On Belyi's Theorem in positive characteristic

Abstract

The famous theorem of Belyi can be viewed as a characterization of compact Riemann surfaces which admit a non-empty open subset uniformized by a subgroup of $SL_2(\mathbb{Z})$ of finite index. I show that if $q\geq 5$, then ${\bf F}_q(T)$ is the one and only function field of positive characteristic for which such an analogous characterization of rigid analytic spaces of dimension one can exist and that Drinfel$'$d modular curves provide examples of rigid analytic spaces of this type.