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$p$, if T$T$ is an algebraic torus of dimension r< p-1$ defined over a number field k$k$, then the local-global divisibility by any power pn$p<^>n$ holds for T(k)$T(k)$. We also show that this bound on the dimension is best possible, by providing a counterexample for every dimension r > p-1$r \geqslant p-1$. Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the pn$p<^>n$-torsion points of T$T$, the local-global divisibility still holds for tori of dimension less than 3(p-1)$3(p-1)$.