Hurwitz–Belyi maps

Roberts, David P.

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

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.

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.

Classification : 11G32, 14H57
Mots clés : Hurwitz variety, Belyi map, ramification

[1] Beukers, Frits; Montanus, Hans 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] Grothendieck, Alexandre 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] Hallouin, Emmanuel Study and computation of a Hurwitz space and totally real ${\mathrm{PSL}}_2\left({𝔽}_8\right)$-extensions of $\mathbb{Q}$, J. Algebra, Volume 292 (2005) no. 1, pp. 259-281 | DOI | MR | Zbl

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

[6] Klug, Michael; Musty, Michael; Schiavone, Sam; Voight, John 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] Krämer, Stefan Numerical calculation of automorphic functions for finite index subgroups of triangle groups, Universität Bonn (Germany) (2015) (Ph. D. Thesis available from http://hss.ulb.uni-bonn.de/2015/4103/4103.htm)

[8] Lando, Sergei K.; Zvonkin, Alexander K. 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] Magaard, Kay; Shpectorov, Sergey; Völklein, Helmut 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] Malle, Gunter Polynomials with Galois groups $\mathrm{Aut}\left(M_{22}\right),M_{22},$ and ${\mathrm{PSL}}_3\left({\mathbf{F}}_4\right)·2_2$ over $\mathbf{Q}$, Math. Comp., Volume 51 (1988) no. 184, pp. 761-768 | DOI | MR | Zbl

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

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

[14] Malle, Gunter; Roberts, David P. Number fields with discriminant $\pm 2^a{3}^b$ and Galois group $A_n$ or $S_n$, LMS J. Comput. Math., Volume 8 (2005), pp. 80-101 | DOI | MR | Zbl

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

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

[17] Roberts, David P. Division polynomials with Galois group $S{U}_3\left(3\right).2\cong G_2\left(2\right)$, 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] Roberts, David P. Polynomials with prescribed bad primes, Int. J. Number Theory, Volume 11 (2015) no. 4, pp. 1115-1148 | DOI | MR | Zbl

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

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

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

[23] Serre, Jean-Pierre Relèvements dans ${\tilde{A}}_n$, C. R. Math. Acad. Sci. Paris, Volume 311 (1990) no. 8, pp. 477-482 | MR | Zbl

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

[25] Zhang, Liangcai; Chen, Guiyun; Chen, Shunmin; Liu, Xuefeng 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