Isogeny components of Jacobians of étale covers Let us work over $\mathbb{C}$. Fix $X$ a smooth projective curve of genus at least $2$. For every simple abelian variety $A$, it is easy to come up with a ramified covering $Y\to X$ with a non-constant morphism $Y\to A$: just take any "diagonal" curve in $X\times A$.
What is known about the set of simple abelian varieties $A$ with an étale covering $Y\to X$ and a non-constant morphism $Y\to A$? Is it possible that it doesn't contain any abelian variety defined over $\overline{\mathbb{Q}}$?
 A: *

*It can certainly happen that this class contains an abelian variety over $\overline{\mathbb{Q}}$ without the curve $X$ being definable over $\overline{\mathbb{Q}}$. Indeed, start with an elliptic curve $E$ over $\overline{\mathbb{Q}}$. Let $Y\to E_{\mathbb{C}}$ be a ramified covering such that $Y$  is not definable over $\overline{\mathbb{Q}}$. (You can achieve this by ramifying in enough transcendental points of $E$.) Now, take $Y=X$ and $A=E_{\mathbb{C}}$.

*Let $k$ be an algebraically closed field. Then, any  smooth proper curve $X$ over $k$ maps  non-trivially to a simple abelian variety $A$ over $k$. Indeed, consider the Abel-Jacobi map $X\to J(X)$. Then there is an isogeny $J(X) \to \prod_i A_i$, where each $A_i$ is a simple abelian variety over $k$. Now, choose $i$ such that the composition $X\to J(X) \to \prod_i A_i \to A_i$ is non-constant. Then $A_i$ is the $A$ you want. 
This shows that, given any etale covering $Y$ of $X$, there is a simple abelian variety $A_Y$ such that $Y$ maps non-trivially to $A_Y$. It is not so clear to me whether you can say much about all these "simple" quotients of the Jacobian of $Y$.  
Final Remark. Note that in $2$ the resulting $A$ might be the Jacobian of $X$. In this case, $A$ is definable over a number field if and only if $X$ is definable over a number field. 
A: Bogomolov and Tschinkel speculated that perhaps, for every pair of curves curve X,X' over $\bar{\mathbf{Q}}$, X admits an unramified cover which dominates X'.  And they prove this when X is hyperelliptic and X' is the curve with equation $y^2 = x^6 - 1$.
https://arxiv.org/abs/math/0202223
