Geometric Modular Forms And Elliptic Curves (2Nd Edition)

Geometric Modular Forms And Elliptic Curves (2Nd Edition)

by Haruzo Hida

Hardcover(Second Edition)

$140.00
Choose Expedited Shipping at checkout for guaranteed delivery by Friday, March 22

Product Details

ISBN-13: 9789814368643
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 12/28/2011
Edition description: Second Edition
Pages: 454
Sales rank: 1,231,993
Product dimensions: 5.80(w) x 9.10(h) x 1.20(d)

Table of Contents

Preface to the second edition v

Preface vii

1 An Algebro-Geometric Tool Box 1

1.1 Sheaves 1

1.1.1 Sheaves and Presheaves 1

1.1.2 Sheafication 3

1.1.3 Sheaf Kernel and Cokernel 4

1.2 Schemes 5

1.2.1 Local Panged Spaces 5

1.2.2 Schemes as Local Ringed Spaces 8

1.2.3 Sheaves over Schemes 9

1.2.4 Topological Properties of Schemes 11

1.3 Projective Schemes 13

1.3.1 Graded Rings 13

1.3.2 Functor Proj 13

1.3.3 Sheaves on Projective Schemes 16

1.4 Categories and Functors 20

1.4.1 Categories 20

1.4.2 Functors 22

1.4.3 Schemes as Functors 23

1.4.4 Abelian Categories 26

1.5 Applications of the Key-Lemma 28

1.5.1 Sheaf of Differential Forms on Schemes 29

1.5.2 Fiber Products 32

1.5.3 Inverse Image of Sheaves 33

1.5.4 Affine Schemes 35

1.5.5 Morphisms into a Projective Space 37

1.6 Group Schemes 38

1.6.1 Group Schemes as Functors 38

1.6.2 Kernel and Cokernel 39

1.6.3 Bialgebras 40

1.6.4 Locally Free Groups 42

1.6.5 Schematic Representations 44

1.7 Cartier Duality 45

1.7.1 Duality of Bialgebras 45

1.7.2 Duality of Locally Free Groups 47

1.8 Quotients by a Group Scheme 50

1.8.1 Naive Quotients 50

1.8.2 Categorical Quotients 52

1.8.3 Geometric Quotients 154

1.9 Morphisms 62

1.9.1 Topological Definitions 62

1.9.2 Diffeo-Geometric Definitions 67

1.9.3 Applications 69

1.10 Cohomology of Coherent Sheaves 73

1.10.1 Coherent Cohomology 73

1.10.2 Summary of Known Facts 77

1.10.3 Cohomological Dimension 78

1.11 Descent 82

1.11.1 Covering Data 82

1.11.2 Descent Data 83

1.11.3 Descent of Schemes 85

1.12 Baxsotti-Tate Groups 88

1.12.1 p-Divisible Abelian Sheaf 88

1.12.2 Connected-Étale Exact Sequence 92

1.12.3 Ordinary Barsotti-Tate Group 93

1.13 Formal Scheme 95

1.13.1 Open Subschemes as Functors 96

1.13.2 Examples of Formal Schemes 97

1.13.3 Deformation Functors 101

1.13.4 Connected Formal Groups 102

2 Elliptic Curves 105

2.1 Curves and Divisors 105

2.1.1 Cartier Divisors 105

2.1.2 Serre-Grothendieck Duality 108

2.1.3 Riemann-Roch Theorem 114

2.1.4 Relative Riemann-Roch Theorem 119

2.2 Elliptic Curves 122

2.2.1 Definition 122

2.2.2 Abel's Theorem 123

2.2.3 Holomorphic Differentials 125

2.2.4 Taylor Expansion of Differentials 126

2.2.5 Weierstrass Equations of Elliptic Curves 127

2.2.6 Moduli of Weierstrass Type 130

2.3 Geometric Modular Forms of Level 1 134

2.3.1 Functorial Definition 134

2.3.2 Coarse Moduli Scheme 136

2.3.3 Fields of Moduli 138

2.4 Elliptic Curves over C 139

2.4.1 Topological Fundamental Groups 140

2.4.2 Classical Weierstrass Theory 142

2.4.3 Complex Modular Forms 143

2.5 Elliptic Curves over p-Adic Fields 145

2.5.1 Power Series Identities 145

2.5.2 Universal Tate Curves 148

2.5.3 Etale Covering of Tate Curves 153

2.6 Level Structures 155

2.6.1 Isogenies 155

