Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $\ell^n$?

share|improve this question
    
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 '11 at 15:18
3  
@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 '11 at 17:25
3  
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 '11 at 17:28
2  
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 '11 at 9:48
add comment

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 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 $A$.

share|improve this answer
    
@Holger: could you say where is the statement you quote from Oort's book ? –  Qing Liu Mar 6 '11 at 22:24
    
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. –  Holger Partsch Mar 7 '11 at 9:44
1  
The statement is at the beginning of section (15.4): Finite subgroups of abelian varieties –  Holger Partsch Mar 8 '11 at 8:30
    
Thanks Holger ! –  Qing Liu Mar 8 '11 at 21:36
2  
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. –  Matthieu Romagny Mar 25 '11 at 22:02
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.