Abstract
As we recalled in the introduction, there are several ways to construct Fuchsian groups of the first kind. Of all of these groups, the most important from the number theoretic viewpoint are the arithmetic groups. The general definition of these groups is a bit technical so we content ourselves for the moment with describing a family of examples: the arithmetic groups coming from a quaternion algebra over Q. Before describing them, we begin by proving several general results concerning the space of lattices.
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Notes
- 1.
We denote by \(\mathrm{SO}_{0}(q)\mathop{\cong}\mathrm{SO}_{0}(2, 1)\) the connected component of the identity of the special orthogonal group of the quadratic form q on P.
- 2.
For a = b = 1 the map obtained in composing Ψ and Φ −1 induces an isomorphism between PSL(2, R) and SO0(2, 1). This is one of the isomorphisms – called exceptional – which exist only between certain “small” Lie groups (see [57, pp. 518–520]).
- 3.
Recall that the norm is F-invariant.
References
Armand Borel, Linear algebraic groups, second ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991.
J.-M. Deshouillers and, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, vol. 34, American Mathematical Society, Providence, RI, 2001, Corrected reprint of the 1978 original.
Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.
T. Y. Lam, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002.
Colin Maclachlan and Alan W. Reid, The arithmetic of hyperbolic 3-manifolds, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003.
Toshitsune Miyake, Modular forms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006, Translated from the 1976 Japanese original by Yoshitaka Maeda.
Rached Mneimné and Frédéric Testard, Introduction à la théorie des groupes de Lie classiques, Collection Méthodes, Hermann, Paris, 1986.
Daniel Perrin, Cours d’algèbre, Collection CAPES/AGREG Mathématiques, Ellipses, Paris, 1996.
G. A. Margulis, Algebraic geometry. An introduction, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2008, Translated from the 1995 French original by Catriona Maclean.
Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press Inc., Boston, MA, 1994, Translated from the 1991 Russian original.
Frigyes Riesz and Béla Sz.-, Cours d’arithmétique, Le Mathématicien, vol. 2, Presses Universitaires de France, Paris, 1977, Deuxième édition revue et corrigée.
Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kano Memorial Lectures, 1.
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Bergeron, N. (2016). Arithmetic Hyperbolic Surfaces. In: The Spectrum of Hyperbolic Surfaces. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27666-3_2
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