Multiplicities in the ordinary part of mod p cohomology for \operatorname{GL}_{n}(\mathbb{Q}_{p})

# Multiplicities in the ordinary part of mod p cohomology for GLn(Qp)

John Enns111The author was supported by an NSERC grant while this work was carried out.
###### Abstract

Given a continuous ordinary Galois representation , Breuil and Herzig constructed an admissible smooth -representation of and showed that it occurs in certain globally defined mod cohomology spaces in [BH]. By applying Taylor-Wiles patching to spaces of ordinary automorphic representations we prove that the indecomposable pieces of each occur with the same multiplicity at a well-chosen tame level.

Let be a continuous local mod Galois representation. It is hoped that there will be a mod local Langlands correspondence associating with a smooth admissible -representation of compatible with the mod cohomology of (globally defined) arithmetic manifolds. Since many aspects of the local correspondence remain mysterious (see [BreICM] for an overview of what is known) and in light of this expected local-global compatibility, much of the work done so far on the topic has involved studying globally defined representations of . This paper continues this theme.

Let be a CM number field in which is totally split, having maximal totally real subfield . Suppose that is a global mod Galois representation arising from an automorphic representation of a definite unitary group that becomes isomorphic to over . Let denote the set of places of dividing , and fix a choice of place of dividing for each . In Section 2.2 it is explained how in this situation the space of mod automorphic forms on gives rise to an admissible -representation of , which we denote depending on a choice of tame level and an auxiliary set of places . It is hoped that this space will realize the local mod Langlands correspondence for , .

We are a long way from understanding the full representation , with the most pressing problem being a lack of understanding of the supersingular representations of . In fact, only recently has there been significant progress in understanding its -socle, which is the modern formulation of the weight part of Serre’s conjecture (see [LLHLM] and other papers by those authors for the most recent work).

When is ordinary (meaning upper-triangular) at places dividing , the ordinary subrepresentation is nonzero and is studied in [BH]. It is defined to be the largest subrepresentation of whose irreducible subquotients are all subquotients of principal series representations of . Given any ordinary local Galois representation , [BH] defines a representation of called (the point being it depends only on ), and one of their main results in the global situation at hand is an inclusion

 ⨂v∈SpΠ(¯r|GF~v)ord↪S(Up,F)[mΣ¯r]ord. (0.0.1)

In fact, by taking into account certain unknown multiplicities of the indecomposable pieces of each , they obtain an essential inclusion in (0.0.1). These multiplicities are indexed by the set of ordinary Serre weights , which is to say the constituents of the -socle of , and are strongly related to the multiplicities with which the appear in this socle. The precise statement of this result is Theorem 4.4.7 in [BH], also summarized in Section LABEL:sec:application below.

The main goal of this paper (Theorem LABEL:thm:application) is to improve the result of [BH] by showing that is independent of for a fixed well-chosen tame level . To do this we show that all the ordinary Serre weights appear in with the same multiplicity (Theorem 3.1.3). This may be viewed as a step towards the structure of the mod local Langlands correspondence.

The main ingredients are a strong link (previously known) between the mod ordinary subspace and spaces of ordinary automorphic forms in characteristic , and Taylor-Wiles patching of spaces of ordinary automorphic representations. In fact we use an adaptation of Diamond’s variant [Dia] of the Taylor-Wiles method, to show freeness over a Hecke algebra in the proof of Theorem 3.1.3. Results on multiplicities of Serre weights have been obtained previously using the formalism of patching functors such as those developed in [CEG+] but these techniques require an understanding of the geometry of potentially crystalline deformation rings which is the subject of the Breuil-Mézard conjecture and is lacking in general. The novelty of patching spaces of ordinary automorphic representations only is that we can use ordinary crystalline deformation rings ([Ger]) at , which are better understood. In fact, since we are only interested in generic Galois representations, it is enough to use the simple “naive” ordinary deformation rings of [CHT].

The paper is structured as follows: in Section 1 we recall results on ordinary deformation rings at places above . In Section 2 we review results on ordinary automorphic representations of . In Section 3 we set up the patching data and show the main results.

### 0.1 Acknowledgements

We thank Florian Herzig for asking about the main result of this note, and for giving some initial suggestions and helpful comments on an earlier draft.

It will be clear to experts that the patching procedure used in this article draws heavily from the work of others, mainly [Ger], [CEG+], [EG] and [Tho].

