On Mordell-Weil group of isotrivial abelian varieties over function fields

Abstract

We show that the Mordell Weil rank of an isotrivial abelian variety with a cyclic holonomy depends only on the fundamental group of the complement to the discriminant provided the discriminant has singularities in the introduced here CM class. This class of singularities includes all unibranched plane curves singularities. As a corollary we give a family of simple Jacobians over field of rational functions in two variable for which the Mordell Weil rank is arbitrary large.