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Modular Forms and Fermat’s Last Theorem

Gary Cornell, Joseph H. Silverman, Glenn Stevens (Herausgeber)

Buch | Softcover

582 Seiten

2000 | 1st ed. 1997. 3rd printing 2000
Springer-Verlag New York Inc.
978-0-387-98998-3 (ISBN)

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Focuses on Andrew Wiles' proof of the Taniyama-Shimura-Weil conjecture for semistable elliptic curves and the works of Frey, Serre, Ribet showing that Wiles' Theorem would complete the proof of Fermat's Last Theorem. This book reflects on the history of the problem. It describes the connections of Wiles' work with other parts of mathematics.

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.

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 Deformationsand 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 andthe Arithmetic of Elliptic Curves.

Zusatzinfo	XIX, 582 p.
Verlagsort	New York, NY
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
ISBN-10	0-387-98998-6 / 0387989986
ISBN-13	978-0-387-98998-3 / 9780387989983
Zustand	Neuware