Section: New Results

Class groups and other invariants of number fields

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

J.-F. Biasse has made practical improvements to the sieving-based algorithm of Jacobson [36] for computing the group structure of the ideal class group of an imaginary-quadratic number field. These improvements, based on the use of large prime variations combined with structured Gaussian elimination, have led to the computation of the class group structure of a number field with a 110-digit discriminant (whereas older techniques were limited to 90-digit discriminants). The resulting article [23] has been accepted for publication in Advances in Mathematics of Communications .

Biasse has also 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)) (whereas the best previously known algorithms have complexity L(1/2, O(1)) ). This class of number fields is analogous to the class of curves described in [27] , cf. 6.1 . The article [24] has been submitted to Mathematics of Computation .

In joint work with Eva Bayer Fluckiger and Jérôme Chaubert (EPF Lausanne), J.-P. Cerri has generalised the notion of norm-Euclideanity to central division algebras, and in particular to quaternion algebras. They have established deep theoretical results in the spirit of Cerri's achievements for number fields (rationality of the minimum, properties of the spectra, ...), and they have obtained good bounds for the Euclidean minimum [12] . This theory should make it possible to formulate algorithms similar to those given by Cerri in the number field case, with the aim of establishing complete lists of Euclidean quaternion algebras over quadratic fields.

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 [25] . 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.

In joint work with Mark van Hoeij (Florida State), Jürgen Klüners (Paderborn), and Allan Steel (Sydney), K. Belabas has proved the polynomial time complexity of the now standard algorithm of van Hoeij (as extended by Belabas) to factor univariate polynomials over number fields, and in particular over the rational numbers [13] . The same approach also yields polynomial time complexity results for bivariate polyomials over a finite field.