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