Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” begins by remarking that when X is the projective line over the complex numbers, minus three points: "every finite cover of X can be described by equations with algebraic number coefficients."


see <http://www.math.ias.edu/files/deligne/GaloisGroups.pdf>


Is the proof something like the Hilbert irreducibility theorem? 


I mean is it like the following? For any cover given by a complex polynomial in two variables, the finitely many complex coefficients can be regarded as variables which can then be specialized to algebraic values which meet whatever rational polynomial conditions as the originals did while avoiding finitely many others, to give an isomorphic cover. Or will I waste my time if I try to formulate such conditions?



Algebraic numbers and the complex projective line minus three points
The comments are all correct. Deligne is citing one half of what is called Belyi's theorem, though experts feel this half was actually clear much earlier due to Weil. Weil used a version of the insight described above by JSE. B Kock gives a nice exposition in "<http://arxiv.org/abs/math/0108222>". 


