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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.05922 |
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Table of Contents:
- 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 $3(p-1)$.