$R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$

Abstract

Abstract. Let F be a field of characteristic not 2 whose virtual cohomological dimension is at most 2. Let G be a semisimple group of adjoint type defined over F . Let RG(F ) denote the normal subgroup of G(F ) consisting of elements R-equivalent to identity. We show that if G is of classical type not containing a factor of type D n , G(F )/RG(F ) = 0. If G is a simple classical adjoint group of type D n , we show that if F and its multi-quadratic extensions satisfy strong approximation property, then G(F )/RG(F ) = 0. This leads to a new proof of the R-triviality of F -rational points of adjoint classical groups defined over number fields.