The circle method and bounds for $L$-functions - I

Abstract

Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $\chi$ be a primitive character of conductor $M$. For the twisted $L$-function $L(s,f\otimes \chi )$ we establish the hybrid subconvex bound $$L(1/2+it,f\otimes \chi )\ll (M(3+|t|){)}^{1/2-1/18+\epsilon },$$ for $t\in \mathbb{R}$. The implied constant depends only on the form $f$ and $\epsilon$.

8 pages