On singular modular forms for principal congruence subgroups
Singular automorphic forms on the symplectic group have the property that they are linear combinations of theta series. This well-known property is due to \textit{R. Howe} [Lect. Notes Math. 880, 211--248 (1981; Zbl 0463.10015)]. It is still an open and delicate task to study the level of the theta series and forms involved. In this paper the author considers vector-valued singular modular forms for principal congruence subgroup of the Siegel modular group \(\text{Sp}(n,\mathbb{Z})\). In this important case the author solves the problem completely [see also \textit{E. Freitag}, Singular modular forms and theta relations, Lecture Notes in Mathematics. 1487. Berlin etc.: Springer-Verlag (1991; Zbl 0744.11024)]. The proof involves techniques related to a certain trace operator. Finally the author indicates several generalizations. He makes remarks towards the Igusa subgroup and the optimal level of the theta series.
scientific article; zbMATH DE number 5071439
scientific article