Artin-Schreier curves given by ${\mathbb{F}}_q$-linearized polynomials

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.