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 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 𝑝𝑛 holds for 𝑇(𝑘). We also show that this bound on the dimension is best possible, by providing a counterexample for every dimension 𝑟 ⩾ 𝑝−1. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the 𝑝𝑛-torsion points of 𝑇, the local– global divisibility still holds for tori of dimension less than 3(𝑝 − 1).