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Canonical Integral Models of Shimura Varieties of Abelian Type

Patrick Daniels, Alex Youcis

Abstract

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when $p>3$ by showing that the Kisin-Pappas-Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models of are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.

Canonical Integral Models of Shimura Varieties of Abelian Type

Abstract

We prove a conjecture of Pappas and Rapoport for all Shimura varieties of abelian type with parahoric level structure when by showing that the Kisin-Pappas-Zhou integral models of Shimura varieties of abelian type are canonical. In particular, this shows that these models of are independent of the choices made during their construction, and that they satisfy functoriality with respect to morphisms of Shimura data.
Paper Structure (47 sections, 52 theorems, 180 equations)

This paper contains 47 sections, 52 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation and conventions
  4. Preliminaries
  5. Some basic Bruhat--Tits theory
  6. The building and its functorialities
  7. Parahoric groups and group schemes
  8. Parahoric models of tori and $R$-smooth tori
  9. Abelianization of parahoric group schemes
  10. Parahoric group schemes and central quotients
  11. Parahoric group schemes and ad-isomorphisms
  12. Fiber products of groups
  13. Smoothness and connectedness of fiber products of group schemes
  14. Some fiber products of parahoric group schemes
  15. Torsors for fiber products of groups
  16. ...and 32 more sections

Key Result

Theorem 1

Let $p >3$. Then the Pappas--Rapoport conjecture holds for Shimura varieties of abelian type with parahoric level structure at $p$.

Theorems & Definitions (107)

  • Theorem 1: See Theorem \ref{['thm:main']}
  • Theorem 2: see Theorem \ref{['thm:main']}
  • Corollary 3: See Corollary \ref{['cor:omnibus-independence']} and Theorem \ref{['thm:functoriality']}
  • Remark 1.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: cf. KalethaPrasad
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • ...and 97 more