Progress in Mathematics 298

Hilbert Modular Forms with Coefficients in Intersection Homology andQuadratic Base Change

Bearbeitet vonJayce Getz, Mark Goresky

1. Auflage 2012. Buch. xiv, 258 S. HardcoverISBN 978 3 0348 0350 2

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Progress in Mathematics

Series EditorsHyman Bass Joseph Oesterlé

Volume 298

Alan WeinsteinYuri Tschinkel

Jayce Getz • Mark Goresky

Hilbert Modular Forms with Coefficients in Intersection

Change and uadratic Base Q

Homology

Jayce GetzDepartment of MathematicsMcGill University

Mark GoreskySchool of MathematicsInstitute for Advanced Study

Montreal, Canada

Princeton, N.J.USA

Québec

Library of Congress Control Number:

© Springer Basel 2012

ISBN 978-3-0348- ISBN 978-3-0348- -DOI 10.1007/978-3-0348-0 -Springer Basel Heidelberg New York Dordrecht London

0350-2 0351351 9

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9 (eBook)

2012936073

Ferran Sunyer i Balaguer (1912–1967) was a self-taught Catalan mathematician who, in spite of aserious physical disability, was very active in researchin classical mathematical analysis, an area in whichhe acquired international recognition. His heirs cre-ated the Fundacio Ferran Sunyer i Balaguer insidethe Institut d’Estudis Catalans to honor the memoryof Ferran Sunyer i Balaguer and to promote mathe-matical research.

Each year, the Fundacio Ferran Sunyer i Balaguerand the Institut d’Estudis Catalans award an in-ternational research prize for a mathematical mono-graph of expository nature. The prize-winning mono-graphs are published in this series. Details about theprize and the Fundacio Ferran Sunyer i Balaguer canbe found at

http://ffsb.iec.cat/EN/

This book has been awarded theFerran Sunyer i Balaguer 2011 prize.

The members of the scientific commiteeof the 2011 prize were:

Alejandro AdemUniversity of British Columbia

Hyman BassUniversity of Michigan

Nuria FagellaUniversitat de Barcelona

Joseph OesterleUniversite de Paris VI

Joan VerderaUniversitat Autonoma de Barcelona

Ferran Sunyer i Balaguer Prize winners since 2001:

2001 Martin Golubitsky and Ian StewartThe Symmetry Perspective, PM 200

2002 Andre UnterbergerAutomorphic Pseudodifferential Analysisand Higher Level Weyl Calculi, PM 209

Alexander Lubotzky and Dan SegalSubgroup Growth, PM 212

2003 Fuensanta Andreu-Vaillo, Vincent Casellesand Jose M. MazonParabolic Quasilinear Equations MinimizingLinear Growth Functionals, PM 223

2004 Guy DavidSingular Sets of Minimizers for theMumford-Shah Functional, PM 233

2005 Antonio Ambrosetti and Andrea MalchiodiPerturbation Methods and SemilinearElliptic Problems on 𝑅𝑛, PM 240

Jose SeadeOn the Topology of Isolated Singularities inAnalytic Spaces, PM 241

2006 Xiaonan Ma and George MarinescuHolomorphic Morse Inequalities andBergman Kernels, PM 254

2007 Rosa Miro-RoigDeterminantal Ideals, PM 264

2008 Luis BarreiraDimension and Recurrence in HyperbolicDynamics, PM 272

2009 Timothy D. BrowningQuantitative Arithmetic of Projective Vari-eties, PM 277

2010 Carlo MantegazzaLecture Notes on Mean Curvature Flow,PM 290

To our families

Contents

1 Introduction

1.1 An observation of Serre . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 First main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Definition of 𝐼𝐻𝜒𝐸𝑛 (𝑋0(𝔠)) and 𝛾𝜒𝐸 (𝔪) . . . . . . . . . . . . . . . 8

1.6 Second main theorem . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7 Explicit cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.8 Finding cycles dual to families of automorphic forms . . . . . . . 13

1.9 Comments on related literature . . . . . . . . . . . . . . . . . . . 14

1.10 Comparison with Zagier’s formula . . . . . . . . . . . . . . . . . . 16

1.11 Outline of the book . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.12 Problematic primes . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Review of Chains and Cochains

