Let be an abelian variety of dimension over a field of characteristic . In the previous posts, we saw that to give a deformation of over a local artinian ring with residue field was the same as giving a continuous morphism of rings : that is, the local deformation space is smooth on parameters.

There is another description of the deformation space in terms of the -divisible group, though. Given , we can form the formal scheme

where each is a finite group scheme of rank over . As a formal scheme, this is *smooth*: that is, given a small extension in and a morphism , there is an extension

In fact, we start by finding an extension (as is smooth), and then observe that this extension must land in some if landed in .

As a result, we can talk about deformations of this formal (group) scheme.

Definition 16Adeformationof a smooth, formal group scheme over over a ring is a smooth, formal group scheme over which reduces mod the maximal ideal (i.e., when restricted to -algebras) to .

Suppose . Using Schlessinger’s criterion, we find that (under appropriate finiteness hypotheses), must be prorepresentable by a power series ring for some . In other words, corresponds to a *formal group* over .

Suppose for instance that was a supersingular elliptic curve. By definition, each of the is a thickening of the zero point, and consequently is the formal completion of at zero. This data is equivalent to the *formal group* of , and as we just saw a deformation of this formal group over a ring is a formal group over which reduces to the formal group of mod the maximal ideal.

The main result is:

Theorem 17 (Serre-Tate)Let . There is an equivalence of categories between abelian schemes over and pairs where is an abelian variety over and a deformation of over .

So, in particular, deformations of over are equivalent to deformations of over .

In this post, I’ll describe an argument due to Drinfeld for this result (which I learned from Katz’s article). (more…)