A subconvex bound for twisted $L$-functions

TL;DR

This paper establishes a subconvex bound for twisted L-functions involving primitive cusp forms and Dirichlet characters, improving the known bounds by employing a novel delta method.

Contribution

It introduces a new delta method to achieve the first subconvex bound for twisted L-functions in the level aspect.

Findings

Proves the bound $L(1/2,f\otimes \chi) \ll \mathfrak{q}^{1/2 - 1/12 + \varepsilon}$.

Develops a modified trivial delta method for analytic number theory.

Advances understanding of subconvexity in the context of automorphic L-functions.

Abstract

Let $\mathfrak{q}>2$ be a prime number, $\chi$ 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 $$ L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1].