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 ℚab of ℚ, and over the field ℚ (A tor) obtained by adjoining to ℚ all torsion points of some abelian variety A over ℚ.