Arithmetic compactifications of integral models of Shimura varieties of abelian type

TL;DR

This paper develops new integral models and compactifications for abelian-type Shimura varieties, extending existing theories and analyzing their boundary structures and nearby cycles.

Contribution

It constructs good toroidal and minimal compactifications for integral models of abelian-type Shimura varieties, extending Kisin-Pappas twisting and analyzing boundary actions.

Findings

Constructed toroidal and minimal compactifications compatible with Hodge-type models.

Extended twisting construction to boundary charts.

Verified Pink's formula for certain level structures.

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.