staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
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

[1] J. W. S. Cassels, An introduction to the Geometry of Numbers. Springer, 1997.  MR 1434478 |  Zbl 0866.11041
[2] R. Dvornicich, U. Zannier, Local-global divisibility of rational points in some commutative algebraic groups. Bull. Soc. Math. France 129 (2001), no. 3, 317–338. Numdam |  MR 1881198 |  Zbl 0987.14016
[3] R. Dvornicich, U. Zannier, On a local-global principle for the divisibility of a rational point by a positive integer. Bull. London Math. Soc. 39 (2007), 27–34.  MR 2303515 |  Zbl 1115.14011
[4] J.-P. Serre, Représentations linéaires des groupes finis. Hermann, 1967.  MR 232867 |  Zbl 0189.02603