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

by Cullinan, John

Let A be an abelian variety over a number field K , and let [cursive l] be a prime number. If A has a K -rational [cursive l]-torsion point, then for almost finite places [Special characters omitted.] of K, A has an [cursive 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 [cursive l]-torsion points; some geometric realizations of these obstructions are provided.