Produktinformationen zu „Modular Forms and Fermat's Last Theorem “
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats 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 and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, 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 an indispensable resource.ity. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats 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 and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine
Klappentext zu „Modular Forms and Fermat's Last Theorem “
This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribet's theorem and ideas of Frey and Serre to prove 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 and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by indepth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theore minto a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.
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.
Inhaltsverzeichnis zu „Modular Forms and Fermat's Last Theorem “
- Contributors
- Schedule of Lectures
- Introduction
- An Overview of the Proof of Fermat's Last Therem.
- A Survey of the Arithmetic Theory of Elliptic Curves
- Modular Functions and Modular Curves
- Galois Cohomology
- Finite Flat Group Schemes
- Three Lectures on the Modularity of xxx and the Langlands Reciprocity Conjecture
- Serre's Conjectures
- An Introduction to the Deformation Theory of Galois Representations
- Explicit Construction of Universal Deformation Rings
- Hecke Algebras and the Gorenstein Property
- Criteria for Complete Intersections
- l-adic Modular Deformations and Wiles's "Main Conjecture"
- The Flat Deformation Functor
- Hecke Rings and Universal Deformation Rings
- Explicit Families of Elliptic Curves with Prescribed Mod N Representations
- Modularity of Mod 5 Representations
- An Extension of Wiles' Results. Appendix to Chapter- Classification of xxx by the j-invariant of E
- Class Field Theory and the First Case of Fermat's Last Theorem
- Remarks on the History of Fermat's Last Theorem 1844 to 1984
- On Ternary Equations of Fermat Type and Relations With Elliptic Curves
- Wiles' Theorem and the Arithmetic of Elliptic Curves
Bibliographische Angaben
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2000, Pr., 582 Seiten, Maße: 15,5 x 23,5 cm, Kartoniert (TB), Englisch
- Herausgegeben:Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn
- Herausgegeben: Gary Cornell, Glenn Stevens, Joseph H. Silverman
- Verlag: Springer, New York
- ISBN-10: 0387989986
- ISBN-13: 9780387989983
- Erscheinungsdatum: 14.01.2000
Rezension zu „Modular Forms and Fermat's Last Theorem “
"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. ...
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