The rational points on certain Abelian varieties over function fields

TL;DR

This paper studies the structure of rational points on certain Abelian varieties over function fields, providing explicit methods to construct high-rank elliptic and hyperelliptic Jacobians using Prym varieties.

Contribution

It introduces a structure theorem for Mordell-Weil groups of twisted Abelian varieties via Prym varieties, enabling explicit construction of high-rank curves over function fields.

Findings

Mordell-Weil groups of twisted Abelian varieties are structured via Prym varieties.

Explicit methods for constructing high-rank elliptic and hyperelliptic Jacobians.

Provides examples of Abelian varieties with large ranks over specific function fields.

Abstract

In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we prove a structure theorem on their Mordell-Weil group. Our results give an explicit method for construction of elliptic curves, hyper- and super-elliptic Jacobians that have large ranks over function fields of certain varieties.