In The Global Langlands Correspondence for Function Fields over a Finite Field, we introduced the global Langlands correspondence for function fields over a finite field, and Vincent Lafforgue’s work on the automorphic to Galois direction of the correspondence. In this post we will discuss the work of Laurent Fargues and Peter Scholze which uses similar ideas but applies it to the local Langlands correspondence (and this time it works not only for “equal characteristic” cases like Laurent series fields but also for “mixed characteristic” cases like finite extensions of ). Note that instead of having complex coefficients like in The Local Langlands Correspondence for General Linear Groups, here we will use -adic coefficients.
I. The Fargues-Fontaine Curve
Let us briefly discuss the idea of “geometrization” and what is meant by Fargues and Scholze making use of V. Lafforgue’s work. Recall that V. Lafforgue’s work concerns the global Langlands correspondence for function fields over a finite field , which on one side concerns the space of cuspidal automorphic forms, which are certain functions on , which in turn parametrizes -bundles on some curve over , and on the other side concerns representations (or more precisely L-parameters) of the etale fundamental group of (which can also be phrased in terms of the Galois group of its function field).
Perhaps the first question that comes to mind is, what is the analogue of the curve in the case of the local Langlands correspondence when the field is not a function field (or more correctly a power series field, since it has to be local) over , but some finite extension of ? Let be this finite extension of . Since the absolute Galois group of is also the etale fundamental group of , perhaps we should take to be our analogue of .
However, in the traditional formulation of the local Langlands correspondence, it is the Weil group that appears instead of the absolute Galois group itself. Considering the theory of the Weil group in Weil-Deligne Representations, this means that we will actually want , where is the maximal unramified extension of and is the Frobenius, instead of .
Now, we want to “relativize” this. For instance, in The Global Langlands Correspondence for Function Fields over a Finite Field, we considered , which parametrizes -bundles on the curve over . But we may also want to consider say , where is some -algebra; this would parametrize -bundles on instead. In fact, we need this “relativization” to properly define as a stack (see also Algebraic Spaces and Stacks).
The problem with transporting this to the case of a finite extension of is that we do not have an “base” like was for the function field case (unless perhaps if we have something like an appropriate version of the titular object in The Field with One Element, which is at the moment unavailable). The solution to this is provided by the theory of adic spaces and perfectoid spaces (see also Adic Spaces and Perfectoid Spaces).
For motivation, let us consider first the case where our field is . Let be a perfectoid space over with pseudouniformizer . Consider the product . We may look at this as the punctured open unit disc over . It sits inside as the locus where the pseudo-uniformizer of and the uniformizer of is invertible (or “nonzero”).
In the case where our field is , a finite extension of , as mentioned earlier we have no “base” like was for . So we cannot form the fiber products analogous to or . However, notice that
This has an analogue in the mixed-characteristic, given by the theory of Witt vectors (compare, for instance and its “mixed-characteristic analogue” )! If is the residue field of , we define the ramified Witt vectors to be . This is the analogue of . Now all we have to do to find the analogue of that we are looking for is to define it as the locus in where both the uniformizer of and the uniformizer of are invertible!
We denote this locus by . But recall again our discussion earlier, that due to the local Langlands correspondence being phrased in terms of the Weil group, we have to quotient out by the powers of Frobenius. Therefore we define the Fargues-Fontaine curve to be .
Aside from our purpose of geometrizing the local Langlands correspondence, the Fargues-Fontaine curve is in itself a very interesting mathematical object. For instance, when is a complete algebraically closed nonarchimedean field over , the classical points of (i.e. maximal ideals of the rings such that is locally ) correspond to untilts of (modulo the action of Frobenius)!
There is also a similar notion for more general . To explain this we need the concept of diamonds, which will also be very important for the rest of the post. A diamond is a pro-etale sheaf on the category of perfectoid spaces over , which is the quotient of some perfectoid space over by a pro-etale equivalence relation (we also say that the diamond is a coequalizer). An example of a diamond is given by . Note that is not perfectoid, but is the quotient of a perfectoid field we denoted in Adic Spaces and Perfectoid Spaces by the action of . Now we can take the tilt and quotient out by (the underline notation will be explained later – for now we think of this as making the group into a perfectoid space) – this is the diamond . More generally, if is an adic space over satisfying certain conditions (“analytic”), we can define the diamond to be such that , for a perfectoid space over , is the set of isomorphism classes of pairs , being the untilt of . If , we also use to denote . Note that if is already perfectoid, is just the same thing as the tilt .