2.6.2 Level N Moduli Problems 157

2.6.3 Generality of Elliptic Curves 163

2.6.4 Proof of Theorem 2.6.8 165

2.6.5 Geometric Modular Forms of Level N 168

2.7 L-Functions of Elliptic Curves 173

2.7.1 L-Functions over Finite Fields 173

2.7.2 Hasse-Weil L-Function 176

2.8 Regularity 180

2.8.1 Regular Rings 180

2.8.2 Regular Moduli Varieties 183

2.9 p-Ordinary Moduli Problems 189

2.9.1 The Hasse Invariant 189

2.9.2 Ordinary Moduli of p-Power Level 193

2.9.3 Irreducibility of p-Ordinary Moduli 195

2.9.4 Moduli Problem of Γ0 and Γ1 Type 196

2.9.5 Moduli Problem of Γ0(p) and Γ 1(p) Type 198

2.10 Deformation of Elliptic Curves 209

2.10.1 A Theorem of Drinfeld 209

2.10.2 A Theorem of Serre-Tate 211

2.10.3 Deformation of an Ordinary Elliptic Curve 214

3 Geometric Modular Forms 223

3.1 Integrality 223

3.1.1 Spaces of Modular Forms 223

3.1.2 Horizontal Control Theorem 236

3.2 Vertical Control Theorem 238

3.2.1 False Modular Forms 240

3.2.2 p-Adic Modular Forms 252

3.2.3 Hecke Operators 257

3.2.4 Families of p-Adic Modular Forms 266

3.2.5 Horizontal Control of p-Power Level 271

3.2.6 Control of Hecke algebra 273

3.2.7 Irreducible Components and Analytic Families 275

3.3 Action of GL(2) on Modular Forms 276

3.3.1 Action of GL2(Z/NZ) 276

3.3.2 Action of GL2(Z) 280

4 Jacobians and Galois Representations 287

4.1 Jacobians of Stable Curves 287

4.1.1 Non-Singular Curves 287

4.1.2 Union of Two Curves 295

4.1.3 Functorial Properties of Jacobians 298

4.1.4 Self-Duality of Jacobian Schemes 302

4.1.5 Generality on Abelian Schemes 304

4.1.6 Endomorphism of Abelian Schemes 313

4.1.7 l-Adic Galois Representations 318

4.2 Modular Galois Representations 322

4.2.1 Hecke Correspondences 323

4.2.2 Galois Representations on Modular Jacobians 326

4.2.3 Ramification at the Level 330

4.2.4 Ramification of p-Adic Representations at p 335

4.2.5 Modular Galois Representations of Higher Weight 337

4.3 Fullness of Big Galois Representations 342

4.3.1 Big I-adic Galois Representations 344

4.3.2 Ramification of I-adic Galois Representations 345

4.3.3 Lie Algebras over p-Adic Ring 346

4.3.4 Lie Algebras of p-Profinite Subgroups of SL(2) 348

4.3.5 Lie Algebra and Lie Group over Zp 355

4.3.6 Arithmetic Galois Characters 359

4.3.7 Fullness of Modular Galois Representation 361

4.3.8 Fullness of Elliptic Curves 365

4.3.9 Fullness of Lie Algebra over Λ 368

4.3.10 Fullness of I-Adic Galois Representation 371

4.3.11 Basic Subgroups 373

4.3.12 Proof of Theorem 4.3.4 380

5 Modularity Problems 383

5.1 Induced and Extended Galois Representations 384

5.1.1 Induction and Extension 385

5.1.2 Automorphic Induction 392

5.1.3 Artin Representations 395

5.2 Some Other Solutions 402

5.2.1 A Theorem of Wiles 402

5.2.2 Modularity of Extended Galois Representations 404

5.2.3 Elliptic Q-Curves 406

5.2.4 Shimura-Taniyama Conjecture 413

5.3 Modularity of Abelian QVarieties 416

5.3.1 Abelian F-varieties of GL(2)-type 417

5.3.2 Endomorphism Algebras of Abelian F-varieties 424

5.3.3 Application to Abelian Q-Varieties 425

5.3.4 Abelian Varieties with Real Multiplication 432

Bibliography 437

List of Symbols 447

Statement Index 449

Index 451

Customer Reviews

Most Helpful Customer Reviews

See All Customer Reviews