Publications Mathématiques de Besançon
Algèbre et Théorie des Nombres
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David P. Roberts
Hurwitz–Belyi maps
(Applications d’Hurwitz–Belyi)
Publications mathématiques de Besançon (2018), p. 25-67, doi: 10.5802/pmb.21
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Class. Math.: 11G32, 14H57
Keywords: Hurwitz variety, Belyi map, ramification


The study of the moduli of covers of the projective line leads to the theory of Hurwitz varieties covering configuration varieties. Certain one-dimensional slices of these coverings are particularly interesting Belyi maps. We present systematic examples of such “Hurwitz–Belyi maps”. Our examples illustrate a wide variety of theoretical phenomena and computational techniques.


[1] Frits Beukers & Hans Montanus, Explicit calculation of elliptic fibrations of $K3$-surfaces and their Belyi-maps, Number theory and polynomials, London Mathematical Society Lecture Note Series 352, Cambridge University Press, 2008, p. 33–51 Article
[2] Alexandre Grothendieck, Esquisse d’un programme, Geometric Galois actions, 1, London Mathematical Society Lecture Note Series 242, Cambridge University Press, 1997, p. 5–48
[3] Emmanuel Hallouin, “Study and computation of a Hurwitz space and totally real ${\rm PSL}_2(\mathbb{F}_8)$-extensions of $\mathbb{Q}$”, J. Algebra 292 (2005) no. 1, p. 259-281 Article
[4] Adam James, Kay Magaard & Sergey Shpectorov, “The lift invariant distinguishes components of Hurwitz spaces for $A_5$”, Proc. Am. Math. Soc. 143 (2015) no. 4, p. 1377-1390 Article
[5] Gareth A. Jones & Alexander K. Zvonkin, “Orbits of braid groups on cacti”, Mosc. Math. J. 2 (2002) no. 1, p. 127-160
[6] Michael Klug, Michael Musty, Sam Schiavone & John Voight, “Numerical calculation of three-point branched covers of the projective line”, LMS J. Comput. Math. 17 (2014) no. 1, p. 379-430 Article
[7] Stefan Krämer, Numerical calculation of automorphic functions for finite index subgroups of triangle groups, Ph. D. Thesis, Universität Bonn (Germany), available from, 2015
[8] Sergei K. Lando & Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences 141, Springer, 2004, With an appendix by Don B. Zagier, Low-Dimensional Topology, II Article
[9] Kay Magaard, Sergey Shpectorov & Helmut Völklein, “A GAP package for braid orbit computation and applications”, Exp. Math. 12 (2003) no. 4, p. 385-393
[10] Gunter Malle, “Polynomials with Galois groups ${\rm Aut}(M_{22}),\;M_{22},$ and ${\rm PSL}_3({\bf F}_4)\cdot 2_2$ over ${\bf Q}$”, Math. Comp. 51 (1988) no. 184, p. 761-768 Article
[11] Gunter Malle, Fields of definition of some three point ramified field extensions, The Grothendieck theory of dessins d’enfants (Luminy, 1993), London Mathematical Society Lecture Note Series 200, Cambridge University Press, 1994, p. 147–168
[12] Gunter Malle, “Multi-parameter polynomials with given Galois group”, J. Symb. Comput. 30 (2000) no. 6, p. 717-731 Article
[13] Gunter Malle & B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer, 1999 Article
[14] Gunter Malle & David P. Roberts, “Number fields with discriminant $\pm 2^a3^b$ and Galois group $A_n$ or $S_n$”, LMS J. Comput. Math. 8 (2005), p. 80-101 Article
[15] David P. Roberts, “Chebyshev covers and exceptional number fields”, in preparation
[16] David P. Roberts, “Fractalized cyclotomic polynomials”, Proc. Am. Math. Soc. 135 (2007) no. 7, p. 1959-1967 Article
[17] David P. Roberts, Division polynomials with Galois group $SU_3(3).2\cong G_2(2)$, Advances in the theory of numbers, Fields Inst. Commun. 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, p. 169–206 Article
[18] David P. Roberts, “Polynomials with prescribed bad primes”, Int. J. Number Theory 11 (2015) no. 4, p. 1115-1148 Article
[19] David P. Roberts, “Lightly ramified number fields with Galois group $S.M_{12}.A$”, J. Théor. Nombres Bordx 28 (2016) no. 2, p. 435-460
[20] David P. Roberts, “Hurwitz number fields”, New York J. Math. 23 (2017), p. 227-272
[21] David P. Roberts, “A three-parameter clan of Hurwitz–Belyi maps”, Publ. Math. Besançon, Algèbre Théorie Nombres 6 (2018), p. 69-83
[22] David P. Roberts & Akshay Venkatesh, “Hurwitz monodromy and full number fields”, Algebra Number Theory 9 (2015) no. 3, p. 511-545 Article
[23] Jean-Pierre Serre, “Relèvements dans $\tilde{A}_n$”, C. R. Math. Acad. Sci. Paris 311 (1990) no. 8, p. 477-482
[24] Jeroen Sijsling & John Voight, “On computing Belyi maps”, Publ. Math. Besançon, Algèbre Théorie Nombres 1 (2014) no. 1, p. 73-131
[25] Liangcai Zhang, Guiyun Chen, Shunmin Chen & Xuefeng Liu, “Notes on finite simple groups whose orders have three or four prime divisors”, J. Algebra Appl. 8 (2009) no. 3, p. 389-399 Article