Field of moduli of generalized Fermat curves of type (k, 3) with an application to non-hyperelliptic dessins d'enfants. (November 2015) - Hidalgo, Ruben A.; Johnson, Pilar
Abstract: A generalized Fermat curve of type ( k, 3 ), where k ≥ 2, is a closed Riemann surface admitting a group H ≅ Z k 3 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 = e 2 i π / 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.