THE MULTIPLICATIVE ARITHMETIC OR SIEGEL MODULAR FORMS

Abstract

CONTENTS Introduction Chapter 1. Hecke rings § 1.1. Definitions and general properties § 1.2. Hecke rings of the groups and Chapter 2. Decompositions of Hecke polynomials § 2.1. Centralizers of Frobenius elements § 2.2. “Negative powers” of Frobenius elements § 2.3. Decompositions of Hecke polynomials § 2.4. Decompositions of the polynomial § 2.5. A symmetric decomposition of Chapter 3. Representations of Hecke rings § 3.1. Siegel modular forms § 3.2. Hecke operators § 3.3. Hecke operators and the Siegel operator § 3.4. The operator polynomial Chapter 4. The action of Hecke operators on theta-series § 4.1. Theta-series § 4.2. Theta-representations of Hecke rings § 4.3. The action of Hecke operators on theta-series § 4.4. Eigenvalues of Hecke operators on spaces of theta-series § 4.5. Invariance of the generic theta-series and Siegel's theorem Chapter 5. The Euler decomposition of theta-transformations of Siegel modular forms § 5.1. The Euler decomposition of theta-transformations § 5.2. On analytic properties of zeta-functions References