The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points
Zhi Qi
TL;DR
The paper advances the mean-Lindelöf analysis for the second moment of ${\mathrm{GL}}_3\times{\mathrm{GL}}_2$ L-functions at special points by focusing on short spectral intervals $T< t_j \le T+\sqrt{T}$. It develops an asymptotic large sieve framework and a SpecialTwisted Large Sieve, combining Kuznetsov trace formula, Voronoï summation for ${\mathrm{SL}}_3({\mathbb Z})$, and stationary-phase analysis to control diagonal and off-diagonal contributions. Importantly, it executes a direct Voronoï-based treatment of the off-diagonal without relying on Poisson summation to detect Eisenstein--Kloosterman cancellation, bounding the sums via a hybrid large sieve and Rankin--Selberg estimates. The main result demonstrates a mean-Lindelöf-type bound in short intervals, $\sum_{T< t_j \le T+M} \omega_j|L(1/2+i t_j, \phi\times u_j)|^2 \ll M T^{1+\varepsilon}$ for $\sqrt{T} \le M \le T$, plus additional control in the full construction, which marks progress toward subconvexity for these Rankin--Selberg L-functions. The techniques have potential implications for subconvexity problems in higher rank and illustrate how a refined spectral large sieve can be combined with Voronoï summation in a short-interval setting.
Abstract
Let $φ$ be a fixed Hecke--Maass form for $\mathrm{SL}_3 (\mathbb{Z})$ and $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb{Z}) $. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j $. In this paper, we prove the mean Lindelöf hypothesis for the second moment of $ L (1/2+it_j, φ\times u_j) $ on $ T < t_j \leqslant T + \sqrt{T} $. Previously, this was proven by Young on $ t_j \leqslant T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation.