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

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 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.