Today we’ll look a little more closely at for abelian varieties and finish up a different sort of classification that I’ve found more useful than the one presented earlier as triples . For safety we’ll assume is algebraically closed of characteristic for the remainder of this post.
First, let’s note that we can explicitly describe all -divisible groups over up to isomorphism (of any dimension!) up to height now. This is basically because height puts a pretty tight constraint on dimension: . If we want to make this convention, we’ll say if and only if , but I’m not sure it is useful anywhere.
For we have two cases: If , then it’s dual must be the unique connected -divisible group of height , namely and hence . The other case we just said was .
For we finally get something a little more interesting, but not too much more. From the height case we know that we can make three such examples: , , and . These are dimensions , , and respectively. The first and last are dual to each other and the middle one is self-dual. Last time we said there was at least one more: for a supersingular elliptic curve. This was self-dual as well and the unique one-dimensional connected height -divisible group. Now just playing around with the connected-étale decomposition, duals, and numerical constraints we get that this is the full list!
If we could get a bit better feel for the weird supersingular case, then we would have a really good understanding of all -divisible groups up through height (at least over algebraically closed fields).
There is an invariant called the -number for abelian varieties defined by . This essentially counts the number of copies of sitting inside the truncated -divisible group. Let’s consider the elliptic curve case again. If is ordinary, then we know explicitly and hence can argue that . For the supersingular case we have that is actually a non-split semi-direct product of by itself and we get that . This shows that the -number is an invariant that is equivalent to knowing ordinary/supersingular.
This is a phenomenon that generalizes. For an abelian variety we get that is ordinary if and only if in which case the -divisible group is a bunch of copies of for an ordinary elliptic curve, i.e. . On the other hand, is supersingular if and only if for supersingular (these two facts are pretty easy if you use the -rank as the definition of ordinary and supersingular because it tells you the étale part and you mess around with duals and numerics again).
Now that we’ve beaten that dead horse beyond recognition, I’ll point out one more type of classification which is the one that comes up most often for me. In general, there is not redundant information in the triple , but for special classes of -divisible groups (for example the ones I always work with explained here) all you need to remember is the to recover up to isomorphism.
A pair of a free, finite rank -module equipped with a -linear endomorphism is sometimes called a Cartier module or -crystal. Every Dieudonné module of a -divisible group is an example of one of these. We could also consider where to get a finite dimensional vector space in characteristic with a -linear endomorphism preserving the -lattice .
Passing to this vector space we would expect to lose some information and this is usually called the associated -isocrystal. But doing this gives us a beautiful classification theorem which was originally proved by Diedonné and Manin. We have that is naturally an -module where is the noncommutative polynomial ring . The classification is to break up into a slope decomposition.
These are just rational numbers corresponding to the slopes of the operator. The eigenvalues of are not necessarily well-defined, but if we pick the normalized valuation , then the valuations of the eigenvalues are well-defined. Knowing the slopes and multiplicities completely determines up to isomorphism, so we can completely capture the information of in a simple Newton polygon. Note that when is the -isocrystal of some Dieudonné module, then the relation forces all slopes to be between 0 and 1.
Unfortunately, knowing up to isomorphism only determines up to equivalence. This equivalence is easily seen to be the same as an injective map whose cokernel is a torsion -module (that way it becomes an isomorphism when tensoring with ). But then by the anti-equivalence of categories two -divisible groups (in the special subcategory that allows us to drop the ) and have equivalent Dieudonné modules if and only if there is a surjective map whose kernel is finite, i.e. and are isogenous as -divisible groups.
Despite the annoying subtlety in fully determining up to isomorphism, this is still really good. It says that just knowing the valuation of some eigenvalues of an operator on a finite dimensional characteristic vector space allows us to recover up to isogeny.