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Let $X$ be a smooth projective geometrically connected curve over a number field $K$.

Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$.

Can we conclude that $X\times_{K,\tau} \mathbf{C}$ is a modular curve for ALL $\tau:K\to \mathbf{C}$?

I'm asking this question out of pure curiosity. I see no reason why all base changes of $X$ to $\mathbf{C}$ should be modular provided one of them is. Then again, I wouldn't know how to construct a counter example. Probably you can do something with elliptic curves.

A modular curve is (to me) an algebraic curve isomorphic to the compactification of $\Gamma\backslash \mathbf{H}$ for some congruence subgroup $\Gamma \subset \mathrm{SL}_2(\mathbf{Z})$.

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up vote 8 down vote accepted

You are asking if any conjugate of a modular curve is again a modular curve, and the answer is yes. This is a very special case of the general theory of conjugation of Shimura varieties, which says that any algebraic conjugate of a Shimura variety is again a Shimura variety, but which in this case can be verified directly.

Firstly, just to explain why I say that you are asking about algebraic conjugates of modular curves: modular curves are defined over $\overline{\mathbb Q}$, so one can replace $\mathbb C$ by $\overline{\mathbb Q}$ in the question (because if two smooth projective curves, both defined over an algebraically closed field $k$ --- in this case $\overline{\mathbb Q}$ --- become isomorphic over a larger algebraically closed field $\Omega$ --- in this case $\mathbb C$ --- then they are already isomorphic over $k$).

Now, regarding fields of definition, we can say more:

The modular curve $X(N)$ is defined over $\mathbb Q(\zeta_N)$, and it has a natural action of $SL_2(\mathbb Z/N)$ which is also defined over that field.

Let $\sigma_a$ denote the automorphism of $\mathbb Q(\zeta_N)$ which maps $\zeta_N$ to $\zeta_N^a$, for $a \in (\mathbb Z/N)^{\times}$. Then one can show that the conjugate of $X(N)$ by $\sigma_a$ is again isomorphic to $X(N)$, and that this isomorphism can be chosen so that it takes the action of $\gamma \in SL_2(\mathbb Z/N)$ to the action of $\begin{pmatrix} a & 1 \\0 & 1\end{pmatrix} \gamma \begin{pmatrix} a^{-1} & 1 \\ 0 & 1 \end{pmatrix}.$

Now any modular curve $X$ is a quotient of $X(N)$ for some level $N$ by some subgroup $H$ of $SL_2(\mathbb Z/N)$, and the preceding paragraph shows that it is also defined over $\mathbb Q(\zeta_N)$, with its conjugate $X^{\sigma_a}$ being isomorphic to the modular curve obtained by taking the quotient of $X(N)$ by the conjugated subgroup $\begin{pmatrix}a & 0 \\ 0 & 1 \end{pmatrix} H \begin{pmatrix} a^{-1} & 0 \\ 0 & 1\end{pmatrix}.$

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