Generalised Fermat Hypermaps and Galois Orbits
Abstract
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 the complete bipartite graph $ K_{n,n} $, where $ n $ is an odd prime power. We will show that all these surfaces, regarded as algebraic curves, are defined over abelian number fields. We will determine the orbits under the action of the absolute Galois group, their minimal fields of definition, and in some easier cases also their defining equations. The paper relies on group and graphtheoretic results by G. A. Jones, R. Nedela and M.Škoviera about regular embeddings of the graphs $K_{n,n}$ [JNŠ] and generalises the analogous question for maps treated in [JStW], partly using different methods.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606712
 Bibcode:
 2006math......6712C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Complex Variables;
 14H45
 EPrint:
 14 pages, new version with extended introduction, minor corrections and updated references