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

Ramified covers of abelian varieties over torsion fields