On the Galois side of Local Langlands, we have four groups whose representations we consider.

1. The Weil group of .
2. The absolute Galois group of .
3. The Weil Deligne group which is given by where acts by .
4. The Langlands Group .

We begin by drawing our focus to the first two entries of this list. When we consider representations of these two groups we consider finite dimensional representations of complex vector spaces as well as -adic ones ( -vector spaces for ). Notice that are non-canonically isomorphic as fields but carry different topologies.

We typically impose one of two continuity conditions on representations.

1. A representation is continuous if it is continuous as a function between topological groups.
2. A representation is smooth if the stabilizer of each vector is an open subgroup.

The second condition doesn’t depend on the topology of the target so smooth               -adic representations are the same thing as smooth complex representations.

A continuous complex representation of has finite image. This result comes from the fact that has no small subgroups: there exists an open subset containing the identity but no nontrivial subgroups. Thus for  , is open so contains an open subgroup whose image is a subgroup of and so is the identity. An open subgroup of has finite index.

For the case any complex representation with finite image is continuous. If a complex representation of is smooth then we take a basis   and an open subgroup fixing . Then is open and fixes the entire representation.

Combining the above: smooth and  continuous complex representations are the same thing.

We now consider complex representations of . The same analysis we applied above to holds for . In particular smooth and continuous representations of have finite image and are trivial on an open subgroup. The pre-image of any subset of the image is an open subset of since it is a union of translates of the kernel of the representation on . Thus smooth and continuous complex representations of are the same.

In conclusion, the eight combinations of the words -adic, complex, smooth, and continuous split into  4 distinct sets:

1. Smooth and continuous complex -representations, smooth -adic representations.
2.  Smooth and continuous complex -representations, smooth -adic representations.
3. Continuous -adic representations of .
4. Continuous -adic representations of .

Notice that restricting an element of 2, 4 to gives an element of 1, 3 respectively. Also notice that 1, 2 are examples of 3, 4 respectively. Thus 3 is the most general type of object and in fact a subcategory of is the object on the Galois side of the Local Langlands correspondence.

Notice that the sets 3, 4 defined above depend apriori on a choice of . But we will see their stucture doesn’t in any particularly important way. Thus it woud be useful to to try to formulate these sets in a way that doesn’t explicitely use the topology of the target. This is accomplished in the literature by the notion of a Weil-Deligne representation. These can be defined as representations of a Weil-Deligne group as mentioned at the start of this post. However, there is an equivalent definition using less machinery and appearing much more frequently in the literature, which we now give.

Definition. A Weil-Deligne representation is an ordered triple consisting of

1. An -adic or complex vector space  (as remarked earlier, smooth representations don’t depend on the topology of the target though one would need to choose an isomorphism between and to show this).
2. A smooth -representation with image in .
3. A nilpotent element such that for all , .

We define a morphism of Weil-Deligne representations to be a morphism of smooth representations so that .

Key Theorem. The category of Weil-Deligne representations  is equivalent ot the category of continuous -adic representations of .

We will not prove this fact here though it comes as a consequence of Grothendieck’s -adic monodromy theorem which states that for a continuous -adic representation of , there exists an open subgroup of so that is unipotent. In particular, is given by for some unique nilpotent operator where is the -adic cyclotomic character. This theorem then gives one a a way of “converting” a continuous -adic representation into a smooth one. Simply make a choice Frobenius and define by . So the equivalence of categories is given by , .

Theorem. The category of continuous -adic representations of is equivalent to the category of Weil-Deligne representations such that has eigenvalues with absolute value (where is a choice of Frobenius).

Semi-Simplifications

I find the various notions of semisimple Galois representations to be quite confusing. Let me try to explain what I think is the correct state of affairs.

