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?

  • $\begingroup$ No, this is Belyi's theorem. He is talking about unramified covers. $\endgroup$ – Felipe Voloch Jul 19 '13 at 1:06
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    $\begingroup$ No, Belyi's theorem is the converse to the OP's statement. (and, yes, I agree Deligne is discussing unramified covers) $\endgroup$ – John Pardon Jul 19 '13 at 1:10
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    $\begingroup$ OK. Belyi's theorem is usually stated as an if and only if. The easier direction was actually proved earlier by Weil. $\endgroup$ – Felipe Voloch Jul 19 '13 at 1:35
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    $\begingroup$ Check out this nice expose: arxiv.org/abs/math/0108222 $\endgroup$ – Igor Rivin Jul 19 '13 at 3:32
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    $\begingroup$ Are you asking "what does the actual proof look like" or "how does one know?" Because in practice, "the moduli space of Xes is zero-dimensional" is the same as "all Xes can be defined over number fields" though one has to be careful to massage this into a provable statement. E.G., rigid things like finite etale covers of a fixed curve (e.g. P^1 - 0,1,oo) or elliptic curves with complex multiplication are defined over number fields. $\endgroup$ – JSE Jul 19 '13 at 4:32

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".


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