Minimal cubic surfaces over finite fields

TL;DR

This paper investigates the Galois action on minimal cubic surfaces over finite fields, classifies possible conjugacy classes in the Weyl group, and provides explicit examples for certain cases.

Contribution

It offers a partial classification of Galois conjugacy classes associated with minimal cubic surfaces over finite fields and supplies explicit examples.

Findings

Identifies 5 conjugacy classes corresponding to minimal cubic surfaces.

Provides explicit examples for some conjugacy classes.

Offers partial answers to the classification problem.

Abstract

Let $X$ be a minimal cubic surface over a finite field $\mathbb{F}_q$. The image $\Gamma$ of the Galois group $\operatorname{Gal}(\overline{\mathbb{F}}_q / \mathbb{F}_q)$ in the group $\operatorname{Aut}(\operatorname{Pic}(\overline{X}))$ is a cyclic subgroup of the Weyl group $W(E_6)$. There are $25$ conjugacy classes of cyclic subgroups in $W(E_6)$, and $5$ of them correspond to minimal cubic surfaces. It is natural to ask which conjugacy classes come from minimal cubic surfaces over a given finite field. In this paper we give a partial answer to this question and present many explicit examples.