Deformations of Prismatic Higher $(G,μ)$-Displays over Quasi-Syntomic Rings

TL;DR

This work develops a deformation-theoretic framework for prismatic higher $(G,\mu)$-displays over quasi-syntomic rings, introducing a G-structure-enhanced prismic analog of displays and their Banality. The authors establish a Grothendieck–Messing style deformation theorem for adjoint-nilpotent prismatic higher $(G,\mu)$-displays, enabling a classification of $p$-divisible groups over a broader class of rings via prismatic $F$-crystals. They construct and study stacks of prismatic higher $(G,\mu)$-displays on multiple sites, relate these objects to Witt displays and Breuil–Kisin–Fargues modules, and connect the theory to shtukas and integral local Shimura varieties through a functor $\text{Fib}_G$. When the integral local Shimura variety is representable by the diamond of a formal scheme, the deformation theory yields an equivalence between prismatic $(G,\mu)$-displays (with quasi-syntomic inputs) and Rapoport–Zink–type moduli, highlighting a deep bridge between prismatic cohomology, $G$-structures, and arithmetic geometry.

Abstract

We prove a deformation theorem for prismatic higher $(G,μ)$-displays over quasi-syntomic rings. As an application, we extend the classification of $p$-divisible groups via prismatic Dieudonné modules to a class of rings properly containing quasi-syntonic rings. Finally, we related the stack of prismatic higher $(G,μ)$-displays to integral local Shimura varieties.

Theorems & Definitions (240)

• Theorem 1: Theorem \ref{['grothendique-messing']}
• Theorem 2: Theorem \ref{['ClasspDivGrps']}
• Theorem 3: Theorem \ref{['equivalencetolocalshimura']}
• Conjecture 4
• Definition 1.1
• Proposition 1.2
• Proposition 1.3
• Definition 1.4
• Remark 1.5
• Proposition 1.6: Rigidity