Although my number theory is really weak, I'm trying to understand the notion of etale coverings in this context. I think this could provide a very interesting point of view.

Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ on $\textrm{Spec} \ \mathbf{Z}$.

Q. Can we classify the etale coverings of $U$? (Say of a given degree...)

This is what I (think I) know.

Given a finite etale morphism $\pi:V\longrightarrow U$, the normalization of $X$ in the function field $L$ of $V$ is $\textrm{Spec} \ O_L$. This is a nontrivial fact, of course.

So, in view of the previous remark, can we also say what $V$ itself should be?

I'm guessing this has to do with the etale fundamental group $\pi_1(U)$. The latter is, I believe, finitely generated. I think that has to do with the fact that there are only finitely many ramified covers of a certain degree, right?

Of course, we can complicate things by replacing $\textrm{Spec} \ \mathbf{Z}$ by $\textrm{Spec} \ O_K$.

Example. Take $U= \textrm{Spec} \ \mathbf{Z} -{ (2)}$ and let $V\rightarrow U$ be an finite etale morphism. Suppose that $V$ is connected and that let $K$ be its function field. The normalization of $\textrm{Spec} \ \mathbf{Z}$ in $K$ is of course $O_K$. The extension $\mathbf{Z}\subset O_K$ is unramified outside $(2)$ and (possibly) ramified at $(2)$. Can one give a description of $V$ here?

EDIT. I just realized one can also ask themselves a similar question for $\mathbf{P}^1_{\mathbf{C}}$. Or even better, for any Riemann surface $X$.

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# Etale coverings of certain open subschemes in Spec O_K

Although my number theory is really weak, I'm trying to understand the notion of etale coverings in this context. I think this could provide a very interesting point of view.

Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ on $\textrm{Spec} \ \mathbf{Z}$.

Q. Can we classify the etale coverings of $U$? (Say of a given degree...)

This is what I (think I) know.

Given a finite etale morphism $\pi:V\longrightarrow U$, the normalization of $X$ in the function field $L$ of $V$ is $\textrm{Spec} \ O_L$. This is a nontrivial fact, of course.

So, in view of the previous remark, can we also say what $V$ itself should be?

I'm guessing this has to do with the etale fundamental group $\pi_1(U)$. The latter is, I believe, finitely generated. I think that has to do with the fact that there are only finitely many ramified covers of a certain degree, right?

Of course, we can complicate things by replacing $\textrm{Spec} \ \mathbf{Z}$ by $\textrm{Spec} \ O_K$.

Example. Take $U= \textrm{Spec} \ \mathbf{Z} -{ (2)}$ and let $V\rightarrow U$ be an finite etale morphism. Suppose that $V$ is connected and that let $K$ be its function field. The normalization of $\textrm{Spec} \ \mathbf{Z}$ in $K$ is of course $O_K$. The extension $\mathbf{Z}\subset O_K$ is unramified outside $(2)$ and (possibly) ramified at $(2)$. Can one give a description of $V$ here?