Modular Forms and Fermat's Last Theorem

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. The 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. Contributors to this volume include: B. Conrad, H. Darmon, E. de Shalit, B. de Smit, F. Diamond, S.J. Edixhoven, G. Frey, S. Gelbart, K. Kramer, H.W. Lenstra, Jr., B. Mazur, K. Ribet, D.E. Rohrlich, M. Rosen, K. Rubin, R. Schoof, A. Silverberg, J.H. Silverman, P. Stevenhagen, G. Stevens, J. Tate, J. Tilouine, and L. Washington. 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 TOC:Preface.- Contributors.- Schedule of Lectures.- Introduction.- An Overview of the Proof of Fermat's Last Theorem.- A Survey of the Arithmetic Theory of Elliptic Curves.- Modular Curves, Hecke Correspondences, and L-Functions.- Galois Cohomology.- Finite Flat Group Schemes.- Three Lectures on the Modularity of PE.3 and the Langlands Reciprocity Conjecture.- Serre's Conjectures.- An Introduction to the Deformation Theory of Galois Representations.- Explicit Construction of Universal Deformation Rings.- Hecke Algebras and the Gorenstein Property.- Criteria for Complete Intersections.- l-adic Modular Deformations and Wiles's "Main Conjecture".- The Flat Deformation Functor.- Hecke Rings and Universal Deformation Rings.- Explicit Families of Elliptic Curves with Prescribed Mod N Representations.- Modularity of Mod 5 Representations.- An Extension of Wiles' Results.- Appendix to Chapter 17: Classification of PE.1 by the j Invariant of E.- Class Field Theory and the First Case of Fermat's Last Theorem.- Remarks on the History of Fermat's Last Theorem 1844 to 1984.- On Ternary Equations of Fermat Type and Relations with Elliptic Curves.- Wiles' Theorem and the Arithmetic of Elliptic Curves.