Grégory Berhuy: Finiteness of $R$-equivalence groups of some adjoint classical groups of type $^2D_3$

Submission: 2006, Jul 7

Let $F$ be a field of charateristic different from $2$. We construct families of adjoint groups $G$ of type $^2D_3$ defined over $F$ (but not over $k$) such that $G(F)/R$ is finite for various fields $F$ which are finitely generated over their prime subfield. We also construct families of examples of such groups $G$ for which $G(F)/R\simeq \zz/2\zz$ when $F=k(t)$, and $k$ is (almost) arbitrary. This gives the first examples of adjoint groups $G$ which are not quasi-split nor defined over a global field, such that $G(F)/R$ is a non-trivial finite group.

2000 Mathematics Subject Classification:

Keywords and Phrases:

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