Non-vanishing of twists of ${\text{GL}}_4({\mathbb{A}}_{\mathbb{Q}})$ $L$-functions

Abstract

Let $\pi$ be a unitary cuspidal automorphic representation of ${\text{GL}}_4({\mathbb{A}}_{\mathbb{Q}})$. Let $f\ge 1$ be given. We show that there exists infinitely many primitive even (resp. odd) Dirichlet characters $\chi$ with conductor co-prime to $f$ such that $L(s,\pi \otimes \chi )$ is non-vanishing at the central point. Our result has applications for the construction of $p$-adic $L$-functions for ${\text{GSp}}_4$ following Loeffler-Pilloni-Skinner-Zerbes, the Bloch-Kato conjecture and the Birch-Swinnerton-Dyer conjecture for abelian surfaces following Loeffler-Zerbes, strong multiplicity one results for paramodular cuspidal representations of ${\text{GSp}}_4({\mathbb{A}}_{\mathbb{Q}})$ and the rationality of the central values of ${\text{GSp}}_4({\mathbb{A}}_{\mathbb{Q}})$ $L$-functions in the remaining non-regular weight case.

45 pages