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

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 $y^q - y= x \cdot F(x) - λ,$ where $F(x)$ is a $\mathbb F_q$-linearized polynomial and $λ\in \mathbb F_q$. 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.

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