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Orders in Quaternion Algebras

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

Abstract

The basic algebraic theory of quaternion algebras was given in Chapter 2. That sufficed for the results obtained so far on deducing information on a Kleinian group Γ from its invariant trace field kΓ and invariant quaternion algebra AΓ. We have yet to expound on the arithmetic theory of quaternion algebras over number fields. This will be essential in extracting more information on the quaternion algebras and, more importantly, in deducing the existence of discrete Kleinian groups of finite covolume. These will be arithmetic Kleinian groups about which a great deal of the remainder of the book will be concerned. All of this is based around the structure of orders in quaternion algebras which encapsulate the arithmetic theory of quaternion algebras. These were introduced in Chapter 2, but we now give a more systematic study, particularly from a local-global viewpoint.

Keywords

Prime Ideal Number Field Division Algebra Maximal Order Valuation Ring 
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.

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Further Reading

  1. Reiner, I. (1975). Maximal Orders. Academic Press, London.MATHGoogle Scholar
  2. Deuring, M. (1935). Algebren. Springer-Verlag, Berlin.CrossRefGoogle Scholar
  3. Weil, A. (1967). Basic Number Theory. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  4. Pierce, R. (1982). Associative Algebra. Graduate Texts in Mathematics Vol. 88. Springer-Verlag, New York.Google Scholar
  5. O’Meara, O. (1963). Introduction to Quadratic Forms. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  6. Vignéras, M.-F. (1980a). Arithmétique des Algèbres de Quaternions..Lecture Notes in Mathematics No. 800. Springer-Verlag, Berlin.Google Scholar
  7. Eichler, M. (1937). Bestimmung der Idealklassenzahl in gewissen normalen einfachen Algebren. J. Reine Angew. Math., 176: 192–202.Google Scholar
  8. Eichler, M. (1938b). Über die Idealklassenzahl hypercomplexer Systeme. Math. Zeitschrift, 43: 481–494.MathSciNetCrossRefGoogle Scholar
  9. Serre, J.-P. (1962). Corps Locaux. Hermann, Paris.MATHGoogle Scholar
  10. Serre, J.-P. (1980). Trees. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  11. Borel, A. (1981). Commensurability classes and volumes of hyperbolic three-manifolds. Ann. Scuola Norm. Sup. Pisa, 8: 1–33.MathSciNetMATHGoogle Scholar
  12. Elstrodt, J., Grunewald, F., and Mennicke, J. (1998). Groups Acting on Hyperbolic Space. Monographs in Mathematics. Springer-Verlag, Berlin.MATHCrossRefGoogle Scholar
  13. Newman, M. (1972). Integral Matrices. Academic Press, NewYork.MATHGoogle Scholar
  14. Vinberg, E., editor (1993b). Geometry II (II). Encyclopaedia of Mathematical Sciences Vol. 29. Springer-Verlag, Berlin.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • Colin Maclachlan
    • 1
  • Alan W. Reid
    • 2
  1. 1.Department of Mathematical Sciences, Kings CollegeUniversity of AberdeenAberdeenUK
  2. 2.Department of MathematicsUniversity of Texas at AustinAustinUSA

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