Local–global divisibility on algebraic tori

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