Integral models of $p$-divisible groups

Question

Let $G$ be a $p$-divisible group over $\mathbb{Q}_p$. Suppose that $G_{\mathbb{C}_p}$ has a model over $\mathcal{O}_{\mathbb{C}_p}$.

Does $G$ have a model over $\mathbb{Z}_p$?

Answer 1

Just to get this off the unanswered list. This is false even if you replace $\mathbb{C}_p$ by a finite extension of $\mathbb{Q}_p$. When I want to think of concrete examples of $p$-divisible groups that are interesting (e.g. of height larger than $1$) then I often times think of abelian varieties.

In this case we can use the following well-known theorem of Grothendieck (see [1, Exp. IX, Thm. 5.13]):

Theorem(Grothendieck): Let $K$ be a $p$-adic field and $A$ an abelian variety over $K$. Then, $A$ has good reduction if and only if $A[p^\infty]$ has a good reduction.

So, in this light, a positive answer to your question would imply that an abelian variety with potentially good reduction has good reduction. But, of course, this is false:

Example: The elliptic curve $E:y^2=x^3+p$ over $\mathbb{Q}_p$ has additive bad reduction but $E_{\mathbb{Q}_p(\sqrt[3]{p})}$ has good reduction. So, $E[p^\infty]$ does not admit a model over $\mathbb{Z}_p$ but $E[p^\infty]_{\mathbb{Q}_p(\sqrt[3]{p})}$ admits a model over $\mathbb{Z}_p[\sqrt[3]{p}]$.

[1] A. Grothendieck, Groupes de monodromie en géometrie algébrique, LNM 288, 340, Springer–Verlag, New York, 1972-3.