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Mathematics of Computation

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On fundamental domains
of arithmetic Fuchsian groups


Author: Stefan Johansson
Journal: Math. Comp. 69 (2000), 339-349
MSC (1991): Primary 11F06, 20H10; Secondary 11R52
Published electronically: September 8, 1999
MathSciNet review: 1665958
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $K$ be a totally real algebraic number field and ${\mathcal{O}}$ an order in a quaternion algebra ${\mathfrak{A}}$ over $K$. Assume that the group ${\mathcal{O}^{1}}$ of units in ${\mathcal{O}}$ with reduced norm equal to $1$ is embedded into $PSL_{2}({\mathbb{R})}$ as an arithmetic Fuchsian group. It is shown how Ford's algorithm can be effectively applied in order to determine a fundamental domain of ${\mathcal{O}^{1}}$ as well as a complete system of generators of ${\mathcal{O}^{1}}$.


References [Enhancements On Off] (What's this?)

  • 1. R. Aurich, E. B. Bogomolny, and F. Steiner, Periodic orbits on the regular hyperbolic octagon, Phys. D 48 (1991), no. 1, 91–101. MR 1098656, 10.1016/0167-2789(91)90053-C
  • 2. Martin Eichler, Über die Idealklassenzahl hyperkomplexer Systeme, Math. Z. 43 (1938), 481-494.
  • 3. Lester R. Ford, The fundamental region for a Fuchsian group, Bull. Amer. Math. Soc. 31 (1925), 531-539.
  • 4. U. Halbritter and M. Pohst, On the computation of the values of zeta functions of totally real cubic fields, J. Number Theory 36 (1990), no. 3, 266–288. MR 1077708, 10.1016/0022-314X(90)90090-E
  • 5. Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. MR 1177168
  • 6. Claiborne G. Latimer, On the fundamental number of a rational generalized quaternion algebra, Duke Math. J. 1 (1935), 433-435.
  • 7. O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Bd. 117, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0152507
  • 8. Marie-France Vignéras, Invariants numériques des groupes de Hilbert, Math. Ann. 224 (1976), no. 3, 189–215. MR 0429755

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Additional Information

Stefan Johansson
Affiliation: Department of Mathematics, Chalmers University of Technology and Göteborg University, S-412 96 Göteborg, Sweden
Address at time of publication: Department of Mathematics, Institute for Advanced Study, Princeton, NJ 08540
Email: sj@math.chalmers.se

DOI: https://doi.org/10.1090/S0025-5718-99-01167-9
Keywords: Arithmetic Fuchsian groups, quaternion orders, fundamental domains
Received by editor(s): June 4, 1997
Published electronically: September 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society