On the number of rational points of Artin-Schreier curves and hypersurfaces

Abstract

Let ${\mathbb{F}}_{q^n}$ denote the finite field with $q^n$ elements. In this paper we determine the number of ${\mathbb{F}}_{q^n}$-rational points of the affine Artin-Schreier curve given by $y^q-y=x(x^{q^i}-x)-\lambda$ and of the Artin-Schreier hypersurface $y^q-y={\sum }_{j=1}^r{a}_j{x}_j(x_j^{q^{i_j}}-x_j)-\lambda .$ Moreover in both cases, we show that the Weil bound is attained only in the case where the trace of $\lambda \in {\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ is zero. We use quadratic forms and permutation matrices to determine the number of affine rational points of these curves and hypersurfaces.