Modular Forms and Fermat's Last Theorem

Overview

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, ...
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0387946098 1998 Springer, Corrected 2nd Printing. This hardcover copy has notes and underlining in pencil through the first 15 pages of the text. A suitable reading copy. "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." 100% customer satisfaction guaranteed. We process and ship orders daily, securely and with delivery confirmation. Read more Show Less

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Overview

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 resource for mastering the epoch-making proof of Fermat's Last Theorem.

"...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: 9780387946092
  • Publisher: Springer-Verlag New York, LLC
  • Publication date: 6/2/1998
  • Edition number: 1
  • Pages: 582
  • Product dimensions: 6.42 (w) x 9.55 (h) x 1.28 (d)

Table of Contents

Preface
Contributors
Schedule of Lectures
Introduction
Ch. I An Overview of the Proof of Fermat's Last Theorem 1
Ch. II A Survey of the Arithmetic Theory of Elliptic Curves 17
Ch. III Modular Curves, Hecke Correspondences, L-Functions 41
Ch. IV Galois Cohomology 101
Ch. V Finite Flat Group Schemes 121
Ch. VI Three Lectures on the Modularity of [rho][subscript E,3] and the Langlands Reciprocity Conjecture 155
Ch. VII Serre's Conjectures 209
Ch. VIII An Introduction to the Deformation Theory of Galois Representations 243
Ch. IX Explicit Construction of Universal Deformation Rings 313
Ch. X Hecke Algebras and the Gorenstein Property 327
Ch. XI Criteria for Complete Intersections 343
Ch. XII l-adic Modular Deformations and Wiles's "Main Conjecture" 357
Ch. XIII The Flat Deformation Functor 373
Ch. XIV Hecke Rings and Universal Deformation Rings 421
Ch. XV Explicit Families of Elliptic Curves with Prescribed Mod N Representations 447
Ch. XVI Modularity of Mod 5 Representations 463
Ch. XVII An Extension of Wiles' Results 475
Appendix to Ch. XVII Classification of [rho][subscript E,l] by the j Invariant of E 491
Ch. XVIII Class Field Theory and the First Case of Fermat's Last Theorem 499
Ch. XIX Remarks on the History of Fermat's Last Theorem 1844 to 1984 505
Ch. XX On Ternary Equations of Fermat Type and Relations with Elliptic Curves 527
Ch. XXI Wiles' Theorem and the Arithmetic of Elliptic Curves 549
Index 573
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