Some time ago, Mark Watkins busted open the “class number n” problem for smallish n, finding all imaginary quadratic fields of class number at most 100 (the original paper is here) Although the paper describes the method in detail, it does not actually give the complete list of imaginary quadratic fields which occur (for fairly obvious reasons given the size of the list). I’ve occasionally wanted to consult the actual list, and most of the time I have just emailed Mark to find out the answer. But now it is available online! Here is the link. (Maybe someone could put this on the LMFDB?)
Consulting the table one immediately notices a number of beautiful facts, such as the fact that (Z/3Z)^3 does not occur as a class group. (Our knowledge of p-parts of class groups, following Gauss, Pierce, Helfgott, Venkatesh, and Ellenberg, is enough to show that (Z/2Z)^n and (Z/3Z)^n for varying n only occur finitely often [similarly these groups plus any fixed group A], but those results are not effective.) One also sees that D = – 5519 and D = -1842523 are the first and last IQF discriminants of class number 97. It’s the type of table that immediately bubbles up interesting questions which one can at least try to give heuristic guesstimates. For example, let mu(A) denote the number of imaginary quadratic fields with class group A. Can one give a plausible guess for the rough size of mu(A)? One roughly wants to combine the Cohen-Lenstra heuristics with the estimate \(h \sim \Delta^{1/2}.\) To do this, I guess one would roughly want to have an estimate for
\(\displaystyle{\sum_{x^{1/2 – \epsilon} < |A| \le x^{1/2 + \epsilon}} \frac{1}{|\mathrm{Aut}(A)|}}.\)
I wouldn't be surprised if someone has already carried out this analysis (thought I don't know any reference). As a specific example, what is the expected growth rate of mu(Z/qZ) over primes q? A related question: is there a finitely generated abelian group which provably does not occur as the first homology of a congruence arithmetic hyperbolic 3-manifold?
At any rate, this is a result that Gauss would have appreciated. Curiously enough, this paper was recently posted as an answer on to the (typically ridiculous as usual) MO question Which results from the last 30 years, in any area of mathematics, do you think are the most important ones? While I wouldn’t quite put it in that class, I do find it curious that it this answer (at the time of writing) has -4 votes on mathoverflow. Given the enormous crap that does receive positive votes, I suppose that such minus votes are not to be taken too seriously. I would, however, make the following claim: Watkins’ result is as least as interesting as any original number theory research that has appeared on MO (at least as far as anything I have seen).
I vaguely remember some work of Sound on the asymptotic number of imaginary quadratic field with given class number, that might have discussed also specifying the group… Searching leads to https://arxiv.org/abs/0707.0237 and then to the follow-up https://arxiv.org/abs/1510.04387 by Holmin, Jones, Kurlberg, McLeman and Petersen that seems to address exactly the question with the group structure taken into account.
Perfect!
And in fact there’s a new preprint this morning by Y. Lamzouri that’s also related (counting imaginary quadratic fields with odd class number <H): http://front.math.ucdavis.edu/1701.05267
Happy to see that both papers cite Watkins!
MathOverflow used to be much better in the good old days.
What would you consider a reasonable answer to that MO question?
In the context of the question, the only reasonable option would be not to answer.
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