Cancellation in the Additive Twists of Fourier Coefficients for $\mathrm{GL}_2$ and $\mathrm{GL}_3$ over Number Fields

TL;DR

This paper investigates the cancellation phenomena in additive twists of Fourier coefficients for automorphic representations of GL2 and GL3 over number fields, establishing uniform bounds in the unramified case.

Contribution

It provides new uniform bounds for additive twists of Fourier coefficients for GL2 and GL3 automorphic forms over arbitrary number fields, extending previous results.

Findings

Proves Wilton type bounds for GL2 additive twists.

Establishes Miller type bounds for GL3 additive twists.

Results are uniform in terms of the additive character.

Abstract

In this article, we study the sum of additively twisted Fourier coefficients of an irreducible cuspidal automorphic representation of $\mathrm{GL}_2$ or $\mathrm{GL}_3$ over an arbitrary number field. When the representation is unramified at all non-archimedean places, we prove the Wilton type bound for $\mathrm{GL}_2$ and the Miller type bound for $\mathrm{GL}_3$ which are uniform in terms of the additive character.