of sections of line bundles over a formal-analytic surface (Theorem 10.7.1) and the lower bound for the θ-invariants of Hermitian vector bundles of arithmetically ample Hermitian line bundles on projective schemes over Spec OK (Theorem 10.3.2 and Corollary 10.3.3). e := (V, b (Vσ , Oσ , iσ )σ:K,→C ) be a smooth formal-analytic surTheorem 10.8.1. Let V face over Spec OK as above. When the “arithmetic pseudo-concavity” condition (10.7.1) d N P V˜ > 0 deg is satisfied, for every OK -morphism e −→ X f := (fˆ, (fσ )σ:K,→C ) : V with range a quasi-projective OK -scheme X , the image of f is algebraic. Recall that the meaning of the algebraicity of the image of f was discussed in paragraph 10.6.4. Proof of Theorem 10.8.1. We have to show that the closed integral subscheme im fˆ satisfies dim im fˆ ≤ 2. (10.8.1) With no restriction of generality, we may assume that X is projective (this follows from Lemma 10.6.3), and moreover, by replacing X by im f , that X = im fˆ.

(10.8.2)

In particular, X is an integral projective scheme over Spec OK ; it is clearly flat over Spec OK (indeed, it contains the OK -point fˆ(P)). Let d := dim X = dim im fˆ. Let us apply Corollary 10.3.3 to X . We shall keep the notation of this corollary, and denote by L and by (E D , iD )D∈N a sequence of Hermitian vector bundles over Spec OK and injections iD : ED ,−→π∗ L⊗D

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satisfying the conclusions (10.3.8) and (10.3.9) of Corollary 10.3.3. In particular, there exists c > 0 such that, for every large enough integer D, hθ0 (E D ) ≥ c · Ddim X .

(10.8.3)

Furthermore, let us choose a family ν := (νσ )σ:K,→C of positive C ∞ volume forms on the Riemann surfaces Vσ that is invariant under complex conjugation and satisfies the normalization conditions Z νσ = 1 for every embedding σ : K ,→ C. (10.8.4) Vσ

For any D ∈ N, we may consider the pro-Hermitian vector bundle over Spec OK e ν; f ∗ L⊗D ) := (Γ(V, b fˆ∗ L⊗D ), (ΓL2 (Vσ , νσ ; fσ∗ L⊗D ΓL2 (V, σ )σ:K,→C )). e implies According to Theorem 10.7.1, the “arithmetic pseudo-concavity” of V e ν; f ∗ L⊗D ) is θ-finite and that as D that the pro-Hermitian vector bundle ΓL2 (V, goes to infinity, e ν; f ∗ L⊗D )) = O(D2 ). h0θ (ΓL2 (V, (10.8.5) Moreover, we may consider the pullback maps ϕˆD : Γ(X , L⊗D ) −→ s 7−→

b fˆ∗ L⊗D ), Γ(V, fˆ∗ s.

For all D ∈ N and s ∈ Γ(X , L⊗D )\{0}, the morphism fˆ does not factorize through the inclusion div s ,→ X (by (10.8.2)). This shows that the maps ϕˆD are injective. We may also consider the maps ϕD,σ : Γ(Xσ , L⊗D σ ) −→ s 7−→

ΓL2 (Vσ , fσ∗ L⊗D σ ), fσ∗ s.

Clearly, they satisfy kϕD,σ skL2 (Vσ ,νσ ) ≤ kϕD,σ skL∞ (Vσ ) ≤ kskL∞ (Xσ ) .

(10.8.6)

(The first inequality holds because of the normalization conditions (10.8.4).) Moreover, the maps ϕˆD and ϕD,σ are compatible — the base change b fˆ∗ L⊗D )σ ϕˆD,σ : Γ(X , L⊗D )σ −→ Γ(V, may be identified with the composition of ϕD,σ and the restriction map ∗ ⊗D ΓL2 (Vσ , fσ∗ L⊗D c σ ) −→ (fσ Lσ )|V σ

,Pσ

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291

— and therefore define a morphism of Hilbertizable vector bundles e f ∗ L⊗D ). ϕD : π∗ L⊗D −→ ΓL2 (V, The norm estimates (10.8.6) and (10.3.8) show that the composite morphism e ν; f ∗ L⊗D ) ϕD ◦ iD : E D −→ ΓL2 (V, ∗ ⊗D e )). Moreover, like ϕˆD , the composite map belongs to Hom≤1 OK (E D , ΓL2 (V, ν; f L ϕˆD ◦ iD is injective. Consequently, we obtain

e ν; f ∗ L⊗D )) h0θ (E D ) ≤ h0θ (ΓL2 (V,

(10.8.7)

(cf. Proposition 6.3.5). From (10.8.3), (10.8.5), and (10.8.7), we derive the desired inequality dim X ≤ 2 by letting D go to infinity.



10.8.2 Proof of Theorem 10.1.2. Let us now explain how, from Theorem 10.8.1, one simply derives the “naive” algebraicity criterion stated at the beginning of this chapter as Theorem 10.1.2. Let us go back to the notation of this theorem, introduced in paragraph 10.1.2, and let us choose an imaginary quadratic field K. We shall denote by {σ0 , σ 0 } the field embeddings of K in C. From the data of ϕ ∈ Z[[X]]N , the pointed Riemann surface with boundary V , and the map j : V −→ PN (C), we may construct a smooth formal-analytic e over Spec OK by letting surface V b := Spf OK [[X]], V Vσ0 := V

and Vσ0 := V c.c. ,

and iσ0 := γc O iσ0 := γc O

−1 −1

∼ : Vbσ0 ' Spf C[[T ]] −→ VbO ∼

: Vbσ0 ' Spf C[[T ]] −→

Then one defines a morphism e −→ PN f := (fˆ, (fσ0 , fσ0 )) : V OK by fˆ := ϕ,

fσ0 := γ,

and

VbOc.c. .

and fσ0 := γ.

292

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization By the definition of the Arakelov degree, we have d N P V˜ = −2 log kDγ(O)−1 ϕ0 (0)kcap . deg V,O

This is positive precisely when the condition (10.1.5) in Theorem 10.1.2 is satisfied. According to Theorem 10.8.1, this condition implies the algebraicity of the image of f . In particular, Y := im fˆK is a closed integral scheme of dimension 1 in PN K such that CˆK ⊂ YˆP and γ(V ) ⊂ Yσ0 (C). (10.8.8) The fact that the series ϕ1 , . . . , ϕN that define fˆ belong to Q[[X]] implies, by a straightforward descent argument, that Y may be written XK for some closed integral curve X in PN Q . From (10.8.8), it follows that the curve X satisfies the conclusion (10.1.6) of Theorem 10.1.2, namely bP CˆQ ⊂ X

and γ(V ) ⊂ X(C).

10.8.3 Borel’s rationality criterion. As a first application of Theorem 10.8.1, let us explain how one may derive Borel’s rationality criterion (Theorem 10.1.1 above) from its naive variant Theorem 10.1.2. We shall actually establish Borel’s criterion in the following more general form: Theorem 10.8.2. Let f be a formal series in Z[[X]]. If the complex radius of convergence of f is positive, and if f extends, as a ˚ R) \ F for some R ∈]1, +∞[ and a finite subset F of C-analytic function, to D(0, ˚ D(0, R) \ {0}, then f is the Taylor expansion at 0 of a rational function in Q(X). Theorem 10.1.1 is the special case of Theorem 10.8.2 in which the singularities F of f are assumed to be (at worst) poles. Theorem 10.8.2 is actually a consequence of the main result of P´olya [90] (see, in particular, Section 11). Let us begin with a simple observation. Lemma 10.8.3. Theorem 10.8.2 holds when f satisfies the additional assumptions that f is meromorphic at each point of F and that F is contained in the algebraic closure Q of Q in C. Proof. When this additional assumption holds, we may find a nonzero polynomial P ∈ Z[X] that admits every point of F as a zero. Then, if N denotes the maximum order of P the poles of f in the closed unit disk D(0; 1), the product P N f is a formal series n∈N an X n in Z[[X]] that is the ˚ R0 ) of radius R0 > 1. expansion of a holomorphic function on an open disk D(0, P This implies that n∈N |an | < +∞, and therefore, since the an are integers, that the an vanish for n large enough. This shows that P N f is a polynomial Q in Z[X], and finally that f = Q/P N belongs to Q(X). 

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293

Let us now consider f as in the statement of Theorem 10.8.2, and let us ˚ R0 ). choose R0 ∈]1, R[ such that F ⊂ D(0, According to 10.5.3, Example (iii), for all ε > 0 small enough, [ ˚ ε) Vε := D(0, R0 ) \ D(a; a∈F

is a compact Riemann surface with boundary, containing 0 in its interior, such that k∂/∂zkcap (10.8.9) Vε ,0 < 1. By construction, f defines an analytic function — which we shall denote by f an — on an open neighborhood of Vε in C. We may apply Theorem 10.1.2 to the following data: V := Vε

and O := 0,

ϕ := (X, f ) ∈ Z[[X]]2 , and γ : V −→ C2

defined by γ(z) := (z, f an (z)).

Condition (10.1.4), ∼

γˆO : Vb0 −→ CˆC , is clearly satisfied. Moreover, ϕ0 (0) = ∂/∂x1 + f 0 (0)∂/∂x2 = Dγ(0)∂/∂t. Therefore, Dγ(0)−1 ϕ0 (0) = ∂/∂t, and according to (10.8.9), condition (10.1.5), kDγ(O)−1 ϕ0 (0)kcap V,O < 1 is satisfied. The conclusion of Theorem 10.1.2 asserts that the image of γ is algebraic, and is actually contained in (the complex points of) an algebraic curve defined over Q. It may be rephrased as the existence of a closed integral curve C in A1Q × P1Q such that γ(V ) ⊂ C(C). By analytic continuation, we have ˚ R) \ F ) ⊂ C(C). γ(D(0, ˚ R) \ F −→ C is contained in the affine In other words, the graph of f an : D(0, 2 complex curve C(C) ∩ A (C).

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This implies that the singularities of f at the points of F are (at worst) poles, and that if a point of F is actually a pole of f , it belongs to Q. Indeed, the projection pr1 : CC −→ A1C is a finite morphism, and by Rie˚ R) \ {0} extends analytically to mann’s theorem, its analytic section γ over D(0, an ˚ ˚ R). Moreover, the poles of D(0, R). This proves that f is meromorphic on D(0, this function are contained in pr1 (C(C) ∩ (C × {∞})), and therefore belong to Q, since C is defined over Q. Finally, we can conclude the proof of Theorem 10.8.2 by applying Lemma 10.8.3.

10.9

The Isogeny Theorem for Elliptic Curves over Q

In this final section, we want to demonstrate that the arithmetic algebraization criterion established in Theorem 10.8.1 admits significant Diophantine applications, in spite of its unsophisticated formulation when compared to the more general and flexible criteria established in [2, Chapter VIII] or in [18]. Namely, we use Theorem 10.8.1 to derive to derive a classical isogeny criterion, due to Serre and Faltings, concerning elliptic curves over Q. The relevance of algebraicity theorems such as Theorems 10.1.2 and 10.8.1 for the construction of isogenies between elliptic curves is one of the striking discoveries presented in the seminal article [34] by D.V. and G.V. Chudnovsky. Like the one in that work, the construction of isogenies presented in this section will rely on a famous result of Honda [66] concerning formal groups of elliptic curves over Q and their models over Z. (A stronger form of the algebraicity criterion, such as those referred to above, would allow one to rely on less precise results concerning these formal groups than Honda’s; see, for instance, [15, Corollary 2.5]. We shall actually not use the full strength of Honda results, and avoid any reference to the theory of minimal and N´eron models of elliptic curves.) In this section, we assume some basic knowledge of the geometry and arithmetic of elliptic curves, say at the level of Robert’s lecture notes [95] or Silverman’s textbook [105]. 10.9.1 The isogeny theorem. Let E be an elliptic curve over Q. If N denotes a “sufficiently divisible” positive integer, then there exists an elliptic curve π : E −→ Spec Z[1/N ] over Z[1/N ] that is a model of E — that is, an elliptic curve whose generic fiber is isomorphic to EQ .24 Moreover, any two such models π : E −→ Spec Z[1/N ] and π 0 : E 0 −→ Spec Z[1/N 0 ] 24 The structure of E as an elliptic curve over Z[1/N ] is determined by its Z[1/N ]-scheme ∼ structure and its zero section ε. Formally, a specific isomorphism ι : EQ −→ E is part of the data that define a model of E over Spec Z[1/N ], and ι maps εQ to the zero element 0E in the abelian group E(Q).

10.9. The Isogeny Theorem for Elliptic Curves over Q

295

˜ ] (where N ˜ denotes of E become isomorphic over an open subscheme Spec Z[1/N ˜ := lcm(N, N 0 )). a common positive multiple of N and N 0 ; one may take N For every prime number p such that (p, N ) = 1, we may consider the elliptic curve EFp over the prime field Fp and its ap invariant ap (EFp ) := p + 1 − |EFp (Fp )| = p + 1 − |E(Fp )|. The integer ap (EFp ) is defined and independently of the chosen model E of E, for every large enough prime number p, and is denoted by ap (E). Our aim in this section is to prove the following theorem on elliptic curves over Q, due to Serre under an additional hypothesis of multiplicative bad reduction [101, IV.2.3] and to Faltings in general [45, §5, Corollary 2]; Faltings actually establishes a considerably more general result concerning abelian varieties over arbitrary number fields). Theorem 10.9.1. Let E and E 0 be two elliptic curves over Q. If ap (E) = ap (E 0 ) for every large enough prime p, then E and E 0 are isogenous over Q. The converse implication is classically known to hold: it follows from the fact that two isogenous elliptic curves over Fp have the same ap invariant. 10.9.2

Honda’s theorem.

Let E be an elliptic curve over Q and let π : E −→ Spec Z[1/N ]

be a model of E as above. We may consider the formal completion Eb of E along its zero section ε. It is a pointed smooth formal curve over Spec Z[1/N ]. Moreover, the group scheme structure of E induces on Eb the structure of a formal group scheme over Spec Z[1/N ]. We shall use Honda’s results on the formal groups associated to elliptic curves over Q and their models in the following form: Theorem 10.9.2 ([66, Section 4, Theorem 5 and Corollary 2]). Let E1 and E2 be two elliptic curves over Q, and let E1 and E2 be elliptic curves over Spec Z[1/N0 ] (for a positive integer N0 ) that are models of E1 and E2 respectively. If ap (E1 ) = ap (E2 ) for every large enough prime p, then there exists a multiple N of N0 such that Eb1,Z[1/N ] and Eb2,Z[1/N ] are isomorphic as formal groups over Spec Z[1/N ]. In fact, Honda establishes a more precise result, concerning normalized isomorphisms of the formal groups of the N´eron models of E1 and E2 . His proof is a beautiful application of basic results concerning minimal models of elliptic curves

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and some classical theorems of Lazard, Lubin, and Tate concerning one-parameter formal group laws (see [76], [79], and [80])25 . Let us also indicate that a motivation behind Honda’s results recalled in Theorem 10.9.2 — which in fact relate the L-functions and the formal groups attached to elliptic curves over Q — was the conjecture of Shimura–Taniyama–Weil on the modularity of elliptic curves over Q (see also [67]), and that this circle of ideas had also been explored by Cartier [27]. 10.9.3 Formal and analytic groups associated to elliptic curves. Besides Honda’s theorem recalled above, we shall also rely on some elementary results concerning the formal and analytic groups associated to an elliptic curve over Q and its base changes to local fields. These results, in contrast to Honda’s theorem, actually admit straightforward generalizations concerning commutative algebraic groups over (local) fields of characteristic zero. For the sake of simplicity, we will state them for elliptic curves only. Let E be an elliptic curve over a field k of characteristic zero, and let ω be a nonzero element in Ω1 (E/k) := Γ(E, Ω1E/k ). b of E at its zero element OE is a formal group The formal completion E scheme of dimension 1 over k. Moreover, if Ga,k ' Spec k[X] denotes the additive b a,k ' Spf k[[X]] the associated formal group (derived from Ga,k group over k and G by completion at 0), then there exists a unique isomorphism ∼ b b a,k −→ d E,ω : G Exp E

of formal groups over k that satisfies the normalization condition ∗

d E,ω ω b = dX. Exp |E d E,ω is compatible with extensions of the base field The construction of Exp 0 d E,ω,k0 of Exp d E,ω coincides with k: if k is an extension of k, the base change Exp ∼ b b d ExpEk0 ,ωk0 : Ga,k0 −→ Ek0 . Moreover, when k is a local field — such as the p-adic field Qp , or R, or C — the formal isomorphism is actually analytic. Let us spell out these analyticity properties in elementary terms. When k = Qp and E is embedded in a projective space PN Qp , say with OE mapped to the origin N O := (0, . . . , 0) ∈ AN Qp ⊂ PQp , the map b a,Q ' Spf Qp [[X]] −→ E b ,−→AN d E,ω : G Exp Qp p 25 The reader familiar with the content of [95] or [105] should have no difficulty in getting acquainted with the content of the classical articles [76], [79], and [80], and then reading Honda’s proof of Theorem 10.9.2 in [66].

