# 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). –  Ariyan Javanpeykar 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. –  Ariyan Javanpeykar Jul 10 '10 at 8:13