Now recall that was defined to be the locus in where the uniformizer of and the uniformizer of were invertible. We actually have that , and, for the Fargues-Fontaine curve , we have that .
Our generalization of the statement that the points of parametrize untilts of is now as follows. There exists a three-way bijection between sections of the map , maps , and untilts over of . Given such an untilt , this defines a closed Cartier divisor on , which in turn gives rise to a closed Cartier divisor on . By the bijection mentioned earlier, these closed Cartier divisors on will be parametrized by maps .
The closed Cartier divisors that arise in this way will be referred to as closed Cartier divisors of degree . We have seen that they are parametrized by the following moduli space we denote by (this will also become important later on):
Now that we have discussed the Fargues-Fontaine curve and some of its properties, we can define as the stack that assigns to any perfectoid space over the groupoid of -bundles on .
When , our -bundles are just vector bundles. In this case we shall also denote by .
II. Vector Bundles on the Fargues-Fontaine Curve
Let us now try to understand a little bit more about vector bundles on the Fargues-Fontaine curve. They turn out to be related to another important thing in arithmetic geometry – isocrystals – and this will allow us to classify them completely.
Let be the completion of the maximal unramified extension of . Letting denote the residue field of , may also be expressed as the fraction field of . It is equipped with a Frobenius lift . An isocrystal over is defined to be a vector space over equipped with a -semilinear automorphism.
Given an isocrystal over , we can obtain a vector bundle on the Fargues-Fontaine curve by defining . It turns out all the vector bundles over can be obtained in this way!
Now the advantage of relating vector bundles on the Fargues-Fontaine curve to isocrystals is that isocrystals are completely classified via the Dieudonne-Manin classification. This says that the category of isocrystals over is semi-simple (so every object is a direct sum of the simple objects), and the form of the simple objects are completely determined by two integers which are coprime, the rank (i.e. the dimension as an -vector space) which must be positive, and the degree (which determines the form of the -semilinear automorphism) . Since these two integers are coprime and one is positive, there is really only one number that completely determines a simple -isocrystal – its slope, defined to be the rational number . Therefore we shall also often denote a simple -isocrystal as . Since isocrystals over and vector bundles on the Fargues-Fontaine curve are in bijection, if we have a simple -isocrystal we shall denote the corresponding vector bundle by . More generally, an isocrystal is a direct sum of simple isocrystals and they can have different slopes. If an isocrystal only has one slope, we say that it is semistable (or basic). We use the same terminology for the corresponding vector bundle.
More generally, for more general reductive groups , we have a notion of -isocrystals; this can also be thought of functors from the category of representations of over to the category of isocrystals over . These are in correspondence with -bundles over the Fargues-Fontaine curve. There is also a notion of semistable or basic for -isocrystals, although its definition involves the Newton invariant (one of two important invariants of a -isocrystal, the other being the Kottwitz invariant).
The set of -isocrystals is denoted and is also called the Kottwitz set. This set is in fact also in bijection with the equivalence classes in under “Frobenius-twisted conjugacy”, i.e. the equivalence relation . Given an element of , we can define the algebraic group to be such that the elements of are the elements of satisfying the condition . If , then .
The groups are inner forms of (see also Reductive Groups Part II: Over More General Fields). More precisely, the are the extended pure inner forms of , which are all the inner forms of if the center of is connected. Groups which are inner forms of each other are in some way closely related under the local Langlands correspondence – for instance, they have the same Langlands dual group. It has been proposed that these inner forms should really be studied “together” in some way, and we shall see that the use of to formulate the local Langlands correspondence provides a realization of this approach.
Let us mention one more important part of arithmetic geometry that vector bundles on the Fargues-Fontaine curve are related to, namely p-divisible groups. A p-divisible group (also known as a Barsotti-Tate group) is an direct limit of group schemes
such that is a finite flat commutative group scheme which is -torsion of order and such that the inclusion induces an isomorphism of with (the kernel of the multiplication by map in ). The number is called the height of the p-divisible group.
An example of a p-divisible group is given by . This is a p-divisible group of height . Given an abelian variety of dimension , we can also form a p-divisible group of height by taking the direct limit of its -torsion.
We can also obtain p-divisible groups from formal group laws (see also The Lubin-Tate Formal Group Law) by taking the direct limit of its -torsion. In this case we can then define the dimension of such a p-divisible group to be the dimension of the formal group law it was obtained from. More generally, for any p-divisible group over a complete Noetherian local ring of residue characteristic , the connected component of its identity always comes from a formal group law in this way, and so we can define the dimension of the p-divisible group to be the dimension of this connected component.
