On the pro-étale cohomology of quotient stacks of Drinfeld spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918153665118208 |
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| author | Yi, Zecheng |
| author_facet | Yi, Zecheng |
| contents | Let $\mathcal{H}^{n-1}_{K}$ denote the $(n-1)$-dimensional Drinfeld space over a $p$-adic field $K$. We give an explicit description of the $\ell$-adic and $p$-adic pro-étale cohomology of quotient stacks $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(\mathcal{O}_K)]$ and $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(K)]$, which are moduli stacks of special formal $\mathcal{O}_D$-modules. The computation makes use of the isomorphism between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze--Weinstein, as well as the $p$-adic pro-étale cohomology of the Drinfeld spaces computed by Colmez--Dospinescu--Niziol. As an application, we also compute the continuous group cohomology of $\operatorname{GL}_n(\mathbb{Q}_p)$ over duals of generalized Steinberg representations over $\mathbb{Q}_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the pro-étale cohomology of quotient stacks of Drinfeld spaces Yi, Zecheng Number Theory Algebraic Geometry Representation Theory 11S31, 11S25, 14F30 (Primary), 22E50, 22E41 (Secondary) Let $\mathcal{H}^{n-1}_{K}$ denote the $(n-1)$-dimensional Drinfeld space over a $p$-adic field $K$. We give an explicit description of the $\ell$-adic and $p$-adic pro-étale cohomology of quotient stacks $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(\mathcal{O}_K)]$ and $[\mathcal{H}^{n-1}_{K}/\operatorname{GL}_n(K)]$, which are moduli stacks of special formal $\mathcal{O}_D$-modules. The computation makes use of the isomorphism between the Lubin-Tate tower and the Drinfeld tower due to Faltings and Scholze--Weinstein, as well as the $p$-adic pro-étale cohomology of the Drinfeld spaces computed by Colmez--Dospinescu--Niziol. As an application, we also compute the continuous group cohomology of $\operatorname{GL}_n(\mathbb{Q}_p)$ over duals of generalized Steinberg representations over $\mathbb{Q}_p$. |
| title | On the pro-étale cohomology of quotient stacks of Drinfeld spaces |
| topic | Number Theory Algebraic Geometry Representation Theory 11S31, 11S25, 14F30 (Primary), 22E50, 22E41 (Secondary) |
| url | https://arxiv.org/abs/2510.02699 |