**Mathematical Surveys and Monographs**

Volume: 82;
2000;
302 pp;
Softcover

MSC: Primary 11; 14; 32;

Print ISBN: 978-0-8218-4961-3

Product Code: SURV/82.S

List Price: $84.00

Individual Member Price: $67.20

**Electronic ISBN: 978-1-4704-1309-5
Product Code: SURV/82.S.E**

List Price: $84.00

Individual Member Price: $67.20

# Arithmeticity in the Theory of Automorphic Forms

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*Goro Shimura*

*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.*

—

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.

#### Table of Contents

# Table of Contents

## Arithmeticity in the Theory of Automorphic Forms

- Table of Contents v6 free
- Preface vii8 free
- Notation and Terminology ix10 free
- Introduction 112 free
- Chapter I. Automorphic Forms and Families of Abelian Varieties 718 free
- Chapter II. Arithmeticity of Automorphic Forms 4556
- Chapter III. Arithmetic of Differential Operators and Nearly Holomorphic Functions 8798
- Chapter IV. Eisenstein Series of Simpler Types 127138
- Chapter V. Zeta Functions Associated with Hecke Eigenforms 159170
- Chapter VI. Analytic Continuation and Near Holomorphy of Eisenstein Series of General Types 185196
- Chapter VII. Arithmeticity of the Critical Values of Zeta Functions and Eisenstein Series of General Types 219230
- Appendix 247258
- A1. The series associated to a symmetric matrix and Gauss sums 247258
- A2. Metaplectic groups and factors of automorphy 251262
- A3. Transformation formulas of general theta series 262273
- A4. The constant term of a theta series at each cusp depends only on the genus 272283
- A5. Theta series of a hermitian form 274285
- A6. Estimate of the Fourier coefficients of a modular form 278289
- A7. The Mellin transforms of Hilbert modular forms 282293
- A8. Certain unitarizable representation spaces 285296

- References 297308
- Index 301312