# Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$

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

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

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

So I don't know a good reference for the proof of $$E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$$.

• The isomorphism between the formal group (the right-hand side in your displayed equation) and the p-adic integers must be in Chapter 4 of the same book, towards the end. It may be false for p = 2, by the way.
– RP_
May 15 '20 at 15:52
• Crossposted at MSE. When cross-posting, it is important to link all versions of the question to prevent needlessly duplicating work. May 22 '20 at 23:52

As RP says, there's a chapter in The Arithmetic of Elliptic Curves that discusses formal groups, and in particular the points of a formal group defined over a complete local ring. The specific result that you want is Chapter IV, Theorem 6.4(b), in the special case that $$K=\mathbb Q_p$$ and $$R=\mathbb Z_p$$ and $$\mathcal M=p\mathbb Z_p$$. That theorem says that there the formal logarithm gives an isomorphism $$\log_{\mathcal F} : \mathcal F(\mathcal M^r) \longrightarrow \hat{\mathbb G}_a(\mathcal M^r),$$ provided that $$r$$ is an integer satisfying $$r>v(p)/(p-1)$$. For your case, $$v(p)=1$$, so the isomorphism is valid for all $$r\ge1$$ except, as noted by RP, when $$p=2$$, in which case you'll need $$r\ge2$$. And indeed, for $$p=2$$ you may need $$r\ge2$$, since there are formal groups over $$\mathbb Z_2$$ in which $$\mathcal F(2\mathbb Z_2)$$ has an element of order 2, hence it cannot possibly be isomorphic to the additive group.