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I have an elliptic curve $E$ defined over a complete discrete-valued field $K$ of characteristc $0$. the residue field $k$ is of positive characteristic $p$.

Then $E[p]=\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$.

What can I say about the height of the formal group associated to $E$?

Seems that there isn't a definition of height of formal group in characteristic $0$.

However I have another question, the formal group law $F$ for $E$ has the coefficients in $\mathcal{O}_K$ so we can consider their reduction modulo the maximal ideal of $\mathcal{O}_K$. Now assume that $E$ has a good reduction $\tilde E$. Is the formal series obtained by reduction the coefficients of $F$ a formal group law on $\tilde E$ ?

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  • $\begingroup$ This isn't possible. The group scheme $E[p]$ is never étale, and so cannot be constant. In more simple terms, if $E[p]$ is as stated, then $[p]:E_k\to E_k$ would be separable, but it's not. $\endgroup$ Aug 11, 2015 at 10:34
  • $\begingroup$ is the height of a formal group defined only in positive characteristic? $\endgroup$
    – user261123
    Aug 11, 2015 at 11:10
  • $\begingroup$ As far as I know, yes. Usually one defines the height of a formal group $\mathbb{G}$ to be the $h$ such that $[p]$ is a powe series in $T^{p^h}$. The proof that such a thing exists (besides for the additive group) uses the Frobenius. What even does the height mean if you're not in positive characteristic? Regardless, you can't have an elliptic scheme over $\mathbb{Z}_p$ with $E[p]$ isomorphic to $\underline{\mathbb{Z}/p\mathbb{Z}}\times\underline{\mathbb{Z}/p\mathbb{Z}}$. $\endgroup$ Aug 11, 2015 at 11:21
  • $\begingroup$ I know that in characteristic $p$ the $p$-torsion points group is not of this form but $K$ is a field of characteristic $0$. I got your reasoning about the height thanks! $\endgroup$
    – user261123
    Aug 11, 2015 at 11:25
  • $\begingroup$ Oh, I apologize. I thought that you were assuming that you had $E/\mathcal{O}_K$—because I thought you were reducing $E$ over $k$. I see now that you were asking about some notion of 'height' over $K$, but yeah, I don't think that exists. $\endgroup$ Aug 11, 2015 at 11:32

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If a formal group is defined over a $p$-adic ring $\mathscr O$, with residue field $k$ of characteristic $p$, then there certainly is a pretty standard definition of the height of the formal group. It’s just the height of the $k$-series. All of this is in the very old text of Fröhlich, I think. As you expect, the height of the formal group of an elliptic curve is $1$, $2$, or $\infty$, but always finite if the reduction is good.

I’m not sure about your second question. What is “a formal group on $\tilde E\,$” ? It’s certainly true, though, that the processes (1) of taking reduction modulo the maximal ideal and (2) localization to get the formal group commute with each other.

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