Belyi's theoerm in characteristic two

TL;DR

This paper establishes an analogue of Belyi's theorem in characteristic two by introducing pseudo-tame morphisms, proving their existence, and constructing tamely ramified functions, expanding understanding of algebraic curves in this characteristic.

Contribution

It introduces the concept of pseudo-tame morphisms in characteristic two and demonstrates their role in constructing tamely ramified functions, filling a gap in algebraic geometry.

Findings

Existence of pseudo-tame rational functions proven.

Construction of tamely ramified functions from pseudo-tame functions.

Extension of Belyi's theorem to characteristic two.

Abstract

We prove an analogue of Belyi's theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called "pseudo-tame" for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a "pseudo-tame" rational function by proving vanishing of an obstruction class. Finally we will construct a tamely ramified rational function from the "pseudo-tame" rational function.