On explicit descent of marked curves and maps
Algebraic Geometry
2018-03-15 v3 Number Theory
Abstract
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 aforementioned results to the wildly ramified case. Moreover, we give explicit counterexamples for singular curves.
Keywords
Cite
@article{arxiv.1504.02814,
title = {On explicit descent of marked curves and maps},
author = {Jeroen Sijsling and John Voight},
journal= {arXiv preprint arXiv:1504.02814},
year = {2018}
}
Comments
35 pages