There is a nice generalization of Belyi's theorem in positive characteristic, proved by M. Saïdi in his paper Revêtements modérés et groupe fondamental de graphe de groupes. (Compositio Math. 107 (1997), no. 3), Théorème 5.6:

Let $C$ be a smooth projective curve defined over a field $K$ of characteristic $p>2$. The following conditions are equivalent:

• The curve $C$ can be defined over $\bar{\mathbf F}_p$,
• There exists a finite cover $C\to\mathbf P^1$ tamely ramified above $\infty,0$ and $1$ (and étale elsewhere).

The proof, very short and elegant, relies on a result of Fulton on the existence of covers with only double ramification. Unfortunately, the argument does not apply in characterisctic $2$, for which the question is still open (some recent progress in this situation can be found in S. Schröer'paper Curves with only triple ramification, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7).

There is a nice generalization of Belyi's theorem in positive characteristic, proved by M. Saïdi in his paper Revêtements modérés et groupe fondamental de graphe de groupes. (Compositio Math. 107 (1997), no. 3), Théorème 5.6:

Let $C$ be a smooth projective curve defined over a field $K$ of characteristic $p>2$. The following conditions are equivalent:

• The curve $C$ can be defined over $\bar{\mathbf F}_p$,
• There exists a finite cover $C\to\mathbf P^1$ tamely ramified above $\infty,0$ and $1$ (and étale elsewhere).

The proof, very short and elegant, relies on a result of Fulton on the existence of covers with only double ramification. Unfortunately, the argument does not apply in characterisctic $2$, for which the question is still open (some recent progress in this situation can be found in S. Schröer's paper Curves with only triple ramification, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7).