# 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 $1997$, 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