staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
John Voight
Computing fundamental domains for Fuchsian groups
Journal de théorie des nombres de Bordeaux, 21 no. 2 (2009), p. 467-489, doi: 10.5802/jtnb.683
Article PDF | Reviews MR 2541438 | Zbl pre05620663 | 1 citation in Cedram

Résumé - Abstract

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group $\Gamma $ with cofinite area. As a consequence, we compute the invariants of $\Gamma $, including an explicit finite presentation for $\Gamma $.

Bibliography

[1] M. Alsina and P. Bayer, Quaternion orders, quadratic forms, and Shimura curves. CRM monograph series, vol. 22, AMS, Providence, 2004.  MR 2038122 |  Zbl 1073.11040
[2] A. Beardon, The geometry of discrete groups. Grad. Texts in Math., vol. 91, Springer-Verlag, New York, 1995.  MR 1393195 |  Zbl 0528.30001
[3] H.-J. Boehm, The constructive reals as a Java library. J. Log. Algebr. Program. 64 (2005), 3–11.  MR 2137732 |  Zbl 1080.68005
[4] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language.. J. Symbolic Comput., 24 (3–4), 1997, 235–265.  MR 1484478 |  Zbl 0898.68039
[5] K. S. Brown, Cohomology of groups. Grad. Texts in Math., vol. 87, Springer-Verlag, New York, 1982.  MR 672956 |  Zbl 0584.20036
[6] H. Cohen, A course in computational algebraic number theory. Grad. Texts in Math., vol. 138, Springer-Verlag, New York, 1993.  MR 1228206 |  Zbl 0786.11071
[7] H. Cohen, Advanced topics in computational algebraic number theory. Grad. Texts in Math., vol. 193, Springer-Verlag, Berlin, 2000.  MR 1728313 |  Zbl 0977.11056
[8] D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra, 2nd ed. Undergrad. Texts in Math., Springer-Verlag, New York, 1997.  MR 1417938 |  Zbl 0861.13012
[9] T. Dokchitser, Computing special values of motivic $L$-functions. Experiment. Math. 13 (2004), no. 2, 137–149. Article |  MR 2068888 |  Zbl 1139.11317
[10] U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comp. 44 (1985), no. 170, 463–471.  MR 777278 |  Zbl 0556.10022
[11] L. R. Ford, Automorphic functions, 2nd. ed. Chelsea, New York, 1972.  JFM 55.0810.04
[12] I.M. Gel’fand, M.I. Graev, and I.I. Pyatetskii-Shapiro, Representation theory and automorphic functions. Trans. K.A. Hirsch, Generalized Functions, vol. 6, Academic Press, Boston, 1990.  MR 1071179 |  Zbl 0718.11022
[13] P. Gowland and D. Lester, A survey of exact computer arithmetic. In Computability and Complexity in Analysis, Lecture Notes in Computer Science, eds. Blanck et al., vol. 2064, Springer, 2001, 30–47.  Zbl 0985.65043
[14] M. Imbert, Calculs de présentations de groupes fuchsiens via les graphes rubanés. Expo. Math. 19 (2001), no. 3, 213–227.  MR 1852073 |  Zbl 0988.20035
[15] S. Johansson, On fundamental domains of arithmetic Fuchsian groups. Math. Comp 69 (2000), no. 229, 339–349.  MR 1665958 |  Zbl 0937.11016
[16] S. Katok, Fuchsian groups. Chicago Lect. in Math., U. of Chicago Press, Chicago, 1992.  MR 1177168 |  Zbl 0753.30001
[17] S. Katok, Reduction theory for Fuchsian groups. Math. Ann. 273 (1986), no. 3, 461–470.  MR 824433 |  Zbl 0561.30036
[18] D. R. Kohel and H. A. Verrill, Fundamental domains for Shimura curves. Les XXIIèmes Journées Arithmetiques (Lille, 2001), J. Théor. Nombres Bordeaux 15 (2003), no. 1, 205–222. Cedram |  MR 2019012 |  Zbl 1044.11052
[19] M.B. Pour-El and J.I. Richards, Computability in analysis and physics. Perspect. in Math. Logic, Springer, Berlin, 1989.  MR 1005942 |  Zbl 0678.03027
[20] H. Shimizu, On zeta functions of quaternion algebras. Ann. of Math. (2) 81 (1965), 166–193.  MR 171771 |  Zbl 0201.37903
[21] H. Verrill, Subgroups of $\operatorname{PSL}_2(\mathbb{R})$. Handbook of Magma Functions, eds. John Cannon and Wieb Bosma, Edition 2.14 (2007).
[22] M.-F. Vignéras, Arithmétique des algèbres de quaternions. Lect. Notes in Math., vol. 800, Springer, Berlin, 1980.  MR 580949 |  Zbl 0422.12008
[23] J. Voight, Quadratic forms and quaternion algebras: algorithms and arithmetic. Ph.D. Thesis, University of California, Berkeley, 2005.
[24] K. Weihrauch, An introduction to computable analysis. Springer-Verlag, New York, 2000.  MR 1795407 |  Zbl 0956.68056