Connected Components of the Moduli of Elliptic K3 Surfaces

Abstract

The combinatorial type of an elliptic $K3$ surface with a zero section is the pair of the $\mathit{ADE}$-type of the singular fibers and the torsion part of the Mordell–Weil group. We determine the set of connected components of the moduli of elliptic $K3$ surfaces with fixed combinatorial type. Our method relies on the theory of Miranda and Morrison on the structure of a genus of even indefinite lattices and on computer-aided calculations of $p$-adic quadratic forms.