×

Dirichlet series and modular forms of the second degree. (Dirichletsche Reihen und Modulformen zweiten Grades.) (German) Zbl 0273.10022

This paper is a contribution to the application of differential operators in the theory of modular forms. The method is treated in the author’s lectures on “Siegel’s modular forms and Dirichlet series” [Lect. Notes Math. 216. Berlin etc.: Springer Verlag (1971; Zbl 0224.10028)]. Let \(\Omega_n\) be the real symplectic group, defined via the \(2n\times 2n\) alternating matrix \(I = \begin{pmatrix} 0 & E \\ -E & 0 \end{pmatrix}\) the \(n\times n\) identity matrix, let \(\Gamma_n\subset\Omega_n\), be the modular group of Siegel, \(A_n\) the subgroup consisting of all matrices in \(\Gamma_n\) whose lower left \(n\times n\) corner is \(0\), and let \(\Delta_n\) be the group generated by \(A_n\) and \(I\).
Define \(\Omega_n^*\) analogously, when \(I\) is replaced by \(I^* = \begin{pmatrix} 0 & E \\ E & 0 \end{pmatrix}\). Then \(\Omega_n^*\) operates an the space \(\mathcal H_n^*\) of all \(n\times n\) real matrices \(W\) satisfying \(T = \tfrac12 (W+W') >0\) in the same way as \(\Omega_n\) operates on the Siegel half space \(\mathcal H_n\). The author considers the Eisenstein series

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
PDFBibTeX XMLCite
Full Text: DOI EuDML Link