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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?

1

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 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?