Modular Forms and Fermat's Last Theorem : Papers from a Conference on Number Theory and Arithmetic Geometry, August 9-18, 1995, at Boston University, USA

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. 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.--book Jacket. An Overview Of The Proof Of Fermat's Last Theorem / Glenn Stevens -- A Survey Of The Arithmetic Theory Of Elliptic Curves / Joseph H. Silverman -- Modular Curves, Hecke Correspondences, And L-functions / David E. Rohrlich -- Galois Cohomology / Lawrence C. Washington -- Finite Flat Group Schemes / John Tate -- Three Lectures On The Modularity Of [rho Epsilon],3 And The Langlands Reciprocity Conjecture / Stephen Gelbart -- Serre's Conjecture / Bas Edixhoven -- An Introduction To The Deformation Theory Of Galois Representations / Barry Mazur -- Explicit Construction Of Universal Deformation Rings / Bart De Smit, Hendrik W. Lenstra, Jr. -- Hecke Algebras And The Gorenstein Property / Jacques Tilouine -- Criteria For Complete Intersections / Bart De Smit, Karl Rubin, René Schoof -- l-adic Modular Deformations And Wiles's Main Conjecture / Fred Diamond, Kenneth A. Ribet -- The Flat Deformation Functor / Brian Conrad -- Hecke Rings And Universal Deformation Rings / Ehud De Shalit -- Explicit Families Of Elliptic Curves With Prescribed Mod N Representations / Alice Silverberg -- Modularity Of Mod 5 Representations / Karl Rubin -- An Extension Of Wiles' Results / Fred Diamond -- Appendix : Classification Of [rho Epsilon],l By The J-invariant Of [epsilon] / Fred Diamond, Kenneth Kramer -- Class Field Theory And The First Case Of Fermat's Last Theorem / H.w. Lenstra, Jr., Peter Stevenhagen -- Remarks On The History Of Fermat's Last Theorem 1844 To 1984 / Michael Rosen -- On Ternary Equations Of Fermat Type And Relations With Elliptic Curves / Gerhard Frey -- Wiles' Theorem And The Arithmetic Of Elliptic Curves / Henri Darmon. Gary Cornell, Joseph H. Silverman, Glenn Stevens, Editors. Papers From A Conference Held Aug. 9-18, 1995, At Boston University. Includes Bibliographical References And Index. 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. The book will focus on two major topics: (1) Andrew Wiles' recent proof of the Taniyama-Shimura-Weil conjecture for semistable elliptic curves; and (2) the earlier works of Frey, Serre, Ribet showing that Wiles' Theorem would complete the proof of Fermat's Last Theorem.