Artin-Schreier curves given by [formula omitted]-linearized polynomials

Abstract

Let Fq 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 yq−y=x⋅F(x)−λ, where F(x) is a Fq-linearized polynomial and λ∈Fq. Our results provide a characterization of the number of affine rational points of this curve in the extension Fqr of Fq, for gcd⁡(q,r)=1. In the case F(x)=xqi−x we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.