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

TL;DR

The paper addresses the problem of simultaneous non-vanishing of central values of $L$-functions of the forms $GL(2)\times GL(3)$ and $GL(3)\times GL(3)$. It develops a method to compute the first moment $\sum_{F} L(1/2,g\times F)L(1/2,\Pi\times F)$ over an orthonormal basis of Hecke-Maaß cusp forms for $\mathrm{SL}(3,\mathbb{Z})$ by leveraging the $GL(3)$ Kuznetsov formula and the GL(3) Voronoi summation. The main result is an asymptotic formula with a power-saving error term, indicating the existence of infinitely many $F$ for which $L(1/2,g\times F)L(1/2,\Pi\times F)\neq 0$, thereby connecting the central values of mixed Rankin–Selberg and self-dual $GL(3)\times GL(3)$ twists. This work advances understanding of how different automorphic representations interact at the central point and contributes to the broader program of non-vanishing results and Ramanujan-type conjecture progress.

Abstract

Let $g$ denotes a fixed holomorphic Hecke cusp form of weight $k\equiv 0$ (mod 4) on SL(2,$\mathbb{Z}$), and $Π$ is a fixed cuspidal automorphic representation on SL(3,$\mathbb{Z}$). In this paper, we give an asymptotic formula for average of $L(1/2,g\times F)L(1/2,Π\times F)$, and device that $L(1/2,g\times F)L(1/2,Π\times F)\neq 0$ for $F$ ranges over an orthogonal basis of the space of Hecke-Maaß cusp forms for SL(3,$\mathbb{Z}$).

Theorems & Definitions (9)

• Theorem 1.1
• Corollary 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7