More Motivation for p-Divisible Groups

Perhaps the last post didn’t provide you with enough motivation to understand deformations of {p}-divisible groups. Today we’ll look at a much more general situation where it is again useful to know the deformation theory. Recall the general setup of trying to understand a class of objects. Say abelian varieties of dimension {g}. You probably want to form some sort of moduli space, so you usually say what the points of the space are and then try to prove that these points form some sort of scheme or stack that isn’t too horrible.

Often times it is horrible, though, so there is another trick to try to understand what is going on. You look at the deformations of some object. This will tell you what is happening in some formal local neighborhood of that object on the moduli space. This means that deformations actually are useful for understanding classification. For instance, all K3 surfaces can be deformed to eachother (analytically, but not algebraically), so this means that as manifolds they are all diffeomorphic, but they are not algebraically equivalent.

Here is the motivation. Recall that one of the easiest examples of a {p}-divisible group is that whenever you have an abelian variety {A} of dimension {g}, you can take the {p^\nu}-torsion points {A[p^\nu]=\mathrm{ker}(A\stackrel{p^\nu}{\rightarrow} A)}. Set {G_{\nu}=A[p^\nu]}, then you have a height {2g} {p}-divisible group {\lim G_{\nu}=A[p^\infty]} (as long as the characteristic of {k} is relatively prime to {p} otherwise the height is different, but you still get a {p}-divisible group).

Here is an absolutely beautiful result of Serre and Tate: Fix some Artin local {k}-algebra {R} and let AbSch({R}) be abelian schemes over {R} and BT({R}) the category with objects {(X,G,\epsilon)} where {X} is an abelian variety over {k}, {G} is a {p}-divisible group over {R} and {\epsilon} is a choice of isomorphism of the special fiber of {G} with the {p}-divisible group associated to {X}, so {\epsilon: G\otimes_R k\stackrel{\sim}{\rightarrow} X[p^\infty]}. The result is that there is an equivalence of categories {F:\mathrm{AbSch}(R)\rightarrow \mathrm{BT}(R)} given simply by {F(A)=(A\otimes_R k, A[p^\infty], \epsilon)}.

In particular, for an abelian variety {X} we get an isomorphism of deformation functors {Def_X\rightarrow Def_{X[p^\infty]}}. So if you want to understand the deformation theory of abelian varieties you could try to understand the deformations of {p}-divisible groups. This is the topic of the great book by Messing The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes. Might I point out that whenever I find a book great and useful it seems to always have been moved to math storage as something no one ever looks at.

A similar thing happens with ordinary K3 surfaces (you can’t quite take the Artin-Mazur formal group, but instead an “enlarged” version of it). Many people are interested in knowing whether certain varieties lift to characteristic {0}, and not only that but in a very nice way (maybe you remember the Hodge filtration or you want it to satisfy the Tate conjecture). These are called “canonical lifts” and a really beautiful way to construct them is by showing that the deformation functor is isomorphic to the deformations of some particular {p}-divisible group of height {1}.

If you know that the deformations of a {p}-divisible group are unobstructed, then this produces a lift. If you know even more, like the deformation functor is represented by a smooth formal group scheme (which in the height {1} case it is) you can take a canonical choice of lift, namely the one you get from the identity element. This is one way to get a canonical lift of ordinary abelian varieties and ordinary K3 surfaces.

Now hopefully everyone has at least a little interest in deformations of {p}-divisible groups. I have to say that I find this idea amazing and quite under-used. You want to know whether there is a lift to characteristic {0}, or maybe just in general you want to understand the deformations of some object. This in general is incredibly hard. Instead, you find a candidate {p}-divisible group associated to the object and show that the deformation functors are related or isomorphic. Then you can extrapolate information about the original deformation functor, since the deformations of {p}-divisible groups are well understood. I have to admit that this technique has not been very useful to me so far in my research, but I’m quite optimistic still…


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