Optimal plane curves of degree $q-1$ over a finite field

Abstract

Let $q\ge 5$ be a prime power. In this note, we prove that if a plane curve $\mathcal{X}$ of degree $q-1$ defined over ${\mathbb{F}}_q$ without ${\mathbb{F}}_q$-linear components attains the Sziklai upper bound $(d-1)q+1=(q-1{)}^2$ for the number of its ${\mathbb{F}}_q$-rational points, then $\mathcal{X}$ is projectively equivalent over ${\mathbb{F}}_q$ to the curve ${\mathcal{C}}_{(\alpha ,\beta ,\gamma )}:\alpha X^{q-1}+\beta Y^{q-1}+\gamma Z^{q-1}=0$ for some $\alpha ,\beta ,\gamma \in {\mathbb{F}}_q^{\ast }$ such that $\alpha +\beta +\gamma =0$. This completes the classification of curves that are extremal with respect to the Sziklai bound. Also, since the Sziklai bound is equal to the Stöhr-Voloch's bound for plane curves of degree $q-1$, our main result classifies the ${\mathbb{F}}_q$-Frobenius classical extremal plane curves of degree $q-1$.