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arXiv · 2510.09356

Fundamental domains for quaternionic S-arithmetic groups over totally real fields

Marc Masdeu·Eloi Torrents
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Abstract

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 $\Gamma$ 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 $\Gamma$ 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 $\mathfrak{p}$.

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Marc Masdeu, Eloi Torrents. 2025-10-10. Fundamental domains for quaternionic S-arithmetic groups over totally real fields. https://arxiv.org/abs/2510.09356

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