模形式与费马大定理 影印本 英文PDF|Epub|txt|kindle电子书版本网盘下载

模形式与费马大定理 影印本 英文PDF格式电子书版下载

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CHAPTER Ⅰ An Overview of the Proof of Fermat's Last Theorem&GLENN STEVENS1

1．A remarkable elliptic curve2

2．Galois representations3

3．A remarkable Galois representation7

4．Modular Galois representations7

5．The Modularity Conjecture and Wiles's Theorem9

6．The proof of Fermat's Last Theorem10

7．The proof of Wiles's Theorem10

References15

CHAPTER Ⅱ A Survey of the Arithmetic Theory of Elliptic Curves&JOSEPH H．SILVERMAN17

1．Basic definitions17

2．The group law18

3．Singular cubics18

4．Isogenies19

5．The endomorphism ring19

6．Torsion points20

7．Galois representations attached to E20

8．The Weil pairing21

9．Elliptic curves over finite fields22

10．Elliptic curves over C and elliptic functions24

11．The formal group of an elliptic curve26

12．Elliptic curves over local fields27

13．The Selmer and Shafarevich-Tate groups29

14．Discriminants，conductors，and L-series31

15．Duality theory33

16．Rational torsion and the image of Galois34

17．Tate curves34

18．Heights and descent35

19．The conjecture of Birch and Swinnerton-Dyer37

20．Complex multiplication37

21．Integral points39

References40

CHAPTER Ⅲ Modular Curves，Hecke Correspondences，and L-Functions&DAVID E．ROHRLICH41

1．Modular curves41

2．The Heckc correspondences61

3．L-functions73

References99

CHAPTER Ⅳ Galois Cohomology&LAWRENCE C．WASHINGTON101

1．H0，H1，and H2101

2．Preliminary results105

3．Local Tate duality107

4．Extensions and deformations108

5．Generalized Selmer groups111

6．Local conditions113

7．Conditions at p114

8．Proof of theorem 2117

References120

CHAPTER Ⅴ Finitc Flat Group Schemes&JOHN TATE121

Introduction121

1．Group objects in a category122

2．Group schemes．Examples125

3．Finite flat group schemes：passage to quotient132

4．Raynaud's results on commutative p-group schemes146

References154

CHAPTER Ⅵ Three Lectures on the Modularity of ？E，3 and the Langlands Reciprocity Conjecture&STEPHEN GELBART155

Lccture Ⅰ．The modularity of ？E，3 and automorphic representations of weight one156

1．The modularity of ？E，3157

2．Automorphic representations of weight one164

Lecture Ⅱ．The Langlands program：Some results and methods176

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

4．The Langlands reciprocity conjecture（LRC）179

5．The Langlands functoriality principle theory and results182

Lecture Ⅲ．Proof of the Langlands-Tunnell theorem192

6．Base change theory192

7．Application to Artin's conjecture197

References204

CHAPTER Ⅶ Serre's Conjectures&BAS EDIXHOVEN209

1．Serre's conjecture：statement and results209

2．The cases we need222

3．Weight two，trivial character and square free level224

4．Dealing with the Langlands-Tunnell form230

References239

CHAPTER Ⅷ An Introduction to the Deformation Theory of Galois Representations&BARRY MAZUR243

Chapter Ⅰ．Galois representations246

Chapter Ⅱ．Group representations251

Chapter Ⅲ．The deformation theory for Galois representations259

Chapter Ⅳ．Functors and representability267

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

Chapter Ⅵ．Back to Galois representations294

References309

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

1．Introduction313

2．Main results314

3．Lifting homomorphisms to matrix groups317

4．The condition of absolute irreducibility318

5．Projective limits320

6．Restrictions on deformations323

7．Relaxing the absolute irreducibility condition324

References326

CHAPTER Ⅹ Hecke Algebras and the Gorenstein Property&JACQUES TILOUINE327

1．The Gorenstein property328

2．Hecke algebras330

3．The main theorem331

4．Strategy of the proof of theorem 3.4334

5．Sketch of the proof335

Appendix340

References341

CHAPTER Ⅺ Criteria for Complete Intersections&BART DE SMIT，KARL RUBIN，AND REN？ SCHOOF343

