Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect

Abstract: In this paper, we study the second moment for $GL(2)\times GL(2)$ $L$-functions $L(\frac{1}{2},f\times g)$, which leads to a uniform subconvexity bound in the spectral aspect. In particular, if either $f$ or $g$ is a dihedral Maass newform, or if one of them has level $1$, we obtain a Burgess-type bound that is uniform in both $t_f$ and $t_g$, where $t_f$, $t_g$ denote the spectral parameters of $f$, $g$. As an application, we also establish a shrinking result for QUE in the case of dihedral Maass newforms.