Geyer–Jarden torsion statements in positive characteristic

Establish, for infinite finitely generated fields K of positive characteristic: (a) when e = 1, that for almost all σ ∈ G_K and any abelian variety A of positive dimension over K(σ), the torsion subgroup A(K(σ))_{tor} is infinite and A(K(σ))[ℓ] ≠ 0 for infinitely many primes ℓ; and (b) when e ≥ 2, that for almost all σ ∈ G_K and any abelian variety A over K(σ), the torsion subgroup A(K(σ))_{tor} is finite.

Background

The paper surveys known results on torsion over fields K(σ). In characteristic zero, statements (a)–(c) are established by works of Jacobson–Jarden, Zywina, and Jarden–Petersen. In positive characteristic, only partial results are known, and statements (a) and (b) remain unresolved.