GL(2) Weyl Bound via a multiplicative character delta method

TL;DR

This paper proves the Weyl bound for GL(2) in the t-aspect using a multiplicative character delta method, offering a new perspective on delta methods and their connection to twisted L-functions.

Contribution

It introduces a novel application of the trivial delta method with multiplicative characters for congruence detection in proving the Weyl bound for GL(2).

Findings

Established the Weyl bound for GL(2) in t-aspect.

Connected multiplicative characters to twisted L-functions.

Provided a new viewpoint on delta methods in analytic number theory.

Abstract

We use a trivial delta method with multiplicative characters for congruence detection to prove the Weyl bound for GL(2) in $t$-aspect for a holomorphic or Hecke-Maass cusp form of arbitrary level and nebentypus. This parallels the work of Aggarwal in 2018, with the difference being multiplicative character has a more natural connection to the twisted $L$-function. This provides another view point to understand and explore the trivial and other delta methods.