L-function for Sp(4)\timesGL(2) via a non-unique model

2021/10/12 by Yan, Pan
#11F70 (Primary) 11F55 #22E50 #22E55 (Secondary) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2110.05693

Abstract

In this paper we prove a conjecture of Ginzburg and Soudry on an integral representation for the L-function L^S(s, π× τ) attached to a pair (π, τ) of irreducible automorphic cuspidal representations of Sp₄(\mathbb A) and GL₂(\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. As applications, we determine the poles of L^S(s, π× τ) and relate the existence of the poles to the non-vanishing of certain period integrals. Moreover, for certain family of cuspidal representations τ of GL₂(\mathbb A), we prove that L^S(s, π× τ) is holomorphic.

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