Small generators for S-unit groups of division algebras
Ted Chinburg, Matthew Stover

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 be a number field, suppose that is a central simple division algebra over , and choose any maximal order of . The object of this paper is to show that the group of -units of is generated by elements of small height once contains an explicit finite set of places of . This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when . Our height bound is an explicit function of the number field and the discriminant of a maximal order in used to define its -units.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Finite Group Theory Research