Introduction343

1．Preliminaries345

2．Complete intersections347

3．Proof of Criterion Ⅰ350

4．Proof of Criterion Ⅱ353

Bibliography355

CHAPTER Ⅻ e-adic Modular Deformations and Wiles's“Main Conjecture'”&FRED DIAMOND AND KENNETH A．RIBET357

1．Introduction357

2．Strategy358

3．The“Main Conjecture”359

4．Reduction to the case ∑＝θ363

5．Epilogue370

Bibliography370

CHAPTER ⅩⅢ The Flat Deformation Functor&BRIAN CONRAD373

Introduction373

0．Notation374

1．Motivation and flat representations375

2．Defining the functor394

3．Local Galois cohomology and deformation theory397

4．Fontaine's approach to finite flat group schemes406

5．Applications to flat deformations412

References418

CHAPTER ⅩⅥ Hecke Rings and Universal Deformation Rings&EHUD DE SHALIT421

1．Introduction421

2．An outline of the proof424

3．Proof of proposition 10-On the structure of the Hecke algebra432

4．Proof of proposition 11-On the structure of the universal deformation ring436

5．Conclusion of the proof：Some group theory442

Bibliography444

CHAPTER ⅩⅤ Explicit Families of Elliptic Curves with Prescribed Mod N Representations&ALICE SILVERBERG447

Introduction447

Part 1．Elliptic curves with the same mod N representation448

1．Modular curves and elliptic modular surfaces of level N448

2．Twists of YN and WN449

3．Model for W when N＝3，4，or 5450

4．Level 4451

Part 2．Explicit families of modular elliptic curves454

5．Modular j invariants454

6．Semistable reduction455

7．Mod 4 representations456

8．Torsion subgroups457

References461

CHAPTER ⅩⅥ Modularity of Mod 5 Representations&KARL RUBIN463

Introduction463

1．Preliminaries：Group theory465

2．Preliminaries：Modular curves466

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

4．Proofofthe modularity theorem（Theorem 2）470

5．Mod 5 representations and elliptic curves471

References473

CHAPTER ⅩⅦ An Extension of Wiles' Results&FRED DIAMOND475

1．Introduction475

2．Local representations mod e476

3．Minimally ramified liftings480

4．Universal deformation rings481

5．Hecke algebras482

6．The main results483

7．Sketch of proof484

References488

APPENDIX TO CHAPTER ⅩⅦ Classification of ？E，e by the j Invariant of E&FRED DIAMOND AND KENNETH KRAMER491

CHAPTER ⅩⅧ Class Field Theory and the First Case of Fermat's Last Theorem&HENDRIK W．LENSTRA，JR．AND PETER STEVENHAGEN499

CHAPTER ⅩⅨ Remarks on the History of Fermat's Last Theorem 1844 to 1984&MICHAEL ROSEN505

Introduction507

1．Fermat's last theorem for polynomials507

2．Kummer's work on cyclotomic fields508

3．Fermat's last theorem for regular primes and certain other cases513

4．The structure of the p-class group517

5．Suggested readings521

Appendix A：Kummer congruence and Hilbert's theorem 94522

Bibliography524

CHAPTER ⅩⅩ On Ternary Equations of Fermat Type and Relations with Elliptic Curves&GERHARD FREY527

1．Conjectures527

2．The generic case540

3．K＝Q542

References548

CHAPTER ⅩⅪ Wiles'Theorem and the Arithmetic of Elliptic Curves&HENRI DARMON549

1．Prelude：plane conics，Fermat and Gauss549

2．Elliptic curves and Wiles' theorem552

3．The special values of L（E/Q，s）at s＝1557

4．The Birch and Swinnerton-Dyer conjecture563

References566

Index573