  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 $k$.
