## Section: New Results

### Number fields

Participants : Razvan Barbulescu, Jean-Marc Couveignes, Jean-Paul Cerri, Pierre Lezowski.

In [30], Jean-Marc Couveignes constructs small models of number fields and deduces a better bound for the number of number fields of given degree $n$ and discriminant bounded by $H$. This work improves on previous results by Schmidt and Ellenberg-Venkatesh. Schmidt obtains a bound ${H}^{\frac{n+2}{4}}$ times a function of $n$. Ellenberg and Venkatesh obtain a bound ${H}^{exp\left(O\left(\sqrt{logn}\right)\right)}$ times a function of $n$. The new idea is to combine geometry of numbers and interpolation theory to produces small projective models and lower the exponent of $H$ down to $O\left({log}^{3}n\right)$. A key point is to look for local equations rather than a full set of generators of the ideal of these models.

In [12], Razvan Barbulescu in a joint work with Jishnu Ray (University of British Columbia, Vancouver) brings elements to support Greenberg's p-rationality conjecture. On the theoretical side, they propose a new family proven to be p-rational. On the algorithmic side, the compare the tools to enumerate number fields of given abelian Galois group and of computing class numbers, and extend the experiments on the Cohen-Lenstra-Martinet conjectures.

In collaboration with Pierre Lezowski, Jean-Paul Cerri has studied in [15] 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. Additionally, 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 non-commutative 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.