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.