Fermat’s Last Theorem: Basic Tools
About this Title
Takeshi Saito, University of Tokyo, Tokyo, Japan. Translated by Masato Kuwata
Publication: Translations of Mathematical Monographs
Publication Year: 2013; Volume 243
ISBNs: 978-0-8218-9848-2 (print); 978-1-4704-1627-0 (online)
MathSciNet review: MR3136492
MSC: Primary 11D41; Secondary 11F11, 11F80, 11G05, 11G18
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 so-called $3$–$5$ 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.
Table of Contents
- Elliptic curves
- Modular forms
- Galois representations
- The 3–5 trick
- Commutative algebra
- Deformation rings