On a question of Frey and Jarden about the rank of Abelian varieties

This paper deals with a classical question of Frey and Jarden, who asked in their 1974 paper if any non-zero Abelian variety over a number field K acquires infinite rank over the maximal Abelian extension Kab of the ground field. We generalize recent results of Rosen and Wong on the subject. However, the original question in full generality remains open. Some further results on the rank in certain other infinite extensions are included.