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

TL;DR

This paper investigates the simultaneous non-vanishing of specific central values of $L$-functions associated with $GL(2)$ and $GL(3)$, providing insights into their behavior when varying over certain automorphic forms.

Contribution

It establishes new results on the simultaneous non-vanishing of $L$-functions for $GL(2)$ and $GL(3)$, advancing understanding of their non-zero values at the critical point.

Findings

Proves non-vanishing of $L(1/2, g imes ext{Di})$ and $L(1/2, ext{Di})$ simultaneously for many forms.

Provides bounds and density results for the non-vanishing set.

Enhances knowledge of the distribution of central $L$-values in automorphic families.

Abstract

We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$ Hecke cusp form of weight $k\equiv 0 \pmod 4$.