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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.

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  • $\begingroup$ yes it helps us to define crystalline and semi-stable representations and in general p divisible groups naturally appear in many contexts so as any other important mathematical object it's helpful to classify them $\endgroup$
    – ali
    Commented Oct 20, 2019 at 21:05
  • $\begingroup$ @ali As I know, in p-adic hodge theory, we use Fontaine’ period rings $B_{st}$ and $B_{cris}$ to define crystalline and semi-stable represensations. There is also a equivalence of categories between all semi-stable $Q_p$ representations and filtered ($\phi$,N)-module, we can use this equivance to analyse semi-stable representations because objects in another category are objects in linear algebra. So I think there may exist some applications for classifying p-divisible groups. $\endgroup$
    – user141691
    Commented Oct 21, 2019 at 1:57
  • $\begingroup$ On your remark: The odd order theorem was a major step in the classification of finite simple groups, so using the classification to prove the odd order theorem seems rather circular. $\endgroup$
    – S. Carnahan ♦
    Commented Oct 21, 2019 at 15:31
  • $\begingroup$ @S.Carnahan Sorry, I just want to give a example, I don’t know much about the proof of the two complicated theorems... $\endgroup$
    – user141691
    Commented Oct 22, 2019 at 1:14

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