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

2022/11/21 by Martínez, Fabio Enrique Brochero, de Oliveira, Daniela Alves
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Primary 12E20 Secondary 11T24

paper · pdf · doi:10.48550/arxiv.2211.11371

openalex publication_date 2022/11/21 · openalex created_date 2022/11/29 · openalex updated_date 2026/07/28

Abstract

Let \mathbb F_qⁿ denote the finite field with qⁿ elements. In this paper we determine the number of \mathbb F_qⁿ-rational points of the affine Artin-Schreier curve given by y^q-y = x(x^qi-x)-λ and of the Artin-Schreier hypersurface y^q-y=∑_j=1^r a_jx_j(x_j^q^i_j-x_j)-λ. Moreover in both cases, we show that the Weil bound is attained only in the case where the trace of λ∈\mathbb F_qⁿ 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.

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