The Galois side of the Local Langlands correspondence deals with Frobenius semisimple Weil-Deligne representations. These are tuples satisfying the Weil-Deligne representation conditions as well as the further condition that if is some choice of Frobenius in , then is semisimple. Since is compact, this is in fact equivalent to the condition that is semisimple. However, (and confusingly!) this is not the same thing as a semisimple object in the category of Weil-Deligne representations. We have an equivalence of categories between Weil-Deligne representations and continuous -adic representations of and this equivalence preserves the categorical notion of semisimplicity. Thus we are free to consider it in the category of continuous -adic representations. In particular, we want to show that if is a continuous -adic representation, then the associated must be . We can assume without loss of generality that is a matrix with for and otherwise and it suffices to show that is a matrix. Observe that the kernel of is one-dimensional in this case. But if , then and so is a scalar multiple of for each . Thus and therefore stabilize the one dimensional subspace spanned by . But this subspace has no complement when we restrict to and so is not semisimple.

To summarize, the following are equivalent:

1. is a semisimple continuous -adic representation of .
2. is a semisimple smooth -adic representation of .
3. is a smooth -adic representation of such that is semisimple (where is a choice of Frobenius)

We complete this section by describing what the representations in question look like.

By our discussions above, a continuous complex representation of has finite image and so in particular factors through some finite extension. This means that all such representations are given by an embedding of  some finite Galois group .

We introduce the character such that and if is a lift of Frobenius then where is the cardinality of the residue field of . More generally we have characters of the form so that . Then any irreducible complex representation of is isomorphic to where is the restriction of an irreducible complex representation of .

We now consider Weil-Deligne representations. We define the special representation by considering vectors and defining for and and . Then any indecomposable Weil-Deligne representation is isomorphic to where is an irreducible complex representation of .

Langlands Group

The above theory enables us to adequately understand the Galois side of the Local Langlands correspondence (at least for the case!). However the situation is slightly unsatisfactoy in that we are forced to deal with Frobenius semisimple representations which form an unusual subcategory of Weil-Deligne representations. The above terminology is the most common in the literature but we can work with a slightly more familiar class of representations by considering the Langlands group .

A representation of is a continuous complex finite dimensional representation of so that the restriction to is complex analytic. Such a representation is semisimple if and only if its restriction to is semisimple.

Theorem. There is a bijection between Frobenius semisimple Weil-Deligne representations and semisimple representations of given by (where is the natural representation of on homogeneous polynomials of degree .

References

Rohrlich’s exposition as posted on Dylan Yott’s website Rohrlich

Toby Gee’s notes from Arizona Winter School Gee

Commelin’s exposition Commelin

Taylor’s paper (page 5) Taylor

The scope of this talk may sound impressive but first a disclaimer which I feel I should make out of honesty. A thorough description of the material in the second two sections requires greater knowledge of representation theory and the Langlands program than I currently possess. For this reason, I have had to be a bit vague in places and rely more heavily than normal on my resources. Moreover, though I have tried to be careful, I would not be surprised if I made some mistakes towards the end. As my understanding of the theory improves in the coming months, I’ll try to update this post.

Local Class Field Theory Via Lubin Tate Formal Groups

In this first section we review the classical construction of a maximal Abelian extension of a local field over using Lubin-Tate formal groups. Let be the residue field of , Let be the complete maximal unramified extension of and let be its completed integer ring with residue field .

In the Lubin-Tate setup, we will consider lifts of one dimensional formal groups over . Take the th power automorphism of  which therefore fixes . If is a formal group over with group law then we can consider the -twist of which we denote with group law . Put another way, is the relative Frobenius twist of corresponding to the ‘Frobenius morphism’ . Thus we have a natural map defined over  and given on rings by  which we call the Frobenius morphism of .

Now we consider lifts of such formal groups to . Notice that lifts uniquely to some automorphism of which fixes .  Thus, given a lift of we can also define its -twist . Furthermore, we are interested in lifts of which are equipped with an -action such that the induced map of on the tangent space of is given by multiplication by . We call such objects formal -modules.

Our analysis of the situation when has dimension one is greatly helped by the following fundamental theorem.

Theorem: Let be a uniformizer of , such that . Then there exists a unique formal group over so that is the Frobenius map . Moreover, carries the structure of a formal module. We call a Lubin-Tate formal group.

The proof of the above is actually quite simple and can be accomplished by determining inductively on rings of the form .

