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

TL;DR

This paper precisely counts the rational points on certain Artin-Schreier curves and hypersurfaces over finite fields, showing the Weil bound is tight only when the trace of a parameter is zero, using quadratic forms and permutation matrices.

Contribution

It provides explicit formulas for the number of rational points on Artin-Schreier curves and hypersurfaces, and characterizes when the Weil bound is attained.

Findings

Number of rational points determined explicitly

Weil bound is attained only when trace of λ is zero

Quadratic forms and permutation matrices are used in the analysis

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 $y^q-y=\sum_{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-\lambda.$ 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.