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

TL;DR

This paper introduces a novel circle method approach to establish a hybrid subconvex bound for twisted L-functions of cusp forms, improving the known bounds in terms of conductor and spectral parameter.

Contribution

It develops a new circle method technique to derive hybrid subconvex bounds for L-functions twisted by characters, applicable to forms of arbitrary level and nebentypus.

Findings

Established a subconvex bound for twisted L-functions involving conductor and spectral parameter.

The bound improves previous results by reducing the exponent in the subconvexity estimate.

The method is applicable to both Hecke-Maass and holomorphic cusp forms.

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 $$ L(1/2+it,f\otimes\chi)\ll (M(3+|t|))^{1/2-1/18+\varepsilon}, $$ for $t\in \mathbb R$. The implied constant depends only on the form $f$ and $\varepsilon$.