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

TL;DR

This paper establishes new subconvex bounds for Rankin-Selberg $L$-functions on $ ext{GL}(n+1) imes ext{GL}(n)$, proves nonvanishing results, and introduces a novel relative trace formula using advanced harmonic analysis techniques.

Contribution

It introduces a new relative trace formula and derives subconvex bounds and nonvanishing results for high-rank automorphic $L$-functions, advancing analytic number theory methods.

Findings

Established subconvex bounds in the $t$-aspect for $ ext{GL}(n+1) imes ext{GL}(n)$

Proved explicit lower bounds for nonvanishing of central $L$-values

Developed a new relative trace formula approach

Abstract

Let $\pi'$ be a fixed unitary cuspidal representation of $\mathrm{GL}(n)/\mathbb{Q}.$ We establish a subconvex bound in the $t$-aspect $$ L(1/2+it,\pi\times\pi')\ll_{\pi,\pi',\varepsilon}(1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4\cdot (4n^2+2n-1)}+\varepsilon}, $$ for any unitary pure isobaric automorphic representation $\pi$ of $\mathrm{GL}(n+1)/\mathbb{Q}.$ Moreover, the bound improves in the standard $L$-function case $$ L(1/2+it, \pi')\ll_{\pi',\varepsilon}(1+|t|)^{\frac{n}{4}-\frac{1}{4(n+1)(4n-1)}+\varepsilon}. $$ We also prove an explicit lower bound for nonvanishing of central $L$-values $$ \sum_{\pi\in\mathcal{A}_0}\textbf{1}_{L(1/2,\pi\times\pi')\neq 0}\gg_{\varepsilon}|\mathcal{A}_0|^{\frac{1}{n(n+1)(4n^2+2n-1)}-\varepsilon}, $$ for a suitable finite family $\mathcal{A}_0$ of unitary cuspidal representations of $\mathrm{GL}(n+1)/\mathbb{Q}.$ 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 $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (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}.