Brauer-Manin obstruction to weak approximation on abelian varieties
type: Item
instance of: scholarly article
title: Brauer-Manin obstruction to weak approximation on abelian varieties
zbMATH Open document ID: 0870.14032
DOI: 10.1007/BF02762704
author: Lan Wang
published in: Israel Journal of Mathematics
publication date: 1996-08-22T00:00:00Z
review text: An algebraic variety \(X\) defined over a number field \(k\) satisfies the weak approximation property if the diagonal map \(X(k)\to \prod_{v\in S}X(k_v)\) has dense image for any finite set \(S\) of places of \(k\). In all known counter-examples it is the Brauer-Manin obstruction which is responsible for failure of weak approximation, and for many classes of varieties it is proven to be the only obstruction [see \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc}, Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)]. In the paper under review it is shown that this is true for abelian varieties under additional assumptions. Namely, one should assume that the Tate-Shafarevich group is finite and the closure of the image of the diagonal map with respect to archimedean places \(A(k)\to \prod_{v\in S_{\infty}}A(k_v)\) contains the connected component of the identity. For some abelian varieties over \(\mathbb Q\), these assumptions are known to hold. On the other hand, it is shown that for abelian varieties over a higher number field the Brauer-Manin obstruction to weak approximation may be not the only one, namely, this is the case for some elliptic curves over quadratic fields.
Mathematics Subject Classification ID: 14K15
Mathematics Subject Classification ID: 14B15
Mathematics Subject Classification ID: 11G99
zbMATH DE Number: 902621
zbMATH Keywords: weak approximation property
zbMATH Keywords: Brauer-Manin obstruction
zbMATH Keywords: abelian varieties
zbMATH Keywords: Tate-Shafarevich group
zbMATH Keywords: elliptic curves over quadratic fields
reviewed by: Boris Kunyavskiĭ
describes a project that uses: ecdata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 902621