On some generalized Rapoport-Zink spaces

TL;DR

This paper extends the class of Rapoport-Zink spaces to include abelian type, establishing their relation to Shimura varieties, and applies these results to the study of supersingular K3 surfaces in mixed characteristic.

Contribution

It introduces Rapoport-Zink spaces of abelian type, enlarging the existing framework, and demonstrates their connection to Shimura varieties and supersingular K3 surfaces.

Findings

Rapoport-Zink spaces of abelian type are strictly larger than of Hodge type.

Shimura varieties of abelian type can be uniformized by these spaces.

Supersingular K3 surfaces' Artin invariants relate to local invariants.

Abstract

We enlarge the class of Rapoport-Zink spaces of Hodge type by modifying the centers of the associated $p$-adic reductive groups. These such-obtained Rapoport-Zink spaces are called of abelian type. The class of Rapoport-Zink spaces of abelian type is strictly larger than the class of Rapoport-Zink spaces of Hodge type, but the two type spaces are closely related as having isomorphic connected components. The rigid analytic generic fibers of formal Rapoport-Zink spaces of abelian type can be viewed as moduli spaces of local $G$-shtukas in mixed characteristic in the sense of Scholze. We prove that Shimura varieties of abelian type can be uniformized by the associated Rapoport-Zink spaces of abelian type. We construct and study the Ekedahl-Oort stratifications for the special fibers of Rapoport-Zink spaces of abelian type. As an application, we deduce a Rapoport-Zink type uniformization for the supersingular locus of the moduli space of polarized K3 surfaces in mixed characteristic. Moreover, we show that the Artin invariants of supersingular K3 surfaces are related to some purely local invariants.