On the Rankin–Selberg method for vector-valued Siegel modular forms

Abstract

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard [Formula: see text]-function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted [Formula: see text]-function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.