In his comment to the question [Algebraic numbers and the complex projective line minus three points](https://mathoverflow.net/questions/137108/algebraic-numbers-and-the-complex-projective-line-minus-three-points#comment353577_137108) JSE says that for algebro-geometric objects defined over the complex numbers "in practice, 'the moduli space of Xes is zero-dimensional' is the same as 'all Xes can be defined over number fields'," so that for example "rigid things like finite etale covers of a fixed curve or elliptic curves with complex multiplication are defined over number fields."


Is it reasonable to fill that in by saying a zero dimensional moduli space sharply constrains the action on Xes of the absolute Galois group of the rationals, so that many field automorphisms of $\mathbb{C}$ are likely to leave a given X unchanged (up to isomorphism), and that large group of field automorphisms is likely to have a finite extension of $\mathbb{Q}$ as fixed field (the moduli field of that given X)? Does it even imply the orbit of each given X is finite?



Rigidity, moduli space, and moduli field
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?](https://mathoverflow.net/questions/1909/what-are-dessins-denfants), especially notice [Leila Schneps - Dessins d'enfants on the Riemann Sphere](http://people.math.jussieu.fr/~leila/Fschneps.pdf).


