Tamely Ramified Morphisms of Curves and Belyi’s Theorem in Positive Characteristic

Abstract

AbstractWe 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 Saïdi, Sugiyama–Yasuda, and Anbar–Tutdere.