Hurwitz–Belyi maps
Publications mathématiques de Besançon. Algèbre et théorie des nombres (2018), pp. 25-67.

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.

L’étude des modules de revêtements de la droite projective conduit à la théorie des variétés de Hurwitz comme revêtements des variétés de configurations. Certaines sections de dimension un des ces revêtements sont des applications de Belyi particulièrement intéressantes. Nous présentons des exemples de telles applications « d’Hurwitz–Belyi » qui illustrent une large variété de phénomènes théoriques et techniques de calculs.

Published online:
DOI: 10.5802/pmb.21
Classification: 11G32,  14H57
Keywords: Hurwitz variety, Belyi map, ramification
David P. Roberts 1

1 Division of Science and Mathematics, University of Minnesota Morris, Morris, Minnesota, 56267, USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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David P. Roberts. Hurwitz–Belyi maps. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2018), pp. 25-67. doi : 10.5802/pmb.21. https://pmb.centre-mersenne.org/articles/10.5802/pmb.21/

[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), Volume 352, Cambridge University Press, 2008, pp. 33-51 | DOI | MR | Zbl

[2] Alexandre Grothendieck Esquisse d’un programme, Geometric Galois actions, 1 (London Mathematical Society Lecture Note Series), Volume 242, Cambridge University Press, 1997, pp. 5-48 (With an English translation on pp. 243–283) | MR | Zbl

[3] Emmanuel Hallouin Study and computation of a Hurwitz space and totally real PSL 2 (𝔽 8 )-extensions of , J. Algebra, Volume 292 (2005) no. 1, pp. 259-281 | DOI | MR | Zbl

[4] Adam James; Kay Magaard; Sergey Shpectorov The lift invariant distinguishes components of Hurwitz spaces for A 5 , Proc. Am. Math. Soc., Volume 143 (2015) no. 4, pp. 1377-1390 | DOI | MR | Zbl

[5] Gareth A. Jones; Alexander K. Zvonkin Orbits of braid groups on cacti, Mosc. Math. J., Volume 2 (2002) no. 1, pp. 127-160 | MR | Zbl

[6] Michael Klug; Michael Musty; Sam Schiavone; John Voight Numerical calculation of three-point branched covers of the projective line, LMS J. Comput. Math., Volume 17 (2014) no. 1, pp. 379-430 | DOI | MR | Zbl

[7] Stefan Krämer Numerical calculation of automorphic functions for finite index subgroups of triangle groups (2015) (Ph. D. Thesis)

[8] Sergei K. Lando; Alexander K. Zvonkin Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004, xvi+455 pages (With an appendix by Don B. Zagier, Low-Dimensional Topology, II) | DOI | MR | Zbl

[9] Kay Magaard; Sergey Shpectorov; Helmut Völklein A GAP package for braid orbit computation and applications, Exp. Math., Volume 12 (2003) no. 4, pp. 385-393 http://projecteuclid.org/euclid.em/1087568015 | MR | Zbl

[10] Gunter Malle Polynomials with Galois groups Aut (M 22 ),M 22 , and PSL 3 (F 4 )·2 2 over Q, Math. Comp., Volume 51 (1988) no. 184, pp. 761-768 | DOI | MR | Zbl

[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), Volume 200, Cambridge University Press, 1994, pp. 147-168 | MR | Zbl

[12] Gunter Malle Multi-parameter polynomials with given Galois group, J. Symb. Comput., Volume 30 (2000) no. 6, pp. 717-731 | DOI | MR | Zbl

[13] Gunter Malle; B. Heinrich Matzat Inverse Galois theory, Springer Monographs in Mathematics, Springer, 1999, xvi+436 pages | DOI | MR | Zbl

[14] Gunter Malle; David P. Roberts Number fields with discriminant ±2 a 3 b and Galois group A n or S n , LMS J. Comput. Math., Volume 8 (2005), pp. 80-101 | DOI | MR | Zbl

[15] David P. Roberts Chebyshev covers and exceptional number fields (in preparation)

[16] David P. Roberts Fractalized cyclotomic polynomials, Proc. Am. Math. Soc., Volume 135 (2007) no. 7, pp. 1959-1967 | DOI | MR | Zbl

[17] David P. Roberts Division polynomials with Galois group SU 3 (3).2G 2 (2), Advances in the theory of numbers (Fields Inst. Commun.), Volume 77, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, pp. 169-206 | DOI | MR | Zbl

[18] David P. Roberts Polynomials with prescribed bad primes, Int. J. Number Theory, Volume 11 (2015) no. 4, pp. 1115-1148 | DOI | MR | Zbl

[19] David P. Roberts Lightly ramified number fields with Galois group S.M 12 .A, J. Théor. Nombres Bordx, Volume 28 (2016) no. 2, pp. 435-460 http://jtnb.cedram.org/item?id=jtnb_2016__28_2_435_0 | Zbl

[20] David P. Roberts Hurwitz number fields, New York J. Math., Volume 23 (2017), pp. 227-272 | Zbl

[21] David P. Roberts A three-parameter clan of Hurwitz–Belyi maps, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 6 (2018), pp. 69-83

[22] David P. Roberts; Akshay Venkatesh Hurwitz monodromy and full number fields, Algebra Number Theory, Volume 9 (2015) no. 3, pp. 511-545 | DOI | MR | Zbl

[23] Jean-Pierre Serre Relèvements dans A ˜ n , C. R. Math. Acad. Sci. Paris, Volume 311 (1990) no. 8, pp. 477-482 | MR | Zbl

[24] Jeroen Sijsling; John Voight On computing Belyi maps, Publ. Math. Besançon, Algèbre Théorie Nombres, Volume 1 (2014) no. 1, pp. 73-131 | MR | Zbl

[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., Volume 8 (2009) no. 3, pp. 389-399 | DOI | MR | Zbl

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