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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2510.09356 |
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| _version_ | 1866912640927793152 |
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| author | Masdeu, Marc Torrents, Eloi |
| author_facet | Masdeu, Marc Torrents, Eloi |
| contents | Let $B$ be a totally-definite quaternion algebra over a totally real field $F$, let $\mathfrak{p}$ be a prime ideal of $F$, and let $Γ$ be the group of reduced norm-$1$ elements of an Eichler $\mathcal{O}_F[1/\mathfrak{p}]$-order $R$ inside $B$. We give an algorithm to compute the fundamental domain for the action of $Γ$ on the Bruhat-Tits tree of $\operatorname{GL}_2(F_\mathfrak{p})$. Using this, we tabulate Shimura curves of genus up to $3$ over any totally real field which can be $\mathfrak{p}$-adically uniformized for some prime $\mathfrak{p}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_09356 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental domains for quaternionic S-arithmetic groups over totally real fields Masdeu, Marc Torrents, Eloi Number Theory 11F06 Primary, 20H10 Let $B$ be a totally-definite quaternion algebra over a totally real field $F$, let $\mathfrak{p}$ be a prime ideal of $F$, and let $Γ$ be the group of reduced norm-$1$ elements of an Eichler $\mathcal{O}_F[1/\mathfrak{p}]$-order $R$ inside $B$. We give an algorithm to compute the fundamental domain for the action of $Γ$ on the Bruhat-Tits tree of $\operatorname{GL}_2(F_\mathfrak{p})$. Using this, we tabulate Shimura curves of genus up to $3$ over any totally real field which can be $\mathfrak{p}$-adically uniformized for some prime $\mathfrak{p}$. |
| title | Fundamental domains for quaternionic S-arithmetic groups over totally real fields |
| topic | Number Theory 11F06 Primary, 20H10 |
| url | https://arxiv.org/abs/2510.09356 |