Connected components of the moduli of elliptic K3 surfaces

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.