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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.

MSC:
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)
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