I’ve posted a new paper with Nigel Boston, “Random pro-p groups, braid groups, and random tame Galois groups.”
The paper proposes a kind of “non-abelian Cohen-Lenstra heuristic.” A typical prediction: if S is a randomly chosen pair of primes, each of which is congruent to 5 mod 8, and G_S(p) is the Galois group of the maximal pro-2 extension of Q unramified away from S, then G_S(p) is infinite 1/16 of the time.
The usual Cohen-Lenstra conjectures — well, there are a lot of them, but the simplest one asks: given an odd prime p and a finite abelian p-group A, what is the probability P(A) that a randomly chosen quadratic imaginary field K has a class group whose p-primary part is isomorphic to A? (Note that the existence of P(A) — which we take to be a limit in X of the corresponding probability as K ranges over quadratic imaginary fields of discriminant at most X — is not at all obvious, and in fact is not known for any p!)
Cohen and Lenstra offered a beautiful conjectural answer to that question: they suggested that the p-parts of class groups were uniformly distributed among finite abelian p-groups. And remember — that means that P(A) should be proportional to 1/|Aut(A)|. (See the end of this post for more on uniform distribution in this categorical setting.)
Later, Friedman and Washington observed that the Cohen-Lenstra conjectures could be arrived at by another means: if you take K to be the function field of a random hyperelliptic curve X over a finite field instead of a random quadratic imaginary field, then the finite abelian p-group you’re after is just the cokernel of F-1, where F is the matrix corresponding to the action of Frobenius on T_p Jac(X). If you take the view that F should be a “random” matrix, then you are led to the following question:
Let F be a random element of GL_N(Z_p) in Haar measure: what is the probability that coker(F-1) is isomorphic to A?
And this probability, it turns out, is precisely the P(A) conjectured by Cohen-Lenstra.
(But now you cry out: but Frobenius isn’t just any old matrix! It’s in the generalized symplectic group! Yes — and Jeff Achter has shown that, at least as far as the probability distribution on A/pA goes, the “right” random matrix model gives you the same answer as the Friedman-Washington quick and dirty model. Phew.)
Now, in place of a random quadratic imaginary field, pick a prime p and a random set S of g primes, each of which is 1 mod p. As above, let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S; this is a pro-p group of rank g. What can we say about the probability distribution on G_S(p)? That is, if G is some pro-p group, can we compute the probability that G_S(p) is isomorphic to G?
Again, there are two approaches. We could ask that G_S(p) be a “random pro-p group of rank g.” But this isn’t quite right; G_S(p) has extra structure, imparted to it by the images in G_S(p) of tame inertia at the primes of S. We define a notion of “pro-p group with inertia data,” and for each pro-p GWID G we guess that the probability that G_S(p) = G is proportional to 1/Aut(G); where Aut(G) refers to the automorphisms of G as GWID, of course.
On the other hand, you could ask what would happen in the function field case if the action of Frobenius on — well, not the Tate module of the Jacobian anymore, but the full pro-p geometric fundamental group of the curve — is “as random as possible.” (In this case, the group from which Frobenius is drawn isn’t a p-adic symplectic group but Ihara’s “pro-p braid group.”)
And the happy conclusion is that, just as in the Cohen-Lenstra setting, these two heuristic arguments yield the same prediction. For the relatively few pro-p groups G such that we can compute Pr(G_S(p) = G), our heuristic gives the right answer. For several more, it gives an answer that seems consistent with numerical experiments.
Maybe it’s correct!