On $p$-gonal fields of definition

Abstract

Let $S$ be a closed Riemann surface of genus $g\ge 2$ and $\ddot{I}†$ be a conformal automorphism of $S$, of prime order $p$ such that $S/⟨\ddot{I}†⟩$ has genus zero. Let $\mathbb{K}\le \mathbb{C}$ 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 $\ddot{I}†$ corresponds to $(x,y)\mapsto (x,e^{2\ddot{I}€i/p}y)$. If, moreover, $\ddot{I}†$ 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 $\ddot{I}†$ is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and Lercier-Ritzenthaler-Sijslingit in the case that $\mathrm{Aut}(S)/\ddot{I}†⟩$ is non-trivial.

The same print as arXiv:2202.12668