In fact, the construction in the theorem shows that for uniformizers and Frobenius lifts , the set corresponds to morphisms such that is the unique morphism where the corresponding ring map satisfies  . Thus  and . Furthermore, we can take Frobenius twists of to define the morphisms determined by so that . One can show that has distinct roots.

Key Example: Let and consider the uniformizer and . Then with group law .

In fact the above example is crucial, because one can use the  description of endomorphisms to construct an isomorphism between any two Lubin-Tate formal groups .

We now restrict to the case where is a monic polynomial. In particular, is of degree .

Let be the collection of zeroes of the polynomial (in particular, for the example we gave above these are just the th roots of unity. Clearly, and one can verify that is Eisenstein. Thus the splitting field of is a totally ramified extension of .

The roots of have a natural -module structure induced from the group law and -action on . One can concretely show that this module is isomorphic to .  Now the key insight is that carries another action, namely that of . The power series involved in the action have coefficients in , thus they commute with the Galois action and so we can concretely see that we have an embedding

.

Since the latter is just , one can compare orders and show that the splitting field of has order over and that the Galois group is Abelian and isomorphic to .

At this point, we’ve constructed a tower of totally ramified Abelien extensions of . In fact the compositum of these will be a maximum Abelian ramified extension. Though this requires some work to prove, we can state that intersecting this with gives the maximal Abelian extension of . Moreover taking care to worry about compatibility issues,  the above identification of and allows us to construct the Artin map .

Connection to Local Langlands

We continue with the notation as before. In this section, we’ll only consider formal groups of dimension but we will vary the height. All the formal groups we considered above were -divisible and so everything we say can equally be translated into the language of -divisible groups.

From the machinery we’ve built up from previous talks, we can easily show that there is a unique formal group over of dimension and height . This is because the corresponding Newton polygon can have no slope zero part because of the assumption that we have a formal group and a single positive slope will increase the dimension to . Thus the formal group has slope . In fact, one can mess around with Dieudonne modules to show that there is a unique formal group of dimension one and slope up to isomorphism. If we restrict ourselves to dimension formal -modules over then there is a unique one of height which we denote .

Now we want to consider the space of deformations of . In particular, we want to consider the functor which assigns to each complete local Noetherian  -algebra with residue field the set of pairs where is a one dimensional formal group over and is an isomorphism.

Theorem (Lubin-Tate): The functor is representable by a formal scheme non-canonically isomorphic to .

We define the Lubin-Tate deformation space of level , , to be the rigid generic fiber (the rigid open ball of radius ) of . We should think of as the local analog of a Shimura variety. In particular one would like to attach level structure to . This was done by Drinfeld, who constructed an adic space for each is which is etale over with Galois group . The points of correspond to triples where and are as before and is an isomorphism of modules.

The system of spaces has natural actions by lots of interesting groups. acts by . The endomorphism ring of is isomorphic to the maximal order of the division ring with invariant over and this acts by . Finally we have an action of the Weil group of which we denote , though this takes a bit to explain. Let and consider the valuation map defined so that is the power of Frobenius that corresponds to restricted to . Let . Then takes to , to where and is the Frobenius twist , and to which we get by twisting by .

This got really complicated all of a sudden so lets take a step back and see what all this implies for the case when . These formal groups are the Lubin-Tate formal groups we constructed earlier. is just with fiber .

Let’s think about what should be in this case. As a scheme it corresponds to the constant -module . We think of as being thought of as a constant group scheme with its -module structure.  We have . The relevant actions are that acts by module automorphisms, acts by multiplying the module by and the inertia group in acts as via the Artin map and Frobenius acts by twisting.

Now we briefly discuss the relation to Local Langlands. The point is that by taking rigid -adic etale cohomology, we can convert the tower of rigid spaces to a direct system of -vector spaces. The group actions on the rigid spaces translate to actions on the cohomology and thus turn the cohomology into an interesting representation of . The key insight is that decomposing this representation into a sum of irreducible representations gives relationships between the irreducible representations of these groups, concretely realizing Local Langlands correspondences.

