Relative representability and parahoric level structures

TL;DR

This paper develops a criterion for representability of certain modifications of formal schemes and applies it to moduli spaces of parahoric level structures on local shtukas, impacting local Shimura varieties.

Contribution

It introduces a new representability criterion for v-sheaf modifications and applies it to study integral models of local Shimura varieties and their morphisms.

Findings

Established local representability of integral models of local Shimura varieties under hyperspecial levels.

Analyzed geometric quotients of perfectoid formal schemes by profinite groups.

Studied forgetful morphisms between integral models of Shimura varieties.

Abstract

We establish a representability criterion of $v$-sheaf theoretic modifications of formal schemes and apply this criterion to moduli spaces of parahoric level structures on local shtukas. In the proof, we introduce nice classes of equivariant profinite perfectoid covers and study geometric quotients of perfectoid formal schemes by profinite groups. As a corollary, we show the local representability of integral models of local Shimura varieties under hyperspecial levels, and study the forgetful morphisms between integral models of Shimura varieties associated with inclusions of parahoric subgroups under hyperspecial levels.