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Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.

Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$?

By a Neron model, I mean a smooth model (not necessarily proper) with the "Neron universal property": for any smooth $O_K$-scheme $\mathcal Y$,

$$\mathrm{Hom}(\mathcal Y, \mathcal X) = X(Y_K).$$

Is the smooth locus of the minimal regular model of $X$ over $O_K$ a Neron model? Does base change help? That is, does there exist a Neron model for $X_L$ after some suitable base change $L/K$?

I'm also interested in knowing if we can just have the nice property $\mathcal{X}(O_K) = X(K)$ for some smooth model $\mathcal X$ of $X$.

Of course, all of this stuff is clear if $X$ has good reduction over $O_K$.

I posted this question on MathOverflow as suggested in the comments:

http://mathoverflow.net/questions/110359/do-all-curves-have-neron-models

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Take any proper regular model $\mathcal Z$ of $X$ over $O_K$. Then $\mathcal Z(O_K)\to X(K)$ is bijective by the valuative criterion of properness. As $\mathcal Z$ is regular, its sections land inside its smooth locus $\mathcal X$ and $\mathcal X(O_K)\to X(K)$ is bijective. When $\mathcal Z$ is the minimal regular model, it is probably true that $\mathcal X$ satisfies the universal Néron mapping property, you might post the question to Mathoverflow. –  user18119 Jul 1 '12 at 11:38
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