A Multi-variable Rankin-Selberg Integral for a Product of $GL_2$-twisted Spinor $L$-functions

TL;DR

This paper introduces a new integral representation for the product of two twisted Spinor L-functions associated with GSp_4 and GL_2 representations, revealing new pole conditions and connections to Fourier coefficients and theta functions.

Contribution

It develops a novel multi-variable Rankin-Selberg integral for specific L-functions, linking pole behavior to Fourier coefficients of residual representations and theta functions.

Findings

New integral representation for L(s_1, Π × τ_1) L(s_2, Π × τ_2)

Identifies conditions for simultaneous poles of the L-functions

Suggests deep connections between Fourier coefficients, residual representations, and theta functions

Abstract

We consider a new integral representation for $L(s_1, \Pi \times \tau_1) L(s_2, \Pi \times \tau_2),$ where $\Pi$ is a globally generic cuspidal representation of $GSp_4,$ and $\tau_1$ and $\tau_2$ are two cuspidal representations of $GL_2$ having the same central character. As and application, we find a new period condition for two such $L$ functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on $GSO(12)$ and a theta function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to unfold completely, but in a way that provides further evidence of a connection.