Counting branched covers of the projective line and Spec Z

I've asked a question like this before, but now I'm more interested in counting the number of covers.

We suppose given the following data.

1. A positive integer $d$

2. A finite set of closed points $B= ( b_1,\ldots,b_n )$ in $\mathbf{P}^1_\mathbf{C}$

3. Branch types $T_1,\ldots, T_n$.

Question. How many branched covers of $\mathbf{P}^1_\mathbf{C}$ exist which are branched only over $b_i$ (with branch type $T_i$ over each $b_i$)?

The answer lies within the Hurwitz number for $(T_1,\ldots,T_n)$. This translates the problem to combinatorial group theory.

Now, for my main question:

Q1. Can one count'' covers of $\textrm{Spec} \mathbf{Z}$ as above? That is, can one count the number of finite field extensions $$\mathbf{Q}\subset K$$ of given degree $d=[K:\mathbf{Q}]$ which are unramified outside a given set of prime numbers $p_1,\ldots,p_n$ with ramification types $T_1,\ldots,T_n$?

I know that one can use Minkowski's Geometry of Numbers to give some nontrivial bounds on the discriminant. Is this the best we can do?

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Special case: what is known about the fundamental group of the complement of one prime $p$? I mean the limit of the Galois group of $K/\mathbb Q$ over all $K$ unramified except at $p$. I know the abelianization, but what about the rest? They say that primes in Spec Z are like knots in 3-space. Does a prime have an Alexander polynomial? –  Tom Goodwillie Jul 9 '10 at 14:14
About the last question, I'm sorry if I misunderstood, but is this a rhetorical question? I have to be honest and say that I don't know any knot theory. So I really don't know. But it is an interesting point of view that one can take. Concerning the fundamental group, I don't really know much more than the fact that it is finitely generated (that's how you can prove that there are only finitely many such covers). –  Ari Jul 9 '10 at 17:09
Not rhetorical, but not comprehensible unless you happen to have the right two points of view. I think I'll post it as a separate question. –  Tom Goodwillie Jul 9 '10 at 21:17
It reminds me of a question I asked the other day: mathoverflow.net/questions/30186/… The answers were just links to online material of J. Baez on the subject and another question asked on MO. I'll be trying to get the right two points of view now. –  Ari Jul 10 '10 at 8:13
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1 Answer

One thing to keep in mind is that the analogue of Spec Z is really P^1 over a finite field k, not P^1/C. And here already one does not have a simple "Hurwitz-type formula" for the number of G-covers with given branching which are defined over k.

Just to give an example which may be illustrative; suppose that G = S_3, and you require that the inertia at the primes p_1, ... p_n is tame and maps to a transposition in G. The extensions of Q of this kind are more or less in bijection with the etale Z/3Z covers of the quadratic field K = Q(sqrt(N)) where N = p_1....p_n. (I am being careless about the real place here.) In any event, to "count" the number of covers is in effect to compute the size of the 3-part of the class group of K. There is not going to be a nice formula for this, and in particular it will depend unpredictably on the primes in question. On the other hand, you can compute the average of this quantity over squarefree integers N, by Davenport-Heilbronn.

So I would say:

"No" to your Q1 as stated. "Yes, for some choices of G," to your Q1 on average -- e.g. for G = S_3 (by Davenport-Heilbronn), for G = D_4 (Cohen-Diaz y Diaz-Olivier), for G = S_4, S_5 (by Bhargava, though perhaps some slight and presumably true refinement of Bhargava to squarefree discriminants is needed), for G = D_p when K is F_ell(T) by work of myself, Venkatesh and Westerland.

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