### 0.2 Notation and conventions

If is a finite extension, denotes a fixed choice of absolute Galois group with inertia subgroup , and denotes the local reciprocity map of class field theory, normalized so that uniformizers correspond to geometric Frobenius elements.

Throughout, denotes a finite extension of that serves as a field of coefficients for our representations. It has ring of integers and residue field . We also fix a choice of algebraic closure having ring of integers and residue field . We always assume that is large enough to contain the image of every embedding . For the purposes of defining deformation problems, denotes the category of local noetherian -algebras having residue field equal to . Given a continuous residual Galois representation we write for the universal (framed) lift of to an object of . In fact we usually write in place of when is clear from the context.

If is a continuous representation, where is a finite-dimensional local -algebra, we have and we use this decomposition to define the set of Hodge-Tate weights of with respect to the embedding . We normalize the definition of Hodge-Tate weights so that the cyclotomic character has weight . Given an element where denotes the set of nonincreasing -tuples of integers, we say that a representation has Hodge type if .

If is a finite free -module then stands for the -module . This is a very special case of the Schikhof duality used in the patching of [CEG+]. On the other hand, if is a vector space over a field then we write for the dual vector space. For example, we would write .

## 1 Ordinary lifting rings

The goal of this section is to recall the “naive” ordinary lifting rings of [CHT] and collect some of their properties. Until the end of the section let be a finite extension of .

### 1.1 Ordinary crystalline representations

###### Definition 1.1.1.

Let . We define associated with an -tuple of characters by setting

 χλi=ϵ−i+1∏τ:K↪E(τ∘Art−1K)−λτ,n+1−i.

A character is crystalline of Hodge type for if and only if it agrees with on . We say that is special if for each there exists such that . We define to be generic if for each and we have .

###### Definition 1.1.2.

Let be a finite extension of and let . A continuous homomorphism is called ordinary of weight if it is conjugate to a homomorphism of the form

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝ψ1∗⋯∗∗ψ2⋯∗∗⋱⋮⋮ψn−1∗ψn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠

where .

We remark that Lemma 3.1.4 of [GG] shows that if is ordinary of weight and is special then is crystalline of Hodge type .

### 1.2 Naive ordinary crystalline lifting rings

In the case of a generic residual representation, the naive ordinary deformation problem that one might write down is representable and one gets a good ordinary deformation ring. This is explained in Section 2.4 of [CHT] and we now review some of the results from there. Let be a continuous homomorphism conjugate to one of the form

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝¯ψ1∗⋯∗∗¯ψ2⋯∗∗⋱⋮⋮¯ψn−1∗¯ψn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (1.2.1)

where the characters are distinct on . This means that has a decreasing -invariant filtration by subspaces such that for , and because the are distinct this filtration is unique.

Fix any choice of characters lifting for . Let denote the set of all lifts of to continuous with in such that has a decreasing filtration by -submodules which are -direct summands such that and acts on by the character for . Again, since the are distinct such a choice of filtration is unique and using this one shows easily that is a deformation problem in the sense of [CHT], Definition 2.2.2. It follows from Lemma 2.2.3 loc. cit. that there exists a quotient of such that a morphism in factors through iff belongs to .

This construction is compatible with finite extension of the coefficient field in the following sense. Let be a finite extension. Then it is well known that , that is, the universal lifting ring of for the category . One checks that .

###### Proposition 1.2.2.

The ring has the following properties:

1. Let be a finite extension of . An -algebra homomorphism factors through iff has a decreasing -invariant filtration such that is the character as a representation of .

2. If in addition we assume that the diagonal characters of obey for each , is a power series ring in variables over .

###### Proof.

The first claim follows from the definition of and the aforementioned compatibility with finite extension, as well as the fact that any such homomorphism lands in and any such filtration on descends to one on . The second claim is Lemmas 2.4.7 and 2.4.8 of [CHT]. ∎

In the special case that for some we write instead of . Thus by the previous proposition is formally smooth if is generic.

###### Definition 1.2.3.

Let be conjugate to an upper triangular representation as in (1.2.1). We call such a representation ordinary. We say that is inertially generic if for each we have

 ¯ψi¯ψ−1j|IK≠1,¯ϵ±1,¯ϵ±2.
###### Remark 1.2.4.

This is stronger than the definition of inertially generic in [BH].

