I've been wondering about the following "finiteness statement" concerning etale covers for a while.


Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ is a smooth quasi-projective geometrically connected scheme over $K$.


Let $X$ be a variety, and let $d$ be an integer.


**Question.** Are there only finitely many varieties $Y$ over $K$ such that there exists a finite etale morphism $Y\to X$ of degree $d$ (up to $K$-isomorphism)?


This is true if $K$ is algebraically closed. In fact, by a standard Lefschetz principle argument, we may and do assume $K$ is the field of complex numbers. Then, the statement follows from GAGA and the fact that the topological fundamental group of $X$ is finitely generated.


In general, there might be problems with twists, but I'm not sure.


Note that if $\dim X = 0$ then the only variety $Y$ satisfying the above hypotheses is $Y= $ Spec $K$, because we require our varieties to be geometrically connected (over $K$).


If it helps, in my set-up, we may assume $K$ is finitely generated over $\mathbf Q$, or that $K$ is even a number field.



Do all varieties have only finitely many etale covers of fixed degree
Just to add one word to ulrich's answer: for a geometrically irreducible $K$-variety $X$, there is a short exact sequence of etale fundamental groups.
$$
0 \to \pi\_1^{\text{et}}(X\otimes\_K \overline{K}) \to \pi\_1^{\text{et}}(X) \to \text{Gal}(\overline{K}/K) \to 0.
$$
An étale degree $d$ cover is equivalent to a homomorphism $\phi:\pi\_1^{\text{et}}(X)\to \mathfrak{S}\_d$. This homomorphism gives an irreducible cover of $X\otimes\_K \overline{K}$ if the image of $\pi\_1^{\text{et}}(X\otimes\_K \overline{K})$ is a transtive subgroup of $\mathfrak{S}\_d$. But for a given homomorphism $\phi\_{\overline{K}}:\pi\_1^{\text{et}}(X\otimes\_K \overline{K}) \to \mathfrak{S}\_d$ with transitive image, there may be many extensions of this to a homomorphism $\phi$. That does not in itself prove that there are infinitely many étale covers of, say $\mathbb{G}\_{m,K}$. But it does help organize the infinitely many covers that ulrich listed.


