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The prismatic realization functor for Shimura varieties of abelian type

Naoki Imai, Hiroki Kato, Alex Youcis

TL;DR

The paper develops a comprehensive prismatic framework for Shimura varieties of abelian type by constructing a prismatic F-gauge realization of the universal $ ext{G}( ext{Z}_p)$-local system and embedding it into a syntomic context. It establishes integral canonical models as syntomic/prismatic objects characterized by a universal deformation space and a Serre--Tate–type deformation theorem, mirroring Drinfeld’s motivic perspective in the abelian-type setting. It then connects these prismatic realizations with existing étale, crystalline, and cohomological structures, proving compatibility with Lovering’s crystalline picture and deriving cohomological consequences such as crystalline lattices and syntomic refinements. The results yield a robust, motivic viewpoint on the $p$-adic geometry of Shimura varieties, provide a syntomic characterization of integral models, and illuminate extensions to G-zips and related moduli problems with potential Langlands-theoretic implications.

Abstract

For the integral canonical model $\mathscr{S}_{\mathsf{K}^p}$ of a Shimura variety $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ of abelian type at hyperspecial level $K_0=\mathcal{G}(\mathbb{Z}_p)$, we construct a prismatic $F$-gauge model for the `universal' $\mathcal{G}(\mathbb{Z}_p)$-local system on $\mathrm{Sh}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$. We use this to obtain several new results about the $p$-adic geometry of Shimura varieties, notably an abelian-type analogue of the Serre--Tate deformation theorem (realizing an expectation of Drinfeld in the abelian-type case) and a prismatic characterization of these models at individual level.

The prismatic realization functor for Shimura varieties of abelian type

TL;DR

The paper develops a comprehensive prismatic framework for Shimura varieties of abelian type by constructing a prismatic F-gauge realization of the universal -local system and embedding it into a syntomic context. It establishes integral canonical models as syntomic/prismatic objects characterized by a universal deformation space and a Serre--Tate–type deformation theorem, mirroring Drinfeld’s motivic perspective in the abelian-type setting. It then connects these prismatic realizations with existing étale, crystalline, and cohomological structures, proving compatibility with Lovering’s crystalline picture and deriving cohomological consequences such as crystalline lattices and syntomic refinements. The results yield a robust, motivic viewpoint on the -adic geometry of Shimura varieties, provide a syntomic characterization of integral models, and illuminate extensions to G-zips and related moduli problems with potential Langlands-theoretic implications.

Abstract

For the integral canonical model of a Shimura variety of abelian type at hyperspecial level , we construct a prismatic -gauge model for the `universal' -local system on . We use this to obtain several new results about the -adic geometry of Shimura varieties, notably an abelian-type analogue of the Serre--Tate deformation theorem (realizing an expectation of Drinfeld in the abelian-type case) and a prismatic characterization of these models at individual level.
Paper Structure (40 sections, 68 theorems, 197 equations)

This paper contains 40 sections, 68 theorems, 197 equations.

Table of Contents

  1. The Tannakian framework for prismatic F-gauges
  2. Some preliminaries on graded and filtered algebra
  3. Basic definitions and results
  4. The Rees algebra construction
  5. Prismatic F-gauges in vector bundles
  6. Relationship to prismatic F-crystals
  7. Prismatic F-gauges with G-structure of type mu
  8. Relationship to the theory of displays
  9. Prismatic Gmu-displays
  10. Relationship to prismatic G-torsors with F-structure
  11. G-F-gauges of type mu
  12. Comparison to G-torsors on the syntomification
  13. Summary of equivalences
  14. Prismatic realization functors on Shimura varieties of abelian type
  15. Notation and basic definitions
  16. ...and 25 more sections

Key Result

Theorem A

Let $(\mathbf{G},\mathbf{X})$ be of abelian type and $p$ be an odd prime. Set $K_0=\mathcal{G}(\mathbb{Z}_p)$ where $\mathcal{G}$ is a reductive $\mathbb{Z}_p$-model of $\mathbf{G}_{\mathbb{Q}_p}$. Then, for the integral canonical model $\mathscr{S}_{\mathsf{K}_0\mathsf{K}^p}(\mathbf{G},\mathbf{X})$ which recovers the universal $\mathcal{G}(\mathbb{Z}_p)$-local system $\omega_{\mathsf{K}_0\mathsf{

Theorems & Definitions (170)

  • Theorem A
  • Remark 1
  • Theorem B: see Theorem \ref{['thm:prismatic-F-gauge-realization']}
  • Remark 2
  • Definition : see Definition \ref{['defn:syntomic-integral-canonical-model']}
  • Theorem C: see Theorem \ref{['thm:Drinfeld-conjecture']} and Theorem \ref{['thm:syntomic-characterization']}
  • Theorem D: Serre--Tate theorem for abelian-type Shimura varieties, see Theorem \ref{['thm:Drinfeld-conjecture']}
  • Remark 3
  • Theorem E: see Theorem \ref{['thm:gauge-propert-coh']}
  • Theorem F: see Theorem \ref{['thm:prismatic-crystalline-comparison']}
  • ...and 160 more