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297

N may be described by an N -tuple of formal series (exp1E,ω , . . . , expE,ω ) in Qp [[X]], d E,ω may be expressed as with vanishing constant terms. Then the analyticity of Exp N the fact that the series exp1E,ω , . . . , expE,ω have positive p-adic radii of convergence. When k = C, a stronger version of this analyticity holds. Namely, there exists a (unique) C-analytic map

ExpE,ω := Ga (C)(:= C) −→ E(C) d E,ω . The fact that over archimedean places, the whose formal germ at 0 is Exp formal isomorphism ExpE,ω not only defines a C-analytic map on an open neighborhood of the origin, but indeed extends analytically to the whole complex line Ga (C) (and not only to an analytic neighborhood of the origin) will play a key role in our proof of the isogeny theorem. In fact, the map ExpE,ω is a surjective morphism of complex analytic Lie groups and satisfies Exp∗E,ω ω = dz. Its kernel is the lattice of periods of ω, Z  Λ := ω ; γ ∈ H1 (E(C), Z) , γ

and ExpE,ω defines an isomorphism of complex analytic Lie groups: ∼

C/Λ −→ E(C). For instance, when E is the complex elliptic curve in P2C with equation X0 X22 = 4X13 − g2 X02 X1 − g3 X03 , with origin OE := (0 : 0 : 1), and when ω := dx/y (where x := X1 /X0 and y := X2 /X0 ), the map ExpE,ω may be expressed in terms of the Weierstrass function ℘Λ associated to the lattice of periods Λ of ω. Namely, for all z ∈ C \ Λ, we have X ℘Λ (z) := z −2 + [(z − λ)−2 − λ−2 ] λ∈Λ\{0}

and ExpE,ω (z) = (1 : ℘Λ (z) : ℘0Λ (z)). 10.9.4 Proof of Theorem 10.9.1 – I. Construction of formal morphisms. In the proof of Theorem 10.9.1, we shall use the following notation. For all n ∈ Z, we denote by b1 b 1 := Spf Z[[T ]] −→ A λn : A Z Z

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the morphism of formal schemes (over Spec Z) attached to “multiplication by n”. Formally, it is defined by λn∗ T = n T. Similarly, for all q ∈ Q, we may consider the “multiplication by q” b a,Q = Spf Q[[X]]. b a,Q = Spf Q[[X]] −→ G λq := G b a,Q These morphisms λq are precisely the endomorphisms of the formal group G over Q. 10.9.4.1. Let E and E 0 be two elliptic curves over Q satisfying the assumptions of Theorem 10.9.1. We may assume that both of them are embedded in a projective space, say 2 , and that their origins OE and OE 0 are mapped to PQ 2 2 0 = (0, 0) ∈ AQ ,→ PQ .

We may take as models of E and E 0 their closures in P2Z[1/N ] , for N sufficiently divisible. We will denote them by π : E −→ Spec Z[1/N ]

and

π 0 : E 0 −→ Spec Z[1/N ].

2 By construction, E and E 0 are embedded in PZ[1/N ] , with their zero sections εE 2 2 and εE 0 sent to the section O = (0, 0) of AZ[1/N ] (,→ PZ[1/N ] ) over Z[1/N ]. After possibly replacing N by a positive multiple, we may also assume that the following two conditions are satisfied:

(1) There exists a uniformizing parameter t at the origin OE of E and a Zariski neighborhood U of the image of εE in E such that t belongs to OE (U ) and 1 its differential dt ∈ Γ(U, ΩE/Z[1/N ] ) does not vanish on U . Then the morphism t : U −→ A1Z[1/N ] is ´etale and defines an isomorphism of pointed smooth formal curves over Spec Z[1/N ]: ∼ b1 tˆ : Eb −→ Spf Z[1/N ][[T ]] =: A Z[1/N ] . This isomorphism induces an isomorphism of formal curves over Q: ∼ b1 . b −→ Spf Q[[T ]] =: A tˆQ : E Q

(2) There exists an isomorphism ∼

b : Eb −→ Eb0 ϕ of formal groups over Spec Z[1/N ].

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299

Assertion (1) follows from basic principles of algebraic geometry, and (2) from b as above have been Honda’s theorem 10.9.2. From now on, we assume that t and ϕ chosen. We also choose differential forms ω ∈ Ω1 (E/Q) \ {0}

and ω 0 ∈ Ω1 (E 0 /Q) \ {0}.

By evaluating ω and dt at the point OE of E, we get nonzero elements ωOE 1 . For some α ∈ Q∗ , we have and dtOE in the fiber at OE of ΩE/Q ωOE = α dtOE . d E,ω (0) of Exp d E,ω maps ∂/∂z to α−1 ∂/∂t. In other words, the differential D Exp b defines an isomorphism The isomorphism of formal groups ϕ ∼

b0 b −→ E ϕ bQ : E between the formal groups of E and E 0 , and there exists a unique β ∈ Q∗ such that the following diagram of (isomorphisms of) formal groups is commutative: b E x  d Exp E,ω 

bQ ϕ

−−−−→

b0 E x d ExpE0 ,ω0

λβ b a,Q −−− b a,Q . −→ G G

10.9.4.2. We may define the following isomorphisms of formal curves over Q: −1 b 1 ∼ b tQ : AQ −→ E ψbQ := b

and 0 b 1 ∼ b0 ψbQ := ϕ t−1 bQ ◦ b Q : AQ −→ E ,

and for all a ∈ N,

∼ b b 1 −→ E ψba,Q := ψbQ ◦ λN a : A Q

and ∼ b0 0 0 b 1 −→ E. := ϕ ◦ λN a : A ψba,Q bQ ◦ ψba,Q = ψbQ Q

For all a ∈ N, we may consider the following commutative diagram of isomorphisms of formal curves over Q: d Exp

ψa,Q E,ω b a,Q b 1 −− b ←−−− −−→ E −− G A Q      λ bQ y ϕ Idy y β ψ0

d Exp

a,Q E 0 ,ω 0 b a,Q . b 1 −−− b 0 ←−−− −−− G −→ E A Q

(10.9.1)

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300

0 Lemma 10.9.3. (1) For all a ∈ N, ψba,Q and ψba,Q extend to isomorphisms of formal curves over Spec Z[1/N ], ∼ b 2 b1 ψba,Z[1/N ] : A Z[1/N ] −→ E (,→ PZ[1/N ] )

and ∼ b0 2 0 b1 b ◦ ψba,Z[1/N ] : A ψba,Z[1/N Z[1/N ] −→ E (,→ PZ[1/N ] ). ] =ϕ 0 (2) There exists a0 ∈ N such that for every integer a ≥ a0 , ψba,Q and ψba,Q extend to morphisms of (formal) schemes over Spec Z,

b 1 −→ A2 ψba : A Z Z

and

b 1 −→ A2 . ψba0 : A Z Z

Proof. Assertion (1) is straightforward. Indeed, all morphisms in the diagram b λN a b 1 t b1 b ϕb b0 A Z[1/N ] −→ AZ[1/N ] ←− E −→ E

are isomorphisms of smooth formal curves over Spec Z[1/N ], and ψba,Z[1/N ] (resp., 0 b−1 ◦ λN a (resp., ϕ t−1 ◦ λN a ). ψba,Z[1/N b◦b ] ) may be defined as t Assertion (2) will follow from (1) and from the analyticity of b tQ , ExpE,ω , and ExpE 0 ,ω0 over the p-adic field Qp , for primes p dividing N . Recall that the Q-morphism tQ : UQ −→ A1Q is ´etale and maps OE to 0. This implies not only that it defines an isomorphism of formal curves over Q, 2 b1 ∼ b b ψbQ = b t−1 Q : AQ −→ UQ,OE = EQ (,→ AQ ),

but also that for every prime number p, it defines an isomorphism between the germ of Qp -analytic curves of A1 (Qp ) and E(Qp ) at 0 and OE . In elementary terms, this means that the power series (ψ 1 , ψ 2 ) with vanishing constant terms in Q[[X]], defined as the two components of ψbQ , have positive padic radii of convergence. In other words, for every prime p, there exists a0 (p) ∈ N such that for every integer a ≥ a0 (p), the coefficients of ψ 1 (pa X) and ψ 2 (pa X) are p-adic integers. Since ψ 1 and ψ 2 belong to Z[1/N ][[X]] by (1), this implies that for every integer a ≥ a0 := maxp|N a0 (p), the series ψ 1 (N a X) and ψ 2 (N a X) belong to b 1 −→ A2 . Z[[X]]. This means precisely that ψba,Q extends to a morphism ψba : A Z Z Similarly, the Qp -analyticity of −1

d E,ω d E 0 ,ω0 ◦ λβ ◦ Exp ϕ bQ = Exp −1 0 0 extends to a bQ ◦ b := ϕ tQ and allows one to prove that ψba,Q implies that of ψbQ 2 0 1 b b  morphism ψa : AZ −→ AZ when a is large enough.

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301

10.9.5 Proof of Theorem 10.9.1 – II. Algebraization. 10.9.5.1. For all a ∈ N, from the isomorphisms in the first row of diagram (10.9.1), we deduce an isomorphism −1 ∼ b b 1 −→ b1 . d E,ω ◦ ψa,Q : A ia := Exp Ga,Q = A Q Q

By construction, d E,ω ◦ ia = ψa,Q . Exp

(10.9.2)

Moreover, from the commutativity of (10.9.1), we deduce 0 d E 0 ,ω0 ◦ λβ ◦ ia = ϕ Exp bQ ◦ ψa,Q = ψba,Q .

(10.9.3)

For all a ∈ N and R ∈ R∗+ , we may define the smooth formal-analytic surface over Spec Z b 1 , D(0, R), ia,C ), ea,R := (A V Z \ b 1 to D(0, b 1 derived from ia by where ia,C denotes the isomorphism from A R)0 = A C C the base field extension Q ,→ C. In other words, b−1 ia,C = Exp−1 E,ω,C ◦ tC ◦ λN a ,C . Observe that a small enough open analytic neighborhood V of 0 in C (= A1 (C)) is contained in the inverse image Exp−1 EC ,ωC (U (C)). Then the composite map tC ◦ ExpEC ,ωC : V −→ C is analytic and ´etale, and maps 0 to 0. In intuitive terms, it describes the “Weierstrass uniformization” ExpEC ,ωC of E(C) in terms of the “local algebraic coordinate” tC on the Zariski open neighborhood U (C) of OEC in E(C). (The function tC ◦ ExpEC ,ωC is actually an elliptic function, in the classical sense, attached to the lattice of periods of (EC , ωC ).) ea,R may be This remark shows that when N = 1, the formal-analytic surface V b b understood as obtained by “glueing” the formal scheme E (identified A1Z though b t) and the complex disk of radius R in its Lie algebra (identified with C via hωC , .i) by means of the complex uniformization of EC . When N > 1, this description remains valid over Spec Z[1/N ]. ea,R is easily described. Indeed, the normal 10.9.5.2. The Hermitian line bundle N P V b bundle NP Va,R is the Z-module Z ∂/∂T .26 . The differential Dia,C (O) = DExpE,ω,C (0)−1 ◦ D b t(OEC )−1 ◦ DλN a ,C (0) 26 The

ba,R := Spf Z[[T ]] is defined by the ideal I := T Z[[T ]], largest ideal of definition I of V 2 ∨ b and NP Va,R by the module (I/I ) . It is a free Z-module with generator the “inverse” ∂/∂T of the class dT of T in I/I 2 .

Chapter 10. Formal-Analytic Arithmetic Surfaces and Algebraization

302

ba,R,C to the vector N a α ∂/∂z in the of ia,C maps the generator ∂/∂T of NP V tangent space at the origin of the disk D(0, R). Moreover, cap k∂/∂zkD(0,R),0 = 1/R

(cf. 10.5.3, Example (i)). These observations establish the following lemma: b1 ba,R = N0 A Lemma 10.9.4. The capacitary metric of the generator ∂/∂T of NP V Z is given by k∂/∂T kcap = N a |α|/R. e V a,R

ea,R is given by The Arakelov degree of N P V d NPV ea,R = log R − a log N − log |α|. deg

(10.9.4) 

10.9.5.3. Let us now assume that a ≥ a0 , so that the formal morphisms b 1 −→ A2 ,→ P2 ψba : A Z Z Z

b 1 −→ A2 ,→ P2 and ψba0 : A Z Z Z

ea,R is defined by the “glueing map” are defined. Since the formal-analytic surface V ia,C , the relations (10.9.2) and (10.9.3), after extending the base field from Q to C, immediately imply the following: Lemma 10.9.5. With the above notation, for all R ∈ R∗+ , if we let ψaan := ExpEC ,ωC |D(0,R) : D(0, R) −→ E(C),−→P2 (C) and ψa0

an

:= ExpEC0 ,ωC0 (β .)|D(0,R) : D(0, R) −→ E 0 (C),−→P2 (C),

an then ψa := (ψba , ψaan ) and ψa0 := (ψba0 , ψa0 ) are morphisms (as defined in 10.6.3) ea,R to P2 . of the formal-analytic surface V  Z

We may consider the product morphism ea,R −→ P2Z × P2Z fa := (ψa , ψa0 ) : V defined by b 1 −→ P2 × P2 fba := (ψba , ψba0 ) : A Z Z Z and an

faan := (ψa an , ψa0 ) : D(0, R) −→ E(C) × E 0 (C),−→P2 (C) × P2 (C). Given a ≥ a0 , let us choose R ∈ R∗+ such that log R > a log N + log |α|.

10.9. The Isogeny Theorem for Elliptic Curves over Q

303

d NPV ea,R , the “arithmetic pseudoThen according to the expression (10.9.4) for deg concavity” condition (10.7.1), d N P V˜ > 0, deg is satisfied. Therefore, according to Theorem 10.1.2, the image of fa is algebraic. Let us consider C := im fQ . It is an integral closed subscheme of dimension 1 of P2Q × P2Q , containing (OE , OE 0 ). According to Proposition 10.6.4, by extending the base field from Q to C, it becomes CC = im faan , the Zariski closure of o n (ExpEC ,ωC (z), ExpEC0 ,ωC0 (βz)); z ∈ D(0, R) . By analytic continuation, the complex curve CC is also the Zariski closure of the map (ExpEC ,ωC (z), ExpEC0 ,ωC0 (βz)) : Ga (C) −→ E(C) × E 0 (C), which is a morphism of C-analytic Lie groups. This shows that C(C) is a subgroup of E(C) × E 0 (C) and therefore that CC is an algebraic subgroup of (E × E 0 )C , and thus an elliptic curve with origin (OE,C , OE 0 ,C )). This implies that C is an elliptic curve over Q (with origin (OE , OE 0 )), and that the projections from C to E and E 0 are dominant (since the projections from C(C) to E(C) and E 0 (C) are surjective), and finally that E, C, and E 0 are isogenous over Q.

