Computing quaternionic representations via twisted forms of Bruhat-Tits trees

TL;DR

The paper develops a framework of twisted Galois forms of Bruhat-Tits trees to study integral representations of quaternionic groups across splitting fields. It builds a bijection between $\mathrm{IF}_{\rho}^{G}(E)$ and orbits on vertex sets of twisted branches, enabling simultaneous local-global analysis and explicit cardinality computations for groups such as $Q_8$, $\mathrm{SL}_2(\mathbb{F}_3)$, and the dicyclic group. Through a careful treatment of maximal orders, orders in division algebras, and explicit cocycle data, the authors obtain both local counts (notably at $p=2$) and global counts over imaginary quadratic fields, revealing how ramification and genus-theoretic data influence the landscape of integral forms. The constructions provide new tools for understanding integral representations in central simple algebras and offer concrete results for classical quaternionic groups with potential extensions to higher rank and other algebraic groups. Overall, the work unifies local twisted-tree methods with global arithmetic to quantify integral representations in a broad, multi-splitting-field setting.

Abstract

This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups.

Figures (3)

• Figure 1: A graph $\mathcal{C}$ (left) and its subdivision $\mathcal{C}(3)$ (right).
• Figure 2: The trees $\mathcal{T}_K \subseteq \mathcal{T}_L^{\mathcal{G}_L} \subset \mathcal{T}_L$. In this figure $w_{\alpha}=B_{\alpha, \frac{1}{2}\nu(\delta_K(\alpha))}$ is the closest point of $\mathcal{T}_K$ to $\mathcal{P}_L(\sqrt{\alpha},-\sqrt{\alpha})$, while $v_\alpha=B_{\sqrt{\alpha}, \nu(2\sqrt{\alpha})}$ is the unique $\mathcal{G}_L$-invariant point of $\mathcal{P}_L(\sqrt{\alpha},-\sqrt{\alpha})$.
• Figure 3: The intersection of the branches $\mathcal{S}_L(\mathbf{u})$ and $\mathcal{S}_L(\mathbf{v})$ can be visualized in Fig. (A), according to the discussion in the text. We assume the field $L$ contains both $\sqrt{-3}$ and $\sqrt{-1}$. In that figure $v_c=B_{0,-1/2}$ is the middle point of the edge in $\widehat{\mathcal{T}}_E$ joining $w_1=B_{0,-1}$ with $w_{0}=B_{1,0}$. Fig. (B) shows the $\mathcal{O}_\Omega$ representations of the quaternion group classified according to the fields where they are defined. See Table \ref{['table1']}.

Theorems & Definitions (66)

• Theorem 2.1
• Corollary 2.1.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Proposition 3.1
• proof
• Lemma 3.2