Connected components of the moduli of elliptic K3 surfaces

TL;DR

This paper classifies the connected components of the moduli space of elliptic K3 surfaces with fixed fiber types and Mordell-Weil torsion, using lattice theory and computational methods.

Contribution

It provides a complete determination of the moduli space components for elliptic K3 surfaces with given combinatorial data, combining lattice theory and computational techniques.

Findings

Classification of moduli space components for elliptic K3 surfaces.

Application of Miranda and Morrison's lattice theory to moduli problems.

Use of computer-aided p-adic quadratic form calculations.

Abstract

The combinatorial type of an elliptic K3 surface with a zero section is the pair of the ADE -type of 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 computer-aided calculations of p-adic quadratic forms.