On a Rankin-Selberg convolution of two variables for Siegel modular forms

Abstract

In this article we study a Rankin-Selberg convolution of two complex variables attached to Siegel modular forms of degree 2. We establish its basic analytic properties, find its singular curves and compute some of its residues. In particular, we show that two known Dirichlet series of Rankin-Selberg type, one introduced by Maass and another by Kohnen and Skoruppa, are obtained as residues from this series of two variables. Furthermore, we define and study a collection of Rankin-Selberg convolutions for Jacobi forms of degree 1.

Rankin-Selberg Method for Jacobi Forms of Integral Weight and of Half-Integral Weight on Symplectic Groups

“…We remark that if n = r = t = 1, then the above residue coincides with Proposition 1(a) in [IM03]. However, Proposition 1 (a) in [IM03] should read…”

On characteristic twists of multiple Dirichlet series associated to Siegel cusp forms

“…Remark This theorem is equivalent to Corollary 1 in [7]. Notice however a counting error in the latter.…”

On convolutions of Siegel modular forms

“…In [6] we studied such a series in more detail and determined its main analytic properties. In particular, we showed its analytic continuation to C 2 , described all its functional equations and found its singular curves.…”

“…In particular, we showed its analytic continuation to C 2 , described all its functional equations and found its singular curves. Moreover, we computed the residue of the function on two specific singular curves and obtained the RankinSelberg convolutions defined by Maass and Kohnen-Skoruppa. In this article the results of [6] are generalized to Siegel cusp forms over Sp n (Z) and Jacobi cusp forms over Sp n (Z) Z j,n × Z j,n . More precisely, we first consider the Selberg Eisenstein series E(Y ; w) associated to the general linear group of degree n. This function depends on the variable w = (w 1 , w 2 , .…”