# Quaternion Algebras II

• Colin Maclachlan
• Alan W. Reid
Part of the Graduate Texts in Mathematics book series (GTM, volume 219)

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

1. Vignéras, M.-F. (1980a). Arithmétique des Algèbres de Quaternions..Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.Google Scholar
2. Borel, A. (1981). Commensurability classes and volumes of hyperbolic three-manifolds. Ann. Scuola Norm. Sup. Pisa, 8: 1–33.
3. Eichler, M. (1938a). Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen. J. Reine Angew. Math., 179: 227–251.Google Scholar
4. 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
5. Platonov, V. and Rapinchuk, A. (1994). Algebraic Groups and Number Theory. Academic Press, London.
6. Weiss, E. (1963). Algebraic Number Theory. McGraw-Hill, New York.
7. Weil, A. (1982). Adèles and Algebraic Groups. Birkhäuser, Boston.
8. Weil, A. (1967). Basic Number Theory. Springer-Verlag, Berlin.
9. Hewitt, E. and Ross, K. (1963). Abstract Harmonic Analysis Volume 1. Grundlehren der Mathematischen Wissenschaften Vol. 115. Springer-Verlag, Berlin.Google Scholar
10. Reiter, H. (1968). Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, Oxford.
11. 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
12. 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