Galois action on families of generalised Fermat curves (Q865490)

\textit{G. A. Jones, R. Nedela} and \textit{M. Škoviera} [Regular embeddings of \(K_{n,n}\) where \(n\) is an odd prime power. European J. Combin., (in press)] classified the regular embeddings of the bipartite graph \(K_{n,n}\) for the case when \(n=p^{e}\), \(p\) is an odd prime, \(e \geq 1\). The classification provides exactly \(p^{e-1}\) such embeddings, which are divided into \(e\) disjoint families, each one containing exactly \(\phi(p^{e-f})\) members, where \(f=1,2,...,e\) and \(\phi\) is Euler's function. Each of the members of these families is a Belyi curve, then defined over \(\overline{\mathbb Q}\). The action of the Galois group \(\text{Gal}(\overline{\mathbb Q}/{\mathbb Q})\) acts on them at the level of the coefficients. In the paper under review, the authors prove that each of these \(e\) families is in fact a complete orbit under such an action. Moreover, in each case, the field of definition is obtained, this being the cyclotomic field \({\mathbb Q}(e^{2\pi i/p^{e-f}})\) for \(f=1,...,e\). Defining equations, in the case \(2f \geq e\) are obtained.