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 Zpn of conformal automorphisms with S/H being the Riemann sphere with exactly n+1cone points, each one of order p. If (p-1)(n-1) ≥ 3, then S is known to be non-hyperelliptic and generically not quasiplat...