The existence of an inertially generic representation implies that .

## 2 Ordinary automorphic forms and Hecke algebras

In Sections 2.1 and 2.2 we recall the necessary background about spaces of automorphic forms on definite unitary groups. In Section 2.3 we review the necessary facts about spaces of ordinary forms.

### 2.1 Representations of GLn

Let denote the standard upper-triangular Borel subgroup of the algebraic group and the diagonal torus. We identify the Weyl group of with and the cocharacter group of with in the usual way. If is a dominant weight we write for the dual Weyl module of of highest weight , defined as the algebraic induction

 M(λ)=IndGLnBn(w0λ)Z

where is the longest element. Then for any ring we write , which is isomorphic to by [RAGS], II.8.8(1). In particular carries an action of and is free over .

A Serre weight of is an absolutely irreducible -representation of or equivalently of . Serre weights are classified by the -restricted set of dominant weights : if then the sub- representation of generated by the highest weight vector is absolutely irreducible and we denote this representation . Each absolutely irreducible representation of arises this way and iff (in particular all Serre weights are defined over and the choice of coefficient field in the definition does not matter). By the remarks above, we may equivalently view as the sub- representation of generated by the highest weight vector, where we think of as a subgroup of acting on .

A Serre weight gives rise to a Hecke algebra

 H(σ):=EndGLn(Qp)(cIndGLn(Qp)GLn(Zp)(σ))

where denotes compact induction. See the beginning of Section 4.3 of [BH] for more details and references concerning what follows. If is a smooth -representation of then acts on via Frobenius reciprocity. Moreover there is a natural identification of with the -algebra of functions such that for and with a convolution product. Concretely, in the case at hand there is an isomorphism of with the polynomial algebra where corresponds to the function having support

 GLn(Zp)(1n−j00p1j)GLn(Zp)

taking to the endomorphism

 σ↠σ¯¯¯¯Uj(Fp)∼←σUj(Fp)↪σ

where denotes the unipotent radical of the standard parabolic subgroup of corresponding to the partition and denotes the opposite unipotent radical. We write for the largest subspace of preserved by on which the operators all act invertibly. This will be referred to as the ordinary subspace.

### 2.2 Automorphic forms on definite unitary groups

Let be an imaginary CM field with maximal totally real subfield and let denote the nontrivial element of . Assume that is unramified at all finite places, and that . Then we can find an algebraic group defined over such that is an outer form of becoming isomorphic to over , is quasisplit at each finite place of , and for each infinite place of , is compact. It is also possible to fix a reductive model for over which we again denote . For each place of that splits as in there is an isomorphism which restricts to an isomorphism . For these statements see e.g. [Tho], Section 6.

We from now on assume that is totally split in . Fix a choice of place of for each . Let denote the set of places of dividing , and the chosen set of lifts. Because is totally split in , there is a bijection between field embeddings and places of dividing (and similarly for ). Let and write for the set of embeddings corresponding to . We write for the embedding corresponding to a place . Given we define a finite free -module with an action of by

 Mλ=⨂˜τ∈˜IpM(λ˜τ)OE

where is the dual Weyl module of Section 2.1 on which we give an action of obtained via the map induced by the embedding that corresponds to .

A Serre weight for is an absolutely irreducible -representation of . As in Section 2.1 the choice of coefficient field does not matter in the definition. If is -restricted then the facts recalled in Section 2.1 imply that the action of on factors through and is absolutely irreducible. We denote this Serre weight by . Every Serre weight for arises in this fashion.

Let be a compact open subgroup. If is an -module with a linear action of where denotes the projection of to , we let denote the space of algebraic automorphic forms on of weight and level , which is defined to be the -module of functions obeying for all and , where is the image of in . By finiteness of class numbers, we may choose finitely many , such that and one easily shows there is an isomorphism of -modules

 S(U,W) ∼→N∏i=1Wt−1iG(F+)ti∩U (2.2.1) f ↦(f(ti))Ni=1.

In particular, is finitely generated (resp. free) over whenever is. We say that is sufficiently small if for some finite place of , the projection of to contains no elements of exact order . This implies that is sufficiently small in the sense of [Tho], which is to say that for each the group has no elements of exact order and one deduces easily from the isomorphism above that if is free over and is sufficiently small then the functor on -modules is exact. See Lemma 6.3 of [Tho]. We will use the following consequence: if is free over and is sufficiently small then the natural map is an isomorphism. Also note that if is free over then without any hypotheses on . This follows directly from (2.2.1).

