A Tannakian framework for $G$-displays and Rapoport-Zink spaces

TL;DR

This paper introduces a Tannakian framework for group-theoretic displays and uses it to define and analyze Rapoport-Zink spaces, generalizing previous definitions and confirming conjectures in specific cases.

Contribution

It develops a new Tannakian approach to group-theoretic displays and defines Rapoport-Zink functors as quotient stacks, extending prior work and verifying a conjecture for unramified EL-type data.

Findings

Rapoport-Zink functors form quotient stacks via Witt vector loop groups

The new framework generalizes existing group-theoretic Rapoport-Zink spaces

Confirmed a conjecture relating different definitions in unramified EL-type cases

Abstract

We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by B\"ultel and Pappas, and further studied by Lau. We use this framework to define Rapoport-Zink functors associated to triples $(G,\{\mu\},[b])$, where $G$ is a flat affine group scheme over $\mathbb{Z}_p$ and $\mu$ is a cocharacter of $G$ defined over a finite unramified extension of $\mathbb{Z}_p$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport-Zink spaces given by B\"ultel and Pappas. As an application, we prove a special case of a conjecture of B\"ultel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data.