Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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
Full-text PDF Free Access

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.


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

  • 1. Yves André, $ G$-functions and geometry, Aspects of Mathematics, E13, Friedr. Vieweg & Sohn, Braunschweig, 1989. MR 990016 (90k:11087)
  • 2. 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 (2008d:11055)
  • 3. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995. MR 1393195 (97d:22011)
  • 4. Frits Beukers and Alexa van der Waall, Lamé equations with algebraic solutions, J. Differential Equations 197 (2004), no. 1, 1-25. MR 2030146 (2004j:34202)
  • 5. 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 (95k:11055)
  • 6. 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 (96a:11056)
  • 7. Jean-Marc Couveignes, Calcul et rationalité de fonctions de Belyĭen genre 0, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 1-38. MR 1262878 (96k:11080)
  • 8. Koji Doi and Hidehisa Naganuma, On the algebraic curves uniformized by arithmetical automorphic functions, Ann. of Math. (2) 86 (1967), 449-460. MR 0219537 (36:2618)
  • 9. Martin Eichler, Zur Zahlentheorie der Quaternionen-Algebren, J. Reine Angew. Math. 195 (1955), 127-151 (1956). MR 0080767 (18,297c)
  • 10. 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 (2001a:11099)
  • 11. -, 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 (2007i:11082)
  • 12. Josep González and Victor Rotger, Non-elliptic Shimura curves of genus one, J. Math. Soc. Japan 58 (2006), no. 4, 927-948. MR 2276174 (2007k:11093)
  • 13. H. Hijikata, A. Pizer, and T. Shemanske, Orders in quaternion algebras, J. Reine Angew. Math. 394 (1989), 59-106. MR 977435 (90d:11128)
  • 14. H. W. Lenstra, Galois theory for schemes, Notes available at http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf.
  • 15. C. Maclachlan and G. Rosenberger, Two-generator arithmetic Fuchsian groups, Math. Proc. Cambridge Philos. Soc. 93 (1983), no. 3, 383-391. MR 698343 (84m:10014)
  • 16. G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825 (92h:22021)
  • 17. Hossein Movasati and Stefan Reiter, Heun equations coming from geometry, Preprint available at http://w3.impa.br/ hossein/myarticles/movasati-reiter-08.pdf.
  • 18. Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1162313 (94d:12006)
  • 19. 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 (2006k:34238)
  • 20. Goro Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 91 (1970), 144-222. MR 0257031 (41:1686)
  • 21. Jeroen Sijsling, Equations for arithmetic pointed tori, Ph.D. thesis, Universiteit Utrecht, 2010.
  • 22. -, Lamé equations and hypergeometric pullbacks, 2010, Webpage at http://sites.google.com/site/sijsling/programs.
  • 23. -, Canonical models for arithmetic $ (1;e)$-curves, In preparation, 2011.
  • 24. David Singerman, Subgroups of Fuschian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319-323. MR 0281805 (43:7519)
  • 25. Kisao Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), no. 4, 600-612. MR 0398991 (53:2842)
  • 26. -, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 1, 201-212. MR 0463116 (57:3077)
  • 27. -, Arithmetic Fuchsian groups with signature $ (1;e)$, J. Math. Soc. Japan 35 (1983), no. 3, 381-407. MR 702765 (84h:10031)
  • 28. 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.
  • 29. Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. MR 580949 (82i:12016)
  • 30. 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 (2008g:11104)
  • 31. -, Shimura curves of genus at most two, Math. Comp. 78 (2009), no. 266, 1155-1172. MR 2476577
  • 32. Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. MR 986252 (90f:32025)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 14H57, 14G35, 14Q05, 34B30

Retrieve articles in all journals with MSC (2010): 14H57, 14G35, 14Q05, 34B30


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
Email: sijsling@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2012-02560-9
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.

American Mathematical Society