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. 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; see MMONO/243). 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. In the first volume the modularity lifting theorem on Galois representations has been reduced to properties of the deformation rings and the Hecke modules. The Hecke modules and the Selmer groups used to study deformation rings are constructed, and the required properties are established to complete the proof. The reader can learn basics on the integral models of modular curves and their reductions modulo $p$ that lay the foundation of the construction of the Galois representations associated with modular forms. More background materials, including Galois cohomology, curves over integer rings, the Néron models of their Jacobians, etc., are also explained in the text and in the appendices. Contents......Page 4 Preface......Page 8 Modular curves over Z......Page 16 Modular forms and Galois representations......Page 76 Hecke modules......Page 122 Selmer groups......Page 158 Curves over discrete valuation rings......Page 194 Finite commutative group scheme over Z_p......Page 206 Jacobian of a curve and its Néron model......Page 214 Bibliography......Page 228 Symbol index......Page 232 Subject index......Page 236 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.