If an algebraic curve over a field of characteristic 0 admits a finite
map to the projective line ramified only over three points, then it must
be definable over some number field. This fact has a famous converse due
to Belyi: any curve over a number field admits such a finite map over
its field of definition.
Similarly, if an algebraic curve over a field of characteristic p>0
admits a finite *tamely ramified* map to the projective line ramified
only over three points, then it must be definable over some finite
field. We prove the converse: any curve over a finite field admits such
a finite map over its field of definition.
A construction of Saidi shows that this reduces to the existence of a
single tamely ramified map. This is easy to establish over an infinite
field of odd characteristic, and only slightly harder (using
Poonen-style probabilistic techniques) over a finite field of odd
characteristic. To handle the case of a finite field of characteristic
2, we use a construction of Sugiyama-Yasuda that they used to establish
existence of tame morphisms over an algebraically closed field of
Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).