Bounds for twisted symmetric square $L$-functions - III

TL;DR

This paper establishes a subconvexity bound for twisted symmetric square L-functions associated with newforms and primitive characters, advancing understanding of their growth and distribution in the conductor aspect.

Contribution

It proves a new subconvex bound for symmetric square L-functions twisted by primitive characters, improving previous bounds in the conductor aspect.

Findings

Proved a subconvexity bound: $L(1/2, ext{Sym} f imes \chi) \,\ll_{f,q,\varepsilon} q^{3\ell(1/4-1/36+\varepsilon)}$

The result compares favorably with recent t-aspect subconvexity results for symmetric square L-functions

Advances the understanding of the size and distribution of twisted symmetric square L-functions in the conductor aspect.

Abstract

Let $f$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is an odd prime. In this paper we prove the subconvex bound $$ L(\t1/2,\Sym f\otimes\chi)\ll_{f,q,\varepsilon} q^{3\ell(1/4-1/36+\varepsilon)} $$ for any $\varepsilon>0$. This can be compared with the recently established $t$-aspect subconvexity of the symmetric square $L$-functions.