Mathematics of Computation

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Arithmetic -curves and Belyĭ maps

Author: Jeroen Sijsling
Journal: Math. Comp. 81 (2012), 1823-1855
MSC (2010): Primary 14H57; Secondary 14G35, 14Q05, 34B30
Posted: January 23, 2012
Full-text PDF

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 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 -groups that are arithmetic.

References

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), http://dx.doi.org/10.4064/aa126-4-3
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 (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), http://dx.doi.org/10.1016/j.jde.2003.10.017
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 Belyĭ en genre 0, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 1, 1–38 (French, with English and French summaries). 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) (German). 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), http://dx.doi.org/10.1007/BFb0054850
11. 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 (2007i:11082), http://dx.doi.org/10.1007/11792086_22
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), http://dx.doi.org/10.1017/S0305004100060709
16. 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 (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. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. 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 -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. Kisao Takeuchi, 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. Kisao Takeuchi, Arithmetic Fuchsian groups with signature (1;𝑒), J. Math. Soc. Japan 35 (1983), no. 3, 381–407. MR 702765 (84h:10031), http://dx.doi.org/10.2969/jmsj/03530381
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 (French). 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), http://dx.doi.org/10.1007/11792086_29
31. John Voight, Shimura curves of genus at most two, Math. Comp. 78 (2009), no. 266, 1155–1172. MR 2476577 (2010h:11099), http://dx.doi.org/10.1090/S0025-5718-08-02163-7
32. Masaaki Yoshida, Fuchsian differential equations, Aspects of Mathematics, E11, Friedr. Vieweg & Sohn, Braunschweig, 1987. With special emphasis on the Gauss-Schwarz theory. MR 986252 (90f:32025)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|