Ramified covers of abelian varieties over torsion fields

Abstract

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