WEIGHT CYCLING AND SERRE-TYPE CONJECTURES FOR UNITARY GROUPS

MATTHEW EMERTON, TOBY GEE, AND FLORIAN HERZIG

Abstract. We prove that for forms of U(3) which are compact at infinity and split at places dividing a prime p, in generic situations the Serre weights of a mod p modular Galois representation which is irreducible when restricted to each decomposition group above p are exactly those previously predicted by the third author. We do this by combining explicit computations in p-adic Hodge theory, based on a formalism of strongly divisible modules and Breuil modules with descent data which we develop in the paper, with a technique that we call “weight cycling”.

1. Introduction

The weight part of (generalisations of) Serre’s conjecture has received considerable attention in recent years. There have been many new conjectures formulated (cf. [BDJ10], [Her09], [Sch08b], [Gee11a]) and several cases of these conjectures have been established (cf. [Gee11b], [GS11a]). However, there have been essentially no theoretical results for groups of semisimple rank greater than one. In this paper we prove the first such results, by showing (under suitable genericity hypotheses) that for a Galois representation which is irreducible when restricted to each decomposition group above p, its Serre weights on forms of U(3) which are compact at infinity and split at places dividing p, and for which it is modular, are precisely those predicted by [Her09]. (The same set of weights is also predicted to be a subset of the modular weights by Doud’s modification [Dou07] of the conjecture of Ash, Doud, Pollack, and Sinnott [ADP02], [AS00].) We remark that other results have recently been obtained in [GG12] (in the ordinary case), and the forthcoming [BLGG12].

We now explain our results in more detail. Let F be an imaginary CM field in which p splits completely, and r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}) be a continuous irreducible representation. Let F^{+} be the maximal totally real subfield of F, and let G be a unitary group over F^{+} which is isomorphic to U(3) at each infinite place and split at each prime above p. In Definition 7.3.5 we define what it means for r̄ to be modular for G (of some specific weight); roughly speaking, it means that the characteristic polynomials of Frobenius elements at unramified places should correspond to Hecke polynomials of some automorphic Hecke eigenform on G. This implies that r̄ is essentially conjugate self-dual. We have a notion of a (strongly) generic Serre weight (see Definition 7.5.4; it is a condition saying that a weight is sufficiently far away from the walls of the alcoves, and implies for example that p is at least 17), and we let W_{gen}(r̄) be the set of generic Serre weights for which r̄ is modular. We also define W^{?}(r̄), the set of conjectural Serre weights for r̄, following the recipe of [Her09]. (See Subsection 7.5.) We remind the reader that W^{?}(r̄) is defined purely locally, and indeed only depends on the restrictions of r̄ to inertia groups at places dividing p. Then our main theorem (Theorem 7.5.6 in the body of the text) is:

The first author was partially supported by NSF grants DMS-0701315 and DMS-1002339.

The second author was partially supported by NSF grant DMS-0841491.

The third author was partially supported by NSF grant DMS-0902044 and by agreement DMS-0635607.

Theorem 1. Suppose that r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}) is a continuous representation. For all places w|p of F suppose that r̄|_{G_{Fw}} is irreducible. If r̄ is modular of some strongly generic Serre weight, then W_{gen}(r̄) = W^{?}(r̄).

This result may be regarded as a generalisation of the results of [Gee11b] and [GS11a] to 3-dimensional representations, but the methods of proof are rather different, and in particular make no use of automorphy lifting theorems. The argument naturally breaks up into two parts. We first prove that any generic weight for which r̄ is modular is one predicted by [Her09]. We then prove that all these predicted weights actually appear.

The first part of the argument makes use of calculations in p-adic Hodge theory. While in essence these are of a similar nature to those of [Gee11b] and [Sch08d] (which in spirit go back at least as far as [Edi92]), they are rather more involved than those earlier calculations, for several reasons. Firstly, the representation theory of GL_{3}(𝔽_{p}) is significantly more complicated than that of GL_{2}(𝔽_{p}), which makes the combinatorics rather more involved. More significantly, the calculations of [Gee11b] and [Sch08d] are for potentially crystalline Galois representations with Hodge–Tate weights in [0, 1], whose reductions mod p correspond to finite flat group schemes with descent data. In our setting we must work instead with 3-dimensional potentially crystalline Galois representations with Hodge–Tate weights in [0, 2], whose reductions no longer correspond to finite flat group schemes. However, thanks to the recent work of Tong Liu ([Liu08]) which established a conjecture of Breuil on the existence of strongly divisible lattices in semistable Galois representations with Hodge–Tate weights in [0, p − 2] (with no restrictions on the ramification), one knows that (the reductions mod p of) the representations we need to study will correspond to certain Breuil modules with descent data.

In fact, we are not aware of any results in the literature covering the extension of Liu’s results to incorporate coefficients and descent data, so we establish the basic results in this paper. We do not attempt a comprehensive treatment, but limit ourselves to allowing coefficients in the rings of integers of finite extensions of ℚ_{p}, and consider only tame descent data. Our treatment follows that of [Sav05], which considers the case of Hodge–Tate weights in [0, 1]. It will not surprise an expert that this extension is possible, and our arguments here are of an essentially formal nature.

Having established these results, we are reduced to performing certain calculations with Breuil modules with descent data. Again, these calculations are far more delicate than those of [Gee11b] and [Sch08d], for several reasons. Firstly, we no longer have the underlying crutch of finite flat group schemes, which means that we cannot immediately make use of scheme-theoretic closure techniques, or make use of properties of étale models. Thankfully, the work of Caruso ([Car11]), and in particular his theory of maximal Breuil modules, provides an adequate substitute for our purposes. In addition, we know of no way to make use of the fact that our representations are potentially crystalline, rather than just potentially semistable, and we do not know of any way to directly transfer the information that the Hodge–Tate weights of our representations are 0, 1, 2 to the Breuil modules we consider. Instead, we assume only that our representations are potentially semistable with Hodge–Tate weights in [0, 2], and use various ad hoc arguments using the assumption that our mod p representations are (locally) irreducible. We also make significant use of the determinant of the representation (in particular, this is how we exploit the fact that our Hodge–Tate weights are 0, 1, 2).

Having in this way established the “easy” direction of the weight part of Serre’s conjecture, we now tackle the “hard” direction of proving that a mod p representation that is modular of some particular weight is actually modular of all the expected weights. In contrast to [Gee11b] and [BLGG12], we do not make use of any automorphy lifting theorems to do this. Instead, we make use of a method which we call “weight cycling”. In the case of the group GL_{2}_{/ℚ_{p}}, this technique was discovered by Kevin Buzzard,^{1} and was first written up in Section 5 of [Tay06]. The argument was extended to GL_{2}_{/F} for F an arbitrary finite extension of ℚ_{p} in [Sch08a], and was also exploited in [GS11a]. We extend it to arbitrary split connected reductive groups over p-adic fields. The result we obtain is of the following form: if a mod p representation r̄ is modular of some given weight, and a certain Hecke operator at p vanishes on the corresponding cohomology class, then r̄ is modular of some other weight in a short list of possibilities. Using the elimination result already proved, this list can often be reduced to a single weight. The argument can then be repeated, and in this way we can “cycle” through all the conjectural weights.

Of course, in order to apply the weight cycling result we need to establish a vanishing result for certain Hecke operators in characteristic p (Corollary 4.5.4). Here we again use our assumption that our mod p Galois representation is irreducible when restricted to any decomposition group above p. We establish a rather general comparison between Hecke operators at p in characteristic zero and characteristic p (Proposition 4.4.2), and use it to show that if the relevant Hecke operators did not vanish, certain (normalised) Hecke operators in characteristic 0 would have to act via p-adic units. Using local-global compatibility one can deduce that the local Galois representation would be reducible, a contradiction. This argument may be regarded as a generalisation of the fact that the Galois representations associated to ordinary modular forms are necessarily reducible at p.

1In fact, the computations that underly the technique seem to go back (at least) to the proofs of Thm. 3.2 of the paper [Hid81] of Hida and Thm (3.4) of the paper [Rib83] of Ribet.

We remark that while automorphy lifting theorem techniques such as those used in [BLGG12] are undeniably a powerful method for proving results about Serre weight conjectures, it does not seem to be possible to prove Theorem 1 by such methods. The problem is that these methods rely on producing automorphic Galois representations in characteristic 0, and deducing information about Serre weights by reducing modulo p. However, irreducible representations of the algebraic group GL_{3} over \overline{𝔽}_{p} cannot always be lifted to characteristic 0, making the link between characteristic 0 and characteristic p far weaker than it is for GL_{2}. Concretely, if we are working over a CM field F of degree 2d over ℚ in which p splits completely, then we generically expect to have 9^{d} Serre weights for a given Galois representation (assumed irreducible when restricted to decomposition groups at places above p), and in this paper we produce all 9^{d} weights (in those cases to which our results apply). By contrast, automorphy lifting techniques appear to be limited to producing 6^{d} weights (the “obvious” weights; for example, in the case that the restriction to the decomposition groups at p was ordinary, these would be the ordinary weights); see the forthcoming paper [BLGG12] for this.

It would be of interest to weaken the hypothesis of (strong) genericity in our results. This would appear to require a significant improvement to our padic Hodge theoretic techniques, in order to prove the “elimination” part of the argument; in particular, it seems likely that one would have to make full use of the precise Hodge–Tate weights, which we do not currently know how to do, or to understand the reductions modulo p of three-dimensional crystalline representations with Hodge–Tate weights in [0, 2p], which also seems to be a hard problem. Even if the “elimination” part can be completed, the conjectural weight set need not consist of 9^{d} weights in degenerate situations and it seems that weight cycling does not necessarily cycle through all these weights.

1.1. The organisation of the paper. In order to make it easier for the reader to see exactly which assumptions are used in each argument, and with an eye to future applications, we have gone to considerable lengths to axiomatise as many of our assumptions as possible, and to argue within our abstract axiomatic framework, rather than with specific unitary groups. For example, where arguments apply for a general GL_{n}, we have presented them in this generality.

In Section 2 we recall the definitions of weights and their associated Hecke algebras, as well as the Satake isomorphism of [Her11b]. We also prove our general weight cycling result for an arbitrary split connected reductive group, and establish some special cases of the tame inertial local Langlands correspondence for GL_{n}.

In Section 3 we establish the necessary results from p-adic Hodge theory; in particular, we generalise some of the results of [Liu08] to allow for coefficients and descent data. We then use these results to give an upper bound for the set of possible reductions mod p of 3-dimensional potentially semistable representations of G_{ℚ_{p}} with Hodge–Tate weights 0, −1, −2 which are of particular tame types, under the assumption that these reductions are irreducible.

In Section 4 we introduce our basic axiomatic framework. We have sets of axioms dealing with both characteristic p and characteristic zero contexts, and we show how the characteristic zero axioms imply the characteristic p ones; this is an abstraction of the results on local-global compatibility and the comparison of Hecke operators in characteristics zero and p discussed above.

In Section 5 we use the calculations of Section 3 to establish the elimination result that we need; that is, to show that in generic situations, if r̄ is modular of some weight then that weight is one of the weights predicted in [Her09]. We carry out these arguments in the setting of Section 4.

In Section 6 we combine the results of Section 5 with our weight cycling technique and the vanishing of Hecke operators proved in Subsection 4.5 to prove an abstract version of Theorem 1 (namely Theorem 6.2.3) in our axiomatic framework.

In Section 7 we begin by recalling some standard material on automorphic forms on compact unitary groups and their associated Galois representations. We then establish that the axioms of Section 4 hold in this setting, and thus establish Theorem 1. Finally, in Subsection 7.6 we use automorphic induction and base change for unitary groups to show that there are many Galois representations which satisfy the hypotheses of our main theorem.

We refer to the beginning of each section for more details regarding its contents.

1.2. Acknowledgements. We would like to thank Xavier Caruso, David Savitt, and Tong Liu for some helpful conversations related to the material in Section 3; in particular, we are grateful to Caruso for the proof of Proposition 3.3.8. We thank Guy Henniart for helpful comments regarding the local Langlands correspondence and Sug Woo Shin for a helpful discussion concerning the proof of Theorem 7.2.1. We thank Paul-James White for providing a more general version of base change for definite unitary groups in [Whi11]. Some of this work was carried out at the IAS (by F.H.) and during a conference visit at the University of Kyoto in November 2008 (by T.G. and F.H.). We thank these institutions for the excellent working conditions they provided. Finally we thank the referees for many helpful comments.

1.3. Notation and terminology. For any field K, we let G_{K} denote an absolute Galois group of K, the precise choice of G_{K} depending on the choice of an algebraic closure \overline{K} of K. In the case of the field ℚ_{p}, it will be convenient to fix once and for all an algebraic closure \overline{ℚ}_{p}, with ring of integers \overline{ℤ}_{p} and residue field \overline{𝔽}_{p}. If K is an algebraic extension of ℚ_{p}, it will always be implicit that K is a subfield of this fixed algebraic closure \overline{ℚ}_{p}, and we may then unambiguously set G_{K} := Gal(\overline{ℚ}_{p}/K). It is also convenient to fix once and for all an isomorphism ı : \overline{ℚ}_{p} ⟶^{∼} ℂ, and we do so.

Let F be a finite extension of ℚ_{l}, for l possibly equal to p. We let Art_{F} be the isomorphism F^{×} ⟶^{∼} W_{F}^{ab} of local class field theory, normalised so that geometric Frobenius elements correspond to uniformisers. Let rec_{F,ℂ} denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of GL_{n}(F) over ℂ to isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of W_{F}. (See [HT01].) We define the local Langlands correspondence rec_{F} over \overline{ℚ}_{p} by ı ∘ rec_{F} = rec_{F,ℂ} ∘ ı. It depends only on ı^{−1}(q^{(n−1)/2}), where q is the cardinality of the residue field. Let N(π) denote the monodromy operator of rec_{F} (π).

We let \overline{ℚ} denote the algebraic closure of ℚ in \overline{ℚ}_{p}. All algebraic extensions F of ℚ are implicitly understood to be subfields of \overline{ℚ}, so that we may unambiguously define G_{F} := Gal(\overline{ℚ}/F). If v is a finite place of F we denote by Frob_{v} ∈ G_{F} a (choice of) geometric Frobenius element at v.

Where possible, we use ρ and \overline{ρ} to denote representations of the absolute Galois groups of local fields, and r and r̄ to denote representations of the absolute Galois groups of number fields. We let ε denote the p-adic cyclotomic character, and we let \overline{ε} denote its reduction modulo p.

If K is a finite extension of ℚ_{p} and ρ is a continuous de Rham representation of G_{K} over \overline{ℚ}_{p}, then we will write WD(ρ) for the corresponding Weil–Deligne representation of W_{K} (defined, for example, in Appendix B.1 of [CDT99]), and if τ : K ↪ \overline{ℚ}_{p} then we will write HT_{τ} (ρ) for the multiset of Hodge–Tate weights of ρ with respect to τ. By definition, if W is a de Rham representation of G_{K} over \overline{ℚ}_{p} and if τ : K ↪ \overline{ℚ}_{p} then the mul-

tiset HT_{τ} (W) contains i with multiplicity dim_{\overline{ℚ}_{p}}(W ⊗_{τ,K} ˆ \overline{K}(i))^{G_{K}}. Thus for example HT_{τ} (ε) = {−1}. We will refer to (the isomorphism class of) WD(ρ)|_{I_{K}} as the inertial type of ρ. We will let WD(ρ)^{F−ss} denote the Frobenius semisimplification of WD(ρ).

We will refer to the mod p Galois representations that are associated to automorphic representations as modular, as this seems to be standard practice in the literature; we remark that one could equally well refer to them as automorphic.

We also signal to the reader that we will frequently use the word weight to refer to an irreducible representation (or an isomorphism class thereof) of the k-valued points of a reductive group over a finite field k. (See Definition 2.1.1 below.) A Serre weight is then a variant of this notion, defined in a suitable global context. (See Definition 7.3.1.) Hopefully no confusion will arise with the other standard use of the word “weight” in representation theory (i.e. in the sense of a character of a maximal torus).

2. Representation theory

This section is devoted to various representation-theoretic preliminaries. In Subsection 2.1 we recall some basic terminology and facts related to weights, and in Subsection 2.2 we recall the Satake isomorphism of [Her11b]. In Subsection 2.3 we then introduce the technique of weight cycling in a general setting. Finally, in Subsection 2.4, which is in a somewhat different vein to the preceding three subsections, we establish some simple instances of the so-called inertial local Langlands correspondence for the group GL_{n}(F), with F a finite extension of ℚ_{p}.

