# Field of definition/moduli through the monodromy

Let $$Y$$ be a smooth projective curve defined over a number field $$K$$, and let $$B$$ be a subset of $$Y(K)$$. It is known that the isomorphism class of a branched cover $$f:X(\Bbb{C})\rightarrow Y(\Bbb{C})$$ which is unramified outside $$B$$ is determined by its monodromy representation $$\rho:\pi_1(Y(\Bbb{C})-B)\rightarrow S_d$$ where $$d=\deg f$$. My question is what can be said about the field of definition or the field of moduli of the smooth projective curve $$X$$ based on the monodromy of $$f:X(\Bbb{C})\rightarrow Y(\Bbb{C})$$? Are there bounds on its degree as an extension of $$K$$? I know that the monodromy group is invariant under the action of $${\rm{Gal}}(\bar{K}/K)$$; so denoting the image of $$\rho:\pi_1(Y(\Bbb{C})-B)\rightarrow S_d$$ by $$H$$ (which is a transitive subgroup of $$S_d$$), the number of group isomorphisms $$\pi_1(Y(\Bbb{C})-B)\stackrel{\cong}{\rightarrow}H$$ should be a bound for the number of non-isomorphic morphisms in the Galois orbit $$\{f^\sigma:X^\sigma\rightarrow Y\}_{\sigma\in{\rm{Gal}}(\bar{K}/K)}$$. I wonder if anything more specific/stronger is known in the literature.

Update: Following the answer by @WillChen, I realized that the previous upper bound on the size of Galois orbit in my answer can be changed from the number of isomorphisms $$\pi_1(Y(\Bbb{C})-B)\stackrel{\cong}{\rightarrow}H$$ to that number divided by $$N_{S_d}(H)$$; because the monodromy, as a subgroup of $$S_d$$, is determined up to conjugacy -- there is no canonical way of numbering the elements of a regular fiber of $$f:X(\Bbb{C})\rightarrow Y(\Bbb{C})$$.

I also came across this article Fields of Definition of Some Three Point Ramified Field Extensions from 1994 which computes the field of definition in some examples with $$|B|=3$$.

Say $$Y$$ has genus $$g$$ and $$B$$ has cardinality $$n$$. Let $$\mathcal{M}_{g,n}$$ be the moduli stack of $$n$$-pointed genus $$g$$ curves. Say the cover $$f : X\rightarrow Y$$ has degree $$d$$ and monodromy group $$H\le S_d$$. There is a moduli stack $$\mathcal{H}$$ whose geometric points classify covers of $$n$$-pointed genus $$g$$ curves with monodromy group $$H$$. It's $$L$$-points (where $$L$$ is a number field) classify covers whose field of moduli is contained in $$L$$. This moduli stack comes with a map $$\mathcal{H}\rightarrow\mathcal{M}_{g,n}$$ which is defined over $$\mathbb{Z}[1/|H|]$$ and finite etale with geometric fiber in bijection with the set $$F := \text{Epi}(\pi_{g,n},H)/N_{S_d}(H)$$ where $$N_{S_d}(H)$$ is the normalizer of $$H$$ inside $$S_d$$ and $$\pi_{g,n}$$ is the fundamental group of an $$n$$-punctured genus $$g$$ surface. If you pick $$F$$ to be the geometric fiber over a $$K$$-rational $$(g,n)$$-curve $$Y$$, then you also get an action of the absolute Galois group $$G_K$$ on $$F$$ which is compatible with the action of $$G_K$$ on covers via the moduli interpretation. Thus you certainly have that the field of moduli of $$X$$ is contained in the field of moduli of the cover, whose degree is then bounded by the size of the corresponding $$G_K$$-orbit on $$F$$, which is of course bounded by $$|F|$$. Your degree can also be further bounded by noting that the objects of $$F$$ often contain covers with different values of certain Galois invariants, the simplest example being the $$K$$-rational conjugacy classes of generators of inertia at the branch points of $$Y$$, though depending on $$g,n,H$$ there are often a host of other more subtle invariants (e.g., the lifting invariant, the spin invariant, the braid invariant just to name a few).