2.1 Cell complexes and orientations . . . . . . . . . . . . . . . . . . . 21

2.2 Subanalytic sets and stratifications . . . . . . . . . . . . . . . . . 22

2.3 Sheaves and the derived category . . . . . . . . . . . . . . . . . . 23

2.4 The sheaf of chains . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Homology manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 Cellular Borel-Moore chains . . . . . . . . . . . . . . . . . . . . . 26

2.7 Algebraic cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Review of Intersection Homology and Cohomology

3.1 The sheaf of intersection chains . . . . . . . . . . . . . . . . . . . 29

3.2 The sheaf of intersection cochains . . . . . . . . . . . . . . . . . . 30

3.3 Homological stratifications . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Products in intersection homology and cohomology . . . . . . . . 35

3.5 Finite mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

ix

x Contents

4 Review of Arithmetic Quotients

4.1 The setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Baily-Borel compactification . . . . . . . . . . . . . . . . . . . . . 42

4.3 𝐿2 differential forms . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4 Invariant differential forms . . . . . . . . . . . . . . . . . . . . . . 46

4.5 Hecke correspondences for discrete groups . . . . . . . . . . . . . 47

4.6 Mappings induced by a Hecke correspondence . . . . . . . . . . . 48

4.7 The reductive Borel-Serre compactification . . . . . . . . . . . . . 49

4.8 Saper’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.9 Modular cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.10 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Generalities on Hilbert Modular Forms and Varieties

5.1 Hilbert modular Shimura varieties . . . . . . . . . . . . . . . . . 58

5.2 Hecke congruence groups . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Hilbert modular forms . . . . . . . . . . . . . . . . . . . . . . . . 62

5.5 Cohomological normalization . . . . . . . . . . . . . . . . . . . . 65

5.6 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.7 The Petersson inner product . . . . . . . . . . . . . . . . . . . . . 68

5.8 Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.9 Fourier series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.10 Killing Fourier coefficients . . . . . . . . . . . . . . . . . . . . . . 74

5.11 Twisting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.12 𝐿-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.12.1 The standard 𝐿-function . . . . . . . . . . . . . . . . . . . 82

5.12.2 Rankin-Selberg 𝐿-functions . . . . . . . . . . . . . . . . . 83

5.12.3 Adjoint 𝐿-functions . . . . . . . . . . . . . . . . . . . . . . 84

5.12.4 Asai 𝐿-functions . . . . . . . . . . . . . . . . . . . . . . . 85

5.13 Relationship with Hida’s notation . . . . . . . . . . . . . . . . . . 89

6 Automorphic Vector Bundles and Local Systems

6.1 Generalities on local systems . . . . . . . . . . . . . . . . . . . . 92

6.2 Classical description of automorphic vector bundles . . . . . . . . 94

6.2.1 Representations of Γ . . . . . . . . . . . . . . . . . . . . . 94

6.2.2 Representations of 𝐾∞ . . . . . . . . . . . . . . . . . . . . 95

6.2.3 Flat vector bundles . . . . . . . . . . . . . . . . . . . . . . 95

6.2.4 Orbifold local systems . . . . . . . . . . . . . . . . . . . . 96

6.3 Classical description of automorphy factors . . . . . . . . . . . . 97

6.4 Adelic automorphic vector bundles . . . . . . . . . . . . . . . . . 98

6.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Contents xi

6.4.2 Flat bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4.3 Orbifold bundles . . . . . . . . . . . . . . . . . . . . . . . 100

6.5 Representations of GL2 . . . . . . . . . . . . . . . . . . . . . . . 100

6.6 Representations of 𝐺 = Res𝐿/ℚGL2 . . . . . . . . . . . . . . . . . 101

6.7 The section 𝑃𝑧 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.8 The local system ℒ(𝜅, 𝜒0) . . . . . . . . . . . . . . . . . . . . . . 103

6.9 Adelic geometric description of automorphic forms . . . . . . . . 105

6.10 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.11 Action of the component group . . . . . . . . . . . . . . . . . . . 109

7 The Automorphic Description of Intersection Cohomology

7.1 The local system ℒ(𝜅, 𝜒0) . . . . . . . . . . . . . . . . . . . . . . 112

7.2 The automorphic description of intersection cohomology . . . . . 114

7.3 Complex conjugation . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.4 Atkin-Lehner operator . . . . . . . . . . . . . . . . . . . . . . . . 117

