Field of moduli of generalized Fermat curves

TL;DR

This paper investigates when the field of moduli of generalized Fermat curves, a special class of Riemann surfaces, is the real numbers and whether it can serve as a field of definition.

Contribution

It characterizes conditions under which the field of moduli of generalized Fermat curves is real and when it can be used as a field of definition.

Findings

Identifies criteria for the field of moduli to be real.

Determines when the field of moduli is also a field of definition.

Analyzes the structure of generalized Fermat curves in this context.

Abstract

As a consequence of the Riemann-Roch theorem, a closed Riemann surface $S$ can be described by a non-singular complex projective algebraic curve $C$. A field of definition for $S$ is any subfield $D$ of $\mathbb{C}$ so that we may choose $C$ to be defined by polynomials in $D[x_0, \ldots, x_n]$. The field of moduli of $S$ is ${\mathbb R}$ if and only if $S$ admits an anticonformal automorphism. In the case that the field of moduli of $S$ is ${\mathbb R}$, then $S$ can be defined over the field of moduli if and only if $S$ admits an anticonformal involution. It may happen that the field of moduli is not a field of definition. In this paper, we consider certain class of closed Riemann surfaces, called generalized Fermat curves. These surfaces are the highest Abelian branched cover of certain orbifolds. In this class of Riemann surfaces, we study the problem of deciding when the field of moduli is ${\mathbb R}$ and when, in such a case, it is a field of definition.