This question is motivated by this one. In that question, there is stated without reference the fact that for an abelian variety over a field of characteristic $p$ (which I assume means the base field is algebraically closed), the only group schemes which appear as the $p$-torsion points are products of $\mathbb{Z}/p\mathbb{Z}$, $\mu_p$, and $\alpha_p$. I have two questions:

  1. Does anyone have a reference for this fact? (I do not believe it follows immediately from the fact that the only group schemes of order $p$ over $k = \bar{k}$ are the ones listed above)
  2. What is known for abelian schemes? Is there a classification of such group schemes?

The statement in the posting you make reference to is false. The $p$-torsion of a supersingular elliptic curve is a group-scheme, call it $\Gamma$, that fits into a nonsplitting exact sequence $$ 0\longrightarrow\alpha_p\longrightarrow\Gamma\longrightarrow\alpha_p\longrightarrow0 \,.$$

  • $\begingroup$ Thank you! Do you know of a good place to read about group schemes appearing in the $p$-torsion of abelian varieties/schemes or elliptic curves? $\endgroup$ – RghtHndSd Jan 25 '14 at 17:49
  • $\begingroup$ Sorry, but it’s been a long long time since I’ve thought about these things, I’m no longer familiar with current research, and I’ve never been comfortable in dimensions greater than one. You should find out about the Dieudonné theory of finite group schemes over a field, and just see what all you can find out. Don’t hesitate to e-mail me if you wish. $\endgroup$ – Lubin Jan 25 '14 at 20:32

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