This is the English translation of the original Japanese book. In this volume, "Fermat's Dream", core theories in modern number theory are introduced. Developments are given in elliptic curves, $p$-adic numbers, the $\zeta$-function, and the number fields. This work presents an elegant perspective on the wonder of numbers. Number Theory 2 on class field theory, and Number Theory 3 on Iwasawa theory and the theory of modular forms, are forthcoming in the series.

Number Theory 1: Fermat's Dream (Translations of Mathematical Monographs) (Vol 1)

Science & Math

Nobushige Kurokawa

Takeshi Saitō

Kazuya Kato

Mathematics

American Mathematical Society

This is the third of three related volumes on number theory. (The first two volumes were also published in the Iwanami Series in Modern Mathematics, as volumes 186 and 240.) The two main topics of this book are Iwasawa theory and modular forms. The presentation of the theory of modular forms starts with several beautiful relations discovered by Ramanujan and leads to a discussion of several important ingredients, including the zeta-regularized products, Kronecker's limit formula, and the Selberg trace formula. The presentation of Iwasawa theory focuses on the Iwasawa main conjecture, which establishes far-reaching relations between a p-adic analytic zeta function and a determinant defined from a Galois action on some ideal class groups. This book also contains a short exposition on the arithmetic of elliptic curves and the proof of Fermat's last theorem by Wiles. Together with the first two volumes, this book is a good resource for anyone learning or teaching modern algebraic number theory.

Number Theory 3: Iwasawa Theory and Modular Forms (Translations of Mathematical Monographs)

Masato Kurihara

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 - 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.

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

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 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.

Fermat's Last Theorem: The Proof (Translations of Mathematical Monographs)

Fermats Last Theorem: Basic Tools Vol 243

UNIVERSITIES PRESS

Fermats Last Theorem: The Proof Vol 245

Number Theory 1

Asian-French Summer School on Algebraic Geometry and Number Theory

Marc Levine

R. Sujatha

Bruno Kahn

Jörg Wildeshaus

Minhyong Kim

Laurent Lafforgue

Alain Genestier

Báo Châu Ngô

Societe Mathematique De France

Fermat expected (2009) ISBN: 4000059580 [Japanese Import]

Iwanami Shoten

The expansion of the scheme theory - <13> 2 [19] number theory foundation of modern mathematics Iwanami Lecture - 3 / [23] algebraic geometry and class field theory (1998) ISBN: 4000106430 [Japanese Import]

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