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Class numbers in the genus of quadratic forms, and algebraic groups
A. S. Rapinchuk
Abstract:
In this paper the class numbers $c(f)$ of quadratic forms $f$ with coefficients in an algebraic number field $K$ are studied by the methods of the theory of algebraic groups. It is shown that if a form $f$ is positive definite, then for any natural number $r$ there exists a quadratic form $g_r$, $K$-equivalent to $f$, such that $c(g_r)$ is divisible by $r$. A generalization of this result to semisimple algebraic $K$-groups of compact type is also obtained.
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English version:
Mathematics of the USSR-Izvestiya, 1982, 19:1, 79–93
Bibliographic databases:
 
UDC:
513.6
MSC: 10C30, 20G25, 20G30
Received: 01.04.1981
Citation:
A. S. Rapinchuk, “Class numbers in the genus of quadratic forms, and algebraic groups”, Izv. Akad. Nauk SSSR Ser. Mat., 45:4 (1981), 775–792; Math. USSR-Izv., 19:1 (1982), 79–93
Citation in format AMSBIB
\Bibitem{Rap81}
\by A.~S.~Rapinchuk
\paper Class numbers in the genus of quadratic forms, and algebraic groups
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1981
\vol 45
\issue 4
\pages 775--792
\mathnet{http://mi.mathnet.ru/izv1585}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=631438}
\zmath{https://zbmath.org/?q=an:0494.20025|0473.20032}
\transl
\jour Math. USSR-Izv.
\yr 1982
\vol 19
\issue 1
\pages 79--93
\crossref{https://doi.org/10.1070/IM1982v019n01ABEH001412}
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http://mi.mathnet.ru/eng/izv1585 http://mi.mathnet.ru/eng/izv/v45/i4/p775
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