The Arithmetic of Hyperbolic 3-Manifolds pp 225-251 | Cite as

# Quaternion Algebras II

## Abstract

One of the main aims of this chapter is to complete the classification theorem for quaternion algebras over a number field by establishing the existence part of that theorem. This theorem, together with other results in this chapter, make use of the rings of adèles and groups of idèles associated to number fields and quaternion algebras. These rings and groups and their component parts are locally compact groups so that some aspects of their Haar measures, duality and abstract harmonic analysis go into this study. The results on adèles and idèles which are discussed here are aimed towards their application, in the next chapter, of producing discrete arithmetic subgroups of finite covolume. They will also enable us to make volume calculations on arithmetic Kleinian and Fuchsian groups in subsequent chapters. For these purposes and other applications subsequently, there are two crucial results here. One is the Strong Approximation Theorem, which is proved in the last section of this chapter. The other, which is central in subsequent results giving the covolume of arithmetic Fuchsian and Kleinian groups in terms of the arithmetic data, is that the Tamagawa number is 1. The Tamagawa number is the volume of a certain quotient of an idèle group measured with respect to its Tamagawa measure. The Tamagawa measures can be invariantly defined on the local components of the rings of adèles and groups of idèles and these are fully discussed here. The relevant quotients are shown to be compact and so will have finite volume. The proof that the Tamagawa volume, which is, by definition, the Tamagawa number, is precisely 1, is not included.

## Keywords

Haar Measure Number Field Compact Subgroup Kleinian Group Fuchsian Group## Preview

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## Further Reading

- Vignéras, M.-F. (1980a).
*Arithmétique des Algèbres de Quaternions*..Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.Google Scholar - Borel, A. (1981). Commensurability classes and volumes of hyperbolic three-manifolds.
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*Algebraic Groups and Discontinuous Subgroups*, volume 9 of*Proceedings of Symposia in Pure Mathematics*. American Mathematical Society, Providence, RI.Google Scholar - Platonov, V. and Rapinchuk, A. (1994).
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*Adèles and Algebraic Groups*. Birkhäuser, Boston.MATHCrossRefGoogle Scholar - Weil, A. (1967).
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*Abstract Harmonic Analysis Volume**1*. Grundlehren der Mathematischen Wissenschaften Vol. 115. Springer-Verlag, Berlin.Google Scholar - Reiter, H. (1968).
*Classical Harmonic Analysis and Locally Compact Groups*. Oxford University Press, Oxford.MATHGoogle Scholar - Borel, A. and Mostow, G., editors (1966).
*Algebraic Groups and Discontinuous Subgroups*, volume 9 of*Proceedings of Symposia in Pure Mathematics*. American Mathematical Society, Providence, RI.Google Scholar - Borel, A. and Mostow, G., editors (1966).
*Algebraic Groups and Discontinuous Subgroups*, volume 9 of*Proceedings of Symposia in Pure Mathematics*. American Mathematical Society, Providence, RI.Google Scholar