2
$\begingroup$

I read an article where it is said: $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$

where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$.

The author says that the proof is in "Arithmetic of elliptic curves" by J.Silverman, at page 191, but here it is said:

If $E$ is an elliptic curve over $\mathbb{Q}_p$ and $\hat{E}$ is the formal group, then:

$$E_1(\mathbb{Q}_p)\approx \hat{E}(p\mathbb{Z}_p)$$

So I do not know a good reference for the proof of $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$.

$\endgroup$
3
  • 1
    $\begingroup$ The isomorphism is given at the beginning of the formal group chapter. $\endgroup$
    – reuns
    Commented May 15, 2020 at 14:59
  • $\begingroup$ @reuns i saw in formal group chapter the isomorphism ${E}_1(\mathbb{Q}_p)\approx \tilde{E}(p\mathbb{Z}_p)$, but dont see the isomorphism $\mathbb{E}_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$. $\endgroup$ Commented May 15, 2020 at 15:18
  • $\begingroup$ Crossposted at MO. When crossposting, it is important to link all versions of the question to prevent needlessly duplicating work. $\endgroup$
    – KReiser
    Commented May 22, 2020 at 23:52

1 Answer 1

2
$\begingroup$
  • At the beginning of the formal group law chapter Silverman shows how to define it from the Weierstrass equation and gives some hints why the obtained series $w\in \Bbb{Z}[[z]]$ will be useful when letting $z\in p\Bbb{Z}_p $ (that he calls $\mathcal{M}$) and that as set $E_{formal}(p \Bbb{Z}_p) = \{ (z,w(z)), z\in p\Bbb{Z}_p\}$

  • In the local field - reduction $\bmod \pi$ - chapter he shows that $E_1(\Bbb{Q}_p) \cong E_{formal}(p \Bbb{Z}_p)$ where $\cong$ is how we constructed the formal group and $E_1(\Bbb{Q}_p)= \{ (x,y) \in \Bbb{Q}_p^2,\not \in \Bbb{Z}_p^2, y^2=x^3+ax+b\} \cup O$ the points whose projective coordinate will be in $[p\Bbb{Z}_p:1:p\Bbb{Z}_p]$ whose reduction will be $[0:1:0]\in E(\Bbb{F}_p)$

  • The group isomorphism $E_{formal}(p \Bbb{Z}_p) \to p \Bbb{Z}_p$ is the formal logarithm, at the end of the formal group law chapter (for finite extensions $K/\Bbb{Q}_p$ the isomorphism is $E_{formal}(\pi_K^r O_K) \to \pi_K^r O_K$ where $r > \frac1{p-1}v(p)/v(\pi_K)$)

$\endgroup$
4
  • $\begingroup$ thank you. So it is false $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$, the correct theorem is $E_1(\mathbb{Q}_p)\approx p\mathbb{Z}_p$? $\endgroup$ Commented May 15, 2020 at 16:44
  • 1
    $\begingroup$ do you really think there is a difference? if you still don't understand how to construct $\Bbb{Z}_p$ then maybe all your questions are a bit.. $\endgroup$
    – reuns
    Commented May 15, 2020 at 16:46
  • $\begingroup$ Yes i am starting with that, i know $p\mathbb{Z}_p=\{x\in\mathbb{Q}_p:v_p(x)>0\}\subset \mathbb{Z}_p=\{x\in\mathbb{Q}_p:v_p(x)\geq 0\}$ and $p\mathbb{Z}_p$ is the unique maximal ideal of $\mathbb{Z}_p$ $\endgroup$ Commented May 15, 2020 at 17:23
  • 2
    $\begingroup$ $p \Bbb{Z}_p = \{ px,x\in \Bbb{Z}_p\}$ $\endgroup$
    – reuns
    Commented May 15, 2020 at 17:24

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .