A remark on the field of moduli of Riemann surfaces

Abstract

Let $S$ be a closed Riemann surface of genus $g\ge 2$ and let $\mathrm{Aut}(S)$ be its group of conformal automorphisms. It is well known that if either: (i) $\mathrm{Aut}(S)$ is trivial or (ii) $S/\mathrm{Aut}(S)$ 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 $S/\mathrm{Aut}(S)$.

arXiv commentActualized version from the old one titled "The Field of Moduli of a Quadrangular Riemann surface"