Trees, quaternion algebras and modular curves

TL;DR

This paper explores the action of unit groups of maximal orders in quaternion algebras over function fields on Bruhat-Tits trees, with applications to arithmetic geometry and group theory.

Contribution

It introduces new insights into the structure of quaternion algebra unit groups and their actions on Bruhat-Tits trees over function fields.

Findings

Characterization of unit group actions on Bruhat-Tits trees

Applications to arithmetic geometry involving modular curves

Connections to group theory via quaternion algebra structures

Abstract

We study the action on the Bruhat-Tits tree of unit groups of maximal orders in certain quaternion algebras over $\mathbb{F}_q(T)$ and discuss applications to arithmetic geometry and group theory.