Section: New Results

Number fields

Participant : Jean-François Biasse.

Jean-François Biasse has made practical improvements to the sieving-based algorithm of Jacobson [51] 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 variants combined with proper structured gaussian elimination, led to the computation of the structure of a class group corresponding to a number field with a 110-digit discriminant (whereas older techniques were limited to 90-digit discriminants). This work has been accepted for publication in the journal Advances In Mathematics of Communications. Biasse is currently working with Jacobson to adapt those techniques to the real quadratic extensions. Biasse also defined a class of number fields for which the 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 known general algorithms have complexity L(1/2, O(1)) ). This class of number fields is analogous to the class of C_ab curves described by Enge and Gaudry [6] . This work has been submited to Mathematics of Computation .