2022/02/25 by Rubén A. Hidalgo, Ruben A. Hidalgo, Hidalgo, Ruben A.
Mathematics · #14H37 #14H55 #30F10 #30F20 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14H37 #msc:14H55 #msc:30F10 #msc:30F20
paper · pdf · doi:10.48550/arxiv.2202.12668
arXiv admin note: text overlap with arXiv:1309.6904
arxiv created 2022/02/25 · openalex publication_date 2022/02/25 · arxiv updated 2022/02/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbb K ≤ \mathbb C be a field of definition of S. We provide an argument for the existence of a field extension \mathbb F of \mathbb K, of degree at most 2(p-1), for which S is definable by a curve of the form yp=F(x) ∈ \mathbb F[x], in which case φ corresponds to (x,y) ↦ (x,e2 πi/p y). If, moreover, φ is also definable over \mathbb K, then \mathbb F can be chosen to be at most a quadratic extension of \mathbb 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 \rm Aut(S)/φ⟩ is non-trivial.