Modular Forms and Fermat's Last Theorem

by Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn

ISBN13: 9780387989983

ISBN10: 0387989986

Format: Paperback

Pub. Date: 2000-01-01

Publisher(s): Springer Nature

Other versions by this Author

Free Shipping On Orders Over $35

Your order must be $35 or more to qualify for free economy shipping. Marketplace items, eBooks and apparel do not qualify towards the $35 purchase minimum.

List Price: ~~$84.99~~

Rent Textbook

Select for Price

Add to Cart

There was a problem. Please try again later.

New Textbook

We're Sorry
Sold Out

Used Textbook

We're Sorry
Sold Out

eTextbook

We're Sorry
Not Available

Buy from our Marketplace starting at $79.20

Summary

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.

Preface

Contributors

xiii

Schedule of Lectures

xvii

Introduction

xix

An Overview of the Proof of Fermat's Last Theorem

(16)

Glenn Stevens

A remarkable elliptic curve

(1)

Galois representations

(4)

A remarkable Galois representation

(1)

Modular Galois representations

(2)

The Modularity Conjecture and Wiles's Theorem

(1)

The proof of Fermat's Last Theorem

(1)

The proog of Wiles's Theorem

(7)

References

(2)

A Survey of the Arithmetic Theory of Elliptic Curves

(24)

Joseph H. Silverman

Basic definitions

(1)

The group law

(1)

Singular cubics

(1)

Isogenies

(1)

The endomorphism ring

(1)

Torsion points

(1)

Galois representations attached to E

(1)

The Weil pairing

(1)

Elliptic curves over finite fields

(2)

Elliptic curves over C and elliptic functions

(2)

The formal group of an elliptic curve

(1)

Elliptic curves over local fields

(2)

The Selmer and Shafarevich-Tate groups

(2)

Discriminants, conductors, and L-series

(2)

Duality theory

(1)

Rational torsion and the image of Galois

(1)

Tate curves

(1)

Heights and descent

(2)

The conjecture of Birch and Swinnerton-Dyer

(1)

Complex multiplication

(2)

Integral points

(2)

References

(1)

Modular Curves, Hecke Correspondences, and L-Functions

(60)

David E. Rohrlich

Modular curves

(20)

The Hecke correspondences

(12)

L-functions

(28)

References

(2)

Galois Cohomology

101

(20)

Lawrence C. Washington

Ho, H1, and H2

101

(4)

Preliminary results

105

(2)

Local Tate duality

107

(1)

Extensions and deformations

108

(3)

Generalized Selmer groups

111

(2)

Local conditions

113

(1)

Conditions at p

114

(3)

Proof of theorem 2

117

(4)

References

120

(1)

Finite Flat Group Schemes

121

(34)

John Tate

Introduction

121

(1)

Group objects in a category

122

(3)

Group schemes, Examples

125

(7)

Finite flat group schemes; passage to quotient

132

(14)

Raynaud's results on commutative p-group schemes

146

(9)

References

154

(1)

Three Lectures on the Modularity of PE, 3 and the Langlands Reciprocity Conjecture

155

(54)

Stephen Gelbart

The modularity of PE, 3 and automorphic representations of weight one

156

(1)

The modularity of PE, 3

157

(7)

Automorphic representations of weight one

164

(12)

The Langlands program: Some results and methods

The local Langlands correspondence for GL(2)

176

(3)

The Langlands reciprocity conjecture (LRC)

179

(3)

The Langlands functoriality principle theory and results

182

(10)

Proof of the Langlands-Tunnell theorem

192

(1)

Base change theory

192

(5)

Application to Artin's conjecture

197

(12)

References

204

(5)

Serre's Conjectures

209

(34)

Bas Edixhoven

Serre's conjecture: statement and results

209

(13)

The cases we need

222

(2)

Weight two, trivial character and square free level

224

(6)

Dealing with the Langlands--Tunnell form

230

(13)

References

239

(4)

An Introduction to the Deformation Theory of Galois Representations

243

(70)

Barry Mazur

Galois representations

246

(5)

Group representations

251

(8)

The deformation theory for Galois representations

259

(8)

Functors and representability

267

(17)

Zariski tangent spaces and deformation problems subject to ``conditions''

284

(10)

Back to Galois representations

294

(19)

References

309

(4)

Explicit Construction of Universal Deformation Rings

313

(14)

Bart de Smit

Hendrik W. Lenstra, Jr.

