Counting extensions of number fields
(Un exposé de Gunter Malle au séminaire de
théorie des nombres de Chevaleret le 8 mars 2004)
Résumé :
Given a number field $k$ and a finite group $G$ we are interested in
the number $N(k,G,x)$ of Galois extensions of $k$ with group $G$ and
norm of the discriminant bounded by $x$. We present a conjecture on
the asymptotic behaviour of this function as $x$ tends to $\infty$ (in
fact even for the case of non-Galois extensions), and give some evidence
for it. We will also report on the proof of a weak form of this conjecture
in the case of nilpotent groups $G$ and related results.