Local–global properties of torsion points on three-dimensional abelian varieties

Let K be a number field and let ℓ>5 be a prime. We classify abelian threefolds A defined over K which have a non-trivial ℓ-torsion point mod p for almost all primes p of K, but are not K-isogenous to any abelian threefold over K with a non-trivial K-rational torsion point.