On explicit descent of marked curves and maps

Jeroen Sijsling, John Voight
2016 Research in Number Theory  
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 D\'ebes 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
more » ... ned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.
doi:10.1007/s40993-016-0057-3 fatcat:q4rqwdi46zcuhfn3nmu3bpivdm