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 $E\_1(\mathbb{Q}\_p)\cong \mathbb{Z}\_p$.



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


