模形式与费马大定理 影印本 英文PDF电子书下载

CHAPTER Ⅰ An Overview of the Proof of Fermat's Last Theorem&GLENN STEVENS 1

1．A remarkable elliptic curve 2

2．Galois representations 3

3．A remarkable Galois representation 7

4．Modular Galois representations 7

5．The Modularity Conjecture and Wiles's Theorem 9

6．The proof of Fermat's Last Theorem 10

7．The proof of Wiles's Theorem 10

References 15

CHAPTER Ⅱ A Survey of the Arithmetic Theory of Elliptic Curves&JOSEPH H．SILVERMAN 17

1．Basic definitions 17

2．The group law 18

3．Singular cubics 18

4．Isogenies 19

5．The endomorphism ring 19

6．Torsion points 20

7．Galois representations attached to E 20

8．The Weil pairing 21

9．Elliptic curves over finite fields 22

10．Elliptic curves over C and elliptic functions 24

11．The formal group of an elliptic curve 26

12．Elliptic curves over local fields 27

13．The Selmer and Shafarevich-Tate groups 29

14．Discriminants，conductors，and L-series 31

15．Duality theory 33

16．Rational torsion and the image of Galois 34

17．Tate curves 34

18．Heights and descent 35

19．The conjecture of Birch and Swinnerton-Dyer 37

20．Complex multiplication 37

21．Integral points 39

References 40

CHAPTER Ⅲ Modular Curves，Hecke Correspondences，and L-Functions&DAVID E．ROHRLICH 41

1．Modular curves 41

2．The Heckc correspondences 61

3．L-functions 73

References 99

CHAPTER Ⅳ Galois Cohomology&LAWRENCE C．WASHINGTON 101

1．H0，H1，and H2 101

2．Preliminary results 105

3．Local Tate duality 107

4．Extensions and deformations 108

5．Generalized Selmer groups 111

6．Local conditions 113

7．Conditions at p 114

8．Proof of theorem 2 117

References 120

CHAPTER Ⅴ Finitc Flat Group Schemes&JOHN TATE 121

Introduction 121

1．Group objects in a category 122

2．Group schemes．Examples 125

3．Finite flat group schemes：passage to quotient 132

4．Raynaud's results on commutative p-group schemes 146

References 154

CHAPTER Ⅵ Three Lectures on the Modularity of ？E，3 and the Langlands Reciprocity Conjecture&STEPHEN GELBART 155

Lccture Ⅰ．The modularity of ？E，3 and automorphic representations of weight one 156

1．The modularity of ？E，3 157

2．Automorphic representations of weight one 164

Lecture Ⅱ．The Langlands program：Some results and methods 176

3．The local Langlands correspondence for GL（2） 176

4．The Langlands reciprocity conjecture（LRC） 179

5．The Langlands functoriality principle theory and results 182

Lecture Ⅲ．Proof of the Langlands-Tunnell theorem 192

6．Base change theory 192

7．Application to Artin's conjecture 197

References 204

CHAPTER Ⅶ Serre's Conjectures&BAS EDIXHOVEN 209

1．Serre's conjecture：statement and results 209

2．The cases we need 222

3．Weight two，trivial character and square free level 224

4．Dealing with the Langlands-Tunnell form 230

References 239

CHAPTER Ⅷ An Introduction to the Deformation Theory of Galois Representations&BARRY MAZUR 243

Chapter Ⅰ．Galois representations 246

Chapter Ⅱ．Group representations 251

Chapter Ⅲ．The deformation theory for Galois representations 259

Chapter Ⅳ．Functors and representability 267

Chapter Ⅴ．Zariski tangent spaces and deformation problems subject to“conditions” 284

Chapter Ⅵ．Back to Galois representations 294

References 309

CHAPTER Ⅸ Explicit Construction of Universal Deformation Rings&BART DE SMIT AND HENDRIK W．LENSTRA，JR． 313

1．Introduction 313

2．Main results 314

3．Lifting homomorphisms to matrix groups 317

4．The condition of absolute irreducibility 318

5．Projective limits 320

6．Restrictions on deformations 323

7．Relaxing the absolute irreducibility condition 324

References 326

CHAPTER Ⅹ Hecke Algebras and the Gorenstein Property&JACQUES TILOUINE 327

1．The Gorenstein property 328

2．Hecke algebras 330

3．The main theorem 331

4．Strategy of the proof of theorem 3.4 334

5．Sketch of the proof 335

Appendix 340

References 341

CHAPTER Ⅺ Criteria for Complete Intersections&BART DE SMIT，KARL RUBIN，AND REN？ SCHOOF 343

