Identify the structural type of Mordell–Weil groups over k(σ) for almost all σ ∈ G_k

Determine, for almost all σ in the absolute Galois group G_k of a number field k, which of the following holds for every abelian variety A over k: (i) A(k(σ)) has a nontrivial divisible subgroup; or (ii) A(k(σ)) has trivial divisible part but A(k(σ))/A(k(σ))_{tor} is not a free Z‑module; or (iii) A(k(σ))/A(k(σ))_{tor} is a free Z‑module.

Background

The paper presents a table of possible structures of Mordell–Weil groups over various large algebraic extensions. For fields of the form k(σ), obtained as fixed fields of single automorphisms of G_k, existing results show that, for almost all σ, the field falls into one of three structural cases for Mordell–Weil groups, but which case actually occurs remains undetermined.