## Modular Forms and Fermat’s Last TheoremGary Cornell, Joseph H. Silverman, Glenn Stevens This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable resource for mastering the epoch-making proof of Fermat's Last Theorem. |

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### Contents

1 | |

7 | |

A Survey of the Arithmetic Theory of Elliptic Curves | 17 |

Elliptic curves over C and elliptic functions | 24 |

Discriminants conductors and Lseries | 31 |

16 Rational torsion and the image of Galois | 34 |

Modular Curves Hecke Correspondences and LFunctions | 41 |

2 The Hecke correspondences | 61 |

Back to Galois representations | 294 |

CHAPTER IX | 313 |

Projective limits | 320 |

CHAPTER X | 327 |

4 Strategy of the proof of theorem 3 4 | 334 |

Criteria for Complete Intersections | 343 |

Introduction | 357 |

The Flat Deformation Functor | 373 |

CHAPTER IV | 101 |

3 Local Tate duality | 107 |

6 Local conditions | 113 |

CHAPTER V | 121 |

3 Finite flat group schemes passage to quotient | 132 |

4 Raynauds results on commutative pgroup schemes | 146 |

CHAPTER VI | 155 |

2 Automorphic representations of weight one | 164 |

Some results and methods | 175 |

5 The Langlands functoriality principle theory and results | 182 |

Proof of the LanglandsTunnell theorem | 192 |

CHAPTER VII | 209 |

2 The cases we need | 222 |

4 Dealing with the LanglandsTunnell form | 230 |

CHAPTER VIII | 243 |

Group representations | 251 |

The deformation theory for Galois representations | 259 |

Functors and representability | 267 |

Zariski tangent spaces and deformation problems | 284 |

Defining the functor | 394 |

Hecke Rings and Universal Deformation Rings | 421 |

Explicit Families of Elliptic Curves | 447 |

Explicit families of modular elliptic curves | 454 |

CHAPTER XVI | 463 |

3 Proof of the irreducibility theorem Theorem 1 | 470 |

2 Local representations mod | 476 |

5 Hecke algebras | 482 |

APPENDIX TO CHAPTER XVII | 491 |

CHAPTER XVIII | 499 |

Introduction | 507 |

3 Fermats last theorem for regular primes and certain other cases | 513 |

5 Suggested readings | 521 |

3 K Q | 542 |

CHAPTER XXI | 549 |

3 The special values of LEQ s at s 1 | 557 |

4 The Birch and SwinnertonDyer conjecture | 563 |

573 | |

### Other editions - View all

Modular Forms and Fermat's Last Theorem Gary Cornell,Joseph H. Silverman,Glenn Stevens No preview available - 2014 |

Modular Forms and Fermat’s Last Theorem Gary Cornell,Joseph H. Silverman,Glenn Stevens No preview available - 2000 |

### Common terms and phrases

A-algebra A-module abelian absolutely irreducible artinian assume automorphic automorphic representation base change character cocycle coefficient-A-algebra coefficient-ring coefficients cohomology commutative condition conductor Conjecture conjugation Corollary corresponding curve over Q cusp cuspidal representation cyclic cyclotomic defined denote diagram dividing eigenform element elliptic curve End A(V equation equivalent exact sequence extension Fermat's Last Theorem finite flat group flat group scheme follows formula function field functor Galois representations given gives group scheme Hecke algebra hence homomorphism implies induced integer isogeny isomorphism kernel L-function Langlands Lemma Math matrix maximal ideal Mazur modular curves modular forms modular of type module morphism multiplicative newform noetherian number field p-adic prime number proof properties Proposition prove quadratic quotient R-group ramified reduction residue field restriction result Ribet satisfies semistable Serre space subgroup Suppose surjective theory topological trivial unique unramified Wiles Z/NZ