Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou)
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917861648236544 |
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| author | van Hoften, Pol |
| author_facet | van Hoften, Pol |
| contents | We study the mod $p$-points of the Kisin--Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin--Madapusi-Pera--Shin. We furthermore show that the mod $p$ isogeny classes are of the form predicted by the Langlands--Rapoport conjecture if either the Shimura variety is proper or if the group at $p$ is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the appendix by Rong Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl--Oort strata. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2010_10496 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou) van Hoften, Pol Number Theory Algebraic Geometry Primary 11G18, Secondary 14G35 We study the mod $p$-points of the Kisin--Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin--Madapusi-Pera--Shin. We furthermore show that the mod $p$ isogeny classes are of the form predicted by the Langlands--Rapoport conjecture if either the Shimura variety is proper or if the group at $p$ is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the appendix by Rong Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl--Oort strata. |
| title | Mod $p$ points on Shimura varieties of parahoric level (with an appendix by Rong Zhou) |
| topic | Number Theory Algebraic Geometry Primary 11G18, Secondary 14G35 |
| url | https://arxiv.org/abs/2010.10496 |