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 $$S/\mathrm{Aut}(S)$$.