Moments of $L$-functions via a relative trace formula

Abstract

We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,Π\otimesπ)$, where $π$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $Π$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $Π$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,Π\otimesπ_1)$ and $L(1/2,Π\otimesπ_2)$ do not vanish simultaneously where $π_1$ and $π_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central characters.