Theta Functions

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Springer Science & Business Media, Dec 6, 2012 - Mathematics - 234 pages
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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.
 

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Contents

Theta Functions from an Analytic Viewpoint
1
2 Plancherel Theorem for R
5
3 The Group AX
8
4 The Irreducibility of U
11
5 Induced Representations
14
6 The Group SpX
20
7 The Group BX
26
8 Fock Representation
31
6 Polarized Abelian Varieties
118
7 Projective Embeddings
125
8 The Field of Abelian Functions
132
Equations Defining Abelian Varieties
136
2 A New Formalism
142
3 Theta Relations Under the New Formalism
146
4 The Ideal of Relations
152
5 Quadratic Equations Defining Abelian Varieties
166

The Discrete Subgroup I
42
Theta Functions from a Geometric Viewpoint
51
Theta Function of a Positive Divisor
58
3 The Automorphy Factor u 2
64
4 The Vector Space LQ l b
71
5 A Change of the Canonical Base
78
Graded Rings of Theta Functions
86
2 Algebraic and Integral Dependence
94
3 Weierstrass Preparation Theorem
99
4 Geometric Lemmas
107
5 Automorphic Forms and Projective Embeddings
112
Chapter W Graded Rings of Theta Constants
173
2 Some Properties of ka
177
3 Holomorphic Mappings by Theta Constants
183
4 The Classical Reduction Theory
189
5 Modular Forms
197
6 The Group of Characteristics
209
7 Modular Varieties
216
Sources
225
Further References and Comments 227
226
Index of Definitions
231
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