Kurokawa Nobushige, Kurihara Masato, Saito Takeshi. Number Theory 3: Iwasawa Theory and Modular Forms

Transl. from the Japan.: Masato Kuwata. — American Mathematical Society, 2012. — xiv, 226 p. — (Translations of Mathematical Monographs. Vol. 242). — ISBN 978-0-8218-1355-3, 978-0-8218-2095-7.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.Preface
Preface to the English Edition
Objectives and Outline of These BooksModular Forms
Ramanujan’s discoveries
Ramanujan’s Δ and holomorphic Eisenstein series
Automorphy and functional equations
Real analytic Eisenstein series
Kronecker’s limit formula and regularized products
Modular forms for SL₂(ℤ)
Classical modular forms
Summary
ExercisesIwasawa Theory
What is Iwasawa theory?
Analytic p-adic zeta functions
Ideal class groups and cyclotomie ℤ_p-extensions
Iwasawa main conjecture
Summary
Exercises
Modular forms (II)
Automorphic forms and representation theory
Poisson summation formula
Selberg trace formula
Langlands conjectures
Summary
Elliptic curves (II)
Elliptic curves over the rational number field
Fermat’s Last Theorem
SummaryBibliography
Answers to Questions
Answers to Exercises
IndexTrue PDF