Central Values of $GL(2)\times GL(3)$ Rankin-Selberg $L$-functions with Applications

TL;DR

This paper computes the average central values of certain $L$-functions associated with $GL(2)$ and $GL(3)$ forms using trace formulas, leading to non-vanishing results and form identification.

Contribution

It introduces a novel application of the Kuznetsov trace formula to evaluate the first moment of $L$-functions for $GL(2) imes GL(3)$, providing new insights into their non-vanishing and uniqueness.

Findings

First moment of $L(1/2,f\otimes \phi)$ computed.

Non-vanishing of $L$-values established.

$f$ is uniquely determined by these $L$-values.

Abstract

Let $f$ be a normalized holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k$ with $k\equiv0\bmod 4$. By the Kuznetsov trace formula for $GL_3(\mathbb R)$, we obtain the first moment of central values of $L(s,f\otimes \phi)$, where $\phi$ varies over Hecke-Maass cusp forms for $SL_3(\mathbb Z)$. As an application, we obtain a non-vanishing result for $L(1/2,f\otimes\phi)$ and show that such $f$ is determined by $\{L(1/2,f\otimes\phi)\}$ as $\phi$ varies.