Introduction

313

(1)

Main results

314

(3)

Lifting homomorphisms to matrix groups

317

(1)

The condition of absolute irreducibility

318

(2)

Projective limits

320

(3)

Restrictions on deformations

323

(1)

Relaxing the absolute irreducibility condition

324

(3)

References

326

(1)

Hecke Algebras and the Gorenstein Property

327

(16)

Jacques Tilouine

The Gorenstein property

328

(2)

Hecke algebras

330

(1)

The main theorem

331

(3)

Strategy of the proof of theorem 3.4

334

(1)

Sketch of the proof

335

(8)

Appendix

340

(1)

References

341

(2)

Criteria for Complete Intersections

343

(14)

Bart De Smit

Karl Rubin

Rene Schoof

Introduction

343

(2)

Preliminaries

345

(2)

Complete intersections

347

(3)

Proof of Criterion I

350

(3)

Proof of Criterion II

353

(4)

Bibliography

355

(2)

l-adic Modular Deformations and Wiles's ``Main Conjecture''

357

(16)

Fred Diamond

Kenneth A. Ribet

Introduction

357

(1)

Strategy

358

(1)

The ``Main Conjecture''

359

(4)

Reduction to the case Σ = ø

363

(7)

Epilogue

370

(3)

Bibliography

370

(3)

The Flat Deformation Functor

373

(48)

Brian Conrad

Introduction

373

(1)

Notation

374

(1)

Motivation and flat representations

375

(19)

Defining the functor

394

(3)

Local Galois cohomology and deformation theory

397

(9)

Fontaine's approach to finite flat group schemes

406

(6)

Applications to flat deformations

412

(9)

References

418

(3)

Hecke Rings and Universal Deformation Rings

421

(26)

Ehud De Shalit

Introduction

421

(3)

An outline of the proof

424

(8)

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

(6)

Conclusion of the proof: Some group theory

442

(5)

Bibliography

444

(3)

Explicit Families of Elliptic Curves with Prescribed Mod N Representations

447

(16)

Alice Silverberg

Introduction

447

(1)

Elliptic curves with the same mod N representation

448

(1)

Modular curves and elliptic modular surfaces of level N

448

(1)

Twists of YN and WN

449

(1)

Model for W when N = 3, 4, or 5

450

(1)

Level 4

451

(3)

Explicit families of modular elliptic curves

454

(1)

Modular j invariants

454

(1)

Semistable reduction

455

(1)

Mod 4 representations

456

(1)

Torsion subgroups

457

(6)

References

461

(2)

Modularity of Mod 5 Representations

463

(12)

Karl Rubin

Introduction

463

(2)

Preliminaries: Group theory

465

(1)

Preliminaries: Modular curves

466

(4)

Proof of the irreducibility theorem (Theorem 1)

470

(1)

Proof of the modularity theorem (Theorem 2)

470

(1)

Mod 5 representations and elliptic curves

471

(4)

References

473

(2)

An Extension of Wiles' Results

475

(24)

Fred Diamond

Introduction

475

(1)

Local representations mod l

476

(4)

Minimally ramified liftings

480

(1)

Universal deformation rings

481

(1)

Hecke algebras

482

(1)

The main results

483

(1)

Sketch of Proof

484

(15)

References

488

(3)

Classification of PE, l by the j Invariant of E

491

(8)

Fred Diamond

Kenneth Kramer

Class Field Theory and the First Case of Fermat's Last Theorem

499

(6)

Hendrik W. Lenstra, Jr.

Peter Stevenhagen

Remarks on the History of Fermat's Last Theorem 1844 to 1984

505

(22)

Michael Rosen

Introduction

507

(1)

Fermat's last theorem for polynomials

507

(1)

Kummer's work on cyclotomic fields

508

(5)

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

513

(4)

The structure of the p-class group

517

(4)

Modular Forms and Fermat's Last Theorem

Rent Textbook

New Textbook

Used Textbook

eTextbook

How Marketplace Works:

Summary

Table of Contents

Digital License