GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

摘要

We consider families of quasiplatonic Riemann surfaces characterisednby the fact that – as in the case of Fermat curves of exponent n – their underlyingnregular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n,wherenn is an odd prime power. We show that these surfaces, regarded as algebraic curves,nare all defined over abelian number fields. We determine their orbits under the actionnof the absolute Galois group, theirminimal fields of definition and in some easier casesntheir defining equations. The paper relies on group – and graph – theoretic results bynG. A. Jones, R. Nedela and M. ˇ Skoviera about regular embeddings of the graphs Kn,nn[7] and generalises the analogous results formaps obtained in [9], partly using differentnmethods.