Modular Forms and Fermat’s Last Theorem

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Gary Cornell, Joseph H. Silverman, Glenn Stevens
Springer Science & Business Media, Dec 1, 2013 - Mathematics - 582 pages
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

An Overview of the Proof of Fermats Last Theorem
1
A remarkable Galois representation
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
357
573
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