# Fundamental domains for Shimura curves

David R. Kohel; Helena A. Verrill

Journal de théorie des nombres de Bordeaux (2003)

- Volume: 15, Issue: 1, page 205-222
- ISSN: 1246-7405

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topKohel, David R., and Verrill, Helena A.. "Fundamental domains for Shimura curves." Journal de théorie des nombres de Bordeaux 15.1 (2003): 205-222. <http://eudml.org/doc/249088>.

@article{Kohel2003,

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.},

author = {Kohel, David R., Verrill, Helena A.},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {Shimura curves; fundamental domains; modular curves; quaternion algebras},

language = {eng},

number = {1},

pages = {205-222},

publisher = {Université Bordeaux I},

title = {Fundamental domains for Shimura curves},

url = {http://eudml.org/doc/249088},

volume = {15},

year = {2003},

}

TY - JOUR

AU - Kohel, David R.

AU - Verrill, Helena A.

TI - Fundamental domains for Shimura curves

JO - Journal de théorie des nombres de Bordeaux

PY - 2003

PB - Université Bordeaux I

VL - 15

IS - 1

SP - 205

EP - 222

AB - 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.

LA - eng

KW - Shimura curves; fundamental domains; modular curves; quaternion algebras

UR - http://eudml.org/doc/249088

ER -

## References

top- [1] M. Alsina, Aritmetica d'ordres quaternionics i uniformitzacio hiperbolica de corbes de Shimura. PhD Thesis, Universitat de Barcelona2000, Publicacions Universitat de Barcelona, ISBN: 84-475-2491-4, 2001.
- [2] W. BOSMA, J. CANNON, eds. The Magma Handbook. The University of Sydney, 2002. http://magma.maths.usyd.edu.au/magma/htmlhelp/MAGMA.htm.
- [3] J. Cremona, Algorithms for modular elliptic curves, Second edition. Cambridge University Press, Cambridge, 1997. Zbl0872.14041MR1628193
- [4] N. Elkies, Shimura Curve Computations. Algorithmic Number Theory, LNCS1423, J. Buhler, ed, Springer (1998), 1-47. Zbl1010.11030MR1726059
- [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. Zbl1040.11044MR1846458
- [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. Tokyo25 (1979), 277-301. Zbl0428.14012MR523989
- [10] J.-F. Michon, Courbes de Shimura hyperelliptiques. Bull. Soc. Math. France109 (1981), no. 2, 217-225. Zbl0505.14024MR623790
- [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, LNM800, Springer-Verlag, 1980. Zbl0422.12008MR580949

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