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