Vanishing of divisible parts over finite extensions of K(σ) when e = 1

Ascertain whether, when e = 1 and K is finitely generated over Q, for almost all σ ∈ G_K, every finite extension M of K(σ) and every abelian variety A over M satisfy A(M)_{div} = 0.

Background

By Proposition 3.3, Kummer-faithfulness reduces to vanishing of divisible parts for abelian varieties and tori. Jarden–Petersen proved the abelian-vanishing part for e ≥ 2, but their proof does not extend to e = 1. The authors state explicitly that whether this vanishing holds for e = 1 is unknown.

References

By Proposition 3.3, we can divide the discussion into two parts; torally Kummer-faithfulness and the vanishing property of the divisible part of the Mordell-Weil groups of abelian varieties. Jarden-Petersen [JP22, Remark 5.5], who proved the latter part when e ≥ 2 (cf. Theorem 3.1 (4)), pointed out that their proof does not work when e = 1. We do not know at the time of writing this paper whether the latter part establishes if e = 1.

— Mordell--Weil groups over large algebraic extensions of fields of characteristic zero  (2408.03495 - Asayama et al., 2024) in Section 5, opening paragraph