On generalizations of Fermat curves over finite fields and their automorphisms

TL;DR

This paper classifies algebraic curves over finite fields with automorphism groups formed by two cyclic groups, where the quotient curves are rational, extending the understanding of Fermat curve generalizations.

Contribution

It provides a complete classification of such curves and characterizes their full automorphism groups, advancing the theory of algebraic curves over finite fields.

Findings

Classified curves with automorphism groups as direct products of two cyclic groups.

Characterized the full automorphism groups of these curves.

Extended the understanding of Fermat curve generalizations over finite fields.

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.