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$2$-adic integral models of some Shimura varieties with parahoric level structure

Jie Yang

TL;DR

The paper extends the construction of integral models for Shimura varieties to the $2$-adic setting, focusing on abelian-type varieties with parahoric level structure. It develops a $p$-adic deformation framework at $p=2$ by adapting Lau’s classification of $2$-divisible groups and introducing very good integral Hodge embeddings to realize versal deformations with crystalline tensors. The main results establish flat normal ${ m O}_E$-schemes extending Shimura varieties, equipped with smooth local-model diagrams and, in the Hodge-type case, canonical models in the sense of Pappas–Rapoport. The paper treats two concrete scenarios (A) and (B): (A) unramified $G$ with hyperspecial containment and tame fixed points, and (B) unitary groups with ramified $p=2$, proving the existence and properties of integral models in both cases and extending the framework to unitary local models. Together, these results broaden the landscape of integral models for Shimura varieties at $p=2$, enabling deeper arithmetic and geometric investigations of their reductions and local structures.

Abstract

We construct integral models over $p=2$ for some Shimura varieties of abelian type with parahoric level structure, extending the previous work of Kim-Madapusi, Kisin, Pappas, and Zhou. For Shimura varieties of Hodge type, we show that our integral models are canonical in the sense of Pappas-Rapoport.

$2$-adic integral models of some Shimura varieties with parahoric level structure

TL;DR

The paper extends the construction of integral models for Shimura varieties to the -adic setting, focusing on abelian-type varieties with parahoric level structure. It develops a -adic deformation framework at by adapting Lau’s classification of -divisible groups and introducing very good integral Hodge embeddings to realize versal deformations with crystalline tensors. The main results establish flat normal -schemes extending Shimura varieties, equipped with smooth local-model diagrams and, in the Hodge-type case, canonical models in the sense of Pappas–Rapoport. The paper treats two concrete scenarios (A) and (B): (A) unramified with hyperspecial containment and tame fixed points, and (B) unitary groups with ramified , proving the existence and properties of integral models in both cases and extending the framework to unitary local models. Together, these results broaden the landscape of integral models for Shimura varieties at , enabling deeper arithmetic and geometric investigations of their reductions and local structures.

Abstract

We construct integral models over for some Shimura varieties of abelian type with parahoric level structure, extending the previous work of Kim-Madapusi, Kisin, Pappas, and Zhou. For Shimura varieties of Hodge type, we show that our integral models are canonical in the sense of Pappas-Rapoport.
Paper Structure (30 sections, 35 theorems, 110 equations)

This paper contains 30 sections, 35 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Strategy of the proof
  5. Organization
  6. $p$-divisible groups and Lau's classification
  7. Dieudonné displays and $p$-divisible groups
  8. The Zink ring
  9. Frames and windows
  10. Dieudonné displays and Dieudonné pairs
  11. Lau's classification
  12. Comparison with Breuil-Kisin's classification
  13. Deformation theory
  14. Versal deformations of $p$-divisible groups
  15. Local models and local Hodge embeddings
  16. ...and 15 more sections

Key Result

Theorem 1.2

Assume that either (A) or (B) holds.

Theorems & Definitions (86)

  • Conjecture 1.1: pappas2021p,daniels2023canonical
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Definition 2.1: zink2001dieudonne
  • Lemma 2.2: lau2014relations
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • ...and 76 more