Geyer–Jarden's torsion conjecture for abelian varieties

Let $K$ be a finitely generated field over its prime field, let $e$ be a positive integer, and let $\sigma$ range over $\mathrm{Gal}(K^\mathrm{sep}/K)^e$. For almost all such $\sigma$, let $A$ be an abelian variety of positive dimension defined over $\tilde{K}(\sigma)$, with $A_\mathrm{tor}(\tilde{K}(\sigma))$ denoting its torsion subgroup and $A(\tilde{K}(\sigma))[n]$ its group of $n$-torsion points. Geyer–Jarden's conjecture. Theorem 1.2 remains true when the elliptic curve $E$ is replaced by an arbitrary abelian variety $A$ of positive dimension: if $e=1$, then $A_\mathrm{tor}(\tilde{K}(\sigma))$ is infinite, and there exist infinitely many primes $l$ such that $A(\tilde{K}(\sigma))[l]\neq 0$; if $e\geq 2$, then $A_\mathrm{tor}(\tilde{K}(\sigma))$ is finite; and if $e\geq 1$, then for every prime $l$, the group $A(\tilde{K}(\sigma))[l^\infty]=\bigcup_{i=1}^\infty A(\tilde{K}(\sigma))[l^i]$ is finite. The conjecture generalizes Geyer and Jarden's theorem on torsion subgroups of elliptic curves. Jacobson and Jarden proved that it is true when $K$ is a finite field; its general status is therefore not fully resolved in the source.