Relative representability and parahoric level structures

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 obtain a construction of (part of) integral models of local and global Shimura varieties under hyperspecial levels from those at hyperspecial levels.