Cancellation in totally definite quaternion algebras

Definite orders with locally free cancellation

“…By first bounding as above and then using the bound in for each such and different possibilities of , Hallouin–Maire show (without using Armitage–Fröhlich). Using more careful arguments, Hallouin–Maire actually show in [, Proposition 11] and for these degrees obtain improved bounds on the discriminant in [, Proposition 12]. Since we leave the ultimate classification up to a computer, a priori we only need bounds that are good enough to ensure that all fields in questions have been tabulated.…”

“…Remark We checked our results against existing lists. The hereditary definite quaternion orders that are Hermite rings have been classified by Vignéras , Hallouin–Maire , and Smertnig . Up to ‐algebra isomorphism there are 168 such orders, of which 149 have locally free cancellation.…”

Notes on Locally Free Groups

“…She showed that the cancellation law holds only when [F : Q] < 33 and that there are only finitely many orders can occur. The bound of degree was reduced significantly to 6 by Hallouin and Marie [6] and they gave a complete list of all definite Eichler orders that satisfy the cancellation property. The list is re-examed by Smertnig [10] due to an overlook on Vignéras' criterion.…”

Computing generators of free modules over orders in group algebras II

“…Hence we may restrict further to the case that A is a totally definite quaternion algebra and use the complete classification of maximal orders in such algebras with locally free cancellation given in [HM06]. However, we take a different approach better suited to consideration of group algebras.…”