On $p$-gonal fields of definition

Abstract

Let $S$ be a closed Riemann surface of genus $g \geq 2$ and $\varphi$ be a conformal automorphism of $S$, of prime order $p$ such that $S/\langle \varphi \rangle$ has genus zero. Let ${\mathbb K} \leq {\mathbb C}$ be a field of definition of $S$, that is, there is an irreducible curve $C$, defined over ${\mathbb K}$, whose Riemann surface structure is biholomorphic to $S$. We provide a simple 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 $y^{p}=F(x) \in {\mathbb F}[x]$, in which case $\varphi$ corresponds to $(x,y) \mapsto (x,e^{2 \pi i/p} y)$. If, moreover, $\varphi$ is also definable over ${\mathbb K}$, then ${\mathbb F}$ can be chosen to be a quadratic extension of ${\mathbb K}$. For $p=2$, that is when $S$ is hyperelliptic and $\varphi$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that ${\rm Aut}(S)/\varphi\rangle$ is non-trivial.