Abstract
The rationality problems for group varieties have a long history. These problems are especially interesting for semisimple group varieties. Let G be a semisimple algebraic group defned over a field K. It is well known that the algebraic variety G is unirational over K, i.e., the field K(G) of rational functions on G is contained in a purely transcendental extension of K (see [DG], Exp.XIV]).
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References
A. Borel & J. Tits, Groupes reductifs, Publ.Math.IHES, 27 (1965), 55–151.
V.I. Chernousov, On the rationality of spinor varieties over the field of rational numbers, Dokl. AN BSSR, 25 (1981), 293–296.
V.I. Chernousov, The group of multipliers of the canonical quadratic form and the stable rationality of the variety PSO, Mat. Zametki, 55 (1994), 114–119.
J.-L. Colliot-Thelene & J.-J. Sansuc, La R-equivalence sur les tores, Ann.Sci. ENS, 10 (1977), 175–230.
M. Demazure et A. Grothendieck, Schemas en groupes, Springer Lecture Notes, v. 152, 1970.
A.S. Merkurjev, Generic element in SK1 for simple algebras, K-theory, 7 (1993), 1–3.
A.S. Merkurjev, R-equivalence and rationality problem for semisimple adjoint classical groups, Publ. Math. IHES (to appear).
V.P. Platonov, Algebraic groups and reduced K-theory, Proc. Internat. Congress Math., Helsinki, v. 1 (1980), 311–317.
V.P. Platonov, On the problem of rationality of spinor varieties, Soviet Math. Dokl., 20 (1979), 1027–1031.
V.P. Platonov, Birational properties of spinor varieties, Proc. Steklov Inst. Math., No. 3 (1983), 173–182.
V.P. Platonov & V.I. Chernousov, On the rationality of canonical spinor varieties, Soviet Math. Dokl., 21 (1980), 830–834.
V.P. Platonov & V.I. Chernousov, The rationality problem for group varieties, C.R. Acad. Sci. Paris, 322 (1996), 245–250.
V.P. Platonov & V.I. Chernousov, The rationality problem for semisimple group varieties, Universität Bielefeld, 37 (1997), pp. 26.
V.P. Platonov & A.S. Rapinchuk, Algebraic groups and number theory, Academic Press, N.Y., 1994, pp. 614.
J.-P. Serre, Cohomologie Galoisienne, Springer, 1994, pp. 181.
R. Steinberg, Lectures on Chevalley groups, Yale University, 1967.
J. Tits, Classification of algebraic semisimple groups, Proc. Symp. Pure Math., 9 (1966), 32–62.
V. Voskresenskii, Algebraic tori, Nauka, Moscow, 1977, pp. 223.
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Platonov, V.P. (1998). Rationality Problems for Group Varieties. In: Balog, A., Katona, G.O.H., Recski, A., Sza’sz, D. (eds) European Congress of Mathematics. Progress in Mathematics, vol 169. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8898-1_8
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DOI: https://doi.org/10.1007/978-3-0348-8898-1_8
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