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This is a very naive question. Fix a prime $p$ and consider the forgetful map from varieties over $\mathbf{Q}$ to varieties over $\mathbf{Q}_p$. Is there a conjectural "purely $p$-adic" characterization of the image of this map, perhaps in terms of some property of the etale $\pi_1$?

Feel free to restrict to whatever subcategories you like in order for this question to make (more?) sense.

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    $\begingroup$ If we restrict to elliptic curves, the image is defined by rationality of $j$, but I don't know a way to distinguish that using the étale fundamental group. $\endgroup$
    – S. Carnahan ♦
    Commented May 23, 2011 at 2:32
  • $\begingroup$ Even the elliptic curve case, you still have to watch out for twists. Scott's answer is correct in this case because the natural map $\mathbb{Q}^{\times}/\mathbb{Q}^{\times \# \operatorname{Aut}(E)} \rightarrow \mathbb{Q}_p^{\times}/\mathbb{Q}_p^{\times \# \operatorname{Aut}(E)}$ is surjective. Maybe you could say something about why you think the etale fundamental group should have anything to do with this? $\endgroup$
    – Pete L. Clark
    Commented May 23, 2011 at 4:42
  • $\begingroup$ Pete: Because it's big and fancy? Presumably it knows a lot more than just the etale cohomology. I promise that I am completely without any intuition here. :) $\endgroup$
    – David Hansen
    Commented May 23, 2011 at 8:27
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    $\begingroup$ As pointed out above, it is very unlikely that the etale fundamental group can say much about the question. For example, blow up a curve in $\mathbf{P}^n_{\mathbf{Q}}$, $n> 2$, defined over $\mathbf{Q}_p$ but not definable over $\mathbf{Q}$. The resulting variety has trivial fundamental group but is not definable over $\mathbf{Q}_p$. A necessary condition would be that all the Galois representations on the etale chomology (or etale homotopy) groups of the variety extend to representations of $Gal_{\mathbf{Q}}$. This is unlikely to be sufficient either. $\endgroup$
    – naf
    Commented May 23, 2011 at 10:55

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I don't know what you think can happen. Take for example curves of genus at least two or some other class of varieties with a moduli space. Restrict yourself to the Zariski open where the varieties have no non-trivial automorphisms. Then the field of moduli is the field of definition, so a variety is defined over $\mathbb{Q}$ if and only if the corresponding point on moduli is. So you might as well be asking, given a variety defined over $\mathbb{Q}$, how do I distinguish its $\mathbb{Q}$-points among its $\mathbb{Q}_p$-points? Why would you do anything else other than looking at the coordinates? Now, if instead you look at adelic points, then there are other things you can do which may be in the direction you want, like descent obstructions and so on. Being a moduli space may help prove something but, ultimately, it's about points on a variety.

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  • $\begingroup$ @Felipe: probably you want the genus to be at least three to be compatible with the rest of your answer... $\endgroup$
    – Pete L. Clark
    Commented May 23, 2011 at 21:19

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