Arithmetic compactifications of PEL-type Shimura varieties / Kai-Wen Lan.

Format:

Book

Author/Creator:

Lan, Kai-Wen.

Series:

London Mathematical Society monographs ; new ser., no. 36.

London Mathematical Society monographs

Language:

English

Subjects (All):

Shimura varieties.

Arithmetical algebraic geometry.

Physical Description:

xxiii, 561 pages : illustrations ; 26 cm.

Place of Publication:

Princeton : Princeton University Press, [2013]

Summary:

By studying the degeneration of abelian varieties with PEL structures, this book explains the compactifications of smooth integral models of all PEL-type Shimura varieties, providing the logical foundation for several exciting recent developments. The book is designed to be accessible to graduate students who have an understanding of schemes and abelian varieties. PEL-type Shimura varieties, which are natural generalizations of modular curves, are useful for studying the arithmetic properties of automorphic forms and automorphic representations, and they have played important roles in the development of the Langlands program. As with modular curves, it is desirable to have integral models of compactifications of PEL-type Shimura varieties that can be described in sufficient detail near the boundary. This book explains in detail the following topics about PEL-type Shimura varieties and their compactifications: A construction of smooth integral models of PEL-type Shimura varieties by defining and representing moduli problems of abelian schemes with PEL structures, An analysis of the degeneration of abelian varieties with PEL structures into semiabelian schemes, over noetherian normal complete adic base rings, A construction of toroidal and minimal compactifications of smooth integral models of PEL-type Shimura varieties, with detailed descriptions of their structure near the boundary, Through these topics, the book generalizes the theory of degenerations of polarized abelian varieties and the application of that theory to the construction of toroidal and minimal compactifications of Siegel moduli schemes over the integers (as developed by Mumford, Faltings, and Chai). Book jacket.

Contents:

