Relatives of the Hermitian curve

Abstract

We introduce the notion of a relative of the Hermitian curve of degree $\sqrt{q}+1$ over $\mathbb{F}_q$, which is a plane curve defined by \[(x^{\sqrt{q}}, y^{\sqrt{q}}, z^{\sqrt{q}})A {}^t \!(x,y,z) =0\] with $A \in GL(3, \mathbb{F}_q)$, and study their basic properties, one of which is that the number of $\mathbb{F}_q$-points of any relative of the Hermitian curve of degree $\sqrt{q}+1$ is congruent to $1$ modulo $\sqrt{q}$. In the latter part of this paper, we classify those curves having two or more rational inflexions.