The defect of strong approximation for commutative algebraic groups
type: Item
instance of: scholarly article
title: The defect of strong approximation for commutative algebraic groups
zbMATH Open document ID: 1194.14067
author: David Harari
published in: Algebra \& Number Theory
publication date: 2009-03-16T00:00:00Z
review text: Let \(k\) be a number field. The first main result of the paper under review is the construction of an exact sequence describing the topological closure of the group of \(k\)-rational points of a 1-motive \(H^0(k,M)\) in its adelic points. The second main result shows that the obstruction for strong approximation to hold for a semiabelian variety is essentially controlled by its algebraic Brauer group. Throughout the paper it is assumed that the Tate-Shafarevich groups of all abelian varieties involved are finite. For a finite place \(v\) of \(k\), let \(\mathcal O_v\) and \(k_v\) be the local ring and the local field at \(v\). For an infinite place \(\mathcal O_v\) and \(k_v\) denote both the completion of \(k\) at \(v\), and cohomology over the fields of real or complex numbers is understood to be Galois cohomology modified à la Tate. Let \(M\) be a 1-motive over \(k\) with dual \(M^\vee\). The first main theorem states that there is an exact sequence of topological groups \[ 0 \to \overline{H^0(k,M)} \to {\prod_{v}}'H^0(k_v, M) \to H^1(k,M^\vee)^D \to \text{III}^1(k,M) \to 0 \] where \((-)^D\) stands for Pontryagin dual, and where the product is a restricted product with respect to maps \(H^0(\mathcal O_v, M) \to H^1(k_v, M)\) for all places \(v\) of \(k\). In the case \(M\) is a semiabelian variety \(G\) and ignoring the real places of \(k\), this restricted product is the group of adelic points of \(G\). Finally, the first group in this sequence is the topological closure of the image of \(H^0(k, M)\) in the restricted product. The second main result, an application of the first, concerns the integral Manin obstruction. Let \(X\) be a flat scheme over the ring of integers of \(k\), whose generic fibre is a principal homogeneous space under a semiabelian variety \(G\) over \(k\). Let \(S\) be a finite set of places of \(k\) containig all infinite places and let \(A_S\) be the ring of \(S\)-adeles and let \(P = (P_v)_{v\in S}\) be an \(A_S\)-point of \(X\). Suppo…
reviewed by: Peter Jossen
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11J61
Mathematics Subject Classification ID: 11R56
zbMATH DE Number: 5529339
zbMATH Keywords: strong approximation
zbMATH Keywords: Brauer group
zbMATH Keywords: 1-motive
MaRDI profile type: Publication
full work available at URL: ant.2008.2.595
OpenAlex ID: W1970460490
DOI: 10.2140/ANT.2008.2.595
description: scientific article; zbMATH DE number 5529339