In Completed Cohomology, we mentioned that the p-adic local Langlands correspondence may be found inside the completed cohomology, and that this is used in the proof of the Fontaine-Mazur conjecture. In this post, we elaborate on these ideas. We shall be closely following the Séminaire Bourbaki article Correspondance de Langlands p-adique, compatibilité local-global et applications by Christophe Breuil.
Let us make the previous statement more precise. Let be a finite extension of , with ring of integers , uniformizer , and residue field . Let us assume that contains the Hecke eigenvalues of a cuspidal eigenform of weight . Consider the etale cohomology of the open modular curve (we will define this more precisely later). Then we have that contains , where is the p-adic Galois representation associated to (see also Galois Representations Coming From Weight 2 Eigenforms), and is the smooth representation of associated to by the local Langlands correspondence (see also The Local Langlands Correspondence for General Linear Groups).
For , if we are given , then we can recover . Therefore the local Langlands correspondence, at least for , can be found inside . This is what is known as local-global compatibility.
If , however, it is no longer true that we can recover from . Instead, the “classical” local Langlands correspondence needs to be replaced by the p-adic local Langlands correspondence (which at the moment is only known for the case of ). The p-adic local Langlands correspondence associates to a p-adic local Galois representation a p-adic Banach space over equipped with a unitary action of . The p-adic local Langlands correspondence is expected to be “compatible” with the classical local Langlands correspondence, in that, if the Galois representation is potentially semistable with distinct Hodge-Tate weights the representation provided by the classical local Langlands correspondence (tensored with an algebraic representation that depends on the Hodge-Tate weights) shows up as the “locally algebraic vectors” of the p-adic Banach space provided by the p-adic local Langlands correspondence (we shall make this more precise later).
In the case of the p-adic local Langlands correspondence we actually have a functor that goes the other way, i.e. from p-adic Banach spaces with a unitary action of to Galois representations . We denote this functor by (it is also known as Colmez’s Montreal functor). In fact the Montreal functor not only works for representations over , but also representations over (hence realizing one direction of the mod p local Langlands correspondence, see also The mod p local Langlands correspondence for GL_2(Q_p)) and more generally over . The Montreal functor hence offers a solution to our problem of the classical local Langlands correspondence being unable to recover back the Galois representation from the -representation.
Therefore, we want a form of local-global compatibility that takes into account the p-adic local Langlands correspondence. In the rest of this post, if we simply say “local-global compatibility” this is what we refer to. We will use “classical” local-global compatibility to refer to the version that only involves the classical local Langlands correspondence instead of the p-adic local Langlands correspondence.
A review of completed cohomology and the statement of local-global compatibility
As may be hinted at by the title of this post and the opening paragraph, the key to finding this local-global compatibility is completed cohomology. Let us review the relevant definitions (we work in more generality than we did in Completed Cohomology). Let be the finite adeles of . For any compact subgroup of we let
Next let be a compact open subgroup of (here the superscript means we omit the factor indexed by in the restricted product) and let be a compact open subgroup of . We define
We let . This is a p-adic Banach space, with unit ball given by . It has a continuous action of which preserves the unit ball. We also let and . We refer to any of these as the completed cohomology. The appearance of Banach spaces should clue us in that this is precisely what we need to formulate a local-global compatibility that includes the p-adic local Langlands correspondence, since the representation of that shows up there is also a Banach space.
Let . We define to be the subspace of consisting of vectors for which there exists a compact open subgroup of such that the representation of generated by in restricted to is the direct sum of algebraic representations of restricted to .
We will work in a more general setting than just weight cuspidal eigenforms (whose associated Galois representations can be found in , as discussed earlier). Therefore, in order to take account cuspidal eigenforms of weight , we will replace with , where is the sheaf on the etale site of that corresponds to the local system on given by
Now , from which we can obtain the “classical” local-global compatibility, is related to the completed cohomology (from which we want to obtain the local-global compatibility that involves the p-adic local Langlands correspondence) via the following -equivariant isomorphism:
where really is shorthand for the character of , and in this last expression is the p-adic cyclotomic character.