Next we recall the relationship between classical automorphic forms on and the algebraic automorphic forms above, following [Tho]. Fix a weight . Let where the limit is with respect to compact open subgroups . This space has a natural action of via . Fix an isomorphism and let denote the space of classical automorphic forms on (over ). For each we have so we may define to be the continuous -representation of of highest weight . If denotes the representation of given by then there is an semilinear isomorphism of -modules

 ι:Sλ(¯¯¯¯Qp)∼→HomG(F+∞)(σ∨λ,A).

In particular is a semisimple admissible -module, and if is an irreducible constituent of with then we view as a constituent of .

Now come the Hecke algebras. Let denote the set of finite places of that split in . If is any compact open subgroup of we say that is unramified at a place if for some . A compact open subgroup is thus unramified at almost all places in . If is any finite set of finite places of containing and all at which is ramified (in this situation we say that is good for ) then we define the universal Hecke algebra (that is, the polynomial algebra in the variables ). It acts on by letting operate as the double coset

where is any choice of uniformizer in (the operator does not depend on this choice).

Next we recall Galois representations associated to automorphic representations of . If is a classical automorphic representation of contributing to as above, there is an associated Galois representation (depending on our fixed isomorphism ) that is de Rham at places diving and has Hodge type . If has level prime to then is crystalline at places diving . Moreover, satisfies local-global compatibility at all finite places (summarized in Theorem 7.2.1 of [HLM]). We say that a semisimple representation is automorphic if there exists a weight and an automorphic form contributing to such that is the semisimplification of the reduction mod of . From now on, when we fix a Galois representation we assume its coefficient field is large enough to contain all its eigenvalues. Saying is automorphic is equivalent to saying that there exists a compact open subgroup as above that is ramified at each place in where is ramified and a finite set of finite places good for such that . Here is the maximal ideal of naturally associated to by defining the image of in to be such that the characteristic polynomial of is equal to . Automorphy of implies an essential conjugate self-duality: .

We define as ranges over compact open subgroups of . This has a smooth action of , and acts on it if is good for . If is a maximal ideal with residue field , we have naturally. We remark that if we fix an automorphic absolutely irreducible representation as above then it’s hoped that will match with under some kind of mod local Langlands correspondence (at least up to some multiplicities depending on ). This is by analogy with Emerton’s mod local-global compatibility theorem described in [BreICM].

If is a Serre weight for it is shown in Lemma 4.4.2 of [BH] that there is an isomorphism which induces . Note that the Hecke algebra acts on this space and we define the ordinary subspace as in Section 2.1.

### 2.3 Ordinary automorphic forms and Galois representations

Let be a continuous representation. Let be a weight. We say that is modular and ordinary of weight if there exists a sufficiently small compact open subgroup such that is a hyperspecial maximal compact subgroup of for all that are inert in and is ramified at all places where is ramified, together with a finite set of finite places good for such that . Equivalently we may say that is an ordinary Serre weight of . We write for the set of ordinary Serre weights of , and sometimes write rather than .

###### Proposition 2.3.1.

Suppose that satisfies the following assumptions:

• is absolutely irreducible, where is a primitive th root of unity,

• is modular and ordinary of some weight,

• is inertially generic for each

• and .

Then iff has an ordinary crystalline lift of weight for each .

The first two assumptions imply is ordinary for each by Proposition 4.4.4 of [BH], so the third assumption makes sense. Also, Proposition 2.3.1 still holds with the weaker definition of inertially generic in [BH].

###### Proof.

The inertial genericity implies that if then is generic for each . We deduce the existence of the lifts from Proposition 4.4.4 of [BH] and Lemma 3.1.5 of [GG].

Conversely, if has an ordinary crystalline lift of weight for each , then is ordinary with diagonal inertial weights

 (−λ˜τ(~v),n,−λ˜τ(~v),n−1,…,−λ˜τ(~v),1−(n−1))

and the inertial genericity implies that the genericity hypothesis in Proposition 4.4.5 of [BH] is satisfied. It immediately follows that . ∎

Next we define a different âordinary subspaceâ with -coefficients, and then relate it to the mod ordinary subspace as well as to automorphic representations. Let . Let be any sufficiently small compact open subgroup and set