I guess this will be the last post on this topic. I’ll explain a tiny bit about what goes into the proof of this theorem and then why anyone would care that such canonical lifts exist. On the first point, there are tons of details that go into the proof. For example, Nick Katz’s article, Serre-Tate Local Moduli, is 65 pages. It is quite good if you want to learn more about this. Also, Messing’s book The Crystals Associated to Barsotti-Tate Groups is essentially building the machinery for this proof which is then knocked off in an appendix. So this isn’t quick or easy by any means.
On the other hand, I think the idea of the proof is fairly straightforward. Let’s briefly recall last time. The situation is that we have an ordinary elliptic curve over an algebraically closed field of characteristic . We want to understand , but in particular whether or not there is some distinguished lift to characteristic (this will be an element of .
To make the problem more manageable we consider the -divisible group attached to . In the ordinary case this is the enlarged formal Picard group. It is of height whose connected component is . There is a natural map just by mapping . Last time we said the main theorem was that this map is an isomorphism. To tie this back to the flat topology stuff, is the group representing the functor .
The first step in proving the main theorem is to note two things. In the (split) connected-etale sequence
we have that is height one and hence rigid. We have that is etale and hence rigid. Thus given any deformation of we can take the connected-etale sequence of this and see that is the unique deformation of over and . Thus the deformation functor can be redescribed in terms of extension classes of two rigid groups .
Now we see what the canonical lift is. Supposing our isomorphism of deformation functors, it is the lift that corresponds to the split and hence trivial extension class. So how do we actually check that this is an isomorphism? Like I said, it is kind of long and tedious. Roughly speaking you note that both deformation functors are prorepresentable by formally smooth objects of the same dimension. So we need to check that the differential is an isomorphism on tangent spaces.
Here’s where some cleverness happens. You rewrite the differential as a composition of a whole bunch of maps that you know are isomorphisms. In particular, it is the following string of maps: The Kodaira-Spencer map followed by Serre duality (recall the canonical is trivial on an elliptic curve) . The hardest one was briefly mentioned a few posts ago and is the dlog map which gives an isomorphism .
Now noting that and that gives us enough compositions and isomorphisms that we get from the tangent space of the versal deformation of to the tangent space of the versal deformation of . As you might guess, it is a pain to actually check that this is the differential of the natural map (and in fact involves further decomposing those maps into yet other ones). It turns out to be the case and hence is an isomorphism and the canonical lift corresponds to the trivial extension.
But why should we care? It turns out the geometry of the canonical lift is very special. This may not be that impressive for elliptic curves, but this theory all goes through for any ordinary abelian variety or K3 surface where it is much more interesting. It turns out that you can choose a nice set of coordinates (“canonical coordinates”) on the base of the versal deformation and a basis of the de Rham cohomology of the family that is adapted to the Hodge filtration such that in these coordinates the Gauss-Manin connection has an explicit and nice form.
Also, the canonical lift admits a lift of the Frobenius which is also nice and compatible with how it acts on the above chosen basis on the de Rham cohomology. These coordinates are what give the base of the versal deformation the structure of a formal torus (product of ‘s). One can then exploit all this nice structure to prove large open problems like the Tate conjecture in the special cases of the class of varieties that have these canonical lifts.