Introduction 343

1．Preliminaries 345

2．Complete intersections 347

3．Proof of Criterion Ⅰ 350

4．Proof of Criterion Ⅱ 353

Bibliography 355

CHAPTER Ⅻ e-adic Modular Deformations and Wiles's“Main Conjecture'”&FRED DIAMOND AND KENNETH A．RIBET 357

1．Introduction 357

2．Strategy 358

3．The“Main Conjecture” 359

4．Reduction to the case ∑＝θ 363

5．Epilogue 370

Bibliography 370

CHAPTER ⅩⅢ The Flat Deformation Functor&BRIAN CONRAD 373

Introduction 373

0．Notation 374

1．Motivation and flat representations 375

2．Defining the functor 394

3．Local Galois cohomology and deformation theory 397

4．Fontaine's approach to finite flat group schemes 406

5．Applications to flat deformations 412

References 418

CHAPTER ⅩⅥ Hecke Rings and Universal Deformation Rings&EHUD DE SHALIT 421

1．Introduction 421

2．An outline of the proof 424

3．Proof of proposition 10-On the structure of the Hecke algebra 432

4．Proof of proposition 11-On the structure of the universal deformation ring 436

5．Conclusion of the proof：Some group theory 442

Bibliography 444

CHAPTER ⅩⅤ Explicit Families of Elliptic Curves with Prescribed Mod N Representations&ALICE SILVERBERG 447

Introduction 447

Part 1．Elliptic curves with the same mod N representation 448

1．Modular curves and elliptic modular surfaces of level N 448

2．Twists of YN and WN 449

3．Model for W when N＝3，4，or 5 450

4．Level 4 451

Part 2．Explicit families of modular elliptic curves 454

5．Modular j invariants 454

6．Semistable reduction 455

7．Mod 4 representations 456

8．Torsion subgroups 457

References 461

CHAPTER ⅩⅥ Modularity of Mod 5 Representations&KARL RUBIN 463

Introduction 463

1．Preliminaries：Group theory 465

2．Preliminaries：Modular curves 466

3．Proof of the irreducibility theorem（Theorem 1） 470

4．Proofofthe modularity theorem（Theorem 2） 470

5．Mod 5 representations and elliptic curves 471

References 473

CHAPTER ⅩⅦ An Extension of Wiles' Results&FRED DIAMOND 475

1．Introduction 475

2．Local representations mod e 476

3．Minimally ramified liftings 480

4．Universal deformation rings 481

5．Hecke algebras 482

6．The main results 483

7．Sketch of proof 484

References 488

APPENDIX TO CHAPTER ⅩⅦ Classification of ？E，e by the j Invariant of E&FRED DIAMOND AND KENNETH KRAMER 491

CHAPTER ⅩⅧ Class Field Theory and the First Case of Fermat's Last Theorem&HENDRIK W．LENSTRA，JR．AND PETER STEVENHAGEN 499

CHAPTER ⅩⅨ Remarks on the History of Fermat's Last Theorem 1844 to 1984&MICHAEL ROSEN 505

Introduction 507

1．Fermat's last theorem for polynomials 507

2．Kummer's work on cyclotomic fields 508

3．Fermat's last theorem for regular primes and certain other cases 513

4．The structure of the p-class group 517

5．Suggested readings 521

Appendix A：Kummer congruence and Hilbert's theorem 94 522

Bibliography 524

CHAPTER ⅩⅩ On Ternary Equations of Fermat Type and Relations with Elliptic Curves&GERHARD FREY 527

1．Conjectures 527

2．The generic case 540

3．K＝Q 542

References 548

CHAPTER ⅩⅪ Wiles'Theorem and the Arithmetic of Elliptic Curves&HENRI DARMON 549

1．Prelude：plane conics，Fermat and Gauss 549

2．Elliptic curves and Wiles' theorem 552

3．The special values of L（E/Q，s）at s＝1 557

4．The Birch and Swinnerton-Dyer conjecture 563

References 566

Index 573