Arithmetic Compactifications of Integral Models of Shimura Varieties of Abelian Type

Abstract

We construct toroidal compactifications of integral models of abelian-type Shimura varieties when the Shimura datum G₂, X₂ satisfies the assumptions in [Kis10] or [KP15], and when the level at 𝑝 is contained in a hyperspecial or a parahoric subgroup. First, we find suitable types of cusp labels and cone decompositions which are compatible with those of toroidal compactifications of associated Hodge-type Shimura varieties. Then, we extend the twisting actions considered in [Kis10] and [KP15] on integral models of Hodge-type Shimura varieties to their toroidal compactifications. Finally, we construct toroidal compactifications up to refinements of given cone decompositions in a suitable cofinal family of cone decompositions, which admits smooth and projective refinements in itself.