Riemann surfaces defined over the reals

TL;DR

This paper provides explicit examples of real Riemann surfaces that cannot be defined over their field of moduli, addressing a gap in the understanding of definability over the reals.

Contribution

It introduces explicit examples of real Riemann surfaces not definable over their field of moduli, expanding the known cases in the field.

Findings

Explicit examples of real Riemann surfaces not definable over their field of moduli.

Addresses a previously unresolved question in the theory of Riemann surfaces.

Contributes to the understanding of the relationship between real structures and field of moduli.

Abstract

The known (explicit) examples of Riemann surfaces not definable over their field of moduli are not real whose field of moduli is a subfield of the reals. In this paper we provide explicit examples of real Riemann surfaces which cannot be defined over the field of moduli.