Arithmetic Geometry over Global Function Fields

350 Pages·2014·2.62 MB·English

by  Bars

Advanced Courses in Mathematics CRM Barcelona
Gebhard Böckle
David Burns
David Goss
Dinesh Thakur
Fabien Trihan
Douglas Ulmer
Arithmetic
Geometry over Global
Function Fields

Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Carles Casacuberta

More information about this series at http://www.springer.com/series/5038
Gebhard Böckle • David Burns • David Goss
Dinesh Thakur • Fabien Trihan • Douglas Ulmer
Arithmetic Geometry over
Global Function Fields
Editors for this volume:
Francesc Bars (Universitat Autònoma de Barcelona)
Ignazio Longhi (Xi’an Jiaotong-Liverpool University)
Fabien Trihan (Sophia University, Tokyo)
Gebhard Böckle David Burns
Interdisciplinary Center for Scientific Computing Department of Mathematics
Universität Heidelberg King’s College London
Heidelberg, Germany London, UK
David Goss Dinesh Thakur
Department of Mathematics Department of Mathematics
The Ohio State University University of Rochester
Columbus, OH, USA Rochester, NY, USA
Fabien Trihan Douglas Ulmer
Department of Information School of Mathematics
and Communication Sciences Georgia Institute of Technology
Sophia University Atlanta, GA, USA
Tokyo, Japan
ISSN 2297-0304 ISSN 2297-0312 (electronic)
ISBN 978-3-0348-0852-1 ISBN 978-3-0348-0853-8 (eBook)
DOI 10.1007/978-3-0348-0853-8
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014955449

Mathematics Subject Classification (2010): Primary: 11R58; Secondary: 11B65, 11G05, 11G09, 11G10, 11G40,
11J93, 11R23, 11R65, 11R70, 14F05, 14F43, 14G10, 33E50