Appendix A

Large Deviations and Cram´er’s Theorem In this appendix, we present some basic results in the theory of large deviations in a form suited to the application to Euclidean lattices discussed in Section 3.4. In particular, we formulate a general version of Cram´er’s theory of large deviations (Theorem A.3.1). Recall that a measurable function H : E −→ R being given on a probability space (E, T , µ), this theory describes the asymptotic behavior, as the positive integer n goes to infinity, of the values of the function Hn : E n −→ R defined by Hn (e1 , . . . , en ) := H(e1 ) + · · · + H(en ).

(A.0.1)

This description is formulated in terms of the integral Z log eξH dµ E

as a function of ξ ∈ R with values in ] − ∞, +∞] and its Legendre–Fenchel transform. In fact, for the application to Euclidean lattices of the theory of large deviations, we rely on an extension of Cram´er’s theorem covering the situation in which µ is an arbitrary σ-finite positive measure, assuming that H is nonnegative.1 Such extensions of Cram´er’s theorem are possibly known to some experts, but for lack of references, in Sections A.4 and A.5 we formulate and establish suitable versions of Cram´er’s theorem (Theorems A.4.4 and A.5.1) covering the case in which µ(E) is possibly +∞. We achieve this by a simple reduction to the case in which µ is a probability measure. In the first two sections (A.1 and A.2) of this appendix, we also extend various preliminary results concerning the asymptotic behavior of the values of Hn on E n ˜ 0 (E, t) in Section 3.4, to investigate the properties of the “asymptotic invariants” h Ar attached to a Euclidean lattice E := (E, k.k), we consider P the situation in which E is the free Zmodule E underlying E, where µ is the counting measure e∈E δe , and where H is the squared Euclidean norm k.k2 . 1 Indeed,

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

305

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Appendix A. Large Deviations and Cram´er’s Theorem

as n goes to infinity to this more general setting in which µ(E) is possibly +∞. Here again, for lack of suitable references, we have included some details concerning these “well-known” results. Then, in Section A.3, we recall various forms of the “classical” Cram´er’s theorem concerning the situation in which µ is a probability measure. Let us finally point out that the generalized version of Cram´er’s theorem presented in this appendix is closely related to the so-called canonical ensembles in statistical mechanics and to the existence of thermodynamic limits. We do not discuss this in detail, but we simply refer the reader to [71] and [98] for presentations of statistical thermodynamics that emphasize the points of contact between statistical mechanics and limit theorems in probability2 , and we indicate that the notation in Section A.5 has been chosen to express these relations. The first paragraph, A.5.1, of the final section A.5, of this appendix summarizes some of its main results in a form suitable for their applications in Section 3.4. Section A.5 has been written to be accessible without a knowledge of the formalism previously developed in this appendix, and could be read immediately after this introduction.

A.1

Notation and Preliminaries

A.1.1 Notation. In this appendix, we consider a measure space (E, T , µ) defined by a set E, a σ-algebra T of subsets of E, and a nonzero σ-finite nonnegative measure µ : T −→ [0, +∞]. Furthermore, we consider a T -measurable function H : E −→ R. We shall denote by inf µ H (resp., supµ H) its essential infimum (resp., supremum) with respect to the measure µ. For every positive integer n, we may introduce the nth power of the measure space (E, T , µ) — it is defined as the product E n of n copies of E, equipped with the σ-algebra T ⊗n := T ⊗ · · · ⊗ T (n-times) over E n and with the product σ-finite measure µ⊗n := µ ⊗ · · · ⊗ µ (n-times) on T ⊗n — and we may consider the function Hn : E n −→ R defined by (A.0.1): Hn (e1 , . . . , en ) := H(e1 ) + · · · + H(en ). 2 See

also [75] and [43] for related mathematical results and references.

A.1. Notation and Preliminaries

307

A.1.2 Log-Laplace transform. Recall that a function p : R −→] − ∞, +∞] is lower semicontinuous and convex if and only if Gr≥ (p) := {(x, y) ∈ R2 | y ≥ p(x)} is a closed convex subset of R2 . One easily shows that this holds precisely when there exists an interval I ⊂ R such that the following conditions are satisfied: (1) p|I is real-valued, continuous, and convex; (2) pR\I ≡ +∞; (3) if ˚ / I ∪ {−∞}, then limx→a+ p(x) = +∞; 6 ∅ and a := inf I ∈ I= I= / I ∪ {+∞}, then limx→b− p(x) = +∞. if ˚ 6 ∅ and b := sup I ∈ If p : R −→] − ∞, +∞] is nondecreasing and convex, then there exists a unique c ∈ [−∞, +∞] such that for all x ∈ R, x < c =⇒ p(x) ∈ R and x > c =⇒ p(x) = +∞, and p is lower semicontinuous if and only if lim p(x) = p(c).

x→c−

Similar remarks apply mutatis mutandis to nonincreasing convex functions and to concave functions from R to [−∞, +∞[. The following proposition is a straightforward consequence of the basic results of Lebesgue integration theory: R Proposition A.1.1. For every ξ ∈ R, the integral X exp(ξH) dµ belongs to ]0, +∞] and the function ` : R −→] − ∞, +∞] defined by Z `(ξ) := log

eξH dµ

E

is lower semicontinuous and convex. −1 ˚ Moreover, the restriction `|˚ (R) I of ` to the interior I of the interval I := ` 00 and > is real analytic. Unless H is constant µ-almost everywhere, it satisfies `|˚ 0, I ∼ I) its derivative defines an increasing real analytic diffeomorphism `0 : ˚ I −→ `0 (˚  I and `0 (˚ I) in R. between the open intervals ˚ The function ` is appears in the litterature under various names. It is sometimes called the log-Laplace transform of the Borel measure H∗ µ on R, which is nothing but the “law” of H when µ is a probability measure. In this situation, it is also called the logarithmic moment generating function of H∗ µ (in [109], for instance).

308

Appendix A. Large Deviations and Cram´er’s Theorem

A.1.3 The functions An> and A< n , and s+ and s− . To every x ∈ R, we may also attach the sequences (An> (x))n≥1 and (A< n (x))n≥1 in [0, +∞] defined by ⊗n A> (Hn−1 ([nx, +∞[) n (x) = µ

and An< (x) = µ⊗n (Hn−1 ( ]− ∞, nx])). We shall be interested in situations in which these sequences have an exponential asymptotic behavior, and accordingly, for every x ∈ R, we consider the following elements of [−∞, +∞]: 1 log A> n (x) n n≥1

(A.1.1)

1 log A< n (x). n≥1 n

(A.1.2)

s+ (x) := sup and s− (x) := sup

Clearly, for every positive integer n, the function A> n : R −→ [0, +∞] is nonincreasing and the function A< n : R −→ [0, +∞] is nondecreasing. Accordingly, the functions s+ and s− : R −→ [−∞, +∞] are respectively nonincreasing and nondecreasing. Observe also that for all x ∈ R, the following alternative holds: either (i) the function H is < x µ-almost everywhere on E, and then A> n (x) = 0 for every positive integer n; or (ii) the function H takes values ≥ x on a subset of positive µ-measure, and then A> n (x) > 0 for every positive integer n. When x = supµ H, either case may occur, but we always have ⊗n A> (H −1 (supµ H)n ) = µ(H −1 (supµ H))n , n (supµ H) = µ

and consequently, s+ (supµ H) = log µ(H −1 (supµ H)). A similar alternative holds for the sequence (A< n (x))n≥1 , and s− (inf µ H) = log µ(H −1 (inf µ H)).

A.2

Lanford’s Inequalities

In the present general framework (which does not require the finiteness of the measure µ and of the log-Laplace transform `), the following inequalities play a < crucial role in the study of the asymptotic behavior of the functions A> n and An as n goes to +∞:

A.2. Lanford’s Inequalities

309

2 Lemma A.2.1. For all (x, y) in R2 and (p, q) in N≥1 , we have

Ap> (x) · Aq> (y) ≤ A> p+q ((px + qy)/(p + q))

(A.2.1)

< < A< p (x) · Aq (y) ≤ Ap+q ((px + qy)/(p + q)).

(A.2.2)

and In (A.2.1) and (A.2.2), we use the standard measure-theoretic convention 0 · (+∞) = (+∞) · 0 = 0. Together with the subadditivity and concavity arguments leading to the proofs of Proposition A.2.2 and Theorem A.2.5 below, this lemma originates in Lanford’s work [75] on the rigorous derivation of “thermodynamic limits” in statistical mechanics. Proof. Observe that the following inclusions hold between T ⊗n -measurable subsets of E n : −1 Hp−1 ([px, +∞[) × Hq−1 ([qy, +∞[) ⊂ Hp+q ([px + qy, +∞[)

(A.2.3)

−1 (] − ∞, px + qy]). Hp−1 (] − ∞, px]) × Hq−1 (] − ∞, qy]) ⊂ Hp+q

(A.2.4)

and

By applying the measure µ⊗p+q to both sides of (A.2.3) (resp., of (A.2.4)),  we get the estimate (A.2.1) (resp., the estimate (A.2.2)). When y = x, Lemma A.2.1 asserts that > > A> p (x)Aq (x) ≤ Ap+q (x)

(A.2.5)

< < A< p (x)Aq (x) ≤ Ap+q (x).

(A.2.6)

and In other words, the sequences (log An> (x))n≥1 and (log A< n (x))n≥1 are superadditive. Combined with the alternative in A.1.3 above concerning the vanishing or nonvanishing of the An> (x) and with Fekete’s lemma on superadditive sequences (Lemma 3.4.3), the estimates (A.2.5) easily imply the following: Proposition A.2.2. For every x ∈ R, precisely one of the following three conditions is satisfied: O> : for every positive integer n, A> n (x) = 0; F> : for every positive integer n, A> n (x) ∈]0, +∞[; then the sequence ((log A> n (x))/n)n≥1 admits a limit: 1 1 log A> log A> lim n (x) = sup n (x) =: s+ (x) ∈ ] − ∞, +∞]. n→+∞ n n n≥1

(A.2.7)

310

Appendix A. Large Deviations and Cram´er’s Theorem

I> : for a positive integer n0 , A> n0 (x) = +∞. Then for every integer n ≥ n0 , An> (x) = +∞. A similar trichotomy holds with the sequence (A> n (x))n≥1 replaced by and s+ (x) by s− (x), and it defines the conditions O< , F< , and  I . (An< (x))n≥1 ,
, IF> , and IO> of R defined by each the conditions I> , F> , and O> . Clearly, they constitute disjoint consecutive intervals, and we have R = II> ∪ IF> ∪ IO> and −1 ([x, +∞[)) = 0}. s−1 + (−∞) = IO> = {x ∈ R | µ(H

Similarly, we define disjoint consecutive intervals IO< , IF< , and II< such that R = IO< ∪ IF< ∪ II< , and we have −1 (−∞) = IO< = {x ∈ R | µ(H −1 (] − ∞, x])) = 0}. s− −1 (]−∞, +∞]) = II> ∪IF> (resp., Lemma A.2.3. For every two points x and y in s+ −1 in s− (] − ∞, +∞]) = IF< ∪ II< ) and every two positive rational numbers α and β such that α + β = 1, we have

αs+ (x) + βs+ (y) ≤ s+ (αx + βy)

(A.2.8)

(resp., αs− (x) + βs− (y) ≤ s− (αx + βy)). Proof. Consider a positive integer n such that p := nα and q := nβ are integers. Clearly, p and q satisfy p + q = n, and from (A.2.1), by applying (1/n) log, we derive 1 1 > 1 α log A> log A> p (x) + β Aq (y) ≤ n (αx + βy). p q n The estimate (A.2.8) follows by taking the limit as n goes to infinity.



−1 The inequality (A.2.8) implies that if s−1 + (+∞) is not empty, then s+ (] − ∞, +∞[) contains at most one point. When one investigates the asymptotic behavior of the numbers (A> n (x))n≥1 , it is sensible to exclude this case and to assume that the following condition is satisfied:

B> : For every x ∈ R, s+ (x) < +∞. Clearly, this condition implies that II> is empty. −1 A similar discussion applies to s−1 − (+∞) and s− (] − ∞, +∞[), and leads one to introduce the following condition:

A.2. Lanford’s Inequalities

311

B< : For every x ∈ R, s− (x) < +∞. Lemma A.2.4. The conditions B> and B< are satisfied if µ(E) < +∞. More generally, if there exists ξ in R+ (resp., in R− ) such that `(ξ) < +∞, then B> (resp., B< ) is satisfied. Proof. When µ(E) < +∞, s+ (x) and s− (x) are bounded from above by log µ(E) for every x ∈ R. Let us assume that ξ is an element of R+ such that `(ξ) < +∞. Then for every positive integer n, we have Z n Z eξH dµ = en`(ξ) < +∞. eξHn dµ⊗n = En

E

Moreover, for every x ∈ R and every P ∈ Hn−1 ([nx, +∞[), eξHn (P ) ≥ enξx . Therefore, µ⊗n (Hn−1 ([nx, +∞[)enξx ≤ en`(ξ) and

1 log A> n (x) ≤ `(ξ) − ξx. n

This shows that s+ (x) ≤ `(ξ) − ξx < +∞.
n and An : Theorem A.2.5. (1) If condition B> is satisfied, then for every x ∈ R, either • µ⊗n (Hn−1 ([nx, +∞[)) = 0 for every positive integer n and s+ (x) = −∞, or • the sequence (log µ⊗n (Hn−1 ([nx, +∞[)))n≥1 lies in R, is superadditive, and satisfies lim

n→+∞

1 log µ⊗n (Hn−1 ([nx, +∞[)) = s+ (x) ∈ R. n

(A.2.9)

The function s+ : R −→ [−∞, +∞[ is nonincreasing and concave. Moreover, for all x ∈ R, x < supµ H =⇒ s+ (x) ∈ R and x > supµ H =⇒ s+ (x) = −∞, and s+ (supµ H) = log µ(H −1 (supµ H)).

312

Appendix A. Large Deviations and Cram´er’s Theorem

(2) Symmetrically, if condition B< is satisfied, then for every x ∈ R, either • µ⊗n (Hn−1 (]−∞, nx])) = 0 for every positive integer n and s− (x) = −∞, or • the sequence (log µ⊗n (Hn−1 (] − ∞, nx])))n≥1 lies in R, is superadditive, and satisfies 1 lim log µ⊗n (Hn−1 (] − ∞, nx])) = s− (x) ∈ R. (A.2.10) n→+∞ n The function s− : R −→ [−∞, +∞[ is nondecreasing and concave. Moreover, for all x ∈ R, x < inf µ H =⇒ s− (x) = −∞ and x > inf µ H =⇒ s− (x) ∈ R, and s− (inf µ H) = log µ(H −1 (inf µ H)). Proof. At this stage, to complete the proof of (1), we simply need to observe that the inequality (A.2.8) holds for every two points x and y in s−1 + (] − ∞, +∞[), not only when the coefficients (α, β) are positive rational numbers such that α+β = 1, but more generally for any two positive real numbers (α, β) such α + β = 1: since s+ is nonincreasing, this follows from a straightforward approximation argument. This establishes the concavity of s+ . The proof of (2) is similar, or follows from (1) applied to −H instead of H. 

A.3

Cram´er’s Theorem

In this section, we assume that the measure µ is a probability measure. Observe that this implies that `(0) = 0. In particular, the interval I := `−1 (R) is nonempty and contains 0. Moreover, the conditions B> and B< are satisfied. A.3.1 The following theorem is the formulation, in measure-theoretic language, of a general version of Cram´er’s theorem concerning the “empirical means” n

1X X n := Xi n i=1 attached to a sequence (Xi )i≥1 of independent and identically distributed realvalued random variables. We refer the reader to the article [28] of Cerf and Petit for a short and elegant proof.3 3 The function s in [28] corresponds to the function s defined above. The assertions concern+ ing s− in Theorem A.3.1 follow from those concerning s+ applied to the function −H instead of H.

