I read through the celebrated paper of Cohen - Lenstra heuristics. But unfortunately, the Cohen - Martinet paper is originally written in French, which I do not understand. So I would like to know if there are any English references for this paper.


Also, may I know in a very short summary what extensions have been added by Cohen - Martinet to the Cohen - Lenstra heuristics.



English references to Cohen - Martinet Heuristics
The ultimate goal of the heuristics I made with Martinet was to generalize
the C-L heuristics to an arbitrary extension $L/K$ of number fields, first
assuming $L/K$ galois, then more generally. Although the basic ideas were
sound, many modifications needed to be made since the original publication.
First, we had to decide what primes were "good" in a suitable sense, and it
seems that our original definition was incorrect. Second, it became apparent
through the work of G. Malle that when the base field $K$ contains $p$-th roots
of unity the predictions must be modified. Also, a very nice paper
of Alex Bartel and H. Lenstra (Alex B. is on this forum, and gave the reference
a few days ago) begins essentially by saying that the goal of their paper is
to show that the C-Martinet heuristics are wrong (of course, with the hope of
correcting it). A lot of work is being done on this, Melanie Wood being indeed
one of the contributors as Will Sawin points out, and pandemic permitting, a whole
week conference should take place next year.


