# Euler Products and Eisenstein Series

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*Goro Shimura*

A co-publication of the AMS and CBMS

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.

#### Reviews & Endorsements

There is plenty in this volume for both the beginner and the expert. Shimura's book is the first to treat the Hecke theory of general non-split classical groups.

-- Bulletin of the London Mathematical Society

Has many didactic features that make it worthy of study by both graduate students and researchers.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Euler Products and Eisenstein Series

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Table of Contents v6 free
- Preface vii8 free
- Notation ix10 free
- Frequently used Symbols xi12 free
- Introduction xiii14 free
- Chapter I. Algebraic and Local Theories of Generalized Unitary Groups 122 free
- 1. Elementary properties of hermitian forms and unitary groups 122
- 2. Parabolic subgroups and some coset decompositions 728
- 3. The denominator ideal of a matrix 1738
- 4. Hermitian forms over a commutative semisimple algebra of rank 2 and quadratic forms 2344
- 5. Quadratic and hermitian forms over a nonarchimedean local field 3051
- 6. Unitary groups over C 3859
- 7. Symplectic groups and split unitary groups over local fields 4869

- Chapter II. Adelization of Algebraic Groups and Auto-morphic Forms 5778
- Chapter III. Euler Factors on Local Groups and Eisenstein Series 101122
- 13. The series α associated with a hermitian matrix 101122
- 14. The series α[sub(ζ)] with nonsingular ζ 109130
- 15. The explicit form of α(0, s) 118139
- 16. The explicit form of a local Euler factor 125146
- 17. Some local group indices 135156
- 18. Eisenstein series on G[sup(n)] 145166
- 19. The poles and residues of Eisenstein series on G[sup(n)] 157178

- Chapter IV. Main Theorems on Euler Products, Eisenstein Series, and the Mass Formula 167188
- Appendix 209230
- A1. Some elementary facts on real and complex analysis 209230
- A2. Bounded domains and kernel functions 214235
- A3. Convergence of Eisenstein series and some related facts 222243
- A4. Fourier expansion of automorphic forms 227248
- A5. Parabolic subgroups and compact subgroups of some classical groups over R 236257
- A6. L-functions of Hecke characters 237258
- A7. Theta functions of hermitian forms 241262
- A8. Central simple algebras over a local field 251272

- References 257278
- Index 259280
- Back Cover Back Cover1282