Geodesic
Belyi functions of the special weighted trees of $(2,3)$-type
Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 17-39 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, we calculate the Belyi functions of the weighted trees of $(2,3)$-type with primitive special edge rotation groups. There are 21 Galois orbits of these trees: 6 rational orbits, 12 orbits over quadratic fields, and 3 orbits over fields of degree $3$, $4$, and $6$. The highest degree of the calculated Belyi function is $32$. The calculations are performed using modular functions techniques. 
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N. M. Adrianov; A. M. Vatuzov. Belyi functions of the special weighted trees of $(2,3)$-type. Fundamentalʹnaâ i prikladnaâ matematika, Tome 25 (2024) no. 2, pp. 17-39. http://geodesic.mathdoc.fr/item/FPM_2024_25_2_a2/

[1] Adrianov N. M., Zvonkin A. K., “Vzveshennye derevya s primitivnymi gruppami vraschenii reber”, Fundament. i prikl. matem., 18:6 (2013), 5–50

[2] Vatuzov A. M., “Vychislenie funktsii Belogo s ispolzovaniem tekhniki modulyarnykh funktsii”, Intellekt. sistemy. Teoriya i prilozheniya, 25:4 (2021), 88–91

[3] Dzhons G. A., Zvonkin A. K. Desyat «detskikh risunkov», “Kleina stepeni 11: tema s variatsiyami”, Fundament. i prikl. matem., 25:2 (2024), 103–175, arXiv: 2104.12015

[4] Zvonkin A. K., Lando S. K., Grafy na poverkhnostyakh i ikh prilozheniya, MTsNMO, M., 2010

[5] Kochetkov Yu. Yu., “Devyatirebernye ploskie derevya. Katalog”, Fundament. i prikl. matem., 13:6 (2007), 159–195

[6] Adrianov N. M., Pakovich F., Zvonkin A. K., Davenport–Zannier Polynomials and Dessins d'Enfants, AMS Math. Surveys Monographs, 249, Providence, RI, 2020 | DOI | MR

[7] Bétréma J., Péré D., Zvonkin A. K., Plane Trees and Their Shabat Polynomials. Catalog: Technical Report LaBRI No 92-75, Bordeaux, 1992

[8] Cassou-Noguès P., Couveignes J.-M., “Factorisations explicites de $g(y)-h(z)$”, Acta Arith., 87:4 (1999), 291–317 | DOI | MR | Zbl

[9] Couveignes J.-M., “Calcul et rationalité de fonctions de Belyi en genre $0$”, Ann. Inst. Fourier, 44:1 (1994), 1–38 | DOI | MR | Zbl

[10] Hoyden-Siedersleben G., Matzat B. H., “Realisierung sporadischer einfacher Gruppen als Galoisgruppen über Kreisteilungskörpern”, J. Algebra, 101 (1986), 273–286 | DOI | MR | Zbl

[11] Klein F., “Über die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades”, Math. Ann., 14 (1878), 111–172 | DOI | MR

[12] Klein F., “Über die Erniedrigung der Modulargleichungen”, Math. Ann., 14 (1878), 417–427 | DOI | MR

[13] Klein F., “Über die Transformation elfter Ordnung der elliptischen Functionen”, Math. Ann., 15 (1879), 533–555 | DOI | MR

[14] Monien H. (Sankt-Peterburg, 2014) http://www.pdmi.ras.ru/EIMI/2014/EG/presentations/Hartmut

[15] Monien H., The sporadic group J2, Hauptmodul and Belyi map, 2017, arXiv: 1703.05200 | MR | Zbl

[16] Monien H., The sporadic group Co3, Hauptmodul and Belyi map, 2018, arXiv: 1802.06923

[17] Musty M., Schiavone S., Sijsling J., Voight J., “A database of Belyi maps”, Proc. of the Thirteenth Algorithmic Number Theory Symposium, The Open Book Series, 2, 2019, 375–392 https://www.lmfdb.org/Belyi/ | DOI | MR | Zbl

[18] Pakovich F., Zvonkin A. K., “Minimum degree of the difference of two polynomials over $\mathbb{Q}$, and weighted plane trees”, Selecta Math. New Ser., 20:4 (2014), 1003–1065 | DOI | MR | Zbl

[19] Pakovich F., Zvonkin A. K., “Davenport–Zannier polynomials over $\mathbb{Q}$”, Int. J. Number Theory, 14:4 (2018), 925–974 | DOI | MR | Zbl

[20] Schwarz H., “Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt”, J. Reine Angew. Math., 75 (1873), 292–335 | MR

[21] Sijsling J., Voight J., “On computing Belyi maps”, Pub. Math. de Besançon, 1 (2014), 73–131 | MR | Zbl

[22] Zvonkin A., How to draw a group?, Discrete Math., 180 (1998), 403–413 | DOI | MR | Zbl