Bhargava and Satriano on Galois closures of rings

Manjul Bhargava and Matt Satriano (starting a postdoc at Michigan this fall) posted a nice paper on the arXIv, “On a notion of “Galois closure” for extensions of rings.” The motivation for this work (I’m guessing) comes from Bhargava’s work on parametrizations of number fields.  Bhargava needs to generalize many classical objects of algebraic number theory from the setting of fields to rings.  For instance, the sextic resolvent of a degree 5 polynomial f(x) (developed by Lagrange, Malfatti, and Vandermonde in the late 18th century) is a degree 6 polynomial g(x) which has a rational root if and only if the equation f(x)=0 is solvable in radicals.

In modern Galois-theoretic language, we would describe the sextic resolvent as follows.  If K/Q is the Galois closure of the field generated by a root of f, then the Galois group Gal(K/Q) acts on the 5 roots of f, and is thus identified with a subgroup of S_5.  On the other hand, S_5 has an index-6 subgroup H, the normalizer of a 5-cycle.  So the action of Gal(K/Q) on S_5 / H gives rise to an extension of Q of degree 6 (maybe a field, maybe a product of fields) and this is Q[x] / g(x).

So far, so good.  But Bhargava needs to define a sextic resolvent which is a rank-6 free Z-module, not a 6-dimensional Q-vector space.  The sticking point is the notion of “Galois closure.”  What is the Galois closure of a rank-5 algebra over Z?  Or, for that matter, over a general commutative ring?

Bhargava gets around this question in his paper on quintic fields by using a concrete construction particular to the case of quintics.  But in the new paper, he and Satriano propose a very nice (“very nice” means “functorial”) completely general construction of a “Galois closure” G(A/B) for any extension of rings A/B such that A is a locally free B-module of rank n.  G(A/B) is a locally free B-module, endowed with an S_n-action, as you might want, and it agrees with the usual definition when A/B is an extension of fields.

But there are surprises — for instance, the Galois closure of the rank 4 algebra C[x,y,z]/(x,y,z)^2 over C is 32-dimensional!  In fact, the authors show that there is no definition of Galois closure which is functorial and for which G(A/B) always has the “expected” rank n! over B.  This might explain why no one has written down this definition before, and I think it is what gives the paper a sort of offbeat charm.  It illustrates a useful point:  you’ve got to know when it’s time to mulch an axiom.

Random pro-p groups, braid groups, and random tame Galois groups

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!