Hilbert symbols, class groups and quaternion algebras. (English) Zbl 0973.11097

Let \(B\) be a quaternion algebra over a number field \(k.\) The main aim of the paper is to examine the connection between two pairs \((a,b)\) and \((c,d)\) of nonzero elements of \(k\) such that \(B\cong(a,b)\cong(c,d).\) With such two pairs the authors associate an invariant \(\rho=\rho_R(|{\mathcal D}(a,b)|,|{\mathcal D}(c,d)|)\) in a quotient of the narrow ideal class group of \(k\), where \(|{\mathcal D}(a,b)|\) and \(|{\mathcal D}(c,d)|\) denote isomorphism types of some maximal orders in \(B\) associated with these pairs. Assuming that \(B\) satisfies the Eichler condition they compute \(\rho_R(|{\mathcal D}(a,b)|,|{\mathcal D}(c,d)|)\) for \(a=c\) in terms of the arithmetic of the field \(k(\sqrt{a}).\) In the paper one can find the application of the main result to the computation of torsion subgroups of arithmetic Kleinian groups which were considered in their earlier paper [Ann. Inst. Fourier 50, No. 6, 1765-1798 (2000; Zbl 0973.20040)].


11R52 Quaternion and other division algebras: arithmetic, zeta functions
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)


Zbl 0973.20040
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[1] Borel, A., Commensumbility classes and volumes of hyperbolic 3-manifolds. Ann. Scuola Norm. Sup. Pisa8 (1981), 1-33. Also in Borel’s Oeuvres, BerlinSpringer (1983). · Zbl 0473.57003
[2] Chinburg, T., Friedman, E., An embedding theorem for quaternion algebras. J. London Math. Soc. (2) 60 (1999), 33-44. · Zbl 0940.11053
[3] Chinburg, T., Friedman, E., The finite subgroups of maximal arithmetic Kleinian groups. Ann. Institut Fourier50 (2000), 1765-1798. · Zbl 0973.20040
[4] Chinburg, T., Friedman, E., Jones, K.N., Reid, A.W., The arithmetic hyperbolic 3-manifold of smallest volume. Ann. Scuola Norm. Sup. Pisa (to appear). · Zbl 1008.11015
[5] Reiner, I., Maximal Orders. Acad. PressLondon (1975). · Zbl 0305.16001
[6] Vigngras, M.-F., Arithmétique des algèbres de Quaternions. , Springer-VerlagBerlin (1980). · Zbl 0422.12008
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