Exponential sums in prime fields for modular forms

TL;DR

This paper investigates exponential sums related to Fourier coefficients of modular forms, focusing on numbers with fixed prime factors, and improves existing bounds for these sums over finite fields.

Contribution

It introduces an improved bound on exponential sums associated with recurrence sequences linked to modular forms over finite fields.

Findings

Enhanced bounds for exponential sums over finite fields.

Better understanding of Fourier coefficients in modular forms.

Implications for number theory and finite field analysis.

Abstract

The main objective of this article is to study the exponential sums associated to Fourier coefficients of modular forms supported at numbers having a fixed set of prime factors. This is achieved by establishing an improvement on Shparlinski's bound for exponential sums attached to certain recurrence sequences over finite fields.