The Number of Isomorphism Classes of Beauville Surfaces with Beauville $p$-Group

TL;DR

This paper extends the enumeration of Beauville surfaces to include cases where the Beauville group is a non-abelian metacyclic p-group or a p-group of nilpotency class 2, beyond previously studied abelian and PSL(2,p) groups.

Contribution

It provides explicit counts of Beauville surfaces for non-abelian metacyclic p-groups and p-groups of nilpotency class 2, expanding known classifications.

Findings

Counts for Beauville surfaces with non-abelian metacyclic p-groups.

Counts for Beauville surfaces with p-groups of nilpotency class 2.

Extension of existing formulas to new classes of p-groups.

Abstract

A Beauville surface is a rigid complex surface of general type, isogenous to a higher product by the free action of a finite group $G$, called a Beauville group. In \cite{GT}, Gonz\'alez-Diez and Torres-Teigell find the number of isomorphism classes of Beauville surfaces for which the group $G$ is $\PSL(2,p)$ with particular types of `Beauville structures'. On the other hand, in \cite{GJT}, Gonz\'alez-Diez, Jones and Torres-Teigell give an explicit formula for this number when the group $G$ is abelian. To the best of the author's knowledge, in the literature, the exact number of isomorphism classes of Beauville surfaces is given only for $\PSL(2,p)$ and for abelian groups. In this paper, we extend the result for Beauville surfaces with abelian $p$-group to Beauville surfaces for which the Beauville group is either a non-abelian metacyclic $p$-group or a $p$-group of nilpotency class $2$.