Arithmeticity in the Theory of Automorphic Forms (Mathematical Surveys and Monographs) - Goro Shimura, American Mathematical Society (2000)
ISBN 0821826719
Subject Automorphic forms
Publisher American Mathematical Society
Publication Date September 2000
Format Hardcover (254 x 184 mm)
Language e
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.

Personal Details
Collection Status In Collection
Index 87
Read It Yes
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Product Details
LoC Classification QA331 .S4645 2000
Dewey 512/.7
Cover Price $72.00
No. of Pages 302