amitk@math.tifr.res.in, parimala@math.tifr.res.in

Submission: 2005, Jul 25

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.

2000 Mathematics Subject Classification: 20G15, 14G05.

Keywords and Phrases: adjoint classical groups, R-equivalence, algebras with involutions, similitudes.

Full text: dvi.gz 62 k, dvi 166 k, ps.gz 776 k, pdf.gz 308 k, pdf 342 k.

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