Hecke orbits on Shimura varieties of Hodge type
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866929758615371776 |
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| author | D'Addezio, Marco van Hoften, Pol |
| author_facet | D'Addezio, Marco van Hoften, Pol |
| contents | We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at odd primes of good reduction. We use a novel result for the local monodromy groups of $F$-isocrystals "coming from geometry", which refines Crew's parabolicity conjecture. In the course of the proof, we also introduce a noncommutative generalisation of Serre--Tate coordinates for formal neighbourhoods of central leaves, built upon the previous work of Caraiani--Scholze and Kim. Using these coordinates, we reinterpret Chai--Oort's notion of strongly Tate-linear subspaces and we establish upper bounds for their monodromy groups. For this step, we employ the notion of Cartier--Witt stacks, as introduced by Drinfeld and Bhatt--Lurie. Another crucial ingredient in the proof is a rigidity result proved by Chai--Oort, which shows that the relevant subspaces are strongly Tate-linear. On the way, we generalise de Jong's full faithfulness theorem for $F$-isocrystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_10344 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Hecke orbits on Shimura varieties of Hodge type D'Addezio, Marco van Hoften, Pol Algebraic Geometry Number Theory 11G18 (Primary) 14G35 (Secondary) We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at odd primes of good reduction. We use a novel result for the local monodromy groups of $F$-isocrystals "coming from geometry", which refines Crew's parabolicity conjecture. In the course of the proof, we also introduce a noncommutative generalisation of Serre--Tate coordinates for formal neighbourhoods of central leaves, built upon the previous work of Caraiani--Scholze and Kim. Using these coordinates, we reinterpret Chai--Oort's notion of strongly Tate-linear subspaces and we establish upper bounds for their monodromy groups. For this step, we employ the notion of Cartier--Witt stacks, as introduced by Drinfeld and Bhatt--Lurie. Another crucial ingredient in the proof is a rigidity result proved by Chai--Oort, which shows that the relevant subspaces are strongly Tate-linear. On the way, we generalise de Jong's full faithfulness theorem for $F$-isocrystals. |
| title | Hecke orbits on Shimura varieties of Hodge type |
| topic | Algebraic Geometry Number Theory 11G18 (Primary) 14G35 (Secondary) |
| url | https://arxiv.org/abs/2205.10344 |