Characterisations and Galois conjugacy of generalised Paley maps

TL;DR

This paper characterizes generalized Paley maps as orientably regular maps with specific automorphism properties, determines their fields of definition, and explores Galois conjugacy, highlighting their uniqueness under certain divisibility conditions.

Contribution

It provides a complete characterization of generalized Paley maps, links their automorphism groups to Galois actions, and establishes their uniqueness among regular embeddings under divisibility criteria.

Findings

Characterization of generalized Paley maps as orientably regular maps with primitive automorphism action.

Determination of fields of definition and Galois orbits for these maps.

Proof that under certain divisibility conditions, these maps are the unique regular embeddings of their graphs.

Abstract

A generalised Paley map is a Cayley map for the additive group of a finite field F, with a subgroup S=-S of the multiplicative group as generating set, cyclically ordered by powers of a generator of S. We characterise these as the orientably regular maps with orientation-preserving automorphism group acting primitively and faithfully on the vertices; allowing a non-faithful primitive action yields certain cyclic coverings of these maps. We determine the fields of definition and the orbits of the absolute Galois group on these maps, and we show that if (q-1)/(p-1) divides |S|, where |F|=q=p^e with p prime, then these maps are the only orientably regular embeddings of their underlying graphs; in particular this applies to the Paley graphs, where |S|=(q-1)/2 is even.