A Local-Global Principle for Torsion points on Abelian Varieties
by
John Cullinan
University of Massachusetts

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 all prime ideals \mathfrakp of K, A has an l-torsion point mod \mathfrakp. Katz has shown that the converse is true if dimA <= 2, and has exhibited specific counterexamples when dimA >= 3. Using the subgroup structure of the symplectic group, we give a complete classification of those abelian threefolds which violate this ``local-global'' principle for l-torsion; some geometric examples will be provided.

Date received: January 16, 2004