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Toric schemes and integral models for Shimura varieties with $Γ_1(p)$-type level

Georgios Pappas, Michael Rapoport

Abstract

We propose a conjectural theory of $p$-integral models of Shimura varieties with level structure at $p$ given by a class of normal subgroups of parahoric subgroups with abelian quotient group. The role of the theory of local models is played in this context by a certain root stack over the local model for parahoric level. The construction of this root stack is based on the "divisor theorem" (a foundational fact about local models) and on the theory of toric varieties in this context, both of which are of independent interest. We prove our conjecture in the case of Shimura varieties of PEL type when the parahoric is an Iwahori (under some additional conditions).

Toric schemes and integral models for Shimura varieties with $Γ_1(p)$-type level

Abstract

We propose a conjectural theory of -integral models of Shimura varieties with level structure at given by a class of normal subgroups of parahoric subgroups with abelian quotient group. The role of the theory of local models is played in this context by a certain root stack over the local model for parahoric level. The construction of this root stack is based on the "divisor theorem" (a foundational fact about local models) and on the theory of toric varieties in this context, both of which are of independent interest. We prove our conjecture in the case of Shimura varieties of PEL type when the parahoric is an Iwahori (under some additional conditions).
Paper Structure (84 sections, 37 theorems, 518 equations)

This paper contains 84 sections, 37 theorems, 518 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notations
  4. Divisors on local models
  5. The pairing
  6. Tori attached to parahorics
  7. Relation to the standard torus
  8. Relation to the cocenter of $G$
  9. The pairing
  10. Divisors on local models and the divisor conjecture
  11. Local models
  12. Statement of the divisor conjecture
  13. The divisor conjecture in the tamely ramified case
  14. Spread-out group schemes
  15. The $\mathcal{G}\xspace_0$-torsor over the local model
  16. ...and 69 more sections

Key Result

Theorem 1.3.1

Let $(G, \{\mu\}, \mathcal{G}\xspace)$ be a local model triple over $\mathbb {Q}\xspace_p$. Suppose that the group $G$ splits over a tamely ramified extension of $\mathbb {Q}\xspace_p$ and that $p$ does not divide the order of the algebraic fundamental group $\pi_1(G_{\rm der})$. There exists a pair over ${\rm M}_{\mathcal{G}\xspace,\mu}\otimes_{O_E}O_{\breve E}$, obtained by pushing out ${\rm P}_

Theorems & Definitions (128)

  • Theorem 1.3.1
  • Remark 1.4.1
  • Remark 1.5.1
  • Remark 1.5.2
  • Conjecture 1.6.1
  • Theorem 1.6.2
  • Remark 2.1.5
  • Conjecture 2.2.3
  • Remark 2.2.4
  • Remark 2.2.5
  • ...and 118 more