Simultaneous non-vanishing of $GL(3)\times GL(2)$ and GL(2) $L$-functions

TL;DR

This paper investigates the simultaneous non-vanishing of certain $L$-functions at the central point, using averages over families of modular forms to establish new non-vanishing results.

Contribution

It introduces a novel average value approach for $L$-functions associated with $GL(3)$ and $GL(2)$, leading to results on their simultaneous non-vanishing.

Findings

Established average values of $L(rac{1}{2},g imes f)L(rac{1}{2},f)$ over level $q$ forms.

Proved non-vanishing of these $L$-functions at the central point for a positive proportion of forms.

Extended understanding of the behavior of automorphic $L$-functions at the critical point.

Abstract

Fix $g$ a Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $q$ be any large prime number. In the family of holomorphic newforms $f$ of level $q$ and fixed weight, we find the average value of the product $L(\half,g\times f)L(\half,f)$. From this we derive a result on the simultaneous non-vanishing of these $L$-functions at the central point.