Quadratic Forms and Hecke Operators - Anatolij N. Andrianov

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Sommaire:
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.