## Theta FunctionsThe 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

1 | |

5 | |

3 The Group AX | 8 |

4 The Irreducibility of | 11 |

5 Induced Representations | 14 |

The Group SpX | 20 |

The Group B | 26 |

8 Fock Representation 9 The Set 3 | 31 |

2 Algebraic and Integral Dependence | 94 |

3 Weierstrass Preparation Theorem | 99 |

4 Geometric Lemmas | 107 |

5 Automorphic Forms and Projective Embeddings | 112 |

Polarized Abelian Varieties | 120 |

Projective Embeddings | 125 |

8 The Field of Abelian Functions | 132 |

Equations Defining Abelian Varieties 1 Theta Relations Classical Forms | 136 |

The Discrete Subgroup I | 42 |

Theta Functions from a Geometric Viewpoint | 52 |

Hodge Decomposition Theorem for a Torus | 54 |

Theta Function of a Positive Divisor | 58 |

3 The Automorphy Factor u | 64 |

4 The Vector Space LQ l y 5 A Change of the Canonical Base | 78 |

Graded Rings of Theta Functions | 86 |

Graded Rings | 90 |

2 A New Formalism | 142 |

3 Theta Relations Under the New Formalism | 146 |

4 The Ideal of Relations | 152 |

Quadratic Equations Defining Abelian Varieties | 166 |

Chapter W Graded Rings of Theta Constants | 173 |

225 | |

231 | |

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algebraic arbitrary element arbitrary point assume automorphism automorphy factor bicontinuous isomorphism biholomorphic C-base canonical base Chap closed set commutative complete set complex manifold contained convergent defined degree g denote an arbitrary denote an element dimension e=0 mod exists an element finite number form on Zx Fourier graded ring Haar measure hence holomorphic function holomorphic mapping homogeneous implies integral domain irreducible isomorphism kernel lattice Lemma Lemma 12 Let a denote Lº(X modular forms monoid Moreover multiplication non-degenerate normal notation observe open neighborhood open subset polarized abelian variety positive divisor positive integer positive-definite projective embedding projective variety prove quotient recall Riemann form satisfying sequence set of representatives Sp(X subring subspace Suppose surjective symmetric theta constants theta function unique unitary representation vector space Weierstrass polynomial Zariski closed set Zºe