Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$. 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 $K$?

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 $K$?