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Euler Products and Eisenstein Series / Edition 1

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Brand new. We distribute directly for the publisher. This volume has three chief objectives: 1) the determination of local Euler factors on classical groups in an explicit ... rational form; 2) Euler products and Eisenstein series on a unitary group of an arbitrary signature; and 3) a class number formula for a totally definite hermitian form. Though these are new results that have never before been published, Shimura starts with a quite general setting. He includes many topics of an expository nature so that the book can be viewed as an introduction to the theory of automorphic forms of several variables, Hecke theory in particular. Eventually, the exposition is specialized to unitary groups, but they are treated as a model case so that the reader can easily formulate the corresponding facts for other groups.There are various facts on algebraic groups and their localizations that are standard but were proved in some old papers or just called "well-known". In this book, the reader will find the proofs of many of them, as well as systematic expositions of the topics. This is the first book in which the Hecke theory of a general (nonsplit) classical group is treated. The book is practically self-contained, except that familiarity with algebraic number theory is assumed. Read more Show Less

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Overview

This volume has three chief objectives: 1) the determination of local Euler factors on classical groups in an explicit rational form; 2) Euler products and Eisenstein series on a unitary group of an arbitrary signature; and 3) a class number formula for a totally definite hermitian form. Though these are new results that have never before been published, Shimura starts with a quite general setting. He includes many topics of an expository nature so that the book can be viewed as an introduction to the theory of automorphic forms of several variables, Hecke theory in particular. Eventually, the exposition is specialized to unitary groups, but they are treated as a model case so that the reader can easily formulate the corresponding facts for other groups. There are various facts on algebraic groups and their localizations that are standard but were proved in some old papers or just called well-known". In this book

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Product Details

Table of Contents

Preface
Notation
Frequently used Symbols
Introduction
Ch. I Algebraic and Local Theories of Generalized Unitary Groups 1
Ch. II Adelization of Algebraic Groups and Automorphic Forms 57
Ch. III Euler Factors on Local Groups and Eisenstein Series 101
Ch. IV Main Theorems on Euler Products, Eisenstein Series, and the Mass Formula 167
App. A1 Some elementary facts on real and complex analysis 209
App. A2 Bounded domains and kernel functions 214
App. A3 Convergence of Eisenstein series and some related facts 222
App. A4 Fourier expansion of automorphic forms 227
App. A5 Parabolic subgroups and compact subgroups of some classical groups over R 236
App. A6 L-functions of Hecke characters 237
App. A7 Theta functions of hermitian forms 241
App. A8 Central simple algebras over a local field 251
References 257
Index 259
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