A remark on the field of moduli of Riemann surfaces

摘要

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