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Theta Functions (Hardcover)
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Theta Functions (Hardcover)
Series: Grundlehren der Mathematischen Wissenschaften, 194
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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" ). 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
. 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 . In 1957/58,
H. Cartan took up this theory in his seminar ; 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
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