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 graph--theoretic results by G. A. Jones, R. Nedela and M.\v{S}koviera about regular embeddings of the graphs $K_{n,n}$ [JN\v{S}] and generalises the analogous question for maps treated in [JStW], partly using different methods.