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Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$ admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$. Is there an analogue of this theorem with $\mathbb{Q}$ replaced by a global function field (i.e. finite extension $\mathbb{F}_q(t)$)?

I am especially interested in the existence of a tamely ramified map.

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Have you seen Wushi Goldring's "Unifying themes suggested by Belyi's theorem"? See link.springer.com/content/pdf/… –  Ari Jun 8 '13 at 8:58
@Ariyan Javanpeykar Yes, I've seen it. It discusses the problem over (the algebraic closure of) finite fields and function fields in characteristic zero. It also discusses characteristic p in general but with wild ramification allowed, while I am more interested in tame ramification. –  Alex Jun 8 '13 at 11:07
(answering your comment below). I think that it is unlikely that this will work if you want the morphism to be tamely ramified. Galois coverings of the affine line in char. p, which are tamely ramified along $\infty$, are trivial. –  Damian Rössler Jun 8 '13 at 20:22
@Damian Rossler But I also allow ramification at 0,1. –  Alex Jun 8 '13 at 20:42
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1 Answer

up vote 5 down vote accepted

Much stronger results are available in positive characteristic. See

Kedlaya, Kiran S.

More étale covers of affine spaces in positive characteristic. J. Algebraic Geom. 14 (2005), no. 1, 187–192.

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What if I want the map to be tamely ramified? –  Alex Jun 8 '13 at 8:25
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