In The Local Langlands Correspondence for General Linear Groups, we introduced some ideas related to what is known as the Langlands program, and discussed in a little more detail the local Langlands correspondence, at least for general linear groups.
In this post, we will discuss the global Langlands correspondence, but we will focus on the case of function fields over a finite field. This will be somewhat easier to state than the case of number fields, and at the same time perhaps give us a bit more geometric intuition. Let us fix a smooth, projective, and irreducible curve , defined over a finite field . We let be its function field. For instance, if is the projective line over , then .
The case of : Global class field theory for function fields over a finite field
To motivate the global Langlands correspondence for function fields, let us first think of the case, which is a restatement of (unramified) global class field theory for function fields. Recall that in Some Basics of Class Field Theory global class field theory tells us that for global field , its maximal unramified abelian extension , also called the Hilbert class field of , has the property that is isomorphic to the ideal class group.
We recall that there is an analogy between the absolute Galois group and the etale fundamental group in the case when there is no ramification. Therefore, in the case of function fields, the corresponding statement of unramified global class field theory may be stated as
where is the etale fundamental group of , a profinite quotient of through which its action factors ( here serves as the basepoint, which is needed to define the etale fundamental group). The Picard scheme is the scheme such that for any scheme its points correspond to the isomorphism classes of line bundles on . This is analogous to the ideal class group. Taking the fiber product with is analogous to taking the Weil group (see also Weil-Deligne Representations and The Local Langlands Correspondence for General Linear Groups).
The global Langlands correspondence, in the case of , is a restatement of this in terms of maps from each side to some field (we will take this field to be ). It states that there is a bijection between characters , and where is any element of of nonzero degree. Again this is merely a restatement of unramified global class field theory, and nothing has changed in its content. However, this restatement points to us the way in which it may be generalized.
Generalizing to , and then to more general reductive groups
To generalize this, we may take maps instead of maps , since is just . To make it look more like the case of number fields, we may also define this same map as a map which factors through for some open dense subset of . This side we call the “Galois side” (as it involves the Galois group).
What about the other side (the “automorphic side”)? First we recall that classifies line bundles on . We shall replace this by , which classifies rank vector bundles on . It was figured out by Andre Weil a long time ago that may also be expressed as the double quotient (this is known as the Weil parametrization). Now functions on this space will give representations of . We will be interested not in all functions on this space, but in particular certain kinds of functions called cuspidal automorphic forms, which gives a representation that decomposes into pieces that we then want to match up with the Galois representations.
In fact we can generalize even further and consider reductive groups (see also Reductive Groups Part I: Over Algebraically Closed Fields and Reductive Groups Part II: Over More General Fields) other than ! Let be such a reductive group over . Instead of we now consider , the moduli stack (see also Algebraic Spaces and Stacks) of -bundles on . As above, we consider the space of cuspidal automorphic forms on , which we shall denote by . Here is a subgroup of finite index in , where is the center of .
As we are generalizing to more general reductive groups than just , we need to modify the other side (the Galois side) as well. Instead of considering Galois representations, which are group homomorphisms , we must now consider L-parameters, which in this context are group homorphisms , where is the dual group of (which as one may recall from Reductive Groups Part II: Over More General Fields, has the roots and coroots of interchanged).
We may now state the “automorphic to Galois” direction of the global Langlands correspondence for function fields over a finite field , which has been proven by Vincent Lafforgue. It says that we have a decomposition
of the space into subspaces indexed by L-parameters . It is perhaps instructive to compare this with the local Langlands correspondence as stated in Reductive Groups Part II: Over More General Fields, to which it should be related by what is known as local-global compatibility.
(The “Galois to automorphic direction” concerns whether an L-parameter is “cuspidal automorphic”, and we will briefly discuss some partial progress by Gebhard Böckle , Michael Harris, Chandrasekhar Khare, and Jack Thorne later at the end of this post.)
Furthermore the decomposition above must respect the action of Hecke operators (analogous to those discussed in Hecke Operators). Let us now discuss these Hecke operators.
Let be two -bundles on . Let be a point of , and let be an isomorphism of bundles over . We say that is a modification of at . A modification can be bounded by a cocharacter, i.e. a homomorphism . This keeps track and bounds the vector bundles’ relative position.
To get an idea of this, we consider the case . Consider the completion of stalk of the vector bundle at . It is a free module over the completion of the structure sheaf at , which happens to be isomorphic to . Let be a modification of at . There is a basis of such that is a basis of , where . But the numbers is the same as a cocharacter , given by .
The Hecke stack is the stack whose points correspond to modifications at whose relative position is bounded by the cocharacter . It has two maps and to , which send the modification to and respectively. The Hecke operator is the composition . In essence what it does is it sends a function in to the function which sends a point in corresponding to the -bundle to the sum of the values of over all modifications of -bundles at bounded by . In this last description one can see that it is in fact analogous to the description of Hecke operators for modular forms discussed in Hecke Operators.
More generally given a representation of , we can obtain a Hecke operator , and these Hecke operators have the property that if , we must have , and if , we must have . If is irreducible, then we can build as a combination of , where the ‘s are the weights of .
