Section: New Results

Class groups and other invariants of number fields

Participants : Jean-François Biasse, Jean-Paul Cerri, Pierre Lezowski.

J.-F.Biasse has determined a class of number fields for which the ideal class group, the regulator, and a system of fundamental units of the maximal order can be computed in subexponential time $L(1/3,O(1\left)\right)$ (whereas the best previously known algorithms have complexity $L(1/2,O(1\left)\right)$). This class of number fields is analogous to the class of curves described in [13] , cf. ref ssec:dlog. The article [18] has been submitted to Mathematics of Computation.

Using new theoretical ideas and his novel algorithmic approach, J.-P. Cerri has discovered examples of generalised Euclidean number fields and of 2-stage norm-Euclidean number fields in degree greater than 2 [11] . These notions, extending the link between usual Euclideanity and principality of the ring of integers of a number field had already received much attention before; however, examples were only known for quadratic fields.

P. Lezowski extended J.-P. Cerri's algorithm, which was restricted to totally real number fields, to decide whether a generic number field is norm-Euclidean. His procedure allowed to find principal and non norm-Euclidean number fields of various signatures and degrees up to 8, but also to give further insight about the norm-Euclideanity of some cyclotomic fields. Besides, many new examples of generalised Euclidean and 2-stage Euclidean number fields were obtained. The article [25] has been submitted to Mathematics of Computation.

In another direction, norm-Euclidean ideal classes have been studied. They generalise the notion of norm-Euclideanity to non principal number fields. Very few such number fields were known before. A modification of the algorithm provided many new examples and allowed to complete the study of pure cubic fields equipped with a norm-Euclidean ideal class. The article [26] has been submitted to International Journal of Number Theory.

With E. Hallouin, J.-M. Couveignes has studied descent obstructions for varieties [21] . Such obstructions play an important role when one studies families of varieties (e.g. curves of a given genus). Obstructions are often measured by elements in groups like class groups. The theory of stacks provides a more general treatment for these obstructions. Couveignes and Hallouin give the first example of a global obstruction for a variety (that is an obstruction that vanishes locally at every place).