Local-global divisibility on algebraic tori

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 $3(p-1)$.

Figures (2)

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Theorems & Definitions (20)

• Theorem 1.2
• Theorem 1.3
• Definition 2.1: Dvornicich and Zannier DZ1
• Proposition 2.2: Dvornicich and Zannier DZ1
• Theorem 2.3: DZ3
• Theorem 2.4
• Lemma 2.5
• proof
• Lemma 2.6: Ill
• Lemma 3.1