2.1. Weights. We introduce notation that will be in force for this subsection and the two that follow. Namely, let F be a finite extension of ℚ_{p} with ring of integers 𝒪_{F}, uniformiser ϖ, and residue field k. Let G_{/𝒪_{F}} be a split connected reductive group and fix a maximal split torus T_{/𝒪_{F}} . Let Φ ⊂ X^{∗}(T) denote the set of roots and choose a system of simple roots Δ ⊂ Φ. Denote by B_{/𝒪_{F}} denote the associated Borel subgroup and by U_{/𝒪_{F}} its unipotent radical. Let W be the Weyl group and K := G(𝒪_{F}), a hyperspecial maximal compact subgroup of G(F).

Definition 2.1.1. A weight is an isomorphism class of irreducible representations V of G(k) over k̄.

Since G(k) is a finite group, there are only finitely many weights.

Remark 2.1.2. We will constantly engage in the following standard abuse of terminology: namely, we will speak of some particular G(k)-representation V being a weight, when we actually mean that V is an isomorphism class representative of a weight.

For a standard parabolic subgroup P = MN we denote by \overline{P} = M\overline{N} the opposite parabolic. The following result is Lemma 2.5 in [Her11b].

Lemma 2.1.3. Suppose that V is a weight and that P = MN is a standard parabolic. Then V^{N(k)} and V_{\overline{N}(k)} are weights for M and the natural, M(k)- equivariant map V^{N(k)} ↪ V ↠ V_{\overline{N}(k)} is an isomorphism. In particular, V^{U(k)} ≅ V_{\overline{U}(k)} is one-dimensional.

Suppose that μ ∈ X_{∗}(T). Let P_{μ} = M_{μ}N_{μ} denote the parabolic subgroup of G defined by μ, where the Levi subgroup M_{μ} contains T. (See [Spr98, Prop. 8.4.5]. In particular, if μ is dominant then P_{μ} is the standard parabolic subgroup defined by the set of simple roots that are orthogonal to μ.) The following lemma is a special case of Proposition 3.8 in [Her11b]. We use the shorthand ^{t}K = tKt^{−1}.

Lemma 2.1.4. Let t = μ(ϖ). Then the image of K ∩^{t}K in G(k) is P_{−μ}(k).

2.2. Hecke operators and the Satake isomorphism. If V is a weight, then we define the Hecke algebra of V as ℋ_{G}(V) := End_{G(F)}(c-Ind_{K}^{G(F)} V). By Frobenius reciprocity we can and usually will think of it as k̄-algebra of compactly supported functions f : G(F) → Endk̄ V satisfying f(k_{1}gk_{2}) = k_{1} ∘ f(g) ∘ k_{2} for all k_{1}, k_{2} ∈ K, g ∈ G(F), where the multiplication is given by convolution. Note that if π is a smooth G(F)-representation, ℋ_{G}(V) naturally acts on the left on (V ⊗_{\overline{𝔽}_{p}} π)^{K}. Explicitly, if ϕ ∈ ℋ_{G}(V) and x ∈ (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then

(2.2.1)  ϕ · x = ∑_{γ∈G(F)/K}(ϕ(γ) ⊗ γ)x.

We now recall some results of [Her11b]. Let T^{−} denote the submonoid of T (F),

T^{−} = {t ∈ T (F) : ord_{F} (α(t)) ≤ 0 ∀α ∈ Δ},

and let ℋ_{T}^{−}(V^{U(k)}) denote the subalgebra of ℋ_{T} (V^{U(k)}) consisting of those φ : T (F) → k̄ that are supported on T^{−}.

Theorem 2.2.2. Suppose that V is a weight. Then

𝒮_{G} : ℋ_{G}(V) → ℋ_{T} (V^{U(k)})
f ↦ (t ↦ ∑_{u∈U(F)/U(𝒪_{F})} f(tu)|_{V^{U(k)}} )

is an injective k̄-algebra homomorphism with image ℋ_{T}^{−}(V^{U(k)}).

In particular, ℋ_{G}(V) ≅ k̄[X_{∗}(T)_{−}] is commutative and noetherian (by Gordan’s lemma). Here, X_{∗}(T)_{−} = {μ ∈ X_{∗}(T) : ⟨μ, α⟩ ≤ 0 ∀α ∈ Δ}. We recall that G(F) = ∐ Kμ(ϖ)K, where μ ranges over X_{∗}(T)_{−} (refined Cartan decomposition). Moreover, ℋ_{G}(V) has a basis (as a \overline{k}-vector space) given by \overline{T}_{μ,ϖ} (μ ∈ X_{∗}(T)_{−}), where \overline{T}_{μ,ϖ} has support Kμ(ϖ)K and sends μ(ϖ) to the endomorphism V ↠ V_{N_{μ}(k)} ⟵^{∼} V^{N_{−μ}(k)} ↪ V. Note that up to scalar multiple, \overline{T}_{μ,ϖ} is independent of the choice of uniformiser ϖ.

2.3. Weight cycling. The following proposition and its corollary encode the basic technique of weight cycling.

Proposition 2.3.1. Suppose that V is a weight and that π is a smooth G(F)-representation. Suppose that μ ∈ X_{∗}(T)_{−} and let t = μ(ϖ). Then we have a commutative diagram as follows, where the map on the right is induced by Frobenius reciprocity and the map on the top is injective.

\xymatrix{(V ⊗_{\overline{𝔽}_{p}} π)^{K} \ar@{^{(}->}[r]^{i} \ar@{-->}[dr]_{_{\overline{T}_{μ,ϖ}}} & ((Ind_{K∩^{t}K}^{K} V^{N_{−μ}(k)}) ⊗_{\overline{𝔽}_{p}} π)^{K} \ar[d] \\  & (V ⊗_{\overline{𝔽}_{p}} π)^{K}}

Note that the natural map V^{N_{−μ}(k)} → V is K∩^{t}K-equivariant by Lemma 2.1.4.

Proof. Suppose that x ∈ (V ⊗_{\overline{𝔽}_{p}} π)^{K}. By (2.2.1) and the definition of \overline{T}_{μ,ϖ} we see that \overline{T}_{μ,ϖ} · x = ∑_{k∈K/K∩^{t}K}(k ⊗ k)y, where

y := (\overline{T}_{μ,ϖ}(t) ⊗ t)x ∈ (V^{N_{−μ}(k)} ⊗_{\overline{𝔽}_{p}} π)^{K∩^{t}K}.

We define the map i by composing the map x ↦ y with the isomorphism

(2.3.2)  (V^{N_{−μ}(k)} ⊗_{\overline{𝔽}_{p}} π)^{K∩^{t}K} ⟶^{∼} ((Ind_{K∩^{t}K}^{K} V^{N_{−μ}(k)}) ⊗_{\overline{𝔽}_{p}} π)^{K},

which holds by Frobenius reciprocity. Explicitly, the isomorphism in (2.3.2) is induced by V^{N_{−μ}(k)} ⊗_{\overline{𝔽}_{p}} π → (Ind_{K∩^{t}K}^{K} V^{N_{−μ}(k)}) ⊗_{\overline{𝔽}_{p}} π sending v ⊗ z to ∑_{k∈K/(K∩^{t}K)}[k, v] ⊗ kz, where [k, v] denotes the element of the induction that is supported on (K ∩^{t}K)k^{−1} and sends k^{−1} to v. Also, the natural map Ind_{K∩^{t}K}^{K} V^{N_{−μ}(k)} → V sends [k, v] to kv. It follows that the above diagram commutes.

Let η : V ↠ V_{N_{μ}(k)} ↠ V_{\overline{U}(k)} ⟶^{∼} \overline{𝔽}_{p} in V^{∨}, where the identification with \overline{𝔽}_{p} is arbitrary. Note that i(x) = 0 implies that η(x) = 0 in π. Since i is K-equivariant and η generates V^{∨} as K-representation, it follows that V^{∨} kills ker(i) ⊂ V ⊗ π, so i is injective. □

Corollary 2.3.3. With the notation as in the proposition, if \overline{T}_{μ,ϖ} fails to be injective on (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then (V′ ⊗_{\overline{𝔽}_{p}} π)^{K} ≠ 0 for at least one of the irreducible constituents V′ of ker(Ind_{K∩^{t}K}^{K} V^{N_{−μ}(k)} → V).

2.4. Inertial local Langlands. The purpose of this subsection is to establish some simple instances of “inertial local Langlands” for GL_{n}(F), where n ≥ 1 is arbitrary and F is a finite extension of ℚ_{p}. (We remark that the n = 2 case is worked out completely in the appendix of [BM02].)

Let E/F denote the unramified extension of degree n. Let 𝒪_{E} (resp. 𝒪_{F}) denote the ring of integers of E (resp. F), and let k_{E} (resp. k_{F}) denote the residue field of E (resp. F). Fix an 𝒪_{F} -basis of 𝒪_{E}. It gives rise to a rational maximal torus 𝕋 of GL_{n}_{/k_{F}} such that k_{E}^{×} = 𝕋(k_{F}). A character θ : k_{E}^{×} → \overline{ℚ}_{p}^{×} is said to be primitive if its Gal(k_{E}/k_{F})-conjugates are all distinct. If this is the case, the Deligne–Lusztig representation (−1)^{n−1}R_{𝕋}^{θ} of GL_{n}(k_{F}) over \overline{ℚ}_{p} is genuine, irreducible, and cuspidal. (See Proposition 7.4 and Theorem 8.3 in [DL76].)

Let B_{n} = T_{n}U_{n} denote the Borel subgroup of GL_{n} of upper triangular matrices. Let I(1) denote the pro-p Iwahori subgroup. It is the inverse image of U_{n}(k_{F}) in GL_{n}(𝒪_{F}). We will sometimes regard GL_{n}(k_{F})-representations as GL_{n}(𝒪_{F})-representations via inflation. If χ is a character of k_{F}^{×} (respectively θ is a character of k_{E}^{×}) we will write χ ∘ Art_{F}^{−1} (respectively θ ∘ Art_{E}^{−1}) for the character of I_{F} given by composition with the homomorphism I_{F} → k_{F}^{×} induced by Art_{F}^{−1} (respectively with the homomorphism I_{E} → k_{E}^{×} induced by Art_{E}^{−1}).

Proposition 2.4.1. Suppose that π is an irreducible admissible smooth representation of GL_{n}(F) over \overline{ℚ}_{p}.

(i) If π|_{GL_{n}(𝒪_{F})} contains the cuspidal representation (−1)^{n−1}R_{𝕋}^{θ} for

some primitive character θ : k_{E}^{×} → \overline{ℚ}_{p}^{×}, then

rec_{F} (π)|_{I_{F}} ≅ ⨁_{σ∈Gal(k_{E}/k_{F})} σ(θ ∘ Art_{E}^{−1})

and N(π) = 0. (ii) If π|_{GL_{n}(𝒪_{F})} contains the principal series representation

Ind_{B_{n}(k_{F})}^{GL_{n}(k_{F})}(χ_{1} ⊗ ⋯ ⊗ χ_{n})

for some distinct characters χ_{i} : k_{F}^{×} → \overline{ℚ}_{p}^{×}, then rec_{F} (π)|_{I_{F}} ≅ ⨁_{i=1}^{n} χ_{i} ∘ Art_{F}^{−1} and N(π) = 0.

Note that the cuspidal representation in part (i) and the principal series representation in part (ii) are irreducible. (In the latter case this is because the χ_{i} are distinct.)

Proof. Using our isomorphism ı : \overline{ℚ}_{p} ⟶^{∼} ℂ we will think of π, θ, χ_{i}, etc., as complex representations for the remainder of this proof. Recall from §1.3 that we have ı ∘ rec_{F} = rec_{F,ℂ} ∘ ı.

For the first part, let τ denote (−1)^{n−1}R_{𝕋}^{θ}. It has central character θ|_{k_{F}^{×}}. Thus there is a unique extension of the inflation of θ to 𝒪_{E}^{×} to a

character θ̃ of E^{×} such that θ̃|_{F^{×}} is the central character of π. Moreover τ extends to a F^{×} GL_{n}(𝒪_{F})-representation by letting F^{×} act via θ̃, and Frobenius reciprocity gives a GL_{n}(F)-equivariant map c-Ind_{F^{×} GL_{n}(𝒪_{F})}^{GL_{n}(F)} τ → π. From Proposition 7.3 in [DL76] and [BH10, (2.1.1)] it follows that τ is uniquely characterised among cuspidal representations by the identity tr τ(x) = (−1)^{n−1} ∑_{σ∈Gal(k_{E}/k_{F})} θ(^{σ}x) for all x ∈ k_{E}^{×} such that all Gal(k_{E}/k_{F})-conjugates of x are distinct. From Theorem 2 in [BH10, §2.4] we see that π ≅ c-Ind_{F^{×} GL_{n}(𝒪_{F})}^{GL_{n}(F)} τ is the supercuspidal representation that is

automorphically induced from μ_{0}θ̃: E^{×} → ℂ^{×}, where μ_{0} is the unramified character sending uniformisers to (−1)^{n−1}. It follows by induction on n from Lemma VII.2.6(6) in [HT01] that rec_{F,ℂ}(π) ≅ Ind_{W_{E}}^{W_{F}}(μ_{0}θ̃∘ Art_{E}^{−1}). Since μ_{0} is unramified and this representation is irreducible, the claim follows.

For the second part, let χ : T_{n}(k_{F}) → ℂ^{×} denote the character χ_{1} ⊗ ⋯ ⊗ χ_{n}. If I ⊂ GL_{n}(𝒪_{F}) denotes the inverse image of B_{n}(k_{F}), then we can define a smooth character ρ : I → ℂ^{×} as the composite of the natural map I ↠ T_{n}(k_{F}) with χ. Theorem 7.7 and Remark 7.8 in [Roc98] show that we have an isomorphism π^{ρ} ⟶^{∼} π_{U_{n}}^{χ} where π_{U_{n}} is the Jacquet module and where the superscripts on either side denote isotypic components. Our assumption, together with Frobenius reciprocity, shows that π^{ρ} ≠ 0, so π_{U_{n}}^{χ} ≠ 0. It follows that there is a T_{n}(F)-equivariant map π_{U_{n}} ↠ χ̃ for some character χ̃ : T_{n}(F) → ℂ^{×} such that χ̃|_{T_{n}(𝒪_{F})} = χ. By Frobenius reciprocity, π is a subrepresentation of n-Ind_{B_{n}(F)}^{G(F)}(χ̃δ_{B_{n}}^{−1/2}) (normalised induction, where δ_{B_{n}} is the modulus character of B_{n}). The latter representation is irreducible by our assumption that the χ_{i}|_{k_{F}^{×}} are distinct. (See Theorem 1.2.1 in [Kud94].)

Thus π ≅ χ̃_{1}|.|^{(1−n)/2} ⊞ ⋯ ⊞ χ̃_{n}|.|^{(n−1)/2}, so rec_{F,ℂ}(π) ≅ (χ̃_{1}|.|^{(1−n)/2} ⊕ ⋯ ⊕ χ̃_{n}|.|^{(n−1)/2}) ∘ Art_{F}^{−1} and N(π) = 0 [HT01, p. 252]. As |.| is unramified, this proves the proposition. □

3. p-adic Hodge theory

In this section we develop the various results in p-adic Hodge theory that will be required in the sequel. In Subsections 3.1 and 3.2 we introduce basic definitions and results related respectively to strongly divisible modules and to Breuil modules; the novelty (to the extent that there is any) is that we allow coefficients and descent data. In fact, as noted in the introduction, we do not allow the most general form of either: rather we restrict to coefficients in the ring of integers 𝒪_{E} of a finite extension E of ℚ_{p} (or its residue field 𝔽 in the case of Breuil modules), and we restrict to descent data for tamely ramified extensions.

In Subsection 3.3 we pass to a detailed study of the particular Breuil modules that are relevant to our applications. This subsection culminates in Theorem 3.3.13, which is the key result that we will apply in the sequel.

3.1. Strongly divisible modules with coefficients and descent data. In this subsection we extend certain results of [Sav05] on the categories of strongly divisible modules with coefficients and descent data to the case of lattices in potentially semistable Galois representations with Hodge–Tate weights in [0, p − 2] ([Sav05] works only with representations with Hodge– Tate weights in [0, 1]). We do not attempt to work in the same level of generality as [Sav05], and in particular we only allow coefficients in the ring of integers of a finite extension of ℚ_{p}.

