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, $r\geq 1$ 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 $$ L (\frac{1}{2}, f \otimes g ) \ll_{g,\epsilon}p^{\frac{23r}{12} +\epsilon}.$$