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

Abstract

We establish hybrid subconvexity bounds and quantitative simultaneous nonvanishing for Rankin--Selberg $L$-functions on $\mathrm{GL}(n+1)\times \mathrm{GL}(n)$ over $\mathbb{Q}$. For a fixed unitary cuspidal representation ${\pi }^{\prime }$ of $\mathrm{GL}(n)$ and any unitary pure isobaric representation $\pi$ of $\mathrm{GL}(n+1)$, we prove the $t$-aspect bound \begin{align*} L(1/2+it,π\timesπ')\ll_{π,π',\varepsilon} (1+|t|)^{\frac{n(n+1)}{4}-\frac{1}{4(3n^2+n-1)}+\varepsilon}. \end{align*} When $\pi$ is tempered, the saving improves to $1/(4(3n-1))$; in particular, the resulting bound for standard $L$-functions improves the general-rank exponent of Nelson. We also obtain, in both the spectral and level aspects, the first quantitative simultaneous nonvanishing result in higher rank for central Rankin--Selberg values in suitable cuspidal families. Our proofs use an amplified, regularized relative trace formula whose spectral expansion retains the full generic spectrum and is expressed directly in terms of central values. On the geometric side, we estimate collective combinations of regular orbital integrals rather than the individual orbits separately. This structure leads to an optimized count of rational parameters, sharper amplification bounds, and a treatment of the level aspect. In rank one, the resulting exponents agree with the Burgess bounds.

67 pages