Last time we saw that if we consider a -divisible group over a perfect field of characteristic , that there wasn’t a whole lot of information that went into determining it up to isomorphism. Today we’ll make this precise. It turns out that up to isomorphism we can translate into a small amount of (semi-)linear algebra.
I’ve actually discussed this before here. But let’s not get bogged down in the details of the construction. The important thing is to see how to use this information to milk out some interesting theorems fairly effortlessly. Let’s recall a few things. The category of -divisible groups is (anti-)equivalent to the category of Dieudonné modules. We’ll denote this functor .
Let be the ring of Witt vectors of and be the natural Frobenius map on . There are only a few important things that come out of the construction from which you can derive tons of facts. First, the data of a Dieudonné module is a free -module, , of finite rank with a Frobenius which is -linear and a Verschiebung which is -linear satisfying .
Fact 1: The rank of is the height of .
Fact 2: The dimension of is the dimension of as a -vector space (dually, the dimension of is the dimension of ).
Fact 3: is connected if and only if is topologically nilpotent (i.e. for ). Dually, is connected if and only if is topologically nilpotent.
Fact 4: is étale if and only if is bijective. Dually, is étale if and only if is bijective.
These facts alone allow us to really get our hands dirty with what these things look like and how to get facts back about using linear algebra. Let’s compute the Dieudonné modules of the two “standard” -divisible groups: and over (recall in this situation that ).
Before starting, we know that the standard Frobenius and Verschiebung satisfy the relations to make a Dieudonné module (the relations are a little tricky to check because constant multiples for involve Witt multiplication and should be done using universal properties).
In this case is bijective so the corresponding must be étale. Also, so is topologically nilpotent which means is connected. Thus we have a height one, étale -divisible group with one-dimensional, connected dual which means that .
Now we’ll do . Fact 1 tells us that because it has height . We also know that must have the property that since has dimension . Thus and hence .
The proof of the anti-equivalence proceeds by working at finite stages and taking limits. So it turns out that the theory encompasses a lot more at the finite stages because are perfectly legitimate finite, -power rank group schemes (note the system does not form a -divisible group because multiplication by is the zero morphism). Of course taking the limit is also a formal -torsion group scheme. If we wanted to we could build the theory of Dieudonné modules to encompass these types of things, but in the limit process we would have finite -module which are not necessarily free and we would get an extra “Fact 5” that is free if and only if is -divisible.
Let’s do two more things which are difficult to see without this machinery. For these two things we’ll assume is algebraically closed. There is a unique connected, -dimensional -divisible of height over . I imagine without Dieudonné theory this would be quite difficult, but it just falls right out by playing with these facts.
Since we can choose a basis, , so that and . Up to change of coordinates, this is the only way that eventually (in fact is the smallest ). This also determines (note these two things need to be justified, I’m just asserting it here). But all the phrase “up to change of coordinates” means is that any other such will be isomorphic to this one and hence by the equivalence of categories .
Suppose that is an elliptic curve. Now we can determine up to isomorphism as a -divisible group, a task that seemed out of reach last time. We know that always has height and dimension . In previous posts, we saw that for an ordinary we have (we calculated the reduced part by using flat cohomology, but I’ll point out why this step isn’t necessary in a second).
Thus for an ordinary we get that by the connected-étale decomposition. But height and dimension considerations tell us that must be the unique height , connected, -dimensional -divisible group, i.e. . But of course we’ve been saying this all along: .
If is supersingular, then we’ve also calculated previously that . Thus by the connected-étale decomposition we get that and hence must be the unique, connected, -dimensional -divisible group of height . For reference, since we see that is also of dimension and height . If it had an étale part, then it would have to be again, so must be connected as well and hence is the unique such group, i.e. . It is connected with connected dual. This gives us our first non-obvious -divisible group since it is not just some split extension of ‘s and ‘s.
If we hadn’t done these previous calculations, then we could still have gotten these results by a slightly more general argument. Given an abelian variety we have that is a -divisible group of height where . Using Dieudonné theory we can abstractly argue that must have height less than or equal to . So in the case of an elliptic curve it is or corresponding to the ordinary or supersingular case respectively, and the proof would be completed because is the unique étale, height , -divisible group.