On Mordell–Weil groups of isotrivial abelian varieties over function fields

Abstract

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