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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
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Abstract

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

References

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