Ramified covers of abelian varieties over torsion fields

Abstract

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

arXiv commentpreliminaries section shortened, minor corrections