Belyi's theorem says that if $X$ is a curve defined over a number field $K$ then $X\_{\overline{K}}$ admits a map to $\mathbb{P}^1\_{\overline{K}}$ which at most 3 branch points. Must there also exist a map to $\mathbb{P}^1\_K$ defined over $K$?


This may be a slightly lazy question, but I figure it's probably got a quick answer.



Does there exist a curve defined over a number field $K$ such that every map to $\mathbb{P}^1_K$ is ramified over at least 4 points?
To record nfdc23's answer:


The following is true:


If $X$ is a curve over a number field $K$, then it admits a map to $\mathbb{P}^1\_K$ defined over $K$ branched over at most 3 points.


This follows from Theorem 2.5 in Mochizuki's 2004 paper [here](http://www.kurims.kyoto-u.ac.jp/~motizuki/Noncritical%20Belyi%20Maps.pdf).


