Über Kernfunktionen für Jacobiformen und Siegelsche Modulformen. (Kernel functions for Jacobi forms and Siegel modular forms) (Q909703)

Let \(H_ n\) be the Siegel half-space and \(\Gamma_ n\) the Siegel modular group of degree n. A Jacobi form \(\Phi\) of degree n is an automorphic form on \(H_ n\times {\mathbb{C}}^ n\), which may also be considered as an automorphic form on \(H_{n+1}\) with respect to the parabolic subgroup \(C_{n+1}\), consisting of all \(m\in \Gamma_{n+1}\), whose last row is the last unit vector, and with respect to the standard factor of automorphy. Given \(\Phi\) one has to assign the function \(\Phi^*(z_ 1,z_ 2)e^{2\pi itz_ 4}\), where t is the index of \(\Phi\) and \(z=\left( \begin{matrix} z_ 1\\ z_ 2\end{matrix} \begin{matrix} z_ 2\\ z_ 4\end{matrix} \right)\in H_{n+1}\) of type (n,1). Extending his earlier paper [Abh. Math. Semin. Univ. Hamb. 57, 165-178 (1987; Zbl 0616.10021)] the author demonstrates how to reformulate the metrization theory for Siegel modular forms of degree \(n+1\) in order to obtain the main results of the metrization theory for the Fourier Jacobi coefficients of degree n. Then an integral representation is derived for all Jacobi cusp forms. The reproducing kernel is related with well known functions from the theory of Siegel modular forms.