A.3. Cram´er’s Theorem

313

Theorem A.3.1. For every x ∈ R, we have s+ (x) = inf (`(ξ) − ξx)

(A.3.1)

s− (x) = inf (`(ξ) − ξx).

(A.3.2)

ξ∈R+

and ξ∈R−

Moreover, for all ξ ∈ R+ (resp., for all ξ ∈ R− ), we have `(ξ) = sup(ξx + s+ (x))

(A.3.3)

x∈R

(resp., `(ξ) = supx∈R (ξx + s− (x))).

(A.3.4)

Observe also that the functions s+ and s− take their values in [−∞, 0]. Moreover, according to (A.3.1) and (A.3.2), they are upper semicontinuous. Moreover, we have lim s+ (x) = 0

x→−∞

(A.3.5)

(indeed, log µ(H −1 ([x, +∞[)) = log A1 (x) ≤ s+ (x) ≤ 0) and lim x→(supµ H)−

s+ (x) = s+ (supµ H) = log µ(H −1 (supµ H))

(since s+ is upper semicontinuous and nonincreasing); similarly, lim x→(inf µ H)+

s− (x) = s− (inf µ H) = log µ(H −1 (inf µ H))

and lim s− (x) = 0.

x→+∞

(A.3.6)

By “Cram´er’s theorem” one usually means the conjunction of the existence of the limits (A.2.9) and (A.2.10) for every x ∈ R that is established in Theorem A.2.5, together with the expressions (A.3.1) and (A.3.2) of these limits in terms of the log-Laplace transform ` stated in Theorem A.3.1. The reader will also refer to [28] for additional references about large deviations and Cram´er’s theorem, notably concerning the successive contributions, starting from Cram´er’s seminal article [36] that have led to the present simple and general formulation of Cram´er’s theorem. Let us also indicate that Chernoff’s version of Cram´er’s theorem [31, Theorem 1] would be enough to derive Theorem A.4.4 below.

314

Appendix A. Large Deviations and Cram´er’s Theorem

A.3.2 The version of Cram´er’s theorem formed by Theorems A.2.5 and A.3.1, when µ is a probability measure, may be supplemented by the following observations, intended to clarify its relation with other presentations of Cram´er’s theory of large deviations (for instance in [109, Section 1.3]). Let us define m+ ∈ [−∞, +∞] and m− ∈ [−∞, +∞] as the “right and left derivatives” m+ := inf∗ `(ξ)/ξ = lim `(ξ)/ξ =: `0r (0) ξ∈R+

ξ→0+

and m− := sup `(ξ)/ξ = lim `(ξ)/ξ =: `0l (0) ξ∈R∗ −

ξ→0−

of the lower semicontinuous convex function ` at 0. We clearly have m− ≤ m+ , and from Theorem A.3.1, we easily get the following corollary: Corollary A.3.2. For all x ∈ R, we have s+ (x) = 0 ⇐⇒ x ≤ m+

(A.3.7)

s− (x) = 0 ⇐⇒ x ≥ m− .

(A.3.8)

s := min(s− , s+ ) : R −→ [−∞, +∞]

(A.3.9)

and Moreover, the function

is upper semicontinuous and concave, and the functions ` and −s may be derived from each other by Legendre–Fenchel duality: for every x ∈ R, s(x) = inf (`(ξ) − ξx), ξ∈R

and for every ξ ∈ R, `(ξ) = sup(ξx + s(x)).



x∈R

Observe finally that if we denote by H + and H − the positive and negative parts4 of H, then the following conditions are equivalent5 : (1) m+ < +∞, (2) I ∩ R∗+ 6= ∅, (3) for some ε ∈ R∗+ , the function exp(εH+ ) is µ-integrable, and the following ones as well: the R+ -valued functions onR E defined by H + := max(H, 0) and H − := max(−H, 0). that I = `−1 (R) = {ξ ∈ R | E eξH dµ < +∞}.

4 Namely 5 Recall

A.4. An Extension of Cram´er’s Theorem

315

(1) m− > −∞, (2) I ∩ R∗− 6= ∅, ∗ , the function exp(εH− ) is µ-integrable. (3) for some ε ∈ R+

In particular, if m+ < +∞ and m− > −∞, then I contains a neighborhood of 0 in R, the function H is µ-integrable, and Z 0 m+ = m− = ` (0) = H dµ. E

More generally, if m+ < +∞ (resp., m− > −∞), then the integral Z Z Z H − dµ H dµ := H + dµ − E

E

E

is well defined in [−∞, +∞[ (resp., in ] − ∞, +∞]) and is easily seen to be equal to m+ (resp., to m− ).

A.4

An Extension of Cram´er’s Theorem Concerning General Positive Measures

In this section, the measure µ is allowed to have a total mass µ(E) different from 1. When µ(E) is finite, one may apply the results in the previous section with µ replaced by the probability measure µ0 := µ(E)−1 µ. In this way, one easily sees that Theorem A.3.1 still holds without alteration, and that its consequences remain valid with minor modifications (in (A.3.5)–(A.3.6) and in (A.3.7)–(A.3.8), 0 has to be replaced by log µ(E)). When the total mass µ(E) is +∞, then for all x ∈ R, A1> (x) + A1< (x) = +∞, and therefore s+ (x) = +∞ or s− (x) = +∞. Therefore, the conditions B> and B< cannot be simultaneously satisfied, and only one of the two functions s+ and s− may have an interesting behavior. In this section, we shall focus on s− , and we will show that the results in Theorem A.3.1 concerning s− admit a sensible generalization when H is bounded from below — say when inf µ H is nonnegative.6 6 The interested reader will have no difficulty in extending the results of this section to the more general situation in which inf µ H > −∞.

316

Appendix A. Large Deviations and Cram´er’s Theorem

A.4.1 In this paragraph, we assume that the interval I := `−1 (R) is not empty, and we choose an element ξ0 in I. Then the integral Z eξ0 H dµ = e`(ξ0 )

E

belongs to ]0, +∞[ and the measure Z −1 ξ0 H µ0 := e dµ eξ0 H µ E

is a probability measure on the measurable space (E, T ). Therefore, we may apply the constructions and the results of the previous sections to the probability space (E, T , µ0 ) and the function H. Let us denote by `0 , s0+ , and s0− the functions `, s+ , and s− associated to these new data. The following lemma is a straightforward consequence of the definitions: Lemma A.4.1. For every positive integer n and every (ξ, x) in R2 , we have µ0⊗n = eξ0 Hn −n`(ξ0 ) µ⊗n ,

(A.4.1)

`0 (ξ) = `(ξ + ξ0 ) − `(ξ0 ),

(A.4.2)

and 1 1 −1 log µ⊗n log 0 (Hn (] − ∞, nx])) = n n

Z −1 Hn (]−∞,nx])

eξ0 Hn dµ⊗n − `(ξ0 ). (A.4.3) 

By means of the relation (A.4.1)–(A.4.3), the assertions involving s0− and `0 in Theorems A.2.5 and A.3.1 applied to the probability space (E, T , µ0 ) and the function H may be reformulated as follows, without reference to the measures µ⊗n 0 : Corollary A.4.2. Let ξ0 be a real number such that `(ξ0 ) < +∞. For every x in R, we have Z 1 lim log eξ0 Hn dµ⊗n = inf (`(ξ + ξ0 ) − ξx). −1 n→+∞ n ξ∈R− Hn (]−∞,nx])

(A.4.4)

Moreover, for every ξ ∈ R− , 1 log `(ξ + ξ0 ) = sup ξx + lim n→+∞ n x∈R

Z

! eξ0 Hn dµ⊗n

.

(A.4.5)

−1 Hn (]−∞,nx])

 Observe that every term of the sequence on the left-hand side of (A.4.4), and consequently its limit, belongs to [−∞, `(ξ0 )].

A.4. An Extension of Cram´er’s Theorem

317

A.4.2 An application. In this paragraph, we apply the previous results to the situation in which H assumes only nonnegative values on E. We shall use the nonnegativity of H through the following simple observation: Lemma A.4.3. Let us assume that H(E) ⊂ R+ , and let us consider a positive integer n and elements x of R+ and η of R∗+ . Then Hn−1 (] − ∞, nx]) is empty if x < 0 and equals Hn−1 ([0, nx]) if x ≥ 0. Therefore, every P ∈ Hn−1 (] − ∞, nx]) satisfies e−ηnx ≤ e−ηHn (P ) ≤ 1, and consequently, 1 1 log A< log n (x) − ηx ≤ n n

Z

e−ηHn dµ⊗n ≤

−1 Hn (]−∞,nx])

1 log A< n (x). n

(A.4.6) 

We may now formulate and establish the main result of this appendix: Theorem A.4.4. Let us assume that the following conditions are satisfied: H(E) ⊂ R+ and for all ξ ∈

R∗− ,

Z

eξH dµ < +∞.

(A.4.7)

E

Then for every x in R, we have s− (x) = inf∗ (`(ξ) − ξx). ξ∈R−

(A.4.8)

In particular, condition B< is satisfied, and the concave function s− : R −→ [−∞, +∞[ is upper semicontinuous. Moreover, for every ξ ∈ R∗− , we have `(ξ) = sup(ξx + s− (x)).

(A.4.9)

x∈R

Proof. Let us consider x in R and η in R∗+ . We may apply Corollary A.4.2 with ξ0 = −η and observe that as n goes to +∞, the inequalities (A.4.6) become Z 1 log e−ηHn dµ⊗n ≤ s− (x). (A.4.10) s− (x) − ηx ≤ lim −1 n→+∞ n Hn (]−∞,nx]) According to (A.4.4), the middle term in (A.4.10) is inf (`(ξ − η) − ξx) =

ξ∈R−

inf

(`(ξ) − ξx) − ηx,

ξ∈]−∞,−η]

318

Appendix A. Large Deviations and Cram´er’s Theorem

so that (A.4.10) may be also be written s− (x) − ηx ≤

inf

(`(ξ) − ξx) − ηx ≤ s− (x).

ξ∈]−∞,−η]

By taking the limit of these inequalities as η goes to 0+, we obtain (A.4.8). ∗ Let ξ be an element of R− . The relation (A.4.5) for ξ0 := −η, together with (A.4.10), shows that sup(ξx + s− (x) − ηx) ≤ `(ξ − η) ≤ sup(ξx + s− (x)). x∈R

x∈R ∗ Since ` is continuous on R− , (A.4.9) follows.



A.5 Reformulation and Complements In this section, for the convenience of the reader, we reformulate some of the results previously established in this appendix without making reference to the formalism introduced in the previous sections. To emphasize the possible thermodynamic interpretation of these results, we will introduce some new notation, related to the previous notation through the following formulas: Ψ(β) = `(−β) and S(x) = s− (x). A.5.1 A scholium. Let us recall that we consider a measure space (E, T , µ) defined by a set E, a σ-algebra T of subsets of E, and a nonzero σ-finite nonnegative measure µ : T −→ [0, +∞]. For every positive integer n, we also consider the product µ⊗n of n copies of the measure µ on E n equipped with the σ-algebra T ⊗n . Furthermore, we consider a T -measurable function H : E −→ R+ . ∗ For every β ∈ R+ such that e−βH is µ-integrable, the integral a positive real number, and we let Z Ψ(β) := log e−βH dµ (∈ R). E

Under this integrability assumption, we also define F (β) := −β −1 Ψ(β). Equivalently, F (β) may be defined by the relation Z e−βF (β) = e−βH dµ. E

R E

e−βH dµ is

(A.5.1)

A.5. Reformulation and Complements

319

The µ-integrability of the function e−βH for every β ∈ R∗+ is easily seen to be equivalent to the subexponential growth of the function N : R+ −→ [0, +∞] defined by N (x) := µ(H −1 ([0, x])), namely to the following condition: SE : For every x ∈ R+ , N (x) is finite, and as x goes to +∞, log N (x) = o(x). We shall also consider the essential infimum inf µ H := inf{x ∈ R+ | N (x) > 0} of H with respect to the measure space (E, T , µ). Theorem A.5.1. Let us keep the above notation and let us assume that condition SE is satisfied and that µ(E) = +∞. (1) For every x in ]inf µ H, +∞[, the limit S(x) := lim

n→+∞

1 log µ⊗n ({(e1 , . . . , en ) ∈ E n | H(e1 ) + · · · + H(en ) ≤ nx}) n

exists in R. The function S : ]inf µ H, +∞[−→ R is non-decreasing and concave, and satisfies lim x→(inf µ H)+

S(x) = log µ(H −1 (inf µ H)) (∈ [−∞, +∞[).

(A.5.2)

(2) The function Ψ : R∗+ −→ R is real analytic and convex. Its derivative up to a sign U := −Ψ0 satisfies, for every β ∈ R∗+ , R

H e−βH dµ e−βH dµ E

U (β) = ER

(A.5.3)

and defines a decreasing real analytic diffeomorphism ∼

U : R∗+ −→ ]inf µ H, +∞[.

(A.5.4)

320

Appendix A. Large Deviations and Cram´er’s Theorem

(3) The functions −S(−.) and Ψ are Legendre–Fenchel transforms of each other. Namely, for every x ∈]inf µ H, +∞[, S(x) = inf∗ (βx + Ψ(β)), β∈R+

(A.5.5)

∗ and for every β ∈ R+ ,

Ψ(β) =

(S(x) − βx).

sup

(A.5.6)

x∈]inf µ H,+∞[

Proof. Assertion (1) follows from Theorem A.4.4 and Theorem A.2.5, (2). Observe that H is not µ-almost everywhere constant — otherwise, the conditions SE and µ(E) = +∞ could not both be satisfied. Therefore, Ψ00 > 0 on R∗+ by Proposition A.1.1. The expression (A.5.3) `a la Gibbs for U := −Ψ0 is a straightforward consequence of Lebesgue integration theory. To complete the proof of (2), we are thus left to show that lim U (β) = +∞ (A.5.7) β→0+

and lim U (β) = inf µ H.

β→+∞

(A.5.8)

Since the function Ψ satisfies Z lim Ψ(β) = log

β→0+

dµ = +∞, E

its derivative cannot remain bounded near zero. This proves (A.5.7). To prove (A.5.8), simply observe that according to (A.5.3), for all β ∈ R∗+ , U (β) ≥ inf µ H and that for all η > inf µ H and β ∈ R∗+ , Z Z −βH dµ ≥ log Ψ(β) = log e E

e−βη dµ ≥ log N (η) − βη,

H −1 ([0,η])

so that limβ→+∞ Ψ0 (β) ≥ −η. Assertion (3) follows directly from Theorem A.4.4.



The following corollary is now a consequence of the elementary theory of Legendre–Fenchel transforms of convex functions of one real variable7 : 7 Basically, it follows from the fact that the Legendre–Fenchel transform g of a real analytic function f with positive second derivative is itself real analytic with positive second derivative, and that f and g are related by the classical Legendre duality relation g(ξ) + f (x) = xξ, where ξ = f 0 (x), or equivalently, x = g 0 (ξ).

A.5. Reformulation and Complements

321

Corollary A.5.2. The function S is increasing and real analytic on ]inf µ H, +∞[. Moreover, its derivative establishes a decreasing real analytic diffeomorphism ∼

∗ , S 0 : ]inf µ H, +∞[−→ R+

the inverse of the diffeomorphism (A.5.4). For all x ∈ ]inf µ H, +∞[, the infimum on the right-hand side of (A.5.5) is ∗ , namely attained at a unique β ∈ R+ β = S 0 (x).

(A.5.9)

Dually, for all β ∈ R∗+ , the supremum on the right-hand side of (A.5.6) is attained at a unique x ∈ ]inf µ H, +∞[, namely x = U (β).

