Geodesic
Belyi Pairs and Fried Families
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 306-318.

Voir la notice de l'article provenant de la source Math-Net.Ru

For the rational functions on algebraic curves, the number of critical values typical for the Belyi functions is three. This number is replaced by four. The emerging objects, unlike the rigid Belyi pairs, can be deformed; they constitute one-parameter families. The main properties of the corresponding curves in the moduli spaces are established, and the relation of these curves to the Belyi pairs is considered. The case of the clean Belyi pairs of genus $2$ of minimal degree $8$ is analyzed.
Keywords: Belyi pairs, Fried families, moduli spaces of curves, Galois theory.
Mots-clés : dessins d'enfants 
@article{TM_2019_307_a16,
     author = {G. B. Shabat},
     title = {Belyi {Pairs} and {Fried} {Families}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {306--318},
     publisher = {mathdoc},
     volume = {307},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2019_307_a16/}
}
TY  - JOUR
AU  - G. B. Shabat
TI  - Belyi Pairs and Fried Families
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2019
SP  - 306
EP  - 318
VL  - 307
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2019_307_a16/
LA  - ru
ID  - TM_2019_307_a16
ER  - 
%0 Journal Article
%A G. B. Shabat
%T Belyi Pairs and Fried Families
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2019
%P 306-318
%V 307
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2019_307_a16/
%G ru
%F TM_2019_307_a16
G. B. Shabat. Belyi Pairs and Fried Families. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 306-318. http://geodesic.mathdoc.fr/item/TM_2019_307_a16/

[1] N. M. Adrianov and G. B. Shabat, “Belyi functions of dessins d'enfants of genus 2 with 4 edges”, Russ. Math. Surv., 60:6 (2005), 1237–1239 | DOI | DOI | MR | Zbl

[2] Bailey P., Fried M.D., “Hurwitz monodromy, spin separation and higher levels of a Modular Tower”, Arithmetic fundamental groups and noncommutative algebra, Proc. 1999 von Neumann Conf. (Berkeley, CA, 1999), Proc. Symp. Pure Math., 70, Amer. Math. Soc., Providence, RI, 2002, 79–220 | DOI | MR | Zbl

[3] Beckmann S., “Ramified primes in the field of moduli of branched coverings of curves”, J. Algebra, 125:1 (1989), 236–255 | DOI | MR | Zbl

[4] G. V. Belyĭ, “On Galois extensions of a maximal cyclotomic field”, Math. USSR, Izv., 14:2 (1980), 247–256 | DOI | MR | Zbl | Zbl

[5] Diaz S., Donagi R., Harbater D., “Every curve is a Hurwitz space”, Duke Math. J., 59:3 (1989), 737–746 | DOI | MR | Zbl

[6] Fried M., “Fields of definition of function fields and Hurwitz families—Groups as Galois groups”, Commun. Algebra, 5 (1977), 17–82 | DOI | MR | Zbl

[7] Girondo E., González-Diez G., Introduction to compact Riemann surfaces and dessins d'enfants, Cambridge Univ. Press, Cambridge, 2012 | MR | Zbl

[8] Grothendieck A., “Esquisse d'un programme (sketch of a programme)”, Geometric Galois actions, v. 1, LMS Lect. Note Ser., 242, Around Grothendieck's “Esquisse d'un programme”, Cambridge Univ. Press, Cambridge, 1997, 5–48; 243–283 | MR | Zbl

[9] A. Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen. Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant, Grundl. Math. Wiss., 3, Springer, Berlin, 1964 | MR | Zbl

[10] Javanpeykar A., Voight J., The Belyi degree of a curve is computable, E-print, 2018, arXiv: 1711.00125v2 [math.NT] | MR

[11] Kazarian M., Zograf P., “Virasoro constraints and topological recursion for Grothendieck's dessin counting”, Lett. Math. Phys., 105:8 (2015), 1057–1084 | DOI | MR | Zbl

[12] Lando S.K., Zvonkin A.K., Graphs on surfaces and their applications, Encycl. Math. Sci., 141, Springer, Berlin, 2004 | MR | Zbl

[13] McMullen C.T., “From dynamics on surfaces to rational points on curves”, Bull. Amer. Math. Soc., 37:2 (2000), 119–140 | DOI | MR | Zbl

[14] A. N. Paršin, “Algebraic curves over function fields”, Sov. Math., Dokl., 9 (1968), 1419–1422 | MR | Zbl

[15] G. B. Shabat, “Unicellular four-edged toric dessins”, J. Math. Sci., 209:2 (2015), 309–318 | DOI | MR | Zbl

[16] Shabat G., “Counting Belyi pairs over finite fields”, 2016 MATRIX annals, MATRIX Book Ser., 1, Springer, Cham, 2018, 305–322 | DOI | MR | Zbl

[17] Shabat G., Belyi pairs in the critical filtrations of Hurwitz spaces, E-print, 2019, arXiv: 1902.02034 [math.AG]

[18] I. R. Šafarevič, “Algebraic number fields”, Proc. Int. Congr. Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 163–176 | MR

[19] Am. Math. Soc. Transl., Ser. 2, 31 (1963), 25–39 | MR

[20] Sijsling J., Voight J., “On computing Belyi maps”, Méthodes arithmétiques et applications: Numéro consacré au trimestre, aut. 2013, Publ. Math. Besançon: Algèbre et Théorie des Nombres; N 2014/1, Pres. Univ. Franche-Comté, Besançon, 2014, 73–131 | MR | Zbl

[21] P. G. Zograf and L. A. Takhtadzhyan, “On Liouville's equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0”, Math. USSR, Sb., 60:1 (1988), 143–161 | DOI | MR | Zbl | Zbl