A subconvex bound for twisted $L$-functions

Abstract

Let $\mathfrak{q}>2$ be a prime number, $\ddot{I}‡$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$L(1/2,f\otimes \ddot{I}‡)\ll {\mathfrak{q}}^{1/2-1/12+\epsilon },$$ where the implicit constant depends only on the archimedean parameter of $f$ and $\epsilon$. The main input is a modifying trivial delta method developed in [1].

12 pages