TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF ℚ

Let K be a finitely generated extension of ℚ, and let A be a nonzero abelian variety over K. Let K̄ be the algebraic closure of K, and let Gal(K) = Gal(K̄/K) be the absolute Galois group of K equipped with its Haar measure. For each σ ∈ Gal(K), let K̄ (σ) be the fixed field of σ in K̄. We prove that for almost all σ ∈ Gal(K), there exist infinitely many prime numbers l such that A has a nonzero K̄(σ)-rational point of order l. This completes the proof of a conjecture of Geyer-Jarden from 1978 in characteristic 0.