In the paper Unifying themes suggested by Belyi's theorem from 2011, the following question is raised:
Let $X$ be a projective non-singular curve over the function field $K:=\overline{\Bbb{Q}}(t)$. Does there exist a morphism $\phi:X\rightarrow\Bbb{P}^1(K)$ which is ramified over at most four points?
I wonder what is the state of the art on this type of questions. To give a context, unlike Belyi's theorem over $\overline{\Bbb{Q}}$, here one cannot take the number of branch points to be three because then the curve will admit a model over $\overline{\Bbb{Q}}$ rather than $\overline{\Bbb{Q}}(t)$. There is also a positive characteristic version discussed both in the aforementioned paper and in this MO question. But I am interested only in the characteristic zero situation.
