On the variation of the rank of Jacobian varieties on unramified abelian towers over number fields

Let $C$ be a smooth projective curve defined over a number field $k$, $X\mathrm{/}k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(\tau ,B)$ the $k(C)\mathrm{/}k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))\mathrm{/}\tau B(k)$ varies when we take an unramified abelian cover $\pi :C^{\mathrm{\prime }}\to C$ defined over $k$.