Subconvex bound for Rankin-Selberg $L$-functions in prime power level

Abstract

Let $f$ be a $p$-primitive cusp form of level $p^{4r}$, where local representation of $f$ be supercuspidal at $p$, $p$ being an odd prime, $r\ge 1$ and $g$ be a Hecke-Maass or holomorphic primitive cusp form for $\mathrm{SL}(2,\mathbb{Z})$. A subconvex bound for the central values of the Rankin-Selberg $L$-functions $L(s,f\otimes g)$ is given by $$L(\frac{1}{2},f\otimes g){\ll }_{g,\epsilon }{p}^{\frac{23r}{12}+\epsilon }.$$