IAS/PARK CITY MATHEMATICS SERIES Volume 18

Arithmetic of L-functions

Cristian Popescu Karl Rubin Alice Silverberg Editors

American Mathematical Society Institute for Advanced Study Arithmetic of L-functions

https://doi.org/10.1090//pcms/018

IAS/PARK CITY MATHEMATICS SERIES Volume 18

Arithmetic of L-functions

Cristian Popescu Karl Rubin Alice Silverberg Editors

American Mathematical Society Institute for Advanced Study John C. Polking, Series Editor Cristian Popescu, Volume Editor Karl Rubin, Volume Editor Alice Silverberg, Volume Editor

IAS/Park City Mathematics Institute runs mathematics education programs that bring together high school mathematics teachers, researchers in mathematics and mathematics education, undergraduate mathematics faculty, graduate students, and undergraduates to participate in distinct but overlapping programs of research and education. This volume contains the lecture notes from the Graduate Summer School program. 2010 Mathematics Subject Classification. Primary 11G40, 11R42, 14G10.

Library of Congress Cataloging-in-Publication Data Arithmetic of L-functions / Cristian Popescu, Karl Rubin, Alice Silverberg, editors. p. cm. — (IAS/Park City mathematics series ; v. 18) Includes bibliographical references. ISBN 978-0-8218-5320-7 (alk. paper) 1. L-functions. 2. Number theory. I. Popescu, Cristian, 1964– II. Rubin, Karl III. Silver- berg, Alice. QA246.A75 2011 512.73—dc23 2011020548

Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2011 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 161514131211 Contents

Preface xiii

Cristian D. Popescu, Karl Rubin, and Alice Silverberg Introduction 1

Part I: Stark’s Conjecture 5

John Tate Stark’s Basic Conjecture 7

1. L-functions. 9 1.1. Ideal class characters. 10 1.2. Classical abelian L-functions. 10 1.3. Representations of finite groups. 12 1.4. Decomposition, inertia, Frobenius. 15 1.5. Artin L-functions. 15 1.6. Reciprocity and the relation between the two kinds of L-functions. 17 1.7. Brauer’s theorem, meromorphicity. 18 1.8. Functional equation. 19

2. Basic Stark Conjecture. 20 2.1. Class number formula. 20 2.2. S-imprimitive L-functions. 21 2.3. Stark regulator. 22 2.4. Stark’s Basic Conjecture. 23 2.5. The case r(χ) = 0. 27 2.6. The case r(χ) = 1 and Stark units. 28

Bibliography 31

Harold Stark The Origin of the “Stark Conjectures” 33

Introduction 35 1. 1966: Class-number one complex quadratic fields 35 2. 1970: The beginnings of the conjectures 37 3. 1975: The first calculations 39 4. The full “over Z” conjecture 40 Appendix 1 Heegner. 41 Appendix 2 An evaluation at s =0. 42 Bibliography 44

v vi CONTENTS

Cristian D. Popescu Integral and p-adic Refinements of the Abelian Stark Conjecture 45

Introduction 47

1. Algebraic preliminaries 50 1.1. Duals 50 1.2. Evaluation maps 52 1.3. Projective modules 54 1.4. Extension of scalars. 56

2. Stark’s Main Conjecture in the abelian case 57 2.1. Notations 57 2.2. The L–functions 58 2.3. Stark’s Main Conjecture (the abelian case) 59 2.4. Idempotents 62

3. Integral refinements of Stark’s Main Conjecture 64 3.1. The Equivariant Tamagawa Number Conjecture – a brief introduction 65 3.2. The integral conjecture of Rubin-Stark 67

4. The Rubin-Stark conjecture in the case r =1 77 4.1. General considerations 77 4.2. Archimedean v1 78 4.3. Non-archimedean v1 – the Brumer-Stark conjecture 80

5. Gross–type (p–adic) refinements of the Rubin-Stark conjecture 91 5.1. The set-up 91 5.2. The relevant graded rings R(Γ) and RΓ 92 5.3. The conjecture 93 5.4. Linking values of derivatives of p–adic and global L–functions 95 5.5. Sample evidence 98

