Geodesic
Class numbers and groups of algebraic groups. II
Izvestiya. Mathematics, Tome 16 (1981) no. 2, pp. 357-372 
V. P. Platonov; A. A. Bondarenko; A. S. Rapinchuk. Class numbers and groups of algebraic groups. II. Izvestiya. Mathematics, Tome 16 (1981) no. 2, pp. 357-372. http://geodesic.mathdoc.fr/item/IM2_1981_16_2_a5/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The central result of this article is a realization theorem, according to which, for a semisimple indefinite algebraic $K$-group $G$ ($K$ is an algebraic number field) an arbitrary finite abelian group of exponent $f$, where $f$ is the index of the kernel $F$ of the universal covering $\widetilde G\to G$, can be realized as a class group $\mathscr G\operatorname{cl}(\varphi(G))$. In the second part of the article the class number of semisimple groups that are not indefinite (groups of compact type) is investigated. The following general theorem is proved: if $G$ is a semisimple group of compact type of degree $n$, then for any natural number $r$ there exists a lattice $M(r)\subset K^{2n}$ such that $\operatorname{cl}(G^{M(r)})$ is divisible by $r$. Bibliography: 12 titles.

[1] Platonov V. P., Bondarenko A. A., Rapinchuk A. S., “Chisla i gruppy klassov algebraicheskikh grupp”, Izv. AN SSSR. Ser. matem., 43 (1979), 603–627 | MR | Zbl

[2] Platonov V. P., “Problema silnoi approksimatsii i gipoteza Knezera–Titsa dlya algebraicheskikh grupp”, Izv. AN SSSR. Ser. matem., 33 (1969), 1211–1220 | MR

[3] Platonov V. P., “Dopolnenie k rabote "Problema silnoi approksimatsii i gipoteza Knezera–Titsa dlya algebraicheskikh grupp”, Izv. AN SSSR. Ser. matem., 34 (1970), 775–777 | MR | Zbl

[4] Tits Zh., “Klassifikatsiya poluprostykh algebraicheskikh grupp”, Matematika, 12:2 (1968), 110–143

[5] Serr Zh.-P., Kogomologii Galua, Mir, M., 1968 | MR

[6] Arifmeticheskie gruppy i avtomorfnye funktsii, Mir, M., 1969 | MR

[7] Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR

[8] Veil A., “Adeli i algebraicheskie gruppy”, Matematika, 8:4 (1964), 3–74

[9] O'Meara O. T., Introduction to quadratic forms, Berlin, Heidelberg, New York, 1963 | MR

[10] Kneser M., “Schwache Approximation in algebraischen Gruppen”, Coll. Groupes Algerbriques, Paris, 1962, 41–52 | Zbl

[11] Borel A., “Some finiteness properties of adele groups over number fields”, Publ. Math. IHES, 1963, no. 16, 101–126 | MR | Zbl

[12] Borel A., “Density and maximality of arithmetic subgroups”, J. reine und angew. Math., 224 (1966), 74–89 | MR