Weak approximation, Brauer and $R$-equivalence in algebraic groups over arithmetical fields

Abstract

We prove some new relations between weak approximation and some rational equivalence relations (Brauer and R-equivalence) in algebraic groups over arithmetical fields. By using weak approximation and local-global approach, we compute completely the group of Brauer equivalence classes of connected linear algebraic groups over number fields, and also completely compute the group of R-equivalence classes of connected linear algebraic groups $G$, which either are defined over a totally imaginary number field, or contains no anisotropic almost simple factors of exceptional type ${}^{3,6}D_{4}$, nor $E_{6}$. We discuss some consequences derived from these, e.g., by giving some new criteria for weak approximation in algebraic groups over number fields, by indicating a new way to give examples of non stably rational algebraic groups over local fields and application to norm principle. Some related questions and relations with groups of Brauer and R-equivalence classes over arbitrary fields of characteristic 0 are also discussed.