Fermat's Last Theorem: Basic Tools (Translations of Mathematical Monographs) (Translations of Mathematical Monographs: IWANAMI Series in Modern Mathematics)

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. Synopsis -- Elliptic Curves -- Modular Forms -- Galois Representations -- The 3-5 Trick -- R = T -- Commutative Algebra -- Deformation Rings -- Appendix A : Supplements To Scheme Theory. Takeshi Saito ; Translated From The Japanese By Masato Kuwata. First Published In Japanese As Feruma Yoso (fermat Conjecture) By Iwanami Shoten, Publishers, Tokyo, 2009, ©2009. Includes Bibliographical References (pages 189-196) And Index. Contents......Page 4 Preface......Page 8 Preface to the English edition......Page 14 Synopsis......Page 16 Elliptic curves......Page 28 Modular forms......Page 50 Galois representations......Page 96 The 3–5 trick......Page 126 R=T......Page 134 Commutative algebra......Page 158 Deformation rings......Page 174 Appendix A......Page 186 Bibliography......Page 204 Symbol index......Page 212 Subject index......Page 214 This is the second volume of the book on the proof of Fermat's Last Theorem by Wiles and Taylor (the first volume is published in the same series). Here the detail of the proof announced in the first volume is fully exposed. The book also includes basic materials and constructions in number theory and arithmetic geometry that are used in the proof.