Cohomology of integer matrices and local-global divisibility on the torus
type: Item
instance of: scholarly article
title: Cohomology of integer matrices and local-global divisibility on the torus
zbMATH Open document ID: 1179.11039
author: Marco Illengo
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/10838
review text: The main result of the paper is the following: Let \(p\neq 2\) be a prime and let \(n<3(p-1)\). For every \(p\)-group \(G\subset \text{SL}_n(\mathbb Z)\) the projection \(H^1(G,{\mathbb F}_p^n) \to \prod H^1(C,{\mathbb F}_p^n)\), the product being taken over all cyclic subgroups \(C\) of \(G\), is injective. It is also proved that the result is best possible with regard to \(n\). The motivation of the result is an application to the problem of local-global divisibility by a prime \(p\) of points on algebraic tori. Namely, let \({\mathcal T}\) be an algebraic torus defined over a number field \(k\) of dimension \(n<3(p-1)\); for any point \(P\in{\mathcal T}(k)\), if for all but finitely many completions \(k_v\) of \(k\) there exists a point \(D_v\in{\mathcal T}(k_v)\) with \(pD_v=P\), then there exists a point \(D\in {\mathcal T}(k)\) such that \(pD=P\). Also, in view of the optimality of the main result, the same statement is no longer true for larger values of \(n\). This improves on a result by the reviewer and \textit{U. Zannier} [Bull. Lond. Math. Soc. 39, No. 1, 27--34 (2007; Zbl 1115.14011)].
reviewed by: Roberto Dvornicich
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 5543164
zbMATH Keywords: cohomology of groups
zbMATH Keywords: algebraic tori
zbMATH Keywords: local-global divisibility
MaRDI profile type: Publication
OpenAlex ID: W2318553126
cites work: An introduction to the geometry of numbers.
cites work: Local-global divisibility of rational points in some commutative algebraic groups
cites work: On a local-global principle for the divisibility of a rational point by a positive integer
cites work: Q5583117
DOI: 10.5802/JTNB.629
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description: scientific article; zbMATH DE number 5543164