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

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 D4, nor E6. 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. Introduction. LetG be a linear algebraic group defined over a field k. There are two closely related questions in the arithmetic theory of algebraic groups over fields : the question of weak approximation and that of rationality of a given G. It is very difficult to study such questions for arbitrary groups over arbitrary fields. One should restrict to some class of groups and fields which ∗Supported by a Lady Davis Fellowship and ICTP