Bibliography 99

Manfred Kolster Special Values of L-functions at Negative Integers 103

Introduction 105

Lecture 1. Iwasawa theory and cohomology 107 1.1. The classical Main Conjecture 107 1.2. Cohomology 111

Lecture 2. Conjectures and results 115 2.1. The Lichtenbaum Conjecture 115 2.2. The Coates-Sinnott Conjecture 118

Bibliography 123 CONTENTS vii

David Burns An Introduction to the Equivariant Tamagawa Number Conjecture: the Relation to Stark’s Conjecture 125

Introduction 127

Lecture 1. Determinant modules 129 1. Free modules 129 2. Modules of finite projective dimension 131 3. Yoneda two-extensions 132

Lecture 2. A particular case of the ETNC 137 1. Canonical classes 137 2. Statement of the conjecture 139 3. The relation to Stark’s Conjecture 140

Lecture 3. Explicit refinements of Stark’s Conjecture 143 1. Congruences for values of L-series 144 2. Higher derivatives of L-series and annihilators of class groups 148

Bibliography 151

Part II: Birch and Swinnerton-Dyer Conjecture 153

Alice Silverberg Introduction to Elliptic Curves 155

Introduction 157 1. Definitions 157 2. Group Law 158 3. N-torsion 159 4. Elliptic Curves over C 160 5. Elliptic Curves over Number Fields 160 6. Elliptic Curves over Finite Fields 162 7. Homomorphisms 162 8. Supersingular and Ordinary 163 9. Mod N representations 164 10. Tate Modules and the Isogeny Theorem 164 11. Reduction of Elliptic Curves 165 12. Conjecture of Birch and Swinnerton-Dyer 165 13. Abelian Varieties 165 Bibliography 167

Benedict H. Gross Lectures on the Conjecture of Birch and Swinnerton-Dyer 169

Introduction 171

Lecture 1. The Mordell-Weil Theorem 173 1. Torsion on elliptic curves 173 2. Galois cohomology 174 3. A 2-descent 175 4. Heights 177 viii CONTENTS

Lecture 2. Conjectures on L-functions 179 1. The incomplete L-function 179 2. The L-function of an elliptic curve 180 3. Periods 182 4. The conjecture of Birch and Swinnerton-Dyer 184 5. Examples of elliptic curves over function fields 185 6. Examples of elliptic curves over number fields 187

Lecture 3. Progress to date 189 1. Results over function fields 189 2. Constant curves 190 3. Non-constant curves 192 4. Results over number fields 193 5. Local and global heights 194

Appendices 197 A. Reduction modulo v 197 B. Descent computations 199 C. The Tate module and the isogeny theorem 202 D. L-functions of elliptic curves over function fields 203

Bibliography 207

Douglas Ulmer Elliptic Curves over Function Fields 211

Introduction 213

Lecture 0. Background on curves and function fields 215 1. Terminology 215 2. Function fields and curves 215 3. Zeta functions 216 4. Cohomology 217 5. Jacobians 217 6. Tate’s theorem on homomorphisms of abelian varieties 219

Lecture 1. Elliptic curves over function fields 221 1. Elliptic curves 221 2. Frobenius 222 3. The Hasse invariant 223 4. Endomorphisms 224 5. The Mordell-Weil-Lang-N´eron theorem 224 6. The constant case 225 7. Torsion 226 8. Local invariants 228 9. The L-function 229 10. The basic BSD conjecture 230 11. The Tate-Shafarevich group 230 12. Statements of the main results 231 13. The rest of the course 232 CONTENTS ix

