The circle method and bounds for $L$-functions - II: Subconvexity for twists of GL(3) $L$-functions

Abstract: Let $\pi$ be a $SL(3,\mathbb Z)$ automorphic form. Let $\chi=\chi_1\chi_2$ be a Dirichlet character with $\chi_i$ primitive modulo $M_i$. Suppose $M_1$, $M_2$ are primes such that $\sqrt{M_2}M^{4\delta}<M_1<M_2M^{-3\delta}$, where $M=M_1M_2$ and $0<\delta<1/28$. In this paper we will prove the following subconvex bound $$ L(\t1/2,\pi\otimes\chi)\ll_{\pi,\varepsilon} M^{3/4}-\delta+\varepsilon}. $$