On p-gonal fields of definition

Let S be a closed Riemann surface of genus g ≥ 2 and φ be a conformal automorphism of S of prime order p such that S/⟨φ⟩ has genus zero. Let K ≤ C be a field of definition of S. We prove the existence of a field extension F of K, of degree at most 2(p−1), for which S is definable by a curve of the form yp = F(x) ∈ F[x], in which case φ corresponds to (x,y) ↦ (x,e2πi/py). If, moreover, φ is also definable over K, then F can be chosen to be at most a quadratic extension of K. For p = 2, that is when S is hyperelliptic and φ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that Aut(S)/⟨φ⟩ is non-trivial.