Field of moduli of generalized Fermat curves of type (k,3) ( k , 3 ) with an application to non-hyperelliptic dessins d'enfants ?
A generalized Fermat curve of type (k,3) ( k , 3 ) , where k?2 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 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) ( 2 , 3 ) and, if k?3 k ? 3 , then a generalized Fermat curve of type (k,3) ( 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=e2i?/p k = e 2 i ? / p , where p?5 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.
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