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 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.

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.

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.

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 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 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?

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 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 an 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.

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.

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.

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 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.

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.

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.

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 $\text{Gal}(\bar{K}/K)$ in $\text{Aut}(V)$ and $\mathfrak{g}$ be the Lie algebra of $G$.

Serre proved in Theorem 5:

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

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

If we assume the following hypotheses on $F$:

• $V$ is an semi-simple $G$-module
• $\text{End}(F)=\mathbb{Z}_p$
• The dimensions of $n_1$ and $n_0$ of $F$ and its dual group $F'$ are $\geq 1$ and coprime.

Then, $G$ is an open subgroup of $\text{Aut}(V)$.

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.

So I think the above Theorem 5 extends to higher-dimensional formal groups $F$ as well, provided $V=T_F \otimes C$ is semi-simple $G$-module or $\mathfrak{g}$-module.

However, not all formal groups a $p$-divisible groups. So we have to make sure the formal group $F$ is indeed a $p$-divisible group.

Thanks

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 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 an 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.

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.

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.