Now let us go back to the decomposition
The statement of the global Langlands correspondence for function fields over a finite field additionally requires that the Hecke operators preserve the subspaces ; that is, they act on each of these subspaces, and do not send an element of such a subspace to another outside of it. Additionally, we require that the action of the Hecke operators are “compatible with the Satake isomorphism”. This means that the action of a Hecke operator is given by multiplication by the scalar . This is somewhat analogous to the Eichler-Shimura relation relating the Hecke operators and the Frobenius briefly mentioned in Galois Representations Coming From Weight 2 Eigenforms.
Ideas related to the proof of the automorphic to Galois direction: Excursion operators and the cohomology of moduli stacks of shtukas
Let us now discuss some ideas related to Vincent Lafforgue’s proof of “automorphic to Galois direction” of the global Langlands correspondence for function fields over a finite field. An important part of these concerns the algebra of excursion operators, denoted by . These are certain endomorphisms of which include the Hecke operators. The idea of the automorphic to Galois direction is that characters correspond uniquely to some L-parameter .
To understand these excursion operators better, we will look at how they are constructed. The construction of the excursion operators involves the cohomology of moduli stacks of shtukas.
A shtuka is a very special kind of a modification of a vector bundle. Given an indexing set , a shtuka over a scheme over consists of the following data:
- A set of points (the are called the “legs” of the shtuka)
- A -bundle over
- An isomorphism
where is the graph of the ‘s. Let us denote the moduli stack of such shtukas by . We take note of the important fact that the moduli stack of shtukas with no legs, , is a discrete set of points and is in fact the same as !
We now want to define sheaves on which will serve as coefficients when we take its etale cohomology, and we want these sheaves to depend on representations of , for the eventual goal of having the cohomology (or appropriate subspaces of it that we want to consider) be functorial in . This is to be accomplished by considering another moduli stack, the moduli stack of modifications over the formal neighborhood of the legs . This parametrizes the following data:
- The set of points
- A pair of -bundles and on the formal completion of along the neighborhood of the union of the the graphs
- An isomorphism
We denote this moduli stack by . The virtue of this moduli stack is that a very important theorem called the geometric Satake equivalence associates to any representation of a certain object called a perverse sheaf on . Now there is a map from to , and pulling back this perverse sheaf associated to we obtain a perverse sheaf on . Now we take the intersection cohomology (we just think of this for now as being somewhat similar to -adic etale cohomology) with compact support of the fiber of over a geometric generic point of with coefficients in . We cut down a “Hecke-finite” (this is a technical condition that we leave to the references for now) subspace of it, and call this subspace . This subspace has an action of .
The above construction is functorial in – that is, a map gives rise to a map . Furthermore, there is this very important phenomenon of fusion. Given a map of sets this is an isomorphism , where is a representation of on the same underlying vector space as , obtained by composing the map from to that sends to with .
Now we can define the excursion operators. Let be a function on . We can then find a representation of and elements , , invariant under the diagonal action of , such that
Let . The excursion operator is defined to be
where is the diagonal representation of on , i.e. we compose the diagonal embedding given by with the representation .
The excursion operators give endomorphisms of . By fusion the subspace is the same as , which, in turn, is the same as (recall that the moduli stack of shtukas with no legs is the same as ). The algebra generated by these endomorphisms as , , and vary is called the algebra of excursion operators, and is denoted by . It is commutative and the different excursion operators satisfy certain natural relations amongst each other.
As stated earlier, the Hecke operators are but particular cases of the excursion operators. Namely, the Hecke operator is just the excursion operator , where sends to .
Now the idea of the decomposition of is as follows. The algebra of excursion operators partitions into eigenspaces , where it acts on each eigenspace as a character . Then as previously mentioned, every character corresponds uniquely to an L-parameter , satisfying
This says therefore that the decomposition of is indexed by L-parameters. Everything we have discussed so far may also be applied with a level structure included, encoded in the form of a finite subscheme of . Then our L-parameters will be maps and we also replace by , which if is split has a Weil parametrization given by , where is the kernel of the map .
Other directions: The Galois to automorphic direction, and the geometric Langlands program
We have so far discussed the “automorphic to Galois direction” of the global Langlands correspondence for function fields over finite fields, and some ideas related to its proof by Vincent Lafforgue. We now briefly discuss the “Galois to automorphic direction” and related work by Gebhard Böckle, Michael Harris, Chandrasekhar Khare, and Jack A. Thorne. This concerns the question of whether a given L-parameter is “cuspidal automorphic”, i.e. it can be obtained from a character of the algebra of excursion operators as stated above.
What Böckle, Harris, Khare, and Thorne do not quite prove this “Galois to automorphic direction” in full. Instead what they prove is that given an everywhere unramified L-parameter with dense Zariski image, then one can find an extension of such that the restriction is cuspidal automorphic. We say that the L-parameter is potentially automorphic.
The way the above “potential automorphy” result is proved is by using techniques similar to that used in modularity (see also Galois Deformation Rings). We recall from our brief discussion in Galois Deformation Rings that the usual approach to modularity has two parts – residual modularity, and modularity lifting. The same is true in potential automorphy. The automorphy lifting part makes use of the same ideas as in the “R=T” theorems in modularity lifting, although in this context, they are called “R=B” theorems instead, since we are considering excursion operators instead of just the Hecke operators.
