On certain vector valued Siegel modular forms of type ($k$,2) over $mathbb{Z}_{(P)}$ and Sturm bound (Automorphic forms, automorphic representations and related topics)

Abstract

We give generators of the module of vector valued Siegel modular forms of type (k, 2) over the ring of scalar valued Siegel modular forms of even weight for Z(p). This gives an example of the positive solution to more general problem whether the module of vector valued modular forms of arbitrary degree is finitely generated over the ring of modular forms for Z(p). Moreover we study analogues of Sturm's bounds for vector valued Siegel modular forms of type (k, 2).