Section: New Results

Class groups and other invariants of number fields

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

In collaboration with E. Friedman, K. Belabas presented in [22] a new algorithm to compute the residue at $s=1$ of the Dedekind zeta function of a number field, conditional on GRH. This improves on previous results of Eric Bach  [31] by a useful constant factor. Such an estimate is one of the two key analytic ingredients to Buchmann's class group algorithm, the other being the existence (under GRH) of an explicit set of small generators  [33] .

In collaboration with F. Thorne, H. Cohen worked on Dirichlet series associated to cubic and quartic fields with given resolvent. In [23] they give an explicit formula for the Dirichlet series ${\sum }_K{\left|\Delta \left(K\right)\right|}^{-s}$, where the sum is over isomorphism classes of all cubic fields whose quadratic resolvent field is isomorphic to a fixed quadratic field $k$. This is a sequel to previous work of Cohen and Morra, where such formulæ are proved in a more general setting, in terms of sums over characters of certain groups related to ray class groups. Here, the analysis is carried further and they prove explicit formulæ for these Dirichlet series over $\mathbb{Q}$. As an application, they compute tables of the number of $S_3$-sextic fields $K$ with discriminant ranging up to ${10}^{23}$. An accompanying Pari/Gp implementation is available.

In [24] , they give an explicit formula for the Dirichlet series ${\sum }_K{\left|\Delta \left(K\right)\right|}^{-s}$, where this time the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to a fixed cubic field $k$. This work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier.

The papers by H. Cohen on Haberland's formula and numerical computation of Petersson scalar products and by A. Angelakis and P. Stevenhagen on imaginary quadratic fields with isomorphic abelian Galois groups, which were presented at the ANTS-X conference, were published in [17] , [16] .