Over the next month, I will be giving a series of four introductory talks at UC Berkeley on -divisible groups and related concepts. I plan to write a post for each talk and include a number of references for those wishing to see more details.
Let be an Abelian variety (a smooth proper group variety over some number field ). Associated with and a prime number is its Tate-module where consists of the -torsion elements of . We recall that in the present case, is a free -module of rank (where is the dimension of ). The Tate-module is a very powerful invariant: for instance we have a theorem of Faltings that for Abelian varieties
The above construction is equally useful when is a finite field. However things break down when the characteristic of is . To see this, lets focus in on the case of an elliptic curve over a field of characteristic . Then the has either points in the ordinary case or just the identity in the supersingular case. The multiplication by map on is a degree -polynomial so somehow just considering geometric points is not capturing everything we’d like.
The missing information in the above picture corresponds to nilpotents so we can remedy the situation by considering the finite flat group scheme which is a kernel of . For instance, if is an ordinary elliptic curve, then ( here is the scheme representing the functor on -algebras given by and is the scheme representing the sheafification of the functor ).
In this case the analog of the Tate module that we want is Unfortunately, direct limits need not exist in the category of finite flat group schemes. There are two (equivalent) ways of resolving this problem. Either we can consider directed systems of group schemes or we can enlarge our category to that of fppf sheaves. This brings us to the first key definition.
Definition: Let be a scheme and consider the site of fppf sheaves of groups on . Such a sheaf is -divisble (often Barsotti-Tate in the literature) if it satisfies each of the following conditions.
- is -torsion, i.e. .
- is an epimorphism
- is a finite, locally free group scheme
Examples and Key Constructions
- If is an Abelian variety, then is a -divisible group.
- We write and .
- Cartier Duality: If is a finite flat group scheme then so is the functor and we denote this . So for instance and .
- We define the Cartier dual of a _divisible group by . So .
- We define the height of a -divisible group to be so that .
- We call a -divisible group etale/connected if it is a limit of etale/connected group schemes. is etale (as a scheme it is just ), is connected when is in characteristic and etale otherwise.
- When is a complete noetherian local ring we get a short exact sequence of group schemes and when is perfect field , the sequence splits and gives a decomposition . In this latter case a -divisible group will likewise split into connected and etale parts.
- Finite flat group schemes over a base of positive characteristic have Frobenius and Verschiebung morphisms and these lift to morphisms of -divisible groups. These morphisms satisfy .
The Associated Formal Group
Definition: A formal group of dimension over a complete Noetherian local ring with residue field of characteristic is the formal scheme with ideal of definition and with a commutative group structure.
Explicitely, if is a formal group of dimension , then the group structure is a family of power series in the variables satisfying
We say a formal group is -divisible if multiplication by is an isogeny. In this case, will be a -divisible group. Conversely given a connected -divisible group , we can take the direct limit in the category of locally ringed spaces and get a formal scheme with a group structure inherited from the . The constructions are inverses and form an equivalence of categories. Note that even if this direct limit exists in the category of schemes, it may not be the same as that in the category of locally ringed spaces. For instance, has one point while has two.
With the above construction in hand, we can define the dimension of a -divisible group . It is the dimension of the connected component . For example, we note that under the above assumptions, and both have height 1 and dimension 1 and 0 respectively.
In fact, we have a result that will often come in handy:
Proposition Let be dual height -divisible groups over as above with respective dimensions . Then .
We are only counting dimensions so we can tensor everything with the residue field of and thus reduce to the case where we are working over a perfect field of characteristic .
are both epimorphisms and since , are both epimorphisms as well. It follows that we have an exact sequence .
Comparing dimensions, the middle is and the left and right are respectively.
One way this result can prove useful is in analyzing -divisible groups corresponding to Abelian varieties. For instance, the -divisible group corresponding to an elliptic curve has height . Moreover, if is the dual Abelian variety to with the corresponding -divisible groups, one can show that . Elliptic curves are self dual so this means the corresponding -divisible group is as well. If is an elliptic curve over a finite field with -divisible group , then the equation yields . Thus we can immediately tell that is not etale since then it would have dimension . If then splits and each has height 1 (of course we’ve stated above explicitely what these groups are so this shouldn’t be surprising). If . Then is a self-dual connected -divisible group.
A Glimpse at Further Motivations
For many questions in number theory, and in particular in global Langlands one considers moduli spaces of Abelian varieties with additional structure, for instance certain Shimura varieties. For local Langlands, the analog we want (in a way that I hope to make slightly clearer in a later talk) are deformation spaces of Abelian varieties.
Let be a ring killed by multiplication by for some . Let be a nilpotent ideal in and let . Consider , the category of Abelian schemes over and the category of triples where is a -divisible group over and is an isomorphism from to the -divisible group of .
Theorem (Serre-Tate) The natural functor is an equivalence of categories.
Intuitively, this identitifies a space of Abelian schemes deformed from to with the deformation space of its -divisible group. Thus if we are interested in local Langlands correspondence or simply want to study local moduli of Abelian varieties, we are naturally drawn to study moduli of -divsible groups.
Tate has an article -Divisible Groups which was historically one of the first introductions to the topic and is very accessible.
Tate has also written an introductory paper Finite Flat Group Schemes which provides the basics of the theory as well as some concrete computations and more advanced topics.
If one wants a more complete treatment of the theory of group schemes and -divisible groups, Demazure’s Lectures on -Divisible Groups is good though a warning that some of the notation is a bit non-standard.
Katz’s Serre-Tate Local Moduli has a nice proof of the Serre-Tate theorem at the beginning.
Messing’s The Crystals associated to Barsotti-Tate Groups is a classic on -divisible groups and their deformation theory but is a harder read than the above.