Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring in the inderminate $u$.

Due to Brueil and Kisin, there is an exact functor between the categories $BT^\phi_{/\Lambda}$ and $BT(O_K)$, $BT(O_K)$ is the category of p-divisible groups over $O_K$(i.e. every $G_n$ is a group scheme over $O_K$) and $BT^\phi_{/\Lambda}$ is the category of finite free $\Lambda$ module $A$ equipped with an injective semi-linear map $\phi: A\rightarrow A$ satisfying some conditions.

When $p>2$, this is an equivance of categroies and when $p=2$, this is an equivance up to isogeny.

Question: What is the motivation of classifying $p$-divisible groups? Does it have some applications on p-adic Hodge theory?

Thanks!

Remark: As classifying other math objects, we can use Tate and Orto's classifications of finte commutative group schemes to prove Mazur's torsion 13 theorem about elliptic curves, we can use classifications of finite simple groups to prove every finite group of odd order is solvable(maybe I am wrong) and we can use classifications of 2-dim manifolds to prove something in topology(I can't remember any examples). So I ask this question.

Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring in the inderminate $u$.

Due to Brueil and Kisin, there is an exact functor between the categories $BT^\phi_{/\Lambda}$ and $BT(O_K)$, $BT(O_K)$ is the category of p-divisible groups over $O_K$(i.e. every $G_n$ is a group scheme over $O_K$) and $BT^\phi_{/\Lambda}$ is the category of finite free $\Lambda$ module $A$ equipped with an injective semi-linear map $\phi: A\rightarrow A$ satisfying some conditions.

When $p>2$, this is an equivance of categroies and when $p=2$, this is an equivance up to isogeny.

Question: What is the motivation of classifying $p$-divisible groups? Does it have some applications on p-adic Hodge theory?

Thanks!

Remark: As classifying other math objects, we can use Tate and Orto's classifications of finte commutative group schemes of prime order to prove Mazur's torsion 13 theorem about elliptic curves, we can use classifications of finite simple groups to prove every finite group of odd order is solvable(maybe I am wrong) and we can use classifications of 2-dim manifolds to prove something in topology(I can't remember any examples). So I ask this question.

Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring in the inderminate $u$.

Due to Brueil and Kisin, there is an exact functor between the categories $BT^\phi_{/\Lambda}$ and $BT(O_K)$, $BT(O_K)$ is the category of p-divisible groups over $O_K$(i.e. every $G_n$ is a group scheme over $O_K$) and $BT^\phi_{/\Lambda}$ is the category of finite free $\Lambda$ module $A$ equipped with an injective semi-linear map $\phi: A\rightarrow A$ satisfying some conditions.

When $p>2$, this is an equivance of categroies and when $p=2$, this is an equivance up to isogeny.

Question: What is the motivation of classifying $p$-divisible groups? Does it have some applications on p-adic Hodge theory?

Thanks!

Let $k$ be a perfect field of characteristic $p>0$ and $W:=W(k)$ is the witt ring. Let $K$ be a totally ramified extension of $K_0:=W(\frac{1}{p})$ and $\Lambda:=W[[u]]$ is the formal series ring in the inderminate $u$.

Due to Brueil and Kisin, there is an exact functor between the categories $BT^\phi_{/\Lambda}$ and $BT(O_K)$, $BT(O_K)$ is the category of p-divisible groups over $O_K$(i.e. every $G_n$ is a group scheme over $O_K$) and $BT^\phi_{/\Lambda}$ is the category of finite free $\Lambda$ module $A$ equipped with an injective semi-linear map $\phi: A\rightarrow A$ satisfying some conditions.

When $p>2$, this is an equivance of categroies and when $p=2$, this is an equivance up to isogeny.

Question: What is the motivation of classifying $p$-divisible groups? Does it have some applications on p-adic Hodge theory?

Thanks!

Remark: As classifying other math objects, we can use Tate and Orto's classifications of finte commutative group schemes to prove Mazur's torsion 13 theorem about elliptic curves, we can use classifications of finite simple groups to prove every finite group of odd order is solvable(maybe I am wrong) and we can use classifications of 2-dim manifolds to prove something in topology(I can't remember any examples). So I ask this question.