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

berhuy@soton.ac.uk

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:

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