# On mirrors of elliptically fibered K3 surfaces

## TL;DR

This paper explores a family of elliptically fibered K3 surfaces with Picard rank 18, establishing their mirror relationship to certain K3 surfaces and explicitly connecting them to double covers of Kummer surfaces from elliptic curves.

## Contribution

We explicitly describe a family of K3 surfaces as hypersurfaces in a torus and establish their mirror relationship to elliptically fibered K3 surfaces, linking them to Kummer surfaces.

## Key findings

- Identified a two-parameter family of K3 surfaces with Picard rank 18.
- Established the explicit connection to double covers of Kummer surfaces.
- Provided a description of mirror families for polarized K3 surfaces of Picard rank one.

## Abstract

We study a two-parameter family of K3 surfaces of (generic) Picard rank $18$ which is mirror to the $18$-dimensional family of elliptically fibered K3 surfaces with a section. Members of this family are given as compactifications of hypersurfaces in three dimensional algebraic torus given by equations $y^2+z+z^{-1} + x^3 + ax +b =0$ where $a$ and $b$ are the parameters of the family. It follows from previous work of Morrison that these surfaces are double covers of Kummer surfaces coming from products of elliptic curves, and we establish this connection explicitly. This in turn provides a description of the one-parameter families of K3 surfaces which are mirror to polarized K3 surfaces of Picard rank one. It has been brought to our attention that the results of this paper have already appeared in the literature. The appropriate references are added at the end of the introduction.