Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912820476510208 |
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| author | Muller, Joseph |
| author_facet | Muller, Joseph |
| contents | In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at maximal level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in a previous paper. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via $p$-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at $p$. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2201_10229 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ Muller, Joseph Number Theory 14G35, 11G18, 14F20 In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at maximal level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in a previous paper. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via $p$-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at $p$. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations. |
| title | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ |
| topic | Number Theory 14G35, 11G18, 14F20 |
| url | https://arxiv.org/abs/2201.10229 |