On Belyi's Theorem in positive characteristic
Kirti Joshi
TL;DR
This work develops a function-field analogue of Belyi's theorem in positive characteristic using Drinfeld modular curves as rigid-analytic uniformizations. It shows that among data $(F,v,B)$ with $F$ a function field over ${\mathbb F}_q$ and $q\ge 5$, the triple $({\mathbb F}_q(T),v,{\mathbb F}_q[T])$ is unique in supporting a natural Belyi-type rigidity for a distinguished, countable set of groups, connected to Drinfeld curves. The author proves a rigidity theorem (arith-rigidity) and derives a corollary (unique-belyi) stating that Drinfeld modular curves $X_{\Gamma}$ realize morphisms to ${\mathbb P}^1$ unramified outside a fixed cusp set $\mathcal{B}$ with $|\mathcal{B}|=q+1$, and these curves are defined over finite separable extensions of ${\mathbb F}_q(T)$. The results highlight a precise, rigid function-field analogue of Belyi’s theorem, clarify the role of $q\ge 5$, and contrast with $q=2$ where ramification phenomena differ.
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.