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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 $\ell^n$?

flag	asked Mar 5 2011 at 13:10 norondion 4,230●6●34

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? – S. Carnahan♦ Mar 5 2011 at 15:18

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@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. – Kevin Buzzard Mar 5 2011 at 17:25

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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. – Kevin Buzzard Mar 5 2011 at 17:28

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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. – Holger Partsch Mar 7 2011 at 9:48

Holger Partsch · Answer 1 · 2011-03-08 08:30:59Z UTC

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 $A$.

Chandan Singh Dalawat · Answer 2 · 2011-03-06 10:19:23Z UTC

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.

finite flat commutative group schemes arising from Abelian varieties

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