Cohomology of integer matrices and local-global divisibility on the torus
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)].
scientific article; zbMATH DE number 5543164
scientific article