Hybrid subconvexity for Maass form symmetric-square $L$-functions

TL;DR

This paper advances subconvexity bounds for symmetric-square Maass form L-functions by establishing mean Lindel"of estimates in shorter intervals, leading to improved subconvexity results within a specific spectral range.

Contribution

It introduces shorter interval mean Lindel"of estimates for symmetric-square L-functions, enabling new subconvexity bounds in a broader spectral range.

Findings

Mean Lindel"of estimate in shorter intervals

Subconvexity bounds for symmetric-square L-functions

Extended spectral range for subconvexity results

Abstract

Recently R. Khan and M. Young proved a mean Lindel\"{o}f estimate for the second moment of Maass form symmetric-square $L$-functions $L(\text{sym}^2 u_{j},1/2+it)$ on the short interval of length $G\gg |t_j|^{1+\epsilon}/t^{2/3}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\text{sym}^2 u_{j},1/2+it)$ as long as $|t_j|^{6/7+\delta} \ll t<(2-\delta)|t_j|$. We obtain a mean Lindel\"{o}f estimate for the same moment in shorter intervals, namely for $G\gg |t_j|^{1+\epsilon}/t$. As a corollary, we prove a subconvexity estimate for $L(\text{sym}^2 u_{j},1/2+it)$ on the interval $|t_j|^{2/3+\delta}\ll t\ll |t_j|^{6/7-\delta}$.