Jun-ichi Igusa

Theta Functions

• Broschiertes Buch

Produktbeschreibung

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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Produktdetails

• Grundlehren der mathematischen Wissenschaften 194
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-65317-9
• Softcover reprint of the original 1st ed. 1972
• Seitenzahl: 252
• Erscheinungstermin: 11. November 2011
• Englisch
• Abmessung: 235mm x 155mm x 14mm
• Gewicht: 387g
• ISBN-13: 9783642653179
• ISBN-10: 3642653170
• Artikelnr.: 36120258

Inhaltsangabe

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.