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Arithmetic compactifications of integral models of Shimura varieties of abelian type

Peihang Wu

Abstract

In this paper, we construct good toroidal and minimal compactifications in the sense of Lan-Stroh for integral models of abelian-type Shimura varieties. We start with finding suitable types of cusp labels and cone decompositions which are compatible with those of the associated Hodge-type Shimura varieties. We then study the action of $\mathbb{Q}$-points of the adjoint group on boundary charts and toroidal compactifications of Hodge-type integral models. In particular, we extend the twisting construction of Kisin and Pappas to boundary charts. Finally, up to taking refinements of cone decompositions, we construct an abelian-type toroidal compactification as an open and closed algebraic subspace of a quotient from a disjoint union of Hodge-type toroidal compactifications and construct minimal compactifications with a similar method. Furthermore, we show results on nearby cycles of these compactifications and verify Pink's formula when the level at $p$ is an intersection of $n$ quasi-parahoric subgroups.

Arithmetic compactifications of integral models of Shimura varieties of abelian type

Abstract

In this paper, we construct good toroidal and minimal compactifications in the sense of Lan-Stroh for integral models of abelian-type Shimura varieties. We start with finding suitable types of cusp labels and cone decompositions which are compatible with those of the associated Hodge-type Shimura varieties. We then study the action of -points of the adjoint group on boundary charts and toroidal compactifications of Hodge-type integral models. In particular, we extend the twisting construction of Kisin and Pappas to boundary charts. Finally, up to taking refinements of cone decompositions, we construct an abelian-type toroidal compactification as an open and closed algebraic subspace of a quotient from a disjoint union of Hodge-type toroidal compactifications and construct minimal compactifications with a similar method. Furthermore, we show results on nearby cycles of these compactifications and verify Pink's formula when the level at is an intersection of quasi-parahoric subgroups.
Paper Structure (35 sections, 146 theorems, 281 equations)

This paper contains 35 sections, 146 theorems, 281 equations.

Table of Contents

  1. Characteristic zero theory
  2. Boundary charts in characteristic zero theory
  3. Boundary mixed Shimura varieties of Hodge type
  4. Compactifications and connected components
  5. $ZP$-cusps
  6. One-motives and the theory of degeneration
  7. Siegel moduli
  8. 1-motives
  9. Theory of degeneration
  10. Boundary mixed Shimura varieties of Hodge type, continued
  11. Toroidal compactifications of integral models of Hodge type
  12. Twisting constructions
  13. Twisting abelian schemes
  14. Twisting degenerating families
  15. Twisting $1$-motives
  16. ...and 20 more sections

Key Result

Proposition 1

Fix a prime $l\neq p$. Let $\mathcal{V}$ be a lisse $\overline{\mathbb{Q}}_l$-sheaf associated with an algebraic representation $\xi$ of $G^c_2$ on a finite-dimensional $\overline{\mathbb{Q}}_l$-vector space $V_\xi$ or a finite $\mathbb{F}_l$-sheaf equipped with an action of an open compact subgroup

Theorems & Definitions (311)

  • Remark
  • Proposition : Proposition \ref{['prop-nearby']} and Corollary \ref{['cor-nearby-intersection']}
  • Proposition : See Proposition \ref{['prop-pink-formula']} for details
  • Lemma : See §\ref{['sss-conclusion']}
  • Definition : Definition \ref{['equi-z']}
  • Remark
  • Proposition : Proposition \ref{['zp-cones']}
  • Proposition : Corollary \ref{['cor-maintheorem']}
  • Proposition : Proposition \ref{['prop-twt-1']}
  • Remark 1.1
  • ...and 301 more