Mathematical Surveys and Monographs 2000; 302 pp; softcover Volume: 82 ISBN10: 0821849611 ISBN13: 9780821849613 List Price: US$80 Member Price: US$64 Order Code: SURV/82.S
 Written by one of the leading experts, venerable grandmasters, and most active contributors \(\ldots\) in the arithmetic theory of automorphic forms \(\ldots\) the new material included here is mainly the outcome of his extensive work \(\ldots\) over the last eight years \(\ldots\) a very careful, detailed introduction to the subject \(\ldots\) this monograph is an important, comprehensively written and profound treatise on some recent achievements in the theory. Zentralblatt MATH The main objects of study in this book are Eisenstein series and zeta functions associated with Hecke eigenforms on symplectic and unitary groups. After preliminariesincluding a section, "Notation and Terminology"the first part of the book deals with automorphic forms on such groups. In particular, their rationality over a number field is defined and discussed in connection with the group action; also the reciprocity law for the values of automorphic functions at CMpoints is proved. Next, certain differential operators that raise the weight are investigated in higher dimension. The notion of nearly holomorphic functions is introduced, and their arithmeticity is defined. As applications of these, the arithmeticity of the critical values of zeta functions and Eisenstein series is proved. Though the arithmeticity is given as the ultimate main result, the book discusses many basic problems that arise in numbertheoretical investigations of automorphic forms but that cannot be found in expository forms. Examples of this include the space of automorphic forms spanned by cusp forms and certain Eisenstein series, transformation formulas of theta series, estimate of the Fourier coefficients of modular forms, and modular forms of halfintegral weight. All these are treated in higherdimensional cases. The volume concludes with an Appendix and an Index. The book will be of interest to graduate students and researchers in the field of zeta functions and modular forms. Table of Contents  Introduction
 Automorphic forms and families of abelian varieties
 Arithmeticity of automorphic forms
 Arithmetic of differential operators and nearly holomorphic functions
 Eisenstein series of simpler types
 Zeta functions associated with Hecke eigenforms
 Analytic continuation and near holomorphy of Eisenstein series of general types
 Arithmeticity of the critical values of zeta functions and Eisenstein series of general types
 Appendix
 References
 Index
