Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-variety in such a way so that the morphism $f: X' \to X$ inducing the topological map is a finite etale morphism over $\overline{K}$. However, the variety $X'$ and the morphism $f$ may not descend to $K$.


My question is as follows: does there always exist a further finite etale covering $f' : X'' \to X'$ such that the composition $X'' \to X$ may be defined over $K$?


EDIT: Just to be clear, I'd like all the covers involved to be geometrically connected to avoid trivial solutions.



Are "large enough" finite etale covers arithmetic?
Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by S. carmeli.


In terms of etale fundamental groups the question can be paraphrased as follows: given an open subgroup $H\subset \pi\_1(X\_{\overline{K}},x)$ does there exist an open subgroup $H'\subset H$ such that the action of the Galois group $G\_K$ on $\pi\_1(X\_{\overline{K}},x)$ preserves $H'$.


This is true and follows from $\pi\_1(X\_{\overline{K}},x)$ being topologically finitely generated. Consider the subgroup $\Gamma\_H\subset G\_K$ consisting of elements $\gamma\in G\_K$ such that $\gamma(H)=H$. Let $h\_1,\dots, h\_n$ be a set of topological generators of $H$ ($H$ is topologically finitely generated because it has finite index in $\pi\_1(X\_{\overline{K}})$). Then $\Gamma\_H$ can be expressed as $\{\gamma\in G\_K|\gamma(h\_i)\in H\}$ so $\Gamma\_H$ is an intersection of finitely many open subset, hence is an open subgroup. In particular, $\Gamma\_H$ has finite index in $G\_K$. Take $\Gamma\subset \Gamma\_H$ to be an open subgroup which is moreover normal in $G\_K$.


Let $g\_1,\dots, g\_m$ be a set of representatives of cosets of $\Gamma$ in $G\_K$. Then $H'=\bigcap g\_i(H)$ is an open subgroup with the desired property. Indeed, suppose that $x\in H'$ and $\gamma g\_i\in G\_K$ are arbitrary elements where $\gamma\in \Gamma$ and $i\in\{1,\dots, m\}$. The result of the action $\gamma \circ g\_i(x)$ lies in $H'$ because for each $k=1,\dots, m$ we have $g\_k^{-1}\gamma g\_i=\gamma'g\_j^{-1}$ for some $\gamma'\in \Gamma$ and $j\in\{1,\dots, m\}$ so $\gamma g\_i(x)\in \gamma g\_ig\_j(H)=g\_k\gamma'(H)=g\_k(H)$.


We can think of this argument as of a generalization of the proof that a compact group acting on a finite-dimensional $\mathbb{Q}\_p$-vector space always preserves some $\mathbb{Z}\_p$-lattice.


