Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups
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| Format: | Preprint |
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2023
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| _version_ | 1866914904643993600 |
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| author | Bravo, Claudio Loisel, Benoit |
| author_facet | Bravo, Claudio Loisel, Benoit |
| contents | Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected semisimple $\mathbb{Z}$-group scheme. Let $\mathcal{S}$ be a finite set of places of $\mathcal{C}$. In this paper, we investigate on the conjugacy classes of maximal unipotent subgroups of $\mathcal{S}$-arithmetic subgroups. These are parameterized thanks to the Picard group of $\mathcal{O}_{\mathcal{S}}$ and the rank of $\mathbf{G}$. Furthermore, these maximal unipotent subgroups can be realized as the unipotent part of natural stabilizer, which are the stabilizers of sectors of the associated Bruhat-Tits building. We decompose these natural stabilizers in terms of their diagonalisable part and unipotent part, and we precise the group structure of the diagonalisable part. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_11193 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups Bravo, Claudio Loisel, Benoit Group Theory Number Theory 20G30, 20E45 (primary) 11R58, 20E42 (secondary) Let $\mathcal{C}$ be a smooth, projective, geometrically integral curve defined over a perfect field $\mathbb{F}$. Let $k=\mathbb{F}(\mathcal{C})$ be the function field of $\mathcal{C}$. Let $\mathbf{G}$ be a split simply connected semisimple $\mathbb{Z}$-group scheme. Let $\mathcal{S}$ be a finite set of places of $\mathcal{C}$. In this paper, we investigate on the conjugacy classes of maximal unipotent subgroups of $\mathcal{S}$-arithmetic subgroups. These are parameterized thanks to the Picard group of $\mathcal{O}_{\mathcal{S}}$ and the rank of $\mathbf{G}$. Furthermore, these maximal unipotent subgroups can be realized as the unipotent part of natural stabilizer, which are the stabilizers of sectors of the associated Bruhat-Tits building. We decompose these natural stabilizers in terms of their diagonalisable part and unipotent part, and we precise the group structure of the diagonalisable part. |
| title | Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups |
| topic | Group Theory Number Theory 20G30, 20E45 (primary) 11R58, 20E42 (secondary) |
| url | https://arxiv.org/abs/2307.11193 |