Ramified covers of abelian varieties over torsion fields

TL;DR

This paper investigates rational points on ramified covers of abelian varieties over specific infinite Galois extensions, establishing the weak Hilbert property for elliptic curves over these fields.

Contribution

It proves that every elliptic curve over Q has the weak Hilbert property over the maximal abelian extension and the field generated by torsion points of an abelian variety.

Findings

Elliptic curves over Q satisfy the weak Hilbert property over Q^{ab} and Q(A_{tor})

The weak Hilbert property holds for these fields for all elliptic curves over Q

Advances understanding of rational points on ramified covers over infinite Galois extensions.

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 $\mathbb{Q}^{\rm ab}$ of $\mathbb{Q}$, and over the field $\mathbb{Q}(A_{\rm tor})$ obtained by adjoining to $\mathbb{Q}$ all torsion points of some abelian variety $A$ over $\mathbb{Q}$.