A theorem of Lutz about the structure of the points of an elliptic curve over a finite extension of $\mathbb{Q}_p$

Question

Reading the article of Greenberg "Iwasawa Theory for Elliptic Curves", he cites (p.13) a theorem of Lutz that says:

Theorem: Let $E/K$ be an elliptic curve defined over a finite extension $K$ of $\mathbb{Q}_p$. Then

\begin{equation} E(K)\cong \mathbb{Z}_p^{[K : \mathbb{Q}_p]}\times U \end{equation}

as a group, with $U=E(K)_{tors}$ finite.

I couldn't find this theorem in books or on the internet. Anyone knows some references or a proof?

Answer 1

Silverman's Arithmetic of elliptic curves Chapter VII Proposition 6.3 states:

Let $K$ be a finite extension of $\mathbb{Q}_{p},$ so in particular $char (K)=0$ and $k$ is a finite field. Then $E(K)$ contains a subgroup of finite index that is isomorphic to $R^{+},$ the additive group of $R$.

Here $R$ is the ring of integers of $K$ and is therefore isomorphic to $\mathbf Z_p^{[K : \mathbf Q_p]}$ as a group.

The preprint https://arxiv.org/pdf/1703.07888.pdf has some more precise results it seems.