Ramified covers of abelian varieties over torsion fields

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of Q . In particular, we prove that every elliptic curve E over Q has the weak Hilbert property of Corvaja and Zannier both over the maximal abelian extension Q(ab) of Q , and over the field Q(A(tor))) obtained by adjoining to Q all torsion points of some abelian variety A over Q .