A remark on the field of moduli of Riemann surfaces

TL;DR

This paper investigates the relationship between the field of moduli and the field of definition for Riemann surfaces, providing bounds on the minimal extension degree needed for definability based on the surface's automorphism quotient.

Contribution

It offers new upper bounds on the minimal degree extension of the field of moduli required for the surface to be definable, depending on the automorphism quotient orbifold.

Findings

Provides explicit bounds for the minimal extension degree

Analyzes cases where the automorphism group is trivial or the quotient is a three-cone-point orbifold

Extends understanding of the field of moduli versus field of definition relationship

Abstract

Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let ${\rm Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) ${\rm Aut}(S)$ is trivial or (ii) $S/{\rm Aut}(S)$ is an orbifold of genus zero with exactly three cone points, then $S$ is definable over its field of moduli ${\mathcal M}(S)$. In the complementary situation, explicit examples for which ${\mathcal M}(S)$ is not a field of definition are known. We provide upper bounds for the minimal degree extension of ${\mathcal M}(S)$ by a field of definition in terms of the quotient orbifold $S/{\rm Aut}(S)$.