Field of Moduli and Generalized Fermat Curves

A generalized Fermat curve of type (p,n) is a closed Riemann surface S admitting a group H \cong Zp n 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 \operatornameAutH(S) the normalizer of H in \operatornameAut(S). If p is a prime, and either (i) n=4 or (ii) n is even and \operatornameAutH(S)/H is not a non-trivial cyclic group or (iii) n is odd and \operatornameAutH(S)/H is not a cyclic group, then we prove that S can be defined over its field of moduli. Moreover, if n ∈ {3,4}, then we also compute the field of moduli of S.