(A.5.10)

Observe that when x ∈ ]inf µ H, +∞[ and β ∈ R∗+ satisfy the equivalent relations (A.5.9) and (A.5.10), then S(x) = βx + Ψ(β),

(A.5.11)

or equivalently, F (β) := −β −1 Ψ(β) = U (β) − β −1 S(x) (“expression of the free energy F in terms of the energy U , the temperature β −1 , and the entropy S”). Observe also that according to (A.5.2), the following two conditions are equivalent: µ({e ∈ E | H(e) = inf µ H}) = 1 and lim S(U (β)) = 0

β→+∞

(“Nernst’s principle”). A.5.2 Products and thermal equilibrium. The formalism summarized in the previous paragraph — which attaches functions Ψ and S to a measure space (E, T , µ) and to a nonnegative function H on E satisfying SE — satisfies a simple but remarkable compatibility with finite products, which we want to discuss briefly. Assume that for every element i in some finite set I, we are given a measure space (Ei , Ti , µi ) and a measurable function Hi : Ei −→ R+ as in the previous paragraph. Then we may form the product measure space Q N (E, T , µ) defined by the set E := Ni∈I Ei equipped with the σ-algebra T := i∈I Ti and the product measure µ := i∈I µi .

322

Appendix A. Large Deviations and Cram´er’s Theorem We may also define a measurable function H : E −→ R+

by the formula H :=

X

pri∗ Hi ,

i∈I

where pri : E −→ Ei denotes the projection on the ith factor. Let us assume that for every i ∈ I, (Ei , Ti , µi ) and Hi satisfy the condition SE, or equivalently that the function e−βHi is µi -integrable for every β ∈ R∗+ . Then (E, T , µ) and H are easily seen to satisfy SE also, as a consequence of ∗ −→ R Fubini’s theorem. Indeed, Fubini’s theorem shows that the function Ψ : R+ attached to (E, T , µ) equipped with the function H, defined as in A.5.1 by the formula Z e−βH dµ,

Ψ(β) := log E

and the “partial functions” Ψi , i ∈ I, attached to each measure space (Ei , Ti , µi ) equipped with the function Hi by the similar formula Z e−βHi dµi , Ψi (β) := log Ei

satisfy the additivity relation Ψ=

X

Ψi .

(A.5.12)

i∈I

Let us also assume that µi (Ei ) = +∞ for every i ∈ I. Then we also have µ(E) = +∞, and we may apply Theorem A.5.1 and Corollary A.5.2 to the data (Ei , Ti , µi , Hi ), i ∈ I, and (E, T , µ, H). In particular, we may define concave functions Si : ]inf µi Hi , +∞[−→ R,

for i ∈ I,

and S : ]inf µ H, +∞[−→ R. Observe also that as a straightforward consequence of the definitions, we have inf µ H =

X

inf µi Hi .

i∈I

The following proposition may be seen as a mathematical interpretation of the second law of thermodynamics:

A.5. Reformulation and Complements

323

Proposition A.5.3. (1) For each i ∈ I, let xi be a real number in ]inf µi Hi , +∞[. Then the following inequality is satisfied: X

X xi ). Si (xi ) ≤ S(

(A.5.13)

i∈I

i∈I

Moreover, equality holds in (A.5.13) if and only if the positive real numbers S 0 (xi ), i ∈ I, are all equal. When this holds, if β denotes their common value, we also have X β = S0( xi ). i∈I

Conversely, for all x ∈]inf µ H, +∞[, there exists a unique family (xi )i∈I ∈ (2) Q i∈I ]inf µi Hi , +∞[ such that x=

X

xi and S(x) =

i∈I

X

Si (xi ).

i∈I

Indeed, if β = S 0 (x), it is given by (xi )i∈I = (Ui (β))i∈I , where Ui = −Ψ0i . Proof. Let (xi )i∈I be an element of A.5.2, for every i ∈ I,

Q

i∈I ]inf µi Hi , +∞[.

According to Corollary

S(xi ) = inf (βxi + Ψi (β)). β>0

(A.5.14)

∗ , namely S 0 (xi ). Moreover, the infimum is attained for a unique β in R+ P Similarly, for x := i∈I xi ,

S(x) = inf (βx + Ψ(β)), β>0

(A.5.15)

and the infimum is attained for a unique positive β, namely S 0 (x). Furthermore, the additivity relation (A.5.12) shows that for every β in R∗+ , βx + Ψ(β) =

X

(βxi + Ψi (β)) .

i∈I

Part (1) of the proposition follows directly from P these observations. Part (2) follows from part (1) and from the relation Ψ0 = i∈I Ψ0i . 

324

Appendix A. Large Deviations and Cram´er’s Theorem

A.5.3 An example: Gaussian integrals and Maxwell velocity distribution. In this paragraph, we discuss a simple but significant instance of the formalism summarized in paragraph A.5.1 and Theorem A.5.1. This example may be seen as a mathematical counterpart of Maxwell’s statistical approach to the theory of ideal gases. It is also included for comparison with the application in Section 3.4 of the above formalism to Euclidean lattices — the present example appears as a “classical limit” of the discussion of Section 3.4. Let V be a finite-dimensional real vector space equipped with a Euclidean norm k.k. We shall denote by λ the Lebesgue measure on V attached to this Euclidean norm. It may be defined as the unique translation-invariant Radon measure on V such that Z 2 e−πkxk dλ(x) = 1. V

(See 2.1.1 and equation (2.1.1).) We may apply the formalism of this appendix to the measure space (V, B, λ), defined by V equipped with the Borel σ-algebra B and with the Lebesgue measure λ, and to the function H := (1/2m)k.k2 , where m denotes a positive real number. ∗ , we have Then for every β in R+ Z 2 e−βkpk /2m dλ(p) = (2πm/β)dim V /2 . V

Therefore, Ψ(β) = (dim V /2) log(2πm/β)

(A.5.16)

U (β) = −Ψ0 (β) = dim V /(2β).

(A.5.17)

and Equations (A.5.9) and (A.5.10), which relates the “energy” x and the “inverse ∗ ∗ and β ∈ R+ temperature” β, take the following form, for all x ∈ ]inf λ H, +∞[= R+ : βx = dim V /2.

(A.5.18)

The function S(x) may be computed directly from its definition. Indeed, for all x ∈ R∗+ and every positive integer n, we have λ⊗n ({(e1 , . . . , en ) ∈ V n |
(1/2m)(ke1 k2 + . . . + ken k2 ) ≤ nx} = vn dim V (2mnx)n(dim V )/2 . (A.5.19) Here vn dim V denotes the volume of the unit ball in the Euclidean space of dimension n dim V . It is given by vn dim V =

π n(dim V )/2 . Γ(1 + n(dim V )/2)

(A.5.20)

A.5. Reformulation and Complements

325

From (A.5.19) and (A.5.20), by a simple application of Stirling’s formula, we get h i 1 log vn dim V (2mnx)n(dim V )/2 = (dim V /2)[1+log(4πmx/ dim V )]. n→+∞ n

S(x) = lim

In particular, S 0 (x) = dim V /(2x), and we recover (A.5.18). Conversely, combined with the expression (A.5.16) for the function Ψ, part (3) of Theorem A.5.1 allows one to recover the asymptotic behavior of the volume vn of the n-dimensional unit ball, in the form p vn1/n ∼ 2eπ/n as n → +∞. Finally, observe that when m = (2π)−1 — the case relevant for the comparison with the application to Euclidean lattices in Section 3.4 — the expressions for Ψ and S take the following simpler forms: Ψ(β) = (dim V /2) log(1/β) and S(x) = (dim V /2)[1 + log(2x/ dim V )]. A.5.4 Relations to probability measures of maximal entropy. In this paragraph, we keep the notation introduced in paragraph A.5.1, and we assume that the hypotheses of Theorem A.5.1 are satisfied — namely, we assume that the growth condition SE holds and that µ(E) = +∞. Let C be the space of probability measures on (E, T ) absolutely continuous with respect to µ. By sending such a measure ν to its Radon–Nikodym derivative f = dν/dµ, one establishes a bijection from C to the convex subset  R f ∈ L1 (E, µ) | f ≥ 0 µ-a.e. and E f dµ = 1 of L1 (E, µ). To every measure ν = f µ in C, we may attach its “energy” Z Z ε(ν) := H dν = Hf dµ ∈ [0, +∞]. E

E

Lemma A.5.4. Let f : E −→ R+ and g : E −→ R∗+ be two T -measurable functions. (1) For every x ∈ E, f (x) log

f (x) ≥ f (x) − g(x). g(x)

Moreover, equality holds in (A.5.21) if and only if f (x) = g(x).

(A.5.21)

326

Appendix A. Large Deviations and Cram´er’s Theorem

(2) If g is µ-integrable, then the negative part (f log(f /g))− of f log(f /g) is µintegrable, and therefore Z f log(f /g) dµ E

is well defined in ] − ∞, +∞]. (3) If f and g belong to C, then Z f log(f /g) dµ E

belongs to [0, +∞] and vanishes if and only if f = g µ-almost everywhere. Proof. For all t ∈ R+ , we have t log t ≥ t − 1, and equality holds if and only if t = 1. Applied to t = f (x)/g(x), this implies (1).  From (1), assertions (2) and (3) follow immediately. Proposition A.5.5. Let ν = f µ be an element of C. (1) If ε(ν) < +∞, then (f log f )− is µ-integrable, and therefore the “informationtheoretic entropy” of ν with respect to µ, Z Z I(ν | µ) := − log(dν/dµ) dν = − f log f dµ, E

E

is well defined in [−∞, +∞[. (2) Let u and β be two positive real numbers such that u = U (β). If ε(ν) = u, then I(ν | µ) ≤ S(u).

(A.5.22)

Moreover, equality is achieved in (A.5.22) for a unique measure ν of C in ε−1 (u), namely for the measure νβ := Z(β)−1 e−βH µ, where

Z Z(β) :=

e−βH dµ.

E

In substance, the content of Proposition A.5.5 goes back to the seminal work of Boltzmann and Gibbs on statistical mechanics8 . Similar results play a central role in information theory and in statistics (see, for instance, [74], in particular Chapter 3). We refer the reader to [48] (in particular Section 3.4) for additional information and references. 8 See, for instance, Boltzmann’s memoirs [9] and [10, Chapter V]. Our presentation is a straightforward generalization, in the framework of general measure theory, of the “axiomatic” approach of Gibbs in [49, Chapter XI]. Lemma A.5.4 above appears in particular in [49, p. 130].

A.5. Reformulation and Complements

327

Proof. Let ν = f µ be an element of C such that u := ε(ν) is finite, and let β := U −1 (u) = S(u). We may consider the measurable function gβ := Z(β)−1 e−βH . It is everywhere positive on E, and satisfies Z gβ dµ = 1

(A.5.23)

E

and log gβ = − log Z(β) − βH = −Ψ(β) − βH. In particular, we have f log f = f log(f /gβ ) + f log gβ = f log(f /gβ ) − Ψ(β)f − βHf. Also recall that

(A.5.24)

Z f dµ = 1 E

and

Z Hf dµ = u = U (β) < +∞. E

Consequently, using (A.5.11), we get Z (Ψ(β)f + βHf ) dµ = Ψ(β) + βu = S(u).

(A.5.25)

E

According to Lemma A.5.4, (2), the negative part (f log(f /gβ ))− of f log(f /gβ ) is µ-integrable. Together with (A.5.24), this shows that the negative part (f log f )− of f log f is µ-integrable. This establishes (1). R Moreover, according to Lemma A.5.4, (3), the integral E f log(f /gβ ) dµ is nonnegative and vanishes if and only if f = gβ µ-almost everywhere. Together  with (A.5.24) and (A.5.25), this establishes (2).

Appendix B

Non-complete Discrete Valuation Rings and Continuity of Linear Forms on Prodiscrete Modules This appendix is devoted to a discussion of the results of “automatic continuity” of Specker [107] and Enochs [44], in a setting adapted to their application to the categories CTCA in Section 4.2.

B.1

Preliminary: Maximal Ideals, Discrete Valuation Rings, and Completions

Let R be a ring, and let m be a maximal ideal of R such that the local ring R(m) := (R \ m)−1 R is a discrete valuation ring. Its maximal ideal is m(m) := (R \ m)−1 m. We shall denote by ι : R −→ R(m) the canonical morphism of rings that maps an element x of R to ι(x) := x/1. We may consider the m-adic completion of R at m: ˆ m := lim R/mn . R ← − n ˆ := lim m/mn . It is a local ring, with maximal ideal m ←−n Since m is a maximal ideal, the localization R(m) may be identified with ˆ m , so that the canonical morphism from R to R ˆ m becomes the a subring of R ι ˆ composition R −→ R(m) ,−→Rm . Moreover, the m-adic (resp., m(m) -adic, resp., ˆ m ) are strictly compatible. ˆ m-adic) filtrations on R (resp., R(m) , resp., R ˆ In particular, Rm may be identified with the m(m) -adic completion of R(m) , and therefore is a complete discrete valuation ring. ˆ m and by |.| = e−v We shall denote by v the m-adic1 valuation on R(m) and R the associated absolute value. For simplicity, we shall also denote by v and |.| their ˆm. composition with the morphism ι : R −→ R 1 Or,

ˆ more correctly, m(m) -adic or m-adic.

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

329

330

Appendix B. Continuity of Linear Forms on Prodiscrete Modules Let us consider ˆ N | lim |λi | = 0}. L = {(λi )i∈N ∈ R m i→+∞

For all λ = (λi )i∈N in L, we may define an R-linear map ˆm Σλ : RN −→ R by letting Σλ ((xi )i∈N ) :=

X

λi ι(xi ).

i∈N

Proposition B.1.1. Let λ := (λi )i∈N be an element of L such that I := {i ∈ N | λi 6= 0} is infinite. If we define n := min v(λi ), i∈N

then we have ˆ n. Σλ (RN ) = m

(B.1.1) ∼

Proof of Proposition B.1.1. Let us choose a bijection ψ : N −→ I. The integer n and the image of Σλ are clearly unchanged if we replace the sequence λ := (λi )i∈N by (λψ(i) )i∈N . Therefore, to establish Proposition B.1.1, we may assume that λ belongs to (R \ {0})N and that the sequence of nonnegative integers ni := v(λi ) (i ∈ N) is such that n := mini∈N ni = n0 . ˆ ni , and a For all (xi )i∈N in RN and i in N, the product λi ι(xi ) belongs to m n N n ˆ . ˆ . This implies that Σλ (R ) is contained in m fortiori to m ˆ n . Observe that for all k ∈ N, λk ι(R) is Conversely, let α be an element of m ˆm = m ˆ nk . Therefore, we may inductively construct a sequence (xi )i∈N dense in λk R such that for all k ∈ N, v(α −

k X

λi ι(xi )) ≥ nk+1 .

i=0

Then α=

X

λi ι(xi ) = Σλ ((xi )i∈N ).



i∈N

Observe that for every natural integer k, by applying Proposition B.1.1 to the sequence (λi+k )i∈N , which still belongs to L, we obtain the following corollary: Corollary B.1.2. Under the assumption of Proposition B.1.1, the map Σλ is continuous and open from RN , equipped with the product topology of the discrete topology ˆ n equipped with the m-adic ˆ on each of the factors R, onto m topology. Indeed, for all k ∈ N, if we define nk := mini∈N≥k v(λi ), then ˆ nk , Σλ ({0}k ⊕ RN≥k ) = m while limk→+∞ nk = +∞.



