Book:Gary Cornell/Modular Forms and Fermat's Last Theorem

Gary Cornell, Joseph H. Silverman and Glenn Stevens: Modular Forms and Fermat's Last Theorem

Published $\text {1997}$, Springer

ISBN 0-387-98998-6

Contents

Preface

Contributors

Schedule of Lectures

Introduction

CHAPTER I: An Overview of the Proof of Fermat's Last Theorem: GLENN STEVENS

$\S 1$. A remarkable elliptic curve

$\S 2$. Galois representations

$\S 3$. A remarkable Galois representation

$\S 4$. Modular Galois representations

$\S 5$. The Modularity Conjecture and Wiles's Theorem

$\S 6$. The proof of Fermat's Lmst Theorem

$\S 7$. The proof of Wiles's Theorem

References

CHAPTER II: A Survey of the Arithmetic Theory of Elliptic Curves: JOSEPH H. SILVERMAN

$\S 1$. Basic definitions

$\S 2$. The group law

$\S 3$. Singular cubics

$\S 4.$ Isogenies

$\S 5$. The endomorphism ring

$\S 6$. Torsion points

$\S 7$. Galois representations attached to $E$

$\S 8$. The Weil pairing

$\S 9$. Elliptic cllrsres over finite fields

$\S 10$. Elliptic curves over $\C$ and elliptic functions

$\S 11$. The formal group of an elliptic curve

$\S 12$. Elliptic curve over local fields

$\S 13$. The Selmer and Shafarevich-Tate groups

$\S 14$. Discriminants, conductors, and $L$-series

$\S 15$. Duality theory

$\S 16$. Rational torsion and the image of Galois

$\S 17$. Tate curves

$\S 18$. Heights and descent

$\S 19$. The conjecture of Birch and Swinnerton-Dyer

$\S 20$. Complex multiplication

$\S 21$. Integral points

References

CHAPTER III: Modular Curves, Hecke Correspondences, and $L$-functions: DAVID E. ROHRLICH

$\S 1$. Modular curves

$\S 2$. The Hecke correspondences

$\S 3$. $L$-functions

References

CHAPTER IV: Galois Cohomology: LAWRENCE C. WASHINGTON

$\S 1$. $H^0$, $H^1$ and $H^2$

$\S 2$. Preliminary results

$\S 3$. Local Tate duality

$\S 4$. Extensions and deformations

$\S 5$. Generalized Selmer groups

$\S 6$. Local conditions

$\S 7$. Conditions at $p$

$\S 8$. Proof of theorem 2

References

CHAPTER V: Finite Flat Group Schemes JOHN TATE

Introduction

$\S 1$. Group objects in a category

$\S 2$. Group schemes. Examples

$\S 3$. Finite flat group schemes; passage to quotient

$\S 4$. Raynaud's results on commutative p-group schemes

References

CHAPTER VI: Three Lectures on the Modularity of $\overline \rho_{E, 3}$ and the Langlands Reciprocity Conjecture: STEPHEN GELBART

Lecture I. The modularity of $\overline \rho_{E, 3}$ and automorphic representations of weight one

$\S 1$. The modularity of $\overline \rho_{E, 3}$

$\S 2$. Automorphic representations of weight one

Lecture II. The Langlands program: Some results and methods

$\S 3$. The local Langlands correspondence for $GL(2)$

$\S 4$. The Langlands reciprocity conjecture (LRC)

$\S 5$. The Langlands functionality principle theory and results

Lecture III. Proof of the Langlands-Tunnell theorem

$\S 6$. Base change theory

$\S 7$. Application to Artin's conjecture

References

CHAPTER VII: Serre's Conjectures: BAS EDIXHOVEN

$\S 1$. Serre's conjecture: statement and results

$\S 2$. The cases we need

$\S 3$. Weight two, trivial character and square free level

$\S 4$. Dealing with the Langlands-Tunnell form

References

CHAPTER VIII: An Introduction to the Deformation Theory of Galois Representations: BARRY MAZUR

Chapter I. Galois representations

Chapter II. Grotlp representations

Chapter III. The deformation theory for Galois representations

Chapter IV. Functors and representatively

Chapter V. Zariski tangent spaces and deformation problems subject to "conditions"

