On generators of arithmetic groups over function fields

TL;DR

This paper explicitly identifies finite generating sets for the unit groups of orders in quaternion division algebras over function fields, advancing understanding of algebraic structures over such fields.

Contribution

It provides an explicit finite generating set for the unit groups of orders in quaternion division algebras over the function field F.

Findings

Explicit finite generating sets for unit groups in quaternion orders

Application to algebraic structures over function fields

Enhanced understanding of arithmetic groups over function fields

Abstract

Let $F=\mathbb{F}_q(T)$ be the field of rational functions with $\mathbb{F}_q$-coefficients, and $A=\mathbb{F}_q[T]$ be the subring of polynomials. Let $D$ be a division quaternion algebra over $F$ which is split at $1/T$. Given an $A$-order $\Lambda$ in $D$, we find an explicit finite set generating $\Lambda^\times$.