Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions

Abstract: Let $f$ and $g$ be two holomorphic or Hecke-Maass primitive cusp forms for $SL(2,\mathbb{Z})$ and $\chi$ be a primitive Dirichlet character of modulus $p$, an odd prime. A subconvex bound for the central values of the Rankin-Selberg $L$-functions is $L(s, f \otimes g \otimes \chi)$ is given by $$L(\frac{1}{2}, f \otimes g \otimes \chi) \ll_{f,g,\epsilon}p^{\frac{27}{28}+\epsilon} ,$$ for any $\epsilon > 0$, where the implied constant depends only on the forms $f,g$ and $\epsilon$. Here the convexity bound has exponent $1+\epsilon$, which was improved to $1-\frac{1}{1324}$ (see \cite{HM}). Our bound reduces it further to $1- \frac{1}{28}$. The main ingredients is to reduce the original problem to a $GL(2) \times GL(2)$ shifted convolution sum problem.