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Full Description
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.
Contents
1. Theta-Series.- §1.1. Definition of Theta-Series.- § 1.2. Symplectic Transformations.- §1.3. Symplectic Transformations of Theta-Series.- §1.4. Computation of the Multiplier.- 2. Modular Forms.- §2.1. Fundamental Domains for Subgroups of the Modular Group.- § 2.2. Definition of Modular Forms.- § 2.3. Fourier Expansions.- § 2.4. Spaces of Modular Forms.- § 2.5. Scalar Product and Orthogonal Decomposition.- 3. Hecke Rings.- §3.1. Abstract Hecke Rings.- §3.2. Hecke Rings of the General Linear Group.- § 3.3. Hecke Rings of the Symplectic Group.- § 3.4. Hecke Rings of the Triangular Subgroup of the Symplectic Group.- §3.5. Factorization of Symplectic Polynomials.- 4. Hecke Operators.- §4.1. Hecke Operators for Congruence Subgroups of the Modular Group.- §4.2. Action of Hecke Operators.- §4.3. Multiplicative Properties of Fourier Coefficients.- 5. The Action of Hecke Operators on Theta-Series.- § 5.1. The Action of Hecke Operators on Theta-Series.- § 5.2. Theta-Matrices of Hecke Operators and Eichler Matrices.- 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.
