I am looking for good references (preferably, books) on Cohen-Lenstra Heuristics (on Real Quadratic fields) which explain in detail the reasons behind its fundamental assumption (higher the cardinality of the automorphism group less likely it is cyclic) and its connection with Dedekind Zeta function.

I have looked at the book 'Computational Algebraic Number Theory' by Cohen but it only states the conjecture and there is not much explanation given.

I would also like to know if there has been any work towards making this probabilistic conjecture more concrete so that we can get an answer for the Gauss' class number conjecture for real quadratic fields.

  • $\begingroup$ There is some paper by Venkatesh that talks about this if I'm not mistaken. $\endgroup$
    – Grothendieck
    Nov 1, 2013 at 12:01
  • 1
    $\begingroup$ Have you looked at the original paper of Cohen and Lenstra? $\endgroup$ Nov 4, 2013 at 5:04
  • 2
    $\begingroup$ Yes, but it is too technical for me, I am looking for some reference which explains it in a relatively simple manner. $\endgroup$ Nov 4, 2013 at 7:16

2 Answers 2


I don't have a book reference, but here are some rambling words about why one might, in general, expect objects $x$ to appear with probability proportional to $\frac{1}{|\text{Aut}(x)|}$. The short version is that this is a very natural number to associate to $x$.

To start with, let $X$ be a finite set. If you wanted to pick a random element of $X$, probably you would do it uniformly, and you expect that in the absence of further structure this is what would happen "in Nature."

Now suppose $X$ comes equipped with the action of a finite group $G$, and you wanted to pick a random orbit of $X$. One way to do this is to pick a random element of $X$ and consider its orbit. The induced probability measure on orbits, rather than assigning each orbit equal weight, assigns each orbit a weight inversely proportional to the size of its stabilizer.

This is one way to motivate the following definition. Let $X$ be a groupoid all of whose objects have a finite automorphism group which is tame in the sense that the sum

$$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Aut}(x)|}$$

converges (where $\pi_0(X)$ is the set of isomorphism classes of objects of $X$). The above sum is called the groupoid cardinality of $X$, and it induces a natural probability measure on $\pi_0(X)$ where an isomorphism class $x$ occurs with probability inversely proportional to $\text{Aut}(x)$.

One way to think about groupoid cardinality is that it is analogous to the Euler characteristic. The basic intuition is that we expect $\chi(X/G) = \frac{\chi(X)}{|G|}$ for a suitably nice group action of a finite group $G$ on a space $X$, and the one-object groupoid associated to a group $G$ can be thought of as a model of the classifying space $BG = EG/G$, where $EG$ is contractible and so in particular $\chi(EG) = 1$. For further discussion of the naturality of groupoid cardinality see this blog post.

Example. Let $X$ be a finite set on which a finite group $G$ acts. Form the action groupoid, whose objects are the elements of $X$ and which has a morphism $s_1 \to s_2$ labeled by $g \in G$ whenever $gs_1 = s_2$. Then the groupoid cardinality of the action groupoid is

$$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Stab}(x)|} = \frac{|S|}{|G|}$$

and the induced probability measure on $\pi_0(X)$ is the one we considered above.

Example. Let $G = \mathbb{Z}/2\mathbb{Z}$. Then $BG \cong \mathbb{RP}^{\infty}$ has a cell decomposition with one cell in every dimension, so its Euler characteristic should be

$$1 - 1 + 1 - 1 \pm ...$$

which has, say, Cesaro sum $\frac{1}{2} = \frac{1}{|G|}$!

Example. Consider the groupoid of finite sets and bijections. Its groupoid cardinality is

$$\sum_{n=0}^{\infty} \frac{1}{n!} = e.$$

With respect to the corresponding probability measure, a random finite set $S$ occurs with probability $\frac{1}{e |S|!}$. The distribution of cardinalities we get this way is Poisson with mean $1$.

What kind of process produces random finite sets? One candidate is to take the fixed point set of a random permutation $\pi \in S_n$ for $n$ large, and in fact one can show that as $n \to \infty$ the distribution of $|\text{Fix}(\pi)|$ approaches a Poisson distribution with mean $1$. See, for example, this blog post for details.

More generally, the distribution of the number of $r$-cycles of a random permutation in $S_n$ as $n \to \infty$ is Poisson with mean $\frac{1}{r}$. This should have an interpretation in terms of random finite sets of $r$-cycles, and indeed it does: a collection of $n$ $r$-cycles should have an automorphism group of size $r^n n!$ because we can both permute the cycles and cyclically permute the elements in each cycle, and this recovers a Poisson distribution with mean $\frac{1}{r}$.

For a related example of a more number-theoretic flavor, one can say the same thing about irreducible factors of degree $r$ in a random monic polynomial of large degree over $\mathbb{F}_q$, except that now one has to let $q \to \infty$ as well: see this blog post for details.

Example. Here is another example from number theory. Recall that by the Chebotarev density theorem, the Frobenius elements associated to primes $p$ in the Galois group $G = \text{Gal}(K/\mathbb{Q})$ of a number field $K$ are equidistributed in $G$ as $p$ varies. But Frobenius elements are not elements of $G$, they are conjugacy classes, hence objects, well-defined up to isomorphism, in the action groupoid associated to the action of $G$ on itself by conjugacy. So the Chebotarev density theorem can be reinterpreted as saying that a given conjugacy class appears as a Frobenius element with probability inversely proportional to the size of its centralizer.

  • 1
    $\begingroup$ The most beautiful and insightful answer I have ever read about Cohen-Lenstra. Still, I fear it might be somehow technical for the OP who was looking for something easier than Cohen-Lenstra. @Qiaochu: do you feel it unfeasable to put some groupoids under the rug? $\endgroup$ Nov 5, 2013 at 22:15
  • 1
    $\begingroup$ @Filippo: Whenever we're talking about the automorphism groups of a collection of mathematical objects we're talking about groupoids. It seems silly to avoid the language that concisely describes what we're actually talking about. $\endgroup$ Nov 5, 2013 at 22:39
  • 1
    $\begingroup$ I completely agree, although you might see my point that if the OP is wondering how to put is hands on C-L heuristics, it might be worth it to put some technology in perspective instead of presenting things "as they are". I say this because I remember things that had scared me for a long time and eventually turned out to be easy, if only I had been smoothly introduced to them. $\endgroup$ Nov 5, 2013 at 23:10

There is among other references the thesis of J. Lengler devoted to the Cohen-lenstra heuristics. This is not too technical and has many details. The author says: " The aim of this thesis is to explain and motivate the fundamental assumption of the Cohen-Lenstra heuristics, namely that the larger the automorphism group of a group is, the less likely it should appear". The author also has a paper on "The Global Cohen-Lenstra Heuristic", see http://arxiv.org/abs/0912.4977.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.