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 K-n,K-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. Skoviera about regular embeddings of the graphs K-n,K-n, [7] and generalises the analogous results for maps obtained in [9], partly using different methods.