# Deformations of p-divisible Groups

I’ve made the official decision to not do a proof of anything with the deformation theory of ${p}$-divisible groups, but now that I’ve motivated it I’ll still state the results. The proof is incredibly long and tedious. It should be interesting to look at what this functor is, since it probably isn’t what you think it is. We’ll construct the moduli functor, but it isn’t just ${p}$-divisible groups up to isomorphism or isogeny, but involves the notion of a quasi-isogeny.

A few definitions are needed. We’ll work in great generality, so let ${S}$ be a base scheme with your favorite properties. Our ${p}$-divisible groups will only be required to be fppf sheaves on ${S}$ (with the ${p}$-divisible group property). An isogeny ${f:G\rightarrow G'}$ of ${p}$-divisible groups is a surjection of sheaves with finite locally free kernel. An example is multiplication by ${p}$ since the kernel is ${G_1}$ which is by definition of a ${p}$-divisible group a finite locally free group scheme (of order ${p^h}$).

The group ${\mathrm{Hom}_S(G,G')}$ is a torsion-free ${\mathbb{Z}_p}$-module. We can make the sheaf version by taking the Zariski sheaf of germs of functions ${\mathcal{H}om_S(G,G')}$. A quasi-isogeny ${G\rightarrow G'}$ is a section ${\rho\in\mathcal{H}om_S(G,G')\otimes_\mathbb{Z} \mathbb{Q}}$ with the property that ${p^n\rho}$ is an isogeny for some integer ${n}$.

Now we have enough to write down the moduli functor that we want. We have everything over an algebraically closed (I think a descent argument allows us to do this all over a perfect field) field, ${k}$, of characteristic ${p>0}$ and ${W}$ its ring of Witt vectors. Consider the category ${\mathrm{Nilp}_W}$ of locally Noetherian schemes, ${S}$, over ${W}$ such that ${p\mathcal{O}_S}$ is locally nilpotent.

Fix a ${p}$-divisible group ${G}$ over ${\mathrm{Spec}(k)}$. Our moduli functor ${\mathcal{M}}$ is a contravariant functor from ${\mathrm{Nilp}_W}$ to the category of of pairs ${(H, \rho)_S}$, where ${H}$ is a ${p}$-divisible group over ${S}$ and ${\rho}$ is a quasi-isogeny ${G_{\overline{S}}\rightarrow H_{\overline{S}}}$. Then we mod out by isomorphism where an isomorphism ${(H_1, \rho_1)\rightarrow (H_2, \rho_2)}$ in this category is given by a lift to an isomorphism of ${\rho_1\circ \rho_2^{-1}}$, i.e. an isomorphism that commutes with the quasi-isogenies.

The theorem is that ${\mathcal{M}}$ is representable by a formal scheme formally locally of finite type over ${\mathrm{Spf}(W)}$. The way to prove representability is the usual way of finding particularly nice open and closed subfunctors. In our case, part of the breakdown is already in this post. Since the definition of quasi-isogeny involves an integer ${n}$ for which ${p^n\rho}$ is an isogeny, you can break ${\mathcal{M}}$ up using this integer. See the book Period Spaces for p-divisible Groups by Rapoport and Zink for more details.

It comes out in the proof that for the nice cases we were considering in the past two motivational posts we get that the functor is representable by ${\mathrm{Spf}(W[[t_1, \ldots, t_d]])}$ where ${d=\dim G \dim G^t}$. One really interesting consequence of this is that if ${G}$ has height ${1}$, then since ${\mathrm{ht}(G)=\dim G + \dim G^t}$ we have that one of ${\dim G}$ or ${\dim G^t}$ is ${1}$ and the other is ${0}$, so in either case the product is ${0}$ and we get that the deformation functor of ${G}$ is representable by ${\mathrm{Spf}(W)}$. In general, it is always smooth and unobstructed.

I’m not sure if I should continue on with ${p}$-divisible groups now that I’ve done this. Maybe I’ll go back to crystalline stuff or move on to something else altogether.