Generalizing to Reductive Groups

Lets end by discussing how one might generalize the above. The key to the Lubin-Tate construction was building a tower of geometric spaces (in this case the ) which carry actions by the relevant groups. In the case we saw that this was accomplished by constructing a deformation space of formal groups of a particular height and dimension. In this case, the formal groups correspond to -divisible groups. Thus, we might hope to build geometric spaces out of moduli of -divisible groups. In fact, this is exactly what is done.

Lets analogize with the case PEL type Shimura varieties. Such a variety is naturally a moduli space for Abelian varieties with extra structure such as polarization and endomorphism data. This data is controlled by a reductive group which in turn acts on the Shimura variety. Similary, one can consider deformation spaces  of -divisible groups with additional structures which will admit actions by reductive groups.  Just as in the case of Shimura varieties, we can add in level structure data to get a tower of spaces. These spaces are called Rapoport-Zink spaces.

Of course many cases of the Local Langlands correspondence are still open. Rapoport-Zink spaces have only been constructed for certain reductive groups and for cases where the correct space is known, the cohomology representations can prove very difficult to analyze.

References

Carayol has an article in Volume II of Automorphic Forms, Shimura Varieties, and L-functions which sketches a proof of Local Langlands for using Lubin-Tate theory.

Yoshida has written a very readable article proving local class field theory using Lubin-Tate formal groups. Yoshida

Mantovan Is a good source for reading about the connections between Lubin-Tate theory and Rapoport-Zink Spaces.

Weinstein has a very readable introduction to many of the ideas surrounding Lubin-Tate theory.

Rapoport and Viehmann have written a survey paper on Rapoport-Zink spaces and connections to Local Langlands titled Towards a Theory of Local Shimura Varieties

Crash Course in Tannakian Categories

In what follows I’m purposely going to be vague in my definitions. The purpose is not to write a 50 treatise on Tannakian formalism: those already exist and by infinitely stronger mathematicians than myself. Rather I want to present some of the key results and discuss when one might expect Tannakian categories to be useful.

Definition: A tensor category is a category  equipped with a bi-functor which is symmetric and associative (i.e. there is a functorial isomorphism . These isomorphisms are also required to satisfy a few compatibility conditions.

Examples: Key examples of the above include categories of vector spaces, -modules, representations of  some group.

Given a tensor category, we can define a number of other structures.

Definition: A unit object  of tensor  category is an object , an isomorphism so that the functor  is an equivalence of categories from to itself.

Definition: Given , we have a functor given by . If this functor is representable, we denote it . If the functor is representable for all then we say that has internals homs. Explicitely, this gives us the formula .

Let be the category of -modules. Then the unit object is and internal homs are given by .

The above definitions let us produce many familiar constructions. For instance, the identity map gives a map . For -modules this is exactly the evaluation map.

If has internal homs and a unit object, we can define for each and a unit object, we define a dual  and the evaluation map is .

Further, by the definition of internal homs we have . Thus the evaluation map gives us a unique map . We call reflexive if this map gives an isomorphism .

Finally given   we can use the universal properties to build a morphism

These constructions motivate the following definition:

Definition: A rigid tensor category is a tensor category equipped with  internal homs and a unit object such that all objects are reflexive and the morphism constructed in the previous paragraph is an isomorphism.

The above is nothing crazy. All we’ve done is axiomatized the notion of a tensor product and shown that the constructions we enjoy for vector spaces extend to this setting.

A tensor functor of rigid tensor categories is a functor   which takes the unit object to a unit object and an isomorphism of functors which satisfies a number of associativity and commutativity axioms. Given two tensor functors one can define the notion of a morphism of tensor functors and denote the class of such objects by .

The case we are most interested in is where is an Abelian category and is -bilinear for some . Note that the isomorphism induces a morphism  which makes -linear.

We now come to a pair of key definitions.

Definition: Let be a field , a -algebra and a rigid abelian -linear tensor category. Then we define a fiber functor valued in -modules to be a functor .

Definition: A Tannakian category is a rigid -linear Abelian tensor category equipped with a fiber functor . If , then we say that the category is neutral Tannakian.

Key Example: If is an affine group scheme over and is the category of finite dimensional representations of , then with the natural fiber functor is a neutral Tannakian category.

