On Kummer‐like surfaces attached to singularity and modular forms

Abstract

We study a family of lattice polarized 𝐾3 surfaces which is an extension of the family of Kummer surfaces derived from principally polarized Abelian surfaces. Our family has two special properties. First, it is coming from a resolution of a simple 𝐾3 singularity. Second, it has a natural parameterization by Hermitian modular forms of four complex variables. In this paper, we show two results: (1) we determine the transcendental lattice and the Néron-Severi lattice of a generic member of our family. (2) We give a detailed description of the double covering structure associated with our 𝐾3 surfaces.