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 S3 , 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.
