Modular Forms and Fermat's Last Theorem
By Gary Cornell, Joseph H. Silverman, Glenn Stevens
* Publisher: Springer
* Number Of Pages: 608
* Publication Date: 2000-01-14
* ISBN-10 / ASIN: 0387989986
* ISBN-13 / EAN: 9780387989983
Product Description:
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.
Summary: Yet another application of elliptic curves...
Rating: 5
The successful proof of Fermat's Last Theorem by Andrew Wiles was probably the most widely publicized mathematical result of the 20th century. And once again, among their numerous other applications, elliptic curves are employed in the proof. The book is a compilation of articles written by first-class mathematicians, the reading of which will give one a thorough understanding of the proof, along with an overview of some very interesting topics in number theory and algebraic geometry. The reader who undertakes an understanding of the proof already no doubt has a substantial amount of 'mathematical maturity', and no review, no matter how complete, would influence greatly such a reader. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book.
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