We begin by recalling some results from Section 2.2 of [Sav05]. Let p be an odd prime and let E and K be finite extensions of ℚ_{p} inside \overline{ℚ}_{p}. (Throughout this section K will be a field, rather than a maximal compact subgroup as in section 2.) Let K_{0} be the maximal unramified subfield of K, so that K_{0} = W (k)[1/p], where k is the residue field of K. Let ϕ ∈ Gal(K_{0}/ℚ_{p}) denote the arithmetic Frobenius. Let K/K′ be a Galois extension, with K′ a field lying between ℚ_{p} and K. Fix the uniformiser p ∈ ℚ_{p}, so we have a fixed embedding B_{st} ↪ B_{dR}.

Definition 3.1.1. A filtered-(φ, N, K/K′, E)-module of rank n is a free K_{0}⊗_{ℚ_{p}} E-module D of rank n together with

∘ a ϕ ⊗ 1-semilinear automorphism φ,

∘ a nilpotent K_{0}⊗_{ℚ_{p}}E-linear endomorphism N such that Nφ = pφN, ∘ a decreasing filtration (Fil^{i} D_{K})_{i∈ℤ} on D_{K} = K ⊗_{K_{0}} D consisting of K ⊗_{ℚ_{p}} E-submodules, which is exhaustive and separated, and ∘ a K_{0}-semilinear, E-linear action of Gal(K/K′) which commutes

with φ and N and preserves the filtration on D_{K}. We say that such a module D is weakly admissible if the underlying filtered (φ, N, K/K, ℚ_{p})-module is weakly admissible.

Given a potentially semistable representation ρ : G_{K′} → GL(V) on an n-dimensional E-vector space V, such that ρ|_{G_{K}} is semistable, we set

D_{st}^{K′}(V) = Hom_{G_{K}}(V, B_{st}),

a weakly admissible filtered (φ, N, K/K′, E)-module of rank n. In the other direction, given a weakly admissible filtered (φ, N, K/K′, E)-module D, we define

V_{st}^{K′}(D) = Hom_{φ,N}(D, B_{st}) ∩ Hom_{Fil}(D_{K}, B_{dR}),

a finite-dimensional E-vector space with an action of G_{K′} via

(gf)(x) = g(f(ḡ^{−1}x))

where g ∈ G_{K′} and ḡ is the image of g in Gal(K/K′).

Proposition 3.1.2. The functors D_{st}^{K′} and V_{st}^{K′} are mutually quasi-inverse, and provide equivalences of categories between the category of weakly admissible filtered (φ, N, K/K′, E)-modules and the category of E-representations of G_{K′} which become semistable upon restriction to G_{K}.

Proof. This follows from Proposition 2.9 of [Sav05]. □

We now wish to consider 𝒪_{E}-lattices inside potentially semistable Galois representations. This entails the consideration of strongly divisible modules. We begin with some basic definitions.

We suppose from now on that K/K′ is a tamely ramified Galois extension with ramification index e(K/K′), and that we have fixed a uniformiser ϖ ∈ K with ϖ^{e(K/K′)} ∈ K′. Write g(ϖ) = h_{g}ϖ for each g ∈ Gal(K/K′). Note that h_{g} ∈ W (k). Let e be the absolute ramification index of K. Let E(u) ∈ W (k)[u] be the (monic) minimal polynomial of ϖ over K_{0}. Let S be the p-adic completion of W (k)[u, \frac{u^{ie}}{i!} ]_{i∈ℕ}. We let φ : S → S denote the unique continuous, ϕ-semilinear map with φ(u) = u^{p} and φ(u^{ie}/i!) = u^{iep}/i!, and we let N be the unique continuous, W (k)-linear derivation of S with N(u) = −u and N(u^{ie}/i!) = −ieu^{ie}/i!, so that Nφ = pφN. There is a descending filtration (Fil^{i} S)_{i≥0} on S given by letting Fil^{i} S be the p-adic completion of the ideal of S generated by E(u)^{j}/j! for j ≥ i for each i ≥ 0. For i ≤ p−1 we have φ(Fil^{i} S) ⊂ p^{i}S, and we let φ_{i} denote φ/p^{i} on Fil^{i} S. We let the group Gal(K/K′) act on S by defining for each g ∈ Gal(K/K′) the continuous ring isomorphism ĝ : S → S with ĝ(w_{i} \frac{u^{i}}{⌊i/e⌋!}) = g(w_{i})h_{g}^{i} \frac{u^{i}}{⌊i/e⌋!} (where w_{i} ∈ W (k)). This action commutes with φ and N and preserves the filtration. (Note that ĝ(E(u)) = E(u) due to our assumption that ϖ^{e(K/K′)} ∈ K′.)

Let S_{𝒪_{E}} = S ⊗_{ℤ_{p}} 𝒪_{E}, and extend the definitions of Fil, φ, φ_{k}, N, and ĝ to S_{𝒪_{E}} in the obvious 𝒪_{E}-linear fashion. Let S_{E} = S_{𝒪_{E}} ⊗_{ℤ_{p}} ℚ_{p}, and again extend the definitions of Fil, φ, N, and ĝ to S_{E} in the obvious fashion.

Let ℳℱ(φ, N, K/K′, E) be the category whose objects are finite free S_{E}- modules 𝒟 with

∘ a φ-semilinear, E-linear morphism φ_{𝒟} : 𝒟 → 𝒟 such that the determinant of φ_{D} with respect to some choice of S_{ℚ_{p}} basis is invertible in S_{ℚ_{p}} (this does not depend on the choice of basis);

∘ a decreasing filtration of 𝒟 by S_{E}-submodules Fil^{i} 𝒟, i ∈ ℤ, with Fil^{i} 𝒟 = 𝒟 for i ≪ 0 and (Fil^{i} S_{E})(Fil^{j} 𝒟) ⊂ Fil^{i+j} 𝒟 for all j and all i ≥ 0;

∘ a K_{0} ⊗_{ℚ_{p}} E-linear map N : 𝒟 → 𝒟 such that

– for all s ∈ S_{E}, x ∈ 𝒟 we have N(sx) = N(s)x + sN(x),
– Nφ = pφN,
– N(Fil^{i} 𝒟) ⊂ Fil^{i−1} 𝒟 for all i;

∘ an S_{E}-semilinear action of Gal(K/K′) on 𝒟 which commutes with φ and N and preserves each Fil^{i} 𝒟.

Fix a positive integer r ≤ p − 2. Then the category 𝒪_{E}-Mod_{dd}^{r} of strongly divisible 𝒪_{E}-modules with descent data is defined to be the category of finitely generated free S_{𝒪_{E}} -modules ℳ̂ with a sub-S_{𝒪_{E}} -module Fil^{r} ℳ̂, additive maps φ, N : ℳ̂ → ℳ̂, and S_{𝒪_{E}} -semilinear bijections ĝ : ℳ̂ → ℳ̂ for each g ∈ Gal(K/K′) such that the following conditions hold.

∘ Fil^{r} ℳ̂ contains (Fil^{r} S_{𝒪_{E}} )ℳ̂,

∘ Fil^{r} ℳ̂ ∩ Iℳ̂ = I Fil^{r} ℳ̂ for all ideals I of 𝒪_{E}, ∘ φ(sx) = φ(s)φ(x) for all s ∈ S_{𝒪_{E}} , x ∈ ℳ̂, ∘ φ(Fil^{r} ℳ̂) is contained in p^{r}ℳ̂ and generates it over S_{𝒪_{E}} , ∘ N(sx) = N(s)x + sN(x) for all s ∈ S_{𝒪_{E}} and x ∈ ℳ̂, ∘ Nφ = pφN, ∘ E(u)N(Fil^{r} ℳ̂) ⊂ Fil^{r} ℳ̂, ∘ for all g ∈ Gal(K/K′), ĝ commutes with φ and N, and preserves

Fil^{r} ℳ̂,

∘ ĝ_{1} ∘ ĝ_{2} = { \atop g_{1} ∘ g_{2}} for all g_{1}, g_{2} ∈ Gal(K/K′).

For any ℳ̂ ∈ 𝒪_{E}-Mod_{dd}^{r}, we define an 𝒪_{E}-module T_{st}^{K′}(ℳ̂) with an action of G_{K′} as follows. Let [\underline{ϖ}] be the element of A_{cris} corresponding to the Teichmüller representatives of a compatible system of p-th power roots of ϖ. We let Â_{st} be the S-algebra with a filtration Fil^{i} Â_{st}, a Frobenius φ, and a monodromy operator N defined in Section 2.2.2 of [Bre99]. This ring has a natural action of G_{K′} (see the second paragraph of Section 3.4 of [Sav05] and the paragraph before Proposition 3.11 of [Sav05]), and the natural map Â_{st}[1/p] → B_{dR} is G_{K′}-equivariant. [From Section 2.2.2 of [Bre99] we have Â_{st} = \{∑_{n=0}^{∞} a_{n} \frac{X^{n}}{n!} : a_{n} ∈ A_{cris}, a_{n} → 0\} and we define an action of G_{K′} on Â_{st} by setting g(∑ a_{n} \frac{X^{n}}{n!} ) = ∑ g(a_{n})\frac{g(X)^{n}}{n!} , where

(3.1.3)  g(X) = \frac{g([\underline{ϖ}])}{[\underline{ϖ}]} \frac{ϖ}{g(ϖ)}(X + 1) − 1.

(Since g(ϖ) = ϖ for g ∈ G_{K}, this extends the natural action of G_{K} on Â_{st}.) From the second paragraph of Section 3.4 of [Sav05] and the paragraph before Proposition 3.11 of loc. cit. we see that this is the unique G_{K′}-action on Â_{st} such that the natural map Â_{cris} ↪ Â_{st} is G_{K′}-equivariant. The second paragraph of Section 3.4 of [Sav05] shows that the map f_{ϖ} : Â_{cris} → B_{dR}^{+} is G_{K′}-equivariant, and as explained in Section 2.2.2 of [Bre99] the map Â_{st}[1/p] → B_{dR} is determined by the restriction of f_{ϖ} to A_{cris} and the fact that it sends the element 1 + X ∈ Â_{st} to [\underline{ϖ}]/ϖ. It follows that the map Â_{st}[1/p] → B_{dR} is G_{K′}-equivariant, as required.] Then we put

T_{st}^{K′}(ℳ̂) := Hom_{Fil^{r},φ,N}(ℳ̂, Â_{st})

(that is, the homomorphisms of S-modules which preserve Fil^{r} and commute with φ and N). This inherits an 𝒪_{E}-module structure from the 𝒪_{E}-module structure on ℳ̂, and G_{K′} acts on T_{st}^{K′}(ℳ̂) by

(gf)(x) = g(f(ˆḡ^{−1}x))

where g ∈ G_{K′} and ḡ is the image of g in Gal(K/K′). Note that this action is well defined since the map S → Â_{st} is G_{K′}-equivariant and since the G_{K′}- action on Â_{st} commutes with φ, N and preserves Fil^{i} Â_{st} for all i. (The first is clear by the definition of the G_{K′}-actions, since the map sends u to [\underline{ϖ}](1 + X)^{−1}. The second follows from the definitions of these actions in Section 2.2.2 of [Bre99], and the fact that the G_{K′}-action commutes with φ and preserves all Fil^{i} on Â_{cris} by the second paragraph of Section 3.4 of [Sav05].)

The main result of this subsection is the following.

Proposition 3.1.4. The functor T_{st}^{K′} provides an anti-equivalence of categories between the category 𝒪_{E}-Mod_{dd}^{r} of strongly divisible 𝒪_{E}-modules with descent data and the category of G_{K′}-stable 𝒪_{E}-lattices in finite-dimensional E-representations of G_{K′} which become semistable over K with Hodge–Tate weights lying in [−r, 0].

When K = K′ and E = ℚ_{p} this is the main result of [Liu08].

Proof. We write ℤ_{p}-Mod^{r} for 𝒪_{E}-Mod_{dd}^{r} and T_{st} for T_{st}^{K′} in the case when K = K′ and E = ℚ_{p}.

Suppose that ℳ̂ is an object of 𝒪_{E}-Mod_{dd}^{r}. Let 𝒟 := ℳ̂[\frac{1}{p}]. We extend φ and N by linearity and we define a filtration on 𝒟 as follows. We let Fil^{r} 𝒟 := (Fil^{r} ℳ̂)[\frac{1}{p}] and

Fil^{i} 𝒟 := \begin{cases}𝒟 & for i ≤ 0, \\ \{x ∈ 𝒟 : E(u)^{r−i}x ∈ Fil^{r} 𝒟\} & for 0 ≤ i ≤ r, \\ ∑_{j=0}^{i−1}(Fil^{i−j} S_{ℚ_{p}})(Fil^{j} 𝒟) & for i > r, by induction. \end{cases}

It is not hard to check that 𝒟 is an object of ℳℱ(φ, N, K/K′, E). (To verify the condition about φ, pick elements f_{i} ∈ Fil^{r} ℳ such that φ_{r}(f_{i}) form an S-basis of ℳ, and let e_{i} := φ_{r}(f_{i}). Write f_{i} = ∑ a_{ij}φ_{r}(f_{j}) and φ(e_{j}) = ∑ b_{jl}e_{l} for some matrices A = (a_{ij}), B = (b_{jl}) with entries in S. We have p^{r}I = φ(A)B, so det(B) ∈ S_{ℚ_{p}}^{×}.) By [Bre97, §6] and Corollary 2.1.4 in [Bre99] we can write 𝒟 = D ⊗_{K_{0}} S_{ℚ_{p}} for some weakly admissible filtered (φ, N, K/K, ℚ_{p})-module D, and we have D ≅ 𝒟⊗_{S_{ℚp},s} K_{0} (where s(h(u)) = h(0)) and D_{K} ≅ 𝒟 ⊗_{S_{ℚp},s_{ϖ}} K (where s_{ϖ}(h(u)) = h(ϖ)), which induce φ, N on D and the filtration on D_{K}. It follows that D inherits an E-action and a semilinear Gal(K/K′)-action and it is straightforward to verify that it becomes a weakly admissible filtered (φ, N, K/K′, E)-module. Note that Fil^{r+1} D_{K} = 0 since Fil^{r+1} 𝒟 ⊂ (Fil^{1} S_{ℚ_{p}})𝒟.

We associate an E-representation V_{st}^{K′}(𝒟) of G_{K′} to 𝒟 by defining

V_{st}^{K′}(𝒟) := Hom_{Fil^{r},φ,N}(𝒟, Â_{st}[1/p])

(that is, the homomorphisms of S-modules which preserve Fil^{r} and commute with φ and N). This inherits an E-module structure from the E-module structure on 𝒟, and G_{K′} acts on V_{st}^{K′}(𝒟) by

(gf)(x) = g(f(ˆḡ^{−1}x)),

where g ∈ G_{K′} and ḡ is the image of g in Gal(K/K′).

We have a natural 𝒪_{E}[G_{K′}]-linear map T_{st}^{K′}(ℳ̂) → V_{st}^{K′}(𝒟). We claim that there is a natural isomorphism of E[G_{K′}]-representations V_{st}^{K′}(𝒟) → V_{st}^{K′}(D) given by f ↦ f|_{D}. In the case that E = ℚ_{p} and K′ = K, this is established in the proof of Proposition 2.2.5 of [Bre02] (which also shows that f(D) ⊂ B_{st}^{+} ↪ Â_{st}[1/p]), so in general this map gives a natural isomorphism V_{st}^{K′}(𝒟) → V_{st}^{K′}(D) of the underlying ℚ_{p}-vector spaces with an action of G_{K}. It is clear that this isomorphism is E-linear, and it remains to check that it is compatible with the action of G_{K′}. In the proof we just cited it is also shown that we have a commutative diagram

\xymatrix{𝒟 \ar[rr]^{f} \ar@{->>}[d]_{f_{ϖ}} &  & Â_{st}[1/p] \ar[d] \\ D_{K} \ar[r]^{f|_{D}⊗1} & B_{st} ⊗_{K_{0}} K \ar@{^{(}->}[r] & B_{dR}}

Since the maps f_{ϖ}, Â_{st}[1/p] → B_{dR}, and B_{st} ⊗_{K_{0}} K → B_{dR} are G_{K′}-linear, so is V_{st}^{K′}(𝒟) → V_{st}^{K′}(D). By Proposition 2.2.5 of [Bre02] we know that T_{st}^{K′}(ℳ̂) → V_{st}^{K′}(𝒟) is a G_{K}-stable ℤ_{p}-lattice. It follows that T_{st}^{K′}(ℳ̂) is a G_{K′}-stable 𝒪_{E}-lattice in V_{st}^{K′}(𝒟), and V_{st}^{K′}(𝒟) ≅ V_{st}^{K′}(D) becomes semistable over K with Hodge–Tate weights in [−r, 0] by Proposition 3.1.2.

