finite flat commutative group schemes arising from Abelian varieties - MathOverflow

finite flat commutative group schemes arising from Abelian varieties

2011-03-05T13:10:04Z

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 $\ell^n$?

Answer by Holger Partsch for finite flat commutative group schemes arising from Abelian varieties

2011-03-05T17:03:40Z

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 $A$.

Answer by Chandan Singh Dalawat for finite flat commutative group schemes arising from Abelian varieties

2011-03-06T10:19:23Z

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.