Now it turns out p-divisible groups can also be classified by a single number, the slope, defined to be the dimension divided by the height. If the terminology appears suggestive of the classification of isocrystals and vector bundles on the Fargues-Fontaine curve, that’s because it is! Isocrystals (and therefore vector bundles on the Fargues-Fontaine curve) and p-divisible groups are in bijection with each other, at least in the case where the slope is between and . This is quite important because the cohomology of deformation spaces of p-divisible groups (such as that obtained from the Lubin-Tate group law) have been used to prove the local Langlands correspondence before the work of Fargues and Scholze! We will be revisiting this later.
III. The Geometry of
Let us now discuss more about the geometry of . It happens that is a small v-sheaf. A v-sheaf is a sheaf on the category of perfectoid spaces over equipped with the v-topology, where the covers of are any maps such that for any quasicompact there are finitely many which cover . A v-sheaf is small if it admits a surjective map from a perfectoid space. In particular being a small v-sheaf implies that has an underlying topological space . The points of this topological space are going to be in bijection with the elements of the Kottwitz set .
If is a locally profinite topological group, we define to be the functor from perfectoid spaces over which sends a perfectoid space over to the set . We let be the classifying stack of -bundles; this means that we can obtain any -bundle on any perfectoid space over by pulling back a universal -bundle on .
We write for the locus in corresponding to the -isocrystals that are semistable. We let the substack of whose underlying topological space is . It turns out that we have a decomposition
More generally, even is is not basic, we have an inclusion
Let us now look at some more of the properties of . In particular, satisfies the conditions for an analogue of an Artin stack (see also Algebraic Spaces and Stacks) but with locally spatial diamonds instead of algebraic spaces and schemes.
A diamond is called a spatial diamond if it is quasicompact quasiseparated, and its underlying topological space is generated by , where runs over all sub-diamonds of which are quasicompact. A diamond is called a locally spatial diamond if it admits an open cover by spatial diamonds.
Now we recall from Algebraic Spaces and Stacks that to be an Artin stack, a stack must have a diagonal that is representable in algebraic spaces, and it has charts which are representable by schemes. It turns out satisfies analogous properties – its diagonal is representable in locally spatial diamonds, and it has charts which are representable by locally spatial diamonds.
We can now define a derived category (see also Perverse Sheaves and the Geometric Satake Equivalence) of sheaves on the v-site of with coefficients in some -algebra . If is torsion (e.g. or ), this can be the category , which is the subcategory of whose pullback to any strictly disconnected perfectoid space lands in (here the subscripts and denote the v-site and the etale site respectively). If is not torsion (e.g. or ) one needs the notion of solid modules (which was further developed in the work of Clausen and Scholze on condensed mathematics) to construct the right derived category.
If is a spatial diamond and is a pro-etale map expressible as a limit of etale maps , we can construct the sheaf as the limit . We say that a sheaf on is solid if is isomorphic to . We can extend this to small v-stacks – if is a small v-stack and is a v-sheaf on , we say that is solid if for every map from a spatial diamond to the pullback of to coincides with the pullback of a solid sheaf from the quasi-pro-etale site of . We denote by the subcategory of whose objects have cohomology sheaves which are solid. Now if we have a solid -algebra , we can consider inside , and we denote by the subcategory of objects of whose image in is solid.
This category is still too big for our purposes. Therefore we cut out a subcategory as follows. If we have a map of v-stacks , we have a pullback map . This pullback map has a left-adjoint . We define to be the smallest triangulated subcategory stable under direct sums that contain , for all which are separated, representable by locally spatial diamonds, and -cohomologically smooth. If is torsion, then coincides with .
Let be the derived category of smooth representations of the group over . We have
Now taking the pushforward of this derived category of sheaves through the inclusion , and using the isomorphism above, we get
Now we can see that this derived category of sheaves on encodes the representation theory of , which is one side of the local Langlands correspondence, but more than that, it encodes the representation theory of all the extended pure inner forms of altogether.
The properties of mentioned earlier, in particular its charts which are representable by locally spatial diamonds, allow us to define properties of objects in which translate into properties of interest in . For example, we have a notion of being compactly generated, and this translates into a notion of compactness for . We also have a notion of Bernstein-Zelevinsky duality for , which translates into Bernstein-Zelevinsky duality for , and finally, we have a notion of universal local acyclicity in , which translates into being admissible for .
