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I'm just now beginning to learn about descending the curves $X(N)$ to $\mathbb{Q}$, and I have a few questions:

  1. Does $X(N)$ have a $\mathbb{Q}$-point for every $N$?

  2. What is $\operatorname{Aut}(X(N))$? I know that $\operatorname{PSL}_2(\mathbb{F}_p)\leq \operatorname{Aut}(X(p))$ for $p$ prime, but can we say more? (for example, is this an equality for $p$ prime; what can we say if $p$ isn't prime?)

  3. Does the map $X(N)\rightarrow \mathbb{P}^1_{\mathbb{C}}$ together with the Galois action descend to $\mathbb{Q}$? How about without it?

  4. Do the curves $X_0(N)$ (and curves given by congruence groups in general) also descend to $\mathbb{Q}$? If so, I ask the above questions for those as well.

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