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

TL;DR

This paper investigates the Mordell-Weil rank of isotrivial abelian varieties over function fields, showing it depends only on the fundamental group of the complement to the discriminant with certain singularities, and constructs Jacobians with arbitrarily large ranks.

Contribution

It establishes a dependence of Mordell-Weil rank on fundamental groups for a new class of singularities and constructs examples of Jacobians with large ranks.

Findings

Mordell-Weil rank depends only on fundamental group for certain singularities.

Introduces a new CM class of singularities including unibranched plane curve singularities.

Constructs Jacobians with arbitrarily large Mordell-Weil rank.

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.