Belyi's theorems in positive characteristic

Official URL: http://dx.doi.org/10.1142/S1793042120500712

There are two types of Belyi’s Theorem 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 [13], which leads to a self-contained proof for the Tame 2-Belyi Theorem. Moreover, we provide unified and simple proofs for Belyi Theorems unlike the known ones that use technical results from Algebraic Geometry.