## Section: New Results

### Computation of Euclidean minima in totally definite quaternion fields

Participant : Jean-Paul Cerri.

In collaboration with Pierre Lezowski, Jean-Paul Cerri has studied norm-Euclidean properties of totally definite quaternion fields over number fields. Building on their previous work about number fields, they have proved that the Euclidean minimum and the inhomogeneous minimum of orders in such quaternion fields are always equal. Besides, they are rational under the hypothesis that the base number field is not quadratic. This single remaning open case corresponds to the similar open case remaining for real number fields.

They also have extended Cerri's algorithm for the computation of the upper part of the norm-Euclidean spectrum of a number field to this noncommutative context. This algorithm has allowed to compute the exact value of the norm-Euclidean minimum of orders in totally definite quaternion fields over a quadratic number field. This has provided the first known values of this minimum when the base number field has degree strictly greater than 1.

Consequently, both theoretical and practical
milestones set in the previous quadrennial report
were reached. These results are
presented in [19], due to appear in
*International Journal of Number Theory*.