## Section: New Results

### Number field enumeration

Participants : Karim Belabas, Henri Cohen, Anna Morra.

In joint work with Étienne Fouvry (Orsay), K. Belabas has proved a new case of
Malle's conjecture, a strong effective form of the inverse Galois problem [22] . They have given an asymptotic enumeration of Galois
sextic fields with group S_{3} , ordered by discriminant, using classical
Davenport-Heilbronn theory in a novel way. The same result was independently
obtained by Bhargava and Wood using a different method. The article [22] will appear in *International Journal of
Number Theory* .

Classical theorems of Davenport and Heilbronn enumerate cubic fields and
estimate the average 3-torsion of class groups of quadratic fields. In
joint work with Manjul Bhargava (Princeton) and Carl Pomerance (Dartmouth
College), K. Belabas has proved the first power-saving error terms for those
results, lending support to a conjecture of Roberts. As a corollary, the
generating Dirichlet series associated to cubic discriminants can be
analytically continued to the left of its simple pole at s = 1 , proving a
conjecture of Cohen. The article [21] will
appear in *Duke Mathematical Journal* .

H. Cohen and A. Morra have obtained an explicit expression for the Dirichlet
generating function associated to cubic extensions of an arbitrary number
field with a fixed quadratic resolvent. As a corollary, they have proved
refinements of Malle's conjecture in this context. The article [26]
has been submitted to the *Journal of Algebra* .

A. Morra has devised and implemented an algorithm to enumerate cubic extensions of principal imaginary quadratic fields, by increasing discriminant. Her algorithm is essentially linear in the output size. The article [30] has been submitted.

These last two results constitute the heart of Morra's thesis [11] , which she has defended in December.