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Let $\overline{\mathbf{Q}}\subset \mathbf{C}$ be the field of algebraic numbers.

Does there exist a number field $K$ with the following property?

There are embeddings $\sigma,\tau:K\to \overline{\mathbf{Q}}$, and a smooth projective geometrically connected curve $X$ over $K$, such that $X_{\overline{\mathbf{Q}},\sigma}$ is not isomorphic to $X_{\overline{\mathbf{Q}},\tau}$ in the category of curves over $\overline{\mathbf{Q}}$.

If yes, the genus of $X$ has to be positive.

Is there a difference between the above question and the following question?

Are there embeddings $\sigma,\tau:K\to \mathbf{C}$, and a smooth projective geometrically connected curve $X$ over $K$, such that $X_{\mathbf{C},\sigma}^{an}$ is not isomorphic to $X_{\mathbf{C},\tau}^{an}$ in the category of Riemann surfaces?

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    @Soka, yes the $j$-invariant of $X_{\bar{\mathbb Q}, \sigma}$ is $\sigma(j)$.2011-11-29

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