Last time we defined -divisible groups and looked at some basic examples and constructions. We saw that the -divisible group of an Abelian variety over a perfect field of positive characteristic encodes similar data to the Tate-module in characteristic . However, the characteristic -case seems to be much more complicated than in characteristic : the Tate module is just a module while a -divisible group can have non-trivial geometric structure. The case of an etale -divisible group defined over a scheme is very similar in practice to a Tate module: one can show that for a choice of geometric point of , is a -module and determines uniquely.

However, the geometry of a geometrically connected -divisible group can be quite complicated, even over an algebraically closed field. What we would desperately like to have, therefore, is a linear algebraic way of understanding a geometrically connected -divisible group. Why would we expect such a thing? Well this isn’t such a strange thing–whenever one computes the cohomology of a space, one is reducing non-linear geometry to linear algebra. Secondly, we already know examples of related objects being characterized by linear algebra. For instance Abelian varieties and their Hodge structures. In practice such a characterization is exactly what we will get via the theory of Dieudonne modules, which we now discuss.

Before we really dive in, some quick remarks. The theory we discuss below can be generalized dramatically. For any commutative ring with a unit, there exists a topological ring so that the category of smooth commutative formal groups over is canonically equivalent to the category of -modules satisfying a few extra conditions. Since we saw last time that the category of geometrically connected -divisible groups is equivalent to the category of “-divisible” formal groups, this general theory completely solves the problem of finding a linear-algebraic characterization of connected -divisible groups. However, the above ring is difficult to define and relatively hard to work with. In practice, the situation we’ll often care about is that of -divisible groups over a perfect field of characteristic . Thus, we specialize to this case now. Finally, to extend our results to all -divisible groups, not just the geometrically connected ones, we’ll have to make extensive use of Cartier duality. I believe a more streamlined approach to the theory exists due to Fontaine, but I have chosen to take the classical approach below.

## The Dieudonne Module of a p-Divisible Group

The general approach is as follows. Take to be a -divisible group over a perfect field of characteristic . Last time we observed that in this case one can decompose into a product of geometrically connected and etale -divisible groups. Taking Cartier duals, one can again decompose into connected and etale pieces. This implies that a -divisible group admits a decomposition where and its dual are connected, is etale and its dual is connected and so on. Over a perfect field, we saw that we have . Thus we cannot have a -divisible group where both it and its dual are etale. From last time we have a formal group associated to a connected -divisible group and as we will see we have a Dieudonne module associated to each formal group. Dieudonne modules will have a duality theory that mirrors that of -divisible groups, and so we will be able construct the Dieudonne module for by constructing those of .

The key fact is that the category of formal groups over is Abelian and so is equivalent to a category of modules. More precisely (see the references for details) one constructs a “universal” injective object and considers . This will be an Abelian group with a right action by and therefore a left action by . This gives a contravariant functor which embeds our category into a module category as desired. Similarly, there is a covariant theory where one takes for some particular object in our category, and the relation between the constructions is that as modules.

Let be the ring of Witt vectors of and the lift of the absolute Frobenius map on . Then the endomorphism ring in question is the non-commutative ring satisfying the relations .

**Definition **A Dieudonne module is a left module which is a free -module of finite rank.

**Theorem **There is an equivalence of categories between -divisible groups over and Dieudonne modules over . If we denote the covariant Dieudonne module of by then is an exact functor.

## Key Properties and Examples

- The height of a _divisible group is the same as the -rank of .
- The dual of is the -module with actions given by for all . .
- is etale if and only if is bijective on and is etale if and only if is bijective on .
- Let be the -divisible group whose Dieudonne module is . Then . .
- Using the above properties, one can show that anf

## Dieudonne-Manin Classification and Slope Decompositions

We now come to the key result in the talk (and a huge motivation for future talks). In what follows we assume that our ground field is algebraically closed. This is a necessary assumption for the theory to go through. The key observation of Dieudonne is that if we only care about isogenies, we can considerably simplify the ring we work over. Instead of -modules, we consider vector spaces. Then becomes invertible and so we can drop the variable . Given a Dieudonne module, we therefore get the following object:

**Definition **An -isocrystal is a finite dimensional -vector space with a bijective – linear endomorphism .

The key fact is that two Dieudonne modules are isogenous if and only if their isocrystals are isomorphic.

**Theorem** (Dieudonne – Manin) Suppose is algebraically closed. Then the category of -isocrystals is semisimple and the simple components are determined by their slopes. In particular each is isomorphic to (which has slope ) for some . The isocrystals coming from Dieudonne modules are precisely those with slopes in .

**Corollary **Any -divisible group over an algebraically closed field is isogenous to a product of .

The above results are a huge leap forward in understanding -divisible groups since they allow us to classify -divisible groups up to isogeny using only simple numerical data.

