Skip to main content

Numerical calculation of three-point branched covers of the projective line

  • Michael Klug (a1), Michael Musty (a2), Sam Schiavone (a3) and John Voight (a4) (a5)

We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.

    • Send article to Kindle

      To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Numerical calculation of three-point branched covers of the projective line
      Available formats
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Numerical calculation of three-point branched covers of the projective line
      Available formats
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Numerical calculation of three-point branched covers of the projective line
      Available formats
Hide All
1. Andrews, G. E., Askey, R. and Roy, R., Special functions , Encyclopedia of Mathematics and its Applications 17 (Cambridge University Press, Cambridge, 1999).
2. Bartholdi, L., Buff, X., von Bothmer, H.-C. G. and Kröker, J., ‘Algorithmic construction of Hurwitz maps’, Preprint, 2013, arXiv:1303.1579v1.
3. Beckmann, S., ‘Ramified primes in the field of moduli of branched coverings of curves’, J. Algebra 125 (1989) no. 1, 236255.
4. Belyĭ, G. V., ‘Galois extensions of a maximal cyclotomic field’, Math. USSR Izv. 14 (1980) no. 2, 247256.
5. Belyĭ, G. V., ‘A new proof of the three-point theorem’, Sb. Math. 193 (2002) no. 3–4, 329332 (translation).
6. Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235265.
7. Clark, P. L. and Voight, J., ‘Congruence subgroups of triangle groups’, Preprint, 2014.
8. Coombes, K. and Harbater, D., ‘Hurwitz families and arithmetic Galois groups’, Duke Math. J. 52 (1985) no. 4, 821839.
9. Cremona, J. E., Algorithms for modular elliptic curves , 2nd edn (Cambridge University Press, Cambridge, 1997).
10. Dèbes, P. and Emsalem, M., ‘On fields of moduli of curves’, J. Algebra 211 (1999) no. 1, 4256.
11. Elkies, N. D., ‘Shimura curve computations’, Algorithmic number theory, Portland, OR, 1998 , Lecture Notes in Computer Science 1423 (Springer, Berlin, 1998) 147.
12. Epstein, D. B. A. and Petronio, C., ‘An exposition of Poincaré’s polyhedron theorem’, Enseign. Math. (2) 40 (1994) no. 1–2, 113170.
13. Fieker, C. and Klüners, J., ‘Computation of Galois groups of rational polynomials’, 2013, arXiv:1211.3588.
14. Ford, L. R., Automorphic functions (McGraw-Hill, New York, 1929).
15. Galbraith, S. D., ‘Equations for modular curves’, PhD Thesis, University of Oxford, 1996.
16. Girondo, E. and González-Diez, G., Introduction to compact Riemann surfaces and dessins d’enfants , London Mathematical Society Student Texts 79 (Cambridge University Press, Cambridge, 2012).
17. Gloub, G. H. and van Loan, C. F., Matrix computations , 3rd edn (Johns Hopkins University Press, Baltimore, MD, 1996).
18. Grothendieck, A., ‘Sketch of a programme (translation into English)’, Geometric Galois actions. 1. Around Grothendieck’s esquisse d’un programme , London Mathematical Society Lecture Note Series 242 (eds Schneps, L. and Lochak, P.; Cambridge University Press, Cambridge, 1997) 243283.
19. Hafner, P. R., ‘The Hoffman–Singleton graph and its automorphisms’, J. Algebraic Combin. 18 (2003) 712.
20. He, Y.-H. and Read, J., ‘Hecke groups, dessins d’enfants and the Archimedean solids’, Preprint, 2013,arXiv:1309.2326v1.
21. Hejhal, D. A., ‘On eigenfunctions of the Laplacian for Hecke triangle groups’, Emerging applications of number theory , IMA Series 109 (eds Hejhal, D., Friedman, J., Gutzwiller, M. and Odlyzko, A.; Springer, 1999) 291315.
22. Holt, D. F., Handbook of computational group theory , Discrete Mathematics and its Applications (Chapman & Hall/CRC, 2005).
23. Hulpke, A., ‘Constructing transitive permutation groups’, J. Symbolic Comput. 39 (2005) no. 1, 130.
24. Ihara, Y., ‘Schwarzian equations’, J. Fac. Soc. Univ. Tokyo 21 (1974) 97118.
25. Javanpeykar, A., ‘Polynomial bounds for Arakelov invariants of Belyi curves’, PhD Thesis, Universiteit Leiden, 2013.
26. Johnson, D. L., Presentations of groups , 2nd edn, London Mathematical Society Student Texts 15 (Cambridge University Press, Cambridge, 1997).
27. Klüners, J., ‘On computing subfields: a detailed description of the algorithm’, J. Théor. Nombres Bordeaux 10 (1998) 243271.
28. Klug, M., ‘Computing rings of modular forms using power series expansions’, Master’s Thesis, University of Vermont, 2013.
29. Köck, B., ‘Belyĭ’s theorem revisited’, Beiträge Algebra Geom. 45 (2004) no. 1, 253265.
30. Kreines, E., ‘On families of geometric parasitic solutions for Belyi systems of genus zero’, Fundam. Prikl. Mat. 9 (2003) no. 1, 103111.
