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I'm currently attempting to understand Galois theory for schemes, largely following the books Galois Theory for Schemes by Henrik Lenstra and Galois Groups and Fundamental Groups by Tamas Szamuely. The main theorem is

Let $X$ be a connected scheme. Then there exists a profinite group $\pi$, uniquely determined up to isomorphism, such that the category $\mathbf{FEt}_X$ of finite \'etale covers of $X$ is equivalent to the category $\pi$-$\mathbf{sets}$ of finite sets on which $\pi$ acts continuously.

I'm attempting to come up with some explicit examples of this equivalence/correspondence for specific schemes. If $X = \operatorname{Spec} k$ for some field $k$, then one simply recovers (Grothendieck's formulation of) Galois theory for fields. If $X = \operatorname{Spec} \mathbf{Z}$, then the finite \'etale covers just look like disjoint unions of $\operatorname{Spec} \mathbf{Z}$, and the profinite group $\pi$ is trivial.

Both of these examples seem pretty boring though. Does anyone know of some interesting examples classifying the finite \'etale covers of some other scheme?

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    $\begingroup$ $\mathbb G_{m,k} \mapsto \hat{\mathbb Z}$ for any algebraically closed field $k$ of characteristic zero? Slightly more interesting is that the same group works for a nodal cubic. $\endgroup$
    – Will Sawin
    Jun 8, 2017 at 22:58
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    $\begingroup$ A famous example is the projective line minus three points; see Belyi's theorem. $\endgroup$
    – Qiaochu Yuan
    Jun 9, 2017 at 0:00
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    $\begingroup$ For varieties over $\mathbb C$, you get the profinite completion of the topological fundamental group. This should give many interesting examples. $\endgroup$
    – R. van Dobben de Bruyn
    Jun 9, 2017 at 0:26
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    $\begingroup$ For a $k$-variety, there is an exact sequence $1 \to \pi_1(\bar{X}) \to \pi_1(X) \to \pi_1(k) \to 1$. It follows that $\pi_1(\mathbf{P}^1_k) = \pi_1(k)$. $\endgroup$
    – user19475
    Jun 9, 2017 at 3:55
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    $\begingroup$ See here. $\endgroup$
    – Armando j18eos
    Jun 10, 2017 at 8:54

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