Field of moduli of generalized Fermat curves of type [formula omitted] with an application to non-hyperelliptic dessins d'enfants

Abstract

A generalized Fermat curve of type (k,3), where k≥2, is a closed Riemann surface admitting a group H≅Zk3 as a group of conformal automorphisms so that the quotient orbifold S/H is the Riemann sphere and it has exactly 4 cone points, each one of order k. Every genus one Riemann surface is a generalized Fermat curve of type (2,3) and, if k≥3, then a generalized Fermat curve of type (k,3) is non-hyperelliptic. For each generalized Fermat curve, we compute its field of moduli and note that it is a field of definition. Moreover, for k=e2iπ/p, where p≥5 is a prime integer, we produce explicit algebraic models over the corresponding field of moduli. As a byproduct, we observe that the absolute Galois group Gal(Q¯/Q) acts faithfully at the level of non-hyperelliptic dessins d'enfants. This last fact was already known for dessins of genus 0, 1 and for hyperelliptic ones, but it seems that the non-hyperelliptic situation is not explicitly given in the existent literature.