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

Kulshrestha, Amit; Parimala, R.

doi:10.1090/S0002-9947-07-04300-0

$R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$
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by Amit Kulshrestha and R. Parimala PDF

Trans. Amer. Math. Soc. 360 (2008), 1193-1221 Request permission

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.

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