Arithmeticity in the Theory of Automorphic Forms
About this Title
Goro Shimura, Princeton University, Princeton, NJ
Publication: Mathematical Surveys and Monographs
Publication Year 2000: Volume 82
ISBNs: 978-0-8218-4961-3 (print); 978-1-4704-1309-5 (online)
MathSciNet review: MR1780262
MSC: Primary 11F67; Secondary 11-02, 11F30, 11F41, 11F55
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.
Graduate students and researchers interested in number theory and modular forms.
Table of Contents
- I. Automorphic forms and families of abelian varieties
- II. Arithmeticity of automorphic forms
- III. Arithmetic of differential operators and nearly holomorphic functions
- IV. Eisenstein series of simpler types
- V. Zeta functions associated with Hecke eigenforms
- VI. Analytic continuation and near holomorphy of Eisenstein series of general types
- VII. Arithmeticity of the critical values of zeta functions and Eisenstein series of general types