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Integral canonical models of exceptional Shimura varieties

Benjamin Bakker, Ananth N Shankar, Jacob Tsimerman

Abstract

We prove that Shimura varieties admit integral canonical models for sufficiently large primes. In the case of abelian-type Shimura varieties, this recovers work of Kisin-Kottwitz for sufficiently large primes. We also prove the existence of integral canonical models for images of period maps corresponding to geometric families. We deduce several consequences from this, including an unramified rigid analogue of Borel's extension theorem, a version of Tate semisimplicity, CM lifting theorems, and a weakened version of Tate's isogeny theorem for ordinary points.

Integral canonical models of exceptional Shimura varieties

Abstract

We prove that Shimura varieties admit integral canonical models for sufficiently large primes. In the case of abelian-type Shimura varieties, this recovers work of Kisin-Kottwitz for sufficiently large primes. We also prove the existence of integral canonical models for images of period maps corresponding to geometric families. We deduce several consequences from this, including an unramified rigid analogue of Borel's extension theorem, a version of Tate semisimplicity, CM lifting theorems, and a weakened version of Tate's isogeny theorem for ordinary points.
Paper Structure (47 sections, 66 theorems, 20 equations)

This paper contains 47 sections, 66 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. The extension property
  3. Main results for Shimura varieties
  4. Applications
  5. Comparison with the extension property of Milne--Moonen
  6. Integral canonical models for images of geometric period maps
  7. An outline of the proof of the Extension Property
  8. Comparison with past work
  9. Acknowledgements
  10. Outline
  11. Notation
  12. The extension property and natural boundary
  13. The extension property
  14. Natural boundary and extension in codimension 1
  15. Natural boundary and local systems
  16. ...and 32 more sections

Key Result

Theorem 1.3

Let $S$ be a Shimura variety over its reflex field $E$. For almost all finite places $v$, there exists a model $\mathcal{S}$ over $\mathcal{O}_v$ which has the extension property, and which admits a log-smooth compactification over $\mathcal{O}_v$. Moreover, for the places $v$ as above, $\mathcal{S}

Theorems & Definitions (133)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.2
  • ...and 123 more