Does a curve over a number field have a finite etale cover of given degree

Question

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.

Does there exist a curve $Y$ over $K$ with a finite etale $K$-morphism $Y\to X$ of degree $d$?

I know how to do this over $\bar{K}$ (and thus over some finite extension of $K$). In fact, it suffices to find a finite degree topological cover $Y_{\mathbf{C}} \to X_{\mathbf{C}}$. This is easy to achieve by embedding $X_{\mathbf{C}}$ into its Jacobian and taking a degree $d$ topological cover of the Jacobian $J = \mathbf{C}^g/\Lambda$ of $X_{\mathbf{C}}$. The latter can be constructed easily by taking a sub-lattice of $\Lambda$ of index $d$.

Two problems arise.

The curve $X$ might not embed into its Jacobian, i.e., it might happen that $X(K)$ is empty. So if it helps, assume $X(K)$ is non-empty.

Also, the etale cover of $J$ constructed over $\mathbf{C}$ might not be defined over $K$ a priori. Can one show that it actually is defined over $K$?

Answer 1

This is not a complete answer, but I am trying to translate your issue in something more tractable.

If you assume there exists a rational point $x$, then the image of the corresponding section $s=s_x$ of $\pi_1(X,\overline x)\to \rm{Gal}_k$ is a closed subgroup of $\pi_1(X,\overline x)$ thus corresponds by Galois theory to a pro-cover $X_s\to X$ which is geometrically connected by construction (this works as well, of course, if you assume only the existence of $s$). So in this case you have plenty of covers (all intermediate covers) but it is not obvious to me how to determine their orders.

Going back to the general formulation of your problem, the usual way to solve your first problem is to use the universal albanese torsor. For a curve, this is just the degree-$1$ part of the Picard scheme of $X/k$, the map $X\to {\rm Pic}^1_{X/k}$ corresponds to the data of the isomorphism class of $\mathcal O_{X\times_k X}(\Delta)$, where $\Delta$ is the diagonal of $X\times_k X$. The albanese torsor is a torsor under the jacobian ${\rm Pic}^0_{X/k}$. Moreover, the morphism $X\to {\rm Pic}^1_{X/k}$ induces on $\pi_1$ an isomorphism $\pi_1(X,\overline x)\to \pi_1(X,\overline x)^{[ab]}$, where the last group is the quotient of $\pi_1(X,\overline x)$ by the kernel of $\pi_1(X_{\overline{k}},\overline x)\to \pi_1(X_{\overline{k}},\overline x)^{ab}$, the abelianization of the geometric fundamental group. (One says that $\pi_1({\rm Pic}^1_{X/k},\overline x)$ is the geometric abelianization of $\pi_1(X,\overline x)$). This enables to reduce your problem to the similar problem for a torsor under an abelian variety. Both problems are equivalent if you want to consider abelian covers only.

I am unsure on how to solve this last issue, but I think this is known - let us hope that an expert of abelian varieties will give the answer.