Belyi's theorems in positive characteristic

Abstract

There are two types of Belyi's Theorems for curves defined over finite fields of characteristic p, namely the Wild and the Tame p-Belyi Theorems. In this paper, we discuss them in the language of function fields. In particular, we provide a constructive proof for the existence of a pseudo-tame element introduced in [Y. Sugiyama and S. Yasuda, Belyi's theorem in characteristic two, Compos. Math. 156(2) (2020) 325-339], which leads to a self-contained proof for the Tame 2-Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi's Theorems unlike the known ones that use technical results from Algebraic Geometry.