Inhalt

I. Theta Functions from an Analytic Viewpoint.- § 1. Preliminaries.- § 2. Plancherel Theorem for Rn.- § 3. The Group A(X).- §4. The Irreducibility of U.- § 5. Induced Representations.- § 6. The Group Sp(X).- §7. The Group B(X).- § 8. Fock Representation.- §9. The Set G(X).- § 10. The Discrete Subgroup ?L.- II. Theta Functions from a Geometric Viewpoint.- § 1. Hodge Decomposition Theorem for a Torus.- § 2. Theta Function of a Positive Divisor.- § 3. The Automorphy Factor u?(z).- §4. The Vector Space L(Q, l, ?).- § 5. A Change of the Canonical Base.- III Graded Rings of Theta Functions.- § 1. Graded Rings.- § 2. Algebraic and Integral Dependence.- § 3. Weierstrass Preparation Theorem.- § 4. Geometric Lemmas.- § 5. Automorphic Forms and Projective Embeddings.- § 6. Polarized Abelian Varieties.- § 7. Projective Embeddings.- §8. The Field of Abelian Functions.- IV. Equations Defining Abelian Varieties.- § 1. Theta Relations (Classical Forms).- § 2. A New Formalism.- § 3. Theta Relations (Under the New Formalism).- §4. The Ideal of Relations.- § 5. Quadratic Equations Defining Abelian Varieties.- V. Graded Rings of Theta Constants.- § 1. Theta Constants.- §2. Some Properties of ?(?)2.- § 3. Holomorphic Mappings by Theta Constants.- § 4. The Classical Reduction Theory.- § 5. Modular Forms.- § 6. The Group of Characteristics.- § 7. Modular Varieties.- Sources.- Further References and Comments.- Index of Definitions.