By taking invariants under the action of , we also have the following -equivariant isomorphism:
Before we give the statement of local-global compatibility let us make one more definition. We first need to revisit the Hecke algebra. Let be a compact open subgroup of . We define to be the -algebra of generated by and . We define
Now let be an absolutely irreducible odd continuous p-adic Galois representation, unramified at all but finitely many places. We say that is promodular if there exists a finite set of places , containing and the places at which is ramified, such that the ideal of generated by and is a maximal ideal of .
We may now give the statement of local-global compatibility. We start with the “weak” version of the statement. Let be a -dimensional odd representation of which is unramified at all but a finite set of places. Assume that the residual representation is absolutely irreducible, and that its restriction to is not isomorphic to a Galois representation of the form .
For ease of notation we also let denote . Then the weak version of local-global compatibility says that, if is promodular, then there exists a finite set of places containing and the places at which is ramified, such that we have the following nonzero continuous -equivariant morphism:
Furthermore, if is not the direct sum of two characters or the extension of a character by itself, all the morphisms will be closed injections.
The strong version of local-global compatibility is as follows. Assume the hypothesis of the weak version and assume further that the restriction of to is not isomorphic to a twist of by some character. Then we have a -equivariant homeomorphism
In this post we will only discuss ideas related to the proof of the weak version of local-global compatibility. It will proceed as follows. First we reduce the problem of showing local-global compatibility to the existence of a map . Then to show that this map exists, we construct, using (completions of) Hecke algebra-valued deformations of the relevant residual representations of and , a module , and showing that, for any maximal ideal , the submodule of annihilated by is nonzero. Initially we shall show this only for “crystalline classical maximal ideals”, but these will turn out to be dense in the completion of the Hecke algebra, which will show that the result is true for all maximal ideals.
A Preliminary Reduction
To show local-global compatibility, it is in fact enough for us to show the existence of a -equivariant map
Let us briefly discuss why this is true. Consider the smooth induced representation with compact support over . We have that . Now let be a smooth representation of over , and let , be in . We have
Now let be such that and , for . It follows from the (classical) local Langlands correspondence that
Let denote the subspace of on which acts by . The results that we have just discussed now tell us that the space
is isomorphic to the space
Furthermore, it follows from Eichler-Shimura relations (which relate the action of and on that the previous space is also isomorphic to
Furthermore, for each of these isomorphisms, a morphism on one side of the isomorphism is a closed injection if and only if the corresponding morphism is also a closed injection. Therefore, as earlier stated, to show local-global compatibility it will be enough for us to show that a -equivariant map exists.
Representations valued in a completion of the Hecke algebra
To show the existence of this map , we will construct a module that we shall denote by . Before we can define this module though, we need to make some definitions involving the Hecke algebra, and representations valued in (completions of) this Hecke algebra.
Let be an absolutely irreducible odd continuous residual Galois representation. Let us suppose furthermore that is modular.
Let be a compact open subgroup of . We let be the completion of with respect to the maximal ideal generated by , , and . We define
Since is absolutely irreducible, for every compact open subgroup of such that the work of Carayol provides us with a unique continuous Galois module unramified outside such that and .
We define . This is a deformation of over the complete Noetherian local -algebra (see also Galois Deformation Rings). After restriction to , we may also look at as a deformation of .
Now let is the representation associated to by the mod p local Langlands correspondence. We also want to construct a deformation of , that is related to by the p-adic local Langlands correspondence.
Let be the deformation ring that represents the functor which assigns to a complete Noetherian local -algebra the set of deformations of over . We define to be the the quotient of by the intersection of all maximal ideals which are kernels of a map for some extension of such that the representation over obtained by base change from the universal representation over is crystalline with distinct Hodge-Tate weights (see also p-adic Hodge Theory: An Overview).
