# English references to Cohen - Martinet Heuristics

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.

• Cohen has two books (in english) on Computational Number Theory (which you can easily find). Perhaps he discusses this in one of them.
– efs
Nov 1, 2020 at 15:56
• Are you interested only in the original work of Cohen and Martinet or also in modern works which may have a different perspective? Nov 1, 2020 at 16:06
• e.g. arxiv.org/pdf/1907.11201.pdf gives, I think, exactly the same predictions as Cohen - Martinet, plus new stuff. Nov 1, 2020 at 16:08
• @WillSawin thanks for the reference. Nov 1, 2020 at 16:11

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.