Additive Twists of Fourier Coefficients of Modular Forms

TL;DR

This paper investigates bounds on sums of additively twisted Fourier coefficients of various modular forms, providing uniform bounds across forms and twists, and improving existing bounds for certain twisted sums.

Contribution

It introduces new uniform bounds for sums of additively twisted Fourier coefficients of modular forms and enhances previous bounds for twisted sums involving nonlinear exponential terms.

Findings

Established bounds uniform in form and twist parameters.

Improved bounds on sums of twisted Fourier coefficients with nonlinear exponential twists.

Demonstrated uniformity and limitations of bounds across different modular forms.

Abstract

We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.