Theta Functions

By Jun-ichi Igusa

Book Overview

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.

• Format: Paperback
• Language: English
• ISBN: 3642653170
• ISBN13: 9783642653179
• Release Date: November 2011
• Publisher: Springer
• Length: 234 Pages
• Weight: 0.79 lbs.
• Dimensions: 0.5" x 6.1" x 9.2"