Softcover ISBN:  9780821898482 
Product Code:  MMONO/243 
List Price:  $52.00 
MAA Member Price:  $46.80 
AMS Member Price:  $41.60 
Electronic ISBN:  9781470416270 
Product Code:  MMONO/243.E 
List Price:  $49.00 
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Book DetailsTranslations of Mathematical MonographsIwanami Series in Modern MathematicsVolume: 243; 2013; 200 ppMSC: Primary 11;
This book, together with the companion volume, Fermat's Last Theorem: The proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics.
Crucial arguments, including the socalled \(3\)–\(5\) trick, \(R=T\) theorem, etc., are explained in depth. The proof relies on basic background materials in number theory and arithmetic geometry, such as elliptic curves, modular forms, Galois representations, deformation rings, modular curves over the integer rings, Galois cohomology, etc. The first four topics are crucial for the proof of Fermat's Last Theorem; they are also very important as tools in studying various other problems in modern algebraic number theory. The remaining topics will be treated in the second book to be published in the same series in 2014. In order to facilitate understanding the intricate proof, an outline of the whole argument is described in the first preliminary chapter, and more details are summarized in later chapters.ReadershipGraduate students and research mathematicians interested in number theory and arithmetic geometry.
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Table of Contents

Chapters

Synopsis

Elliptic curves

Modular forms

Galois representations

The 3–5 trick

$R=T$

Commutative algebra

Deformation rings


Additional Material

Reviews

This book can serve as an introduction to the world of modularity results and will prove valuable for anyone willing to invest some work  which of course one has to do in order to understand interesting mathematics. In the opinion of the reviewer, the author found a good balance between unavoidable omissions and desirable contents of a book like this.
Zentralblatt fur Mathematik


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This book, together with the companion volume, Fermat's Last Theorem: The proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics.
Crucial arguments, including the socalled \(3\)–\(5\) trick, \(R=T\) theorem, etc., are explained in depth. The proof relies on basic background materials in number theory and arithmetic geometry, such as elliptic curves, modular forms, Galois representations, deformation rings, modular curves over the integer rings, Galois cohomology, etc. The first four topics are crucial for the proof of Fermat's Last Theorem; they are also very important as tools in studying various other problems in modern algebraic number theory. The remaining topics will be treated in the second book to be published in the same series in 2014. In order to facilitate understanding the intricate proof, an outline of the whole argument is described in the first preliminary chapter, and more details are summarized in later chapters.
Graduate students and research mathematicians interested in number theory and arithmetic geometry.

Chapters

Synopsis

Elliptic curves

Modular forms

Galois representations

The 3–5 trick

$R=T$

Commutative algebra

Deformation rings

This book can serve as an introduction to the world of modularity results and will prove valuable for anyone willing to invest some work  which of course one has to do in order to understand interesting mathematics. In the opinion of the reviewer, the author found a good balance between unavoidable omissions and desirable contents of a book like this.
Zentralblatt fur Mathematik