Computing quaternionic representations via twisted forms of Bruhat-Tits trees

TL;DR

This paper develops a new approach using twisted Galois forms of Bruhat-Tits trees to analyze and compute the set of integral forms of quaternionic representations over various fields, extending previous methods.

Contribution

It introduces a theory of twisted Galois forms of Bruhat-Tits trees to study integral representations across different splitting fields, enabling explicit cardinality computations.

Findings

Explicitly computed the cardinality of integral forms for specific quaternionic groups

Developed a unified framework for studying integral representations over multiple fields

Extended previous methods to a broader class of quaternionic representations

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.