¹Universidad Técnica Federico Santa María, Valparaíso, Chile. Email: ruben.hidalgo@usm.cl
²Universidad Autónoma de Madrid, Madrid, España. Email: sebastian.reyes@uam.es
³Universidad de Concepción, Concepción, Chile. Email: mariaevaldes@udec.cl

A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Z_pⁿ of conformal automorphisms with S/H being the Riemann sphere with exactly n+1 cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by Aut_H(S) the normalizer of H in Aut(S). If p is a prime, and either (i) n=4 or (ii) n is even and Aut_H(S)/H is not a non-trivial cyclic group or (iii) n is odd and Aut_H(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ε {3,4}, then we also compute the field of moduli of S.

Key words: Algebraic curves, Riemann surfaces, Field of moduli, Field of definition.

Una curva de Fermat generalizada de tipo (p,n) es una superficie de Riemann cerrada S la cual admite un grupo H \cong Z_pⁿ de automorfismos conformales de manera que S/H sea de género cero y tenga exactamente n+1 puntos cónicos, cada uno de orden p. Si (p-1)(n-1) ≥ 3, entonces se sabe que S no es hiperelíptica y genéricamente no es casiplatónica. Denotemos por Aut_H(S) el normalizador de H en Aut(S). Si p es primo y tenemos que (i) n=4 o bien (ii) n es par y Aut_H(S)/H no es un grupo cíclico no trivial o bien (iii) n es impar y Aut_H(S)/H no es un grupo cíclico, entonces verificamos que S se puede definir sobre su cuerpo de moduli. Más aún, si n ε {3,4}, entonces determinamos tal cuerpo de moduli.

Palabras clave: Curvas algebraicas, superficies de Riemann, cuerpo de moduli, cuerpo de definición.

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