Characterization of a Hermitian curve by Galois point

TL;DR

This paper characterizes Hermitian (Fermat) curves in characteristic p>2 by the number of Galois points on the curve, providing a new perspective on their algebraic structure.

Contribution

It offers a novel characterization of Hermitian curves based on Galois points, linking geometric properties to algebraic function field extensions.

Findings

Hermitian curves have a specific number of Galois points.

The characterization applies when degree minus one is a power of p.

Provides a new criterion for identifying Hermitian curves.

Abstract

For a plane curve, a point in the projective plane is said to be Galois when the point projection induces a Galois extension of function fields. We give a new characterization of a Fermat curve whose degree minus one is a power of $p$ in characteristic $p>2$, which is sometimes called Hermitian, by the number of Galois points lying on the curve.