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 $p$?

whichdescents!), the answer can be positive or negative. For example, for $\Gamma_0(N)$ I could choose the moduli problem based on embeddings of $\mu_N$ and for $\Gamma_0(M)$ I could choose the one based on embeddings of $\mu_M$ or of $\mathbf{Z}/M\mathbf{Z}$. These all give moduli schemes over $\mathbf{Q}$. For the first choice of the latter the answer is affirmative, and for the 2nd choice of the latter the answer is negative (when $N > 2$). – user29283 May 22 '13 at 1:43