A key object of consideration when studying neutral Tannakian categories will be . Definitionally, this is a collection indexed over of elements of . However, we will prefer to think to think of this as a group-valued functor on -algebras so that  consists of collections of elements of indexed by and the group operation is composition.

Note that for we have a natural functor (since each object of is definitionally a -rep). The key theorem is that we can reconstruct from the fiber functor.

Theorem: Let be the category of finite dimensional -representations of . Then the morphism is an isomorphism.

Now we have the fundamental theorem of Tannakian categories.

Theorem: Let be a neutral Tannakian category over . Then is representable by some affine group scheme over and there is a unique tensor equivalence of categories .

These results together are pretty crazy. The first says that since we can recover an affine group scheme from its category of finite dimensional representations, the two categories contain equivalent data. The second says that any neutral Tannakian category can be thought of as representations of a group scheme. This is cool because often times a Tannakian category won’t obviously come from representations. For instance categories of Hodge structures are often Tannakian. As we shall see, the same is true for isocrystals.

Before discussing isocrystals, we briefly look at how gradings factor into the Tannakian picture. Given an Abelian group and a field , we can produce the constant group scheme over asociated with and we have its Cartier dual . For instance, and .

The category of -graded -vector spaces is  neutral Tannakian and has a natural fiber functor. Thus it must be equivalent to the category of representations of some group scheme. It turns out that the correct group scheme is . So for example, the category of -graded -vector spaces is equivalent to the category . As a corollary, we see that if the fiber functor of factors through the category of -graded vector spaces, then we get a map of group schemes.

Isocrystals with Additional Structure

I’ve spent a lot of time talking about Tannakian categories. Lets see how this shows up with regards to isocrystals. Recall that we defined an isocrystal last time as follows. If is a perfect field of characteristic with Witt vectors , the lift of Frobenius and , then an isocrystal is a finite dimensional -vector space equipped with an automorphism which is -linear. This is a -linear Tannakian category which we denote .

In practice it’s good to remember that this formalism arose because we wanted to understand -divisible groups. These -divisible groups will show up in our studies of Abelian varieties. However, we typically study Abelian varieties keeping track of some extra structure such as endomorphisms or polarizations or level structure. For Shimura varieties this is encoded in the choice of a reductive group in the Shimura data. Analogously we want to consider categories of isocrystals with “-data”.

Suppose is a reductive group over and that embeds naturally into for some finite -vector space . Then given an element , we can take the space and equip it with the automorphism . This is an isocrystal and we have encoded -structure via . One can check that if there exists so that , then give isomorphic isocrystals. Thus, the collection of isocrystals so that is a -twist of an element of is parametrised by the set where is the aforementioned -conjugacy relation.

The above is somewhat awkward since it relies on a natural faithful representation of our reductive group. For many classical groups this is fine but for some reductive groups the definition is unclear. We can improve things by realizing that for each representation of , an element of gives an isocrystal structure on the vector space underlying that representation. This motivates the following definition.

Definition: A -isocrystal is a functor such that the natural fiber functors commute: .

Note that a -isocrystal is the same thing as an isocrystal.

We now think of as a category whose objects are elements and whose morphisms are so that .

Proposition: The natural functor we constructed above is an isomorphism of categories.

This proof is an exercise in applying the first key theorem of Tannakian categories given above.

Warning: The above construction is the correct notion for an isocrystal with -structure only in certain cases. These are:

1. is algebraically closed and is connected.
2. 
3. is semi-simple simply connected and is finite.

In general, one drops the compatibility condition on fiber functors in the definition but this makes the theory more difficult as (for instance) the above proposition is no longer true.

The category is Tannakian and so is equivalent to the category of representations of some group scheme . Now suppose is an isocrystal. Then by the Tannakian formalism we get a homomorphism . has a natural -grading given by the slope decomposition. Thus we get a slope map . The composition of these two maps allows us to define a functorial Newton map .

The Newton map is crucial in further constructions. Intuitively it should be thought of as a generalization of the slope decomposition of an isocrystal to the case of -structure.  For instance, in the case of -isocrystals and algebraically closed, each isocrystal produces a map . This will factor through some maximal torus and so becomes a map which is the same data as an -tuple of rationals. These are precisely the slopes of the isocrystal.

