# A reference for the proof of $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$

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 $$E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$$.

• The isomorphism is given at the beginning of the formal group chapter. – reuns May 15 '20 at 14:59
• @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 $\mathbb{E}_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$. – danihelovij May 15 '20 at 15:18
• Crossposted at MO. When crossposting, it is important to link all versions of the question to prevent needlessly duplicating work. – KReiser May 22 '20 at 23:52

• 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 $$r > \frac1{p-1}v(p)/v(\pi_K)$$)
• thank you. So it is false $E_1(\mathbb{Q}_p)\approx \mathbb{Z}_p$, the correct theorem is $E_1(\mathbb{Q}_p)\approx p\mathbb{Z}_p$? – danihelovij May 15 '20 at 16:44
• 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.. – reuns May 15 '20 at 16:46
• 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 $\mathbb{Z}_p$ – danihelovij May 15 '20 at 17:23
• $p \Bbb{Z}_p = \{ px,x\in \Bbb{Z}_p\}$ – reuns May 15 '20 at 17:24