This volume has three chief objectives: the determination of local Euler factors on classical groups in an explicit rational form; Euler products and Eisenstein series on a unitary group of an arbitrary signature; and 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.
Algebraic and local theories of generalized unitary groups (Chapter I) Adelization of algebraic groups and automorphic forms (Chapter II) Euler factors on local groups and Eisenstein series (Chapter III) Main theorems on Euler products, Eisenstein series, and the mass formula (Chapter IV) Appendix References Index.