Lattice polarized K3 surfaces and Siegel modular forms

The goal of the present paper is two-fold. First, we present a classification of algebraic K3 surfaces polarized by the lattice H⊕E8⊕E7. Key ingredients for this classification are as follows: 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, the details of which are described by the authors in Clingher and Doran (2011) [7].