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

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