2025/09/18 by Leolin Nkuete, Nkuete, Leolin, Antigona Pajaziti +5
Computer Science · Mathematics · #14H25 (Primary) 11G20 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2509.14871
openalex publication_date 2025/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A maximal curve over a finite field \mathbb Fq is a curve whose number of points reaches the upper Hasse-Weil-Serre bound. We define the discriminant of \mathbb Fq as d(\mathbb Fq):= \lfloor2√(q)\rfloor2-4q, which arises as the discriminant of the characteristic polynomial of the Frobenius for a maximal elliptic curve defined over \mathbb Fq. In this article we investigate the existence of a maximal curve of genus 5 defined over a finite field \mathbb Fq of discriminant -19. Using the knowledge on the automorphism group of such a curve, we prove that such curve does not exist when q≡ 2,3,4 \mod 5. In the case q≡ 1\mod 5 we give models of the potential maximal curve. Finally, for the case q≡ 0\bmod 5, we prove that such a curve might exist only for q=57.