Consider the Belyi's theorem:
If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$ branch points.
For the proof, one takes a morphism $f:X\longrightarrow \mathbb P^1(\mathbb C)$ defined over $\overline{\mathbb Q}$ and then uses some "techniques" to reduce the branch locus to a set of three points. These "techniques" involve only morphisms from $\mathbb P^1(\mathbb C)$ to $\mathbb P^1(\mathbb C)$ so they don't use the fundamental hypothesis "$X$ is defined over $\overline{\mathbb Q}$". Now I don't understand where this hypothesis is really used, maybe in choosing the morphism $f$?
Look for example at the following excerpts taken respectively from Serre's "Lectures on the Mordell-Weil theorem" and Szamuely's "Galois Groups and Fundamental Groups":
My "conjecture": If a curve is defined over $\overline{\mathbb Q}$ then we can construct a morphism to $\mathbb P^1(\mathbb C)$ defined over $\overline{\mathbb Q}$. However this point is not clear to me.
Thanks in advance
First of all, of course if $X$ is defined over $\overline{\mathbb Q}$ we can construct a morphism to $\mathbb P^1$ defined over $\overline{\mathbb Q}$ --- simply take any non-constant element of the rational function field of $X$ over $\overline{\mathbb Q}$.
Secondly, since there are curves that cannot be defined over $\overline{\mathbb Q}$, and so by Belyi these curves do not admit morphisms to $\mathbb P^1$ ramified only over three points. So certainly, the reduction of the branch locus in $\mathbb P^1$ to three points uses the hypothesis that the curve is defined over $\overline{\mathbb Q}$.