On computing Belyi maps
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 73-131.

We survey methods to compute three-point branched covers of the projective line, also known as Belyĭ maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and $p$-adic methods. Along the way, we pose several questions and provide numerous examples.

Nous donnons un aperçu des méthodes actuelles pour le calcul des revêtements de la droite projective ramifiés en au plus trois points, connus sous le nom de morphismes de Belyĭ. Ces méthodes comprennent une approche directe, se ramenant à la solution d’un système d’équations polynomiales ainsi que des méthodes analytiques complexes, de formes modulaires et $p$-adiques. Ce faisant, nous posons quelques questions et donnons de nombreux exemples.

Published online:
DOI: 10.5802/pmb.5
Classification: 11G32,  11Y40
Keywords: Belyi maps, dessins d’enfants, covers, uniformization, computational algebra
Sijsling, J. 1; Voight, J. 2

1 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK
2 Department of Mathematics and Statistics, University of Vermont, 16 Colchester Ave, Burlington, VT 05401, USA; Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA
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Sijsling, J.; Voight, J. On computing Belyi maps. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2014), pp. 73-131. doi : 10.5802/pmb.5. http://www.numdam.org/articles/10.5802/pmb.5/`

[1] William W. Adams and Philippe Loustaunau, An introduction to Gröbner bases, Grad. Studies in Math., 3, Amer. Math. Soc., Providence, RI, 1994. | DOI | Zbl

[2] N. M. Adrianov, N. Ya. Amburg, V. A. Dremov, Yu. A. Levitskaya, E. M. Kreines, Yu. Yu. Kochetkov, V. F. Nasretdinova, and G. B. Shabat, Catalog of dessins d’enfants with $\le 4$ edges, arXiv:0710.2658v1, 2007. | DOI | Zbl

[3] Ane S. I. Anema and Jaap Top, Explicit algebraic coverings of a pointed torus, in Arithmetic and geometry of K3 surfaces and Calabi-Yau threefolds, Fields Inst. Commun., vol. 67, Springer, New York, 143–152. | DOI | Zbl

[4] Elizabeth A. Arnold, Modular algorithms for computing Gröbner bases, J. Symbolic Comput. 35 (2003), no. 4, 403–419. | DOI | Zbl

[5] A. O. L. Atkin, H. P. F. Swinnerton-Dyer, Modular forms on noncongruence subgroups, in Combinatorics, (Proc. Sympos. Pure Math., vol. XIX, Univ. California, Los Angeles), Amer. Math. Soc., Providence, 1968, 1–25. | DOI | Zbl

[6] Ali Ayad, A survey on the complexity of solving algebraic systems, International Math. Forum 5 (2010), no. 7, 333–353. | Zbl

[7] Laurent Bartholdi, Xavier Buff, Hans-Christian Graf von Bothmer, and Jakob Kröker, Algorithmic construction of Hurwitz maps, arXiv:1303.1579v1, 2013. | DOI | MR | Zbl

[8] Alan F. Beardon and Kenneth Stephenson, The uniformization theorem for circle packings, Indiana Univ. Math. J. 39 (1990), no. 4, 1383–1425. | DOI | Zbl

[9] Arnaud Beauville, Les familles stables de courbes elliptiques sur ${\mathbf{P}}^{1}$ admettant quatre fibres singulières, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 19, 657–660. | Zbl

[10] Daniel J. Bates, Jonathan D. Hauenstein, Andrew J Sommese, and Charles W. Wampler, Bertini: Software for numerical algebraic geometry, available at bertini.nd.edu with permanent doi: dx.doi.org/10.7274/R0H41PB5.

[11] Sybilla Beckmann, Ramified primes in the field of moduli of branched coverings of curves, J. Algebra 125 (1989), no. 1, 236–255. | DOI | MR | Zbl

[12] G.V. Belyĭ, Galois extensions of a maximal cyclotomic field, Math. USSR-Izv. 14 (1980), no. 2, 247–256. | DOI | Zbl

[13] G.V. Belyĭ, A new proof of the three-point theorem, translation in Sb. Math. 193 (2002), no. 3–4, 329–332. | DOI | Zbl

[14] Kevin Berry and Marvin Tretkoff, The period matrix of Macbeath’s curve of genus seven, in Curves, Jacobians, and abelian varieties, Amherst, MA, 1990, Providence, RI: Contemp. Math., vol. 136, Amer. Math. Soc., 31–40. | DOI | MR | Zbl

