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How are the finite flat group schemes $\mathcal{A}[\ell^n]$ arising from an Abelian scheme $\mathcal{A}/S$ singled out among other finite flat commutative group schemes of exponent $\ell^n$?

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  • $\begingroup$ Do you happen to have an example of a finite flat group scheme of exponent $\ell^n$ that doesn't come from the $\ell^n$-torsion in an abelian variety? $\endgroup$
    – S. Carnahan
    Mar 5, 2011 at 15:18
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    $\begingroup$ @Scott: a group scheme killed by $\ell$ won't be the $\ell$-torsion in an abelian variety if it has order $\ell^n$ with $n$ odd. And there are other obstructions other than this: e.g. a group scheme of order $\ell^2$ and killed by $\ell$ won't be the $\ell$-torsion in an elliptic curve if it's not self-dual (by the Weil pairing); but even this isn't enough; e.g. $\alpha_\ell\times\alpha_\ell$ isn't the $\ell$-torsion in a characteristic $\ell$ elliptic curve either, because of a formal group argument. My gut feeling is that there is unlikely to be a simple neat criterion. $\endgroup$ Mar 5, 2011 at 17:25
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    $\begingroup$ If there were a simple criterion which told you precisely when a 2-dimensional mod $\ell$ Galois representation of the absolute Galois group of the rationals were the $\ell$-torsion in an elliptic curve, then we would have known lots of new cases of Serre's conjecture the moment Taniyama-Shimura-Weil was proved---and this didn't happen, so this is more evidence that the question may well not really admit a satisfactory answer. $\endgroup$ Mar 5, 2011 at 17:28
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    $\begingroup$ By the way, there is also an article by Christian Liedtke, entitle "The p-torsion subgroup scheme of elliptic curves" arxiv.org/abs/0904.1307 in which he studies what kind of twisted forms of $\mathbb Z/p \oplus \mu_p$ can be realized as p-torsion of elliptic curves over fields of characteristic p. $\endgroup$ Mar 7, 2011 at 9:48

2 Answers 2

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Is the abelian scheme you consider a fixed one?

If the base is of characteristic $p$, and $l = p$ then the Lie algebra of $A[l]$ is isomorphic to $Lie(A)$, so you get a dimension condition.

For example, $\alpha_p \oplus \alpha_p$ cannot be embedded in an elliptic curve.

There are also more general condition your group scheme $G$ should satisfy. Assume that $S$ is the spectrum of an Artinian algebra and that $G$ is $p$-torsion with a trivial bi-nilpotent part. We have an exact sequence: $$ 0 \to G^{mult} \to G \to G^{et} \to 0$$

then the orders of $G^{mult}$ and $G^{et}$ must be equal if $G$ is the $p$-torsion of an abelian scheme.

On the other hand, in section (15.4) of the book "Commutative group schemes" of F. Oort, there is the following result:

Every finite flat commutative group scheme is a subgroup scheme of some abelian variety $A$.

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  • $\begingroup$ @Holger: could you say where is the statement you quote from Oort's book ? $\endgroup$
    – Qing Liu
    Mar 6, 2011 at 22:24
  • $\begingroup$ Excuse me for not giving the exact reference. Today I am nowhere near the library so I cannot fix it right now, but it's on my list for tomorrow. $\endgroup$ Mar 7, 2011 at 9:44
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    $\begingroup$ The statement is at the beginning of section (15.4): Finite subgroups of abelian varieties $\endgroup$ Mar 8, 2011 at 8:30
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    $\begingroup$ Here is a related result, due to Raynaud : every finite locally free commutative group scheme $G$ over a scheme $S$ can be embedded into a projective abelian scheme, Zariski locally over $S$. This is in Berthelot, Breen, Messing, Théorie de Dieudonné cristalline, th. 3.1.1. $\endgroup$ Mar 25, 2011 at 22:02
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I'm reminded of two papers by Maja Volkov, an erstwhile student of Jean-Marc Fontaine. They are

MR2148801 (2006a:14027) Volkov, Maja A class of $p$-adic Galois representations arising from abelian varieties over $\Bbb Q_p$. Math. Ann. 331 (2005), no. 4, 889–923.

MR1837096 (2002d:11067) Volkov, Maja Les représentations $l$-adiques associées aux courbes elliptiques sur ${\Bbb Q}_p$. J. Reine Angew. Math. 535 (2001), 65–101.

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