Moduli Stacks of Polarized K3 Surfaces in Mixed Characteristic

TL;DR

This paper constructs and analyzes moduli stacks of polarized K3 surfaces in mixed characteristic, establishing their representability and smoothness, which lays groundwork for further arithmetic and geometric studies of K3 surfaces.

Contribution

It proves that moduli functors of polarized K3 surfaces with level structures are representable by smooth algebraic spaces over parts of Spec(Z).

Findings

Moduli functors are representable by Deligne-Mumford stacks.

Moduli spaces with level structures are smooth algebraic spaces.

Results enable further study of K3 surfaces in mixed characteristic.

Abstract

In this note we define moduli functors of (primitively) polarized K3 spaces. We show that they are representable by Deligne-Mumford stacks over Spec(Z). Further, we look at K3 spaces with a level structure. Our main result is that the moduli functors of K3 spaces with a primitive polarization of degree 2d and a level structure are representable by smooth algebraic spaces over open parts of Spec(Z). To do this we use ideas of Grothendieck, Deligne, Mumford, Artin and others. These results are the starting point for the theory of complex multiplication for K3 surfaces and the definition of Kuga-Satake abelian varieties in positive characteristic given in our Ph.D. thesis.