Freeness of Mordell–Weil groups over finite extensions of K[σ] when e = 1

Determine whether, for a finitely generated field K over Q and e = 1, for almost all σ ∈ G_K, any finite extension L of K[σ] and any semiabelian variety A of positive dimension over L satisfy that A(L)/A(L)tor is a free abelian group of countably infinite rank.

Background

Theorem 4.1 proves that for e ≥ 2, and almost all σ ∈ G_K^e, the Mordell–Weil group modulo torsion over finite extensions of K[σ] is free of countably infinite rank for all positive-dimensional semiabelian varieties. The case e = 1 is not accessible by the methods used, primarily because torsion in tori and abelian varieties can be infinite over K(σ) when e = 1.