\(R\)-equivalence and rationality problem for semisimple adjoint classical algebraic groups
Given an algebraic variety \(X\) defined over a non-closed field \(k\), one can try to study its rational points introducing some equivalence relations on this set. This approach was fruitfully used by \textit{Yu. I. Manin} [Cubic forms, North-Holland (1986; Zbl 0582.14010)] for cubic hypersurfaces. Two points are said to be \(R\)-equivalent if one can join them by a chain of \(k\)-rational curves. The set of classes \(X(k)/R\) is an important birational invariant of \(X\). The reviewed paper treats the case of a linear algebraic group \(G\) which was first studied by \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 175-229 (1977; Zbl 0356.14007)]. A result by \textit{V. E. Voskresenskij} [Algebraic Tori, Nauka, Moscow (1977; Zbl 0499.14013)] showed that for \(G=SL(1,A)\), \(A\) a central simple \(k\)-algebra, the above invariant \(G(k)/R\) coincides with the reduced Whitehead group \(SK_1(A)\), the invariant used by \textit{V. P. Platonov} [Proc. Int. Congr. Math., Helsinki 1978, vol. 1, 311-317 (1980; Zbl 0425.16018)] for the proof of non-rationality of \(G\). Platonov's examples of non-rational semisimple groups refer to the class of simply connected groups. Earlier arithmetic considerations allowed Serre to construct non-rational groups which are, by construction, neither simply connected, nor adjoint. It was a long-standing problem whether there exist non-rational adjoint groups. The reviewed paper answers this question. Namely, there are given examples of adjoint simple \(k\)-groups \(G\) of type \(D_n\), \(n\geq 3\), such that \(G(E)/R\neq 1\) for some extension \(E\) of \(k\) which implies that \(G\) is not \(k\)-rational. The ground field \(k\) can be taken to be an arbitrary number field for odd \(n\), and the field of rational functions in one variable over a number field for even \(n\). The proof uses techniques from the cohomology of quadratic forms (\textit{J. Arason} [J. Algebra 36, 448-491 (1975; Zbl 0314.12104)]) and Severi-Brauer varieties (\textit{A. S. Merkurjev} and \textit{A. A. Suslin} [Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011-1046 (1982; Zbl 0525.18008)]), and, more specifically, the index reduction formula (\textit{A. S. Merkurjev} [ibid. 55, No. 1, 218-224 (1991; Zbl 0763.12003)]; \textit{A. S. Merkurjev, I. A. Panin}, and \textit{A. Wadsworth} [K-theory 10, 517-596 (1996; Zbl 0874.16012)]; \textit{A. Schofield} and \textit{M. Van den Bergh} [Trans. Am. Math. Soc. 333, No. 2, 729-739 (1992; Zbl 0778.12004)]).
scientific article; zbMATH DE number 1057024
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