The theory of theta functions has a long history; for this, we refer A. Krazer and W. Wirtinger the reader to an encyclopedia article by ("Sources" [9]). We shall restrict ourselves to postwar, i. e. , after 1945, periods. Around 1948/49, F. Conforto, c. L. Siegel, A. Well reconsidered the main existence theorems of theta functions and found natural proofs for them. These are contained in Conforto: Abelsche Funktionen und algebraische Geometrie, Springer (1956); Siegel: Analytic functions of several complex variables, Lect. Notes, I. A. S. (1948/49); Well: Theoremes fondamentaux de la theorie des fonctions theta, Sem. Bourbaki, No. 16 (1949). The complete account of Weil's method appeared in his book of 1958 [20]. The next important achievement was the theory of compacti- fication of the quotient variety of Siegel's upper-half space by a modular group. There are many ways to compactify the quotient variety; we are talking about what might be called a standard compactification. Such a compactification was obtained first as a Hausdorff space by I. Satake in "On the compactification of the Siegel space", J. Ind. Math. Soc. 20, 259-281 (1956), and as a normal projective variety by W. L. Baily in 1958 [1]. In 1957/58, H. Cartan took up this theory in his seminar [3]; it was shown that the graded ring of modular forms relative to the given modular group is a normal integral domain which is finitely generated over C.

Table of Contents:
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.