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A remark on the field of moduli of Riemann surfaces

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

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The author would like to express his deep gratitude to the referee for supplying very useful comments, suggestions, and corrections.

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Correspondence to Rubén A. Hidalgo.

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Hidalgo, R.A. A remark on the field of moduli of Riemann surfaces. Arch. Math. 114, 515–526 (2020).

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