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David R. Kohel; Helena A. Verrill
Fundamental domains for Shimura curves
Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 205-222, doi: 10.5802/jtnb.398
Article PDF | Reviews MR 2019012 | Zbl 1044.11052 | 1 citation in Cedram

Résumé - Abstract

We describe a process for defining and computing a fundamental domain in the upper half plane $\mathcal{H}^$ of a Shimura curve $X^D_0 (N)$ associated with an order in a quaternion algebra $A/ \mathbf{Q}$. A fundamental domain for $X^D_0 (N)$ realizes a finite presentation of the quaternion unit group, modulo units of its center. We give explicit examples of domains for the curves $X^6_0(1), \, X^{15}_0(1), \text{and} X^{35}_0 (1)$. The first example is a classical example of a triangle group and the second is a corrected version of that appearing in the book of Vignéras [13], due to Michon. These examples are also treated in the thesis of Alsina [1]. The final example is new and provides a demonstration of methods to apply when the group action has no elliptic points.


[1] M. Alsina, Aritmetica d'ordres quaternionics i uniformitzacio hiperbolica de corbes de Shimura. PhD Thesis, Universitat de Barcelona 2000, Publicacions Universitat de Barcelona, ISBN: 84-475-2491-4, 2001.
[2] W. BOSMA, J. CANNON, eds. The Magma Handbook. The University of Sydney, 2002.
[3] J. Cremona, Algorithms for modular elliptic curves, Second edition. Cambridge University Press, Cambridge, 1997.  MR 1628193 |  Zbl 0872.14041
[4] N. Elkies, Shimura Curve Computations. Algorithmic Number Theory, LNCS 1423, J. Buhler, ed, Springer (1998), 1-47.  MR 1726059 |  Zbl 1010.11030
[5] D. Kohel, Endomorphism rings of elliptic curves over finite fields. Thesis, University of California, Berkeley, 1996.
[6] D. Kohel, Hecke module structure of quaternions. Class field theory—its centenary and prospect (Tokyo, 1998), K. Miyake, ed, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo (2001), 177-195.  MR 1846458 |  Zbl 1040.11044
[7] D. Kohel, Brandt modules. Chapter in The Magma Handbook, Volume 7, J. Cannon, W. Bosma Eds., (2001), 343-354.
[8] D. Kohel, Quaternion Algebras. Chapter in The Magma Handbook, Volume 6, J. Cannon, W. Bosma Eds., (2001), 237-256.
[9] A. Kurihara, On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo 25 (1979), 277-301.  MR 523989 |  Zbl 0428.14012
[10] J.-F. Michon, Courbes de Shimura hyperelliptiques. Bull. Soc. Math. France 109 (1981), no. 2, 217-225. Numdam |  MR 623790 |  Zbl 0505.14024
[11] D. Roberts, Shimura curves analogous to X0(N). Ph.D. thesis, Harvard, 1989.
[12] H. Verrill, Subgroups of PSL2(R), Chapter in The Magma Handbook, Volume 2, J. Cannon, W. Bosma Eds., (2001), 233-254.
[13] M.-F. Vignéras, Arithmétiques des Algèbres de Quaternions, LNM 800, Springer-Verlag, 1980.  MR 580949 |  Zbl 0422.12008