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

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 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 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.

Homological stability for Hurwitz spaces and the Cohen-Lenstra conjecture over function fields

Now I’ll say a little bit about the actual problem treated by the new paper with Venkatesh and Westerland.  It’s very satisfying to have an actual theorem of this kind:  for years now we’ve been going around saying “it seems like asymptotic conjectures in analytic number theory should have a geometric reflection as theorems about stable cohomology of moduli spaces,” but for quite a while it was unclear we’d ever be able to prove something on the geometric side.

The new paper starts with the question: what do ideal class groups of number fields tend to look like?

That’s a bit vague, so let’s pin it down:  if you write down the ideal class group of the quadratic imaginary number fields , as d ranges over squarefree integers in [0..X],  you get a list of about finite abelian groups.

The ideal class group is the one of the most basic objects of algebraic number theory; but we don’t know much about this list of groups!  Their orders are more or less under control, thanks to the analytic class number formula.  But their structure is really mysterious.

The braid group, analytic number theory, and Weil’s three columns

This post is about a new paper of mine with Akshay Venkatesh and Craig Westerland; but I’m not going to mention that paper in the post. Instead, I want to explain why topological theorems about the stable homology of moduli spaces are relevant to analytic number theory.  If you’ve seen me give a talk about this stuff, you’ve probably heard this spiel before.

We start with Weil’s famous quote about “the three columns”:

“The mathematician who studies these problems has the impression of deciphering a trilingual inscription. In the first column one finds the classical Riemannian theory of algebraic functions. The third column is the arithmetic theory of algebraic numbers.  The column in the middle is the most recently discovered one; it consists of the theory of algebraic functions over finite fields. These texts are the only source of knowledge about the languages in which they are written; in each column, we understand only fragments.”

Let’s see how a classical question of analytic number theory works in Weil’s three languages.  Start with the integers, and ask:  how many of the integers between X and 2X are squarefree?  This is easy:  we have an asymptotic answer of the form

(In fact, the best known error term is on order X^{17/54}, and the correct error term is conjectured to be X^{1/4}; see Pappalardi’s “Survey on k-freeness” for more on such questions.)

So far so good.  Now let’s apply the popular analogy between number fields and function fields, going over to Weil’s column 3, and ask: what’s the analogous statement when Z is replaced by F_q[T]?