IV. The Hecke Correspondence and Excursion Operators
Now let us look at how the strategy in The Global Langlands Correspondence for Function Fields over a Finite Field works for our setup. We will be working in the “geometric” setting (i.e. sheaves or complexes of sheaves instead of functions) mentioned at the end of that post, so there will be some differences from the work of Lafforgue that we discussed there, although the motivations and main ideas (e.g. excursion operators) will be somewhat similar.
Just like in The Global Langlands Correspondence for Function Fields over a Finite Field, we will have a Hecke stack that parametrizes modifications of -bundles over the Fargues-Fontaine curve. This means that is the groupoid of triples where and are -bundles over and is an isomorphism of vector bundles meromorphic on some degree Cartier divisor on (which is part of the data of the modification). Note that we have maps and which sends the triple to and respectively.
Now we need to bound the relative position of the modification. Recall that this is encoded via (conjugacy classes of) cocharacters . The way this is done in this case is via the local Hecke stack , which parametrizes modifications of -bundles on the completion of along (compare the moduli stacks denoted inThe Global Langlands Correspondence for Function Fields over a Finite Field). The local Hecke stack admits a stratification into Schubert cells labeled by conjugacy classes of cocharacters . We can now pull back a Schubert cell to the global Hecke stack to get a substack with maps and , and define a Hecke operator as
More generally, to consider compositions of Hecke operators we need to consider modifications at multiple points. For this we will need the geometric Satake equivalence.
Let be an affinoid perfectoid space over . For each in some indexing set , we let be a Cartier divisor on . Let be the completion of along the union of the , and let be the localization of obtained by inverting the . For our reductive group , we define the positive loop group to be the functor which sends an affinoid perfectoid space to , and we define the loop group to be the functor which sends to .
We define the Beilinson-Drinfeld Grassmannian to be the quotient . We further define the local Hecke stack to be the quotient .
The geometric Satake equivalence tells us that the category of perverse sheaves on satisfying certain conditions (quasicompact over , flat over , universally locally acyclic) is equivalent to the category of representations of on finite projective -modules.
Let be such a representation of representations of . Let be the corresponding object of . The global Hecke stack has a map to the local Hecke stack . It also has maps to to and respectively. We can now define the Hecke operator as follows:
Once we have the Hecke operators, we can then consider excursion operators and apply the strategy of Lafforgue discussed in The Global Langlands Correspondence for Function Fields over a Finite Field. We set to be . Let be an excursion datum, i.e. is a finite set, is a representation of , , , and for all . An excursion operator is the following composition:
Now this composition turns out to be the same as multiplication by the scalar determined by the following composition:
And the that appears here is precisely the L-parameter that we are looking for. This therefore gives us the “automorphic to Galois” direction of the local Langlands correspondence.
V. Relation to Local Class Field Theory
It is interesting to look at how this all works in the case , i.e. local class field theory. There is historical precedent for this in the work of Pierre Deligne for what we might now call the case of the (geometric) global Langlands correspondence for function fields over a finite field, but which might also be called geometric class field theory.
Let us go back to the setting in The Global Langlands Correspondence for Function Fields over a Finite Field, where we are working over a function field of some curve over the finite field . Since we are considering , our in this case will be the Picard group , which parametrizes line bundles on . The statement of the geometric Langlands correspondence in this case is that there is an equivalence of character sheaves on (see the discussion of Grothendieck’s sheaves to functions dictionary at the end of The Global Langlands Correspondence for Function Fields over a Finite Field) and -local systems of rank on (these are the same as one-dimensional representations of ).
We have an Abel-Jacobi map , sending a point of to the corresponding divisor in . More generally we can define , where is the quotient of by the symmetric group on its factors, and is the degree part of .
Now suppose we have a rank -local system on , which we shall denote by . We can form a local system on . We can push this forward to and get a sheaf on . What we hope for is that this sheaf is the pullback of the character sheaf on that we are looking for via . This is in fact what happens, and what makes this possible is that the fibers of are simply connected for , by the Riemann-Roch theorem. So for this , by taking fundamental groups of the fiber sequence, we have that . So representations of give rise to representations of , and since representations of the fundamental group are the same as local systems, we see that there must be a local system on , and furthermore the sheaf is the pullback of this local system. There is then an inductive method to extend this to , and we can check that the local system is a character sheaf.
Now let us go back to our case of interest, the local Langlands correspondence. Instead of the curve we will use , the moduli of degree Cartier divisors. It will be useful to have an alternate description of in terms of Banach-Colmez spaces.
For any perfectoid space over and any vector bundle over , the Banach-Colmez space is the locally spatial diamond such that . We define to be such that are the sections in which are nonzero fiberwise on .
