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 $\ell^n$?

$\begingroup$ 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

5$\begingroup$ @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 selfdual (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. $\endgroup$– Kevin BuzzardMar 5, 2011 at 17:25

5$\begingroup$ If there were a simple criterion which told you precisely when a 2dimensional 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 TaniyamaShimuraWeil was provedand this didn't happen, so this is more evidence that the question may well not really admit a satisfactory answer. $\endgroup$– Kevin BuzzardMar 5, 2011 at 17:28

3$\begingroup$ By the way, there is also an article by Christian Liedtke, entitle "The ptorsion 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 ptorsion of elliptic curves over fields of characteristic p. $\endgroup$– Holger PartschMar 7, 2011 at 9:48
2 Answers
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 binilpotent 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 $A$.

$\begingroup$ @Holger: could you say where is the statement you quote from Oort's book ? $\endgroup$– Qing LiuMar 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

2$\begingroup$ The statement is at the beginning of section (15.4): Finite subgroups of abelian varieties $\endgroup$ Mar 8, 2011 at 8:30

3$\begingroup$ 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
I'm reminded of two papers by Maja Volkov, an erstwhile student of JeanMarc 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.