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Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 2, Pages 395–414 (Mi izv1671)  

This article is cited in 3 scientific papers (total in 4 papers)

Class numbers and groups of algebraic groups. II

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


Abstract: 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.

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English version:
Mathematics of the USSR-Izvestiya, 1981, 16:2, 357–372

Bibliographic databases:

UDC: 513.6
MSC: 20G30, 12A85
Received: 13.11.1979

Citation: V. P. Platonov, A. A. Bondarenko, A. S. Rapinchuk, “Class numbers and groups of algebraic groups. II”, Izv. Akad. Nauk SSSR Ser. Mat., 44:2 (1980), 395–414; Math. USSR-Izv., 16:2 (1981), 357–372

Citation in format AMSBIB
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\by V.~P.~Platonov, A.~A.~Bondarenko, A.~S.~Rapinchuk
\paper Class numbers and groups of algebraic groups.~II
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1980
\vol 44
\issue 2
\pages 395--414
\mathnet{http://mi.mathnet.ru/izv1671}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=571102}
\zmath{https://zbmath.org/?q=an:0464.20031|0453.20033}
\transl
\jour Math. USSR-Izv.
\yr 1981
\vol 16
\issue 2
\pages 357--372
\crossref{https://doi.org/10.1070/IM1981v016n02ABEH001312}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981MK40900006}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. S. Rapinchuk, “Class numbers in the genus of quadratic forms, and algebraic groups”, Math. USSR-Izv., 19:1 (1982), 79–93  mathnet  crossref  mathscinet  zmath
    2. V. P. Platonov, “The arithmetic theory of algebraic groups”, Russian Math. Surveys, 37:3 (1982), 1–62  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. S. I. Adian, E. I. Zel'manov, G. A. Margulis, S. P. Novikov, A. S. Rapinchuk, L. D. Faddeev, V. I. Yanchevskii, “Vladimir Petrovich Platonov (on his 60th birthday)”, Russian Math. Surveys, 55:3 (2000), 601–610  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. A. A. Bondarenko, “Class numbers of classical groups”, Math. Notes, 68:1 (2000), 43–49  mathnet  crossref  mathscinet  zmath  isi
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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