Rapoport-Zink uniformization for the moduli space of polarized K3 surfaces

TL;DR

This paper computes the p-adic period map for polarized supersingular K3 surfaces, leading to a Rapoport-Zink type uniformization of their moduli space via a local Shimura variety, with a novel discrete group action.

Contribution

It introduces a new uniformization of the moduli space of polarized supersingular K3 surfaces using explicit local Shimura varieties, differing from classical cases by the discrete group action.

Findings

Explicit description of the p-adic period map image.

Construction of a Rapoport-Zink type uniformization.

Application to moduli of supersingular cubic fourfolds.

Abstract

We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local Shimura variety of orthogonal type. In contrast to the case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the complex case, the uniformizing domain does not carry an action of a $p$-adic Lie group, but only of a discrete subgroup. We briefly sketch how the same arguments can be applied to obtain a uniformization for the moduli space of smooth cubic fourfolds with supersingular reduction.