To obtain an analogue of the residual modularity part, Böckle, Harris, Khare, and Thorne make use of results of Alexander Braverman and Dennis Gaitsgory from what is known as the geometric Langlands correspondence (for function fields over a finite field). Although we will not discuss the work of Braverman and Gaitsgory here, we will end this post with a rough idea of what the geometric Langlands correspondence is about.
The geometric Langlands correspondence replaces the cuspidal automorphic forms (which as we recall are -valued functions on ) with -valued sheaves (actually a complex of -valued sheaves, or more precisely an object of the category the “derived category of -valued sheaves with constructible cohomologies”) via Grothendieck’s sheaves to functions dictionary.
Suppose we have some scheme over . First let us suppose that . Then since is just a point, a complex of sheaves on is just a complex of vector spaces (we shall take the sheaves to be -valued, so this complex is a complex of -vector spaces). This complex has an action of . Now we take the alternating sum of the traces of Frobenius acting on this complex, and this gives us an element of . For more general , for every point we apply this same construction to the sheaf which is the pullback of the sheaf on to via the morphism . This provides us with a -valued function on .
One can also go in the other direction constructing complexes of sheaves given certain functions. Suppose we have a commutative connected algebraic group and suppose we have a character of . Then we can associate to this character an element of as follows. We have the Lang isogeny given by for some element of . The Lang isogeny defines a covering map of whose group of deck transformations is the group . But because we have a character (a -valued function on ), we can take the composition
This gives us a -dimensional representation of . This in turn gives us a -dimensional local system, which is known by the theory of constructible sheaves to be an object of . This resulting sheaf is also called a character sheaf. In the case when it is called the Kummer sheaf, and when it is called the Artin-Schreier sheaf.
Grothendieck’s sheaves to functions dictionary is the inspiration for the geometric Langlands correspondence, which is stated entirely in terms of sheaves. We consider the same setting as before, but we now define a slightly modified version of the Hecke stack where aside from parametrizing modifications we also include in the data being parametrized the point being removed to make the modification. Let be the map that gives us this point on . Given a representation of we let be the perverse sheaf on given by geometric Satake as discussed earlier, and we define the Hecke functor that sends an object of to an object of follows
Then the geometric Langlands correspondence (for function fields over a finite field) states that given an L-parameter , one can find a Hecke eigensheaf, i.e. a sheaf such that applying the Hecke functor to it we have where is the local system associated to the representation .
A version of the geometric Langlands correspondence has also been formulated for function fields over instead of . Many things have to be modified, since in this case there is no Frobenius, and instead the theory of “D-modules” takes its place. This version of the geometric Langlands correspondence has found some fascinating connections to mathematical physics as well.
More recently, a very general and abstract formulation of the geometric Langlands correspondence has been formulated by replacing L-parameters by coherent sheaves on the moduli stack of L-parameters (a single L-parameter corresponding instead to a skyscraper sheaf on the corresponding point). This allows one to have the entire formulation be stated as an equivalence of categories between derived categories of constructible sheaves on on one side, and coherent sheaves on the moduli stack of L-parameters. This conjectural statement, appropriately modified to be made more precise (i.e. the moduli stack on the Galois side needs to be modified to parametrize “local systems with restricted variation” while the sheaves on both sides need to be ind-constructible, resp. ind-coherent, with nilpotent singular support), is also known as the categorical geometric Langlands correspondence.
We have given a rough overview of the ideas involved in the global Langlands correspondence for function fields over a finite field. Hopefully we will be able to dive deeper into the finer aspects of the theory, as well as discuss other closely related aspects of the Langlands program (for example the global Langlands correspondence for number fields) in future posts on this blog.
Shtukas for reductive groups and Langlands correspondence for function fields by Vincent Lafforgue
Global Langlands parameterization and shtukas for reductive groups by Vincent Lafforgue (plenary lecture at the 2018 International Congress of Mathematicians)
Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale by Vincent Lafforgue
Potential automorphy of -local systems by Jack A. Thorne
Potential automorphy of -local systems by Jack A. Thorne (invited lecture at the 2018 International Congress of Mathematicians)
-local systems are potentially automorphic by Gebhard Böckle, Michael Harris, Chandrasekhar Khare, and Jack A. Thorne
Geometrization of the local Langlands program (notes by Tony Feng from a workshop at McGill University)
The geometric Langlands conjecture (notes from Oberwolfach Arbeitsgemeinschaft)
Recent progress in geometric Langlands theory by Dennis Gaitsgory
The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support by Dima Arinkin, Dennis Gaitsgory, David Kazhdan, Sam Raskin, Nick Rozenblyum, and Yakov Varshavsky
An Introduction to the Langlands Program by Daniel Bump, James W. Cogdell, Ehud de Shalit, Dennis Gaitsgory, Emmanuel Kowalski, and Stephen S. Kudla (edited by Joseph Bernstein and Stephen Gelbart)