Local-global divisibility on algebraic tori
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916147002081280 |
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| author | Alessandrì, Jessica Chirivì, Rocco Paladino, Laura |
| author_facet | Alessandrì, Jessica Chirivì, Rocco Paladino, Laura |
| 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 $3(p-1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05922 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Local-global divisibility on algebraic tori Alessandrì, Jessica Chirivì, Rocco Paladino, Laura Number Theory 11E72, 14G05, 11G35 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 $3(p-1)$. |
| title | Local-global divisibility on algebraic tori |
| topic | Number Theory 11E72, 14G05, 11G35 |
| url | https://arxiv.org/abs/2301.05922 |