Singular Frobenius operators on Siegel modular forms with characters, and zeta functions. (Q2704035)

Let \(k,q\) be natural numbers and \(\chi\) a Dirichlet character modulo \(q\). Let \({\mathfrak M}_k^n(q,\chi)\) be the \(\mathbb C\)-linear space of the Siegel modular forms of weight \(k\) and character \(\chi\) for \(\Gamma_0^n(q)\) and \({\mathfrak N}_k^n(q,\chi) \) the subspace of cusp forms. Let NEWLINE\[NEWLINE{\mathfrak M}_k^n(q,q)=\sum_{\chi\bmod q}{\mathfrak M}_k^n(q,\chi),\quad {\mathfrak N}_k^n(q,q)=\sum_{\chi\bmod q}{\mathfrak N}_k^n(q,\chi).NEWLINE\]NEWLINE The authors show that the singular Frobenius operators for the groups \(\Gamma^2(q,q)\) of prime level \(q\) can be diagonalized simultaneously together with the regular Hecke operators on certain subspaces of the space \({\mathfrak N}_k^2(q,q).\) An application is given to Euler factorization of radial Dirichlet series associated with eigenfunctions of the operators under consideration.