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Vanishing of the divisible part for abelian varieties 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 M of K(σ) and any abelian variety A over M satisfy A(M)div = 0.

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Background

By Proposition 3.3, Kummer-faithfulness is equivalent to the combination of toral Kummer-faithfulness and vanishing of A(K)div for all abelian varieties A over the base. The paper proves toral Kummer-faithfulness for K(σ) when e = 1 (Theorem 5.4), and Jarden–Petersen established A(M)div = 0 when e ≥ 2, but their argument does not extend to e = 1. The vanishing for e = 1 thus remains unsettled.

References

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., 7 Aug 2024) in Section 5 (before Theorem 5.4)