Elliptic curves and finitely generated Galois groups

Let $K$ be an extension of $\mathbb{Q}$ and $A/K$ an elliptic curve.

If $\mathrm{Gal}(\bar K/K)$ is finitely generated, then $A$ is of infinite rank over $K$.

In particular, this implies the $g=1$ case of the Junker-Koenigsmann conjecture.

This "anti-Mordellic'' result follows from a new "Mordellic'' theorem, which asserts that if $K_0$ is finitely generated over $\mathbb{Q}$, the points of an abelian variety $A_0/K_0$ over the compositum of all bounded-degree Galois extensions of $K_0$ form a virtually free abelian group.

This, in turn, follows from a second Mordellic result, which asserts that the group of $A_0$ over the extension of $K_0$ defined by the torsion of $A_0(\bar K_0)$ is free modulo torsion.