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Canonical integral models for Shimura varieties of toral type

Patrick Daniels

TL;DR

This work proves the Pappas–Rapoport conjecture for canonical integral models of Shimura varieties with parahoric level in the toral setting. It builds a bridge between crystalline $ ext{G}$-representations and $ ext{G}$-shtukas via Bhatt–Scholze prismatic $F$-crystals and uses this to construct shtukas on integral models that extend to the generic fiber. The authors establish a fully faithful prismatic realization functor into $ ext{G}$-shtukas and verify compatibility with the generic fiber, enabling the canonical construction of integral models for toral Shimura varieties and verifying PR’s conjecture in this case. The results illuminate the role of prismatic methods in $p$-adic Hodge theory for toral Shimura varieties and pave the way for extensions to broader abelian-type cases via Lubin–Tate and local-model techniques.

Abstract

We prove the Pappas-Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt-Scholze theory of prismatic $F$-crystals, that there is a fully faithful functor from $\mathcal{G}$-valued crystalline representations of Gal$(\bar{K}/K)$ to $\mathcal{G}$-shtukas over Spd$(\mathcal{O}_K)$, where $\mathcal{G}$ is a parahoric group scheme over $\mathbb{Z}_p$ and $\mathcal{O}_K$ is the ring of integers in a $p$-adic field $K$.

Canonical integral models for Shimura varieties of toral type

TL;DR

This work proves the Pappas–Rapoport conjecture for canonical integral models of Shimura varieties with parahoric level in the toral setting. It builds a bridge between crystalline -representations and -shtukas via Bhatt–Scholze prismatic -crystals and uses this to construct shtukas on integral models that extend to the generic fiber. The authors establish a fully faithful prismatic realization functor into -shtukas and verify compatibility with the generic fiber, enabling the canonical construction of integral models for toral Shimura varieties and verifying PR’s conjecture in this case. The results illuminate the role of prismatic methods in -adic Hodge theory for toral Shimura varieties and pave the way for extensions to broader abelian-type cases via Lubin–Tate and local-model techniques.

Abstract

We prove the Pappas-Rapoport conjecture on the existence of canonical integral models of Shimura varieties with parahoric level structure in the case where the Shimura variety is defined by a torus. As an important ingredient, we show, using the Bhatt-Scholze theory of prismatic -crystals, that there is a fully faithful functor from -valued crystalline representations of Gal to -shtukas over Spd, where is a parahoric group scheme over and is the ring of integers in a -adic field .
Paper Structure (16 sections, 23 theorems, 137 equations)

This paper contains 16 sections, 23 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Notation and conventions
  4. Preliminaries
  5. Recollections on shtukas
  6. Shtukas and local systems
  7. Crystalline representations and prismatic $F$-crystals
  8. Prismatic $F$-crystals and shtukas
  9. From prismatic $F$-crystals to shtukas
  10. $\mathcal{G}(\mathbb{Z}_p)$-valued crystalline representations and $\mathcal{G}$-shtukas
  11. The generic fiber
  12. Shimura varieties of toral type
  13. Shimura varieties and crystalline representations
  14. The shtuka on the generic fiber
  15. The conjecture of Pappas and Rapoport
  16. ...and 1 more sections

Key Result

Theorem 1

The collection of integral models $(\mathscr{S}_{\sf K})_{{\sf K}^p}$ defined above is canonical in the sense of Pappas and Rapoport.

Theorems & Definitions (55)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.6
  • ...and 45 more