Lecture 2. Surfaces and the Tate conjecture 235 1. Motivation 235 2. Surfaces 235 3. Divisors and the N´eron-Severi group 236 4. The Picard scheme 237 5. Intersection numbers and numerical equivalence 237 6. Cycle classes and homological equivalence 238 7. Comparison of equivalence relations on divisors 239 8. Examples 239 9. Tate’s conjectures T1 and T2 241 10. T1 and the Brauer group 242 11. The descent property of T1 244 12. Tate’s theorem on products 244 13. Products of curves and DPC 245 Lecture 3. Elliptic curves and elliptic surfaces 247 1. Curves and surfaces 247 2. The bundle ω and the height of E 250 3. Examples 250 4. E and the classification of surfaces 252 5. Points and divisors, Shioda-Tate 253 6. L-functions and Zeta-functions 254 7. The Tate-Shafarevich and Brauer groups 255 8. The main classical results 256 9. Domination by a product of curves 257 10. Four monomials 257 11. Berger’s construction 258 Lecture 4. Unbounded ranks in towers 261 1. Grothendieck’s analysis of L-functions 261 2. The case of an elliptic curve 264 3. Large analytic ranks in towers 265 4. Large algebraic ranks 268 Lecture 5. More applications of products of curves 271 1. More on Berger’s construction 271 2. A rank formula 272 3. First examples 273 4. Explicit points 274 5. Another example 275 6. Further developments 275 Bibliography 277 Bryan Birch Heegner’s Proof 281 Preface 283 1. Preliminaries 284 1.1. Step 1. Fundamental region 284 1.2. Step 2. j(z) 284 1.3. Step 3. Γ0(N) 284 1.4. Step 4. X0(N)andFN (X, Y ) 285 1.5. Step 5. X0(2) and X0(3). Integrality 285 xCONTENTS

2. Weber 286 2.1. Step 6. Quotation 286 2.2. Step 7. Complex multiplication 287 2.3. Step 8. The class polynomial 287 3. Heegner 288 3.1. Application 1. Heegner points of X0(N) 288 3.2. Application 2. Weber’s Zoo 288 3.3. Application 3. Gauss’ conjecture 289 3.4. Application 4. Points on an elliptic curve 290 Vinayak Vatsal Complex Multiplication: a Concise Introduction 293 1. Introduction 295 2. Preliminaries 296 3. The basic definition 297 4. Localization and class field theory 300 5. The modular equation 300 6. The key ingredient: Kronecker congruence 303 7. Proof of the Kronecker congruence 304 8. The ray class field 306 9. Shimura’s adelic formulation 308 10. CM points on modular curves 308 Bibliography 313

Guido Kings The Equivariant Tamagawa Number Conjecture and the Birch– Swinnerton-Dyer Conjecture 315 Introduction 317 Lecture 1. Motives, cohomology and determinants 319 1. Motives 319 2. Realizations 321 3. Motivic cohomology 323 4. L-functions 324 5. Determinants 326 Lecture 2. The equivariant Tamagawa number conjecture 329 1. Rationality conjecture 329 2. Local unramified cohomology 331 3. Global unramified cohomology 332 4. The equivariant Tamagawa number conjecture 334 5. Unramified cohomology for elliptic curves 336 Lecture 3. The relation to the Birch-Swinnerton-Dyer conjecture 339 1. The Tamagawa number conjecture and the Birch-Swinnerton-Dyer conjecture 339 2. The Selmer group and RΓc(ZS,TpE) 340 3. Local Tamagawa numbers 343 4. Proof of the Main Theorem 345 5. Appendix: Review of some results on elliptic curves 346 Bibliography 349 CONTENTS xi

Part III: Analytic and Cohomological Methods 351

David E. Rohrlich Root Numbers 353

Introduction 355

Lecture 1. Trivial central zeros 357 1. Nonexistence of trivial central zeros for Dirichlet L-functions 358 2. Hecke characters and Hecke L-functions 359 3. A family of Hecke L-functions with trivial central zeros 363 4. An open problem 369 5. Evaluation of the quadratic Gauss sum 371 6. Exercises 374

Lecture 2. Local formulas 377 1. The idele class group 377 2. Idele class characters 378 3. The functional equation 382 4. Quadratic root numbers 384 5. Local root numbers 386 6. An open problem 388 7. Epsilon factors 390 8. Exercises 397