There is a map from to which sends a section to , which in turn induces an isomorphism . A more explicit description of this map is given by Lubin-Tate theory (see also The Lubin-Tate Formal Group Law). After choosing a coordinate, the Lubin-Tate formal group law with an action of , over , is isomorphic to . We can form the universal cover which is isomorphic to . Now let be a perfectoid space with tilt . We have , where is the set of topologically nilpotent elements in , and the map which sends a topologically nilpotent element to the power series gives a map to , which upon quotienting out by the action of Frobenius gives an isomorphism between and .
What this tells us is that . Defining to be the completion of the union over all of the -torsion points of in , we have that . This is an -torsor over , and then quotienting out by the action of Frobenius we obtain our map to .
More generally, we have an isomorphism , where parametrized degree relative Cartier divisors on .
Now that we have our description of (and more generally ) in terms of Banach-Colmez spaces, let us now see how we can translate the strategy of Deligne to the local case. Once again we have an Abel-Jacobi map
Given a local system on , we want to have a character sheaf on whose pullback to is precisely this local system. Again what our strategy hinges will be whether will be simply connected. And in fact this is true for , and by a result called Drinfeld’s lemma for diamonds this will actually be enough to prove the local Langlands correspondence for (i.e. it is not needed for – in fact this is false for !). The fact that is simply connected for is a result of Fargues, and, at least for the characteristic case, follows from expressing , whose category of etale covers is the same as that of . Then Zariski-Nagata purity allows one to reduce this to showing that is simply connected, which it is by Hensel’s lemma.
VI. The Cohomology of Local Shimura Varieties
Many years before the work of Fargues and Scholze, the case of the local Langlands correspondence (see also The Local Langlands Correspondence for General Linear Groups) was originally proven using the cohomology of the Lubin-Tate tower (which we shall denote by ) which parametrizes deformations of the Lubin-Tate formal group law (see also The Lubin-Tate Formal Group Law) with level structure, together with the cohomology of Shimura varieties. Let us now investigate how the cohomology of the Lubin-Tate tower can be related to what we have just discussed.
It turns out that because of the relationship between Lubin-Tate formal group laws, p-divisible groups, and vector bundles on the Fargues-Fontaine curve, the Lubin-Tate tower is also a moduli space of modifications of vector bundles on the Fargues-Fontaine curve, but of a very specific kind! Namely, it parametrizes modifications where we fix the two vector bundles, and furthermore one has to be the trivial bundle and the other a degree bundle , and so the only thing that varies is the isomorphism between them (as opposed to the Hecke stack, where the vector bundles can also vary) away from a point. So we see that the Lubin-Tate tower is a part of the Hecke stack (we can think of it as the fiber of the Hecke stack above ).
More generally, the Lubin-Tate tower is a special case of a local Shimura variety at infinite level, which is itself related to a special case of a moduli stack of local shtukas. These parametrize modifications of -bundles and , which are bounded by some cocharacter . This moduli stack of local shtukas, denoted , is an inverse limit of locally spatial diamonds with “level structure” given by some compact open subgroup of . In the case where the cocharacter is miniscule, the data is called a local Shimura datum, and we define the local Shimura variety at infinite level, denoted , to be such that . It is similarly a limit of local Shimura varieties at finite level , denoted , and for each we have .
Let us now see how the cohomology of the moduli stack of local shtukas is related to our setup. We will consider the case of finite level, i.e. , since the cohomology at infinite level may be obtained as a limit. Consider the inclusion . Now consider the object of . Now for our cocharacter , we have a Hecke operator , and we apply this Hecke operator to obtain . Now we pull this back through the inclusion , to get an object of . We can think of all this happening not on the entire Hecke stack, but only on , since we are specifically only considering this very special kind of modification parametrized by . But the derived pushforward from to a point gives (from which we can compute the cohomology).
This relationship between the cohomology of the moduli stack of local shtukas and sheaves on , as we have just discussed, has been used to obtain new results. For instance, David Hansen, Tasho Kaletha, and Jared Weinstein used this formulation together with the concept of the categorical trace to prove the Kottwitz conjecture.