Similarly, we have a deformation ring Let that represents the functor which assigns to a complete Noetherian local -algebra the set of deformations of over . Recall that the p-adic local Langlands correspondence provides us with the Montreal functor from representations of to representations of , which means we have a map . We let be the quotient of parametrizing deformations of whose central character corresponds to under local class field theory. We define
Now it turns out that the surjection is actually an isomorphism. A consequence of this is that, if we have a complete Noetherian local -algebra that is a quotient of , any deformation of over comes from a deformation of via the Montreal functor .
Now all we need to do to construct is to find an appropriate complete Noetherian local -algebra . We recall that is a deformation of over , so we want to find inside of , apply the discussion in the previous paragraph, and then we can extend scalars to obtain the deformation over . To do this we need to show to discuss crystalline classical maximal ideals, and show that they are Zariski dense inside (this fact will also be used again to achieve the goal we stated earlier of showing the existence of a map ).
We say that a maximal ideal of is classical if the system of Hecke eigenvalues associated to comes from a cuspidal eigenform of weight .
Let be a classical maximal ideal of . Then we have a representation
which is potentially semistable with distinct Hodge-Tate weights. We say that the classical maximal ideal is crystalline if the associated Galois representation is crystalline.
Let us now outline the argument showing that the crystalline classical maximal ideals are dense in . This is the same as the statement that the intersection of all crystalline classical maximal ideals is zero. And so our strategy will be to show that any element in this intersection acts by on .
Let be a sufficiently small compact open subgroup of . Then the -representation is a topological direct factor of for some , where is the -representation provided by the continuous -valued functions on .
Now it happens that the polynomial functions of are dense inside the continuous functions . This implies that the vectors in for which acts algebraically are dense in . Since, by the previous paragraph, is a topological direct factor of for sufficiently small, this implies that a similar result holds for . Taking limits over , we obtain that the vectors in for which acts by an algebraic representation of are dense in .
If is a maximal ideal of , we write to denote the submodule of annihilated by . We now have that is contained in , where the direct sum is over all classical maximal ideals of . Furthermore, the subrepresentation of generated by the vectors for which acts by an algebraic representation is contained in , where the direct sum is now over all crystalline classical maximal ideals of . Now it turns out that, if is the Galois representation associated to some cuspidal eigenform of weight , the representation contains a vector fixed under the action of if and only if is crystalline. If is an element in the intersection of all the crystalline classical maximal ideals, it annihilates , and therefore also the subrepresentation of generated by the vectors for which acts by an algebraic representation. But this subrepresentation is dense in and by continuity acts by zero on . This shows that the intersection of all the crystalline classical maximal ideals is zero and that they are Zariski dense in .
Since the crystalline classical maximal ideals are dense in in , we have that the map factors through . Now we find our complete Noetherian local -algebra mentioned earlier as the image of the map , so that we can obtain a deformation of that gives rise to via the Montreal functor . Then we extend scalars to to obtain .
Existence of the map
Now that we have the -valued representations and , we may now define the module which as we said will help us prove the existence of a map . It is defined as follows:
Let be a maximal ideal of . We let denote the set of elements of that are annihilated by the elements of . Our aim is to show that for all maximal ideals. As we shall show later, applying this to the maximal ideal generated by , , and will give us our result. Our approach will be to show first that for “crystalline” maximal ideals, then, using the fact that the crystalline classical maximal ideals are Zariski dense in , show that this is true for all maximal ideals of .
Let be a crystalline classical maximal ideal of . Then . To show this, we choose some field that contains . Now recall again that we have
Now since contains , we find that inside there lies a tensor product of and some locally algebraic representation of . What the crystalline condition on does is it actually provides us with at most one equivalence class of invariant norms on this locally algebraic representation of , which must be the one induced by on . It turns out that after completion, the representation of on the resulting p-adic Banach space is precisely if is irreducible, and a closed subrepresentation of if is reducible (here and , and these two correspond to each other under the p-adic local Langlands correspondence).
Now we know that if is a crystalline classical maximal ideal. Now we want to extend this to all the maximal ideals of by making use of the fact that the set of crystalline classical maximal ideals is Zariski dense inside .
