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Can you classify all conditions for $p$-divisible groups arising from an abelian variety? For example, $2\dim = \mbox{height}$.

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  • $\begingroup$ I believe it's known, attributed to Serre and to Oort, that for an algebraically closed field k of characteristic p, the map from abelian varieties to quasi-isogeny classes of p-divisible groups which are self-dual (also understood up to quasi-isogeny) is essentially surjective. Beyond that, I don't know what's known; it'd be nice to hear something more nuanced. $\endgroup$
    – Eric Peterson
    Commented Jul 24, 2019 at 21:36

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