Goro Shimura 
Arithmeticity in the Theory of Automorphic Forms 

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 preliminaries-including 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 CM-points 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 number-theoretical 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 half-integral weight. All these are treated in higher-dimensional 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.