  In this paper we construct abelian varieties of large Mordell-Weil rank over
function fields. We achieve this by using a generalization of the notion of
Prym variety to higher dimensions and a structure theorem for the Mordell-Weil
group of abelian varieties over function fields proven in our previous works.
We consider abelian and dihedral covers of the projective space and apply the
above results to the twists of their Albanese varieties.
