Normal points on Artin-Schreier curves over finite fields

TL;DR

This paper extends the concept of freeness from multiplicative groups to additive module structures over finite fields, applying it to study the existence of normal points on Artin-Schreier curves with coordinates that are normal over the prime field.

Contribution

It introduces the notion of $(f,g)$-freeness in additive modules over finite fields and applies this to analyze rational points on Artin-Schreier curves with normal coordinates.

Findings

Established conditions for the existence of normal points on Artin-Schreier curves.

Developed a new framework for $(f,g)$-freeness in additive module structures.

Provided concrete results on points with normal coordinates over finite fields.

Abstract

In 2022, S.D. Cohen and the two authors introduced and studied the concept of $(r, n)$-freeness on finite cyclic groups $G$ for suitable integers $r, n$, which is an arithmetic way of capturing elements of special forms that lie in the subgroups of $G$. Combining this machinery with some character sum techniques, they explored the existence of points $(x_0, y_0)$ on affine curves $y^n=f(x)$ defined over a finite field $\mathbb F$ whose coordinates are generators of the multiplicative cyclic group $\mathbb F^*$. In this paper we develop the natural additive counterpart of this work for finite fields. Namely, any finite extension $\mathbb E$ of a finite field $\mathbb F$ with $Q$ elements is a cyclic $\mathbb F[x]$-module induced by the Frobenius automorphism $\alpha\mapsto \alpha^{Q}$, and any generator of this module is said to be a normal element over $\mathbb F$. We introduce and study the concept of $(f, g)$-freeness on this module structure for suitable polynomials $f, g\in \mathbb F[x]$. As a main application of the machinery developed in this paper, we study the existence of $\mathbb F_{p^n}$-rational points in the Artin-Schreier curve $\mathfrak A_f : y^p-y=f(x)$ whose coordinates are normal over the prime field $\mathbb F_p$ and establish concrete results.