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

Abstract

Recently R. Khan and M. Young proved a mean Lindelöf 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 Lindelöf 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 $|t_j{|}^{2/3+\delta }\ll t\ll |t_j{|}^{6/7-\delta }$.