Let be a smooth irreducible representation of over . We define
Let be the centralizer of in . Given a representation in the L-packet and a representation in the L-packet , the refined local Langlands correspondence gives us a representation of . We let be the extension of the highest-weight representation of to . The Kottwitz conjecture states that
The approach of Hansen, Kaletha, and Weinstein involve first using a generalized Jacquet-Langlands transfer operator . We define the regular semisimple elements in to be the semisimple elements whose connected centralizer is a maximal torus, and we define the strongly regular semisimple elements to be the regular semisimple elements whose centralizer is connected. We denote their corresponding open subvarieties in by and respectively. The generalized Jacquet-Langlands transfer operator is defined to be
Here the set is the set of all triples where , , and is a certain specially defined element of ( being the centralizer of in ) that depends on and . When applied to the Harish-Chandra character , we have
Next we have to relate this to the cohomology of the moduli stack of local shtukas. We first need the language of distributions. We define
To any object of , we can associate an object of . We also have “elliptic” versions of these constructions, i.e. an object of the category . Now we can define the action of the generalized Jacquet-Langlands transfer operator on . The hope will be that we will have the following equality:
Proving this equality is where the geometry of (and the Hecke stack) and the trace formula come into play. The action of the generalized Jacquet-Langlands transfer operator on can be described in a similar way to a Hecke operator where we pull back to the moduli of local Shtukas, multiply by a kernel function, and then push forward.
On the other side, one needs to compute . Here we use that . This is a version of the expression of the cohomology of the moduli stack of local shtukas that we previously discussed where is the pullback to the Hecke stack of the sheaf corresponding to provided by the geometric Satake equivalence and before pushing forward via we are pulling back to the degree part of the Hecke stack, which is why we have (the embedding of this degree part) and denotes that we are pushing forward from this degree part.
Hansen, Kaletha, and Weinstein then apply a categorical version of the Lefschetz-Verdier trace formula (using a framework developed by Qing Lu and Weizhe Zheng) to be able to relate to .
Let us discuss briefly the setting of this categorical trace. We consider a category whose objects are pairs where is an Artin v-stack over and . The morphisms in this category are given by a pair of maps where is smooth-locally representable in diamonds, together with a map . We also write for the pair . Given an endomorphism the categorical trace of is given by where is the pullback of and and (here is the dualizing sheaf, which may obtained as the right-derived pullback of via the structure morphism of ). In the special case where the correspondence arises form an automorphism of , and , then one may think of as the fixed points of and the categorical trace gives an element of (the local term) for each fixed point.
For Hansen, Kaletha, and Weinstein’s application, they consider to be the identity. The categorical trace is then given by , where is the inertia stack, classifying pairs with an automorphism of , and is called the characteristic class.
The idea now is that certain properties of the setting we are considering (such as universal local acyclicity) allow us to identify the trace distribution as a characteristic class . From there we can use properties of the abstract theory to relate it to (for instance, we can use a Kunneth formula for the characteristic class to decouple the parts involving and , and relate the former to pulling back to the moduli stack of local shtukas, and relate the part involving the latter to multiplication by the kernel function).
VII. The Spectral Action
We have seen that the machinery of excursion operators gives us the automorphic to Galois direction of the local Langlands correspondence. We now describe one possible approach to obtain the other (Galois to automorphic) direction. We are going to use the language of the categorical geometric Langlands correspondence mentioned at the end of in The Global Langlands Correspondence for Function Fields over a Finite Field.
Recall our construction of the moduli stack of local -adic Galois representations in Moduli Stacks of Galois Representations. Using the same strategy we can construct a moduli stack of L-parameters, which we shall denote by . This notation comes from the fact that in Fargues and Scholze’s work the L-parameters can be viewed as 1-cocycles.
Let denote the subcategory of compact objects in . The categorical local Langlands correspondence in this case is the following conjectural equivalence of categories:
Here the right-hand side is the derived category of bounded complexes on with quasicompact support, coherent cohomology, and nilpotent singular support. We will leave the definition of these terms to the references, but we will think of as being a derived category of coherent sheaves on .
We now outline an approach to proving the categorical local Langlands correspondence. Let be the category of perfect complexes on . Then there is an action of on , called the spectral action, such that composing with the map gives us the action of the Hecke operator.
The idea is that the spectral action gives us a functor from to , sending an object of to the object of , where is the Whittaker sheaf (the sheaf on corresponding to the representation , where is a Borel subgroup of , is the unipotent radical of , and is a character of ). The hope is then that this functor can be extended from to all of , and that it will provide the desired equivalence of categories.
Now we discuss how this spectral action is constructed. Let us first consider the following more general situation. Let be a field of characteristic , let be a split reductive group, and let be a discrete group. We write and for their corresponding classifying spaces. Let be an idempotent-complete, -linear stable -category.
For all , a -equivariant, exact tensor action of on is a functor
natural in , exact as an action of after forgetting the -equivariance, and such that the action of is compatible with the tensor product.
