We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the presence of a smooth point, and in fact these results imply more generally that a marked curve descends to its field of moduli. We give a constructive version of their results, based on an algebraic version of the notion of branches of a morphism and allowing us to extend the aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.
CITATION STYLE
Sijsling, J., & Voight, J. (2016). On explicit descent of marked curves and maps. Research in Number Theory, 2(1). https://doi.org/10.1007/s40993-016-0057-3
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