Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin-Selberg $L$-functions for $\mathrm{G}\mathrm{L}(n+1)\times \mathrm{G}\mathrm{L}(n)$

Let ${\pi }^{\mathrm{\prime }}$ be a fixed unitary cuspidal representation of $\mathrm{G}\mathrm{L}(n)\mathrm{/}\mathbb{Q}\mathrm{.}$ We establish a subconvex bound in the $t$-aspect $$L(1\mathrm{/}2+it,\pi \times {\pi }^{\mathrm{\prime }}){\ll }_{\pi ,{\pi }^{\mathrm{\prime }},\epsilon }(1+\mathrm{\mid }t\mathrm{\mid }{)}^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\epsilon },$$ for any unitary pure isobaric automorphic representation $\pi$ of $\mathrm{G}\mathrm{L}(n+1)\mathrm{/}\mathbb{Q}\mathrm{.}$ Moreover, the bound improves in the standard $L$-function case $$L(1\mathrm{/}2+it,{\pi }^{\mathrm{\prime }}){\ll }_{{\pi }^{\mathrm{\prime }},\epsilon }(1+\mathrm{\mid }t\mathrm{\mid }{)}^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\epsilon }\mathrm{.}$$ We also prove an explicit lower bound for nonvanishing of central $L$-values $$\sum _{\pi \in {\mathcal{A}}_0}{\mathbf{\text{1}}}_{L(1\mathrm{/}2,\pi \times {\pi }^{\mathrm{\prime }})\mathrm{\ne }0}{\gg }_{\epsilon }\mathrm{\mid }{\mathcal{A}}_0{\mathrm{\mid }}^{\frac{1}{n(n+1)(4n^2+2n-1)}-\epsilon },$$ for a suitable finite family ${\mathcal{A}}_0$ of unitary cuspidal representations of $\mathrm{G}\mathrm{L}(n+1)\mathrm{/}\mathbb{Q}\mathrm{.}$ More generally, we address the spectral side subconvexity in the case of uniform parameter growth, and a quantitative form of simultaneous nonvanishing of central $L$-values for $\mathrm{G}\mathrm{L}(n+1)\times \mathrm{G}\mathrm{L}(n)$ (over $\mathbb{Q}$) in both level and eigenvalue aspects. Among other ingredients, our proofs employ a new relative trace formula in conjunction with P. Nelson’s construction of archimedean test functions in \cite{Nel21} and volume estimates in \cite{Nel20}.