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While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math.ias.edu/files/deligne/GaloisGroups.pdf) and are not working with someone who already understands it, then you need to look at the later literature on Dessins d'enfants. Good references are given in What are dessins d'enfants?, especially notice Leila Schneps - Dessins d'enfants on the Riemann Sphere.

While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math.ias.edu/files/deligne/GaloisGroups.pdf) and are not working with someone who already understands it, then you need to look at the later literature on Dessins d'enfants. Good references are given in What are dessins d'enfants?, especially notice Leila Schneps - Dessins d'enfants on the Riemann Sphere.

While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math.ias.edu/files/deligne/GaloisGroups.pdf) and are not working with someone who already understands it, then you need to look at the later literature on Dessins d'enfants. Good references are given in What are dessins d'enfants?

While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math.ias.edu/files/deligne/GaloisGroups.pdf) and are not working with someone who already understands it, then you need to look at the later literature on Dessins d'enfants. Good references are given in What are dessins d'enfants?, especially notice Leila Schneps - Dessins d'enfants on the Riemann Sphere.

User36938 gives several answers (and could post an answer if s/he wants to). Contrary to using Galois theory the question is really about algebraically closed fields $\mathbb{C,\overline{Q}}$ and has little to do with moduli fields. A structure defined by finitely many polynomials over $\mathbb{\overline{Q}}$ obviously has all the coefficients in some number field -- what user36938 generalizes under the name direct limit games. Stating the theorem in terms of dual numbers, user36938 can use Nakayama to show each point of the moduli space is etale over the algebraically closed field $\mathbb{\overline{Q}}$, so each point is just a copy of Spec($\mathbb{\overline{Q}}$), so each $X$ classified by the moduli space is defined over $\mathbb{\overline{Q}}$. There are serious details as user says, notably in defining the moduli space. But this outlines a proof.

While there are a lot of ideas in the comments, I think the upshot is that if you want to read Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” (http://www.math.ias.edu/files/deligne/GaloisGroups.pdf) and are not working with someone who already understands it, then you need to look at the later literature on Dessins d'enfants. Good references are given in What are dessins d'enfants?