The faithfulness of T_{st}^{K′} follows immediately from the case when E = ℚ_{p} and K′ = K. Suppose now that ℳ̂_{1}, ℳ̂_{2} are objects of 𝒪_{E}-Mod_{dd}^{r} and that we are given an 𝒪_{E}[G_{K′}]-linear map θ : T_{st}^{K′}(ℳ̂_{1}) → T_{st}^{K′}(ℳ̂_{2}). From fullness in the case when E = ℚ_{p} and K′ = K we get a map η : ℳ̂_{1} → ℳ̂_{2} in ℤ_{p}-Mod^{r} such that T_{st}(η) = θ. We need to show that η is 𝒪_{E}-linear and compatible with the Gal(K/K′)-actions. If λ ∈ 𝒪_{E}, then T_{st}(λη) = T_{st}(η)λ = λT_{st}(η) = T_{st}(ηλ), so λη = ηλ, as required. If g ∈ G_{K′}, then ˆḡ^{−1}ηˆḡ is a map in ℤ_{p}-Mod^{r} and it is straightforward to show that T_{st}(η)g = gT_{st}(η) is equivalent to T_{st}(ˆḡ^{−1}ηˆḡ) = T_{st}(η), so ˆ ḡ^{−1}ηˆḡ = η, as required.

It remains to check that T_{st}^{K′} is essentially surjective. Suppose that V is a finite-dimensional E-representation of G_{K′}, which becomes semistable over K with Hodge–Tate weights in [−r, 0], and suppose that V_{𝒪_{E}} is a G_{K′}- stable 𝒪_{E}-lattice in V. By Proposition 3.1.2 there is a weakly admissible filtered (φ, N, K/K′, E)-module D with Fil^{0} D_{K} = D_{K}, Fil^{r+1} D_{K} = 0, and V_{st}^{K′}(D) ≅ V as E-representations of G_{K′}.

Let 𝒟 := S_{E} ⊗_{E} D, a free S_{E}-module of finite rank. We endow 𝒟 with the structure of an object of ℳℱ(φ, N, K/K′, E) as follows:

∘ φ_{𝒟} is given by the tensor product of φ on S_{E} and φ on D.

∘ N := N ⊗ Id + Id ⊗N : 𝒟 → 𝒟.

∘ Fil^{i} 𝒟 := 𝒟 for i ≤ 0, and by induction for i ≥ 0

Fil^{i+1} 𝒟 := {x ∈ 𝒟|N(x) ∈ Fil^{i} 𝒟 and f_{ϖ}(x) ∈ Fil^{i+1} D_{K}}

where f_{ϖ} : 𝒟 → D_{K} is defined by λ ⊗ x ↦ s(λ)x, where s : S_{E} → K ⊗_{ℚ_{p}} E is the unique continuous K_{0} ⊗_{ℚ_{p}} E-linear map sending u^{ie}/i! to ϖ^{ie}/i!.

∘ The action of Gal(K/K′) is given by the tensor products of the actions on S_{E} and D.

By Theorem 2.3.5 of [Liu08], there is a strongly divisible lattice ℳ̂ ⊂ 𝒟 of weight r with T_{st}(ℳ̂) corresponding to V_{𝒪_{E}} under the isomorphism V_{st}^{K′}(𝒟) ≅ V_{st}^{K′}(D) ≅ V. Note that ℳ̂ ∈ ℤ_{p}-Mod^{r}. We now show that ℳ̂ is stable under the actions of 𝒪_{E} and Gal(K/K′) on 𝒟. Since 𝒪_{E} is generated by 𝒪_{E}^{×} as a ℤ_{p}-module, to check that ℳ̂ is 𝒪_{E}-stable it is enough to check that xℳ̂ = ℳ̂ for all x ∈ 𝒪_{E}^{×}. Certainly x^{−1}ℳ̂ is a strongly divisible lattice in 𝒟. We also have

T_{st}(ℳ̂) = xT_{st}(ℳ̂) = T_{st}(x^{−1}ℳ̂),

so it suffices to show that if ℳ̂ and ℳ̂′ are strongly divisible lattices in 𝒟 with T_{st}(ℳ̂) = T_{st}(ℳ̂′), then ℳ̂ = ℳ̂′. But this is a formal consequence of Theorem 2.3.5 of [Liu08]. The proof that ℳ̂ is Gal(K/K′)-stable is very similar: if g ∈ G_{K′} then ˆ ḡ(ℳ̂) is also a strongly divisible lattice in 𝒟, and

T_{st}(ℳ̂) = gT_{st}(ℳ̂) = T_{st}(ˆḡ(ℳ̂)),

so ˆ ḡ(ℳ̂) = ℳ̂, as required.

It remains to check that if I is an ideal of 𝒪_{E} then I Fil^{r} ℳ̂ = Fil^{r} ℳ̂∩Iℳ̂, and that ℳ̂ is free as an S_{𝒪_{E}} -module. The first follows since Fil^{r} ℳ̂ = ℳ̂ ∩ Fil^{r} 𝒟, by definition. The second follows as in the proof of Proposition 3.2.3.2 of [BM02]: let J denote the kernel of the ring homomorphism S ↠ W (k), h(u) ↦ h(0). Then ℳ̂/Jℳ̂ ↪ D is a finitely generated W (k) ⊗_{ℤ_{p}} 𝒪_{E}- module which is a ℤ_{p}-lattice. Since D is a finite free K_{0} ⊗_{ℚ_{p}} E-module and K_{0}/ℚ_{p} is unramified one can check that ℳ̂/Jℳ̂ is a finite free W (k)⊗_{ℤ_{p}}𝒪_{E}- module. (The point is that W (k) ⊗_{ℤ_{p}} 𝒪_{E} is a product of discrete valuation rings.) As in loc. cit. we see that any lift of a basis to ℳ̂ is a basis of ℳ̂ as S_{𝒪_{E}} -module. □

Remark 3.1.5. One of the referees has pointed out to us that it is possible to prove Proposition 3.1.4 without using the formalism of S_{ℚ_{p}}-modules and how they relate to weakly admissible modules, as an essentially formal consequence of the results of [Liu08]. However, it is convenient to have the relationship with S_{ℚ_{p}}-modules available in calculations, such as those of Lemma 3.3.2 below, and so we have let the proof of the proposition stand in its current form, as an illustration of that relationship.

3.2. Breuil modules with descent data: part one. As its title indicated, in this subsection we introduce Breuil modules with descent data, which morally speaking are the reductions mod p of strongly divisible modules with descent data, and which are the basic tool for studying the reductions mod p of lattices in potentially semistable Galois representations.

We carry over the running hypotheses of the preceding subsection, namely we consider a tamely ramified Galois extension K/K′, with K finite over ℚ_{p}, and with ramification index e(K/K′), and we furthermore assume given a uniformiser ϖ ∈ K such that ϖ^{e(K/K′)} ∈ K′. We let e denote the absolute ramification index of K.

Let 𝔽 be a finite extension of 𝔽_{p}. Recall that the category 𝔽-BrMod_{dd}^{r} of Breuil modules of weight r with descent data from K to K′ and coefficients in 𝔽 consists of quintuples (ℳ, ℳ_{r}, φ_{r}, ĝ, N) where:

∘ ℳ is a finitely generated (k⊗_{𝔽_{p}}𝔽)[u]/u^{ep}-module, free over k[u]/u^{ep}. ∘ ℳ_{r} is a (k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}-submodule of ℳ containing u^{er}ℳ. ∘ φ_{r} : ℳ_{r} → ℳ is 𝔽-linear and φ-semilinear (where φ : k[u]/u^{ep} → k[u]/u^{ep} is the p-th power map) with image generating ℳ as a (k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}-module.

∘ N : ℳ → ℳ is k ⊗_{𝔽_{p}} 𝔽-linear and satisfies N(ux) = uN(x) − ux for all x ∈ ℳ, u^{e}N(ℳ_{r}) ⊂ ℳ_{r}, and φ_{r}(u^{e}N(x)) = cN(φ_{r}(x)) for all x ∈ ℳ_{r}. Here, c ∈ (k[u]/u^{ep})^{×} is the image of φ_{1}(E(u)) under the natural map S → k[u]/u^{ep}.

∘ ĝ : ℳ → ℳ are additive bijections for each g ∈ Gal(K/K′), preserving ℳ_{r}, commuting with the φ_{r}- and N-actions, and satisfying ĝ_{1} ∘ ĝ_{2} = { \atop g_{1} ∘ g_{2}} for all g_{1}, g_{2} ∈ Gal(K/K′). Furthermore, if a ∈ k ⊗_{𝔽_{p}} 𝔽, m ∈ ℳ then ĝ(au^{i}m) = g(a)((g(ϖ)/ϖ)^{i} ⊗ 1)u^{i}ĝ(m).

Remark 3.2.1. It follows from the assumption that ℳ is a finitely generated (k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}-module, which is free over k[u]/u^{ep}, that ℳ is in fact a free (k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}-module. (The proof of Proposition 1.2 of [Sav08] goes through, with ℳ_{r}, φ_{r} replacing ℳ_{1}, φ_{1}.)

Suppose that 𝔽 is the residue field of a finite extension E of ℚ_{p}. If ℳ̂ is an object of 𝒪_{E}-Mod_{dd}^{r}, then ℳ := (ℳ̂/𝔪_{E}ℳ̂) ⊗_{S_{𝒪_{E}} /𝔪_{E}S_{𝒪_{E}}} (k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep} is naturally an object of 𝔽-BrMod_{dd}^{r}, where we define ℳ_{r} to be the image of Fil^{r} ℳ̂ in ℳ, the map φ_{r} is induced by (1/p^{r})φ|_{Fil_{r} ℳ̂}, and N and ĝ are those coming from ℳ̂. (To see that φ_{r} is well defined on ℳ comes down to the fact that ker(S/pS → k[u]/u^{ep}) = Fil^{p}(S/pS).)

We define a functor T_{st} from the category 𝔽-BrMod_{dd}^{r} to the category of finite-dimensional 𝔽-representations of G_{K′} as follows: we let Â denote the k[u]/u^{ep}-algebra defined by the pushout (i.e. tensor product) square

\xymatrix{S \ar[r] \ar@{->>}[d] & Â_{st} \ar@{->>}[d] \\ k[u]/u^{ep} \ar[r] & Â}

The vertical arrows induce a filtration Fil^{r}, maps φ_{r} and N, and an action of G_{K′} on the bottom row. (See §2.3 of [Car06] or §2.1 of [Car11].) Explicitly,

for g ∈ G_{K′} we have g(u) = (\frac{g(ϖ)}{ϖ})^{q} u and g(X) = (\frac{g(ϖ_{1})}{ϖ_{1}})^{f} (1 + X) − 1, where ϖ_{1} ∈ 𝒪\overline{K} denotes the p-th root of ϖ used in (3.1.3), where pq ≡ 1 (mod e(K/K′)), and where f ≡ 1 (mod p) is a multiple of e(K/K′). (See also §4.2 in [Car11].) For ℳ ∈ 𝔽-BrMod_{dd}^{r} we define

T_{st}(ℳ) = Hom_{BrMod}(ℳ, Â),

(that is, the k[u]/u^{ep}-homomorphisms which are compatible with the filtration, φ_{r} and N). We let G_{K′} act by

(gf)(x) = gf(ˆḡ^{−1}x).

Lemma 3.2.2. The functor T_{st} is faithful. For any ℳ ∈ 𝔽-BrMod_{dd}^{r} we have

dim_{𝔽} T_{st}(ℳ) = rank_{(k⊗_{𝔽p}𝔽)[u]/u^{ep}} ℳ.

If ℳ̂ ∈ 𝒪_{E}-Mod_{dd}^{r} and ℳ := (ℳ̂/𝔪_{E}ℳ̂)⊗_{S_{𝒪_{E}} /𝔪_{E}S_{𝒪_{E}}} (k⊗_{𝔽_{p}}𝔽)[u]/u^{ep} denotes

the Breuil module corresponding to the reduction of ℳ̂, then

T_{st}^{K′}(ℳ̂) ⊗_{𝒪_{E}} 𝔽 ≅ T_{st}(ℳ).

Proof. The first two parts follow immediately from the corresponding results without coefficients and descent data (use Lemme 2.3.4 in [Car06] together with Corollary 2.3.3 of [CL09], respectively Lemme 2.3.1.2 in [Bre99]).

For the last part, we first define φ_{r} : Fil^{r} ℳ̂ → ℳ by (1/p^{r})φ. Then for any ideal I ⊂ 𝒪_{E}, the S/I-module ℳ̂/I inherits a filtration (the image of Fil^{r} ℳ̂) and maps φ_{r}, N, ĝ such that the natural map ℳ̂ → ℳ̂/I is compatible with all structures. Similarly, Â_{st}/p^{n} inherits from Â_{st} a filtration, maps φ_{r}, N, and a G_{K′}-action. The natural map T_{st}^{K′}(ℳ̂) → \underleftarrow{lim} Hom(ℳ̂/p^{n}, Â_{st}/p^{n}) is an isomorphism, as explained just after Définition 4.1.1.1 in [Bre98]. (Here and in the following, we consider S-linear homomorphisms that respect filtration, φ_{r}, N.) We consider the following three 𝒪_{E}[G_{K′}]-linear maps:

(3.2.3)  T_{st}^{K′}(ℳ̂)/p → Hom(ℳ̂/p, Â_{st}/p)
(3.2.4)  Hom(ℳ̂/p, Â_{st}/p)/𝔪_{E} → Hom(ℳ̂/𝔪_{E}, Â_{st}/p)
(3.2.5)  Hom(ℳ̂/𝔪_{E}, Â_{st}/p) → Hom(ℳ, Â) = T_{st}(ℳ)

The first two are isomorphisms by the argument used in the proof of Proposition 2.3.2.4 in [Bre99]. The last is an isomorphism by Lemme 2.3.4 of [Car06]. □

Proposition 3.2.6. Suppose that 𝒩 ∈ 𝔽-BrMod_{dd}^{r}, and that T′ is a G_{K′}- subrepresentation of T_{st}(𝒩) (so that in particular T′ has the structure of an 𝔽-vector space). Then there is a unique quotient 𝒩′ of 𝒩 in 𝔽-BrMod_{dd}^{r} such that if f : 𝒩 → 𝒩′ is the quotient map, then T_{st}(f) is identified with the inclusion T′ ↪ T_{st}(𝒩).

Proof. We write 𝔽_{p}-BrMod^{r} for the category 𝔽-BrMod_{dd}^{r} in case K′ = K and 𝔽 = 𝔽_{p}. In this case the proposition is proved in Proposition 2.2.5 of [Car11]. Thus we see that in general there is a unique quotient 𝒩′ of 𝒩 in 𝔽_{p}-BrMod^{r} such that if f : 𝒩 ↠ 𝒩′ is the quotient map, then T_{st}(f) is the inclusion T′ ↪ T_{st}(𝒩).