[15] Jean Bétréma, Danielle Péré, and Alexander Zvonkin, Plane trees and their Shabat polynomials, Laboratoire Bordelais de Recherche en Informatique, Université Bordeaux I, 1992. | DOI | MR

[16] Frits Beukers and Hans Montanus, Explicit calculation of elliptic fibrations of $K3$-surfaces and their Belyi-maps, in Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 33–51. | DOI | Zbl

[17] F. Beukers and C. L. Stewart, Neighboring powers, J. Number Theory 130 (2010), 660–679. | DOI | MR | Zbl

[18] Bryan Birch, Noncongruence subgroups, covers and drawings, in The Grothendieck theory of dessins d’enfants, London Math. Soc. Lecture Note Ser., vol. 200, Cambridge University Press, 1994, 25–46. | DOI | Zbl

[19] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (3–4), 1997, 235–265. | DOI | MR | Zbl

[20] Nigel Boston, On the Belgian chocolate problem and output feedback stabilization: Efficacy of algebraic methods, 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton) 2012, October 2012, 869–873. | DOI

[21] Florian Bouyer and Marco Streng, Examples of CM curves of genus two defined over the reflex field, arxiv:1307.0486v1, 2013. | DOI | MR | Zbl

[22] Philip L. Bowers and Kenneth Stephenson, Uniformizing dessins and Belyĭ maps via circle packing, Mem. Amer. Math. Soc. 170 (2004), no. 805. | DOI | Zbl

[23] V. Braungardt, Covers of moduli surfaces, Compositio Math. 140 (2004), no. 4, 1033–1036. | DOI | MR | Zbl

[24] C. Chevalley, A. Weil and E. Hecke, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers, Abh. Math. Sem. Univ. Hamburg 10 (1934), no. 1, 358–361. | DOI | Zbl

[25] Pete L. Clark and John Voight, Algebraic curves uniformized by congruence subgroups of triangle groups, preprint at http://www.math.dartmouth.edu/~jvoight/articles/triangle-072011.pdf. | DOI | MR | Zbl

[26] Charles R. Collins and Kenneth Stephenson, A circle packing algorithm, Comput. Geom. 25 (2003), no. 3, 233–256. | DOI | MR | Zbl

[27] Marston D. E. Conder, Gareth A. Jones, Manfred Streit and Jürgen Wolfart, Galois actions on regular dessins of small genera, Rev. Mat. Iberoam. 29 (2013), no. 1, 163–181. | DOI | MR | Zbl

[28] Kevin Coombes and David Harbater, Hurwitz families and arithmetic Galois groups, Duke Math. J. 52 (1985), no. 4, 821–839. | DOI | MR | Zbl

[29] Jean-Marc Couveignes, Calcul et rationalité de fonctions de Belyi en genre 0, Annales de l’Institut Fourier (Grenoble) 44 (1994), no. 1, 1–38. | DOI | Zbl

[30] Jean-Marc Couveignes, Quelques revêtements definis sur $ℚ$, Manuscripta Math. 92 (1997), no. 4, 409–445. | DOI | Zbl

[31] Jean-Marc Couveignes, A propos du théorème de Belyi, J. Théor. Nombres Bordeaux 8 (1996), no. 1, 93–99. | DOI | Zbl

[32] Jean-Marc Couveignes, Tools for the computation of families of coverings, in Aspects of Galois theory, London Math. Soc. Lecture Notes Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, 38–65. | Zbl

[33] Jean-Marc Couveignes and Louis Granboulan, Dessins from a geometric point of view, in The Grothendieck theory of dessins d’enfants, London Math. Soc. Lecture Note Ser., vol. 200, Cambridge University Press, 1994, 79–113. | DOI | Zbl

[34] David A. Cox, John B. Little, Donal O’Shea, Ideals, varieties, and algorithms, 2nd ed., Springer-Verlag, New York, 1996.

