The Belyi Characterization of a Class of Modular Curves
Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 409-429.

A class of modular curves is characterized by the existence of certain pairs of Belyi functions which generate their function fields. Applications to the modular equation and the computation of special values of the j-function are given.

Une classe de courbes modulaires est caractérisée par l’existence de certains couples de fonctions de Belyi qui engendrent leurs corps des fonctions. Des applications à l’équation modulaire et au calcul de valeurs spéciales de la fonction j sont données.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1031
Classification: 14H57, 11F03, 14Q05
Keywords: dessins d’enfants, Belyi functions, modular curves, modular equation
Filom, Khashayar 1

1 Department of Mathematical Sciences Sharif University of Technology Tehran, Iran
@article{JTNB_2018__30_2_409_0,
     author = {Filom, Khashayar},
     title = {The {Belyi} {Characterization} of a {Class} of {Modular} {Curves}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {409--429},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {2},
     year = {2018},
     doi = {10.5802/jtnb.1031},
     mrnumber = {3891319},
     zbl = {1453.14088},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.1031/}
}
TY  - JOUR
AU  - Filom, Khashayar
TI  - The Belyi Characterization of a Class of Modular Curves
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 409
EP  - 429
VL  - 30
IS  - 2
PB  - Société Arithmétique de Bordeaux
UR  - http://www.numdam.org/articles/10.5802/jtnb.1031/
DO  - 10.5802/jtnb.1031
LA  - en
ID  - JTNB_2018__30_2_409_0
ER  - 
%0 Journal Article
%A Filom, Khashayar
%T The Belyi Characterization of a Class of Modular Curves
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 409-429
%V 30
%N 2
%I Société Arithmétique de Bordeaux
%U http://www.numdam.org/articles/10.5802/jtnb.1031/
%R 10.5802/jtnb.1031
%G en
%F JTNB_2018__30_2_409_0
Filom, Khashayar. The Belyi Characterization of a Class of Modular Curves. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 2, pp. 409-429. doi : 10.5802/jtnb.1031. http://www.numdam.org/articles/10.5802/jtnb.1031/

[1] Filom, Khashayar; Kamalinejad, Ali Dessins on Modular Curves (2016) (https://arxiv.org/abs/1603.01693)

[2] Girondo, Ernesto; Gonzàlez-Diez, Gabino Introduction to Compact Riemann Surfaces and Dessins d’Enfants, London Mathematical Society Student Texts, 79, Cambridge University Press, 2012, xii+298 pages | MR | Zbl

[3] He, Yang-Hui; McKay, John; Read, James Modular subgroups, dessins d’enfants and elliptic K3 surfaces, LMS J. Comput. Math., Volume 16 (2013), pp. 271-318 | MR | Zbl

[4] Lando, Sergei K.; Zvonkin, Alexander K. Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences, 141, Springer, 2004, xv+455 pages | MR | Zbl

[5] McKay, John; Sebbar, Abdellah J-invariants of arithmetic semistable elliptic surfaces and graphs, Proceedings on Moonshine and related topics (Montréal, 1999) (CRM Proceedings & Lecture Notes), Volume 30, American Mathematical Society, 2001, pp. 119-130 | DOI | MR | Zbl

[6] Milne, James Stuart Modular Functions and Modular Forms (Elliptic Modular Curves) (2012) (http://www.jmilne.org/math/CourseNotes/MF.pdf)

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

Cited by Sources: