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Marco Illengo
Cohomology of integer matrices and local-global divisibility on the torus
Journal de théorie des nombres de Bordeaux, 20 no. 2 (2008), p. 327-334, doi: 10.5802/jtnb.629
Article PDF | Reviews MR 2477506 | Zbl pre05543164

Résumé - Abstract

Let ${p\ne 2}$ be a prime and let $G$ be a $p$-group of matrices in ${{{\mathrm{SL}}}_n(\mathbb{Z})}$, for some integer $n$. In this paper we show that, when ${n<3(p-1)}$, a certain subgroup of the cohomology group ${H^1(G,{{\mathbb{F}}}_p^n)}$ is trivial. We also show that this statement can be false when ${n\ge 3(p-1)}$. Together with a result of Dvornicich and Zannier (see [2]), we obtain that any algebraic torus of dimension ${n<3(p-1)}$ enjoys a local-global principle on divisibility by $p$.

Bibliography

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