[35] David A. Cox, John B. Little, Donal O’Shea, Using algebraic geometry, Springer-Verlag, New York, 2005. | Zbl

[36] J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. | Zbl

[37] Michael Stoll and John E. Cremona, On the reduction theory of binary forms, J. Reine Angew. Math. 565 (2003), 79–99. | DOI | MR | Zbl

[38] Pierre Dèbes and Jean-Claude Douai, Algebraic covers: field of moduli versus field of definition, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 3, 303–338. | DOI | Numdam | MR | Zbl

[39] Pierre Dèbes and Michel Emsalem, On fields of moduli of curves, J. Algebra 211 (1999), no. 1, 42–56. | DOI | MR | Zbl

[40] V. A. Dremov, Computation of two Belyi pairs of degree 8, Russian Math. Surveys 64 (2009), no. 3, 570–572. | DOI | Zbl

[41] Virgile Ducet, Cnstruction of algebraic curves with many rational points over finite fields, Ph.D. thesis, Université d’Aix-Marseille, 2013.

[42] Clifford J. Earle, On the moduli of closed Riemann surfaces with symmetries, in Advances in the theory of riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studies, vol. 66, Princeton Univ. Press, Princeton, 119–130. | DOI

[43] Robert W. Easton and Ravi Vakil, Absolute Galois acts faithfully on the components of the moduli space of surfaces: A Belyi-type theorem in higher dimension, Int. Math. Res. Notices 2007, no. 20, Art. ID rnm080. | DOI | Zbl

[44] W. L. Edge, Fricke’s octavic curve, Proc. Edinburgh Math. Soc. 27 (1984), 91–101. | DOI | MR | Zbl

[45] Noam D. Elkies, $ABC$ implies Mordell, Internat. Math. Res. Notices 1991, no. 7, 99–109. | DOI | Zbl

[46] Noam D. Elkies, Shimura curve computations, Algorithmic number theory (Portland, OR, 1998), Lecture notes in Comput. Sci., vol. 1423, 1–47. | DOI | Zbl

[47] Noam D. Elkies, Shimura curves for level-3 subgroups of the $\left(2,3,7\right)$ triangle group, and some other examples, in Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, 302–316. | DOI | Zbl

[48] Noam D. Elkies, The complex polynomials $P\left(x\right)$ with $Gal\left(P\left(x\right)-t\right)={M}_{23}$, in ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, eds. Everett Howe and Kiran Kedlaya, Open Book Series 1, Math. Science Publishers, 2013, 359–367. | DOI | Zbl

[49] Noam D. Elkies, Explicit modular towers, in Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (1997), eds. T. Basar and A. Vardy, Univ. of Illinois at Urbana-Champaign, 1998, 23–32, arXiv:math.NT/0103107.

[50] Noam D. Elkies and Mark Watkins, Polynomial and Fermat-Pell families that attain the Davenport-Mason bound, preprint at http://magma.maths.usyd.edu.au/~watkins/papers/hall.ps.

[51] Arsen Elkin, Belyi Maps, http://homepages.warwick.ac.uk/ masjaf/belyi/.

[52] Christophe Eyral and Mutsuo Oka, Fundmental groups of join-type sextics via dessins d’enfants, Proc. London Math. Soc. (3) 107 (2013), 76–120. | DOI | MR | Zbl

[53] Helaman R. P. Ferguson, David H. Bailey, and Steve Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comp. 68 (1999), 351–369. | DOI | MR | Zbl

[54] Claus Fieker and Jürgen Klüners, Computation of Galois groups of rational polynomials, Galois group, arxiv:1211.3588v2, 2012. | DOI | MR | Zbl

[55] Machiel van Frankenhuysen, The ABC conjecture implies Vojta’s height inequality for curves, J. Number Theory 95 (2002), 289–302. | DOI | MR | Zbl

[56] Robert Fricke and Felix Klein, Vorlesungen über die Theorie der automorphen Funktionen. Band 1: Die gruppentheoretischen Grundlagen. Band II, Bibliotheca Mathematica Teubneriana, Bände 3, 4 Johnson, New York, 1965.

[57] M. Fried, Fields of definition of function fields and Hurwitz families—groups as Galois groups, Comm. Algebra 5 (1977), no. 1, 17–82. | DOI | MR | Zbl

[58] Ernesto Girondo and Gabino González-Diez, Introduction to compact Riemann surfaces and dessins d’enfants, Cambridge University Press, Cambridge, 2012. | DOI | Zbl

[59] V. D. Goppa, Codes that are associated with divisors, Problemy Peredači Informacii 13 (1977), no. 1, 33–39. | Zbl

[60] Louis Granboulan, Calcul d’objets géométriques à l’aide de méthodes algébriques et numériques: dessins d’enfants, Ph.D. thesis, Université Paris 7, 1997.

