Class numbers and groups of algebraic groups.~II

V. P. Platonov ; A. A. Bondarenko ; A. S. Rapinchuk

Izvestiya. Mathematics , Tome 16 (1981) no. 2, pp. 357-372

Résumé

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.