Local-global divisibility on algebraic tori

Abstract

We give a complete answer to the local-global divisibility problem for algebraic tori. In particular, we prove that given an odd prime p, if T is an algebraic torus of dimension r < p − 1 defined over a number field k, then the local-global divisibility by any power p n holds for T (k). We also show that this bound on the dimension is best possible, by providing a counterexample of every dimension r p − 1. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the p n -torsion point of T , the local-global divisibility still holds for tori of dimension less than 3(p − 1).