finite flat commutative group schemes arising from Abelian varieties - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:47:28Z http://mathoverflow.net/feeds/question/57456 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57456/finite-flat-commutative-group-schemes-arising-from-abelian-varieties finite flat commutative group schemes arising from Abelian varieties norondion 2011-03-05T13:10:04Z 2011-03-08T08:30:59Z <p>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 $\ell^n$?</p> http://mathoverflow.net/questions/57456/finite-flat-commutative-group-schemes-arising-from-abelian-varieties/57472#57472 Answer by Holger Partsch for finite flat commutative group schemes arising from Abelian varieties Holger Partsch 2011-03-05T17:03:40Z 2011-03-08T08:30:59Z <p>Is the abelian scheme you consider a fixed one?</p> <p>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.</p> <p>For example, $\alpha_p \oplus \alpha_p$ cannot be embedded in an elliptic curve.</p> <p>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: $$0 \to G^{mult} \to G \to G^{et} \to 0$$</p> <p>then the orders of $G^{mult}$ and $G^{et}$ must be equal if $G$ is the $p$-torsion of an abelian scheme.</p> <p>On the other hand, in section (15.4) of the book "Commutative group schemes" of F. Oort, there is the following result:</p> <p>Every finite flat commutative group scheme is a subgroup scheme of some abelian variety $A$.</p> http://mathoverflow.net/questions/57456/finite-flat-commutative-group-schemes-arising-from-abelian-varieties/57553#57553 Answer by Chandan Singh Dalawat for finite flat commutative group schemes arising from Abelian varieties Chandan Singh Dalawat 2011-03-06T10:19:23Z 2011-03-06T10:19:23Z <p>I'm reminded of two papers by Maja Volkov, an erstwhile student of Jean-Marc Fontaine. They are </p> <p>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. </p> <p>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.</p>