B.2. Continuity of Linear Forms on Prodiscrete Modules

331

B.2 Continuity of Linear Forms on Prodiscrete Modules We keep the notation of the previous section, and we consider a topological module M over the ring R equipped with the discrete topology. The topological R-module M is complete and prodiscrete, with a countable basis of neighborhoods of 0, if and only if it isomorphic (as a topological module) to the projective limit limk Mk of a projective system ←− q0

q1

q2

M0 ←− M1 ←− M2 ←− · · · of discrete R-modules. (The maps qi may actually be assumed surjective.) Then if we denote by pk : M −→ Mk the canonical projection maps, a sequence (mi )i∈N ∈ M N converges to zero in M if and only if for every k ∈ N, the projection pk (mi ) vanishes for i large enough (depending on k). N Moreover, for every such sequence P(mi )i∈N in M that converges to zero and for every sequence (ri )i∈N , the series i∈N ri mi converges in M . Proposition B.2.1. Let M be a complete prodiscrete topological R-module, with a ˆ m be an R-linear map. countable basis of neighborhoods of 0, and let ϕ : M −→ R If ϕ is not continuous when M is equipped with its prodiscrete topology and ˆ m with the discrete topology, then there exists k ∈ N such that ϕ(M ) = m ˆ k. R Proof. Let us denote by π an element of m \ m2 . (The set m \ m2 is indeed not 2 empty, since m(m) 6= m(m) and therefore m 6= m2 .) When ϕ is not continuous, there exists a sequence (mi )i∈N in M N that converges to 0 in M and such that (ϕ(mi ))i∈N has an infinity of nonzero terms. Then the family λ = (λi )i∈N := (ι(π)i ϕ(mi ))i∈N satisfies the assumptions of Proposition B.1.1, and therefore, for some nonnegative ˆ n. integer n, Σλ (RN ) = m We are going to prove that every element of Σλ (RN ) belongs to the image of ˆm, ˆ n . Since ϕ(M ) is an R-submodule of R ϕ. This will show that ϕ(M ) contains m this will complete the proof. To achieve this, let (xi )i∈N be a sequence in RN . We may form the following sum, convergent in M : X m := π i xi mi . i∈N

For every nonnegative integer k, we have X π i xi mi + π k+1 rk , m= 0≤i≤k

(B.2.1)

332

Appendix B. Continuity of Linear Forms on Prodiscrete Modules

where rk is defined as the following convergent sum in M : X π i xi+k+1 mi+k+1 . rk := i∈N

By applying ϕ to the relation (B.2.1), we see that for every k ∈ N, X ˆ k+1 . ι(π i xi )ϕ(mi ) ∈ m ϕ(m) − 0≤i≤k

Finally, Σλ ((xi )i∈N ) =

X

ι(π)i ϕ(mi )ι(xi ) = ϕ(m).



i∈N

Observe that when R is a domain, the morphism ι : R −→ R(m) is injective, ˆ k for some k ∈ N if and only if R is m-adically and its image ι(R) contains m complete. Therefore, from Proposition B.2.1, we immediately derive the following corollary: Corollary B.2.2. Let M be a complete prodiscrete topological R-module, with a countable basis of neighborhoods of 0. If R is a domain and is not m-adically complete, then every R-linear map ϕ : M −→ R is continuous when M is equipped  with its prodiscrete topology and R with the discrete topology.

Appendix C

Measures on Countable Sets and Their Projective Limits This appendix is devoted to various results concerning measure theory on Polish spaces defined as projective limits of countable systems of countable discrete sets that are used in the proofs of Chapter 7.

C.1

Finite Measures on Countable Sets

Let D be a (possibly finite) countable set. The real vector space Mb (D) of real bounded measures on D (equipped with the σ-algebra P(D) of all subsets of D) may be identified with the vector space l1 (D) of absolutely summable elements of RD : ∼ Mb (D) −→ l1 (D), (C.1.1) µ 7−→ (µ({x})x∈D . This isomorphism maps the cone Mb+ (D) of positive bounded measures onto 1 l+ (D) = l1 (D) ∩ RD +.

Moreover, the total mass of a measure µ ∈ Mb (D) coincides with the l1 -norm of its image by the isomorphism (C.1.1): kµk =

X

|µ({x})|.

x∈D

On the real vector space Mb (D), or equivalently on l1 (D), we may consider the following separated locally convex topologies: (i) the topology of vague convergence of measures in Mb (D), or equivalently the topology on l1 (D) induced by the topology of pointwise convergence on RD , or the σ(l1 (D), R(D) )-topology on l1 (D); (ii) the topology of narrow convergence of measures in Mb (D), that is, the σ(l1 (D), l∞ (D))-topology on l1 (D); (iii) the topology defined by the “total mass norm” on Mb (D) (which coincides with the l1 -norm on l1 (D)). © Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

333

334

Appendix C. Measures on Countable Sets and Their Projective Limits

The first of these topologies is strictly finer than the second, and the second one strictly finer than the third. The following proposition compares the induced topologies on the cone Mb+ (D), and in particular asserts that the topology of narrow convergence and the topology of norm convergence on Mb+ (D) coincide: Proposition C.1.1. Let (µi ) be a sequence, or more generally a net, of elements of Mb+ (D) that converges vaguely to an element µ of Mb+ (D). Then the following conditions are equivalent: (1) (µi ) converges to µ in the topology of narrow convergence on Mb (D); (2) limi µi (D) = µ(D); (3) the total mass µi (D) remains bounded (for i large enough); moreover, for ∗ every ε ∈ R+ , there exists a finite subset F of D such that, for i large enough, µi (D \ F ) < ε; (4) limi kµi − µk = 0. This is well known, and the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (1) are indeed easily established. b (D) converges In this monograph, we shall say that a sequence (µi ) in M+ b to a measure µ ∈ M+ (D) when the above equivalent conditions are satisfied. The following convergence criterion plays a central role in our study of the θ-invariants of infinite-dimensional Hermitian vector bundles. Proposition C.1.2. Let (µi )i∈N be a sequence in Mb+ (D). If there exists a sequence (ti )i∈N in l1 (N) such that for every i ∈ N, µi+1 ≤ eti µi ,

(C.1.2)

then the sequence (µi )i∈N converges to some µ ∈ Mb+ (D). Proof. When the condition (C.1.2) is satisfied by ti = 0 for every i ∈ N — that is, when µi+1 ≤ µi for every i ∈ N — then to every subset X of D we may associate the limit µ(X) := lim µi (X) i→+∞

of the nonincreasing sequence (µi (X))i∈N in R+ . It is straightforward that this b (D) and that (µi )i∈N converges to µ. defines an element µ of M+ One reduces the general case of the proposition to this special case by letting, for every i ∈ N, P νi := e− 0≤j 0 and j ∈ N≥i(ε) , kγj − 1pj (Fε ) γj k < ε.

(C.2.10)

Moreover, for all x in the finite set Fε , lim γj (pj {x}) − µ({x}) = lim γj (pj {x}) − γ(x) = 0.

j→+∞

j→+∞

Therefore, lim k1pj (Fε ) γj − pj∗ (1Fε µ)k = 0.

(C.2.11)

k1Fε µ − µk = µ(C \ Fε ) ≤ ε.

(C.2.12)

j→+∞

Finally, For all (i, j) ∈ N2 such that i ≤ j, we have kpij∗ γj − pi∗ µk ≤ kpij∗ (γj − 1pj (Fε ) γj )k + kpij∗ (1pj (Fε ) γj − pj∗ (1Fε µ))k + kpi∗ (1Fε µ − µ)k ≤ kγj − 1pj (Fε ) γj k + k1pj (Fε ) γj − pj∗ (1Fε µ)k + k1Fε µ − µk. From (C.2.10–C.2.12), we deduce that as j goes to +∞, the upper limit of ∗ the last sum is ≤ 2ε. Since ε ∈ R+ is arbitrary, this shows that lim kpij∗ γj − pi∗ µk = 0.

j→+∞



Proof of Lemma C.2.4. We first choose a finite subset Fε of C so large that X (C.2.13) γ(x) ≤ ε/2. x∈C\Fε

Then, since Fε is finite, if the integer j is large enough — say j ≥ i(ε) — the map pj|Fε : Fε −→ Dj is injective and X |γj ({pj (x)}) − γ(x)| ≤ ε/4. (C.2.14) x∈Fε

340

Appendix C. Measures on Countable Sets and Their Projective Limits

According to (C.2.7), we may also assume that when j ≥ i(ε), X γj (Dj ) ≤ γ(x) + ε/8.

(C.2.15)

x∈C

Then the estimate (C.2.8) follows from (C.2.13). To establish (C.2.9), write γj (Dj \ pj (Fε )) = γj (Dj ) − γj (pj (Fε )) and observe that γj (pj (Fε )) ≥ =

X

γ(x) −

X

x∈Fε

x∈Fε

X

X

x∈C

γ(x) −

|γj ({pj (x)}) − γ(x)|

x∈C\Fε

γ(x) −

X

|γj ({pj (x)}) − γ(x)|.

x∈Fε

Together with the estimates (C.2.13–C.2.15), this shows that γj (Dj \ pj (Fε )) ≤ ε/8 + ε/2 + ε/4.



Appendix D

Exact Categories Exact categories play a key role in Quillen’s foundational work [92] on higher algebraic K-theory, which introduced this terminology. As shown in particular by Deligne and Keller, exact categories admit a formalism of derived categories that includes the formalism of derived categories of abelian categories, and is especially convenient for applications. This appendix gathers the basic definitions and summarizes, without proof, some basic properties of exact categories in the sense of Quillen, for the convenience of the reader of Section 8.5. We refer the reader to the expositions by Keller [70] and B¨ uhler [24] for the relevant proofs and for references. Let us indicate that the historical development of the formalism of exact categories has been rather intricate. We note in particular that the axioms defining exact categories in [92] may be significantly streamlined, as shown by the works of Yoneda and Keller (see [24, Section 2]). Moreover, an important forerunner of Quillen’s notion of exact category was the theory of “abelian categories” developed by Heller [63]. These categories are precisely the exact categories, in Quillen’s sense, whose underlying additive categories are idempotent complete (cf. [24, Appendix B]), and Heller had already shown that these idempotent complete exact categories constituted a convenient framework for homological algebra.

D.1

Definitions and Basic Properties

D.1.1 Definitions. Let A be an additive category. A kernel–cokernel pair (i, p) in A is a pair of composable morphisms i

p

A0 −→ A −→ A00 in A such that i is a kernel of p and p is a cokernel of i. Let E be a class of kernel–cokernel pairs in A. An allowable monomorphism (with respect to E) is a morphism i in A for which there exists a morphism p in A such that (i, p) belongs to E. An allowable epimorphism (with respect to E) is a morphism p in A for which there exists a morphism i in A such that (i, p) belongs to E. An exact structure on A is a class E of kernel–cokernel pairs that is closed under isomorphisms and satisfies the following conditions: E0 : For every object A of A, the identity morphism 1A is an allowable monomorphism. © Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

341

Appendix D. Exact Categories

342

E0op : For every object A of A, the identity morphism 1A is an allowable epimorphism. E1 : The class of allowable monomorphisms is closed under composition. E1op : The class of allowable epimorphisms is closed under composition. E2 : The pushout of an allowable monomorphism along an arbitrary morphism exists and yields an allowable monomorphism. E2op : The pullback of an allowable epimorphism along an arbitrary morphism exists and yields an allowable epimorphism. An exact category is a pair (A, E) consisting of an additive category and an exact structure E in A. Elements (i, p) of E are often displayed as i

p

0 −→ A0 −→ A −→ A00 −→ 0 and called allowable short exact sequences of the exact category, or simply short exact sequences when no confusion can arise. D.1.2 Some properties of allowable monomorphisms and epimorphisms. In every exact category A as above, the following properties are satisfied (see [24, Section 2]): (i) Every split exact sequence in A — namely every diagram in A isomorphic to a diagram of the form (1

0 ,0)

pr

2 M M 00 −→ 0 M 0 ⊕ M 00 −→ 0 −→ M 0 −→

for objects M 0 and M 00 in A — is a short exact sequence. (ii) The direct sum of two short exact sequences is a short exact sequence. (iii) The pullback of an allowable monomorphism along an allowable epimorphism yields an allowable monomorphism. Dually, the pushout of an allowable epimorphism along an allowable monomorphism yields an allowable epimorphism. (iv) Suppose that i : A −→ B is a morphism in A admitting a cokernel. If there exists a morphism j : B −→ C in A such that ji : A −→ C is an allowable monomorphism, then i is an allowable monomorphism. Dually, let p : B 0 −→ A0 be a morphism in A admitting a kernel. If there exists a morphism q : C 0 −→ B 0 in A such that pq : C 0 −→ A0 is an allowable epimorphism, then p is an allowable epimorphism. In Quillen’s definition of exact categories [92, §2], property (iv) appears as (op) an extra axiom in addition to (a simple variant of) axioms E0−2 .

D.2. The Derived Category of an Exact Category

343

D.1.3 Allowable morphisms and exactness. In an exact category A, a morphism f : A −→ B is said to be an allowable morphism if it admits a factorization e

m

f = m e : A −→ I −→ B,

(D.1.1)

where e is an allowable epimorphism and m an allowable monomorphism. The factorization (D.1.1) is then unique, up to unique isomorphism. A diagram of allowable morphisms f0

f

A0 −→ A −→ A00 is called exact (at A) when the factorizations e

e0

m

f = m e : A0 −→ I −→ A

m0

and f 0 = m0 e0 : A −→ I 0 −→ A00

of f and f 0 as products of allowable monomorphisms and epimorphisms are such that the diagram e0

m

0 −→ I −→ A −→ I 0 −→ 0 is a short exact sequence.

D.2

The Derived Category of an Exact Category

D.2.1 Definitions and notation. Let us recall that an additive category A is said to be idempotent complete if it satisfies the following equivalent conditions (see, for instance, [24, 6.1–2]): IC : For every object A of A and every endomorphism p : A −→ A that is idempotent — that is, that satisfies p2 = p — there is decomposition ∼

ϕ : A −→ K ⊕ I of A in A such that ϕpϕ

−1

 0 = 0

 0 . IdI

IC0 : Every idempotent endomorphism in A has a kernel. For every additive category A, we shall denote the additive category of complexes and chain maps in A by Ch(A), and the homotopy category of A by K(A). The category K(A) has the same objects as Ch(A), and its morphisms are the chain maps modulo the chain maps homotopic to zero. It is equipped with the natural structure of a triangulated category, with suspension functor the “shift” functor ·[1], and with exact triangles the “strict triangles” associated to the mapping cones of chain maps in Ch(A), up to isomorphisms in K(A).

Appendix D. Exact Categories

344

D.2.2 The derived category D(A). We finally review the definition and some basic properties of the derived category of an exact category. We refer the reader to [70] and [24, Section 10] for details of this construction. Let A be an exact category.

A complex (A• , d• ) :

dn−1

dn

· · · −→ An−1 −→ An −→ An+1 −→ · · ·

(D.2.1)

in Ch(A) is called acyclic, or exact, if the morphisms (dn )n∈Z are admissible and if for every n ∈ Z, the diagram dn−1

dn

An−1 −→ An −→ An+1 is exact (at An ). Equivalently, the complex (D.2.1) is acyclic if it may be obtained by “splicing together” a sequence of short exact sequences in A: 0 −→ Z n A −→ An −→ Z n+1 A −→ 0,

n ∈ Z.

The mapping cone of a chain map between acyclic complexes is acyclic, and the class Ac(A) of acyclic complexes in A is a triangulated subcategory of K(A). A chain map in Ch(A) is called a quasi-isomorphism if its mapping cone is homotopy equivalent to an acyclic complex.

The derived category D(A) of the exact category A is defined as the Verdier quotient D(A) := K(A)/Ac(A). If ∗ is an element of {+, −, b}, one may also define the full subcategory D∗ (A) of D(A) formed by the complexes that are acyclic in degree n for all n  0, resp., all n  0, resp., all n  0 and all n  0. It may be identified with the Verdier quotient K∗ (A)/Ac∗ (A), where K∗ (A) (resp., Ac∗ (A)) is the full subcategory of K(A) (resp., Ac(A)) defined by complexes satisfying the boundedness condition ∗.

When the exact category A is idempotent complete, the following properties hold: (i) a retract in K(A) of an acyclic complex is acyclic;

D.2. The Derived Category of an Exact Category

345

(ii) the class of acyclic complexes is closed under isomorphisms in K(A); in particular, every null homotopic complex in Ch(A) is acyclic; (iii) a chain map in Ch(A) becomes an isomorphism in D(A) if and only if it is a quasi-isomorphism, and if and only if its mapping cone is acyclic.

