On Mordell-Weil groups of Jacobians over function fields

TL;DR

This paper investigates the Mordell-Weil groups of Jacobians over function fields, establishing connections to homomorphisms over the base field, and provides explicit points and constructions of high-rank elliptic curves.

Contribution

It introduces a novel relationship between Mordell-Weil groups of Jacobians over function fields and homomorphisms over the base field, with explicit point constructions and new elliptic curve examples.

Findings

Explicit points on elliptic curves with unbounded rank over bar(t)

New construction of elliptic curves with high rank over (t)

Relation between Mordell-Weil groups and Jacobian homomorphisms

Abstract

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over $\Fpbar(t)$ and a new construction of elliptic curves with moderately high rank over $\C(t)$.