Arithmetic $(1;e)$-curves and Belyĭ maps
Author:
Jeroen Sijsling
Journal:
Math. Comp. 81 (2012), 1823-1855
MSC (2010):
Primary 14H57; Secondary 14G35, 14Q05, 34B30
DOI:
https://doi.org/10.1090/S0025-5718-2012-02560-9
Published electronically:
January 23, 2012
MathSciNet review:
2904604
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Abstract | References | Similar Articles | Additional Information
Abstract: Using the theory of Belyĭ maps, we calculate the algebraic curves associated to the Fuchsian groups of signature $(1;e)$ that are commensurable with a triangle group, along with the Picard-Fuchs differential equations on these curves, which are related to pullbacks of hypergeometric differential equations. We focus particularly on the $(1;e)$-groups that are arithmetic.
- Yves André, $G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989. MR 990016
- P. Bayer and A. Travesa, Uniformizing functions for certain Shimura curves, in the case $D=6$, Acta Arith. 126 (2007), no. 4, 315–339. MR 2289964, DOI https://doi.org/10.4064/aa126-4-3
- Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. Corrected reprint of the 1983 original. MR 1393195
- Frits Beukers and Alexa van der Waall, Lamé equations with algebraic solutions, J. Differential Equations 197 (2004), no. 1, 1–25. MR 2030146, DOI https://doi.org/10.1016/j.jde.2003.10.017
- Bryan Birch, Noncongruence subgroups, covers and drawings, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 25–46. MR 1305392
- Paula Beazley Cohen, Claude Itzykson, and Jürgen Wolfart, Fuchsian triangle groups and Grothendieck dessins. Variations on a theme of Belyĭ, Comm. Math. Phys. 163 (1994), no. 3, 605–627. MR 1284798
- Jean-Marc Couveignes, Calcul et rationalité de fonctions de Belyĭ en genre $0$, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 1–38 (French, with English and French summaries). MR 1262878
- Koji Doi and Hidehisa Naganuma, On the algebraic curves uniformized by arithmetical automorphic functions, Ann. of Math. (2) 86 (1967), 449–460. MR 219537, DOI https://doi.org/10.2307/1970610
- Martin Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), 127–151 (1956) (German). MR 80767, DOI https://doi.org/10.1515/crll.1955.195.127
- Noam D. Elkies, Shimura curve computations, Algorithmic number theory (Portland, OR, 1998) Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, pp. 1–47. MR 1726059, DOI https://doi.org/10.1007/BFb0054850
- Noam D. Elkies, Shimura curves for level-3 subgroups of the $(2,3,7)$ triangle group, and some other examples, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 302–316. MR 2282932, DOI https://doi.org/10.1007/11792086_22
- Josep González and Victor Rotger, Non-elliptic Shimura curves of genus one, J. Math. Soc. Japan 58 (2006), no. 4, 927–948. MR 2276174
- H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59–106. MR 977435
- H. W. Lenstra, Galois theory for schemes, Notes available at http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf.
- C. Maclachlan and G. Rosenberger, Two-generator arithmetic Fuchsian groups, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 383–391. MR 698343, DOI https://doi.org/10.1017/S0305004100060709
- G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825
- Hossein Movasati and Stefan Reiter, Heun equations coming from geometry, Preprint available at http://w3.impa.br/ hossein/myarticles/movasati-reiter-08.pdf.
- Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. MR 1162313
- Hironori Shiga, Toru Tsutsui, and Jürgen Wolfart, Triangle Fuchsian differential equations with apparent singularities, Osaka J. Math. 41 (2004), no. 3, 625–658. With an appendix by Paula B. Cohen. MR 2107667
- Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144–222. MR 257031, DOI https://doi.org/10.2307/1970604
- Jeroen Sijsling, Equations for arithmetic pointed tori, Ph.D. thesis, Universiteit Utrecht, 2010.
- ---, Lamé equations and hypergeometric pullbacks, 2010, Webpage at http://sites.google.com/site/sijsling/programs.
- ---, Canonical models for arithmetic $(1;e)$-curves, In preparation, 2011.
- David Singerman, Subgroups of Fuschian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323. MR 281805, DOI https://doi.org/10.1112/blms/2.3.319
- Kisao Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600–612. MR 398991, DOI https://doi.org/10.2969/jmsj/02740600
- Kisao Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201–212. MR 463116
- Kisao Takeuchi, Arithmetic Fuchsian groups with signature $(1;e)$, J. Math. Soc. Japan 35 (1983), no. 3, 381–407. MR 702765, DOI https://doi.org/10.2969/jmsj/03530381
- M. Van Hoeij and R. Vidunas, Transformations between the Heun and Gauss hypergeometric functions of the hyperbolic type, Paper in progress; data available at http://www.math.fsu.edu/ hoeij/files/Heun/TextFormat, 2011.
- Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949
- John Voight, Computing CM points on Shimura curves arising from cocompact arithmetic triangle groups, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, pp. 406–420. MR 2282939, DOI https://doi.org/10.1007/11792086_29
- John Voight, Shimura curves of genus at most two, Math. Comp. 78 (2009), no. 266, 1155–1172. MR 2476577, DOI https://doi.org/10.1090/S0025-5718-08-02163-7
- Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. MR 986252
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Additional Information
Jeroen Sijsling
Affiliation:
Mathematisch Instituut Universiteit Utrecht, Postbus 80010, 3508TA Utrecht, The Netherlands
Address at time of publication:
IRMAR–Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cédex, France
MR Author ID:
974789
ORCID:
0000-0002-0632-9910
Email:
sijsling@gmail.com
Received by editor(s):
October 13, 2010
Received by editor(s) in revised form:
March 24, 2011
Published electronically:
January 23, 2012
Article copyright:
© Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.