Chapter VI. Back to Galois representations

References

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

$\S 1$. Introduction

$\S 2$. Main results

$\S 3$. Lifting homomorphisms to matrix groups

$\S 4$. The condition of absolute irreducibility

$\S 5$. Projective limits

$\S 6$. Restrictions on deformations

$\S 7$. Relaxing the absolute irreducibility condition

References

CHAPTER X: Hecke Algebras and the Gorenstein Property: JACQUES TILOUINE

$\S 1$. The Gorenstein property

$\S 2$. Hecke algebras

$\S 3$. The main theorem

$\S 4$. Strategy of the proof of theorem 3.4

$\S 5$. Sketch of the proof

Appendix

References

CHAPTER XI: Criteria for Complete Intersections: BART DE SMIT, KARL RUBIN, AND RENÉ SCHOOF

Introduction

$\S 1$. Preliminaries

$\S 2$. Complete intersections

$\S 3$. Proof of Criterion I

$\S 4$. Proof of Criterion II

Bibliography

CHAPTER XII: $\ell$-adic Modular Deformations and Wiles's "Main Conjecture": FRED DIAMOND AND KENNETH A. RIBET

$\S 1$. Introduction

$\S 2$. Strategy

$\S 3$. The "Main Coniecture"

$\S 4$. Reduction to the case \Sigma = \varnothing

$\S 5$. Epilogue

Bibliography

CHAPTER XIII: The Flat Deformation Functor: BRIAN CONRAD

Introduction

$\S 0$. Notation

$\S 1$. Motivation and flat representations

$\S 2$. Defining the functor

$\S 3$. Local Galois cohomology and deformation theory

$\S 4$. Fontaine's approach to finite flat group schemes

$\S 5$. Applications to flat deformations

References

CHAPTER XIV: Hecke Rings and Universal Deformation Rings: EHUD DE SHALIT

$\S 1$. Introduction

$\S 2$. An outline of the proof

$\S 3$. Proof of proposition 10 - On the structure of the Hecke algebra

$\S 4$. Proof of proposition 11 - On the structure of the universal deformation ring

$\S 5$. Conclusion of the proof: Some group theory

Bibliography

CHAPTER XV: Explicit Families of Elliptic Curves with Prescribed Mod $N$ Representations: ALICE SILVERBERG

Introduction

Part 1. Elliptic curves with the same mod $N$ representation

$\S 1$. Modular curves and elliptic modular surfaces of level $N$

$\S 2$. Twists of $Y_N$ and $W_N$

$\S 3$. Model for $W$ when $N = 3$, $4$, or $5$

$\S 4$. Level 4

Part 2. Explicit families of modular elliptic curves

$\S 5$. Modular $j$ invariants

$\S 6$. Semistable reduction

$\S 7$. Mod 4 representations

$\S 8$. Torsion subgroups

References

CHAPTER XVI: Modularity of Mod 5 Representations: KARL RUBIN

Introduction

$\S 1$. Preliminaries: Group theory

$\S 2$. Preliminaries: Modular curves

$\S 3$. Proof of the irreducibility theorem (Theorem 1)

$\S 4$. Proof of the modularity theorem (Theorem 2)

$\S 5$. Mod 5 representations and elliptic curves

References

CHAPTER XVII: An Extension of Wiles' Results: FRED DIAMOND

$\S 1$. Introduction

$\S 2$. Local representations mod $\ell$

$\S 3$. Minimally ramified liftings

$\S 4$. Universal deformation rings

$\S 5$. Hecke algebras

$\S 6$. The main results

$\S 7$. Sketch of proof

References

APPENDIX TO CHAPTER XVII: Classification of $\overline \rho_{E, \ell}$ by the $j$ Invariant of $E$: FRED DIAMOND AND KENNETH KRAMER

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

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

Introduction

$\S 1$. Fermat's last theorem for polynomials

$\S 2$. Kummer's work on cyclotomic melds

$\S 3$. Fermat's last theorem for regular primes and certain other cases

$\S 4$. The structure of the $p$-class group

$\S 5$. Suggested readings

Appendix A: Kummer congruence and Hilbert's theorem 94

Bibliography

CHAPTER XX: On Ternary Equations of Fermat Type and Relations with Elliptic Curves: GERHARD FREY

$\S 1$. Conjectures

$\S 2$. The generic case

$\S 3$. $K = \Q$

References

CHAPTER XXI: Wiles' Theorem and the Arithmetic of Elliptic Curves: HENRI DARMON

$\S 1$. Prelude: plane conics, Fermat and Gauss

$\S 2$. Elliptic curves and Wiles' theorem

$\S 3$. The special values of $L(E / \Q, s)$ at $s = 1$

$\S 4$. The Birch and Swinnerton-Dyer conjecture

References

Index