How can we justify the use of Example 5,4 (of Cohen, Lenstra) assuming their heuristics

Question

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!

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.