The idea is that, if for all maximal ideals that belong to some set that is Zariski dense in , then for all maximal ideals in . Let us consider first the simpler case of a module of finite type over . We want to show that if for all then for all maximal ideals in .
Since is maximal, is a field, and acts faithfully on . If some element acts by zero on it must act by zero on for all . If for all , then this element must be in the intersection of all the in , but since is Zariski dense in , this intersection is zero and has to be zero.
Suppose for the sake of contradiction that for all but for some maximal ideal in . Then Nakayama’s lemma says that there exists some nonzero element such that . But this contradicts the above paragraph, so we must have for all maximal ideals in .
Now let be a compact open subgroup of , and let be defined similarly to but with in place of . We apply the above argument to , which is a -module of finite type. Then it is a property of (which is ) that if for sufficiently small .
Now that we know that for all maximal ideals of , we apply this to the particular maximal ideal generated by and . But we have
where again and . Since we have just shown that the left-hand side of the above isomorphism is nonzero, then so must the right hand-side, which means there is map .
Furthermore this map is a closed injection if is not a direct sum of two characters or an extension of a character by itself. In the case that is absolutely irreducible, this follows from the fact that is topologically irreducible and admissible. If is reducible and indecomposable, then is also reducible and indecomposable and one needs to show that a nonzero morphism cannot be factorized by a strict quotient of . We leave further discussion of these to the references.
Application to the Fontaine-Mazur conjecture
Let us now discuss the application of local-global compatibility to (a special case of) the Fontaine-Mazur conjecture, whose statement is as follows.
Let be an absolutely irreducible odd continuous p-adic Galois representation, unramified at all but finitely many places, and whose restriction to is potentially semistable with distinct Hodge-Tate weights. Then the Fontaine-Mazur conjecture states that there exists some cuspidal eigenform of weight such that is the twist of (the Galois representation associated to ) by some character.
The Fontaine-Mazur conjecture is also often stated in the following manner. Let be as in the previous paragraph. Then can be obtained as the subquotient of the etale cohomology of some variety. This statement in fact follows from the previous one, because if is a Galois representation obtained from some cuspidal eigenform of weight , then it may be found as the subquotient of the etale cohomology of what is known as a Kuga-Sato variety.
Now let us discuss how local-global compatibility figures into the proof (due to Matthew Emerton) of a special case of the Fontaine-Mazur conjecture. This special case is when and we have the restriction of the corresponding residual Galois representation to is absolutely irreducible, and the restriction of to is not isomorphic to a Galois representation of the form twisted by a character for , or twisted by a character for .
In this case it follows from the work of Böckle, Diamond-Flach-Guo, Khare-Wintenberger, and Kisin that is promodular. Then the local-global compatibility that we have discussed tells us that we have a closed injective map . The condition of the restriction being potentially semistable with distinct Hodge-Tate weights guarantees that (here is defined exactly the same as except with in place of ). This follows from the compatibility of the p-adic local Langlands correspondence and the “classical” local Langlands correspondence, which says that if is potentially semistable with distinct Hodge-Tate weights then we have the following isomorphism:
The closed injective map then tells us that, since , we must have as well. But we have the isomorphism
and the Galois representations that show up on the left hand side of this isomorphism are associated to cuspidal eigenforms of weights . This completes our sketch of the proof of the special case of the Fontaine-Mazur conjecture.
We have discussed here the ideas involved in Emerton’s proof of a special case of the Fontaine-Mazur conjecture. There is also another proof due to Mark Kisin that makes use of a different approach, namely, ideas related to the Breuil-Mezard conjecture (a version of which was briefly discussed in Moduli Stacks of (phi, Gamma)-modules) and the method of “patching” (originally developed as part of the approach to proving Fermat’s Last Theorem). This approach will be discussed in future posts on this blog.
Correspondance de Langlands p-adique, compatibilité local-global et applications by Christophe Breuil
Local-global compatibility in the p-adic Langlands programme for GL_2/Q by Matthew Emerton
Completed cohomology and the p-adic Langlands program by Matthew Emerton
The Breuil-Schneider conjecture, a survey by Claus M. Sorensen