On curves over finite fields with many rational points

TL;DR

This paper investigates maximal curves over finite fields, proving under certain conditions that they are isomorphic to specific Artin-Schreier curves, thereby advancing understanding of their structure and properties.

Contribution

It establishes a classification of maximal curves over finite fields as isomorphic to particular Artin-Schreier curves under a non-gap hypothesis.

Findings

Maximal curves are isomorphic to y^q + y = x^m under certain conditions.

The study provides a characterization of maximal curves reaching the Hasse-Weil bound.

Results contribute to the understanding of rational points on algebraic curves over finite fields.

Abstract

We study arithmetical and geometrical properties of {\it maximal curves}, that is, curves defined over the finite field $\mathbb F_{q^2}$ whose number of $\mathbb F_{q^2}$-rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are $\mathbb F_{q^2}$-isomorphic to $y^q+y=x^m$ for some $m\in \mathbb Z^+$.