Modular Forms and Fermat’s Last Theorem by Gary Cornell (PDF)

Description

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.

User’s Reviews

Editorial Reviews: Review “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)

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Let a,b,c,n be positive whole numbers where n > 1c=a+bc^n =(a+b)^n = a^n + b^n + rem(a,b,n) by binomial expansionc^n = a^n + b^n iff rem(a,b,n)=0rem(a,b,n) >0c^n <> a^n + b^nQEDNow think about the equation of a circle… 🙂

⭐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.

⭐This item is very instructively, not only for “real” mathematicians. Of course, sometimes it’s very difficult to “read”. It gives me pleasure to own the proof of FLT.

⭐As you’d expect from a Springer work of this nature, there’s not a lot of basics before you’re up there swimming with the best of them. This is a meaty and fascinating tome that expects you to know what’s what before you start.This is not a text book – it’s more for the (pretty advanced) mathematician to find out exactly what’s behind Wiles’ famous proof, complete with the work that buttressed it all that had been done in the previous four centuries.Extensive references give you everything you need to read up on the background to anything you need to study in further depth.In short, this is all you need to understand this vital and highly important intellectual achievement.

⭐フェルマー向けの本はいろいろありますが、これは完全に専門家向けフェルマーが解決した年に行われた会議の報告集です。ワイルズ氏の論文でつかわれている理論の解説が主です。理解するのに整数論専門の大学院レベルは必要でしょう。米国有数の学者たちが関連する内容の講義をしてそれを記録したもの。シルバーマン氏の楕円曲線概説、テイト氏の有限平坦群スキーム論,…..ワイルズ氏の論文を読む大変さが一部分わかる気がする。

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