Timeline for How do you calculate the group scheme of E[p] for a an elliptic curve E in characteristic p?
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10 events
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| Dec 19, 2014 at 13:53 | comment | added | Matthieu Romagny | In your first sentence, the p-torsion kernel $E[p]$ of a supersingular elliptic curve is not $\alpha_p$. Indeed, the former has order $p^2$. | |
| Aug 18, 2010 at 9:17 | comment | added | Torsten Ekedahl | A further remark on the extensions of $\mathbb Z/p$ by $\mu_p$; in general for an ordinary elliptic and by Serre-Tate theory (which maybe is earlier in dimension 1) the extension for $E[p]$ is (essentially) equal to the Kodaira-Spencer map so is usually non-trivial (for some definition of usually). | |
| Aug 18, 2010 at 7:16 | answer | added | Emerton | timeline score: 33 | |
| Aug 18, 2010 at 6:41 | comment | added | Max Flander | Thanks for the advice guys, I'll have a look at those chapters. | |
| Aug 18, 2010 at 6:32 | comment | added | BCnrd | Kevin: one small correction is that $p$-torsion commutative extensions of $\mathbf{Z}/(p)$ by $\mu_p$ over a field $k$ of char. $p > 0$ are classified by ${\rm{H}}^1_ {\rm{fppf}}(k, \mu_ p) = k^{\times}/(k^{\times})^p$, so if $k$ is imperfect and separably closed then this is non-trivial (and so such non-split extension structures exist: nice example is $p$-torsion on Tate curve over sep. closure of $\mathbf{F}_p((q))$, classified by $1 + q$, which is not a $p$-power). This is also discussed within Example A.8.3 in the book "Pseudo-reductive groups"... | |
| Aug 18, 2010 at 6:26 | comment | added | BCnrd | Kevin's advice is right (as usual). The question has the same flavor as trying to prove general theorems about Lie groups without exploiting Lie algebras, just bare hands. Bad idea; the reason Dieudonne modules were invented was exactly to render these kinds of questions straightforward to figure out by a little computation. You also seem to have a shaky grasp of finite flat group schemes; read Tate's article in the big FLT book to get a better grip on that. $p$-torsion in char. $p$ & $p$-div. groups are subtle things, don't expect to find "dumbed down" explanations: need some real theory. | |
| Aug 18, 2010 at 6:20 | comment | added | Kevin Buzzard | PS I think this isn't a real question. I don't think you have a precise question; I think your question is really just asking for a basic reference, and I don't know the answer to this. | |
| Aug 18, 2010 at 6:18 | comment | added | Kevin Buzzard | ...and ignores the non-reduced structure on $E[p]$ which presumably is precisely what you're asking about (I'm talking about your statement in the 2nd para). In the 3rd para you talk about $\alpha_p(C_p)$ but this doesn't make any sense because $\alpha_p$ is a group scheme in characteristic $p$ so you can't evaluate it at a field of characteristic zero (assuming $C_p$ is what I think it means). Unfortunately I can't remember how I learnt this stuff myself---perhaps from talking to my advisor :-/ Did you try Conrad's paper in the Boston Fermat proceedings? LEARN DIEUDONNE MODULES they're easy! | |
| Aug 18, 2010 at 6:14 | comment | added | Kevin Buzzard | What you say seems a bit confused to me. I think your claim in the ordinary case is only true if, for example, the base is a separably closed field. Is this what you mean by "characteristic p"? In the supersingular case the answer can't be $\alpha_p$ because $\alpha_p$, assuming you're using the same notation as the standard notation, is a group scheme of order $p$, and $E[p]$ has kernel of order $p^2$. In the supersingular case, over a sep closed field, $E[p]$ is a non-split extension of $\alpha_p$ by itself. I don't know what you mean by "$E[p]=Z/pZ$"; this is only true "on points"... | |
| Aug 18, 2010 at 6:06 | history | asked | Max Flander | CC BY-SA 2.5 |