Appendix E

Upper Bounds on the Dimension of Spaces of Holomorphic Sections of Line Bundles over Compact Complex Manifolds E.1

Spaces of Sections of Analytic Line Bundles and Multiplicity Bounds

In this appendix, we present two proofs of Proposition 10.2.3, which we rephrase as follows: Proposition E.1.1. For every analytic line bundle L over a compact complex manifold M of complex dimension n, there exists C in R∗+ such that for every positive integer D, the dimension of the vector space Γ(M, L⊗D ) of analytic sections of L⊗D satisfies dimC Γ(M, L⊗D ) ≤ C · Dn . The first proof follows the arguments of Serre [100] and Siegel [104], and relies on the use of Schwarz’s lemma, after the introduction of suitable coordinate charts on M . The second proof will assume that the manifold M is K¨ahler and relies on some simple computations of intersection numbers. To establish Proposition E.1.1, we may clearly — and we shall — assume that M is connected. For all R ∈ R+ , we define Bn (R) := {(z1 , . . . , zn ) ∈ Cn | |z1 |2 + · · · + |zn |2 < R2 }. By compactness, there exists a finite family ∼

ϕα : Uα −→ Bn (1), 1 ≤ α ≤ A of analytic charts on M — the Uα are open subsets of M, and the ϕα analytic diffeomorphisms — such that [ X= Uα . 1≤α≤A

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

347

348

Appendix E. Holomorphic Sections of Line Bundles

For every α in {1, . . . , A}, we may consider the “center of the chart ϕα ”: Pα := ϕ−1 α (0). Proposition E.1.1 will be a straightforward consequence of the following result, of independent interest: Proposition E.1.2. With the above notation, for every holomorphic line bundle L over M , there exists c in R+ such that for every nonnegative integer D and every holomorphic section s of L⊗D over M , if the vanishing multiplicity of s at Pα satisfies multPα s > c · D for every α in {1, . . . , A}, then s vanishes everywhere on M. When M is K¨ahler, the following stronger variant of Proposition E.1.2 holds: Proposition E.1.3. Let us assume that the compact complex manifold M is connected and K¨ ahler. For every point P of M and every holomorphic line bundle L over M , there exists c in R+ such that for every nonnegative integer D and every holomorphic section s of L⊗D over M , if multP s > c · D, then s vanishes everywhere on M. Proof of Proposition E.1.1 from Proposition E.1.2. Again, we may introduce charts (ϕα )1≤α≤A and their centers (Pα )1≤α≤A as above. For all i ∈ N, let us consider the infinitesimal neighborhood MPα ,i of order i of Pα in M and the space Γ(MPα ,i , L⊗D ) of sections of L⊗D over MPα ,i — that is, the space of jets of order i of sections of L⊗D at Pα . Proposition E.1.2 precisely asserts the existence of c ∗ such that the natural evaluation map in R+ i : Γ(M, L⊗D ) −→ ηD

M

Γ(MPα ,i , L⊗D )

1≤α≤A

is injective if i ≥ bcDc. Consequently, if we let i(D) := bcDc, we get dimC Γ(M, L

⊗D

)≤

X 1≤α≤A

Since 

⊗D

dimC Γ(MPα ,i(D) , L



 n + i(D) )=A . n

 cn n + i(D) 1 ∼ i(D)n ∼ Dn , n! n! n

as D goes to +∞, this establishes the announced upper bound.



E.2. Proof of Proposition E.1.2

E.2

349

Proof of Proposition E.1.2

We will rely on the following higher-dimensional version of the well-known Schwarz lemma, combined with compactness arguments: Lemma E.2.1. Let R be a positive real number and i a nonnegative integer. For every holomorphic function f on Bn (R) such that mult0 f ≥ i and for all z in Bn (R), we have  |f (z)| ≤

kzk R

i |f (w)|.

sup w∈Bn (R)

Proof. Let f be such a holomorphic function satisfying mult0 f ≥ i, and let z be a point in Bn (R). When z = 0, the inequality is immediate. Let us therefore assume z nonzero, and let us introduce the function gz of one complex variable t defined by gz (t) := t−i f (t · z/kzk). It is a priori defined and analytic on the punctured disk D(0, R)\{0}. According to the assumption on mult0 f, it actually extends to an analytic function on D(0, R). Consequently, using the maximum modulus principle and the definition of gz , we obtain for every t in D(0, R), |gz (t)| ≤ lim sup |gz (w)| ≤ R−i

sup

|f (w)|.

w∈Bn (R)

|w|→R

The required upper bound on |f (z)| follows from this inequality applied to t = kzk.  Let us choose a continuous Hermitian metric k.kL on L, and for every α in {1, . . . , A}, a nonvanishing holomorphic section sα of L over Uα .1 1 The existence of such a section is equivalent to the triviality of the holomorphic line bundle L over Uα . This follows from the isomorphism

ϕ∗α : H 1 (Uα , O∗ ) ' H 1 (Bn , O∗ ) and from the vanishing of H 1 (Bn , O∗ ), itself a straightforward consequence of the vanishing of H 1 (Bn , O) and H 2 (Bn , Z), thanks to the short exact sequence 0 −→ Z −→ O

exp(2πi.)

−→

O∗ −→ 0

of sheaves on M and the associated long exact sequence of cohomology groups. Observe also that for a given line bundle L, it is straightforward that such sα exist if the domains Uα of the charts ϕα are small enough. This observation makes particularly elementary the proof of the vanishing statement in Proposition E.1.2 for one specific line bundle L and for a suitable choice of the Pα . This weaker version is sufficient to derive Corollary 10.2.3 below and Chow’s theorem.

350

Appendix E. Holomorphic Sections of Line Bundles By compactness of M again, there exists r in ]0, 1[ such that [ M= Uα (r), 1≤α≤A

n where Uα (r) := ϕ−1 α (B (r)). Finally, let us choose a real number r0 in the interval ]r, 1[. Let D be a nonnegative integer, and s a holomorphic section of L⊗D over L. We shall denote by k.kL⊗D the continuous Hermitian metric on L⊗D defined as the Dth tensor power of the metric k.kL on L. For all α in {1, . . . , A}, we may write

s|Uα = fα ◦ ϕα · s⊗D α , where fα denotes a holomorphic function on Bn (1). The following estimates are straightforward: for all P in Uα (r), !D ks(P )kL⊗D ≤

sup Q∈Uα (r)

ksα (Q)kL

|fα (ϕα (P ))|,

(E.2.1)

and for all w in Bn (r0 ), −D

 |fα (w)| ≤

inf Q∈Uα

(r 0 )

ksα (Q)kL

−1

sα (ϕα (w)) L⊗D .

(E.2.2)

Moreover, if mult0 fα — or equivalently multPα s — is at least i, the Schwarz lemma, Lemma E.2.1, shows that |fα (ϕα (P ))| ≤

 r i r0

sup

|fα (w)|.

w∈Bn (r 0 )

Combining this inequality with (E.2.1) and (E.2.2), we obtain that if multPα s ≥ i, then D  r i  sup Q∈Uα (r) ksα (Q)kL sup ks(Q)kL⊗D . sup ks(P )kL⊗D ≤ 0 r inf Q∈Uα (r0 ) ksα (Q)kL Q∈Uα (r 0 ) P ∈Uα (r) This shows that if multPα s ≥ i for every α in {1, . . . , A}, then max ks(P )kL⊗D ≤ λi M D max ks(P )kL⊗D ,

P ∈M

P ∈M

where λ := r/r0 and M := max

1≤α≤A

supQ∈Uα (r) ksα (Q)kL . inf Q∈Uα (r0 ) ksα (Q)kL

(E.2.3)

E.3. Proof of Proposition E.1.3

351

By definition, λ belongs to ]0, 1[, and M to [1, +∞[, and c :=

log M log λ−1

is a well-defined nonnegative real number. The upper bound (E.2.3) implies the vanishing of s when λi M D < 1, that is, when i > c.D.

E.3

Proof of Proposition E.1.3

Let M be a compact connected complex manifold of dimension n, equipped with a K¨ ahler form ω. Let P be a point in M and let ˜ −→ M ν:M denote the blow up of P in M , and E := ν −1 (P ) its exceptional divisor. Lemma E.3.1. There exist a C ∞ Hermitian metric k.k on the line bundle O(−E) ˜ and a positive real number λ such the first Chern form over M α := c1 (O(−E), k.k) satisfies α + λ ν ∗ ω ≥ 0.

(E.3.1)

Proof. Let us consider the blow up fn −→ An ν0 : A C C at the origin 0 of the n-dimensional complex affine space AnC . fn may be identified with the smooth subscheme of An × Pn−1 The variety A C C C defined by fn ⇐⇒ for every 1 ≤ i < j ≤ n, zi wi = 0. ((z1 , . . . , zn ), (w1 : · · · : wn )) ∈ A C zj wj The map ν0 coincides with the restriction of the first projection pr1 : AnC ×Pn−1 −→ C Pn−1 , and the exceptional divisor E := ν0−1 (0) with {0} × Pn−1 . Moreover, the C C line bundle OAfn (−E) is canonically isomorphic to the pullback pr∗ fn OPn−1 (1) of 2|AC

C

C

the tautological line bundle OPn−1 (1) by the restriction of the second projection C

pr2 : AnC × Pn−1 −→ AnC . (The inverse isomorphism maps the section pr∗ C

n f 2|A C

Xi of

352

Appendix E. Holomorphic Sections of Line Bundles

pr∗

to the pullback under pr1|Afn of the ith coordinate on AnC , for every

OPn−1 (1) n f 2|A C C

C

i ∈ {1, . . . , n}.) Consequently, like OPn−1 (1) , the line bundle OAfn (−E) may be endowed with C C fn . a C ∞ Hermitian metric k.k0 such that c1 (O fn (−E), k.k0 ) ≥ 0 over A C

AC

Let finally k.k1 be a C ∞ metric on OAfn (−E) that coincides with k.k0 over C

n \ Bn (3/4)). ν0−1 (Bn (1/2)) and satisfies k1OAgn (−E) k1 = 1 over ν0−1 (AC C

∼

Let us choose a C-analytic chart ϕ : U −→ Bn (1) on M such that ϕ−1 (0) = P. The chart ϕ induces an analytic isomorphism between blow ups: ∼

ϕ˜ : ν −1 (U ) −→ ν0−1 (Bn (1)). By means of ϕ, ˜ we may transport the metric k.k1 on OAfn (−E) over ν0−1 (Bn (1)) C

to a metric on OM˜ (−E) over ν −1 (U ). The metric k.k satisfies k1OM˜ (−E) k = 1

(E.3.2)

over ν −1 (U \ ϕ−1 (Bn (3/4))), and therefore may be extended to a metric (still ˜ by requiring (E.3.2) to hold over M ˜ \ ν −1 (ϕ−1 (Bn (3/4))). denoted by k.k) over M ˜ \ν −1 (K), By construction, the first Chern form c1 (OM˜ (−E), k.k) is ≥ 0 on M where K := ϕ−1 (Bn (3/4) \ Bn (1/2)). Since ν is an isomorphism over M \ {P } — which contains the compact K — and ω is everywhere > 0, the condition (E.3.1) holds for every λ large enough.  Let us consider the cohomology class [ω] of ω. For every line bundle L over M , we may consider its first Chern class c1 (L) in the real cohomology group H 2 (M, R) and form the intersection number c1 (L) · [ω]n−1 ∈ H 2n (M, R) ' R. We may now establish Proposition E.1.3 in the following more precise form: Lemma E.3.2. Let λ be as in Lemma E.3.1. For every analytic line bundle L over M , every positive integer D, and every analytic section s of L⊗D over M that does not vanish identically, the order of vanishing multP s of s at P satisfies the following upper bound: multP s ≤ λd−1 (c1 (L).[ω]d−1 ) D.

(E.3.3)

Proof. Consider a section s in Γ(M, L⊗D ) \ {0} and its divisor div s on M . It is effective, and multP is the multiplicity of the exceptional divisor E in its inverse ˜. image ν ∗ div s on M ˜ The divisor on M Z := ν ∗ div s − multP s · E

E.3. Proof of Proposition E.1.3 is therefore effective. Together with (E.3.1), this shows that Z (α + λν ∗ ω)d−1 δZ ≥ 0.

353

(E.3.4)

˜ M

˜ , R) of the divisors E If [E] and [Z] denote the cohomology classes in H 2 (M and Z, the following equality of cohomology classes holds: [α + λν ∗ ω] = −[E] + λν ∗ [ω] and [δZ ] = ν ∗ c1 (L) − multP s · [E], and the integral in (E.3.4) may be written as an intersection number: Z (α + λν ∗ ω)d−1 δZ = (−[E] + λν ∗ [ω])d−1 .(D ν ∗ c1 (L) − multP s · [E]). (E.3.5) ˜ M

Since ν is a birational morphism, and ν∗ [E]i = 0 if 0 < i < d and (−1)d−1 [E]d = c1 (O(−E))d−1 · [E] = 1, the intersection number on the right-hand side of (E.3.5) is also equal to λd−1 (c1 (L) · [ω]d−1 ) D − multP s. Together with (E.3.4), this establishes the upper bound (E.3.3).



ahler geometry, of The previous proof is a counterpart, in the framework of K¨ the derivation of the basic properties of the Seshadri constants of nef line bundles on algebraic varieties (see [77, Chapter 5, in particular Proposition 5.1.9] and [16, Lemma 2.3]).

Appendix F

John Ellipsoids and Finite-Dimensional Normed Spaces F.1

John Ellipsoids and John Euclidean Norms

Let E be a finite-dimensional real vector space equipped with a norm k.k, and let B := {x ∈ E | kxk ≤ 1} be its closed unit ball. A simple compactness argument shows that among the ellipsoids with center 0 in E contained in B, there is one of maximal volume. As shown by John [68], this ellipsoid J satisfies B ⊂ (dimR E)1/2 J. Moreover, this ellipsoid is unique1 . It is called the John ellipsoid of B. These results may be translated in terms of Euclidean norms as follows: Proposition F.1.1. Among the Euclidean norms |.| on E such that k.k ≤ |.|, there exists a unique one — the John Euclidean norm k.kJ attached to k.k — for which the associated Euclidean norm on ∧dim E E is minimal. The norm k.kJ satisfies k.k ≤ k.kJ ≤ (dimR E)1/2 k.k.

F.2

Properties of the John Norm

Let us indicate a few properties of the John norm that are direct consequences of Proposition F.1.1. F.2.1 Invariance under automorphisms. Let K := Isom(E, k.k) be the set of elements of GLR (E) that preserve the norm k.k. It is a compact subgroup of GLR (E). The Euclidean norm k.kJ is fixed under the action of K. This follows from the uniqueness assertion in Proposition F.1.1 and from the fact that the action of the compact group K on ∧dim E E factors through {1, −1} ,→ R∗ = GLR (∧dim E E). 1 This uniqueness result appears to have been observed independently by Loewner (in a dual form; see [25, pp. 159–160], Danzer, Laugwitz, and Lenz [37], and Zaguskin [120]).

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

355

356

Appendix F. John Ellipsoids and Finite-Dimensional Normed Spaces

F.2.2 The John norm attached to a normed complex vector spaces. Assume that E is a complex vector space and that k.k is a norm on this complex vector space. Then the John norm k.kJ associated to (E, k.k) seen as a normed real vector space is a Hermitian norm on E. Indeed, in this situation, the group Isom(E, k.k) contains the operation [i] of multiplication by i on E, and therefore k.kJ is invariant under the action of [i], hence a Hermitian norm on E. F.2.3 Compatibility with finite products. Let (E1 , k.k1 ), . . . , (EN , k.kN ) be a finite family of finite-dimensional normed vector spaces (over R or over C). Let us consider the direct sum E := E1 ⊕ · · · ⊕ EN and the norm k.k on E defined by: k(e1 , . . . , en )k := max kei ki 1≤i≤n

for every (e1 , . . . , en ) ∈ E1 ⊕ · · · ⊕ EN .

