On transformation laws for theta functions
The results by Siegel on the transformation laws for theta functions (depending on a single complex variable) attached to quadratic forms are well known. The results have been generalized for theta functions of higher degree. \textit{A. N. Andrianov} and \textit{G. N. Maloletkin} [Math. USSR, Izv. 9 (1975), 227--241 (1976); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 39, 243-258 (1975; Zbl 0316.10017) and Proc. Steklov Inst. Math. 148, 1--12 (1980); translation from Tr. Mat. Inst. Steklova 148, 5--15 (1978; Zbl 0443.10023)] studied the transformation properties of theta series depending on one complex \(n\times n\) matrix variable corresponding to positive definite and also indefinite quadratic forms. \textit{C. Ziegler} [Abh. Math. Semin. Univ. Hamb. 59, 191--224 (1989; Zbl 0707.11035)] developed a theory of holomorphic Jacobi forms of higher degree and showed that theta functions (depending on two complex matrix variables) attached to positive definite quadratic forms are examples of such forms. The present paper generalizes Ziegler's results and defines a theta function of higher degree depending on a complex \(n\times n\) matrix variable \(Z\) and a complex \(j\times n\) matrix variable \(W\), attached to an indefinite quadratic form and determines the behavior of the theta function under modular transformations. As an immediate consequence the paper obtains the transformation law of the function under modular transformations.
scientific article; zbMATH DE number 2143236
scientific article