Work

Maximal curves with respect to quadratic extensions over finite fields

Year: 2026

Type: article

Abstract: In this paper, we propose a detailed study of a canonical bound which relates the numbers of rational points of a curve over a finite field with that over its quadratic extension. Alternative proofs which make a connection with the variance enable to obtain optimal refinements. [Formula: see text]We focus on the curves reaching the bound, which we call Hallouin–Perret-maximal curves. We provide different characterizations and stress natural links with the curves which attain the Ihara bound. As consequences, we establish the list of such curves with low genus and we outline a maximality result which involves the Suzuki curves. [Formula: see text]At last, we determine which polynomials correspond to the Jacobian of a Hallouin–Perret-maximal curve of genus 2.

Source: International Journal of Number Theory

DOI: 10.1142/s1793042126501101

Authors: Yves Aubry, Fabien Herbaut, Julien Monaldi