Hostname: page-component-74656f9884-g96k6 Total loading time: 0 Render date: 2026-11-10T13:38:56.407Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

Published online by Cambridge University Press:  01 May 2009

ANTOINE D. COSTE ,
GARETH A. JONES ,
MANFRED STREIT  and
JÜRGEN WOLFART
ANTOINE D. COSTE
Affiliation:
CNRS, UMR 8627, Building 210, Laboratory of Theoretical Physics, F–91405 Orsay Cedex, France e-mail: antoine.coste@m4x.org
GARETH A. JONES
Affiliation:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK e-mail: G.A.Jones@maths.soton.ac.uk
MANFRED STREIT
Affiliation:
Usinger Str. 56, D-61440 Oberursel, Germany e-mail: Manfred.Streit@bahn.de
JÜRGEN WOLFART
Affiliation:
Math. Sem. der Univ., Postfach 111932, D–60054 Frankfurt a.M., Germany e-mail: wolfart@math.uni-frankfurt.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

Keywords

Primary 14H45secondary 14H2514H3014H5505C1005C2530F1030F35

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 2 , May 2009 , pp. 289 - 299
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

REFERENCES

1.Bauer, M., Coste, A., Itzykson, Cl. and Ruelle, P., Comments on the links between su(3) modular invariants, simple factors in the Jacobians of Fermat curves, and rational triangular billards, J. Geom. Phys. 22 (2) (1997), 134–189.CrossRefGoogle Scholar
2.Belyĭ, G. V., On Galois extensions of a maximal cyclotomic field, Math. USSR Izv. 14 (1980), 247–256.CrossRefGoogle Scholar
3.Bujalance, E., Cirre, F. J. and Conder, M., On extendability of group actions on compact Riemann surfaces, Trans. AMS 355 (4) (2002), 1537–1557.CrossRefGoogle Scholar
4.Cohen, P., Itzykson, C. and Wolfart, J., Fuchsian triangle groups and Grothendieck dessins: variations on a theme of Belyi, Comm. Math. Phys. 163 (1994), 605–627.CrossRefGoogle Scholar
5.Girondo, E. and Wolfart, J., Conjugators of Fuchsian groups and quasiplatonic surfaces, Quart. J. Math. 56 (2005), 525–540.CrossRefGoogle Scholar
6.Grothendieck, A., Esquisse d'un Programme, in Geometric Galois Actions 1: Around Grothendieck's Esquisse d'un Programme (Lochak, P. and Schneps, L., Editors), London Mathematical Society Lecture Note Series 242 (Cambridge University Press, Cambridge, 1997), 5–84.Google Scholar
7.Jones, G. A., Nedela, R. and Škoviera, M., Regular embeddings of K n,n where n is an odd prime power, Euro. J. Combinatorics 28 (2007), 1595–1609.CrossRefGoogle Scholar
8.Jones, G. A. and Singerman, D., Belyi functions, hypermaps and Galois groups, Bull. Lond. Math. Soc. 28 (1996), 561–590.CrossRefGoogle Scholar
9.Jones, G. A., Streit, M. and Wolfart, J., Galois action on families of generalised Fermat curves, J. of Algebra 307 (2007), 829–840.CrossRefGoogle Scholar
10.Singerman, D., Finitely maximal Fuchsian groups, J. Lond. Math. Soc. (2) 6 (1972), 29–38.CrossRefGoogle Scholar
11.Streit, M., Field of definition and Galois orbits for the Macbeath–Hurwitz curves, Arch. Math. 74 (2000), 342–349.CrossRefGoogle Scholar
12.Streit, M. and Wolfart, J., Characters and Galois invariants of regular dessins, Revista Mat. Complutense 13 (2000), 49–81.Google Scholar
13.Streit, M. and Wolfart, J., Cyclic projective planes and Wada dessins, Doc. Math. 6 (2001), 39–68.CrossRefGoogle Scholar
14.Voevodsky, V. A. and Shabat, G., Equilateral triangulations of Riemann surfaces and curves over algebraic number fields, Soviet Math. Dokl. 39 (1989), 38–41.Google Scholar
15.Wolfart, J., ABC for polynomials, dessins d'enfants, and uniformization – a survey, in Elementare und Analytische Zahlentheorie (Tagungsband) (Schwarz, W. and Steuding, J., Editors) (Steiner, Stuttgart, 2006), 313–345. Available at http://www.math.uni-frankfurt.de/~wolfart/.Google Scholar
16.Wolfart, J., The ‘obvious’ part of Belyi's Theorem and Riemann surfaces with many automorphisms, in Geometric Galois Actions 1: Around Grothendieck's Esquisse d'un Programme (Lochak, P. and Schneps, L., Editors), London Mathematical Society Lecture Note Series 242 (Cambridge University Press, Cambridge, 1997), 97–112.CrossRefGoogle Scholar