Lecture 3. Motivic L-functions 401 1. Artin representations and Artin L-functions 401 2. The functional equation 405 3. Compatible families 407 4. Premotives 413 5. Uniqueness of the functional equation 414 6. An open problem 415 7. Local factors for Artin L-functions 415 8. Exercises 417

Lecture 4. Local formulas in arbitrary dimension 419 1. The local Weil and Weil-Deligne groups 419 2. From Galois representations to Weil-Deligne representations 423 3. An open problem 427 4. Local factors 428 5. Normalizations of the root number in the literature 433 6. Exercises 433

Lecture 5. The minimalist dichotomy 435 1. Elliptic curves 436 2. The minimalist trichotomy 438 3. Elliptic curves revisited 440 4. An open problem 443 5. Exercises 444

Bibliography 445 xii CONTENTS

Karl Rubin Euler Systems and Kolyvagin Systems 449

Introduction 451 Lecture 1. Galois cohomology 453 1.1. G-modules. 453 1.2. Characterization of the cohomology groups. 453 1.3. Continuous cohomology. 455 1.4. Change of group. 455 1.5. Selmer groups. 457 1.6. Kummer theory. 458 1.7. The Brauer group. 460 1.8. Local fields and duality. 460 1.9. Unramified and transverse cohomology groups. 461 1.10. Comparing Selmer groups. 463 Lecture 2. Kolyvagin systems 465 2.1. Varying the Selmer structure. 465 2.2. Kolyvagin systems. 466 2.3. Sheaves and monodromy. 467 2.4. Hypotheses on A and F. 468 2.5. The core Selmer group. 469 2.6. Stepping along the Selmer graph. 470 2.7. Choosing useful primes. 471 2.8. The stub subsheaf. 472 2.9. Recovering the group structure of the dual Selmer group. 474 Lecture 3. Kolyvagin systems for p-adic representations. 477 3.1. Cohomology groups for p-adic representations. 477 3.2. Kolyvagin systems for A. 479 3.3. Example: units and ideal class groups. 480 3.4. Example: Tate modules of elliptic curves. 485 Lecture 4. Euler systems 487 4.1. Definition of an Euler system. 487 4.2. Example: the cyclotomic unit Euler system 487 4.3. Euler systems and Kolyvagin systems. 489

Lecture 5. Applications 493 5.1. Cyclotomic units and ideal class groups. 493 5.2. Elliptic curves. 493 5.3. General speculation. 496 Bibliography 499 Preface

The IAS/Park City Mathematics Institute (PCMI) was founded in 1991 as part of the “Regional Geometry Institute” initiative of the National Science Foundation. In mid 1993 the program found an institutional home at the Institute for Advanced Study (IAS) in Princeton, New Jersey. The IAS/Park City Mathematics Institute encourages both research and ed- ucation in mathematics and fosters interaction between the two. The three-week summer institute offers programs for researchers and postdoctoral scholars, gradu- ate students, undergraduate students, high school teachers, undergraduate faculty, and researchers in mathematics education. One of PCMI’s main goals is to make all of the participants aware of the total spectrum of activities that occur in math- ematics education and research: we wish to involve professional mathematicians in education and to bring modern concepts in mathematics to the attention of educators. To that end the summer institute features general sessions designed to encourage interaction among the various groups. In-year activities at the sites around the country form an integral part of the High School Teachers Program. Each summer a different topic is chosen as the focus of the Research Program and Graduate Summer School. Activities in the Undergraduate Summer School deal with this topic as well. Lecture notes from the Graduate Summer School are being published each year in this series. The first eighteen volumes are:

