## 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.

## Modeling lambda-invariants by p-adic random matrices

The paper “Modeling λ-invariants by p-adic random matrices,” with Akshay Venkatesh and Sonal Jain, just got accepted by Comm. Pure. Appl.  Math. But I forgot to blog about it when we finished it!  (I was a little busy at the time with the change in my personal circumstances.)

Anyway, here’s the idea.  As I’ve already discussed here, one heuristic for the Cohen-Lenstra conjectures about the p-rank of the class group of a random quadratic imaginary field K is to view this p-part as the cokernel of g-1, where g is a random generalized symplectic matrix over Z_p.  In the new paper, we apply the same philosophy to the variation of the Iwasawa p-adic λ-invariant.

The p-adic λ-invariant of a number field K is closely related to the p-rank of the class group of K; in fact, Iwasawa theory more or less gets started from the theorem that the p-rank of the class group of $K(\zeta_{p^n})$ is

$\lambda n + \mu p^n + \nu$

for some constants $\lambda, \mu, \nu$ when n is large enough, with $\mu$ expected to be 0 (and proved to be 0 when K is quadratic.)  On the p-adic L-function side, the λ-invariant is (thanks to the main conjecture) related to the order of vanishing of a p-adic L-function.  On the function field side, the whole story is told by the action of Frobenius on the p-torsion of the Jacobian of a curve, which is specified by some generalized symplectic matrix g over F_p.  The p-torsion in the class group is the dimension of the fixed space of g, while the λ-invariant is the dimension of the generalized 1-eigenspace of g, which might be larger.  It’s also in a sense more natural, depending only on the characteristic polynomial of g (which is exactly what the L-function keeps track of.)

So in the paper we do two things.  On the one hand, we study the dimension of the generalized 1-eigenspace of a random generalized symplectic matrix, and from this we derive the following conjecture: for each p > 2 and r >= 0,  the probability that a random quadratic imaginary field K has p-adic λ-invariant r is

$p^{-r} \prod_{t > r} (1-p^{-t})$.

Note that this decreases like $p^{-r}$ with r, while the p-rank of the class group is supposed to be r with probability more like $p^{-r^2}$.  So large λ-invariants should be substantially more common than large p-ranks.

The second part of the paper tests this conjecture numerically, and finds fairly good agreement with the data. A novelty here is that we compute p-adic  λ-invariants of K for small p and large disc(K); previous computational work has held K fixed and considered large p.  It turns out that you can do these computations reasonably efficiently by interpolation; you can compute special values L(s,chi_K) transcendentally for many s; given a bunch of these values, determined to a certain p-adic precision, you can compute the initial coefficients of the p-adic L-function with some controlled p-adic precision as well, and, in particular, you can provably locate the first coefficient which is nonzero mod p.  The location of this coefficient is precisely the λ-invariant.  This method shows that, indeed, large λ-invariants do pop up!  For instance, the 3-adic λ-invariant of $Q(\sqrt{-956238})$ is 14, which I think is a record.

Some questions still floating around:

• Should one expect an upper bound $\lambda \ll_\epsilon D_K^\epsilon$ for each odd p?  Certainly such a bound is widely expected for the p-rank of the class group.
• In the experiments we did, the convergence to the conjectural asymptotic appears to be from below.  For the 3-ranks of class groups of quadratic imaginary fields, this convergence from below was conjectured by Roberts to be explained by a secondary main term with negative coefficient.  Roberts’ conjecture was proved this year — twice!  Bhargava, Shankar, and Tsimerman gave a proof along the lines of Bhargava’s earlier work (involving thoughful decompositions of fundamental domains into manageable regions, and counting lattice points therein) and Thorne and Taniguchi have a proof along more analytic lines, using the Shintani zeta function.  Anyway, one might ask (prematurely, since I have no idea how to prove the main term correct!) whether the apparent convergence from below for the statistics of the λ-invariant is also explained by some kind of negative secondary term.

## 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 $\mathbf{Q}(\sqrt{-d})$, as d ranges over squarefree integers in [0..X],  you get a list of about $\zeta(2)^{-1} X$ 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.

$\frac{6}{\pi^2}X + O(X^{1/2}) = \zeta(2)^{-1} X + O(X^{1/2}).$