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 , if is an algebraic torus of dimension defined over a number field , then the local–global divisibility by any power holds for . We also show that this bound on the dimension is best possible, by providing a counterexample for every dimension . Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the ‐torsion points of , the local–global divisibility still holds for tori of dimension less than .