Lattice Polarized K3 Surfaces and Siegel Modular Forms

TL;DR

This paper classifies lattice polarized K3 surfaces, describes their moduli space and period map using Siegel modular forms, and establishes a Hodge correspondence linking these surfaces to abelian surfaces via a geometric two-isogeny.

Contribution

It provides a detailed classification, explicit descriptions of the moduli space, and a novel Hodge correspondence relating K3 surfaces to abelian surfaces.

Findings

Classification of H+E_8+E_7 polarized K3 surfaces

Explicit formulas for the inverse period map using Siegel modular forms

A geometric two-isogeny linking K3 surfaces to abelian surfaces

Abstract

The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H+E_8+E_7. Key ingredients for this classification are: a normal form for these lattice polarized K3 surfaces, a coarse moduli space and an explicit description of the inverse period map in terms of Siegel modular forms. Second, we give explicit formulas for a Hodge correspondence that relates these K3 surfaces to principally polarized abelian surfaces. The Hodge correspondence in question underlies a geometric two-isogeny of K3 surfaces.