Theta Functions [Paperback]

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.Springer Book Archives