It remains to verify that the Gal(K/K′)- and 𝔽-actions descend to 𝒩′ via f. (It is then automatic that 𝒩′ becomes an object of 𝔽-BrMod_{dd}^{r} and the result follows.) Suppose for the moment that ℳ ∈ 𝔽_{p}-BrMod^{r} and that g ∈ Gal(K/K′). We define the g-twist of ℳ as follows: we let ℳ^{(g)} := ℳ ⊗_{k[u]/u^{ep},g} k[u]/u^{ep}. Then we have a g-semilinear bijection i : ℳ → ℳ^{(g)} sending x to x ⊗ 1. There is a unique way to make ℳ^{(g)} into an object of 𝔽_{p}-BrMod^{r} such that the map i induces a bijection on filtrations and commutes with φ_{r} and N. (Note that c is fixed by Gal(K/K′).) It follows that for our 𝒩 ∈ 𝔽-BrMod_{dd}^{r} the g-semilinear bijection ĝ : 𝒩 → 𝒩 induces an isomorphism ĝ : 𝒩^{(g)} ⟶^{∼} 𝒩 in 𝔽_{p}-BrMod^{r}. The assumption that T′ is G_{K′}-stable implies that there is a unique 𝔽_{p}[G_{K}]-linear isomorphism θ(g) : T_{st}(𝒩′) ⟶^{∼} T_{st}(𝒩′^{(g)}) such that T_{st}(ĝ) ∘ T_{st}(f) = T_{st}(f^{(g)}) ∘ θ(g). By the uniqueness of 𝒩′ we get a unique isomorphism ĝ_{𝒩′} : 𝒩′^{(g)} ⟶^{∼} 𝒩′ in 𝔽_{p}-BrMod^{r} such that f ∘ ĝ = ĝ_{𝒩′} ∘ f.

Finally, a similar but easier argument shows that 𝒩′ inherits a natural action of 𝔽^{×} and thus (by 𝔽_{p}-linearity) of 𝔽, as required. □

Remark 3.2.7. As already remarked in the proof, 𝒩 is actually a quotient in the naive sense that 𝒩′ = 𝒩/𝒩′′ for some 𝒩′′ ∈ 𝔽-BrMod_{dd}^{r} (this follows easily from the proof of Proposition 2.2.5 of [Car11] and the exactness of the functor M𝔖̃, for which see Theorem 2.1.2 of [Car11]; we thank Xavier Caruso for explaining this to us).

It will be convenient for us to be able to employ duality in the calculations that follow. To this end, we briefly recall the duality theory for 𝔽-BrMod_{dd}^{r} developed in chapter V of [Car05] (see also Section 2.1 of [Car11]).

Definition 3.2.8. If ℳ is an object of 𝔽-BrMod_{dd}^{r}, then there is an object ℳ^{∗} of 𝔽-BrMod_{dd}^{r} such that

T_{st}(ℳ^{∗}) = T_{st}(ℳ)^{∨}(r),

(i.e. T_{st}(ℳ^{∗}) is the r-th Tate twist of the contragredient of T_{st}(ℳ)). Explicitly, ℳ^{∗} is defined as follows.

(i) ℳ^{∗} := Hom_{k[u]/u^{ep}−Mod}(ℳ, k[u]/u^{ep}), with 𝔽-structure inherited from that of ℳ.

(ii) ℳ_{r}^{∗} := {f ∈ ℳ^{∗}|f(ℳ_{r}) ⊂ u^{er}k[u]/u^{ep}}.

(iii) Let φ_{r} : u^{er}k[u]/u^{ep} → k[u]/u^{ep} be the unique semilinear map sending u^{er} to c^{r}. Then we define φ_{r}(f) by φ_{r}(f)(φ_{r}(x)) = φ_{r}(f(x)) for all x ∈ ℳ_{r}, f ∈ ℳ_{r}^{∗}.

(iv) N(f) := N ∘ f − f ∘ N, where N : k[u]/u^{ep} → k[u]/u^{ep} is the unique

k-linear derivation such that N(u) = −u.

(v) Let g ∈ Gal(K/K′) act on ℳ^{∗} by (ĝf)(x) = g(f(ĝ^{−1}x)), where Gal(K/K′) acts on k[u]/u^{ep} by g(au^{i}) = g(a)(g(ϖ)/ϖ)^{i}u^{i} for a ∈ k.

(A priori we have a perfect, 𝔽_{p}-bilinear pairing ⟨·, ·⟩ : T_{st}(ℳ)×T_{st}(ℳ^{∗}) → 𝔽_{p}(r) that is G_{K}-equivariant. Explicitly, ⟨f, f′⟩ = ∑ f(e_{i})f′(e_{i}^{∗}), where (e_{i}) is any k[u]/u^{ep}-basis of ℳ and (e_{i}^{∗}) is the dual basis of ℳ^{∗}, by the proof of Theorem V.4.3.1 of [Car05]. It follows that ⟨·, ·⟩ is G_{K′}-equivariant and that ⟨λf, f′⟩ = ⟨f, λf′⟩ for all λ ∈ 𝔽.) We have ℳ^{∗∗} ≅ ℳ. We define the covariant functor T_{st}^{∗,r} from 𝔽-BrMod_{dd}^{r} to the category of 𝔽-representations of G_{K′} by

T_{st}^{∗,r}(ℳ) := T_{st}(ℳ)^{∨}(r),

so that

T_{st}^{∗,r}(ℳ) = T_{st}(ℳ^{∗}).

We then have the following immediate corollary of Proposition 3.2.6.

Corollary 3.2.9. Suppose that ℳ ∈ 𝔽-BrMod_{dd}^{r}, and that T′ is a G_{K′}- subrepresentation of T_{st}^{∗,r}(ℳ) (so that in particular T′ has the structure of an 𝔽-vector space). Then there is a unique subobject ℳ′ of ℳ in 𝔽-BrMod_{dd}^{r} such that if f : ℳ′ → ℳ is the inclusion map, then T_{st}^{∗,r}(f) is identified with the inclusion T′ ↪ T_{st}^{∗,r}(ℳ).

Proof. Let 𝒩 := ℳ^{∗}, so that T′ is a sub-G_{K′}-representation of T_{st}(𝒩) = T_{st}^{∗,r}(ℳ). Then by Proposition 3.2.6, there is a unique quotient 𝒩′ of 𝒩 such that applying T_{st} to the natural projection map 𝒩 → 𝒩′ gives the inclusion T′ ↪ T_{st}(𝒩). Then ℳ′ := (𝒩′)^{∗} is the required subobject of ℳ, and uniqueness is clear. □

Remark 3.2.10. Just as in Remark 3.2.7, ℳ′ is a subobject of ℳ in the naive sense that it is a sub-(k[u]/u^{ep} ⊗ 𝔽)-module of ℳ, which inherits the structure of an object of 𝔽-BrMod_{dd}^{r} from ℳ in the obvious way.

Finally, we recall something of the theory of maximal objects from [Car11] (see Sections 4.1 and 4.2 of [Car11] for a proof of the existence of maximal Breuil modules with descent data and coefficients). If ℳ is an object of 𝔽-BrMod_{dd}^{r}, the corresponding maximal object Max(ℳ) is an object of 𝔽-BrMod_{dd}^{r} together with a morphism ι_{Max}^{ℳ} : ℳ → Max(ℳ) such that T_{st}^{∗,r}(ι_{Max}^{ℳ}) is an isomorphism. Furthermore, it satisfies the following universal property: if f : ℳ → ℳ′ is such that T_{st}^{∗,r}(f) is an isomorphism, then there exists a unique morphism g : ℳ′ → Max(ℳ) such that g ∘ f = ι_{Max}^{ℳ}.

3.3. Breuil modules with descent data: part two. We now specialise to the particular situation of interest to us in this paper. Let K_{0} be the unique unramified extension of ℚ_{p} of degree d, and let K = K_{0}((−p)^{1/(p^{d}−1)}). Let K′ = K_{0}. Fix ϖ = (−p)^{1/(p^{d}−1)}. We write ω̃_{d} : Gal(K/K_{0}) → K_{0}^{×} for the character g ↦ g(ϖ)/ϖ, and we let ω_{d} be the reduction of ω̃_{d} modulo ϖ. (By inflation we can also think of ω̃_{d} and ω_{d} as characters of I_{K_{0}} = I_{ℚ_{p}}. Note that ω_{d} is a tame fundamental character of niveau d and that ω̃_{d} is the Teichmüller lift of ω_{d}.)

Let E/ℚ_{p} be a finite extension such that E contains the images of all embeddings K ↪ \overline{ℚ}_{p}.

If ρ : G_{K_{0}} → GL_{n}(E) is a potentially semistable representation which becomes semistable over K, then the associated inertial type is a representation of I_{K_{0}} which becomes trivial when restricted to I_{K}, so we may think of it as a representation of Gal(K/K_{0}) ≅ I_{K_{0}}/I_{K}. Since this group is abelian, this representation is isomorphic to a direct sum χ_{1} ⊕ ⋯ ⊕ χ_{n} where each χ_{i} : Gal(K/K_{0}) → 𝒪_{E}^{×} is a character.

Let 𝔽 be the residue field of 𝒪_{E}. Let ϕ be the arithmetic Frobenius on k, and let σ_{0} : k ↪ 𝔽 be a fixed embedding. Inductively define σ_{1}, …, σ_{d−1} by σ_{i+1} = σ_{i} ∘ ϕ^{−1}; we will often consider the numbering to be cyclic, so that σ_{d} = σ_{0}. There are idempotents e_{i} ∈ k ⊗_{𝔽_{p}} 𝔽 such that if M is any k ⊗_{𝔽_{p}} 𝔽-module, then M = ⨁_{i} e_{i}M, and e_{i}M is the subset of M consisting of elements m for which (x ⊗ 1)m = (1 ⊗ σ_{i}(x))m for all x ∈ k. Note that (ϕ ⊗ 1)(e_{i}) = e_{i+1} for all i.

Proposition 3.3.1. Maintain our current assumptions on K_{0}, K and K′, so that K′ = K_{0} and K = K_{0}((−p)^{1/(p^{d}−1)}. Suppose that ρ : G_{K_{0}} → GL_{n}(E) is a continuous representation such that ρ|_{G_{K}} is semistable with Hodge–Tate weights contained in [−r, 0], with r ≤ p − 2, and that ρ has inertial type χ_{1} ⊕ ⋯ ⊕ χ_{n} as above. Let \overline{ρ} be the reduction mod 𝔪_{E} of a G_{K_{0}}-stable 𝒪_{E}-lattice in ρ. Then there is an element ℳ of 𝔽-BrMod_{dd}^{r} with

T_{st}^{∗,r}(ℳ) ≅ \overline{ρ},

and ℳ has a (k[u]/u^{ep} ⊗_{𝔽_{p}} 𝔽)-basis v_{1}, …, v_{n} such that ĝ(v_{i}) = (1 ⊗ \overline{χ}_{i}(g))v_{i} for all g ∈ Gal(K/K_{0}).

Proof. By Proposition 3.1.4, there is an element ℳ̂ of 𝒪_{E}-Mod_{dd}^{r} such that T_{st}^{K′}(ℳ̂) is the given lattice in ρ. Then we let \overline{𝒩} := (ℳ̂/𝔪_{E}ℳ̂) ⊗_{S_{E}/𝔪_{E}S_{E}} k[u]/u^{ep}, and ℳ := \overline{𝒩}^{∗}, so that T_{st}^{∗,r}(ℳ) ≅ \overline{ρ}. In the case r = 1, the claim about the form of the descent data is proved in Section 5 of [GS11b]; but the proof given there makes no use of the fact that r = 1, and extends immediately to this more general situation. □

Lemma 3.3.2. Maintain our current assumptions on K_{0}, K and K′, so e = p^{d} − 1. Then every rank one object of 𝔽-BrMod_{dd}^{r} may be written in the form:

∘ ℳ = ((k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}) · m,
∘ e_{i}ℳ_{r} = u^{r_{i}}e_{i}ℳ,
∘ φ_{r}(∑_{i=0}^{d−1} u^{r_{i}}e_{i}m) = λm for some λ ∈ (k ⊗_{𝔽_{p}} 𝔽)^{×},
∘ ĝ(m) = (∑_{i=0}^{d−1}(ω_{d}(g)^{k_{i}} ⊗ 1)e_{i})m for all g ∈ Gal(K/K_{0}), and
∘ N(m) = 0.

Here the integers 0 ≤ r_{i} ≤ er and k_{i} satisfy k_{i} ≡ p(k_{i−1} + r_{i−1}) (mod e) for all i. Conversely, any module ℳ of this form is a rank one object of 𝔽-BrMod_{dd}^{r}. Furthermore,

T_{st}^{∗,r}(ℳ)|_{I_{K_{0}}} ≅ σ_{0} ∘ ω_{d}^{κ_{0}},

where κ_{0} ≡ k_{0} + s_{0} (mod e), and

s_{0} := p(r_{0}p^{d−1} + r_{1}p^{d−2} + ⋯ + r_{d−1})/(p^{d} − 1).

Proof. If ℳ is a rank one object of 𝔽-BrMod_{dd}^{r}, then it is of the claimed form by Theorem 3.5 of [Sav08], except for the statement that N(m) = 0. (Note that the cited proof only uses that u^{e}ℳ ⊂ ℳ_{1} at one point, once all the work is done: to know that r_{i} ≤ e in the statement of the theorem. So we can just replace ℳ_{1} by ℳ_{r} throughout.) Now it is easy to see that N(ℳ_{r}) ⊂ ℳ_{r}. Then N(m) = 0 by using that φ_{r}(u^{e}N(x)) = cN(φ_{r}(x)) for all x ∈ ℳ_{r}.

Conversely it is straightforward to verify that any module ℳ of the above form is a rank one object of 𝔽-BrMod_{dd}^{r}. The calculation of T_{st}^{∗,r}(ℳ)|_{I_{K_{0}}} may be performed in the same way as in Example 3.7 of [Sav08] or Corollaire 4.4.2 of [Car11]; we sketch the details of an approach following Example 3.7 of [Sav08].

Given ℳ as in the statement of the lemma, define another element ℳ′ of 𝔽-BrMod_{dd}^{r} by

∘ ℳ′ = ((k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}) · m′,

∘ e_{i}ℳ′_{r} = e_{i}ℳ′,
∘ φ_{r}(∑_{i=0}^{d−1} e_{i}m′) = λm′,
∘ ĝ(m′) = (∑_{i=0}^{d−1}(ω_{d}(g)^{p^{i}κ_{0}} ⊗ 1)e_{i})m′ for all g ∈ Gal(K/K_{0}), and
∘ N(m′) = 0.

We can define a morphism ℳ′ → ℳ by mapping e_{i}m′ ↦ u^{s_{i}}e_{i}m, where

s_{i} := p(r_{i}p^{d−1} + r_{i+1}p^{d−2} + ⋯ + r_{i+d−1})/(p^{d} − 1).

One easily checks that this is a morphism in the category 𝔽-BrMod_{dd}^{r}; for example, s_{i} ≥ r_{i}, so the map is compatible with the filtration, and since s_{i+1} = p(s_{i} − r_{i}) the morphism commutes with φ_{r}. Since T_{st}^{∗,r} is faithful we have a nonzero map T_{st}^{∗,r}(ℳ′) → T_{st}^{∗,r}(ℳ), which must be an isomorphism because both source and target are one-dimensional 𝔽-vector spaces.

The calculation of T_{st}^{∗,r}(ℳ′) may now be carried out in the same fashion as in Example 3.7 of [Sav08]. We sketch the argument. Denote a Teichmüller lift by a tilde. One checks without difficulty that ℳ′ may be lifted to a strongly divisible module with descent data, contained inside the object of ℳℱ(φ, N, K/K′, W (𝔽)[1/p]) associated to the weakly admissible filtered (φ, N, K/K′, W (𝔽)[1/p])-module D = (K_{0} ⊗_{ℚ_{p}} W (𝔽)[1/p]) · v, where N = 0, φ(v) = λ̃p^{r}v, g(v) = (1 ⊗ σ̃_{0} ∘ ω̃_{d}(g)^{κ_{0}})v, and Fil^{i}(K ⊗_{K′} D) is 0 for i ≥ r + 1 and K ⊗_{K′} D for i ≤ r. The corresponding Galois representation can then be computed as in Example 2.13 of [Sav05]: we find V_{st}^{K′}(D)^{∨}(r)|_{I_{K_{0}}} ≅ σ̃_{0} ∘ω_{d}^{κ_{0}}, which indeed reduces to σ_{0} ∘ ω_{d}^{κ_{0}}. □

Lemma 3.3.3. Maintain our assumptions on K_{0}, K and K′, so e = p^{d} −1. Suppose that ℳ is the rank one object of 𝔽-BrMod_{dd}^{r} of Lemma 3.3.2. Then there exists a basis m̃ of Max(ℳ) such that

∘ e_{i} Max(ℳ)_{r} = u^{er}e_{i} Max(ℳ),

∘ φ_{r}(∑_{i=0}^{d−1} u^{er}e_{i}m̃) = λm̃,
∘ ĝ(m̃) = ∑_{i=0}^{d−1}(ω_{d}(g)^{p^{i}s} ⊗ 1)e_{i}m̃ for all g ∈ Gal(K/K_{0}), and
∘ N(m̃) = 0,

with s ≡ κ_{0} − \frac{er}{p−1} (mod e), where we keep the notation of Lemma 3.3.2.

