$L$-function for $\mathrm{Sp}(4)\times\mathrm{GL}(2)$ via a non-unique model
Pan Yan
TL;DR
The paper proves Ginzburg–Soudry’s conjecture by constructing a global integral for $L^S(s, \pi\times\tau)$ on $\mathrm{Sp}_4\times\mathrm{GL}_2$ via Cai–Friedberg–Ginzburg–Kaplan’s generalized doubling method, which unfolds to a non-unique local model and is analyzed with the New Way method. It first establishes an unramified local identity, then extends to ramified and archimedean places, and finally assembles a global Eulerian identity: $\mathcal{Z}(\varphi, \theta_{\psi}^{\Phi}, E^{*,S}(\cdot, f)) = L^S(s+\tfrac{1}{2}, \pi\times\tau) \cdot \mathcal{Z}_S(\varphi, \Phi, f_{\mathcal{W}(\tau\otimes\chi_T,2,\psi_{2T}),s}^*)$. The authors derive pole–period correspondences and prove holomorphy of $L^S(s, \pi\times\tau)$ for certain self-dual $\tau$ when $L(\tfrac{1}{2}, \tau\otimes\chi_T)=0$, thereby contributing to the understanding of tensor product $L$-functions via non-unique models and theta/Eisenstein machinery. The work advances integral representations for $G\times\mathrm{GL}_k$ with $k>1$ and connects analytic properties of $L$-functions to explicit automorphic periods and residues.
Abstract
In this paper we prove a conjecture of Ginzburg and Soudry on an integral representation for the $L$-function $L^S(s, π\times τ)$ attached to a pair $(π, τ)$ of irreducible automorphic cuspidal representations of $\mathrm{Sp}_4({\mathbb A})$ and $\mathrm{GL}_2({\mathbb A})$, which is derived from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis. Two applications are given. First, we relate the existence of the poles of $L^S(s,π\timesτ)$ to the non-vanishing of certain period integrals. Second, for certain family of cuspidal representations, we prove that $L^S(s, π\times τ)$ is holomorphic.