[61] L. Granboulan, Construction d’une extension régulière de $ℚ\left(T\right)$ de groupe de Galois ${M}_{24}$, Experimental Math. 5 (1996), 3–14. | DOI | Zbl

[62] Alexandre Grothendieck, Sketch of a programme (translation into English), in Geometric Galois actions 1, eds. Leila Schneps and Pierre Lochak, London Math. Soc. Lect. Note Series, vol. 242, Cambridge University Press, Cambridge, 1997, 243–283. | DOI | Zbl

[63] Leon Greenberg, Maximal Fuchsian groups, Bull. Amer. Math. Soc. 69 (1963), 569–573. | DOI | MR | Zbl

[64] Gert-Martin Greuel and Gerhard Pfister, A Singular introduction to commutative algebra, Springer, Berlin, 2002. | DOI | Zbl

[65] Emmanuel Hallouin, Computation of a cover of Shimura curves using a Hurwitz space, J. of Algebra 321 (2009), no. 2, 558–566. | DOI | MR | Zbl

[66] Emmanuel Hallouin and Emmanuel Riboulet-Deyris, Computation of some moduli spaces of covers and explicit ${𝒮}_{n}$ and ${𝒜}_{n}$ regular $ℚ\left(T\right)$-extensions with totally real fibers, Pacific J. Math. 211 (2003), no. 1, 81–99. | DOI | Zbl

[67] Emmanuel Hallouin, Study and computation of a Hurwitz space and totally real ${PSL}_{2}\left({𝔽}_{8}\right)$-extensions of $ℚ$, J. Algebra 292 (2005), no. 1, 259–281. | Zbl

[68] Amihay Hanany, Yang-Hui He, Vishnu Jejjala, Jurgis Pasukonis, Sanjaye Ramgoolam, and Diego Rodriguez-Gomez, The beta ansatz: a tale of two complex structures, J. High Energy Physics 6 (2011), arXiv:1104.5490. | DOI | MR | Zbl

[69] Yang-Hui He and John McKay, $𝒩=2$ gauge theories: congruence subgroups, coset graphs and modular surfaces, arXiv:1201.3633v1, 2012. | DOI | MR | Zbl

[70] Yang-Hui He and John McKay, Eta products, BPS states and K3 surfaces, arXiv:1308.5233v1, 2013. | DOI

[71] Yang-Hui He, John McKay, and James Read, Modular subgroups, dessins d’enfants and elliptic K3 surfaces, arXiv:1211.1931v1, 2012. | DOI | MR | Zbl

[72] Dennis A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, Emerging Applications of Number Theory, eds. D. Hejhal, J. Friedman, M. Gutzwiller and A. Odlyzko, IMA Series No. 109, Springer-Verlag, 1999, 291–315. | DOI | Zbl

[73] Joachim A. Hempel, Existence conditions for a class of modular subgroups of genus zero, Bull. Austral. Math. Soc. 66 (2002), 517–525. | DOI | MR | Zbl

[74] Frank Herrlich and Gabriela Schmithüsen, Dessins d’enfants and origami curves, in Handbook of Teichmüller theory, Vol. II, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009, 767–809. | DOI | Zbl

[75] Kenji Hoshino, The Belyi functions and dessin d’enfants corresponding to the non-normal inclusions of triangle groups, Math. J. Okayama Univ. 52 (2010), 45–60. | Zbl

[76] Kenji Hoshino and Hiroaki Nakamura, Belyi function on ${X}_{0}\left(49\right)$ of degree 7, Math. J. Okayama Univ. 52 (2010), 61–63. | Zbl

[77] A. Hurwitz, Über Riemannsche Flächen mit gegebenen Verzweigungspunkten, Math. Ann. 39 (1891), 1–61. | DOI | Zbl

[78] Yasutaka Ihara, Some remarks on the number of rational points of algebraic curves over finite fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 721–724 (1982). | Zbl

[79] Ariyan Javanpeykar, Polynomial bounds for Arakelov invariants of Belyi curves, with an appendix by Peter Bruin, Ph.D. thesis, Universiteit Leiden, 2013.