In particular, the Breuil module in Lemma 3.3.2 is maximal if and only if r_{i} = er for all i.

Proof. Let ℳ′ denote the Breuil module defined in the statement of the lemma. Let

t_{i} = \frac{erp}{p − 1} − \frac{r_{i}p^{d} + r_{i+1}p^{d−1} + ⋯ + r_{i+d−1}p}{e}.

It is an integer, and since 0 ≤ r_{i} ≤ er, it lies in the interval [0, \frac{erp}{p−1}]. Define a map f : ℳ → ℳ′ by f(e_{i}m) := u^{t_{i}}e_{i}m̃. Then f ≠ 0 since r < p − 1. To see that f commutes with φ_{r}, use that t_{i+1} = p(r_{i} + t_{i} − er). Since p divides t_{i}, it follows that f commutes with N. It is straightforward to check that f commutes with descent data. Since T_{st}^{∗,r} is faithful, it follows that T_{st}^{∗,r}(f) is an isomorphism.

It remains to verify that ℳ′ is maximal. If we have a map of Breuil modules h : ℳ′ → 𝒩 such that T_{st}^{∗,r}(h) is an isomorphism, 𝒩 has to be of rank one and we can find a basis n such that

∘ e_{i}𝒩_{r} = u^{r′_{i}}e_{i}𝒩,

∘ φ_{r}(∑_{i=0}^{d−1} u^{r′_{i}}e_{i}n) = μn,
∘ ĝ(n) = ∑_{i=0}^{d−1}(ω_{d}(g)^{k′_{i}} ⊗ 1)e_{i}n for all g ∈ Gal(K/K_{0}), and
∘ N(n) = 0,

for some μ ∈ (k ⊗ 𝔽)^{×} and integers r′_{i} ∈ [0, er] and k′_{i} such that k′_{i} ≡ p(k′_{i−1} + r′_{i−1}) modulo e. We can write h(e_{i}m̃) = ν_{i}(u^{m_{i}} + O(u^{m_{i}+1}))e_{i}n for some ν_{i} ∈ 𝔽^{×} and m_{i} ≥ 0. Since h commutes with φ_{r} we find that

(3.3.4)  m_{i+1} = p(m_{i} + er − r′_{i}),

unless both sides are at least ep. But h ≠ 0 implies that m_{i} < ep for some i and if m_{i} < C for some constant C ≤ ep, then (3.3.4) shows that m_{i−1} < C/p. Inductively we thus find that m_{i} = 0 for all i, so r′_{i} = er for all i and h is an isomorphism. □

Remark 3.3.5. Similarly (or by duality) we find that ℳ in Lemma 3.3.2 is minimal if and only if r_{i} = 0 for all i.

The following lemma works more generally than for our particular choice of extension K/K′ above, and so just for the duration of its statement and proof, we suppress our running definitions of K and K′, and return to the more general context of Subsection 3.2.

Lemma 3.3.6. Suppose that K/K′ is totally ramified and that 𝔽 contains an embedding of k. Suppose that ℳ is a rank d object of 𝔽-BrMod_{dd}^{r}. Then ℳ has a basis m^{(j)} (1 ≤ j ≤ d) such that for all j we have m^{(j)} ∈ φ_{r}(ℳ_{r}) and (k ⊗_{𝔽_{p}} 𝔽) · m^{(j)} is Gal(K/K′)-stable.

Proof. We let d = [k : 𝔽_{p}]. Since we are assuming that 𝔽 contains an embedding of k, we may define the idempotents e_{i} as above. To prove the lemma, it suffices to show that e_{0}ℳ has an 𝔽[u]/u^{ep}-basis n^{(j)} such that 𝔽 · n^{(j)} is Gal(K/K′)-stable. Thus it is enough to show that there are generators α^{(j)} (1 ≤ j ≤ d) of e_{d−1}ℳ_{r} as an 𝔽[u]/u^{ep}-module such that 𝔽 · α^{(j)} is Gal(K/K′)-stable. (Note that (φ_{r}(α^{(j)}))_{j=1}^{d} is then a basis of e_{0}ℳ as 𝔽[u]/u^{ep}-module.)

Note that Gal(K/K′) is abelian of order e(K/K′), which is prime to p. It acts trivially on k. Moreover k contains the e(K/K′)-th roots of unity. Thus e_{d−1}ℳ_{r} decomposes as a direct sum of one-dimensional 𝔽-subspaces that are Gal(K/K′)-stable. Pick an 𝔽-basis β_{l} of e_{d−1}ℳ_{r} such that 𝔽 · β_{l} is Gal(K/K′)-stable for all l. Pick a subset α^{(j)} (1 ≤ j ≤ d) of that basis that forms a basis inside e_{d−1}(ℳ_{r}/uℳ_{r}) ≅ 𝔽^{d}. (The latter isomorphism follows from the theorem on elementary divisors, since r < p.) By Nakayama’s lemma the α^{(j)} generate e_{d−1}ℳ_{r}, as required. □

Lemma 3.3.7. Suppose that M and M′ are free 𝔽[u]/u^{ep}-modules of rank d and that f : M → M′ is an 𝔽[u]/u^{ep}-linear map. Let (e_{j})_{j=1}^{d} be a basis of M. If f(e_{1}) ∈ u^{n+1}M′ for some positive integer n+1 ≤ ep, then im(f) ⊅ u^{n}M′.

Proof. Suppose to the contrary that we have the containment im(f) ⊃ u^{n}M′. Let I denote the 𝔽-vector space im(f)/u im(f). On the one hand, the images of the f(e_{j}) span I, and by assumption f(e_{1}) is zero in I. Thus dim_{𝔽} I < d. On the other hand, by the theorem on elementary divisors we can pick a basis (e′_{j})_{j=1}^{d} of M′ and non-negative integers a_{j} (possibly greater than or equal to ep) such that the (u^{a_{j}}e′_{j})_{j=1}^{d} span im(f) as an 𝔽[u]/u^{ep}-module. Since u^{n}M′ is spanned by the (u^{n}e′_{j})_{j=1}^{d}, we see that a_{j} ≤ n < ep for all j. Therefore dim_{𝔽} I = d and we get a contradiction, as required. □

We owe the following argument to Xavier Caruso. We would like to thank him for letting us include it here. Without it, we would need a slightly stronger condition on (a, b, c) in Theorem 3.3.13 below. (Namely we would need to demand that a − b > 3, b − c > 3, a − c < p − 4 and that none of 2a−b−c, 2b−a−c, 2c−a−b are congruent to an element of [−3, 3] modulo p − 1.) In the statement, note that, as gcd(p, e) = 1, we have a natural ring homomorphism ℤ[1/p] → ℤ/e (“reduction modulo e”).

Proposition 3.3.8. Maintain our assumptions on K_{0}, K and K′, so e = p^{d} − 1. Assume that r < \frac{p−1}{2}. Suppose that ℳ is a rank d object of 𝔽-BrMod_{dd}^{r}. Suppose that T_{st}^{∗,r}(ℳ) extends to an irreducible representation of G_{ℚ_{p}}. Suppose that 𝒩 is a rank one object of 𝔽-BrMod_{dd}^{r} and that we are given a map 𝒩 → ℳ of Breuil modules that is injective (as an 𝔽-linear map).

Suppose that ℳ has a basis m^{(j)} (0 ≤ j ≤ d−1) with ĝ(m^{(j)}) = (ω_{d}(g)^{a_{j}} ⊗ 1)m^{(j)} for all j and all g ∈ Gal(K/K_{0}), where the integers a_{j} satisfy the condition that (a_{j} − a_{l})/p is congruent to an element of the interval (\frac{er}{p−1}, \frac{e(p−1−r)}{p−1}) modulo e whenever j ≠ l. Then there exists a basis n of 𝒩 and a permutation π of {0, 1, …, d − 1} such that

ĝ(n) = ∑_{i=0}^{d−1}(ω_{d}(g)^{a_{π(i)}} ⊗ 1)e_{i}n.

Proof. Since 𝒩 → ℳ is nonzero and T_{st}^{∗,r} is faithful we see that T_{st}^{∗,r}(𝒩) is a one-dimensional subrepresentation 𝔽(χ) of T_{st}^{∗,r}(ℳ). As T_{st}^{∗,r}(ℳ) extends to an irreducible representation of G_{ℚ_{p}} we have T_{st}^{∗,r}(ℳ) ≅ ⨁_{j=0}^{d−1} 𝔽(χ^{p^{j}}) and the characters χ^{p^{j}} (0 ≤ j ≤ d − 1) are pairwise distinct. By Lemma 3.2.9 there are rank one Breuil modules 𝒩^{(j)} for 0 ≤ j ≤ d−1 together with maps 𝒩^{(j)} → ℳ that are injective as 𝔽-linear maps and such that T_{st}^{∗,r}(𝒩^{(j)}) ≅ 𝔽(χ^{p^{j}}). We can take 𝒩^{(0)} = 𝒩. It follows that we have a map α : ⨁_{j} 𝒩^{(j)} → ℳ such that T_{st}^{∗,r}(α) is an isomorphism. By the universal property of maximal objects and Lemma 1.5.1 of [Car11] we have a map β : ℳ → ⨁_{j} Max(𝒩^{(j)}) such that β ∘ α is a direct sum of the natural maps 𝒩^{(j)} → Max(𝒩^{(j)}).

By Lemma 3.3.2 there is a basis n^{(j)} of 𝒩^{(j)}, integers r_{i}^{(j)} ∈ [0, er] and x_{i}^{(j)} ∈ {a_{1}, …, a_{d}}, and λ^{(j)} ∈ (k ⊗_{𝔽_{p}} 𝔽)^{×} such that

∘ e_{i}𝒩_{r}^{(j)} = u^{r_{i}^{(j)}}e_{i}𝒩^{(j)},
∘ φ_{r}(∑_{i=0}^{d−1} u^{r_{i}^{(j)}}e_{i}n^{(j)}) = λ^{(j)}n^{(j)},
∘ ĝ(n^{(j)}) = ∑_{i=0}^{d−1}(ω_{d}(g)^{x_{i}^{(j)}} ⊗ 1)e_{i}n^{(j)} for all g ∈ Gal(K/K_{0}), and
∘ N(n^{(j)}) = 0.

(For the descent data note that e_{i}(𝒩/u𝒩) injects into e_{i}(ℳ/uℳ).) By Lemma 3.3.3 we know that there is a basis n^{(j)} of Max(𝒩^{(j)}) and an integer s such that

∘ e_{i} Max(𝒩^{(j)})_{r} = u^{er}e_{i} Max(𝒩^{(j)}),

∘ φ_{r}(∑_{i=0}^{d−1} u^{er}e_{i}n^{(j)}) = λ^{(j)}n^{(j)},

∘ ĝ(n^{(j)}) = ∑_{i=0}^{d−1}(ω_{d}(g)^{p^{i+j}s} ⊗ 1)e_{i}n^{(j)} for all g ∈ Gal(K/K_{0}), and

∘ N(n^{(j)}) = 0.

(Here we use that T_{st}^{∗,r}(𝒩^{(j)}) ≅ 𝔽(χ^{p^{j}}).) The same lemma shows that

(3.3.9) p^{i+j}s ≡ x_{i}^{(j)} − \frac{er}{p − 1} + \frac{r_{i}^{(j)}p^{d} + r_{i+1}^{(j)}p^{d−1} + ⋯ + r_{i+d−1}^{(j)}p}{e} (mod e).

Recall from the proof of Lemma 3.3.3 that the natural map

𝒩^{(j)} → Max(𝒩^{(j)})

sends n^{(j)} to ∑_{i} u^{t_{i}^{(j)}}e_{i}n^{(j)} for certain t_{i}^{(j)} ∈ [0, \frac{erp}{p−1}]. In particular, it follows that

(3.3.10)  im(β ∘ α) ⊃ ⨁_{j=0}^{d−1} u^{erp/(p−1)} Max(𝒩^{(j)}).

By Lemma 3.3.6 we may assume that m^{(j)} ∈ φ_{r}(ℳ_{r}). (Note that we may arrange that ĝ(m^{(j)}) = (ω_{d}(g)^{a_{j}} ⊗ 1)m^{(j)}: for each i we have e_{i}(ℳ/uℳ) ≅ ⨁_{j} 𝔽(ω_{d}(g)^{a_{j}}) as 𝔽[Gal(K/K_{0})]-module by our assumption on the descent data of ℳ, so we may permute the e_{i}m^{(j)} such that ĝ(e_{i}m^{(j)}) = ω_{d}(g)^{a_{j}}e_{i}m^{(j)} for all j.) It follows that

β(m^{(j)}) ∈ ⨁_{l} φ_{r}((Max 𝒩^{(l)})_{r}) = ⨁_{l}(k ⊗_{𝔽_{p}} 𝔽)[u^{p}]/u^{ep} · n^{(l)}.

Since β commutes with descent data, it furthermore follows that

β(e_{0}m^{(j)}) = ∑_{l=0}^{d−1} γ_{jl}u^{b_{jl}}e_{0}n^{(l)},

where γ_{jl} ∈ 𝔽 and 0 ≤ b_{jl} < ep is uniquely determined by the congruences

\begin{aligned}b_{jl} & ≡ 0 (mod p), \\ b_{jl} & ≡ a_{j} − p^{l}s (mod e). \end{aligned} ; (3.3.11)

It will be convenient to assume in the following that the a_{j}, and hence the x_{i}^{(j)}, are divisible by p. (Note that so far that only their values modulo e mattered and that (e, p) = 1.) Since r_{i}^{(j)} ∈ [0, er] it follows from (3.3.9) that (3.3.12)

p^{i+j−1}s is congruent to an element of [\frac{x_{i}^{(j)}}{p} − \frac{er}{p − 1}, \frac{x_{i}^{(j)}}{p} ] modulo e.

Our assumption on the a_{j} implies that two such intervals are either equal or disjoint. In particular, x_{i}^{(j)} only depends on i + j modulo d.

Suppose now that the (x_{i}^{(0)})_{i=0}^{d−1} are not all distinct. (This is the descent data on 𝒩.) Then the (x_{0}^{(l)})_{l=0}^{d−1} are not all distinct. So there is j such that x_{0}^{(l)} ≠ a_{j} for all l. From equations (3.3.11) and (3.3.12) we find that \frac{b_{jl}}{p} is congruent to an element of

[\frac{a_{j} − x_{0}^{(l−1)}}{p} , \frac{a_{j} − x_{0}^{(l−1)}}{p} + \frac{er}{p − 1}]

modulo e. Our assumption on the a_{j} then implies that b_{jl} > \frac{erp}{p−1} for all l. But, by applying Lemma 3.3.7 to the map β : e_{0}ℳ → ⨁_{j} e_{0} Max(𝒩^{(j)}) and using that r < p − 1, we see that this contradicts (3.3.10). □

Theorem 3.3.13. Let ρ : G_{ℚ_{p}} → GL_{3}(\overline{ℚ}_{p}) be a potentially semistable representation with Hodge–Tate weights −2, −1, and 0 such that \overline{ρ} is irreducible.

In the following we assume that a, b, c are integers satisfying a − b > 2, b − c > 2, a − c < p − 3.

(i) Suppose that WD(ρ)|_{I_{ℚp}} ≅ ω^{a} ⊕ω^{b} ⊕ω^{c}. Then \overline{ρ}|_{I_{ℚp}} ≅ ψ ⊕ψ^{p} ⊕ψ^{p^{2}},

where

ψ = ω_{3}^{(a+a_{0})+p(c+a_{2})+p^{2}(b+a_{1})}

or

ψ = ω_{3}^{(a+2−a_{2})+p(b+2−a_{1})+p^{2}(c+2−a_{0})}

with (a_{0}, a_{1}, a_{2}) ∈ {(1, 1, 1), (1, 2, 0), (2, 1, 0)}.

