Let $\Gamma\subseteq \Gamma'\subset SL\_2(\mathbb Z)$ be congruence subgroups, and 
$X(\Gamma)$, $X(\Gamma')$ be the associated smooth projective modular curves over $\mathbb C$. The inclusion $\Gamma\subseteq \Gamma'$ induces a (canonical) non-constant morphism $p:X(\Gamma)\to X(\Gamma')$ of curves over $\mathbb C$.


Question: 1) Suppose $N,M$ are positive integers with $N\mid M$. Then $\Gamma=\Gamma\_0(M)\subseteq \Gamma'=\Gamma\_0(N)$ and there exist canonical $\mathbb Q$-models 
$X$ and $X'$ of $X(\Gamma)$ and $X(\Gamma')$ respectively. Does there exist a $\mathbb Q$-morphism $p\_{\mathbb Q}:X\to X'$ whose base change to $\mathbb C$ is $p$?


2) More generally: Let $F$ be a number field such that $X(\Gamma)$ and $X(\Gamma')$ have models which are defined over $F$ (one can always find such a number field). Does there exist $F$-schemes $X$ and $X'$ whose base change to $\mathbb C$ (with respect to an embedding $\sigma:F\hookrightarrow \mathbb C$) are $X(\Gamma)$ and $X(\Gamma')$ respectively, and an $F$-morphism $p\_F:X\to X'$ such that the base change to $\mathbb C$ (with respect to $\sigma$) of $p\_F$ is $p$?



Field of definition of canonical morphism between (congruence) modular curves
Yes. Please see Theorem 7.1.3 of Katz-Mazur.


