On $p$-gonal fields of definition

TL;DR

This paper investigates the fields over which p-gonal Riemann surfaces can be defined, providing bounds on the degree of field extensions needed for such definitions, especially when automorphisms are also definable over the base field.

Contribution

It establishes bounds on the degree of field extensions required to define p-gonal surfaces and automorphisms, generalizing known results for hyperelliptic cases to higher primes.

Findings

Existence of a field extension of degree at most 2(p-1) for defining the surface as y^p=F(x)

If automorphism is definable over the base field, extension degree can be at most 2

Special case for p=2 recovers known hyperelliptic results

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 $S$. 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 $y^{p}=F(x) \in {\mathbb F}[x]$, 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 at most 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.