(ii) Suppose that WD(ρ)|_{I_{ℚp}} ≅ ω_{3}^{a+pb+p^{2}c} ⊕ω_{3}^{b+pc+p^{2}a} ⊕ω_{3}^{c+pa+p^{2}b}. Then

\overline{ρ}|_{I_{ℚp}} ≅ ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where
ψ = ω_{3}^{(a+a_{0})+p(c+a_{2})+p^{2}(b+a_{1})}
with (a_{0}, a_{1}, a_{2}) ∈ {(0, 2, 1), (1, 1, 1), (1, 2, 0)}, or
ψ = ω_{3}^{(a+a_{0})+p(b+a_{2})+p^{2}(c+a_{1})}
with a_{0}, a_{1}, a_{2} ∈ [0, 2] such that a_{1} + a_{2} + a_{3} = 3.

(iii) Suppose that WD(ρ)|_{I_{ℚp}} ≅ ω_{3}^{c+pb+p^{2}a} ⊕ω_{3}^{b+pa+p^{2}c} ⊕ω_{3}^{a+pc+p^{2}b}. Then

\overline{ρ}|_{I_{ℚp}} ≅ ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where
ψ = ω_{3}^{(c+2−a_{0})+p(a+2−a_{2})+p^{2}(b+2−a_{1})}
with (a_{0}, a_{1}, a_{2}) ∈ {(0, 2, 1), (1, 1, 1), (1, 2, 0)}, or
ψ = ω_{3}^{(c+2−a_{0})+p(b+2−a_{2})+p^{2}(a+2−a_{1})}
with a_{0}, a_{1}, a_{2} ∈ [0, 2] such that a_{1} + a_{2} + a_{3} = 3.

Here we write ω for ω_{1}, the mod p cyclotomic character, and ω̃ for its Teichmüller lift. Note that we consider ω̃_{3} as \overline{ℚ}_{p}-valued character and ω_{3} as \overline{𝔽}_{p}-valued character. This depends on a choice of embedding K_{0} → \overline{ℚ}_{p}, but all the statements in the theorem are independent of that choice.

Proof. As before we let K′ = K_{0} be the unique unramified cubic extension of ℚ_{p} and let K = K_{0}((−p)^{1/(p^{3}−1)}). We may assume (by a standard Baire category argument) that ρ : G_{ℚ_{p}} → GL_{3}(𝒪_{E}) for some finite extension E/ℚ_{p}. (Take E large enough so as to contain the images of all embeddings K → \overline{ℚ}_{p}.) The assumptions on a, b, c show that p > 7, so we may apply Proposition 3.3.1 with n = d = 3 and r = 2 < p − 1 to ρ|_{G_{K_{0}}} , we see that there is an object ℳ of 𝔽-BrMod_{dd}^{2} with T_{st}^{∗,2}(ℳ) ≅ \overline{ρ}, and that Gal(K/K_{0}) acts on some basis of ℳ via the characters ω^{a} ⊗ 1, ω^{b} ⊗ 1, ω^{c} ⊗ 1 in the first case, or ω_{3}^{a+pb+p^{2}c} ⊗ 1, ω_{3}^{b+pc+p^{2}a} ⊗ 1, ω_{3}^{c+pa+p^{2}b} ⊗ 1 in the second case, or ω_{3}^{c+pb+p^{2}a} ⊗ 1, ω_{3}^{b+pa+p^{2}c} ⊗ 1, ω_{3}^{a+pc+p^{2}b} ⊗ 1 in the third case. (Note that, for example in the first case, we at first get a basis on which Gal(K/K_{0}) acts via 1 ⊗ σ_{0}ω^{a}, 1 ⊗ σ_{0}ω^{b}, 1 ⊗ σ_{0}ω^{c}, but then we can just permute the induced basis on each e_{i}ℳ to get the desired basis for ℳ.)

Since \overline{ρ} is assumed to be irreducible, \overline{ρ}|_{G_{K_{0}}} ≅ χ ⊕ χ^{p} ⊕ χ^{p^{2}} for some character χ : G_{K_{0}} → 𝔽^{×} (e.g. by Lemma 2.16 of [ADP02]). By Corollary 3.2.9, there is a rank one submodule 𝒩 of ℳ with T_{st}^{∗,2}(𝒩) ≅ χ. Then by Lemma 3.3.2 𝒩 has the form

∘ 𝒩 = ((k ⊗_{𝔽_{p}} 𝔽)[u]/u^{ep}) · n,

∘ e_{i}𝒩_{2} = u^{r_{i}}e_{i}𝒩,
∘ φ_{2}(∑_{i=0}^{2} u^{r_{i}}e_{i}n) = λn for some λ ∈ (k ⊗_{𝔽_{p}} 𝔽)^{×}, and

∘ ĝ(n) = (∑_{i=0}^{2}(ω_{3}(g)^{k_{i}} ⊗ 1)e_{i})n for all g ∈ Gal(K/K_{0}),

where each k_{i} ∈ ℤ, r_{i} ∈ [0, 2e], and r_{i} ≡ p^{2}k_{i+1} − k_{i} (mod e). Note that e = p^{3} −1. We now consider the three cases in the statement of the theorem separately.

First case: if WD(ρ)|_{I_{ℚp}} ≅ ω^{a} ⊕ ω^{b} ⊕ ω^{c}, then WD(det ρ)|_{I_{K_{0}}} = ω^{a+b+c}, so that (WD(det ρ)ω^{−(a+b+c)})|_{I_{K_{0}}} = 1. Thus (det ρ)|_{G_{K_{0}}} ω^{−(a+b+c)} is a crystalline character with all Hodge–Tate weights equal to −3 (since ρ is assumed to have Hodge–Tate weights −2, −1 and 0), so we see that (det ρ)ω^{−(a+b+c)}|_{I_{K_{0}}} = ε^{3}, and in particular

det \overline{ρ}|_{I_{K_{0}}} = χ^{1+p+p^{2}}|_{I_{K_{0}}} = ω^{a+b+c+3}.

By Proposition 3.3.8 and the assumptions on a, b, c we may assume that the k_{i} are a permutation of (1+p+p^{2})a, (1+p+p^{2})b, (1+p+p^{2})c. (Consider 𝒩/u𝒩 to compare the k_{i} with the a_{π(i)} in the proposition.) Write k_{i} = (1 + p+p^{2})x_{i} with x_{i} ∈ {a, b, c}, so that r_{i} ≡ p^{2}k_{i+1} −k_{i} ≡ (1+p+p^{2})(x_{i+1} −x_{i}) (mod e). Since r_{i} ∈ [0, 2e] and by our conditions on (a, b, c), we can write r_{i} = (1 + p + p^{2})(x_{i+1} − x_{i}) + a_{i}e with a_{i} ∈ [0, 2]. By Lemma 3.3.2, we see that one of χ|_{I_{K_{0}}} , χ^{p}|_{I_{K_{0}}} and χ^{p^{2}}|_{I_{K_{0}}} is equal to

ω_{3}^{(x_{0}+a_{0})+p(x_{2}+a_{2})+p^{2}(x_{1}+a_{1})}.

Since χ^{1+p+p^{2}}|_{I_{K_{0}}} = ω^{a+b+c+3}, we see that

x_{0} + x_{1} + x_{2} + a_{0} + a_{1} + a_{2} ≡ a + b + c + 3 (mod p − 1),

so a_{0} + a_{1} + a_{2} = 3. By cyclic symmetry (our choice of σ_{0}) we may assume that either x_{0} = a, x_{1} = b, x_{2} = c or x_{0} = a, x_{1} = c, x_{2} = b. Our conditions on (a, b, c) and that each r_{i} ∈ [0, 2e] then give the claimed result.

Second case: if WD(ρ)|_{I_{ℚp}} ≅ ω_{3}^{a+pb+p^{2}c} ⊕ ω_{3}^{b+pc+p^{2}a} ⊕ ω_{3}^{c+pa+p^{2}b}, then arguing as in the previous case we see that χ^{1+p+p^{2}}|_{I_{K_{0}}} = ω^{a+b+c+3}, and by Proposition 3.3.8 and the assumptions on a, b, c we may assume that the k_{i} are a permutation of a + pb + p^{2}c, b + pc + p^{2}a, c + pa + p^{2}b. (To check the condition on the a_{j} in the proposition, it suffices to show that 2(p^{2} + p + 1) < p^{2}x + py + z < (p − 3)(p^{2} + p + 1) whenever x, y, z are in {±(a−b), ±(b−c), ±(c−a)} and x > 0. The second inequality is obvious. For the first, note that x ≥ 3 and that y and z are bounded below by −(p − 4).) Write k_{i} = x_{i} + px′_{i} + p^{2}x′′_{i}, where (x_{i}, x′_{i}, x′′_{i}) is a cyclic permutation of (a, b, c). So by our conditions on (a, b, c) and since r_{i} ∈ [0, 2e], we can write r_{i} = (x′_{i+1} + px′′_{i+1} + p^{2}x_{i+1}) − (x_{i} + px′_{i} + p^{2}x′′_{i}) + a_{i}e with a_{i} ∈ [0, 2]. By Lemma 3.3.2, we see that one of χ|_{I_{K_{0}}} , χ^{p}|_{I_{K_{0}}} and χ^{p^{2}}|_{I_{K_{0}}} is equal to

ω_{3}^{(x_{0}+a_{0})+p(x_{2}+a_{2})+p^{2}(x_{1}+a_{1})}.

Just as above we see that a_{0} + a_{1} + a_{2} = 3. Again we may assume that either x_{0} = a, x_{1} = b, x_{2} = c or x_{0} = a, x_{1} = c, x_{2} = b. The possibilities for the a_{i} are determined by the signs of x_{i+1} − x′′_{i}. So in the first case we find that a_{0} ≠ 2, a_{1} ≠ 0, and a_{2} ≠ 2, whereas in the second case we get no restriction on the a_{i} (because we have r_{i} = a_{i}e in this case).

Third case: this is analogous to the second case, or can be deduced from the second case by duality. □

4. Abstract framework

In order to avoid too much notational complexity, and with an eye to future applications and possible generalisations of our work, in this section we set up an axiomatic framework for the general setting in which our results will apply. In Section 7 we will establish that our axioms hold in the particular case of spaces of automorphic forms on definite unitary groups.

It will be convenient to introduce our axiomatic framework in two stages. Since our weight cycling arguments take place in the context of representations over \overline{𝔽}_{p} of the group GL_{n}(ℚ_{p}), it is most natural to place our framework in the setting of such \overline{𝔽}_{p}-representations. This is what we do in Subsection 4.2 below. On the other hand, in practice, our axioms will be established by a comparison between a mod p and a characteristic zero setting, and it will be convenient to axiomatise this argument as well. The relevant axiomatic framework for this argument is the subject of Subsection 4.3, and the argument itself is the subject of Subsections 4.4 and 4.5. In Subsection 4.1 we recall some basic facts and notation related to representations of GL_{n}.

4.1. Some representations of GL_{n}. Let F be a number field. From now on until Section 6 we fix a place w of F lying over p. Fix a uniformiser ϖ of F_{w} and write k_{w} for the residue field of F_{w}.

4.1.1. Dual Weyl modules. Let ℤ_{+}^{n} denote the set of tuples (λ_{1}, …, λ_{n}) of integers with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}. Let B_{n} ⊂ GL_{n} be the Borel subgroup of upper-triangular matrices and let T_{n} ⊂ B_{n} be the diagonal maximal torus. We can then view any λ ∈ ℤ_{+}^{n} as a dominant element of X^{∗}(T_{n}) = Hom(T_{n}, 𝔾_{m}), and let M_{λ} be the algebraic ℤ-representation of GL_{n} given by

M_{λ} := Ind_{B_{n}}^{GL_{n}}(w_{0}λ)_{/ℤ},

where w_{0} is the longest element of the Weyl group (see [Jan03] for more details of these notions). Then for any commutative ring A we have that M_{λ}(A) ≅ M_{λ}(ℤ) ⊗_{ℤ} A is a finite free A-module with a natural action of GL_{n}(A) that is functorial in A. For any λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}, let W_{λ} be the finite free \overline{ℤ}_{p}-module with an action of GL_{n}(𝒪_{F_{w}}) given by

W_{λ} := ⨂_{τ:F_{w}→\overline{ℚ}_{p}} (M_{λ_{τ}} (𝒪_{F_{w}}) ⊗_{𝒪_{Fw} ,τ} \overline{ℤ}_{p}).

Note that W_{λ}⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} has a natural GL_{n}(F_{w})-action, as M_{λ_{τ}} (𝒪_{F_{w}})⊗_{𝒪_{Fw}} F_{w} ≅ M_{λ_{τ}} (F_{w}).

4.1.2. Weights of GL_{n}(k_{w}). We now explicitly describe all weights (i.e. irreducible \overline{𝔽}_{p}-representations) of GL_{n}(k_{w}).

Given any a ∈ ℤ_{+}^{n} we define the algebraic 𝔽_{p}-representation N_{a} of GL_{n} to be the subrepresentation of M_{a}_{/𝔽_{p}} that is generated by the highest weight vector. (It is in fact the unique irreducible subrepresentation of M_{a}_{/𝔽_{p}}; see §II.2 in [Jan03].)

We say that an element a ∈ (ℤ_{+}^{n})^{Hom(k_{w},\overline{𝔽}_{p})} is a restricted weight if for each σ ∈ Hom(k_{w}, \overline{𝔽}_{p}) and each 1 ≤ i ≤ n − 1 we have a_{σ,i} − a_{σ,i+1} ≤ p − 1. For a restricted weight a ∈ (ℤ_{+}^{n})^{Hom(k_{w},\overline{𝔽}_{p})}, we define an \overline{𝔽}_{p}-representation F_{a} of GL_{n}(k_{w}) by

(4.1.3)  F_{a} := ⨂_{σ:k_{w}→\overline{𝔽}_{p}} (N_{a_{σ}}(k_{w}) ⊗_{k_{w},σ} \overline{𝔽}_{p}).

It is irreducible, and every irreducible \overline{𝔽}_{p}-representation of GL_{n}(k_{w}) is of the form F_{a} for some a. For restricted weights a and b we have F_{a} ≅ F_{b} as representations of GL_{n}(k_{w}) if and only if for each σ ∈ Hom(k_{w}, \overline{𝔽}_{p}), we have

a_{σ,i} − a_{σ,i+1} = b_{σ,i} − b_{σ,i+1}

for each 1 ≤ i ≤ n − 1 and the character k_{w}^{×} → \overline{𝔽}_{p}^{×} given by

x ↦ ∏_{σ∈Hom(k_{w},\overline{𝔽}_{p})} σ(x)^{a_{σ,n}−b_{σ,n}}

is trivial. (For all this, see for example Theorems 3.9 and 3.10 of and the appendix to [Her09].)

If F_{w}/ℚ_{p} is unramified, the sets Hom(F_{w}, \overline{ℚ}_{p}) and Hom(k_{w}, \overline{𝔽}_{p}) are in natural bijection. We can thus define λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})} by demanding that λ_{τ} = a\overline{τ} for all τ and we call it the lift of a. Since N_{a} is by definition a subrepresentation of M_{a}_{/𝔽_{p}}, it follows that we have a natural GL_{n}(k_{w})-linear map N_{a}(k_{w}) ↪ M_{a}(𝒪_{F_{w}}) ⊗_{𝒪_{Fw}} k_{w} and hence a natural GL_{n}(k_{w})-linear map F_{a} ↪ W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}.

4.1.4. Locally algebraic modules. If G is an open subgroup of GL_{n}(F_{w}), we say that a function f : G → \overline{ℚ}_{p} is locally algebraic if each element of G has a neighbourhood U such that the restriction f_{|U} of f to U is of the form f_{|U} : g ↦ ϕ(g), for some element ϕ of the affine coordinate ring of the affine algebraic group (Res_{F_{w}/ℚ_{p}} GL_{n})_{/\overline{ℚ}_{p}}.

If V is a finite free \overline{ℤ}_{p}-module equipped with an action of an open subgroup G of GL_{n}(F_{w}), we say that the G-action on V is locally algebraic if each of the matrix coefficients of V is a locally algebraic function on G. (Recall that the matrix coefficients of V are the functions on G of the form g ↦ ⟨gv, v^{∨}⟩, where v ∈ V and v^{∨} ∈ V^{∨}.)

As an example, note that the GL_{n}(𝒪_{F_{w}})-action on each of the dual Weyl modules W_{λ} introduced in §4.1.1 is algebraic and thus locally algebraic.

4.2. Axioms for mod p representations. We keep our notation of the preceding subsection. In particular, recall that we assume that w is a place of the number field F that lies over p. From now on until Section 6 we will assume moreover that F_{w}/ℚ_{p} is unramified. (The only exception is Proposition 4.5.2.)

