Rational points and Galois points for a plane curve over a finite field

TL;DR

This paper explores the connection between rational points and Galois points on plane curves over finite fields, proposing a conjecture and providing partial affirmative results for certain low-genus curves.

Contribution

It introduces a new problem regarding the equivalence of rational and Galois points on plane curves and offers partial solutions for genus at most one curves with rational points.

Findings

Galois points coincide with rational points for specific curves like Hermitian and Klein quartic.

The paper proposes a conjecture about the converse relationship.

Partial affirmative results are obtained for genus ≤ 1 curves with rational points.

Abstract

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico-Hefez curves. We propose a problem: Does the converse hold true? When the curve of genus at most one has a rational point, we will have an affirmative answer.