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.

We know, for instance, that infinitely many of these groups have order prime to l, where l is a fixed odd prime.  We can even make this quantitative:  a result of Kohnen and Ono shows that, of the ~X groups above, at least have order prime to l.

But an old conjecture of Cohen and Lenstra, well-supported by numerical evidence, suggests much more is true:  that in fact a positive proportion of class groups have order prime to l.  You might not find this surprising — after all, if we think of class numbers as random numbers, (1-1/l) of them should be indivisible by l.  So maybe it’s a bit more striking that Cohen-Lenstra predict that the proportion of class groups of order prime to l is the infinite product

In this post there’s a bit more about why this odd-looking expression is by far the most natural choice.  Natural or not, when you look at millions of quadratic imaginary fields, the Cohen-Lenstra heuristic looks really good.

Still, I used to be a bit skeptical about this conjecture.  After all, there are lots of contexts in number theory where numerical experiment gives misleading results.  Is there another reason to believe the Cohen-Lenstra heuristic?

To get some ideas, we might follow the analogy of the previous post and ask: what is the analogy of the Cohen-Lenstra conjecture over the function field k(t), where k is a finite field or C?  To fix ideas, let’s focus on the “first moment” aspect of the heuristic:

Question (Q version):  What is the average size of Cl(K)/ l Cl(K) as K ranges over imaginary quadratic fields of discriminant between X and 2X?  Is this asymptotically of the form aX for some constant a?

By class field theory, a surjection from Cl(K) to Z/lZ is naturally identified with an unramified Z/lZ-extension L of K, which is in turn a quadratic extension of Q.  The total extension L/Q has discriminant D_{K/Q}^l (at least away from 2 and l)  and its Galois group G is dihedral of order 2l.  So the Cohen-Lenstra conjecture can be rephrased in the following way:

Question(Q version, 2)  How many G-extensions of Q are there with discriminant between X^l and (2X)^l?  Is this asymptotically of the form aX for some constant a?

But now we get to use the fact that k(t) isn’t just a field; it’s the function field of the curve P^1/k.  And G-extensions of k(t) are in bijection with branched G-covers of P^1; or, equivalently, G-covers of P^1 with some number of punctures.  Actually, not all branched covers appear this way:  only those where the monodromy around each branch points is in the conjugacy class of an involution.  Let’s call such branch points simple.  When k is a finite field F_q (and at this point, if not earlier, we’d better insist that q be prime to 2l) a G-cover with n simple branch points has discriminant q^{nl}.  So in the F_q(t) column we are asking:

Question (F_q(t) version):  How many G-covers of P^1/F_q are there with n simple branch points?  Is this asymptotically of the form aq^n for some constant a?

What makes this tractable is that there’s a moduli space, called a Hurwitz space, for G-covers of P^1 with n simple branch points:  following the notation of the paper, I’ll call it Hur_{G,n}^c.  It’s a smooth scheme over Spec Z[1/2l] of dimension n.  (This paper of Romagny and Wewers provides a nice summary of its properties.)   So let’s rephrase the question yet again:

Question(F_q(t) version, 2):  How many points does Hur_{G,n}^c have over F_q?  Is this asymptotically of the form aq^n for some constant a?

And now you say — well, of course!  By the Weil conjectures, the answer is yes, and a is the number of connected components of Hur_{G,n}^C.  But careful:  in this question, q is FIXED.  The Weil bounds, on the other hand, come into play when q is allowed to go to infinity with n fixed; in the latter situation, one already has a result towards Cohen-Lenstra, which you can find in the work  of Jeff Achter.  To get the result we’re after, with F_q a fixed finite field, we need more.  This brings us to the third column, where the question is:

Question(C(t) version):  What is the cohomology of Hur_{G,n}^c(C)?

The complex manifold Hur_{G,n}^c(C) has a purely topological description; it’s a K(pi,1) where pi is the stabilizer in the natural action of the n-strand braid group on the set

{surjections from the free group on n letters f_1, … f_n to G, sending each f_i to an involution}

So the study of the cohomology of Hur_{G,n}^c(C) is a topological question about the cohomology of a certain finite index subgroup of the braid group (a congruence subgroup in the terminology of this previous post.)  And now I can state the main theorem of this paper, which is that the cohomology of Hurwitz spaces is stable:  for some positive real numbers A,B and positive integer D there’s a natural map

Hur_{G,n}^c(C) -> Hur_{G,n+D}^c(C)

such that the induced map on homology

H_p(Hur_{G,n}^c(C)) -> Hur_{G,n+D}^c(C)

is an isomorphism for all p < An + B.

This turns out to imply something close to a positive answer for the F_q(t) version of our Question — it doesn’t quite give the existence of the desired limit a, but it does give the existence of a liminf and a limsup, which approach a as q gets large.  This implication from a theorem in topology to a statement in analytic number theory over F_q(t) is exactly analogous to the method by which we counted squarefree polynomials over F_q in the previous post.

In a later post, maybe a bit about the proof of the stable cohomology theorem, which is, after all, the main point of the paper.