Cohomology of integer matrices and local-global divisibility on the torus

Marco Illengo[1]

  • [1] Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa, Italia

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 2, page 327-334
  • ISSN: 1246-7405

Abstract

top
Let p ≠ 2 be a prime and let  G be a p -group of matrices in SL n ( 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 , 𝔽 p n ) is trivial. We also show that this statement can be false when n ≥ 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 .

How to cite

top

Illengo, Marco. "Cohomology of integer matrices and local-global divisibility on the torus." Journal de Théorie des Nombres de Bordeaux 20.2 (2008): 327-334. <http://eudml.org/doc/10838>.

@article{Illengo2008,
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 $\{n&lt;3(p-1)\}$, 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 $\{n&lt;3(p-1)\}$ enjoys a local-global principle on divisibility by $p$.},
affiliation = {Scuola Normale Superiore Piazza dei Cavalieri 7 56126 Pisa, Italia},
author = {Illengo, Marco},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cohomology of groups; algebraic tori; local-global divisibility},
language = {eng},
number = {2},
pages = {327-334},
publisher = {Université Bordeaux 1},
title = {Cohomology of integer matrices and local-global divisibility on the torus},
url = {http://eudml.org/doc/10838},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Illengo, Marco
TI - Cohomology of integer matrices and local-global divisibility on the torus
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 2
SP - 327
EP - 334
AB - 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 ${n&lt;3(p-1)}$, 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 ${n&lt;3(p-1)}$ enjoys a local-global principle on divisibility by $p$.
LA - eng
KW - cohomology of groups; algebraic tori; local-global divisibility
UR - http://eudml.org/doc/10838
ER -

References

top
  1. J. W. S. Cassels, An introduction to the Geometry of Numbers. Springer, 1997. Zbl0866.11041MR1434478
  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. Zbl0987.14016MR1881198
  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. Zbl1115.14011MR2303515
  4. J.-P. Serre, Représentations linéaires des groupes finis. Hermann, 1967. Zbl0189.02603MR232867

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.