Bounds for twisted symmetric square $L$-functions

Abstract

Let $f\in S_k(N,\psi )$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell }$. Assume that $q$ is a prime and $\ell >1$. In this paper we describe a method to establish convexity breaking bounds of the form $$ L(\tfrac{1}{2},\Sym f\otimesχ)\ll_{f,\varepsilon} q^{3/4\ell-δ_{\ell}+\varepsilon} $$ for some ${\delta }_{\ell }>0$ and any $\epsilon >0$. In particular, for $\ell =3$ we show that the bound holds with ${\delta }_{\ell }=1/4$.

19 pages, (Extensively revised version)

Cited by in corpus (1)

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