ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.05
https://doi.org/10.26493/1855-3974.2764.fe5
(Also available at http://amc-journal.eu)
Regular dessins with moduli fields of the form
Q(ζp, p
√
q)*
Nicolas Daire
Département de Mathématiques et Applications, École Normale Supérieure,
45 rue d’Ulm, 75005 Paris, France
Fumiharu Kato , Yoshiaki Uchino
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro,
Tokyo 152-8551, Japan
Received 10 December 2021, accepted 29 March 2023, published online 27 September 2023
Abstract
Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins
with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli
fields are nonabelian Galois extensions of the form Q(ζp, p
√
q), where p is an odd prime
and ζp is a pth root of unity and q ∈ Q is not a pth power, and we then show that their
regular closures have the same moduli fields. Finally, in the special case p = q = 3 we
give another example of a regular dessin of degree 219 ·34 and genus 14155777 with moduli
field Q(ζ3, 3
√
3).
Keywords: Dessins d’enfants, coverings.
Math. Subj. Class. (2020): 14H57, 14H30
*The authors are grateful to Professor Jürgen Wolfart for valuable comments.
E-mail addresses: nicolas.daire@ens.psl.eu (Nicolas Daire), bungen@math.titech.ac.jp (Fumiharu Kato),
uchino.y.ab@m.titech.ac.jp (Yoshiaki Uchino)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
ISSN 1855-3966 (tiskana izd.), ISSN 1855-3974 (elektronska izd.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P2.05
https://doi.org/10.26493/1855-3974.2764.fe5
(Dostopno tudi na http://amc-journal.eu)
Regularne risbe z modulskimi obsegi oblike
Q(ζp, p
√
q)*
Nicolas Daire
Département de Mathématiques et Applications, École Normale Supérieure,
45 rue d’Ulm, 75005 Paris, France
Fumiharu Kato , Yoshiaki Uchino
Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro,
Tokyo 152-8551, Japan
Prejeto 10. decembra 2021, sprejeto 29. marca 2023, objavljeno na spletu 27. septembra 2023
Povzetek
Gareth Jones je na konferenci SIGMAP 2014 zastavil vprašanje iskanja primerov reg-
ularnih risb z neabelskimi modulskimi obsegi. V tem članku najprej konstruiramo risbe,
katerih modulski obsegi so neabelske Galoisove razširitve oblike Q(ζp, p
√
q), kjer je p liho
praštevilo in je ζp p-ti koren enote in q ∈ Q ni p-ta potenca, potem pa pokažemo, da imajo
njihova regularna zaprtja iste modulske obsege. Nazadnje, v posebnem primeru p = q = 3
podamo še en primer regularne risbe stopnje 219 · 34 in reda 14155777 z modulskim ob-
segom Q(ζ3, 3
√
3).
Ključne besede: Otroške risbe, krovi.
Math. Subj. Class. (2020): 14H57, 14H30
*Avtorji so hvaležni profesorju Jürgenu Wolfartu za dragocene komentarje.
E-poštni naslovi: nicolas.daire@ens.psl.eu (Nicolas Daire), bungen@math.titech.ac.jp (Fumiharu Kato),
uchino.y.ab@m.titech.ac.jp (Yoshiaki Uchino)
cb To delo je objavljeno pod licenco https://creativecommons.org/licenses/by/4.0/