• Volume 1: Geometry and Quantum Field Theory (1991) • Volume 2: Nonlinear Partial Differential Equations in Differential Geom- etry (1992) • Volume 3: Complex Algebraic Geometry (1993) • Volume 4: Gauge Theory and the Topology of Four-Manifolds (1994) • Volume 5: Hyperbolic Equations and Frequency Interactions (1995) • Volume 6: Probability Theory and Applications (1996) • Volume 7: Symplectic Geometry and Topology (1997) • Volume 8: Representation Theory of Lie Groups (1998) • Volume 9: Arithmetic Algebraic Geometry (1999) • Volume 10: Computational Complexity Theory (2000) • Volume 11: Quantum Field Theory, Supersymmetry, and Enumerative Geometry (2001) • Volume 12: Automorphic Forms and their Applications (2002) • Volume 13: Geometric Combinatorics (2004) • Volume 14: Mathematical Biology (2005) • Volume 15: Low Dimensional Topology (2006) • Volume 16: Statistical Mechanics (2007) • Volume 17: Analytic and Algebraic Geometry: Common Problems, Dif- ferent Methods (2008) • Volume 18: Arithmetic of L-functions

Volumes are in preparation for subsequent years.

xiii xiv PREFACE

Some material from the Undergraduate Summer School is published as part of the Student Mathematical Library series of the American Mathematical Soci- ety. We hope to publish material from other parts of the IAS/PCMI in the future. This will include material from the High School Teachers Program and publications documenting the interactive activities which are a primary focus of the PCMI. At the summer institute late afternoons are devoted to seminars of common interest to all participants. Many deal with current issues in education: others treat math- ematical topics at a level which encourages broad participation. The PCMI has also spawned interactions between universities and high schools at a local level. We hope to share these activities with a wider audience in future volumes.

John C. Polking Series Editor July 2011

Titles in This Series

18 Cristian Popescu, Karl Rubin, and Alice Silverberg, Editors, Arithmetic of L-functions, 2011 17 Jeffery McNeal and Mircea Mustat¸˘a, Editors, Analytic and Algebraic Geometry, 2010 16 Scott Sheffield and Thomas Spencer, Editors, Statistical Mechanics, 2009 15 Tomasz S. Mrowka and Peter S. Ozsv´ath, Editors, Low Dimensional Topology, 2009 14 Mark A. Lewis, Mark A. J. Chaplain, James P. Keener, and Philip K. Maini, Editors, Mathematical Biology, 2009 13 Ezra Miller, Victor Reiner, and Bernd Sturmfels, Editors, Geometric Combinatorics, 2007 12 Peter Sarnak and Freydoon Shahidi, Editors, Automorphic Forms and Applications, 2007 11 Daniel S. Freed, David R. Morrison, and Isadore Singer, Editors, Quantum Field Theory, Supersymmetry, and Enumerative Geometry, 2006 10 Steven Rudich and Avi Wigderson, Editors, Computation Complexity Theory, 2004 9 Brian Conrad and Karl Rubin, Editors, Arithmetic Algebraic Geometry, 2001 8 Jeffrey Adams and David Vogan, Editors, Representation Theory of Lie Groups, 2000 7 Yakov Eliashberg and Lisa Traynor, Editors, Symplectic Geometry and Topology, 1999 6 EltonP.HsuandS.R.S.Varadhan,Editors, Probability Theory and Applications, 1999 5 Luis Caffarelli and Weinan E, Editors, Hyperbolic Equations and Frequency Interactions, 1999 4 Robert Friedman and John W. Morgan, Editors, Gauge Theory and the Topology of Four-Manifolds, 1998 3 J´anos Koll´ar, Editor, Complex Algebraic Geometry, 1997 2 Robert Hardt and Michael Wolf, Editors, Nonlinear Partial Differential Equations in Differential Geometry, 1996 1 Daniel S. Freed and Karen K. Uhlenbeck, Editors, Geometry and Quantum Field Theory, 1995 The overall theme of the 2009 IAS/PCMI Graduate Summer School was connec- tions between special values of L-functions and arithmetic, especially the Birch and Swinnerton-Dyer Conjecture and Stark’s Conjecture. These conjectures are introduced and discussed in depth, and progress made over the last 30 years is described. This volume contains the written versions of the graduate courses delivered at the summer school. It would be a suitable text for advanced graduate topics courses on the Birch and Swinnerton-Dyer Conjecture and/or Stark’s Conjecture. The book will also serve as a reference volume for experts in the field.

PCMS/18 AMS on the Web www.ams.org