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Deformation theory for prismatic $G$-displays

Kazuhiro Ito

TL;DR

This work develops a Grothendieck–Messing style deformation theory for G-μ-displays over the Bhatt–Scholze prismatic site, establishing universal deformation objects over the ring R_{G,μ} and connecting deformations to prismatic F-gauges. It shows that deformations over Breuil–Kisin prisms and over perfectoid prisms are captured by bijective evaluation maps, with a Kodaira–Spencer criterion characterizing universality. The theory yields two major applications: first, formal smoothness and local representability of integral local Shimura varieties with hyperspecial level, proving conjectures of Pappas–Rapoport; second, a refined perspective on p-divisible group classifications via prismatic Dieudonné crystals and their BK_min realizations. Overall, the paper integrates prismatic geometry with display theory to provide a versatile deformation toolkit applicable to integral models of Shimura varieties and p-divisible groups, and it clarifies the interaction with prismatic G–F-gauges in the GL_N setting. The results bridge Grothendieck–Messing deformation theory, Breuil–Kisin theory, and prismatic Dieudonné crystals, yielding a robust framework for studying deformations in p-adic Hodge-theoretic contexts.

Abstract

For a smooth affine group scheme $G$ over the ring of $p$-adic integers and a cocharacter $μ$ of $G$, we develop the deformation theory for $G$-$μ$-displays over the prismatic site of Bhatt-Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic $F$-gauges introduced by Drinfeld and Bhatt-Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of $p$-divisible groups.

Deformation theory for prismatic $G$-displays

TL;DR

This work develops a Grothendieck–Messing style deformation theory for G-μ-displays over the Bhatt–Scholze prismatic site, establishing universal deformation objects over the ring R_{G,μ} and connecting deformations to prismatic F-gauges. It shows that deformations over Breuil–Kisin prisms and over perfectoid prisms are captured by bijective evaluation maps, with a Kodaira–Spencer criterion characterizing universality. The theory yields two major applications: first, formal smoothness and local representability of integral local Shimura varieties with hyperspecial level, proving conjectures of Pappas–Rapoport; second, a refined perspective on p-divisible group classifications via prismatic Dieudonné crystals and their BK_min realizations. Overall, the paper integrates prismatic geometry with display theory to provide a versatile deformation toolkit applicable to integral models of Shimura varieties and p-divisible groups, and it clarifies the interaction with prismatic G–F-gauges in the GL_N setting. The results bridge Grothendieck–Messing deformation theory, Breuil–Kisin theory, and prismatic Dieudonné crystals, yielding a robust framework for studying deformations in p-adic Hodge-theoretic contexts.

Abstract

For a smooth affine group scheme over the ring of -adic integers and a cocharacter of , we develop the deformation theory for --displays over the prismatic site of Bhatt-Scholze, and discuss how our deformation theory can be interpreted in terms of prismatic -gauges introduced by Drinfeld and Bhatt-Lurie. As an application, we prove the local representability and the formal smoothness of integral local Shimura varieties with hyperspecial level structure. We also revisit and extend some classification results of -divisible groups.
Paper Structure (33 sections, 54 theorems, 217 equations)

This paper contains 33 sections, 54 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Strategy of the proof
  5. The structure of this paper
  6. Review of prismatic $G$-$\mu$-displays
  7. ${\mathcal{O}}_E$-prisms
  8. Prismatic sites
  9. Display groups
  10. $G$-$\mu$-displays
  11. Hodge filtrations and $G$-$\phi$-modules
  12. Prismatic $G$-$\mu$-displays over complete regular local rings
  13. The Grothendieck--Messing deformation theory for $G$-$\mu$-displays
  14. Deformations
  15. Special nilpotent thickenings
  16. ...and 18 more sections

Key Result

Theorem 1.1.3

Let $\mathcal{Q}$ be a $G$-$\mu$-display over $(W(k), (p))$. There exists a universal deformation $\mathfrak{Q}^{\mathrm{univ}}$ of $\mathcal{Q}$ over $R_{G, \mu}$.

Theorems & Definitions (203)

  • Definition 1.1.1
  • Definition 1.1.2
  • Theorem 1.1.3: Theorem \ref{['Theorem:Existence of universal deformation']}
  • Remark 1.1.4
  • Definition 1.1.5
  • Definition 1.1.6
  • Theorem 1.1.7: Theorem \ref{['Theorem:Existence of universal deformation']}
  • Theorem 1.1.8: Theorem \ref{['Theorem:characterization of universal family']}
  • Remark 1.1.9
  • Theorem 1.1.10: Theorem \ref{['Theorem:deformation theory for prismatic G-F-gauges of type mu']}
  • ...and 193 more