Tamely ramified morphisms of curves and Belyi's theorem in positive characteristic

TL;DR

This paper proves that all smooth projective curves over finite fields have tame morphisms to the projective line, refines the tame Belyi theorem in positive characteristic, and constructs a counterexample over certain infinite fields.

Contribution

It establishes the existence of tame morphisms for curves over finite fields and refines Belyi's theorem in positive characteristic, also providing a counterexample over infinite perfect fields of characteristic two.

Findings

Every smooth projective curve over a finite field admits a tame morphism to the projective line.

A curve with no such tame morphism exists over an infinite perfect field of characteristic two.

Refinement of the tame Belyi theorem in positive characteristic.

Abstract

We show that every smooth projective curve over a finite field k admits a finite tame morphism to the projective line over k. Furthermore, we construct a curve with no such map when k is an infinite perfect field of characteristic two. Our work leads to a refinement of the tame Belyi theorem in positive characteristic, building on results of Sa\"idi, Sugiyama-Yasuda, and Anbar-Tutdere.