Brauer-Manin obstruction to weak approximation on abelian varieties
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.
scientific article; zbMATH DE number 902621
scientific article