On the rank of abelian varieties over function fields

Let $\text{cac}$ be a smooth projective curve defined over a number field $k$, $A\mathrm{/}k(\text{cac})$ an abelian variety and $(\tau ,B)$ the $k(\text{cac})\mathrm{/}k$-trace of $A$. We estimate how the rank of $A(k(\text{cac}))\mathrm{/}\tau B(k)$ varies when we take a finite cover $\pi :{\text{cac}}^{\mathrm{\prime }}\to \text{cac}$ defined over $k$ geometrically abelian.