Kummer sandwiches and Greene-Plesser construction

TL;DR

This paper explores the construction of K3 surfaces via Greene-Plesser orbifolding, extending known mirror symmetry relations, and provides explicit formulas for point-counts and isogeny transformations in related families.

Contribution

It generalizes the relation between Dwork pencils and quartic mirror families to Kummer surfaces from a three-parameter family, including explicit point-count formulas and isogeny behavior.

Findings

Derived a formula for rational point-counts of the three-parameter family.

Established the relation between Kummer surfaces and mirror symmetry.

Analyzed the transformation behavior under $(2,2)$-isogenies.

Abstract

In the context of K3 mirror symmetry, the Greene-Plesser orbifolding method constructs a family of K3 surfaces, the mirror of quartic hypersurfaces in $\mathbb{P}^3$, starting from a special one-parameter family of K3 varieties known as the quartic Dwork pencil. We show that certain K3 double covers obtained from the three-parameter family of quartic Kummer surfaces associated with a principally polarized abelian surface generalize the relation of the Dwork pencil and the quartic mirror family. Moreover, for the three-parameter family we compute a formula for the rational point-count of its generic member and derive its transformation behavior with respect to $(2,2)$-isogenies of the underlying abelian surface.