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Modular Forms and Fermat’s Last Theorem

  • Book
  • © 1997

Overview

Editors:
  1. Gary Cornell

    1. Department of Mathematics, University of Connecticut, Storrs, USA

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  2. Joseph H. Silverman

    1. Department of Mathematics, Brown University, Providence, USA

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    You can also search for this editor in PubMed   Google Scholar

  3. Glenn Stevens

    1. Department of Mathematics, Boston University, Boston, USA

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. An Overview of The Proof of Fermat’s Last Theorem

    • Glenn Stevens
    Pages 1-16

  3. A Survey of the Arithmetic Theory of Elliptic Curves

    • Joseph H. Silverman
    Pages 17-40

  4. Modular Curves, Hecke Correspondences, and L-Functions

    • David E. Rohrlich, George F. Rohrlich
    Pages 41-100

  5. Galois Cohomology

    • Lawrence C. Washington
    Pages 101-120

  6. Finite Flat Group Schemes

    • John Tate
    Pages 121-154

  7. Three Lectures on the Modularity of \(\bar{\rho }\) E,3 and the Langlands Reciprocity Conjecture

    • Stephen Gelbart
    Pages 155-207

  8. Serre’s Conjecture

    • Bas Edixhoven
    Pages 209-242

  9. An Introduction to the Deformation Theory of Galois Representations

    • Barry Mazur
    Pages 243-311

  10. Explicit Construction of Universal Deformation Rings

    • Bart de Smit, Hendrik W. Lenstra Jr.
    Pages 313-326

  11. Hecke Algebras and the Gorenstein Property

    • Jacques Tilouine
    Pages 327-342

  12. Criteria for Complete Intersections

    • Bart De Smit, Karl Rubin, René Schoof
    Pages 343-356

  13. ℓ-adic Modular Deformations and Wiles’s “Main Conjecture”

    • Fred Diamond, Kenneth A. Ribet
    Pages 357-373

  14. The Flat Deformation Functor

    • Brian Conrad
    Pages 373-420

  15. Hecke Rings and Universal Deformation Rings

    • Ehud De Shalit
    Pages 421-445

  16. Explicit Families of Elliptic Curves with Prescribed Mod N Representations

    • Alice Silverberg
    Pages 447-461

  17. Modularity of Mod 5 Representations

    • Karl Rubin
    Pages 463-474

  18. An Extension of Wiles’ Results

    • Fred Diamond
    Pages 475-498

  19. Class Field Theory and the First Case of Fermat’s Last Theorem

    • H. W. Lenstra, P. Stevenhagen Jr.
    Pages 499-503

  20. Remarks on the History of Fermat’s Last Theorem 1844 to 1984

    • Michael Rosen
    Pages 505-525
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Keywords

About this book

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. Contributor's includeThe 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. 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.

Reviews

"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)

Editors and Affiliations

  • Department of Mathematics, University of Connecticut, Storrs, USA

    Gary Cornell

  • Department of Mathematics, Brown University, Providence, USA

    Joseph H. Silverman

  • Department of Mathematics, Boston University, Boston, USA

    Glenn Stevens

Bibliographic Information

  • Book Title: Modular Forms and Fermat’s Last Theorem

  • Editors: Gary Cornell, Joseph H. Silverman, Glenn Stevens

  • DOI: https://doi.org/10.1007/978-1-4612-1974-3

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York, Inc. 1997

  • Hardcover ISBN: 978-0-387-94609-2Published: 03 October 1997

  • Softcover ISBN: 978-0-387-98998-3Published: 14 January 2000

  • eBook ISBN: 978-1-4612-1974-3Published: 01 December 2013

  • Edition Number: 1

  • Number of Pages: XIX, 582

  • Topics: Number Theory, Algebraic Geometry

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