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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Amit Kulshrestha and R. Parimala
Journal: Trans. Amer. Math. Soc. 360 (2008), 1193-1221
MSC (2000): Primary 20G15, 14G05
Published electronically: October 23, 2007
MathSciNet review: 2357694
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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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Additional Information

Amit Kulshrestha
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India 400005
Email: amitk@math.tifr.res.in

R. Parimala
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, India 400005
Address at time of publication: Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email: parimala@mathcs.emory.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04300-0
PII: S 0002-9947(07)04300-0
Keywords: Adjoint classical groups, $R$-equivalence, algebras with involutions, similitudes
Received by editor(s): July 31, 2005
Published electronically: October 23, 2007
Dedicated: Dedicated to our teacher Professor R. Sridharan on his seventieth birthday.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.