In these days, I'm studying Cohen-Lenstra heuristics to understand the paper of Rene Schoof "Class Numbers of Real Cyclotomic Fields of Prime Conductor".

On page 932 of Schoof's paper, there is a sentence "According to Cohen-Lenstra, the probability that M does not occur in a "random $\mathbb{Z}[\zeta_{d}]$-module modulo a random principal ideal" is equal to $\prod_{2\leq k}(1-q^{-k})$."

First of all, I don't understand the exact meaning of the sentence. So I hope someone can explain it to me.

I checked the Example 5.4 by myself, and tried to understand the sentence as "The probability that an arbitrary $\mathbb{Z}[\zeta_{d}]$-module has trivial $\rho$-component ($\rho$ is the prime ideal of $\mathbb{Z}[\zeta_{d}]$ corresponding to $M$) is $\eta_{\infty}(\rho)/\eta_{1}(\rho)$."

Even if this interpretation is true, I don't know how to deduce this from the Fundamental Assumption 8.1 of Cohen-Lenstra paper. $d$ is not necessarily a prime, so there might not be an abelian group $\Gamma$ with $A_{\Gamma}$ isomorphic to $\mathbb{Z}[\zeta_{d}]$.

There is a line in Schoof's paper "the Cohen-Lenstra heuristics do not really apply to our situation." I hope someone can ask my question and explain the point of Schoof's paper.

Thank you very much!


1 Answer 1


$\DeclareMathOperator\Aut{Aut}$I infer from the context that the precise meaning of "a random $\mathbb{Z}[\zeta_d]$-module modulo a random principal ideal" means that you start by producing a random $\mathbb{Z}[\zeta_d]$-module with respect to the Cohen--Lenstra probability distribution (i.e. by definition such a "random" module is, vaguely speaking, isomorphic to a given module $M$ with probability proportional to $1/\#\Aut M$), and then you pick an element of this module uniformly at random, and quotient out the submodule that this element generates.

This results in what Cohen–Lenstra call the $1$-probability, and the event that Schoof is talking about is the complement of the event that Cohen–Lenstra are talking about in Example 5.10 (not 5.4), so the probability of that event is, according to that example, $\eta_\infty(q)/\eta_1(q) = \prod_{k\geq 2}(1-q^{-k})$.

So far, this has nothing to do with the Fundamental Assumption 8.1 in Cohen–Lenstra. The latter is a conjecture that says that class groups of varying number fields behave like "random" modules in a suitable sense, whereas the statement from Schoof you are quoting is just a statement about the particular probability distribution in question (as, indeed, is most of the paper of Cohen–Lenstra), it has, up to that point, no number theoretic content.

Eventually, Schoof does use the Cohen–Lenstra heuristics to justify his claims, but only very loosely. The Cohen–Lenstra heuristics make predictions about the behaviour of class groups of varying number fields, where the number fields vary in "horizontal" families: one fixes a finite group $G$, and runs through Galois fields, say, whose Galois group is isomorphic to $G$. The class groups of these fields are all modules over the group ring $\mathbb{Z}[G]$, and Cohen–Lenstra define a suitable probability distribution on the collection of isomorphism classes of such modules. In contrast Schoof lets his fields vary in "vertical" families: the degree of the fields he considers goes to infinity. That is why he says that Cohen–Lenstra do not directly apply to his situation. In a way, it is meaningless to compare the class group of $\mathbb{Q}(\zeta_l+\zeta_l^{-1})$ with that of $\mathbb{Q}(\zeta_{l'}+\zeta_{l'}^{-1})$ for distinct primes $l$ and $l'$, because the Cohen–Lenstra–Martinet philosophy says that you should not think of them just as groups, but as modules over the respective Galois group, but here they are modules over different, incompatible rings (the Galois groups are not isomorphic). So Schoof is appealing to the general "Cohen–Lenstra principle", rather than any precise conjecture.

  • $\begingroup$ To Alex B . Thank you very much! So as I understand, the General Cohen-Lenstra principle means that a random module $M$ appears with probability proportional to $1/ \sharp Aut M$. May I ask, why Schoof considered a random module modulo a random principal ideal? I also want to ask what's the meaning of 1-probability in general? $\endgroup$
    – gualterio
    Sep 17, 2019 at 10:19
  • $\begingroup$ I haven't studied much about the Cohen-Lenstra Heuristics. But I want to study the distribution of $p$-ranks of ideal class groups of abelian extensions of conductor $pq$. I've only read the first part of Cohen-Lenstra paper, and the paper of Schoof. And I'm also studying the thesis of Johannes Lengler. So I haven't seen how the heuristics is used yet. Today I was surprised to know that many papers just use the 'general' Cohen-Lenstra principle. And according to Malle's paper, the Cohen-Lenstra heuristics might not work in some case. So as a beginner in this field, I want to get some intuition $\endgroup$
    – gualterio
    Sep 17, 2019 at 10:28
  • $\begingroup$ about when I can use the heuristics. So can you recommend some references? Thank you very much again! $\endgroup$
    – gualterio
    Sep 17, 2019 at 10:32
  • 1
    $\begingroup$ @gualterio: you should probably ask those as separate questions, rather than attempting to have a conversation in the comments section. Briefly: the vanilla C-L distribution seems to explain well the behaviour of class groups of imaginary quadratic fields, but fails already for real quadratics, apparently due to the presence of units of infinite order. C-L try to justify the use of "1-probability" in their paper. Lenstra and I offer an explanation of this phenomenon and a reformulation of the C-L heuristics here: arxiv.org/abs/1803.06903, see also arxiv.org/abs/1510.02758. $\endgroup$
    – Alex B.
    Sep 17, 2019 at 11:25

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