[80] Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials: Constructive aspects of the inverse Galois problem, Cambridge University Press, Cambridge, 2002. | Zbl

[81] Gareth A. Jones, Congruence and noncongruence subgroups of the modular group: a survey, Proceedings of groups–St. Andrews 1985, London Math. Soc. Lecture Note Ser., vol. 121, Cambridge, 1986, 223–234. | DOI

[82] Gareth Jones and David Singerman, Belyĭ functions, hypermaps and Galois groups, Bull. London Math. Soc. 28 (1996), no. 6, 561–590. | DOI | MR | Zbl

[83] Gareth Jones and David Singerman, Maps, hypermaps, and triangle groups, in The Grothendieck theory of dessins d’enfants, London Math. Soc. Lecture Note Ser., vol. 200, Cambridge University Press, 1994, 115–145. | DOI | Zbl

[84] Gareth A. Jones and Manfred Streit, Galois groups, monodromy groups and cartographic groups, in Geometric Galois actions 2., eds. Leila Schneps and Pierre Lochak, London Math. Soc. Lect. Note Series, vol. 243, Cambridge University Press, Cambridge, 1997, 25–65 | DOI | Zbl

[85] Gareth A. Jones, Manfred Streit and J. Wolfart, Wilson’s map operations on regular dessins and cyclotomic fields of definition, Proc. Lond. Math. Soc. (3) 100 (2010), no. 2, 510–532. | DOI | MR | Zbl

[86] John W. Jones and David P. Roberts, Galois number fields with small root discriminant, J. Number Theory 122 (2007), 379–407. | DOI | MR | Zbl

[87] Gaston Julia, Étude sur les formes binaires non quadratiques à indéterminées réelles ou complexes, Mémoires de l’Académie des Sciences de l’Institut de France 55, 1–296 (1917).

[88] Nicholas M. Katz, Travaux de Laumon, Séminaire Bourbaki 691 (1987–1988), 105–132. | Numdam

[89] A. V. Kitaev, Dessins d’enfants, their deformations and algebraic the sixth Painlevé and Gauss hypergeometric functions, arXiv:nlin/0309078v3, 2003. | Zbl

[90] Felix Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, reprint of the 1884 original, Birkhäuser, Basel, 1993. | Zbl

[91] Michael Klug, Michael Musty, Sam Schiavone, and John Voight, Numerical computation of three-point branched covers of the projective line, arxiv:1311.2081, 2014. | DOI | MR | Zbl

[92] Bernhard Köck, Belyĭ’s theorem revisited, Beiträge Algebra Geom. 45 (2004), no. 1, 253–265. | Zbl

[93] Paul Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leizig, Math.-Phys. Kl. 88 (1936), 141–164. | Zbl

[94] Joachim König, A family of polynomials with Galois group ${PSL}_{5}\left(2\right)$ over $ℚ\left(t\right)$, arXiv:1308.1566v1, 2013.

[95] E. M. Kreĭnes, On families of geometric parasitic solutions for Belyi systems of genus zero, Fundamentalnaya i Priklandaya Matematika 9 (2003), no. 1, 103–111. | DOI | Zbl

[96] E. M. Kreĭnes, Equations determining Belyi pairs, with applications to anti-Vandermonde systems, Fundamentalnaya i Priklandaya Matematika 13 (2007), no. 4, 95–112. | DOI

[97] Martin Kreuzer and Lorenzo Robbiano, Computational commutative algebra 1, Springer-Verlag, New York, 2000. | Zbl

[98] Chris A. Kurth and Ling Long, Computations with finite index subgroups of ${PSL}_{2}\left(ℤ\right)$ using Farey symbols, arXiv:0710.1835, 2007. | DOI

[99] Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, with an appendix by D. Zagier, Encyclopaedia of Mathematical Sciences, Low-Dimensional Topology, II, Springer-Verlag, Berlin, 2004. | DOI

[100] Finnur Larusson and Timur Sadykov, Dessins d’enfants and differential equations, arXiv:math/0607773, 2006. | DOI | MR | Zbl

[101] H.W. Lenstra, Galois theory for schemes, online notes at http://websites.math.leidenuniv.nl/algebra/GSchemes.pdf.

