Relative Trace Formula and $L$-functions for $\mathrm{GL}(n+1)\times \mathrm{GL}(n)$

Abstract: We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and the geometric side consists of Rankin-Selberg $L$-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphic functions. The formula yields new results towards central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second moment evaluation, and simultaneous nonvnaishing in the level aspect.