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

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 $2$.