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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

characteristic 2.

Joint work with Daniel Litt (Georgia) and Jakub Witaszek (Michigan).