Small generators for S-unit groups of division algebras

TL;DR

This paper proves that the $S$-unit group of a division algebra over a number field can be generated by elements of small height, extending Lenstra's theorem from fields to division algebras with explicit bounds.

Contribution

It generalizes Lenstra's theorem to division algebras, providing explicit height bounds for generators of $S$-unit groups in this broader setting.

Findings

$S$-unit groups are generated by small height elements

Explicit height bounds depend on the number field and discriminant

Extension of Lenstra's theorem to division algebras

Abstract

Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is generated by elements of small height once $S$ contains an explicit finite set of places of $k$. This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when $B = k$. Our height bound is an explicit function of the number field and the discriminant of a maximal order in $B$ used to define its $S$-units.