\(L\)-functions of \(S_{3}(\Gamma (4,8))\)
Let \(S_3(\Gamma(4, 8))\) denote the space of Siegel modular cusp forms of weight 3 on the congruence subgroup \(\Gamma(4,8)\subset\text{Sp}_2(\mathbb{Z})\) containing all matrices \(\left(\begin{smallmatrix} A & B\\ C & D\end{smallmatrix}\right)\) such that \(\text{diag\,}B\equiv \text{diag\,}C\equiv 0\bmod 8\). In previous work, \textit{B. van Geemen} and \textit{D. van Straten} [Math. Comput. 61, No. 204, 849--872 (1993; Zbl 0787.11018)] and \textit{B. van Geemen} and \textit{N. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)] determined the Hecke eigenforms belonging to \(S_3(\Gamma(4, 8))\), and computing some Hecke operators for these forms they stated a list of seven conjectures for the Andrianov \(L\)-functions of these forms (outside of 2). The main theorem of the paper under review says that six of these conjectures are true. The corresponding formulae express the Andrianov \(L\)-functions in terms of \(L\)-functions of certain elliptic cusp forms. The Yoshida lift is a crucial tool in the author's arguments. -- Concerning the last of the aforementioned conjectures the author expresses his belief that this one will also be true though he expects that a proof will require more preparations.
scientific article; zbMATH DE number 5161938