Now what we want to show is that a -equivariant, exact tensor action of on is the same as an -linear action of on .
To prove the above statement, Fargues and Scholze use the language of higher category theory. Let be the -category of anima, which is obtained from simplicial sets by inverting weak equivalences. The specific anima that we are interested in is , which is obtained by taking the nerve of the category . An important property of is that it is freely generated under sifted colimits by the full subcategory of finite sets.
We now define two functors and from to . The functor sends a finite set to the exact -linear actions of on , which is equivalent to the exact -linear monoidal functors from to . The functor sends a finite set to the -equivariant exact actions of on , which is equivalent to natural transformations from the functor to the functor .
There is a natural transformation from to that happens to be an isomorphism on finite sets. Now since the category is generated by finite sets under sifted colimits, all we need is for the functors and to preserve sifted colimits.
For this follows from the fact that preserves sifted colimits. For , this comes from the fact that for some animated -algebra and some set , and then looking at the structure of and .
Now that we have our abstract theory let us go back to our intended application. Let be the Weil group of . It turns out that every L-parameter factors through a quotient , where is some open subgroup of the wild inertia. This means that is the union of all over all such (compare also with the construction in Moduli Stacks of Galois Representations), and this also means that we can focus our attention on .
We can actually go further and replace with its subgroup generated by the elements and satisfying , together with the wild inertia (we have also already considered this in Moduli Stacks of Galois Representations, where we called it ), and get the same moduli space, i.e. .
Let be the free group on generators. For every map , we have a map
The category is a sifted category, and upon taking sifted colimits, we obtain an isomorphism
There is also a version of this statement that involves higher category theory. It says that the map
is an isomorphism in the stable -category . Furthermore the category generates under cones and retracts, and identifies with the -category of -modules inside .
If we take invariants under the action of , we then have
Note that is precisely the same data as the algebra of excursion operators. We can see this using the fact that is isomorphic to , and is functions on which are invariant under the action of . But this is the same as the data of an excursion operator ( here has elements), because such a function is of the form .
Now that we have our description of as , we can now apply the abstract theory developed earlier to obtain our spectral action.
Let us now focus on the case of and relate the spectral action to the more classical language of Hecke eigensheaves (see also The Global Langlands Correspondence for Function Fields over a Finite Field). Let be an algebraically closed field over . Given an L-parameter , we have an inclusion and a sheaf on . For any we can take the spectral action . This turns out to be a Hecke eigensheaf! However, it is often going to be zero. Still, in work by Johannes Anschütz and Arthur-César Le Bras, they show that the above construction can give an example of a nonzero Hecke eigensheaf, by relating the spectral action to an averaging functor, which is an idea that comes from the work of Edward Frenkel, Dennis Gaitsgory, and Kari Vilonen on the geometric Langlands program.
VIII. The p-adic local Langlands correspondence
The work of Fargues and Scholze deals with the “classical” (i.e. ) local Langlands correspondence. As we have seen for example in Completed Cohomology and Local-Global Compatibility, the p-adic local Langlands correspondence (i.e. ) is much more complicated and mysterious compared to the classical case. Still, one might wonder whether the machinery we have discussed here can be suitably modified to obtain an analogous “geometrization” of the p-adic local Langlands correspondence.
Since we are dealing with what we might call p-adic, instead of -adic, Galois representations, we would have to replace with the moduli stack of -modules (also known as the Emerton-Gee stack, see also Moduli Stacks of (phi, Gamma)-modules).
We still would like to work with the derived category of some sort of sheaves on . This is because, in work of Pierre Colmez, Gabriel Dospinescu, and Wieslawa Niziol (and also in related work of Peter Scholze which uses a different approach), the p-adic etale cohomology of the Lubin-Tate tower has been used to realize the p-adic local Langlands correspondence, and we have already seen that the Lubin-Tate tower is related to and the Hecke stack. Since p-adic etale cohomology is the subject of p-adic Hodge theory (see also p-adic Hodge Theory: An Overview), we might also expect ideas from p-adic Hodge theory to become relevant.
So now have to find some sort of p-adic replacement for . It is believed that the correct replacement might be the derived category of almost solid modules, whose theory is currently being developed by Lucas Mann. Some of the ideas are similar to that used by Peter Scholze to formulate p-adic Hodge theory for rigid-analytic varieties (see also Rigid Analytic Spaces), but also involves many new ideas. Let us go through each of the meanings of the words in turn.
