Quantitative non-vanishing of central values of certain $L$-functions on ${\rm GL}(2)\times {\rm GL}(3)$

TL;DR

This paper derives an exact average formula for triple products involving a fixed Hecke-Maass form and elliptic cusp forms, leading to quantitative non-vanishing results for certain degree 6 $L$-functions at their central point.

Contribution

It provides a new exact formula for triple product averages and applies it to establish non-vanishing of central $L$-values for a family of degree 6 $L$-functions.

Findings

Exact average formula for triple products involving $ m SL_2(b Z)$ cusp forms.

Quantitative non-vanishing results for degree 6 $L$-functions at the center.

Non-vanishing holds for a large family of $L$-functions as the weight grows.

Abstract

Let $\phi$ be an even Hecke-Maass cusp form on ${\rm SL}_2(\mathbb{Z})$ whose $L$-function does not vanish at the center of the functional equation. In this article, we obtain an exact formula of the average of triple products of $\phi$, $f$ and $\bar f$, where $f$ runs over an orthonormal basis $H_k$ of Hecke eigen elliptic cusp forms on ${\rm SL}_2(\mathbb{Z})$ of a fixed weight $k\geq 4$. As an application, we prove a quantitative non-vanishing results on the central values for the family of degree $6$ $L$-functions $L(s,\phi \times {\rm Ad}\,f)$ with $f$ in the union of $H_k$ $({\rm K} \leq k < 2{\rm K})$ as ${\rm K}\rightarrow \infty$.