Today we’ll try to answer the question: What is Serre-Tate theory? It’s been a few years, but if you’re not comfortable with formal groups and -divisible groups, I did a series of something like 10 posts on this topic back here: formal groups, p-divisible groups, and deforming p-divisible groups.
The idea is the following. Suppose you have an elliptic curve where is a perfect field of characteristic . In most first courses on elliptic curves you learn how to attach a formal group to (chapter IV of Silverman). It is suggestively notated , because if you unwind what is going on you are just completing the elliptic curve (as a group scheme) at the identity.
Since an elliptic curve is isomorphic to it’s Jacobian there is a conflation that happens. In general, if you have a variety you can make the same formal group by completing this group scheme and it is called the formal Picard group of . Although, in general you’ll want to do this with the Brauer group or higher analogues to guarantee existence and smoothness. Then you prove a remarkable fact that the elliptic curve is ordinary if and only if the formal group has height . In particular, since the -divisible group is connected and -dimensional it must be isomorphic to .
It might seem silly to think in these terms, but there is another “enlarged” -divisible group attached to which always has height . This is the -divisible group you get by taking the inductive limit of the finite group schemes that are the kernel of multiplication by . It is important to note that these are non-trivial group schemes even if they are “geometrically trivial” (and is the reason I didn’t just call it the “-torsion”). We’ll denote this in the usual way by .
I don’t really know anyone that studies elliptic curves that phrases it this way, but since this theory must be generalized in a certain way to work for other varieties like K3 surfaces I’ll point out why this should be thought of as an enlarged -divisible group. It is another standard fact that is ordinary if and only if . In fact, you can just read off the connected-etale decomposition:
We already noted that , so the -divisible group is a -dimensional, height formal group whose connected component is the first one we talked about, i.e. is an enlargement of . For a general variety, this enlarged formal group can be defined, but it is a highly technical construction and would take a lot of work to check that it even exists and satisfies this property. Anyway, this enlarged group is the one we need to work with otherwise our deformation space will be too small to make the theory work.
Here’s what Serre-Tate theory is all about. If you take a deformation of your elliptic curve say to , then it turns out that is a deformation of the -divisible group . Thus we have a natural map . The point of the theory is that it turns out that this map is an isomorphism (I’m still assuming is ordinary here). This is great news, because the deformation theory of -divisible groups is well-understood. We know that the versal deformation of is just . The deformation problem is unobstructed and everything lives in a -dimensional family.
Of course, let’s not be silly. I’m pointing all this out because of the way in which it generalizes. We already knew this was true for elliptic curves because for any smooth, projective curve the deformations are unobstructed since the obstruction lives in . Moreover, the dimension of the space of deformations is given by the dimension of . But for an elliptic curve , so by Serre duality this is one-dimensional.
On the other hand, we do get some actual information from the Serre-Tate theory isomorphism because carries a natural group structure. Thus an ordinary elliptic curve has a “canonical lift” to characteristic which comes from the deformation corresponding to the identity.