Artin-Schreier curves given by $\mathbb F_q$-linearized polynomials

TL;DR

This paper studies Artin-Schreier curves defined by $F_q$-linearized polynomials, providing formulas for counting affine rational points over extensions, with explicit results for certain polynomial cases using quadratic characters.

Contribution

It introduces a novel approach linking circulant matrices and quadratic forms to analyze Artin-Schreier curves, offering explicit point count formulas over finite field extensions.

Findings

Characterization of affine rational points in terms of quadratic forms and circulant matrices.

Explicit point counts for specific linearized polynomials using Legendre symbols.

Complete description of rational points for $F(x) = x^{q^i}-x$ cases.

Abstract

Let $\mathbb F_q$ be a finite field with $q$ elements, where $q$ is a power of an odd prime $p$. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve $y^q - y= x \cdot F(x) - \lambda,$ where $F(x)$ is a $\mathbb F_q$-linearized polynomial and $\lambda \in \mathbb F_q$. Our results provide a characterization of the number of affine rational points of this curve in the extension $\mathbb F_{q^r}$ of $\mathbb F_q$, for $\gcd(q,r)=1$. In the case $F(x) = x^{q^i}-x$ we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.