In The Local Langlands Correspondence for General Linear Groups, we stated the local Langlands correspondence for where is a finite extension of . We recall that it states that there is a one-to-one correspondence between irreducible admissible representations of and F-semisimple Weil-Deligne representations of . Here all the relevant representations are over .
In this post, we will consider representations over a finite field of characteristic . While we may hope that some sort of “mod p local Langlands correspondence” will also hold in this case, at the moment all we know is the case of , which we will discuss in this post. It is a sort of stepping stone to the “p-adic local Langlands correspondence” (where the representations are over some extension of ), which, as in the mod p case, is only known for the case of . The p-adic local Langlands correspondence for plays an important role in the proofs of the known cases of the Fontaine-Mazur conjecture, which concerns when a Galois representation comes from the etale cohomology of some variety.
Let us start by discussing some representation theory of over . This will be somewhat similar to our discussion in The Local Langlands Correspondence for General Linear Groups, as we will see later when we state the classification of the irreducible admissible smooth representations of , but we will also need some new ingredients.
A Serre weight is an absolutely irreducible representation (absolutely irreducible means it is irreducible over the algebraic closure of ) of over . This is the same as an absolutely irreducible smooth representation of over .
Serre weights are completely classified and can be explicitly described. Let and let . Then a Serre weight is always of the form .
The name “Serre weight” originates from its relationship to Serre’s modularity conjecture, which is a conjecture about when a residual representation comes from a modular form, and what the level and the weight of the modular form should be. Avner Ash and Glenn Stevens made the observation that a residual representation is modular of weight () and level if and only if (here is a certain maximal ideal of the Hecke algebra associated to ) is nonzero.
For convenience, from here on in this post we shall consider Serre weights not just as representations of but as representations of , which sends the uniformizer of to .
Serre weights are important because from them we can obtain induced representations of . Let be the representation of coming from compactly supported functions from to which satisfy .
The endomorphisms of form the Hecke algebra, which is isomorphic to . In other words, we can consider as a module over , and we can take the quotient . This quotient is irreducible, and it is an important class of absolutely irreducible admissible smooth representations of called the supersingular representations.
The rest of the absolutely irreducible admissible smooth representations of are very similar to what we discussed in The Local Langlands Correspondence for General Linear Groups. Namely, they can be obtained from principal series representations, which are induced representations of characters from the Borel subgroup of (i.e. the upper-triangular matrices in ).
To recall, the principal representations are , which means they come from functions such that for and , where and are characters of , and as a function on means it sends an element of to .
In the case that , the principal series representations will be absolutely irreducible, in which case we obtain another class of absolutely irreducible admissible smooth representations. Otherwise, we may obtain absolutely irreducible representations as a quotient. These will be twists (this means a tensor product by a character) of the Steinberg representation, which is defined as the quotient (here is the trivial representation of ). This gives a third class of absolutely irreducible admissible representations. Finally we have the characters, which give a fourth class.
In summary, the absolutely irreducible admissible smooth representations of over can be classified into the following four kinds as follows:
- One-dimensional representations (characters)
- Principal series representations for
- Twists of Steinberg representations for
- Supersingular representations for a Serre weight
Let us now discuss the other side of the correspondence, the “Galois side”. For ease of notation let us also denote by .
Let be an element of the inertia subgroup of . Serre’s level 2 fundamental character is given by composing the map
which takes values in with the isomorphism .
Let be a natural number which is mutually prime to . We have that is an irreducible 2-dimensional representation of . In fact, any absolutely irreducible representation of over is of this form, possibly tensored with the unramified character which takes the inverse of the Frobenius to .
The mod p local Langlands correspondence is now the bijection described explicitly as follows:
To the supersingular representation where , we associate the Galois representation .
To the representation which is obtained as the extension we associate the Galois representation which is obtained as an extension . Here is the reduction mod p of the p-adic cyclotomic character, and and are characters which are not equal to each other nor to the product of the other by the p-adic cyclotomic character or its inverse.
The p-adic local Langlands correspondence, which, as stated earlier concerns representations over some finite extension of and is important in the Fontaine-Mazur conjecture, needs to be compatible with the mod p local Langlands correspondence as well. Its statement is more involved than the mod p local Langlands correspondence, and its proof involves -modules. We reserve further discussion of the p-adic local Langlands correspondence to future posts.
The emerging p-adic Langlands programme by Christophe Breuil
Representations of Galois and of GL_2 in characteristic p by Christophe Breuil
Towards a modulo p Langlands correspondence for GL_2 by Christophe Breuil and Vytautas Paskunas