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

TL;DR

The paper develops an explicit algorithm to compute fundamental domains for S-arithmetic groups arising from definite quaternion algebras over totally real fields, acting on the Bruhat-Tits tree $\mathcal{T}_{\mathfrak{p}}$. By reducing the equivalence problem for vertices and edges to lattice problems and employing LLL alongside a boundary-data technique, the method yields concrete fundamental domains and enables the p-adic uniformization of Shimura curves with bad reduction. This culminates in a tabulation of Shimura curves of genus up to $3$ that admit $\mathfrak{p}$-adic uniformizations for some prime $\mathfrak{p}$, with explicit examples across several totally real fields. The work bridges algorithmic quaternion algebras with arithmetic geometry, providing tools to compute reduction graphs, apply Čerednik–Drinfel'd uniformizations, and explore the arithmetic of Shimura curves in a computationally practical setting.

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 $Γ$ 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 $\mathfrak{p}$.

Figures (3)

• Figure 1:
• Figure 2: Fundamental domain and boundary data (left), and the quotient of the Bruhat-Tits tree (right).
• Figure 3: Fundamental domain of genus $65$, with $16$ vertices and $80$ edges.

Theorems & Definitions (13)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Lemma 3.4