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

Abstract

Let $\pi$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form satisfying the Ramanujan conjecture and the Selberg-Ramanujan conjecture, and let $\chi$ be a primitive Dirichlet character modulo $M$, which we assume to be prime for simplicity. We will prove the following subconvex bound $$ L\left(\tfrac{1}{2},\pi\otimes\chi\right)\ll_{\pi,\varepsilon} M^{\frac{3}{4}-\frac{1}{1612}+\varepsilon}. $$