Weak approximation, Brauer and \(R\)-equivalence in algebraic groups over arithmetical fields. II. (Q1411725)

The main object of the paper under review is a connected linear algebraic group \(G\) defined over a number field \(k\). The author shows that the finite abelian groups \(G(k)/\text{Br}\) (the group of Brauer equivalence classes) and \(A(S,G)\) (the defect of weak approximation with respect to a finite set \(S\) of valuations of \(k\)) are birational invariants of \(G\) {as groups}. He also presents some formulas for \(G(k)/\text{Br}\) and proves the following bound for the defect of weak approximation: if \(H\) runs over connected \(k\)-subgroups of \(G\), then the order of \(A(S,H)\) is bounded by a constant depending only on \(G\), \(k\), \(S\). A similar result holds for the order of the Tate-Shafarevich group of \(H\). For Part I, cf. ibid. 40, 247--291 (2000; Zbl 1014.20019).