The “almost” refers to theory of almost rings and almost modules developed by Gerd Faltings (see also the discussion at the end of Adic Spaces and Perfectoid Spaces). For an -module over a local ring , we say that is almost zero if it is annihilated by some element of the maximal ideal of . We define the category of almost -modules (or -modules) to be the category of -modules modulo the category of almost zero modules.
The “solid” refers to the theory of solid rings and solid modules discussed earlier, although we will use the later language developed by Dustin Clausen and Peter Scholze. Let be a ring. We define the category of condensed -modules, denoted , to be the category of sheaves of -modules on the category of profinite sets. Given a profinite set , we define to be the limit , where runs over all finite-type -algebras contained in , and we define the category of solid -modules, denoted , to be the subcategory of generated by . The idea of condensed mathematics is to incorporate topology – for instance the category of compactly generated weak Hausdorff spaces, which forms most of the topological spaces we care about, embeds fully faithfully into the category of condensed sets. On the other hand, condensed abelian groups, rings, modules, etc. have nice algebraic properties, for instance when it comes to forming abelian categories, which topological abelian groups, rings, modules, etc. do not have. The solid rings and solid modules corresponds to “completions”, and in particular they have a reasonable “completed tensor product” that will become useful to us later on when forming derived categories.
Finally, the “derived category” refers to the same idea of a category of complexes with morphisms up to homotopy and quasi-isomorphisms inverted, as we have previously discussed, except, however, that we need to actually not completely forget the homotopies; in fact we need to remember not only the homotopies but the “homotopies between homotopies”, and so on, and for this we need to formulate derived categories in the language of infinity category theory. The reason why we need to this is because our definition will involve “gluing” derived categories, and for this we need to remember the homotopies, including the higher ones.
Let us now look at how Mann constructs this derived category of almost solid modules. Let be the category of affinoid perfectoid spaces together with a pseudouniformizer of . We define a functor as the sheafification of the functor (the derived category of almost solid -modules) on equipped with the pro-etale topology.
If is weakly perfectoid of finite type over some totally disconnected space, then is just . More generally, will gave a pro-etale cover by some which is weakly perfectoid of finite type over some totally disconnected space, and can be expressed as the limit , where , and runs over is the degree part of the Cech nerve of .
Now let be a small v-stack. There is a unique hypercomplete (this means it satisfies descent along all hypercovers, which are generalizations of the Cech nerve) sheaf on that agrees with the functor for every affinoid perfectoid space in . We define to be the global sections of this sheaf. This is the construction that we want to apply to .
The derived category of almost solid modules comes with a six-functor formalism (see also Perverse Sheaves and the Geometric Satake Equivalence). Let be a map. The derived pullback is the restriction map of the sheaf . The derived pushforward is defined to be the right adjoint to the derived pullback. The derived tensor product and derived Hom are inherited from .
The remaining two functors in the six-functor formalism are the “shriek” functors and . If is a “nice” enough map, we have a factorization of into a composition where is etale and is proper, and we define
where is the right-adjoint to . We then define to be the right-adjoint to . The six-functor formalism satisfies certain important properties, such as functoriality of , proper base change for , and a projection formula for . In Lucas Mann’s thesis, he uses the six-functor formalism he has developed to prove Poincare duality for a rigid-analytic variety of pure dimension over an algebraically closed nonarchimedean field of mixed characteristic:
As of the moment, there are still many questions regarding a possible geometrization of the p-adic local Langlands program. As more developments are worked out, we hope to be able to discuss them in future posts on this blog, together with the different aspects of the theory that has already been developed, and the many other different future directions that it may lead to.
Geometrization of the Local Langlands Correspondence by Laurent Fargues and Peter Scholze
Geometrization of the Local Langlands Program (notes by Tony Feng from a workshop at McGill University)
The Geometric Langlands Conjecture (notes from Oberwolfach Arbeitsgemeinschaft)
Berkeley Lectures on p-adic Geometry by Peter Scholze and Jared Weinstein
Etale Cohomology of Diamonds by Peter Scholze
On the Kottwitz Conjecture for Local Shtuka Spaces by David Hansen, Tasho Kaletha, and Jared Weinstein
Averaging Functors in Fargues’ Program for GL_n by Johannes Anschütz and Arthur-César Le Bras
Cohomologue p-adique de la Tour de Drinfeld: le Cas de la Dimension 1 by Pierre Colmez, Gabriel Dospinescu, and Wiesława Nizioł
A p-adic 6-Functor Formalism in Rigid-Analytic Geometry by Lucas Mann
Lectures on Condensed Mathematics by Peter Scholze