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An arithmetic finiteness theorem on reductive groups. (Un théorème de finitude arithmétique sur les groupes réductifs.) (French) Zbl 0806.20036
Given an algebraic variety defined over a non-closed field, one can try to study its rational points introducing some equivalence relations on this set. This approach was fruitfully used by 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 reductive algebraic group \(G\) defined over a number field \(k\). Here one can pose two natural questions: is the group \(G(k)/R\) finite, and is it always abelian? The author gives a positive answer to the first one.
It is worth noting several important steps towards his theorem made before. First, J.-L. Colliot-Thélène and J.-J. Sansuc [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 175-229 (1977; Zbl 0356.14007)] treated the cases of algebraic tori and quasi-split groups reducing the problem to the case of an anisotropic group. Later on they remarked that an ergodic theorem by G. A. Margulis [Funkts. Anal. Prilozh. 13, 28-39 (1979; Zbl 0423.22015)] implies the result for semisimple simply connected groups. One can also mention the case \(G = SL(1,D)\), \(D\) denoting a skew-field, where \(G(k)/R\) happens to coincide with the reduced Whitehead group \(SK_ 1(D)\); the result obtained by V. E. Voskresenskij [Usp. Mat. Nauk 32, No. 6, 247-248 (1977; Zbl 0379.16005)] explains most phenomena of the reduced \(K\)-theory by V. P. Platonov [Izv. Akad. Nauk SSSR, Ser. Mat. 40, 227-261 (1976; Zbl 0338.16005)].
The author proves his theorem by reducing the general case to the simply connected one studying how \(R\)-equivalence behaves under isogeny. The main ingredient is “the norm principle” used before by K. Kato and S. Saito [Ann. Math., II. Ser. 118, 241-275 (1983; Zbl 0562.14011)] and M. Rost [C. R. Acad. Sci., Paris, Sér. I 310, 189-192 (1990; Zbl 0711.14004)]. Calculations with finite Galois modules complete the proof.

20G30 Linear algebraic groups over global fields and their integers
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14L35 Classical groups (algebro-geometric aspects)