Improvements of The Weil Bound For Artin-Schreier Curves

TL;DR

This paper improves the Weil bound for counting rational points on Artin-Schreier curves over finite fields, significantly reducing the error term in the estimate.

Contribution

It introduces a method to enhance the Weil bound, effectively removing a large square root factor in the error term for these curves.

Findings

Weil bound can be substantially improved for Artin-Schreier curves.

Error term in point count estimates is reduced.

Enhanced bounds apply over extension fields of finite fields.

Abstract

For Artin-Schreier curve y^q -y = f(x) defined over a finite field F_q of q elements, we show that the Weil bound for the number of the rational points over extension fields of F_q can often be greatly improved, essentially removing an extra factor of size about the square root of q in the error term.