Local-global properties of torsion points on three-dimensional abelian varieties
Let A be an abelian variety over a number field K, and let ℓ be a prime number. If A has a K-rational ℓ-torsion point, then for almost finite places [special characters omitted] of K, A has an ℓ-torsion point mod [special characters omitted]. Katz has shown that the converse is true if the dimension of A is less than three, and has exhibited specific counterexamples when A has dimension greater than or equal to three. Using the subgroup structure of the finite symplectic group, we classify those abelian threefolds which violate this local-global principle for ℓ-torsion points; some geometric realizations of these obstructions are provided.