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.
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.
In a comment on the accepted answer, the OP asks about tame ramification. Saidi shows in Theorem 5.6 here that a smooth projective curve $C$ over a field $k$ of characteristic $p>2$ is defined over $\overline{\mathbb{F}_p}$ if and only if $C$ admits a map $C\to \mathbb{P}^1$ with only tame ramification over $\{0,1,\infty\}$.
The proof is rather easy. I'll first sketch the argument that a curve defined over $\overline{F_p}$ admits a map as claimed. A result of Fulton shows that any curve admits a map $g: C\to \mathbb{P}^1$ with ramification indices at most two, hence a tamely ramified map if the characteristic is different from $2$. Let $q$ be such that the ramification values are defined over $\mathbb{F}_q$. Then composing $g$ with the map $$z\mapsto z^{q-1}$$ gives a map $C\to \mathbb{P}^1$ with the desired property.
To see the other direction, one may use Riemann existence or just observe that maps with the desired property don't deform and the moduli space of such maps is locally finite type, hence any $k$-point comes from an $\overline{\mathbb{F}_p}$-point.