[102] A.K. Lenstra, H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), 513–534. | DOI | MR | Zbl

[103] Reynald Lercier, Christophe Ritzenthaler, and Jeroen Sijsling, Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group, arXiv:1301.0695, 2013. | DOI | MR | Zbl

[104] Wen-Ching Winnie Li, Ling Long, and Zifeng Yang, Modular forms for noncongruence subgroups, Q. J. Pure Appl. Math. 1 (2005), no. 1, 205–221. | DOI | MR | Zbl

[105] Wilhelm Magnus, Noneuclidean tesselations and their groups, Pure and Applied Mathematics, vol. 61, Academic Press, New York, 1974. | DOI | Zbl

[106] Nicolas Magot and Alexander Zvonkin, Belyi functions for Archimedean solids, Discrete Math. 217 (2000), no. 1–3, 249–271. | DOI | MR | Zbl

[107] Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer, Berlin, 1999. | DOI | MR | Zbl

[108] G. Malle and B. H. Matzat, Realisierung von Gruppen ${PSL}_{2}\left({𝔽}_{p}\right)$ als Galoisgruppen über $ℚ$, Math. Ann. 272 (1985), 549–565. | DOI | Zbl

[109] Gunter Malle, Polynomials with Galois groups $Aut\left({M}_{22}\right)$, ${M}_{2}2$, and ${PSL}_{3}\left({𝔽}_{4}\right)·2$ over $ℚ$, Math. Comp. 51 (1988), 761–768. | DOI | Zbl

[110] Gunter Malle, Polynomials for primitive nonsolvable permutation groups of degree $d\le 15$, J. Symbolic Comput. 4 (1987), no. 1, 83–92. | DOI | Zbl

[111] Gunter Malle, Fields of definition of some three point ramified field extensions, in The Grothendieck theory of dessins d’enfants, London Math. Soc. Lecture Note Ser., vol. 200, Cambridge University Press, 1994, 147–168. | DOI | Zbl

[112] Gunter Malle and David P. Roberts, Number fields with discriminant $±{2}^{a}{3}^{b}$ and Galois group ${A}_{n}$ or ${S}_{n}$, LMS J. Comput. Math. 8 (2005), 1–22. | DOI | Zbl

[113] G. Malle and W. Trinks, Zur Behandlung algebraischer Gleichungssysteme mit dem Computer, Mathematisches Institut, Universität Karlsruhe, 1984, unpublished manuscript.

[114] Al Marden and Burt Rodin, On Thurston’s formulation and proof of Andreev’s theorem, Lecture Notes in Math., vol. 1435, Springer, 1989, 103–164. | DOI | Zbl

[115] Donald E. Marshall, Numerical conformal mapping software: zipper, http://www.math.washington.edu/ marshall/zipper.html.

[116] Ernst W. Mayr, Some complexity results for polynomial ideals, J. Complexity 13 (1997), no. 3, 303–325. | DOI | MR | Zbl

[117] Donald E. Marshall and Steffen Rohde, The zipper algorithm for conformal maps and the computation of Shabat polynomials and dessins, in preparation.

[118] Donald E. Marshall and Steffen Rohde, Convergence of a variant of the zipper algorithm for conformal mapping, SIAM J. Numer. Anal. 45 (2007), no. 6, 2577–2609. | DOI | MR | Zbl

[119] Yu. V. Matiyasevich, Computer evaluation of generalized Chebyshev polynomials, Moscow Univ. Math. Bull. 51 (1996), no. 6, 39–40.

[120] B. Heinrich Matzat, Konstructive Galoistheorie, Lect. Notes in Math., vol. 1284, Springer, Berlin, 1987. | DOI

[121] A. D. Mednykh, Nonequivalent coverings of Riemann surfaces with a prescribed ramification type, Siberian Math. J. 25 (1984), 606–625. | DOI | MR | Zbl

[122] Rick Miranda and Ulf Persson, Configurations of ${\mathrm{I}}_{n}$ fibers on elliptic $K3$ surfaces, Math. Z. 201 (1989), no. 3, 339–361. | DOI | MR | Zbl

[123] Bojan Mohar, A polynomial time circle packing algorithm, Discrete Math. 117 (1993), no. 1-3, 257–263. | DOI | MR | Zbl

[124] Hans Montanus, Hall triples and dessins d’enfant, Nieuw Arch. Wiskd. (5) 7 (2006), no. 3, 172–176. | Zbl

[125] Hossein Movasati and Stefan Reiter, Heun equations coming from geometry, Bull. Braz. Math. Soc. (N.S.) 43 (2012), no. 3, 423–442. | DOI | MR | Zbl

[126] Andrew Obus, Good reduction of three-point Galois covers, arXiv:1208.3909, 2012. | DOI | MR | Zbl

[127] Jennifer Paulhus, Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups, in ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, eds. Everett Howe and Kiran Kedlaya, the Open Book Series 1, Mathematical Science Publishers, 2013, 487–505. | DOI | MR | Zbl

