GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

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...

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Published inGlasgow mathematical journal Vol. 51; no. 2; pp. 289 - 299
Main Authors COSTE, ANTOINE D., JONES, GARETH A., STREIT, MANFRED, WOLFART, JÜRGEN
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.05.2009
Subjects
05C10
05C25
14H30
14H55
30F10
30F35
Mathematics
Primary 14H45
secondary 14H25
05C10
Primary 14H45
secondary 14H25
14H30
05C25
30F10
14H55
30F35
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Summary: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.
Bibliography:PII:S0017089509004972
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content type line 23
ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089509004972