Frey–Jarden conjecture on infinite rank over the maximal abelian extension

Establish that for any 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 opens by recalling a classical conjecture of Frey and Jarden from 1974 concerning the size of Mordell–Weil groups over the maximal abelian extension of a number field. This conjecture has motivated extensive work on ranks over large algebraic extensions, and the present paper studies related structural properties (e.g., Kummer-faithfulness and freeness modulo torsion) for broader classes of extensions.