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 \[ L\left(\frac{1}{2},\,\mathrm{Sym}^2π\right)\ll_{\varepsilon, π_\infty} q(\mathrm{Sym}^2π)^{\frac{1}{4}-\frac{1}{168}+o(1)}, \] 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.

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