Arithmetic compactifications of integral models of Shimura varieties of abelian type
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912728457674752 |
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| author | Wu, Peihang |
| author_facet | Wu, Peihang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Arithmetic compactifications of integral models of Shimura varieties of abelian type Wu, Peihang Number Theory Algebraic Geometry 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. |
| title | Arithmetic compactifications of integral models of Shimura varieties of abelian type |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2505.09135 |