Let X be a smooth, projective curve over the algebraic closure of ℚ. Let f:X->ℙ1 be a meromorphic function. Assume that the zeros and the poles are defined over some number field, K. Then does this imply that cf is defined over K, for some c?
If so, do we have to assume that the zeros and poles are individually defined over K, or would this work if they are collectively defined over K as well?
By being collectively defined I mean that there's some K-model of X, XK, and a closed subscheme YK of XK, such that after base change YK becomes the ramification locus of f.
Let D:=(f). Obviously, H0(O(D),X) is 1 dimensional. DK:=(fK) will also be degree 0, so all that's left to show is that H0(O(DK),XK) is also nonzero. This smacks of some invariance of cohomology theorem. I couldn't quite find the right one to use.
If this line of argument works, this seems to imply that the ramification locus may be defined merely collectively.