Modular Forms and Fermat's Last Theorem / Edition 1

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This volume contains expanded versions of lectures given at an instruc tional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's include The pu rpose 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 Wile sF result can be combined with Ribet's theorem and ideas of Frey and S erre to show, at long last, that Fermat's Last Theorem is true.

"...introduces & explains Wiles ideas, techinques & proof of Fermat's theorem & explains how Wiles' result can be combined with Ribet's theorem of Frey & Serre to prove, at last, Fermat's Last Theorem."

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Editorial Reviews

From the Publisher
"The story of Fermat's last theorem (FLT) and its resolution is now well known. It is now common knowledge that Frey had the original idea linking the modularity of elliptic curves and FLT, that Serre refined this intuition by formulating precise conjectures, that Ribet proved a part of Serre's conjectures, which enabled him to establish that modularity of semistable elliptic curves implies FLT, and that finally Wiles proved the modularity of semistable elliptic curves.

The purpose of the book under review is to highlight and amplify these developments. As such, the book is indispensable to any student wanting to learn the finer details of the proof or any researcher wanting to extend the subject in a higher direction. Indeed, the subject is already expanding with the recent researches of Conrad, Darmon, Diamond, Skinner and others. ...

FLT deserves a special place in the history of civilization. Because of its simplicity, it has tantalized amateurs and professionals alike, and its remarkable fecundity has led to the development of large areas of mathematics such as, in the last century, algebraic number theory, ring theory, algebraic geometry, and in this century, the theory of elliptic curves, representation theory, Iwasawa theory, formal groups, finite flat group schemes and deformation theory of Galois representations, to mention a few. It is as if some supermind planned it all and over the centuries had been developing diverse streams of thought only to have them fuse in a spectacular synthesis to resolve FLT. No single brain can claim expertise in all of the ideas that have gone into this "marvelous proof". In this age of specialization, where "each one of us knows more and more about less and less", it is vital for us to have an overview of the masterpiece such as the one provided by this book." (M. Ram Murty, Mathematical Reviews)

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Product Details

  • ISBN-13: 9780387989983
  • Publisher: Springer New York
  • Publication date: 1/14/2000
  • Edition description: 1st ed. 1997. 3rd printing 2000
  • Edition number: 1
  • Pages: 582
  • Product dimensions: 1.38 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

I An Overview of the Proof of Fermat’s Last Theorem.- II A Survey of the Arithmetic Theory of Elliptic Curves.- III Modular Curves, Hecke Correspondences, and L-Functions.- IV Galois Coharnology.- V Finite Flat Group Schemes.- VI Three Lectures on the Modularity of

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and the Langlands Reciprocity Conjecture.- VII Serre’s Conjectures.- VIII An Introduction to the Deformation Theory of Galois Representations.- IX Explicit Construction of Universal Deformation Rings.- X Hecke Algebras and the Gorenstein Property.- XI Criteria for Complete Intersections.- XII—-adic Modular Deformations and Wiles’s “Main Conjecture”.- XIII The Flat Deformation Functor.- XIV Hecke Rings and Universal Deformation Rings.- XV Explicit Families of Elliptic Curves with Prescribed Mod NRepresentations.- XVI Modularity of Mod 5 Representations.- XVII An Extension of Wiles’ Results.- Appendix to Chapter XVII Classification of

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by the jInvariant of E.- XVIII Class Field Theory and the First Case of Fermat’s Last Theorem.- XIX Remarks on the History of Fermat’s Last Theorem 1844 to 1984.- XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves.- XXI Wiles’ Theorem and the Arithmetic of Elliptic Curves.

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