[128] Heinz-Otto Peitgen (ed.), Newton’s method and dynamical systems, Kluwer Academic, Dordrecht, 1989. | DOI | Zbl

[129] Kevin Pilgrim, Dessins d’enfants and Hubbard trees, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 5, 671–693. | DOI | Numdam | MR | Zbl

[130] Michel Raynaud, Spécialisation des revêtements en charactéristique $p>0$, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 87–126. | DOI | Numdam | Zbl

[131] David P. Roberts, Nonsolvable polynomials with field discriminant ${5}^{A}$, Int. J. Number Theory 7 (2011), no. 2, 289–322. | DOI | Zbl

[132] David P. Roberts, An ABC construction of number fields, in Number theory, CRM Proc. Lecture Notes, vol. 36, Amer. Math. Soc., Providence, 2004, 237–267. | DOI | Zbl

[133] David P. Roberts, Lightly ramified number fields with Galois group $S.M12.A$, preprint at http://cda.morris.umn.edu/~roberts/research/m12.pdf. | DOI | MR | Zbl

[134] Matthieu Romagny and Stefan Wewers, Hurwitz spaces, in Groupes de Galois arithmétiques et différentiels, Sémin. Congr., vol. 13, Soc. Math. France, Paris, 313–341. | Zbl

[135] Simon Rubinstein-Salzedo, Totally ramified branched covers of elliptic curves, arxiv:1210.3195, 2012. | DOI | MR | Zbl

[136] Simon Rubinstein-Salzedo, Period computations for covers of elliptic curves, arxiv:1210.4721, 2012. | DOI | MR | Zbl

[137] William Stein, SAGE Mathematics Software (version 4.3), The SAGE Group, 2013, http://www.sagemath.org/.

[138] Leila Schneps, Dessins d’enfants on the Riemann sphere, in The Grothendieck theory of dessins d’enfants, London Math. Soc. Lecture Note Ser., vol. 200, Cambridge University Press, 1994, 47–77. | DOI | Zbl

[139] Leila Schneps, ed. The Grothendieck theory of dessins d’enfants, London Mathematical Society Lecture Note Series, vol. 200, Cambridge University Press, Cambridge, 1994. | DOI | Zbl

[140] René Schoof, Counting points on elliptic curves over finite fields, J. Théorie Nombres Bordeaux 7 (1995), 219–254. | DOI | MR | Zbl

[141] Björn Selander and Andreas Strömbergsson, Sextic coverings of genus two which are branched at three points, preprint at http://www2.math.uu.se/~astrombe/papers/g2.ps.

[142] Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics 1, Jones and Bartlett, 1992. | DOI | Zbl

[143] G. Shabat, On a class of families of Belyi functions, in Formal power series and algebraic combinatorics, eds. D. Krob, A. A. Mikhalev and A. V. Mikhalev, Springer-Verlag, Berlin, 2000, 575–581. | DOI | Zbl

[144] G.B. Shabat and V. Voevodsky, Drawing curves over number fields, in The Grothendieck Festschrift, vol. III, Birkhauser, Boston, 1990, 199–227. | DOI | Zbl

[145] Gorō Shimura, On the field of rationality for an abelian variety, Nagoya Math. J. 45 (1972), 167–178. | DOI | MR | Zbl

[146] David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. | DOI | Zbl

[147] D. Singerman and R.I. Syddall, Belyĭ uniformization of elliptic curves, Bull. London Math. Soc. 139 (1997), 443–451. | DOI | MR | Zbl

[148] Andrew J. Sommese, Charles W. Wampler II, The numerical solution of systems of polynomials arising in engineering and science, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. | DOI | Zbl

[149] William Stein, Modular forms: a computational approach, Grad. Studies in Math., vol. 79, American Mathematical Society, Providence, RI, 2007. | DOI | Zbl

[150] Manfred Streit, Homology, Belyĭ functions and canonical curves, Manuscripta Math. 90 (1996), 489–509. | DOI | Zbl

[151] Manfred Streit, Field of definition and Galois orbits for the Macbeath-Hurwitz curves, Arch. Math. (Basel) 74 (2000), no. 5, 342–349. | DOI | MR | Zbl

[152] Manfred Streit and Jürgen Wolfart, Characters and Galois invariants of regular dessins, Rev. Mat. Complut. 13 (2000), no. 1, 49–81. | DOI | MR | Zbl

[153] Kisao Takeuchi, Commensurability classes of arithmetic triangle groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 201-212. | Zbl

[154] W. Thurston, The geometry and topology of $3$-manifolds, Princeton University Notes, Princeton, 1982.