Then the John norms k.kJ and k.k1,J , . . . , k.kN,J associated to k.k and k.k1 , . . . , k.kN satisfy the relation X kei k2i,J . (F.2.1) k(e1 , . . . , en )k2J := 1≤i≤n

Indeed, the group Isom(E, k.k) contains the group {1, −1}N acting diagonally on E1 ⊕ · · · ⊕ EN . Therefore the (Euclidean or Hermitian) norm kk is invariant under the action of this subgroup, and therefore may be written 1/2

 k(e1 , . . . , en )kJ := 

X

kei k2J 

.

1≤i≤n

The expression (F.2.1) follows readily from this observation.

F.3 Application to Lattices in Normed Real Vector Spaces F.3.1 Let Γ be a free Z-module of finite rank, and let k.k be a norm on the finite-dimensional real vector space ΓR := Γ ⊗Z R. Let covolk.k Γ denote the covolume of Γ in ΓR with respect to the Lebesgue measure2 on ΓR that gives the volume 1 to the unit ball B := {x ∈ ΓR | kxk ≤ 1}. 2A

Lebesgue measure on ΓR is a nonzero translation-invariant Radon measure on ΓR .

F.3. Application to Lattices in Normed Real Vector Spaces

357

In other words, for every Lebesgue measure λ on ΓR and every Borel subset ∆ of ΓR that is a fundamental domain for the action of Γ, we have covolk.k Γ =

λ(∆) . λ(B)

(F.3.1)

To the pair (Γ, k.k) we may associate the real number χk.k (Γ) := − log covolk.k Γ. When the norm k.k is Euclidean, the pair (Γ, k.k) define a Euclidean lattice Γ, and the expression (F.3.1), applied to the Lebesgue measure λ := λΓ that gives the volume 1 to the unit cube in the Euclidean space (ΓR , k.k) (cf. Section 2.1.1), shows that −1 covolk.k Γ = vrk Γ . covol Γ, where vrk E denotes the volume of the unit ball in dimension rk E. Therefore, d Γ + log vrk Γ . χk.k (Γ) = deg

(F.3.2)

In general, we may consider the John Euclidean norm k.kJ on ΓR associated to k.k and form the Euclidean lattice ΓJ := (Γ, k.kJ ). If B and J denote the unit balls in ΓR associated to k.k and k.kJ respectively, we have J ⊂ B ⊂ (rk Γ)1/2 J. From these inclusions, we derive the following estimates, valid for every Lebesgue measure λ on ΓR : λ(J) ≤ λ(B) ≤ (rk Γ)rk Γ/2 λ(J). Consequently, we have χk.kJ (Γ) ≤ χk.k (Γ) ≤ (rk Γ/2) log rk Γ + χk.kJ (Γ). Using the relation (F.3.2), these estimates may also be written d ΓJ + log vrk Γ + (rk Γ/2) log rk Γ. d ΓJ + log vrk Γ ≤ χk.k (Γ) ≤ deg deg

(F.3.3)

F.3.2 Let us now assume that the norm k.k on ΓR is the restriction to ΓR of a norm k.kC , invariant under complex conjugation, on the complex vector space ΓC . Then we may consider the John norm k.kJ˜ on ΓC associated to k.kC . It is a Hermitian norm on ΓC , invariant under complex conjugation. We shall still denote by k.kJ˜ its restriction to ΓR .

358

Appendix F. John Ellipsoids and Finite-Dimensional Normed Spaces

Since dimR ΓC = 2rk Γ, the unit balls BC and J˜C in ΓC associated to k.kC and k.kJ˜ satisfy J˜C ⊂ BC ⊂ (2 rk Γ)1/2 J˜C . Consequently, the unit balls B and J˜ := J˜C ∩ ΓR in ΓR associated to k.k and k.kJ˜ satisfy ˜ J˜ ⊂ B ⊂ (2 rk Γ)1/2 J. If we let ΓJ˜ := (Γ, k.kJ˜), we now obtain, instead of (F.3.3), d Γ ˜ + log vrk Γ ≤ χk.k (Γ) ≤ deg d Γ ˜ + log vrk Γ + (rk Γ/2) log(2 rk Γ). deg J J

(F.3.4)

F.3.3 In practice, in using the above estimates, it is useful to notice that as a consequence of Stirling’s formula, as n goes to infinity, log vn = log

π n/2 = −(n/2) log n + (n/2)(1 + log 2π) + O(log n). (F.3.5) Γ(1 + n/2)

Indeed, for every positive integer n, we have −(n/2) log n + n log 2 ≤ log vn ≤ −(n/2) log n + (n/2)(1 + log 2π).

(F.3.6)

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Index A< n (x), 314 A> n (x), 314 ⊥ . , 98 .∨ , 89, 129 B(R), 133 ERHilb , 117 EU , xxiv ERHilb , xxi, 2 Eσ , 11 EσHilb , 114 ECHilb , 117 E• , 163 EσHilb , 115 F (β), 324 Fσtop , 118 H 2 (R), 132 ME • , 164, 165 N (x), 325 S(x), 325 U (β), 325 b B(R), 133

Homcont A , 89 Hom≤λ C , 121 Homcont C , 121 Hom≤λ OK (E, F ), 16 NE (x), 38 Ψ(β), 324 Sum(E • ), 164 Sum, xxix Sum+ , xxix k.kcap V,O , 270 k.kcap V,O , 245 AE003 , 213 AE03 , 213 AE1−3 , 212, 220 AM01−3 , 211, 212 AM1−3 , 211, 220 AMor1−4 , 222 IndSE1−2 , 185 ProSE1−2 , 184 D(proVectOK ), 223 ˜ 281 E,

CPA , 7, 79, 80 CTCA , 7, 79, 82 E, xii, 1, 11 ∨ E , xii, 12 E U , 113, 116 E • , xxii, 1, 162 b xxi, 112 E,

F(M ), 80 FS(M ), 80 b H(Ω), xxii

b ind , 179 E b 2 E, bR , xx, 2 E ΓL2 (V, ν; E, k.k), 272 b 283 ˜ ν; E), ΓL2 (V, ˜ E), ˜ 247 ΓL2 (V; GextE,G (T ), 46 b H(R), 133 Hilbcont , 207

Mb (D), 339 Mb+ (D), 339 O, 13 O(δ), 13 O(λ), 7 OSpec OK , 13 OSpec OK (δ), 13 U(N ), 83 ·∨ , 90, 127 d E, xvii, 3, 15 deg d L, 14 deg d degn L, 42 η, 34 η(t), 34

© Springer Nature Switzerland AG 2020 J.-B. Bost, Theta Invariants of Euclidean Lattices and Infinite-Dimensional Hermitian Vector Bundles over Arithmetic Curves, Progress in Mathematics 334, https://doi.org/10.1007/978-3-030-44329-0

369

Index

370 γE , 6 γn , 74 γV , 163 bσ , 113 E h0Ar , 51 h0Ar (E), xvii, 4 h0Ar (E, t), 6, 63 h0Ar− , 51 ˜ 0 , 50, 64, 67 h Ar ˜ 0 (E, t), 6 h Ar h0θ (E), xv, 5 h1θ (E), 5 indVectOK , 122 ≤1 indVectOK , 122 ] OK , 122 indVect λn (E), 41 λ1 (E), xii, 38 b xxviii, 4 h0 (E), θ

Ded1−3 , 86 StS1−2 , 177 Ac(A), 350 Ac∗ (A), 351 Amp, 202, 241 An1−2 , 203 B< , 316 B> , 316 Ch(A), 350 Ch(proVectOK ), 223 Cons1−2 , 236 Conv1−2 , 343 D(A), 350 D∗ (A), 350 Eop 0−2 , 348 E0−2 , 348 Fin, 202, 240 Gr1−3 , 270 IC0 , 350 IC, 349 K(A), 350 K∗ (A), 351 PSFC02−3 , 267 PSFC1−3 , 266

Pro1−2 , 226 SE, 325 θ-Fin1−3 , 178 proVectC , 226, 228, 229 Hom(E, F ), xvi HomOC (E, F), xvi µmax , 233 µi , 166 µE • , 166 µ b , 167 E

µ b(E), 16 µ bmax (E), 16 µ bmin (E), 16 multPα s, 354 ω, 34 ωOK /Z , 14 Zar

V , 249 ω OK /Z , 14 0 b 231 h (C, E), 0 b 225 h (C, E), π∗ E, 13 b 117 π∗ E, proVectOK , 8, 124 ≤1

proVectOK , 124 ≤1

proVectZ , xx ] O , 125 proVect K ψ(t), 58 ρ(E), xii rk E, xii τ (x), 33 θE , xv θE , 6, 37 ˜ 279 V, 0 b hθ (E), xxviii lim E • , xxiv ←− ∧k E, 12 f ∗ E, 12 fη , 164 fi,η , 163 gV,O , 270 gΩ,0 , xxvi

Index b 231 h0 (C, E), h0 (C, E), xv h0 (C, E), 225 b 225 h0 (C, E), 0 h (C, E), 3 0 hBl , 52 hθ (E), 44 k(C), 225 l(U ), 169 m+ , 320 m− , 320 pi , 163 pij , 163 s(x), 320 s+ (x), 314 s− (x), 314 vn , xiii, 73 F< , 316 F> , 315 I< , 316 I> , 316 O< , 316 O> , 315 inf µ H, 312 covol(E) , 3 coF(M ), 81 coFS(M ), 81 supµ H, 312 admissible extension, 18 morphism injective, 17 surjective, 17 short exact sequence, 183 of Euclidean lattices, xvi, 4 of Hermitian vector bundles, 18 of ind-Hermitian vector bundles, 186 of pro-Hermitian vector bundles, 185, 187, 194 algebraicity criterion, 244, 245 Diophantine, 246

371 allowable epimorphism, 348 monomorphism, 348 morphism, 207, 349 in proVectOK , 211–223 short exact sequences, 348 ample, arithmetically, 260–261 analytic topology, 252 analytic vector bundle, 269 Hermitian, 269 Andreotti, A., 251 Andreotti–Hartshorne, algebraicity theorem of, 246 Arakelov degree, xvii, 14, 32 normalized, 42 Arakelov divisor, 32 effective, 32 arithmetic extension, 19–23 pseudo-concavity, 289, 295 arithmetically ample, 260–261 Artin, E., 3 B¨ uhler, Th., 347 Banaszczyk’s measure, xv, 6, 7, 9, 163 Banaszczyk, W., xiv, xv, xix, 6, 9, 36, 49, 50, 54, 56–58, 68, 157, 163 Behnke–Stein, theorem of, 273 Bergman space, 133 Blichfeldt pair, 60–61 Boltzmann, L., 332 Borel’s rationality criterion, 298–300 canonical Hermitian line bundle, 14 capacitary metric, 245, 270 capacity, xxvii Chevalley, C., 92 Chow’s theorem, 246, 248 Chudnovsky, D.V. and G.V., xxxi, 5, 244, 245, 300 closed graph theorem, 101, 107 completed tensor product, 84 convergence of measures

Index

372 narrow, 340 vague, 339 covering radius, xii, 57 covolume, xii, 3 Cram´er’s theorem, 49, 311, 318, 319 Dedekind ring, 79 defining filtration, 83, 227 derived category of an exact category, 350–351 descent properties of CP. and CTC. , 102, 229 direct image, 15, 21–23, 117 of a Hermitain vector bundle, 13 dual Euclidean lattice, xii Hermitian vector bundle, 12 elliptic curve, 300–309 Enochs, E.E., 335 entropy, 331–333 Euclidean lattice, xi, 1, 12 exact category, 207, 215, 348 derived category of an, 350–351 idempotent complete, 223, 349, 351 exact structure, 348 extension admissible, 18 arithmetic, 19–23 exterior power, 12 Fekete, M., 64 filtration defining, 83 filtration, defining, 227 first main theorem of Nevanlinna theory, 248 first minimum, xii, 38–42, 51, 55, 58, 74 formal curve, 243, 244, 266–268 formal geometry, 243 formal groups of elliptic curves, 301– 303

formal suface, 253 formal surface, 257 formal-analytic surface, 247, 279 Gaussian integral, 26, 330–331 geometry of numbers, xii, xiii Gibbs, J.W., 332 Green’s function, 244, 270 Groenewegen, R. P., xix, 4, 25, 36, 39, 56 Hardy space, 132 Hasse, H., 3 Hecke–Serre duality, 29 height, 144 of morphisms, 17–18 Heller, A., 347 Hermite constant, xiv, 75 Hermite, C., xiii, xiv Hermitian analytic vector bundle, 269 Hermitian line bundle, 12 Hermitian vector bundle, 11 infinite-dimensional, xi over a formal-analytic surface, 281 Hilbertizable ind-vector bundle, 118 Hilbertizable pro-vector bundle, 118, 247 Hlawka, E., xiv Honda’s theorem, 301 idempotent complete exact category, 223, 349, 351 ind-Euclidean lattice, 117, 138–139 ind-Hermitian vector bundle, 112 isogeny theorem, 300–309 isometric isomorphism, 12, 113 Jensen formula, 276 John ellipsoid, 361 John norm, 258, 361 K¨othe, G., 92 Kaplansky, I., 81, 91 Keller, B., 347 Kolmogorov, A., 342

Index Laplace transform, 38, 313 large deviations, 6, 49, 311–324 last minimum, 41 Lefschetz, S., 92 Legendre transform, 6, 65, 70, 311, 320, 326 linearly compact vector space, 92 localization properties of CP. and CTC. , 102 maximal slope, 233 Maxwell velocity distribution, 330– 331 Minkowski’s First Theorem, xiii Minkowski, H., xii–xiv morphism from a formal-analytic surface to a scheme, 285–289 schematic image of, 286–289 narrow convergence of measures, 340 Nevanlinna theory, 248 open mapping theorem, 101, 107 Poincar´e, H., 251 Poisson formula, 3, 26–29 Poisson-Jensen formula, 276 pro-Euclidean lattice, 1, 117 pro-Hermitian vector bundle, 111, 112 pro-vector bundle, 226 wild, 237–240 pseudo-concave, 246, 251 pseudo-concavity, arithmetic, 289, 295 Quillen, D., xix, 3, 4, 8, 44, 183, 347, 349 rank of a Euclidean lattice, xii of a Hermitian vector bundle, 12 rationality criterion, 244, 298–300 relative duality, 14, 15 Riemann inequality, 29 Riemann surface with boundary, 268 Riemann–Roch formula, 3, 28, 29

373 Robin constant, xxvii Roessler, D., 4 schematic image, 286–289 Schoof, R., 4, 10, 25 Schwarz lemma, xxvii, 248, 277, 355 second law of thermodynamics, 67, 328 Serre duality, 28, 29 Serre, J.-P., 251, 353 Seshadri constant, 359 short exact sequence, 183 admissible, 183 of Euclidean lattices, xvi, 4 of ind-Hermitian vector bundles, 185 admissible, 186 of pro-Hermitian vector bundles, 184, 208 admissible, 185, 187, 194 Siegel’s mean value theorem, 71 Siegel, C.L., 353 slope, 16 maximal, 16, 233 minimal, 16 Specker, E., 335 strict morphism, 92–102, 107–109 strict short exact sequence, 95–99 strongly summable pro-Hermitian vector bundle, 8, 177 projective system of Hermitian vector bundles, 164 subadditivity, 44–47, 60–62 summable projective system of Hermitian vector bundles, 164 superadditive, 63, 64, 66 Tate, J., 3 thermodynamic formalism, 66, 312, 324–333 theta invariants, xi, xv theta series, xv

374 θ-finite pro-Hermitian vector bundle, xxx, 8, 178 Toeplitz, O., 92 topological module, 79 transference estimates, xiv vague convergence of measures, 339 van der Geer, G., 4, 25 vanishing criterion, 201–207, 240–242 vanishing multiplicity, 354 wild pro-vector bundles, 237–240 Zhang, S., 260

Index