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I'm puzzled by a part of the construction of the canonical subgroup of a "not too supersingular" elliptic curve. In Katz's paper, one produces a subgroup of the formal group of the elliptic curve but to me it is not clear how that subgroup "algebraize" to a subgroup of the elliptic curve.

Our set up is the following: we work over $R_0$ a complete DVR of mixed characteristic ($0,p$), $R$ is a $p$-adically complete $R_0$-algebra and $E/R$ is an elliptic curve. If it helps we can suppose that $p$ is nilpotent in $R$.

  • First of all. What is the formal group Katz is considering? The completion of $E$ at its zero section or along the zero section mod $p$?

  • In Tate's paper on $p$-divisible groups an equivalence of categories between formal Lie groups and connected $p$-divisible groups is stated over complete local noetherian rings. Does it holds over general $p$-adically complete rings? I don't know how to use this result, because even if I can assume that $R$ is noetherian and complete (abelian schemes are always defined over a noetherian ring) I don't think it would be enough to work over the localizations at primes to produce the canonical subgroup.

Any idea? Thanks.

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  • $\begingroup$ You can have a look to this paper by Robert Coleman projecteuclid.org/download/pdf_1/euclid.ajm/1144070588 He computes an explicit equation of the canonical subgroup $\endgroup$
    – Ricky
    Commented May 18, 2016 at 16:15
  • $\begingroup$ I don't think my question is addressed there, but thanks anyway! $\endgroup$
    – Bear
    Commented May 18, 2016 at 16:24
  • $\begingroup$ First question, at the zero section of the original, not mod $p$. I do know a little about this topic, so I’d be happy to discuss things by e-mail. You may find my language impenetrably old-fashioned, though. $\endgroup$
    – Lubin
    Commented May 19, 2016 at 18:13

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