An improvement of the Hasse-Weil bound for Artin-Schreier curves via cyclotomic function fields

TL;DR

This paper improves the classical Hasse-Weil bound for Artin-Schreier curves by connecting their function fields to cyclotomic function fields and using class field theory to derive tighter bounds based on linear code minimum distances.

Contribution

It introduces a novel approach linking Artin-Schreier curves to cyclotomic function fields and uses this connection to refine bounds on the number of rational points.

Findings

Improved bounds for the number of rational points on Artin-Schreier curves.

Established a relationship between function fields and cyclotomic fields.

Enhanced the Serre bound for specific classes of curves.

Abstract

The corresponding Hasse-Weil bound was a major breakthrough in history of mathematics. It has found many applications in mathematics, coding theory and theoretical computer science. In general, the Hasse-Weil bound is tight and cannot be improved. However, the Hasse-Weil bound is no longer tight when it is applied to some specific classes of curves. One of the examples where the Hasse-Weil bound is not tight is the family of Artin-Schreier curves. Due to various applications of Artin-Schreier curves to coding, cryptography and theoretical computer science, researchers have made great effort to improve the Hasse-Weil bound for Artin-Schreier curves. In this paper, we focus on the number of rational places of the Artin-Schreier curve defined by $y^p-y=f(x)$ over the finite field $\mathbb{F}_q$ of characteristic $p$, where $f(x)$ is a polynomial in $\mathbb{F}_q[x]$. Our road map for attacking this problem works as follows. We first show that the function field $E_f:=\mathbb{F}_q(x,y)$ of the Artin-Schreier curve $y^p-y=f(x)$ is a subfield of some cyclotomic function field. We then make use of the class field theory to prove that the number of points of the curve is upper bounded by a function of a minimum distance of a linear code. By analyzing the minimum distance of this linear code, we can improve the Hasse-Weil bound and Serre bound for Artin-Schreier curves.