Section: New Results

Class groups and other invariants of number fields

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

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 [31] will appear in 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 [15] .

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 [10] . The article [22] has been submitted to Mathematics of Computation.

Assuming the GRH, Bach proved that one can calculate the residue of the Dedekind zeta function of a number field $K$ from the knowledge of the splitting of primes $p<X$, with an error bounded explicitly in terms of $X$ and the field discriminant. This is a crucial ingredient in all algorithms used to compute class groups and unit groups in subexponential time (under GRH). Using Weil's explicit formula, K. Belabas improved on Bach's bound, speeding up by a sizable constant factor this part of the class group algorithm. The article has been submitted to Mathematics of Computation.