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On a conjecture of Pappas and Rapoport

Patrick Daniels, Pol van Hoften, Dongryul Kim, Mingjia Zhang

TL;DR

The paper resolves the Pappas–Rapoport conjecture for canonical integral models of Shimura varieties by proving existence for Hodge-type varieties with quasi-parahoric level at $p$ using a shtuka/v-sheaf framework. It develops a detailed theory of moduli stacks of quasi-parahoric shtukas, establishes a finite étale descent mechanism, and demonstrates a natural open–closed decomposition relating parahoric and quasi-parahorics. It then proves scheme-theoretic local model diagrams in many cases and confirms the Kisin–Pappas conjecture on local models, culminating in a Rapoport–Zink type uniformization of isogeny classes. The results provide a robust, uniform approach to canonical integral models, connect local and global structures, and have potential implications for CM lifts and reduction theory in broader Shimura varieties contexts.

Abstract

We prove a conjecture of Pappas and Rapoport about the existence of ''canonical'' integral models of Shimura varieties of Hodge type with quasi-parahoric level structure at a prime $p$. For these integral models, we moreover show uniformization of isogeny classes by integral local Shimura varieties, and prove a conjecture of Kisin and Pappas on local model diagrams.

On a conjecture of Pappas and Rapoport

TL;DR

The paper resolves the Pappas–Rapoport conjecture for canonical integral models of Shimura varieties by proving existence for Hodge-type varieties with quasi-parahoric level at using a shtuka/v-sheaf framework. It develops a detailed theory of moduli stacks of quasi-parahoric shtukas, establishes a finite étale descent mechanism, and demonstrates a natural open–closed decomposition relating parahoric and quasi-parahorics. It then proves scheme-theoretic local model diagrams in many cases and confirms the Kisin–Pappas conjecture on local models, culminating in a Rapoport–Zink type uniformization of isogeny classes. The results provide a robust, uniform approach to canonical integral models, connect local and global structures, and have potential implications for CM lifts and reduction theory in broader Shimura varieties contexts.

Abstract

We prove a conjecture of Pappas and Rapoport about the existence of ''canonical'' integral models of Shimura varieties of Hodge type with quasi-parahoric level structure at a prime . For these integral models, we moreover show uniformization of isogeny classes by integral local Shimura varieties, and prove a conjecture of Kisin and Pappas on local model diagrams.
Paper Structure (27 sections, 41 theorems, 160 equations)

This paper contains 27 sections, 41 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main Results
  4. A sketch of the proof of Theorem \ref{['Thm:IntroMain']}
  5. Moduli stacks of quasi-parahoric shtukas
  6. Outline of the paper
  7. Acknowledgments
  8. Declarations
  9. Preliminaries
  10. Recollections from Fargues--Scholze
  11. Some Bruhat--Tits theory
  12. Finite étale covers of v-sheaves
  13. The moduli stack of quasi-parahoric shtukas
  14. Newton strata in the moduli stack of quasi-parahoric shtukas
  15. A group action on the moduli stack of parahoric shtukas
  16. ...and 12 more sections

Key Result

Theorem 1

If ${(\mathsf{G}, \mathsf{X})}$ is of Hodge type, then there exists a system $\{\mathscr{S}_{K}{(\mathsf{G}, \mathsf{X})}\}_{K^p}$ satisfying PappasRapoportShtukas.

Theorems & Definitions (100)

  • Theorem 1: Theorem \ref{['Thm:Main']}
  • Theorem 2: Theorem \ref{['Thm:WhyDidWeCheckThis?']}, Theorem \ref{['Thm:SchemeTheoreticLocalModelExists']}
  • Theorem 3: Theorem \ref{['Thm:QuasiParahoricShtukas']}
  • Corollary 4: Corollary \ref{['Cor:QuasiParahoricShtukas']}
  • Theorem 2.1.4
  • Definition 2.1.7
  • Definition 2.1.8
  • Lemma 2.1.9
  • proof
  • Lemma 2.1.10
  • ...and 90 more