New York Journal of Mathematics
Volume 20 (2014) 1175-1202


Ted Chinburg and Matthew Stover

Small generators for S-unit groups of division algebras

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Published: December 1, 2014
Keywords: Division algebras, S-unit groups, S-arithmetic lattices, heights on algebras, generators for S-unit groups, geometry of numbers
Subject: 17A35, 20H10, 22E40, 11F06, 16H10, 16U60, 20F05, 11H06

Let k be a number field, suppose that B is a central simple division algebra over k, and choose any maximal order D of B. The object of this paper is to show that the group DS* 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.


Chinburg partially supported by NSF Grant DMS 1100355.
Stover partially supported by NSF RTG grant DMS 0602191 and NSF grant DMS 1361000.

Author information

Ted Chinburg:
Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 10104

Matthew Stover:
Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122