Subconvexity for $GL(3)\times GL(2)$ twists (with an appendix by Will Sawin)

Abstract: Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass cusp form and $\chi$ be any non-trivial character $\bmod \, p$, where $p$ is prime. We show that the $L$-function associated with this triplet satisfy \begin{equation*} L\left(\frac{1}{2},\pi\times f\times\chi\right)\ll_{\pi,f,\epsilon} p^{\frac{3}{2}-\frac{1}{16}+\epsilon}. \end{equation*} The method also yields the subconvex bound \begin{equation*} L\left(\frac{1}{2},\pi\otimes \chi\right)\ll_{\pi,\epsilon }p^{\frac{3}{4}-\frac{1}{32}+\epsilon }. \end{equation*}