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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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For $X(N)$, one can give a $\mathbb{Q}$-model, but there's a sense in which doing so is "cheating".

The moduli interpretation of $X(N)$ only makes sense over $\mathbb{Q}(\zeta_N)$; if $R$ is an algebra over this field, then $X(N)$ classifies triples $(E, P_1, P_2)$ where $E$ is an elliptic curve over $R$ and $P_1, P_2$ are $R$-points of $E$ order $N$ which pair to $\zeta_N$ under the Weil pairing.

You can choose a model for $X(N)$ over $\mathbb{Q}$ in various ways, but you lose the moduli interpretation above. The usual way of doing this is such that the rational functions defined over $\mathbb{Q}$ are those whose $q$-expansions at the cusp $\infty$ have coefficients in $\mathbb{Q}$. This is done in, for instance, Stevens' book "Arithmetic on modular curves". With this choice, it is essentially tautological that the cusp $\infty$ is a $\mathbb{Q}$-point, and that the $j$-invariant map $X(N) \to X(1) \cong \mathbf{P}^1$ is defined over $\mathbb{Q}$. But this $\mathbb{Q}$-structure is rather artificial, and doesn't interact well with the moduli interpretation; e.g. sometimes one encounters modular curves $X$ which are quotients of $X(N)$, and which have a natural $\mathbb{Q}$-structure which is compatible with the moduli space interpretation of $X$, but where the map $X(N) \to X$ isn't defined over $\mathbb{Q}$ when we give $X(N)$ the $\mathbb{Q}$-structure described above. (See e.g. Elkies' paper on mod 3 and mod 9 representations of elliptic curves.)

I hope this at least partially answers your questions (1) and (3). I don't know the answer to (2); but for $N < 6$ we have $X(N) \cong \mathbf{P}^1$, so the automorphism group of $X(N)$ is infinite in these cases (in particular it is much larger than $PSL_2(\mathbf{Z} / N)$).

For $X_0(N)$ and $X_1(N)$ life is much easier than $X(N)$: the moduli space interpretations of these curves make sense over $\mathbb{Q}$, and they have rational models compatible with this moduli space structure, and the cusp $\infty$ is always a $\mathbb{Q}$-point.

(EDIT: Actually the last sentence is wrong, sorry! This works for $X_0(N)$ but not for $X_1(N)$. The canonical $\mathbb{Q}$-structure on $X_1(N)$ is the one that makes $\mathbb{Q}$-expansions at the cusp 0, not the cusp $\infty$, rational.)

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Are you saying $X(N)\rightarrow \mathbb{P}^1$ is always defined together with the group action over $\mathbb{Q}$? This surprises me. I thought $PSL_2(\mathbb{F}_p)$ is not known (for a general prime $p$) to be realized as a Galois group over $\mathbb{Q}$. I think Shih proved it was just for some of the primes.... Are you sure you didn't mean defined without the group action over $\mathbb{Q}$? That would make more sense. (P.S. Thanks a lot for the answer) – Nicole Sep 4 '11 at 20:55
I'm not sure I quite understand your question -- what do you mean by "defined without the group action"? – David Loeffler Sep 6 '11 at 11:40
The cover is defined, but it is not nec. Galois. – Nicole Sep 13 '11 at 0:11

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