Theta functions and Siegel-Jacobi forms (Q1372992)

The author introduces a more general type of theta functions with characteristics than the classical ones. This allows him to compute certain graded rings of Siegel modular forms of genus \(g\) for theta-type subgroups of arbitrary level. In particular, this result generalizes \textit{J. I. Igusa's} fundamental lemma [Am. J. Math. 86, 219-246 (1964; Zbl 0146.31704)]. Considering theta functions instead of theta nullwerte, the author obtains similar results for Jacobi forms of genus \(g\), where the special Eichler/Zagier-case \(g=1\) is pointed out in greater detail. As an application, he gives a description of the Shioda surfaces and a compactification of the universal abelian variety.