As the title says, if $\mathcal{A}$ is an abelian variety over $\mathbb{Q}_p$, is there a criterion as to if I should expect there to exist $A$ over $\mathbb{Q}$ such that $$\mathcal{A}\cong A\times_{\mathbb{Q}}\mathbb{Q}_p?$$ I'm not aware of even a necessary condition, sorry for the naive question. I assume this has no good answer, but I would like to check.
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1$\begingroup$ For an elliptic it is simply the question if $j(\mathcal{A})\in \mathbb{Q}$. $\endgroup$– Chris WuthrichCommented 5 hours ago
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3$\begingroup$ @ChrisWuthrich: This condition is necessary, but if $j(\mathcal{A})\in\mathbb{Q}$ I can only conclude, a priori, that there is $A$ over $\mathbb{Q}$ that becomes isomorphic to $\mathcal{A}$ over $\overline{\mathbb{Q}_p}$. $\endgroup$– Laurent Moret-BaillyCommented 5 hours ago
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$\begingroup$ You might be able to say something about Brauer-Manin obstructions for various moduli spaces of abelian varieties, especially modular curves. See math.mit.edu/~poonen/papers/heuristic.pdf $\endgroup$– Vik78Commented 4 hours ago
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$\begingroup$ @LaurentMoret-Bailly If $j(\mathcal{A})$ is in $\mathbb{Q}$ there is an elliptic curve $E/\mathbb{Q}$ such that $E \times \mathbb{Q}_p$ is a twist of $\mathcal{A}$. If $j\not \in \{0,1728\}$, there is a $D\in\mathbb{Q}_p^{\times}$ such that $\mathcal{A}$ is the quadratic twist $E^D$. Now approximate $D$ by a rational $D'$ such that $\mathbb{Q}(\sqrt{D}) = \mathbb{Q}(\sqrt{D'})$. Then $A=E^{D'}$ will do. Similar with sextic, cubic and quartic twists. $\endgroup$– Chris WuthrichCommented 37 mins ago
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1 Answer
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As phrased, this problem looks just as difficult as determining whether a given element of $\mathbb{Q}_p$ lies in $\mathbb{Q}$. There are real numbers which are unknown to be rational (e.g. $\pi + e$); I don't see why the $p$-adic case should be substantially easier.
In light of this, it is more natural to ask rather that you can approximate an abelian variety over $\mathbb{Q}_p$ by one over $\mathbb{Q}$ (by approximate, I mean that the reductions modulo $p^n$ are isomorphic for some large $n$).
This problem is considered in the following paper, at least for principally polarised abelian varieties:
Daniel Loughran, Gregory Sankaran- Rationality and arithmetic of the moduli of abelian varieties, https://arxiv.org/abs/2310.01244
The answer is: You can approximate with something over $\mathbb{Q}$ in dimensions 1,2,3. It is unknown if it is possible in dimensions $4,5,6$. It is not possible in dimensions at least $7$ assuming the Bombieri-Lang conjecture.
Without any kind of polarisation I really have no idea. It is difficult to attack such problems without a reasonable moduli space parametrising the objects of interest.
answered 3 hours ago
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$\begingroup$ Thanks, also that's a very nice paper! $\endgroup$ Commented 8 mins ago