Resources

The well-read reader will notice that I have heavily borrowed the above from Dat, Orlik, and Rapoport’s Period Domains over Finite and p-adic Fields. I haven’t read this whole text but I intend to soon.

The original references for this material are Kottwitz’s Isocrystals with Additional Structure I, II. Isocrystals I Isocrystals II

The following paper by Rappoport-Richartz discusses the newton map in some detail. Rapoport – Richartz

Milne has written an extensive introduction to Tannakian categories with Deligne.

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.

Motivation

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.

1. is -torsion, i.e. .
2. is an epimorphism
3. 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

1. 
2. 
3. .

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 .

Proof Sketch

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.

Resources

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.

We are interested in studying smooth proper curves over an algebraically closed field and it turns out that an interesting invariant (especially for curves of higher genus) is whether or not a curve admits a nontrivial degree two morphism to . We call such curves hyperelliptic if their genus is at least .

One the one hand, if a curve is hyperelliptic we get an interesting map to work with which among other things immediately tells us that the functional field is a degree two extension of . On the other, if is not hyperelliptic, then the canonical divisor of is very ample and so gives a canonical embedding .

Lets see how this second property comes about. Recall that a divisor on is very ample precisely when for any two points . In particular, removing a point from a divisor drops the dimension of the local system by at most and very ampleness is equivalent to the dimension drop being the maximum possible for every pair of points. Picking any two   and applying Riemann-Roch gives

Since , we see that is zero for all exactly when is very ample and that if this is not the case, we have for some pair of points and this gives a degree two morphism to .

Now, since the canonical divisor of an elliptic curve is degree it’s clearly impossible for it to be very ample so all elliptic curves admit a morphism to . In fact, we knew this anyway because Weierstrass form gives a very explicit degree extension of .

For genus and so very ampleness would imply that which is absurd. Thus all genus two curves are hyperelliptic.

We’d like to find a non-hyperelliptic curve and so the next place to look is genus three. Here, the situation is not immediately impossible, and in fact we will find examples. The image of a canonical embedding would have to be a quartic curve in and so we consider such curves. In fact, we will see that if is a nonsingular quartic, then and so all such curves are non-hyperelliptic.

Since is nonsingular, we have an exact sequence of sheaves on :

Then here is a nice observation. If is an exact sequence  of locally sheaves of rank respectively, then splits up into a filtration so that . Thus, if we take the top exterior power our filtration has only one nonzero term, and this gives .

Tensoring both sides with gives

But the ideal sheaf  and so  and so our final formula is

The right-hand side is and so we get what we wanted.

Let’s do the same thing for schemes over . We have that is proper and therefore separated and is proper by assumption. Therefore by the cancellation property,   is proper and so is a clopen subscheme of , which contradicts connectedness.

Today’s simple concept is how a group action is the same thing as a grading on . To be clear, this is a map of schemes such that:

• If is the multiplication map for , then .
• is the identity section, then is the identity map on .

We translate these conditions  into maps of global sections. If takes to , then the condition we get is   . This equality implies that if and .

The second condition implies that .

Thus, the map splits into infinitely many subsets corresponding to the images of the different . Since is a homomorphism of additive groups, is a group, and since is a homomorphism of algebras, . If for , then applying to both sides, we get that . Thus . Finally and so has the structure of a graded ring. Conversely projection maps from a graded ring to its components give maps satisfying the above properties.

Galois Theory over a Field

We’ll be working in the category of finite separable extensions of . If we pick some algebraic closure then we can define a functor given by . (the group of natural isomorphisms of ) is a profinite group and the fundamental theorem is that gives an equivalence of categories between and transitive -sets.

Maybe the above description is a bit unfamiliar so lets break it down super carefully. Let be a finite separable extension. Then if we know that has precisely embeddings into and as defined above, is precisely this set of embeddings. is definitionally the set of natural isomorphisms of the functor which is the same thing as a collection of automorphisms of , one for each , which form a compatible system in the sense that if we have a map of -Algebras then

is a commutative diagram. In particular, for any , the automorphisms of an embedding are all induced by automorphisms of (exercise) and so a compatible system of such automorphisms is exactly an element of .

