Let $K$ be a finite extension of $\mathbb{Q}_p$ with ring of integers $A$. Let $F$ be a $p$-divisible group and $T$ be the Tate module. Consider the vector space $V=T \otimes_{\mathbb{Q}_p} C$, where $C=\widehat{\bar{K}}$. 

Let $G$ be the image of the Galois group $\operatorname{Gal}(\bar{K}/K)$ in $\operatorname{Aut}(V)$ and $\mathfrak{g}$ be the Lie algebra of $G$.


[Serre](https://doi.org/10.1007/978-3-642-87942-5_10 "Sur les groupes de Galois attachés aux groupes \$p\$-divisibles") proved in **Theorem 5**:

> *If $F$ is a 1-dimensional formal group, then the image of $\operatorname{Gal}(\bar{K}/K)$ is open in $\operatorname{Aut}(V)$, where $V=T_pF \otimes C$.*

This is a consequence of **Theorem 4** of the same paper, which says:

> If we assume the following hypotheses on $F$:
> 
> a) $V$ is a semi-simple $G$-module.
>
> b) $\operatorname{End}(F)=\mathbb{Z}_p$.
> 
> c) The dimensions $n_1$ and $n_0$ of $F$ and its dual group $F'$ are $\geq 1$ and coprime.
> 
> Then, $G$ is an **open**
> subgroup of $\operatorname{Aut}(V)$.

**My question:**

Can we extend the Theorem 5 (the first result above) to formal groups of dimension more than $1$, provided enough conditions? 

---

Serre proved in **Proposition 8** that if $F$ is of dimension 1, then $V$ is a simple $\mathfrak{g}$-module. Thus, the *hypothesis* of Theorem 4 (the second result above) is automatically satisfied because $G$ is simple if its Lie algebra $\mathfrak{g}$ is simple.

However, not all formal groups are $p$-divisible groups. So we have to make sure the formal group $F$ is indeed a $p$-divisible group. According to [this post](https://math.stackexchange.com/questions/3153491/lubin-tate-formal-groups-are-p-divisible-groups#:~:text=Let%20A%20be%20a%20complete,be%20a%20formal%20group%20law.&text=is%20injective%20and%20makes%20A,F%20being%20p%2Ddivisible.) a formal group is a $p$-divisible group if and only if it is of finite height. 

So I think the above Theorem 5 extends to higher-dimensional formal groups $F$ as well, provided 

- $V=T_F \otimes C$ is a semi-simple $G$-module or $\mathfrak{g}$-module. 

- $\gcd(n,n')=1$, where $n$ and $n'$ are the dimensions of $F$ and its dual $F'$ respectively, as $p$-divisible groups.

Thus the conditions of Theorem 5 is satisfied. The answer of my question  is YES, under the two assumptions.

Am I doing correct?