Masato Kurihara - Böcker

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.

Kenkichi Iwasawa was one of the most original and influential mathematicians of the twentieth century. He made a number of fundamental contributions in group theory and algebraic number theory. In group theory, he created the theory of (L)-groups (including the structure theorem called "Iwasawa decomposition"), which played an important role in the solution of Hilbert's Fifth Problem. In number theory, he constructed a beautiful theory on Zp-extensions, now called "Iwasawa theory", realizing the deep analogy between number fields and algebraic function fields. Iwasawa theory has had a strong influence on the recent development of arithmetic algebraic geometry, including the solution of Fermat's Last Theorem. This volume of the collected papers of K. Iwasawa contains all 66 of his published papers, including 11 papers in Japanese, for which English abstracts by the editors are attached. In addition, the volume contains 5 papers unpublished until 2001. Also included is a masterly summary of Iwasawa theory by J. Coates (The University of Cambridge).