Homological Stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II

Akshay Venkatesh, Craig Westerland, and I, recently posted a new paper, “Homological Stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields, II.” The paper is a sequel to our 2009 paper of the same title, except for the “II.”  It’s something we’ve been working on for a long time, and after giving a lot of talks about this material it’s very pleasant to be able to show it to people at last!

The main theorem of the new paper is that a version of the Cohen-Lenstra conjecture over F_q(t) is true.  (See my blog entry about the earlier paper for a short description of Cohen-Lenstra.)

For instance, one can ask: what is the average size of the 5-torsion subgoup of a hyperelliptic curve over F_q? That is, what is the value of

$\lim_{n \rightarrow \infty} \frac{\sum_C |J(C)[5](\mathbf{F}_q)|}{\sum_C 1}$

where C ranges over hyperelliptic curves of the form y^2 = f(x), f squarefree of degree n?

We show that, for q large enough and not congruent to 1 mod 5, this limit exists and is equal to 2, exactly as Cohen and Lenstra predict. Our previous paper proved that the lim sup and lim inf existed, but didn’t pin down what they were.

In fact, the Cohen-Lenstra conjectures predict more than just the average size of the group $J(C)[5](\mathbf{F}_q)$ as n gets large; they propose a the isomorphism class of the group settles into a limiting distribution, and they say what this distribution is supposed to be! Another way to say this is that the Cohen-Lenstra conjecture predicts that, for each abelian p-group A, the average number of surjections from $J(C)(\mathbf{F}_q)$ to A approaches 1. There are, in a sense, the “moments” of the Cohen-Lenstra distribution on isomorphism classes of finite abelian p-groups.

We prove that this, too, is the case for sufficiently large q not congruent to 1 mod p — but, it must be conceded, the value of “sufficiently large” depends on A. So there is still no q for which all the moments are known to agree with the Cohen-Lenstra predictions. That’s why I call what we prove a “version” of the Cohen-Lenstra conjectures. If you think of the Cohen-Lenstra conjecture as being about moments, we’re almost there — but if you think of it as being about probability distributions, we haven’t started!

Naturally, we prefer the former point of view.

This paper ended up being a little long, so I think I’ll make several blog posts about what’s in there, maybe not all in a row.