Quadratic Forms and Hecke Operators

1. Theta-Series.- x1.1. Definition of Theta-Series.- 1.1.1. Representations of Quadratic Forms by Quadratic Forms.- 1.1.2. Definition of Theta-Series.- x 1.2. Symplectic Transformations.- 1.2.1. The Symplectic Group.- 1.2.2. The Siegel Upper Half-Plane.- x1.3. 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.- x1.4. 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.- x2.1. 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.- x 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.- x 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.- x 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.- x 2.5. Scalar Product and Orthogonal Decomposition.- 2.5.1. Scalar Product.- 2.5.2. Orthogonal Decomposition.- 3. Hecke Rings.- x3.1. 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.- x3.2. Hecke Rings of the General Linear Group.- 3.2.1. Global Rings.- 3.2.2. Local Rings.- 3.2.3. The Spherical Mapping.- x 3.3. Hecke Rings of the Symplectic Group.- 3.3.1. Global Rings.- 3.3.2. Local Rings.- 3.3.3. The Spherical Mapping.- x 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.- x3.5. 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.- x4.1. Hecke Operators for Congruence Subgroups of the Modular Group.- 4.1.1. Hecke Operators.- 4.1.2. Invariant Subspaces and Eigenfunctions.- x4.2. 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).- x4.3. 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.- x 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.- x 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.