Introduction to Siegel modular forms and Dirichlet series / Anatoli Andrianov.

Format:

Book

Author/Creator:

Andrianov, A. N. (Anatoliĭ Nikolaevich)

Series:

Universitext

Language:

English

Subjects (All):

Siegel domains.

Dirichlet series.

Physical Description:

xi, 182 pages : illustrations ; 24 cm.

Place of Publication:

New York ; London : Springer, 2009.

Summary:

Introduction to Siegel Modular Forms and Dirichlet Series gives a concise and self-contained introduction to the multiplicative theory of Siegel modular forms, Hecke operators, and zeta functions, including the classical case of modular forms in one variable. It serves to attract young researchers to this beautiful field and makes the initial steps more pleasant. It treats a number of questions that are rarely mentioned in other books. It is the first and only book so far on Siegel modular forms that introduces such important topics an analytic continuation and the functional equation of spinor zeta functions of Siegel modular forms of genus two.

Contents:

Introduction: The Two Features of Arithmetic Zeta Functions 1

1 Modular Forms 7

1.1 The Symplectic Group and the Upper Half-Plane 7

1.2 Fundamental Domains for the Modular Group 11

1.3 Modular Forms 19

2 Dirichlet Series of Modular Forms 41

2.1 Radial Dirichlet Series 41

2.2 Mellin Transform of Cusp Forms in One Variable 44

2.3 Transformations of Lobachevsky Half-Spaces 47

2.4 Radial Series and Eisenstein Series 55

2.5 Properties of Radial Series for Sums of Two Squares 58

3 Hecke-Shimura Rings of Double Cosets 63

3.1 An Approach to Multiplicativity 63

3.2 Abstract Rings of Double Cosets 66

3.3 Rings of Double Cosets of the General Linear Group 75

3.4 Rings of Double Cosets of the Symplectic Group 85

3.5 Rings of Triangular-Symplectic Double Cosets 106

4 Hecke Operators 119

4.1 Hecke Operators for Congruence Subgroups 119

4.2 Action of Hecke Operators for [Gamma subscript 0 superscript n](q) 125

4.3 Hecke Operators and Siegel Operator 130

5 Euler Factorization of Radial Series 137

5.1 Radial Series of Genus One and Zeta Functions 138

5.2 Binary Radial Series and Gaussian Composition 145

5.3 Zeta Functions of Eigenforms for Genus 2 165

Conclusion: Other Groups, Other Horizons 169.

Notes:

Includes bibliographical references (pages 175-177) and index.

ISBN:

9780387787527

0387787526

OCLC:

288986689