Open torsion statements over K(σ) in positive characteristic (Geyer–Jarden program)

Determine whether the following hold when K is an infinite field of positive characteristic: (a) for e = 1, for almost all σ ∈ G_K, and for 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; (b) for e ≥ 2, for almost all σ ∈ G_K^e and any abelian variety A over K(σ), the torsion subgroup A(K(σ))tor is finite.

Background

The paper summarizes known results on torsion of abelian varieties over fields of the form K(σ) and K[σ], including proofs in characteristic zero. Geyer and Jarden conjectured that certain torsion finiteness/infinity statements hold over large algebraic fields for all finitely generated K. While parts are known in characteristic zero and for elliptic curves, the cases (a) and (b) remain unresolved in positive characteristic for infinite fields.