For certain special extensions we have that the elements of restricted to their actions on are also induced via pre-composition by automorphisms of . These are precisely the Galois extensions and of note is the fact that they are cofinal in the category in the sense that any finite separable extension of is contained within some Galois extension. If one sits down and traces through what this implies about , it is that can be computed as an inverse limit of the automorphism groups of all the Galois extensions of .

The reader who was excited by my first paragraph but immediately disgruntled by my second should have started to feel happier now.

Galois Theory over a Scheme

By being rather abstract in my constructions above, my aim was to hint that this theory might be generalized to other contexts. This can be very satisfactorily done and anyone interested in the details should read Lenstra’s online notes Galois Theory for Schemes.

The point is that one can define a special type of category called a Galois category equipped with a “fiber” functor . The fundamental theorem one proves is that gives an equivalence of categories between and -sets. The astute reader will notice that in the case of fields we got an equivalence between and transitive -sets. This came about because we didn’t consider the full Galois category associated to a field . This is the category of finite separable -algebras which we call . So for instance the full category also contains products of fields and this is how we get actions with multiple orbits. As in the case of fields we can consider Galois objects in a general Galois category and will be the limit of the automorphism groups of such objects.

Before I delve into schemes, I want to briefly give another example of a Galois category. Given a sufficiently nice topological space we can consider the category of covering spaces of . In this case the fiber functor is given by picking a base point and then mapping a cover of to its fiber over . Any undergrad has probably noticed the intense similarities between the theory of covering spaces and Galois theory. These similarities can be explained by the fact that both categories are Galois categories. Now onto schemes!

The field should be thought of geometrically as a single point corresponding to its unique prime ideal. Then the Galois category which we have constructed is what is called the finite etale site over . More generally, given some scheme we define the finite etale site over to be the category whose object are pairs where is finite etale and morphisms are maps satisfying . A finite etale morphism is a morphism which is both finite and etale. An etale morphism is defined to be flat and unramified where we remind the reader that an unramified morphism is defined to be locally of finite type and for each , we have is a finite separable field extension of where is the maximal ideal in the stalk . Maybe on a first pass, though, it’s best not to worry too much about the technical definition. The point is that an etale map is the geometric analog of a finite cover of topological spaces. In other words if is etale, then can’t look so different. In particular, at the very least they have to be the same dimension.

So lets check that this agrees with our understanding of . In particular, lets figure out for which schemes we have an etale map for a perfect field. is finite so for starters is some finite -algebra. Consider . It has one point and the local ring there is (since this ring is already local!). But the maximal ideal of is just and so the relevant ring in the definition of unramified is which is not even a field! A similar argument shows that will have to be reduced and so will be a product of fields.

So to recap, what we’ve been talking about is how, given a scheme , we can construct a Galois category . In the Galois category of covers of a topological space, the group was exactly the fundamental group of . So in the case of a scheme , we intuitively think of similarly as some sort of fundamental group and so we call the “etale fundamental group of ”.

In fact one can make a concrete connection between the two concepts. If is a complex variety, then we can give the analytic topology coming from the natural topology on . We call the resulting topological space . Then one can prove that the profinite completion of the fundamental group of is exactly the etale fundamental group of .

Galois Groups acting on Geometric Spaces

We are now going to temporarily switch gears. (Don’t worry everything will come together in the end).  Take a field and pick an algebraic closure . Let be two schemes over and let be respectively. Given a map defined over we can uniquely lift to a map defined over . Explicitly, a map over is the same thing as maps   which agree over . comes  equipped with projections to and so composing the first of these with gives the desired maps.

The question we’re interested in is when we can do the converse. In other words, given a map over , when does it descend to a map over ? To answer this question, we consider the action of on . Take . Then acting with on on the right component gives us a map and so composing with we get a different map . Careful though! This new map is not defined over . To get this, we also need to have act on . So we define as

Then the maps that descend are precisely those that are invariant under this Galois action. After all, any map that descends will clearly have to be invariant under the Galois action, but we can argue that an invariant map will locally be defined by equations with coefficients in and so this will allow us to construct .

