R-equivalence and rationality problem for semisimple adjoint classical algebraic groups

Abstract

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

“…The main ingredients in proofs of our results are Merkurjev's computation of G(F )/R for all adjoint groups of classical type [Me2,Th. 1], as well as results on the classification of hermitian forms over division algebras with involution over fields of virtual cohomological dimension two [BP2].…”

Cohomological invariants and R-triviality of adjoint classical groups

“…The algebraic group G is called R-trivial if G(L)/R = 1 for every field extension L/F . It was established by ColliotThélène and Sansuc in [2] (see also [7,Proposition 1]) that the group G is R-trivial if the variety of G is stably rational.…”

“…of Th. 8] proved that adjoint groups of type 2 A n are rational if n is even, and Merkurjev [7,Prop. 4] showed that adjoint groups of type C n are stably rational for n odd.…”

“…4] showed that adjoint groups of type C n are stably rational for n odd. On the other hand, Merkurjev also produced in [7] examples of adjoint groups of type 2 A 3 (= 2 D 3 ) and of type 2 D n for any n ≥ 4 which are not R-trivial, hence not stably rational. Examples of adjoint groups of type 1 D 4 which are not R-trivial were constructed by Gille in [3].…”

“…Our constructions are based on Merkurjev's computation in [7] of the group of R-equivalence classes of adjoint classical groups, which we now recall briefly. According to Weil (see [4, §26]), every absolutely simple classical group of adjoint type over a field F of characteristic different from 2 can be obtained as the connected component of the identity in the automorphism group of a central 1 simple algebra with involution (A, σ) over F .…”