Local-global divisibility on algebraic tori

TL;DR

This paper completely characterizes when local-global divisibility holds for algebraic tori over number fields, establishing dimension bounds related to prime p and providing counterexamples beyond these bounds.

Contribution

It proves the dimension bounds for local-global divisibility on algebraic tori are optimal and extends results under specific field hypotheses.

Findings

Local-global divisibility holds for tori of dimension less than p-1.

Counterexamples exist for dimensions greater or equal to p-1.

Under certain field conditions, divisibility holds up to dimension 3(p-1).

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 $r \geq p-1$. 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 $3(p-1)$.