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

We prove an asymptotic formula for the second moment of the $\mathrm{G}\mathrm{L}(n)\times \mathrm{G}\mathrm{L}(n+1)$ Rankin–Selberg central $L$-values $L(1\mathrm{/}2,\mathrm{\Pi }\otimes \pi )$, where $\pi$ is a fixed cuspidal representation of $\mathrm{G}\mathrm{L}(n)$ that is tempered and unramified at every place, while $\mathrm{\Pi }$ varies over a family of automorphic representations of $\mathrm{P}\mathrm{G}\mathrm{L}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $\mathrm{\Pi }$ of $\mathrm{P}\mathrm{G}\mathrm{L}(n+1)$ such that $L(1\mathrm{/}2,\mathrm{\Pi }\otimes {\pi }_1)$ and $L(1\mathrm{/}2,\mathrm{\Pi }\otimes {\pi }_2)$ do not vanish simultaneously where ${\pi }_1$ and ${\pi }_2$ are cuspidal representations of $\mathrm{G}\mathrm{L}(n)$ that are unramified and tempered at every place and have trivial central characters.