7.5 Pairings of vector bundles . . . . . . . . . . . . . . . . . . . . . . 122

7.6 Generalities on Hecke correspondences . . . . . . . . . . . . . . . 126

7.7 Hecke correspondences in the Hilbert modular case . . . . . . . . 129

7.8 Atkin-Lehner-Hecke compatibility . . . . . . . . . . . . . . . . . . 131

7.9 Integral coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8 Hilbert Modular Forms with Coefficients in a Hecke Module

8.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.2 Base change for the Hecke algebra . . . . . . . . . . . . . . . . . 136

8.3 Hilbert modular forms with coefficients in a Hecke module . . . . 141

8.4 Hilbert modular forms with coefficients in intersection homology 143

8.5 Proof of Theorem 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.6 The Fourier coefficients of [𝛾(𝔪),Φ𝛾,𝜒𝐸 ]𝐼𝐻∗ . . . . . . . . . . . . 146

9 Explicit Construction of Cycles

9.1 Notation for the quadratic extension 𝐿/𝐸 . . . . . . . . . . . . . 151

9.2 Canonical section over the diagonal . . . . . . . . . . . . . . . . . 152

9.3 Homological properties of 𝑍0(𝔠𝐸) . . . . . . . . . . . . . . . . . . 157

9.4 The twisting correspondence . . . . . . . . . . . . . . . . . . . . . 160

9.5 Twisting the cycle 𝑍0(𝔠𝐸) . . . . . . . . . . . . . . . . . . . . . . 165

10 The Full Version of Theorem 1.3

10.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . 167

10.2 Rankin-Selberg integrals . . . . . . . . . . . . . . . . . . . . . . . 170

xii Contents

11 Eisenstein Series with Coefficients in Intersection Homology

11.1 Eisenstein series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

11.2 Invariant classes revisited . . . . . . . . . . . . . . . . . . . . . . 180

11.3 Definition of the 𝑉𝜒𝐸 (𝔪) . . . . . . . . . . . . . . . . . . . . . . . 181

11.4 Statement and proof of Theorem 2 . . . . . . . . . . . . . . . 181

Appendices

A Proof of Proposition 2.4

A.1 Cellular cosheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A.2 Proof of Lemma 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 184

A.3 Proof of Proposition 2.4 . . . . . . . . . . . . . . . . . . . . . . . 185

B Recollections on Orbifolds

B.1 Effective actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

B.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

B.3 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

B.4 Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

B.5 Sheaves and cohomology . . . . . . . . . . . . . . . . . . . . . . . 194

B.6 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

B.7 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

C Basic Adelic Facts

C.1 Adeles and ideles . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

C.2 Characters of 𝐿∖𝔸𝐿 . . . . . . . . . . . . . . . . . . . . . . . . . . 200

C.3 Characters of GL1(𝐿)∖GL1(𝔸𝐿) . . . . . . . . . . . . . . . . . . . 201

C.4 Haar measure on the adeles . . . . . . . . . . . . . . . . . . . . . 203

D Fourier Expansions of Hilbert Modular Forms

D.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . 205

D.2 Fourier analysis on GL2(𝐿)∖GL2(𝔸𝐿) . . . . . . . . . . . . . . . . 206

D.3 Whittaker models . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

D.4 Decomposition of 𝑊ℎ . . . . . . . . . . . . . . . . . . . . . . . . . 208

D.5 Computing 𝑊𝜙∞ and 𝑊ℎ0 . . . . . . . . . . . . . . . . . . . . . . 209

D.6 Final steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

E Review of Prime Degree Base Change for GL2

E.1 Automorphic forms and automorphic representations . . . . . . . 214

E.2 Hecke operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

E.3 Agreement of 𝐿-functions . . . . . . . . . . . . . . . . . . . . . . 221

E.4 Langlands functoriality . . . . . . . . . . . . . . . . . . . . . . . . 222

E.5 Prime degree base change for GL1 . . . . . . . . . . . . . . . . . 228

E.6 Conductors of admissible representations of GL2 . . . . . . . . . 229

11.

Contents xiii

E.7 The archimedean places . . . . . . . . . . . . . . . . . . . . . . . 232

E.8 Global base change . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Index of Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253