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Let $E/ \Bbb{Q}_p$ be an elliptic curve over $ \Bbb{Q}_p$. $\hat{E}$ denote the corresponding formal group of $E$. I want to know what $\Bbb{Q}_p(\hat{E}[p])/\Bbb{Q}_p$ is .

At first I tried to prove $\hat{E}$ is Lubin Tate formal group, but it wasn't. I might be missing some basic approach to this problem.

P.S At first I wrote it is degree $p-1$ extension, but Chris Wuthrich pointed out that it does not hold in general(when $E$ is super singular, it is true), and I edited.

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  • $\begingroup$ That edit made my comment absurd, so I delete it. $\endgroup$
    – Chris Wuthrich
    Commented Aug 3, 2022 at 19:20
  • $\begingroup$ Sorry to bother ,but could you tell me why $[p]t＝0,t≠0$ implies extension degree is a divisor of $p-1$(Where did you used the condition, $E$ is super singular at $p$ ?) $\endgroup$
    – Duality
    Commented Aug 3, 2022 at 19:20
  • $\begingroup$ The equation $[p](t) = 0$ defines you the extension. You can factor the power series as $t\cdot f(t) \cdot u$ with $f$ a polynomial and $u$ a unit. Sagemath can calculate that easily. $\endgroup$
    – Chris Wuthrich
    Commented Aug 3, 2022 at 19:21
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    $\begingroup$ I suggest you read more on the formal group of elliptic curves. Section 1.11 in Serre's "Propriétés galoisiennes des points d'ordre fini des courbes elliptiques" is a good read. $\endgroup$
    – Chris Wuthrich
    Commented Aug 3, 2022 at 19:29
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    $\begingroup$ Your P.S. is wrong. I did not say that it holds for supersingular curves. $\endgroup$
    – Chris Wuthrich
    Commented Aug 3, 2022 at 20:04

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