31. Kreines, E. M., ‘Equations determining Belyi pairs, with applications to anti-Vandermonde systems’, Fundam. Prikl. Mat. 13 (2007) no. 4, 95112.
32. Lenstra, A. K., Lenstra, H. W. Jr and Lovász, L., ‘Factoring polynomials with rational coefficients’, Math. Ann. 261 (1982) 513534.
33. Linton, S. A., ‘Double coset enumeration’, J. Symbolic Comput. 12 (1991) 415426.
34. Malle, G. and Matzat, B. H., Inverse Galois theory , Springer Monographs in Mathematics (Springer, Berlin, 1999).
35. Magnus, W., Noneuclidean tesselations and their groups , Pure and Applied Mathematics 61 (Academic Press, New York, 1974).
36. Magot, N. and Zvonkin, A., ‘Belyi functions for Archimedean solids’, Discrete Math. 217 (2000) no. 1–3, 249271.
37. Maskit, B., ‘On Poincaré’s theorem for fundamental polygons’, Adv. Math. 7 (1971) 219230.
38. Miranda, R., Algebraic curves and Riemann surfaces , Graduate Studies in Mathematics 5 (American Mathematical Society, Providence, RI, 1995).
39. Petersson, H., ‘Über die eindeutige Bestimmung und die Erweiterungsfähigkeit von gewissen Grenzkreisgruppen’, Abh. Math. Semin. Univ. Hambg. 12 (1937) no. 1, 180199.
40. Ratcliffe, J. G., Foundations of hyperbolic manifolds , 2nd edn (Springer, New York, 2005).
41. Rotman, J. J., An introduction to the theory of groups , 4th edn (Springer, New York, 1995).
42. Schneps, L., ‘Dessins d’enfants on the Riemann sphere’, The Grothendieck theory of dessins d’enfants , Lecture Notes in Mathematics 200 (Cambridge University Press, Cambridge, 1994) 4777.
43. Selander, B. and Strömbergsson, A., ‘Sextic coverings of genus two which are branched at three points’, Preprint, 2002,
44. Serre, J.-P., A course in arithmetic , Graduate Texts in Mathematics 7 (Springer, New York–Heidelberg, 1973).
45. Serre, J.-P., Topics in Galois theory , Research Notes in Mathematics 1 (Jones and Bartlett, Boston–London, 1992).
46. Shimura, G., ‘On some arithmetic properties of modular forms of one and several variables’, Ann. of Math. (2) 102 (1975) 491515.
47. Shimura, G., ‘On the derivatives of theta functions and modular forms’, Duke Math. J. 44 (1977) 365387.
48. Shimura, G., ‘Automorphic forms and the periods of abelian varieties’, J. Math. Soc. Japan 31 (1979) no. 3, 561592.
49. Shimura, G., Arithmeticity in the theory of automorphic forms , Mathematical Surveys and Monographs 82 (American Mathematical Society, Providence, RI, 2000).
50. Sijsling, J., ‘Arithmetic (1; e)-curves and Belyĭ maps’, Math. Comp. 81 (2012) no. 279, 18231855.
51. Sijsling, J. and Voight, J., ‘On computing Belyĭ maps’, Preprint, 2013, arXiv:1311.2529.
52. Silverman, J., Advanced topics in the arithmetic of elliptic curves , Graduate Texts in Mathematics 151 (Springer, New York, 1994).
53. Silverman, J., The arithmetic of elliptic curves , 2nd edn, Graduate Texts in Mathematics 106 (Springer, Dordrecht, 2009).
54. Singerman, D. and Syddall, R. I., ‘Belyĭ uniformization of elliptic curves’, Bull. Lond. Math. Soc. 29 (1997) 443451.
55. Slater, L. J., Generalized hypergeometric functions (Cambridge University Press, Cambridge, 1966).
56. Takeuchi, K., ‘Arithmetic triangle groups’, J. Math. Soc. Japan 29 (1977) no. 1, 91106.
57. Takeuchi, K., ‘Commensurability classes of arithmetic triangle groups’, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) no. 1, 201212.
58. Voight, J., ‘Quadratic forms and quaternion algebras: algorithms and arithmetic’, PhD Thesis, University of California, Berkeley, CA, 2005.
59. Voight, J., ‘Computing fundamental domains for Fuchsian groups’, J. Théor. Nombres Bordeaux 21 (2009) no. 2, 467489.
60. Voight, J. and Willis, J., ‘Computing power series expansions of modular forms’, Computations with modular forms , Mathematical Computer Science 6 (eds Boeckle, G. and Wiese, G.; Springer, Berlin, 2014) 331361.
61. Voight, J. and Zureick-Brown, D., ‘The canonical ring of a stacky curve’, Preprint, 2014.
62. Wolfart, J., ‘The ‘obvious’ part of Belyi’s theorem and Riemann surfaces with many automorphisms’, Geometric Galois actions, Vol. 1 , London Mathematics Society Lecture Note Series 242 (Cambridge University Press, Cambridge, 1997) 97112.
63. Wolfart, J., ‘ABC for polynomials, dessins d’enfants, and uniformization — a survey’, Elementare und analytische Zahlentheorie , Schriften der Wissenschaftliche Gesellschaft Johann Wolfgang Goethe Universität Frankfurt am Main 20 (Franz Steiner, Stuttgart, 2006) 313345.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 50 *
Loading metrics...

Abstract views

Total abstract views: 164 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st March 2018. This data will be updated every 24 hours.