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Contents
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Cohomological Theory of Crystals over Function Fields
and Applications
Gebhard B¨ockle
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 First Basic Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1 τ-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 (Algebraic) Drinfeld A-modules . . . . . . . . . . . . . . . . . . . 11
1.3 A-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 A-crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Motivation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Localization for abelian categories. . . . . . . . . . . . . . . . . . 20
2.4 Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 A-crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Functors on τ-sheaves and A-crystals . . . . . . . . . . . . . . . . . . . 25
3.1 Inverse image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Further functors deduced from functors on
quasi-coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Extension by zero . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Derived Categories and Derived Functors . . . . . . . . . . . . . . . . . 33
5 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.1 Basics on flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Flatness under functors . . . . . . . . . . . . . . . . . . . . . . . 38
5.3 Representability of flat crystals . . . . . . . . . . . . . . . . . . . 39
v
vi Contents
6 The L-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.1 Naive L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 Crystalline L-functions . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 Trace formulas for L-functions. . . . . . . . . . . . . . . . . . . . 44
7 Proof of Anderson’s Trace Formula and a
Cohomological Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 47
7.1 The Cartier operator . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Cartier sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
7.3 Operators of trace class . . . . . . . . . . . . . . . . . . . . . . . 50
7.4 Anderson’s trace formula . . . . . . . . . . . . . . . . . . . . . . . 51
7.5 Proof of Theorem 6.13 . . . . . . . . . . . . . . . . . . . . . . . . 53
7.6 The crystalline trace formula for general (good) rings A . . . . . 54
8 Global L-functions for A-motives . . . . . . . . . . . . . . . . . . . . . . 57
8.1 Exponentiation of ideals . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Definition and basic properties of the global L-function . . . . . . 60
8.3 Global L-functions at negative integers . . . . . . . . . . . . . . . 62
8.4 Meromorphy and entireness . . . . . . . . . . . . . . . . . . . . . 63
8.5 The global Carlitz–Goss L-function of the affine line . . . . . . . 64
9 Relation to E´tale Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 73
9.1 An equivalence of categories . . . . . . . . . . . . . . . . . . . . . 73
9.2 A result of Goss and Sinnott. . . . . . . . . . . . . . . . . . . . . 77
10 Drinfeld Modular Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.1 A moduli space for Drinfeld modules . . . . . . . . . . . . . . . . 86
10.2 An explicit example . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.3 Drinfeld modular forms via cohomology . . . . . . . . . . . . . . 91
10.4 Galois representations associated to Drinfeld modular forms . . . 96
10.5 Ramification of Galois representations associated to Drinfeld
modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
10.6 Drinfeld modular forms and Hecke characters . . . . . . . . . . . 100
10.7 An extended example. . . . . . . . . . . . . . . . . . . . . . . . . 102
Appendix: Further Results on Drinfeld Modules . . . . . . . . . . . . . . . 108
A.1 Drinfeld A-modules over C∞ . . . . . . . . . . . . . . . . . . . . 108
A.2 Torsion points and isogenies of Drinfeld modules . . . . . . . . . 110
A.3 Drinfeld–Hayes modules . . . . . . . . . . . . . . . . . . . . . . . 111
A.4 Torsion points of Drinfeld–Hayes modules . . . . . . . . . . . . . 114
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Contents vii
On Geometric Iwasawa Theory and Special Values
of Zeta Functions
David Burns and Fabien Trihan
with an appendix by Francesc Bars
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
1.1 Relative algebraic K-theory and Iwasawa algebras . . . . . . . . 123
1.2 Pro-coverings,pro-sheavesand perfect complexes . . . . . . . . . 126
2 Higher direct images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
3 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.1 Finite generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2 The Hochschild–Serre exact triangle . . . . . . . . . . . . . . . . 129
3.3 An important reduction step. . . . . . . . . . . . . . . . . . . . . 131
3.4 The abelian case . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.5 Strategy of Burns and Kato . . . . . . . . . . . . . . . . . . . . . 133
3.6 Application to Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . 134
3.7 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . 136
4 Semistable abelian varieties over unramified extensions . . . . . . . . . 137
4.1 Hypotheses and notations . . . . . . . . . . . . . . . . . . . . . . 137
4.2 Statement of the main results . . . . . . . . . . . . . . . . . . . . 138
5 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.1 The complexes N and S . . . . . . . . . . . . . . . . . . . . . . 141
0 0
5.2 Extending to the complexes S∞ and N∞ . . . . . . . . . . . . . . 143
5.3 The complex N . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
ar
5.4 The complex N∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.5 Main Conjecture for A over K∞ . . . . . . . . . . . . . . . . . . . 146
6 Constant ordinary abelian varieties over abelian extensions . . . . . . . 147
6.1 The p-adic L-function of A/L . . . . . . . . . . . . . . . . . . . . 148
6.2 The interpolation formula . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Main Conjecture for A over L . . . . . . . . . . . . . . . . . . . . 150
7 Proof of Theorem 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.1 Frobenius-Verschiebung decomposition . . . . . . . . . . . . . . . 151
7.2 Selmer modules and class groups . . . . . . . . . . . . . . . . . . 152
7.3 Functional equations . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.4 Completion of the proof . . . . . . . . . . . . . . . . . . . . . . . 155
viii Contents
8 Explicit consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.1 Weil-´etale cohomology . . . . . . . . . . . . . . . . . . . . . . . . 156
8.2 Leading term formulas for affine curves . . . . . . . . . . . . . . . 158
9 Proofs of Theorem 8.1 and Corollary 8.2 . . . . . . . . . . . . . . . . . . 161
10 Fitting invariants and annihilation results . . . . . . . . . . . . . . . . . 165
Appendix: On Non-Noetherian Iwasawa Theory
A.1 The general setting . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.2 Divisor class groups. . . . . . . . . . . . . . . . . . . . . . . . . . 171
A.2.1 General observations . . . . . . . . . . . . . . . . . . . . . 171
A.2.2 Pro-characteristicideals . . . . . . . . . . . . . . . . . . . 172
A.2.3 The main conjecture . . . . . . . . . . . . . . . . . . . . . 172
A.3 Selmer groups in the p-cyclotomic extension . . . . . . . . . . . . 173
A.3.1 Pro-Fitting ideals . . . . . . . . . . . . . . . . . . . . . . . 173
A.3.2 Pro-characteristicideals . . . . . . . . . . . . . . . . . . . 174
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
The Ongoing Binomial Revolution
David Goss
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
2 Newton, Euler, Abel, and Gauss . . . . . . . . . . . . . . . . . . . . . . 186
3 The pth power mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4 The Theorem of Lucas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
5 The Theorem of Mahler . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
6 Measure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7 The group S(p) and binomial symmetries in finite characteristic . . . . 191
8 The future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Contents ix
Arithmetic of Gamma, Zeta and Multizeta Values
for Function Fields
Dinesh S. Thakur
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
1 Gamma: Definitions, Properties and Functional Equations . . . . . . . . 201
1.1 Arithmetic gamma for F [t]: Definitions and analogies . . . . . . 201
q
1.2 Arithmetic gamma for sF [t]: Interpolations . . . . . . . . . . . . 204
q
1.3 Arithmetic gamma for general A: Definitions and
interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
1.4 Functional equations for arithmetic gamma . . . . . . . . . . . . 206
1.5 Geometric gamma: Definitions and interpolations . . . . . . . . . 208
1.6 Functional equations for geometric gamma . . . . . . . . . . . . . 209
2 Special Γ-values, Relations with Drinfeld Modules
and Uniform Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
2.1 Arithmetic gamma: F [t] case . . . . . . . . . . . . . . . . . . . . 211
q
2.2 Arithmetic gamma: General A case . . . . . . . . . . . . . . . . . 213
2.3 Special values of arithmetic Γ . . . . . . . . . . . . . . . . . . . 213
v
2.4 Geometric gamma values . . . . . . . . . . . . . . . . . . . . . . . 215
2.5 Uniform framework . . . . . . . . . . . . . . . . . . . . . . . . . . 217
3 Solitons, t-motives and Complete Gamma Relations for Fq[t] . . . . . . 221
3.1 Anderson’s solitons: General overview . . . . . . . . . . . . . . . 221
3.2 t-modules, t-motives and dual t-motives . . . . . . . . . . . . . . 223
3.3 Period recipe and examples . . . . . . . . . . . . . . . . . . . . . 225
3.4 Explicit construction . . . . . . . . . . . . . . . . . . . . . . . . . 226
3.5 Analog of Gross–Koblitz for geometric gamma:
The F [t] case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
q
3.6 Fermat t-motives . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
3.7 ABP criterion: Period relations are motivic . . . . . . . . . . . . 228
3.8 Complete determination of geometric gamma
relations for F [t] . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
q
4 Automata Method, General A, the v-adic Situation and
Another Gamma Mystery . . . . . . . . . . . . . . . . . . . . . . . . . . 231
4.1 Automata and transcendence for arithmetic gamma. . . . . . . . 231
4.2 General A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.3 Another gamma function coming from the exponential
analogy: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 235
4.4 Some details: Drinfeld correspondence in the simplest case . . . . 236
4.5 Some details: Definition of new gamma . . . . . . . . . . . . . . . 238