A common notation is for the decomposition of an -isocrystal to be represented graphically as a Newton polygon. We define the slope of to be and represent it graphically as a line segment in the plane with slope and projection onto the -axis of length . Given an isocrystal, we can order the slopes from smallest to biggest and represent the isocrystal as a convex polygon starting at the origin. Notice that the endpoint of the Newton polygon of a -divisible group has -coordinate equal to and -coordinate equal to .

## Examples

Note that the following is just for fun and not especially important in anything that follows.

Let be an Abelian variety over a finite field . Then a theorem of Tate’s says that the characteristic polynomial of the Frobenius morphism on over determines up to isogeny. If we take the algebra then the slope appears in the decomposition of the -divisible group of if there exists a place over in where . Then the multiplicity of is .

Now we have the following result.

**Theorem (Honda-Tate) **Isogeny classes of simple Abelian varieties over a finite field of order correspond bijectively with algebraic integers all of whose complex absolute values are . Moreover and its conjugates are precisely the eigenvalues of the Frobenius endomorphism acting on the Tate module.

So pick positive satisfying . Let , be a root of so that both complex absolute values of are . Then corresponds to an isogeny class of simple Abelian varieties over and by the theory of the Newton polygon (from undergraduate number theory!) the valuations of the roots are and so the slopes are .

## Resources

First I recommend looking at the following course notes by Ching-Li Chai and Frans Oort. They contain a veritable gold mine of theory if one is willing to accept the black boxes. Moreover there are a huge number of useful references in this text. Course Notes

I personally enjoyed the original paper *The Theory of Commutative Formal Groups over Fields of Finite Characteristic *by Manin where many of these results were originally proven. The reader should be warned that this is a fairly old paper and some of the notation is dated.

Demazure’s *Lectures on p-Divisible Groups* has a careful construction of the Dieudonne module theory via Witt vectors. Personally I find the Witt vector formalism hard to deal with but if you want it, it’s here.

Ehud de Shalit has written a very readable introduction to the theory of -isocrystals. This was one of the first things I read and though it is a bit steep in places, makes a good first text. F – Isocrystals .

Finally a remark. A more modern treatment of this theory is in terms of crystalline cohomology. The interested reader can look at the notes of Chai and Oort above and I think they generally point one in the right direction.

There is a typo in the formula for the dual of a Dieudonné module.

P.S. This course is (so far) an excellent summary of this material.

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Many thanks for pointing this out and for your kind words!

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Hey Bertie, here are some comments:

1) Connected and geometrically connected are the same thing for group schemes, so you can drop that pesky adjective when discussing them. In fact, if is, say, finite type and has a -point then connected implies geometrically connected. This is a nice exercise if you haven’t seen it before.

2) Minor pedantic point. The equivalence between connected -divisible groups and -divisible formal groups only holds (as far as I know) over a Noetherian local ring with with residue characteristic . You probably only were thinking over , but I’ll add this regardless.

3) A minor typo or, rather, ambiguity. You say that “Thus we cannot have a p-divisible group where both it and its dual are etale.”. Whereas the previous sentence as just talking about perfect fields in general, this sentence is only talking about perfect fields of characteristic . This observation also follows from the fact that if is étale and -torsion then, geometrically, it’s some product of ‘s whose Cartier dual is a product of some ‘s which are not étale.

4) Again, pedantic point since you sort of fix this in the next sentence, but, of course, being abelian is not equivalent to being a category of modules. An abelian category is a category of modules if and only if it has a ‘compact progenerator’ which, I believe, is somehow the ‘dual’ of your .

5) As we talked about in the seminar, I think you really want to change your (the Dieudonné ring) to be modulo relations. I think that the makes sense if and are supposed to be nilpotent on . This is the case if is a finite flat connected group scheme over . Since your goal is to build , for a -divisible group, as a limit of such things this is, perhaps, where the two notions meet.

6) I think that your third property of the Dieudonne module is slightly confusing (although correct!) and is one reason to use the contravariant Dieudonne module. Namely, it’s not hard to show that a scheme (with ) is étale if and only if the relative Frobenius map is an isomorphism. Thus, one would expect that is étale if is bijective. This is what does happen for the contravariant theory, but and get switched when one goes to the covariant theory. Just a thought.

7) To answer something we talked about in the lecture, we can explicitly describe the such that . Namely, they should be the truncated Witt scheme (or maybe the indices reversed since we’re covariant). You can read about them here: https://people.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf

8) Silly typo: in the last properties bullet the phrase ‘anf’ is, what I can only assume, a jive way of saying ‘and’. 🙂

9) With regards to your references, the all-encompassing theory of Fontaine is found in *Groupes -divisibles sur les corps locaux’.

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