Frey–Jarden conjecture on infinite rank over maximal abelian extensions

Determine whether, for every abelian variety A defined over an algebraic number field k, the Mordell–Weil group A(k^ab) over the maximal abelian extension k^ab of k has infinite rank.

Background

The paper studies Mordell–Weil groups of semiabelian varieties over large algebraic extensions. A classical conjecture of Frey and Jarden predicts that passing to the maximal abelian extension greatly enlarges the Mordell–Weil group, giving infinite rank. Many partial results are known, but the full conjecture remains unresolved.