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 . [ABSTRACT FROM AUTHOR]