The Etale Fundamental Group of an Elliptic Curve

Lets work through a careful example of the above ideas. Let be an elliptic curve defined over some number field . If we then base change to , we get some elliptic curve which we can think of as a torus. Since tori have fundamental group , we expect to have a etale fundamental group that is . In fact, we recall that has a group structure and that the multiplication by map is etale.  The kernel of has exactly elements and for any , we have that the translation by map is an automorphism of which commutes with . Thus we have found automorphisms of over itself via . It follows that the pairs are all Galois over and the inverse limit of their automorphism groups is . To confirm that this is in fact the full etale fundamental group of we need to show that the maps are cofinal in the etale site over . This can be done by showing that the etale maps are isogenies and noting that given an isogeny , we get a dual isogeny and their composition is .

Now, fix some and consider the automorphisms for . From our discussion in the previous section, we know that admits a action. But is defined over and therefore so are . Thus, is fixed by this Galois action and so the upshot is that acts on the set of automorphisms , . It’s clear that these actions are compatible over all and so we get a action on the etale fundamental group.  The etale fundamental group we’ve constructed is precisely the product of the Tate module of over each prime, and the Galois action is indeed the usual one.

Lets conclude by sketching a slightly different way to see this action. Returning to defined over , we now try to compute its etale fundamental group. The map gives a functor of their etale sites and therefore etale fundamental groups. Similarly, we have the structure map . One shows that inside , one still has all the multiplication by maps and also a new class of etale morphisms given by base-changing by some finite extension of . Using these morphisms, one shows that the maps and (where denotes the etale fundamental group of ) are injective and surjective respectively. Thus we have an exact sequence

But in any short exact sequence of groups , we can take lifts of into and these lifts give an action of on by conjugation. This is again our desired action.

If we take abelian varieties then (i.e. the morphisms of varieties that preserve the group structures) is a a finitely generated free abelian group. In particular, has the natural structure of a -algebra.

Of course by Yoneda, studying is essentially the same thing as studying and so understanding the is crucial. Vector spaces are generally easier to understand than modules, and so it’s reasonable to consider . Of course, we lose something by passing to . To give a concrete example, the endomorphism ring of an elliptic curve with complex multiplication is given by an order in a quadratic imagery extension of , but there are multiple non-isomorphic orders within a given field extension. However, we expect (and are correct in our expecations) that studying -algebras and vector spaces is easier than working over .

So now we are working in a new category where the objects are still abelian varieties over but the morphisms are given by .  If has an inverse then and so (where denotes multiplication by in X). The converse is also clear.

Since each is surjective and has finite kernel,   gives is surjective and has finite kernel and gives is surjective and has finite kernel.

Conversely, if is surjective and has finite kernel then we see that is a finite group scheme and so killed by some . Then  and so there exists some giving . Similarly, we get .

So to recap, we’ve now shown that the invertible morphisms in this new category are precisely those that are surjective with finite kernel. But this is the ‘traditional’ definition of isogeny, and we have arrived at it through purely wholesome ideas! Another way to look at this is that we’ve ‘localized’ our original category to force the morphisms to be invertible and have shown that in this new category, the isomorphisms are precisely the isogenies.

I already find this quite satisfying, but I should mention some benefits of working up to isogeny.  One can prove the Poincare reducibility theorem which says that if is a subvariety of then we can find a complement of in . i.e there exists a so that is finite and is isogenous (isomorphic in our current category!) with . Using the standard sorts of arguments one can make with any ‘simple’ object, we can actually show that any abelian variety is isogenous to a product where and are not isogenous for . Further, one shows that (the matrix ring over some division algebra).  Thus, classifying abelian varieties up to isogeny becomes an issue of understanding which division algebras can appear.

Furthermore (and perhaps most importantly), we have for finite fields and number fields Tate’s Isogeny theorem which allows us to conclude that abelian varieties are isogenous if and only if their rational Tate modules (i.e. ) are isomorphic as Galois modules.