On generalizations of Fermat curves over finite fields and their automorphisms

Abstract

Let $\mathcal{X}$ be an irreducible algebraic curve defined over a finite field ${\mathbb{F}}_q$ of characteristic $p>2$. Assume that the ${\mathbb{F}}_q$-automorphism group of $\mathcal{X}$ admits as an automorphism group the direct product of two cyclic groups $C_m$ and $C_n$ of orders $m$ and $n$ prime to $p$ such that both quotient curves $\mathcal{X}/C_n$ and $\mathcal{X}/C_m$ are rational. In this paper, we provide a complete classification of such curves, as well as a characterization of their full automorphism groups.