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Larsen’s question on the rank of the free part over k(A(k)tor)

Determine whether, for a general abelian variety A of dimension g over a finitely generated field k of characteristic zero and K = k(A(k)tor), the free Z-module M in the decomposition A(K) ≅ M ⊕ (Q/Z)^{2g} has infinite rank.

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Background

For K = k(A(k)tor), Larsen proved that A(K) decomposes as a direct sum of a free abelian group and a divisible part isomorphic to (Q/Z){2g}, and he showed that the free part has rank 0 when A is an elliptic curve. The general behavior of the rank of the free part for higher-dimensional abelian varieties remains unclear.

References

Larsen [Lar05] proved that A(K) = M + (Q/Z)+29 for any abelian variety A over k of dimension g, where K = k(A(k)tor) and M is a free Z-module, and proved that rank(M) = o if A is an elliptic curve (the result for the case where k = Q and A is an elliptic curve was independently obtained by Habegger [Hab13]). He asked whether rank(M) is infinite for a general A, but it remains open.

— Mordell--Weil groups over large algebraic extensions of fields of characteristic zero (2408.03495 - Asayama et al., 7 Aug 2024) in Remark 1.6 (Table 1, footnote c)