Field of definition/moduli through the monodromy

Question

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 $|B|=3$.

Answer 1

Quick answer:

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).