On mirrors of elliptically fibered K3 surfaces

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.