Hybrid subconvexity bounds for twists of $\rm GL (3)\times GL(2)$ $L$-functions

Abstract

In this paper, we solve the hybrid subconvexity problem for $\rm GL (3)\times GL (2)$ $L$-functions twisted by a primtive Dirichlet charater modulo $M$ (prime) in the $M$- and $t$-aspects. We also improve hybrid subconvexity bounds for twists of $\rm GL(3)$ $L$-functions in the $M$- and $t$-aspects.