[155] M. A. Tsfasman, S. G. Vlăduţ and Th. Zink, Modular curves, Shimura curves, and Goppa codes, better than Varshamov-Gilbert bound, Math. Nachr. 109 (1982), 21–28. | DOI | MR | Zbl

[156] Mark van Hoeij, algcurves package, available at http://www.math.fsu.edu/ hoeij/maple.html.

[157] Mark van Hoeij and Raimundas Vidunas, Belyi functions for hyperbolic hypergeometric-to-Heun transformations, arxiv:1212.3803v2, 2013. | DOI | MR | Zbl

[158] Mark van Hoeij and Raimundas Vidunas, Algorithms and differential relations for Belyi functions, arxiv:1305.7218v1, 2013.

[159] Jan Verschelde, Algorithm 795: PHCpack: A General-Purpose Solver for Polynomial Systems by Homotopy Continuation, ACM Trans. Math. Softw. 25 (1999), no. 2, 251–276, http://www.math.uic.edu/ jan/PHCpack/phcpack.html. | DOI | Zbl

[160] Raimundas Vidunas, Transformations of some Gauss hypergeometric functions, J. Comp. Appl. Math. 178 (2005), 473–487. | DOI | MR | Zbl

[161] Raimundas Vidunas and Galina Filipuk, A classification of coverings yielding Heun-to-hypergeometric reductions, arXiv:1204.2730v1, 2012.

[162] Raimundas Vidunas and Alexander V. Kitaev, Computation of highly ramified coverings, arxiv:0705.3134v1, 2007. | DOI | MR | Zbl

[163] S. G. Vlăduţ and V. G. Drinfel’d, The number of points of an algebraic curve, Funktsional. Anal. i Prilozhen. 17 (1983), no. 1, 68–69. | DOI | MR

[164] John Voight and John Willis, Computing power series expansions of modular forms, in Computations with modular forms, eds. Gebhard Boeckle and Gabor Wiese, Contrib. Math. Comput. Sci., vol. 6, Springer, Berlin, 2014, 331–361. | DOI | Zbl

[165] Helmut Völklein, Groups as Galois groups. An introduction, Cambridge Studies in Advanced Mathematics, vol. 53, Cambridge University Press, Cambridge, 1996. | Zbl

[166] Mark Watkins, A note on integral points on elliptic curves, with an appendix by N. D. Elkies, J. Théor. Nombres Bordeaux 18 (2006), no. 3, 707–719. | DOI | MR | Zbl

[167] André Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509–524. | DOI | MR | Zbl

[168] Bruce Westbury, Circle packing, available at https://github.com/BruceWestbury/Circle-Packing, 2013.

[169] Franz Winkler, A $p$-adic approach to the computation of Gröbner bases, J. Symb. Comp. 6 (1988), no. 2–3, 287–304. | DOI | Zbl

[170] Klaus Wohlfahrt, An extension of F. Klein’s level concept, Illinois J. Math. 8 (1964), 529–535. | DOI | MR | Zbl

[171] Jürgen Wolfart, Triangle groups and Jacobians of CM type, preprint at http://www.math.uni-frankfurt.de/~wolfart/Artikel/jac.pdf.

[172] Jürgen Wolfart, ABC for polynomials, dessins d’enfants, and uniformization – a survey, in Elementare und analytische Zahlentheorie, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006, 313–345. | Zbl

[173] Jürgen Wolfart, The “obvious” part of Belyi’s theorem and Riemann surfaces with many automorphisms, Geometric Galois actions 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 97–112. | DOI | Zbl

[174] Melanie Wood, Belyi-extending maps and the Galois action on dessins d’enfants, Publ. RIMS, Kyoto Univ. 42 (2006), 721–737. | DOI | MR | Zbl

[175] Leonardo Zapponi, Fleurs, arbres et cellules: un invariant galoisien pour une famille d’arbres, Compositio Math. 122 (2000), no. 2, 113–133. | DOI | Zbl

[176] Alexander Zvonkin, Belyi functions: examples, properties, and applications, http://www.labri.fr/perso/zvonkin/Research/belyi.pdf. | DOI | Zbl

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