The numerous explicit formulae of the classical theory of quadratic forms revealed remarkable multiplicative properties of the numbers of integral representations of integers by positive definite integral quadratic forms. These properties were explained by the original theory of Hecke operators. As regards the integral representations of quadratic forms in more than one variable by quadratic forms, no multiplicative properties were known at that time, and so there was nothing to explain. However, the idea of Hecke operators was so natural and attractive that soon attempts were made to cultivate it in the neighbouring field of modular forms of several variables. The approach has proved to be fruitful; in particular, a number of multiplicative properties of integral representations of quadratic forms by quadratic forms were eventually discovered. By now the theory has reached a certain maturity, and the time has come to give an up-to-date report in a concise form, in order to provide a solid ground for further development. The purpose of this book is to present in the form of a self-contained text-book the contemporary state of the theory of Hecke operators on the spaces of hoi om orphic modular forms of integral weight (the Siegel modular forms) for congruence subgroups of integral symplectic groups. The book can also be used for an initial study of modular forms of one or several variables and theta-series of positive definite integral quadratic forms.

Table of Contents:
1. Theta-Series.- 1.1. Definition of Theta-Series.- 1.1.1. Representations of Quadratic Forms by Quadratic Forms.- 1.1.2. Definition of Theta-Series.- 1.2. Symplectic Transformations.- 1.2.1. The Symplectic Group.- 1.2.2. The Siegel Upper Half-Plane.- 1.3. Symplectic Transformations of Theta-Series.- 1.3.1. Transformations of Theta-Functions.- 1.3.2. The Siegel Modular Group and the Theta-Group.- 1.3.3. Symplectic Transformations of Theta-Series.- 1.4. Computation of the Multiplier.- 1.4.1. Factors of Automorphy.- 1.4.2. Quadratic Forms of Level 1.- 1.4.3. The Multiplier as a Gaussian Sum.- 1.4.4. Quadratic Forms in an Even Number of Variables.- 1.4.5. Quadratic Forms in an Odd Number of Variables.- 2. Modular Forms.- 2.1. Fundamental Domains for Subgroups of the Modular Group.- 2.1.1. The Modular Triangle.- 2.1.2. The Minkowski Domain.- 2.1.3. The Fundamental Domain for the Siegel Modular Group.- 2.1.4. Subgroups of Finite Index.- 2.2. Definition of Modular Forms.- 2.2.1. Congruence Subgroups of the Modular Group.- 2.2.2. Modular Forms of Integral Weight.- 2.2.3. Modular Forms of Half-Integral Weight.- 2.2.4. Theta-Series as Modular Forms.- 2.3. Fourier Expansions.- 2.3.1. Modular Forms for Triangular Subgroups.- 2.3.2. Koecher's Effect.- 2.3.3. Fourier Expansions of Modular Forms.- 2.3.4. The Siegel Operator.- 2.3.5. Cusp Forms.- 2.3.6. Singular Forms.- 2.4. Spaces of Modular Forms.- 2.4.1. Zeroes of Modular Forms for ?1.- 2.4.2. Modular Forms Whose Initial Fourier Coefficients Equal Zero.- 2.4.3. Dimension of Spaces of Modular Forms.- 2.5. Scalar Product and Orthogonal Decomposition.- 2.5.1. Scalar Product.- 2.5.2. Orthogonal Decomposition.- 3. Hecke Rings.- 3.1. Abstract Hecke Rings.- 3.1.1. Averaging over Double Cosets.- 3.1.2. Hecke Rings.- 3.1.3. The Imbedding i.- 3.1.4. The Anti-Isomorphism j.- 3.1.5. Representations on Automorphic Functions.- 3.1.6. Hecke Rings over a Commutative Ring.- 3.2. Hecke Rings of the General Linear Group.- 3.2.1. Global Rings.- 3.2.2. Local Rings.- 3.2.3. The Spherical Mapping.- 3.3. Hecke Rings of the Symplectic Group.- 3.3.1. Global Rings.- 3.3.2. Local Rings.- 3.3.3. The Spherical Mapping.- 3.4. Hecke Rings of the Triangular Subgroup of the Symplectic Group.- 3.4.1. Global Rings.- 3.4.2. Local Rings.- 3.4.3. Decomposition of Elements Tn(a) for n = 1, 2.- 3.5. Factorization of Symplectic Polynomials.- 3.5.1. "Negative Powers" of Frobenius Elements.- 3.5.2. Factorization of Symplectic Polynomials.- 3.5.3. A Symmetrie Factorization of Polynomials Qpn(v) for n = 1, 2.- 3.5.4. Coefficients of Factors of Polynomials Rpn(v).- 3.5.5. A Symmetrie Factorization of Polynomials Rpn(v).- 4. Hecke Operators.- 4.1. Hecke Operators for Congruence Subgroups of the Modular Group.- 4.1.1. Hecke Operators.- 4.1.2. Invariant Subspaces and Eigenfunctions.- 4.2. Action of Hecke Operators.- 4.2.1. Hecke Operators for ?0n(q).- 4.2.2. Hecke Operators for ?0n.- 4.2.3. Relations with Hecke Operators for GLn.- 4.2.4. Hecke Operators and the Siegel Operator.- 4.2.5. The Action of the Middle Factor of the Symmetrie Factorization of Polynomials Rpn(v).- 4.3. Multiplicative Properties of Fourier Coefficients.- 4.3.1. Modular Forms of One Variable.- 4.3.2. Modular Forms of Genus 2, Gaussian Composition, and Zeta-Functions.- 4.3.3. Modular Forms of Arbitrary Genus and Even Zeta-Functions.- 5. The Action of Hecke Operators on Theta-Series.- 5.1. The Action of Hecke Operators on Theta-Series.- 5.1.1. Theta-Series and ?-Series.- 5.1.2. The Basic Case.- 5.1.3. The General Case.- 5.2. Theta-Matrices of Hecke Operators and Eichler Matrices.- 5.2.1. The Comparison of Fourier Coefficients.- 5.2.2. Theta-Matrices of Elements Tn(p).- 5.2.3. Theta-Matrices of Coefficients of Even Local Zeta-Functions.- 5.2.4. Theta-Matrices of Generators of Hecke Rings.- 5.2.5. Relations, Relations.- Appendix 1. Symmetrie Matrices over Fields.- A.1.1. Arbitrary Fields.- A.1.2. The Field ?.- Appendix 2. Quadratic Spaces.- A.2.1. The Geometrie Language.- A.2.2. Non-Degenerate Spaces.- A.2.3. Gaussian Sums.- A.2.5. Non-Singular Spaces over Residue Class Rings.- A.2.6. The Genus of Quadratic Spaces over ?.- Appendix 3. Modules in Quadratic Fields and Binary Quadratic Forms.- A.3.1 Modules of Algebraic Number Fields.- A.3.2 Modules in Quadratic Fields and Prime Numbers.- A.3.3 Modules in Imaginary Quadratic Fields and Quadratic Forms.- Notes.- On Chapter 1.- On Chapter 2.- On Chapter 3.- On Chapter 4.- On Chapter 5.- References.- Index of Terminology.- Index of Notation.