1 Definition of Moduli Problems 1

1.1 Preliminaries in Algebra 1

1.1.1 Lattices and Orders 1

1.1.2 Determinantal Conditions 6

1.1.3 Projective Modules 13

1.1.4 Generalities on Pairings 16

1.1.5 Classification of Pairings by Involutions 21

1.2 Linear Algebraic Data 26

1.2.1 PEL-Type O-Lattices 26

1.2.2 Torsion of Universal Domains 34

1.2.3 Self-Dual Symplectic Modules 37

1.2.4 Gram-Schmidt Process 47

1.2.5 Reflex Fields 50

1.2.6 Filtrations 55

1.3 Geometric Structures 57

1.3.1 Abelian Schemes and Quasi-Isogenies 57

1.3.2 Polarizations 62

1.3.3 Endomorphism Structures 66

1.3.4 Conditions on Lie Algebras 69

1.3.5 Tate Modules 69

1.3.6 Principal Level Structures 71

1.3.7 General Level Structures 75

1.3.8 Rational Level Structures 76

1.4 Definitions of Moduli Problems 79

1.4.1 Definition by Isomorphism Classes 79

1.4.2 Definition by Z₍₎^×-Isogeny Classes 83

1.4.3 Comparison between Two Definitions 84

1.4.4 Definition by Different Sets of Primes 88

2 Representability of Moduli Problems 91

2.1 Theory of Obstructions for Smooth Schemes 92

2.1.1 Preliminaries 92

2.1.2 Deformation of Smooth Schemes 95

2.1.3 Deformation of Morphisms 99

2.1.4 Base Change 101

2.1.5 Deformation of Invertible Sheaves 103

2.1.6 De Rham Cohomology 108

2.1.7 Kodaira-Spencer Morphisms 114

2.2 Formal Theory 117

2.2.1 Local Moduli Functors and Schlessinger's Criterion 117

2.2.2 Rigidity of Structures 120

2.2.3 Prorepresentability 125

2.2.4 Formal Smoothness 128

2.3 Algebraic Theory 135

2.3.1 Grothendieck's Formal Existence Theory 135

2.3.2 Effectiveness of Local Moduli 136

2.3.3 Automorphisms of Objects 137

2.3.4 Proof of Representability 137

2.3.5 Properties of Kodaira-Spencer Morphisms 139

3 Structures of Semi-Abelian Schemes 143

3.1 Groups of Multiplicative Type, Tori, and Their Torsors 143

3.1.1 Groups of Multiplicative Type 143

3.1.2 Torsors and Invertible Sheaves 144

3.1.3 Construction Using Sheaves of Algebras 148

3.1.4 Group Structures on Torsors 151

3.1.5 Group Extensions 154

3.2 Biextensions and Cubical Structures 155

3.2.1 Biextensions 155

3.2.2 Cubical Structures 157

3.2.3 Fundamental Example 158

3.2.4 The Group G(L) for Abelian Schemes 159

3.2.5 Descending Structures 160

3.3 Semi-Abelian Schemes 161

3.3.1 Generalities 161

3.3.2 Extending Structures 163

3.3.3 Raynaud Extensions 165

3.4 The Group K(L) and Applications 167

3.4.1 Quasi-Finite Subgroups of Semi-Abelian Schemes over Henselian Bases 167

3.4.2 Statement of the Theorem on the Group K(L) 169

3.4.3 Dual Semi-Abelian Schemes 171

3.4.4 Dual Raynaud Extensions 172

4 Theory of Degeneration for Polarized Abelian Schemes 175

4.1 The Setting for This Chapter 175

4.2 Ample Degeneration Data 175

4.2.1 Main Definitions and Main Theorem of Degeneration 176

4.2.2 Equivalence between ι and τ 181

4.2.3 Equivalence between ψ and Actions on L_η^h 184

4.2.4 Equivalence between the Positivity Condition for ψ and the Positivity Condition for τ 189

4.3 Fourier Expansions of Theta Functions 190

4.3.1 Definition of ψ and τ 190

4.3.2 Relations between Theta Representations 197

4.3.3 Addition Formulas 202

4.3.4 Dependence of τ on the Choice of L 208

4.4 Equivalences of Categories 211

4.5 Mumford's Construction 215

4.5.1 Relatively Complete Models 216

4.5.2 Construction of the Quotient 226

4.5.3 Functoriality 234

4.5.4 Equivalences and Polarizations 242

4.5.5 Dependence of τ on the Choice of L, Revisited 254

4.5.6 Two-Step Degenerations 257

4.6 Kodaira-Spencer Morphisms 262

4.6.1 Definition for Semi-Abelian Schemes 262

4.6.2 Definition for Periods 265

4.6.3 Compatibility with Mumford's Construction 270

5 Degeneration Data for Additional Structures 285

5.1 Data without Level Structures 285

5.1.1 Data for Endomorphism Structures 285

5.1.2 Data for Lie Algebra Conditions 287

5.2 Data for Principal Level Structures 292

5.2.1 The Setting for This Section 292

5.2.2 Analysis of Principal Level Structures 293

5.2.3 Analysis of Splittings for G[η]_η 305

5.2.4 Weil Pairings in General 314

5.2.5 Splittings of G[n]_η in Terms of Sheaves of Algebras 320

5.2.6 Weil Pairings for G[n]_η via Splittings 326

5.2.7 Construction of Principal Level Structures 337

5.3 Data for General PEL-Structures 345

5.3.1 Formation of Étale Orbits 345

5.3.2 Degenerating Families 353

5.3.3 Criterion for Properness 354

5.4 Notion of Cusp Labels 355

5.4.1 Principal Cusp Labels 355

5.4.2 General Cusp Labels 361

5.4.3 Hecke Actions on Cusp Labels 364

6 Algebraic Constructions of Toroidal Compactifications 373

6.1 Review of Toroidal Embeddings 373

6.1.1 Rational Polyhedral Cone Decompositions 373

6.1.2 Toroidal Embeddings of Torsors 375

6.2 Construction of Boundary Charts 376

6.2.1 The Setting for This Section 376

6.2.2 Construction without the Positivity Condition or Level Structures 378

6.2.3 Construction with Principal Level Structures 384

6.2.4 Construction with General Level Structures 397

6.2.5 Construction with the Positivity Condition 400

6.2.6 Identifications between Parameter Spaces 409

6.3 Approximation and Gluing 411

6.3.1 Good Formal Models 411

6.3.2 Good Algebraic Models 420

6.3.3 Étale Presentation and Gluing 427

6.4 Arithmetic Toroidal Compactifications 437

6.4.1 Main Results on Toroidal Compactifications 437

6.4.2 Towers of Toroidal Compactifications 441

6.4.3 Hecke Actions on Toroidal Compactifications 444

7 Algebraic Constructions of Minimal Compactifications 447

7.1 Automorphic Forms and Fourier-Jacobi Expansions 447

7.1.1 Automorphic Forms of Naive Parallel Weights 447

7.1.2 Fourier-Jacobi Expansions 449

7.2 Arithmetic Minimal Compactifications 454

7.2.1 Positivity of Hodge Invertible Sheaves 454

7.2.2 Stein Factorizations and Finite Generation 455

7.2.3 Main Construction of Minimal Compactification 458

7.2.4 Main Results on Minimal Compactifications 464

7.2.5 Hecke Actions on Minimal Compactifications 472

7.3 Projectivity of Toroidal Compactifications 474

7.3.1 Convexity Conditions on Cone Decompositions 474

7.3.2 Generalities on Normalizations of Blowups 477

7.3.3 Main Result on Projectivity of Toroidal Compactifications 478.

Notes:

Includes bibliographical references and index.

ISBN:

9780691156545

0691156549

OCLC:

802102903