This book, together with the companion volume, Fermat's Last 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 - trick, 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.

Takeshi Saito, professor of mathematics at the University of Tokyo, is an expert in arithmetic geometry, known especially for his fundamental contributions to ramification theory of local fields and I-adic sheaves, étale cohomology, and Galois representations.

Takeshi Saito studied at the University of Tokyo under Kazuya Kato and received his doctorate in 1989. He began his career at the University of Tokyo as an assistant in 1987, was promoted to lecturer in 1990, assistant professor in 1992, and full professor in 1999.

Among his most influential works are a series of articles with Kazuya Kato on the Bloch conductor formula and ramification theory of varieties over perfect or local fields and a series of articles with Ahmed Abbes on ramification groups of local fields. His research on ramification theory culminated in 2017 with a milestone article establishing an algebraic theory of characteristic cycles in positive characteristic.

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