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:
 * $\S 1$. 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:
 * $\S 1$. 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:
 * $\S 1$. Modular curves
 * $\S 2$. The Hecke correspondences
 * $\S 3$. $L$-functions
 * References


 * CHAPTER IV: Galois Cohomology:
 * $\S 1$. $H^0$, $H^1$ 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
 * 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:
 * 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:
 * $\S 1$. 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:
 * 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: AND
 * $\S 1$. 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:
 * $\S 1$. 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:, , AND
 * 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": AND
 * $\S 1$. 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:
 * 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:
 * $\S 1$. 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:
 * 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:
 * 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:
 * $\S 1$. 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 $E$: AND


 * CHAPTER XVIII: Class Field Theory and the First Case of Fermat's Last Theorem: AND


 * CHAPTER XIX: Remarks on the History of Fermat's Last Theorem 1844 to 1984:
 * 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:
 * $\S 1$. Conjectures
 * $\S 2$. The generic case
 * $\S 3$. $K = \Q$
 * References


 * CHAPTER XXI: Wiles' Theorem and the Arithmetic of Elliptic Curves:
 * $\S 1$. 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