Section: New Results

Quaternion algebras

Participants : Jean-Paul Cerri, Pierre Lezowski, Aurel Page.

With J. Chaubert, J.-P. Cerri and P. Lezowski have studied whether some quaternion fields over number fields are Euclidean, that is to say whether they admit a left or right Euclidean order. In particular, they have established the complete list of totally definite and Euclidean quaternion fields over the rationals or a quadratic number field. Moreover, they have proved that every field in this list is in fact norm Euclidean. The proofs are both theoretical and algorithmic. The article [23] will appear in International Journal of Number Theory.

Starting with an order in a suitable quaternion algebra over a number field $F$ with exactly one complex place, one can construct discrete subgroups of ${\mathrm{PSL}}_2\left(ℂ\right)$. These groups, called arithmetic Kleinian groups, act properly discontinuously with finite covolume on the hyperbolic 3-space. In [34] , A. Page designs an efficient algorithm which computes a fundamental domain and a presentation for such a group. It is a generalization to the dimension 3 of an algorithm of J. Voight's [44] together with a new, nondeterministic, but faster enumeration procedure. A public implementation is available in KleinianGroups (see  5.8 ).