The subconvexity problem for symmetric square $L$-functions in level aspect

Abstract

In this paper, we address the subconvexity problem in level aspect for symmetric square $L$-functions for cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{Q})$ with a prescribed local ramification at prime $p$. To be more precise, let $\pi$ be a tempered cuspidal automorphic representation of conductor $q(\pi)=p^2$ with a non-quadratic central character of conductor $p$. We prove that if the corresponding local representation $\pi_p$ belongs to a suitable class of representations $\mathcal S$, then \[ L\left(\frac{1}{2},\,\mathrm{Sym}^2\pi\right)\ll_{\varepsilon, \pi_\infty} q(\mathrm{Sym}^2\pi)^{\frac{1}{4}-\frac{1}{168}+o(1)}, \] where implied constant depends polynomially on the spectral parameters of $\pi_\infty$. This is the first instance of level-aspect subconvex bound for $L$-functions of a $\mathrm{GL}_3(\mathbb Q)$ automorphic representation. Our approach is based on the delta-symbol method. Apart from some standard analytic number theoretic tools, Katz's theory of hypergeometric sums, and Deligne's proof of Weil-conjectures play an important role in the proof.