Let r̄: G_{F} → GL_{n}(\overline{𝔽}_{p}) be a continuous representation such that r̄|_{G_{Fw}} is irreducible. Let 𝒫 be a set of finite places of F which lie above split places of F^{+} and do not lie over p, and let 𝕋^{𝒫} denote the commutative \overline{ℤ}_{p}-polynomial algebra generated by formal variables T_{v}^{(j)} where 1 ≤ j ≤ n and v ∈ 𝒫. We will write 𝕋 for 𝕋^{𝒫} when the choice of 𝒫 is clear.

Assume that r̄ is unramified at each place v ∈ 𝒫. We define a maximal ideal 𝔪 of 𝕋 with residue field \overline{𝔽}_{p} by demanding that for each v ∈ 𝒫, the characteristic polynomial of r̄^{∨}(Frob_{v}) is equal to the reduction modulo 𝔪 of

X^{n} + ⋯ + (−1)^{j}(Nv)^{j(j−1)/2}T_{v}^{(j)}X^{n−j} + ⋯ + (−1)^{n}(Nv)^{n(n−1)/2}T_{v}^{(n)}.

(By definition, 𝔪 is determined by r̄, but in the applications 𝒫 will be chosen so that conversely r̄ is determined by 𝔪.)

We let V denote a weight of the group GL_{n}(k_{w}). (Recall that when k_{w} = 𝔽_{p} such a weight is determined by an n-tuple of integers a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} satisfying a_{1}−a_{2}, …, a_{n−1}−a_{n} ≤ p−1 and we will write V = F (a_{1}, …, a_{n}). For more details of this, see §4.1.2.)

Write G = GL_{n}, and consider the Hecke algebra ℋ_{G}(V) defined in Section 2.2. In Section 2.2 we defined the elements \overline{T}_{μ,ϖ} ∈ ℋ_{G}(V) for μ ∈ X_{∗}(T_{n})_{−}, which we can think of as a non-decreasing sequence of n integers. For 1 ≤ j ≤ n we let \overline{T}_{j} = \overline{T}_{μ^{(j)},ϖ}, where μ^{(j)} = (0, …, 0, 1, …, 1), with n − j zeroes followed by j ones.

Now let S be an \overline{𝔽}_{p}-vector space with an action of 𝕋, and suppose that S additionally carries a smooth action of GL_{n}(F_{w}), and that the actions of 𝕋 and GL_{n}(F_{w}) commute. If V is a weight of GL_{n}(k_{w}), we write S(V) for (V ⊗_{\overline{𝔽}_{p}} S)^{GL_{n}(𝒪_{Fw})} , so that S(V) carries commuting actions of 𝕋 and ℋ_{G}(V). We additionally make the following assumptions, letting F_{w,n} denote the unramified extension of F_{w} of degree n and k_{w,n} its residue field.

A1 Hecke operators at p act by zero. For all weights V of GL_{n}(k_{w}) the vector space S(V) is finite dimensional, and each of the Hecke operators \overline{T}_{1}, …, \overline{T}_{n−1} acts nilpotently on the localisation S(V)_{𝔪}. A2 Lifts of specified weight and type. Suppose that F_{w} = ℚ_{p}. Suppose

that S(V)_{𝔪} ≠ 0 for some V = F (a_{1}, …, a_{n}). Then

∘ r̄|_{G_{Fw}} has a crystalline lift with Hodge–Tate weights −(a_{1} +

n − 1), …, −(a_{n−1} + 1), −a_{n}. ∘ Suppose that F (a_{1}, …, a_{n}) is a Jordan–Hölder factor of the reduction mod p of the cuspidal representation R_{𝕋}^{θ} of GL_{n}(k_{w}) as in Subsection 2.4, for some primitive character θ : k_{w,n}^{×} → \overline{ℚ}_{p}^{×}. Then r̄|_{G_{Fw}} has a potentially semistable lift with Hodge– Tate weights −(n − 1), …, −1, 0 and inertial type

⨁_{σ∈Gal(k_{w,n}/k_{w})} σ(θ ∘ Art_{F_{w,n}}^{−1}).

∘ Suppose that F (a_{1}, …, a_{n}) is a Jordan–Hölder factor of the reduction mod p of the principal series representation

Ind_{B_{n}(k_{w})}^{GL_{n}(k_{w})}(χ_{1} ⊗ χ_{2} ⊗ ⋯ ⊗ χ_{n})

of GL_{n}(k_{w}) as in Subsection 2.4, for some distinct characters χ_{i} : k_{w}^{×} → \overline{ℚ}_{p}^{×}. Then r̄|_{G_{Fw}} has a potentially semistable lift with Hodge–Tate weights −(n − 1), …, −1, 0 and inertial type

⨁_{i=1}^{n} χ_{i} ∘ Art_{F_{w}}^{−1} .

Remark 4.2.1. We could have stated Axiom A2 in greater generality, but we decided to give the simplified statement in the case that F_{w} = ℚ_{p} because this is all we need in our main theorems.

Definition 4.2.2. We will say that r̄ is modular of weight V if S(V)_{𝔪} ≠ 0. We let W_{w}(r̄) denote the set of weights V for which r̄ is modular of weight V.

In the eventual application to automorphic forms on definite unitary groups, Axioms A1 and A2 will be consequences of local-global compatibility at places dividing p for the p-adic Galois representations associated to automorphic forms.

The following lemma will be useful later.

Lemma 4.2.3. Suppose that Axiom A1 holds. Then the smooth GL_{n}(F_{w})- representation S is admissible, and whenever V is a finite free \overline{𝔽}_{p}-module with a smooth action of GL_{n}(𝒪_{F_{w}}), we have a natural isomorphism

S(V)_{𝔪} ≅ (V ⊗_{\overline{𝔽}_{p}} S_{𝔪})^{GL_{n}(𝒪_{Fw})} .

Proof. Suppose that U is any compact open subgroup of GL_{n}(𝒪_{F_{w}}). Then

S^{U} ≅ Hom_{U}(1, S) ≅ Hom_{GL_{n}(𝒪_{Fw})} (Ind_{U}^{GL_{n}(𝒪_{Fw})} 1, S),

so dim S^{U} is bounded above by the sum of the dimensions Hom_{GL_{n}(𝒪_{Fw})} (V, S) ≅

S(V^{∨}), where V runs over all irreducible constituents of Ind_{U}^{GL_{n}(𝒪_{Fw})} 1. By Axiom A1 it follows that dim S^{U} < ∞, so S is admissible.

Now for any V as in the statement of the lemma, we have that V ⊗_{\overline{𝔽}_{p}} S ≅ \underrightarrow{lim}_{U}(V ⊗_{\overline{𝔽}_{p}} S^{U}), where U runs through compact open subgroups that are normal in GL_{n}(𝒪_{F_{w}}). Note that the transition maps of the inductive limit are injective. It follows that

S(V)_{𝔪} ≅ \underrightarrow{lim}_{U}((V ⊗_{\overline{𝔽}_{p}} S^{U})^{GL_{n}(𝒪_{Fw})} )_{𝔪},
(V ⊗_{\overline{𝔽}_{p}} S_{𝔪})^{GL_{n}(𝒪_{Fw})} ≅ \underrightarrow{lim}_{U}((V ⊗_{\overline{𝔽}_{p}} S^{U})_{𝔪})^{GL_{n}(𝒪_{Fw})} .

As S is admissible, we have that V ⊗_{\overline{𝔽}_{p}} S^{U} is finite-dimensional for all U. For any finite free \overline{𝔽}_{p}-module M, M_{𝔪} is naturally a direct summand of M: it is the largest subspace on which all T ∉ 𝔪 act invertibly. It follows that

((V ⊗_{\overline{𝔽}_{p}} S^{U})^{GL_{n}(𝒪_{Fw})} )_{𝔪} ⟶^{∼} ((V ⊗_{\overline{𝔽}_{p}} S^{U})_{𝔪})^{GL_{n}(𝒪_{Fw})} . □

4.3. Axioms for characteristic zero representations. We maintain the notation of the preceding subsection. In this subsection we introduce an axiomatic framework that relates certain characteristic p and characteristic 0 representations of GL_{n}(F_{w}).

To this end, we suppose given a \overline{ℤ}_{p}-module S̃ and an \overline{𝔽}_{p}-vector space S, each equipped with commuting actions of GL_{n}(F_{w}) and 𝕋, together with an embedding

(4.3.1)  S ↪ S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}.

that is equivariant for the two actions. Suppose that the GL_{n}(F_{w})-action on S is smooth, so S is as in the preceding section. If Ṽ is a \overline{ℤ}_{p}-module equipped with a GL_{n}(𝒪_{F_{w}})-action, then we write S̃(Ṽ) = (Ṽ ⊗_{\overline{ℤ}_{p}} S̃)^{GL_{n}(𝒪_{Fw})} . Hence

if Ṽ is an A-module for a \overline{ℤ}_{p}-algebra A equipped with a GL_{n}(𝒪_{F_{w}})-action, then S̃(Ṽ) is an A-module as well.

For λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}, we defined in §4.1.1 a dual Weyl module W_{λ}. It is a finite free \overline{ℤ}_{p}-module with a (locally) algebraic GL_{n}(𝒪_{F_{w}})-action, and W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} has a natural GL_{n}(F_{w})-action. Thus the \overline{ℚ}_{p}-vector space

S̃(W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) ≅ ((W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) ⊗_{\overline{ℤ}_{p}} S̃)^{GL_{n}(𝒪_{Fw})}

carries a natural action of the (commutative) double coset algebra

\overline{ℤ}_{p}[GL_{n}(𝒪_{F_{w}})\ GL_{n}(F_{w})/ GL_{n}(𝒪_{F_{w}})]

that commutes with the 𝕋-action. Explicitly, suppose that g ∈ GL_{n}(F_{w}) and f ∈ ((W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) ⊗_{\overline{ℤ}_{p}} S̃)^{GL_{n}(𝒪_{Fw})} . Write GL_{n}(𝒪_{F_{w}})g GL_{n}(𝒪_{F_{w}}) = ∐_{i} k_{i}g GL_{n}(𝒪_{F_{w}}) as a finite disjoint union with k_{i} ∈ GL_{n}(𝒪_{F_{w}}). Then

(4.3.2)  [GL_{n}(𝒪_{F_{w}})g GL_{n}(𝒪_{F_{w}})] (f) = ∑_{i} k_{i}gf,

where k_{i}g acts diagonally.

Definition 4.3.3. For μ ∈ X_{∗}(T_{n})_{−} denote by T_{μ,w} the following double coset operator on S̃(W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}):

T_{μ,w} := [GL_{n}(𝒪_{F_{w}})μ(ϖ) GL_{n}(𝒪_{F_{w}})] .

Moreover, let T_{j} = T_{μ^{(j)},w}, where μ^{(j)} = (0, …, 0, 1, …, 1), with n−j zeroes followed by j ones.

We introduce the following axioms for S̃ related to the above set-up.

Ã1 Finiteness and extension of scalars. If Ṽ is a free finite-rank \overline{ℤ}_{p}- module equipped with a locally algebraic action of GL_{n}(𝒪_{F_{w}}), then S̃(Ṽ) is a finite free \overline{ℤ}_{p}-module and for A = \overline{ℚ}_{p} and A = \overline{𝔽}_{p} the natural map

S̃(Ṽ) ⊗_{\overline{ℤ}_{p}} A → S̃(Ṽ ⊗_{\overline{ℤ}_{p}} A)

is an isomorphism. Ã2 Crystalline lifts. Suppose that λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}. If S̃(W_{λ} ⊗_{\overline{ℤ}_{p}}

\overline{ℚ}_{p})_{𝔪} ≠ 0, then \overline{r}|_{G_{Fw}} has a crystalline lift ρ : G_{F_{w}} → GL_{n}(\overline{ℚ}_{p}) such that for all τ : F_{w} → \overline{ℚ}_{p} we have HT_{τ} (ρ) = −(λ_{τ,1} +n−1, …, λ_{τ,n}).

If moreover T_{j} has eigenvalue t_{j} ∈ \overline{ℚ}_{p} on S̃(W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔪} for 1 ≤ j ≤ n, then we can demand that φ^{[F_{w}:ℚ_{p}]} acting on the \overline{ℚ}_{p}- vector space D_{cris}(ρ) has characteristic polynomial

(X^{n} + ⋯ + (−1)^{j}q_{w}^{j(j−1)/2}t_{j}X^{n−j} + ⋯ + (−1)^{n}q_{w}^{n(n−1)/2}t_{n})^{[F_{w}:ℚ_{p}]},

where q_{w} = #k_{w} = p^{[F_{w}:ℚ_{p}]}. Ã3 Lifts of tame type. Suppose that R is a cuspidal representation

(−1)^{n−1}R_{𝕋}^{θ} of GL_{n}(k_{w}) for some primitive character θ : k_{w,n}^{×} → \overline{ℚ}_{p}^{×}, respectively a principal series representation Ind_{B_{n}(k_{w})}^{GL_{n}(k_{w})}(χ_{1} ⊗ ⋯ ⊗

χ_{n}) of GL_{n}(k_{w}) for some distinct characters χ_{i} : k_{w}^{×} → \overline{ℚ}_{p}^{×}. (See Subsection 2.4 for definitions.)

If S̃(R)_{𝔪} ≠ 0, then \overline{r}|_{G_{Fw}} has a potentially semistable lift ρ : G_{F_{w}} → GL_{n}(\overline{ℚ}_{p}) such that for all τ : F_{w} → \overline{ℚ}_{p} we have HT_{τ} (ρ) = −(n − 1, …, 1, 0) and whose inertial type is

⨁_{σ∈Gal(k_{w,n}/k_{w})} σ(θ ∘ Art_{F_{w,n}}^{−1}),

respectively

⨁_{i=1}^{n} χ_{i} ∘ Art_{F_{w}}^{−1} .

Theorem 4.5.5 below shows that (in an obvious sense) Axioms Ã1–Ã3 imply Axioms A1–A2.

4.4. Hecke action at p. We suppose that S and S̃ are as in the preceding subsection. In this subsection we will only require Axiom Ã1.

Let a ∈ (ℤ_{+}^{n})^{Hom(k_{w},\overline{𝔽}_{p})} be a restricted weight and let λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})} denote its lift as in §4.1.2. From the natural injections F_{a} ↪ W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} and S ↪ S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} we get a 𝕋-equivariant embedding S(F_{a}) ↪ S̃(W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}). By Axiom Ã1 we can think of it as 𝕋-equivariant embedding

(4.4.1)  S(F_{a}) ↪ S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}.

The goal of this subsection is to compare the action of the Hecke operators \overline{T}_{μ,ϖ} on S(F_{a}) with the action of suitably normalised Hecke operators on S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}. Note that this last space is isomorphic to S̃(W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) by Axiom Ã1, so that the Hecke operators T_{μ,w} of Definition 4.3.3 act on it.

Proposition 4.4.2. Suppose that λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})} is the lift of the restricted weight a, and suppose that Axiom Ã1 holds. The action of the Hecke operator (∏_{τ:F_{w}→\overline{ℚ}_{p}} τ(ϖ)^{−⟨μ,λ_{τ} ⟩})T_{μ,w} on S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} has the following properties:

(i) It stabilises the \overline{ℤ}_{p}-lattice S̃(W_{λ}).

(ii) The induced action on S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} stabilises the subspace S(F_{a}). (iii) The induced action on S(F_{a}) coincides with the action of the oper-

ator \overline{T}_{μ,ϖ} in ℋ(F_{a}).

Proof. Consider equation (4.3.2) for g = μ(ϖ). To establish part (i) it suffices to show that (∏_{τ:F_{w}→\overline{ℚ}_{p}} τ(ϖ)^{−⟨μ,λ_{τ} ⟩})μ(ϖ) stabilises W_{λ} ⊂ W_{λ} ⊗\overline{ℚ}_{p}.

Recall that W_{λ} ≅ ∏_{τ:F_{w}→\overline{ℚ}_{p}} (M_{λ_{τ}} (𝒪_{F_{w}}) ⊗_{𝒪_{Fw} ,τ} \overline{ℤ}_{p}). For ν_{τ} ∈ X^{∗}(T_{n}) the

element μ(ϖ) ∈ T_{n}(F_{w}) acts on the ν_{τ} -weight space of M_{λ_{τ}} (𝒪_{F_{w}}) ⊗_{𝒪_{Fw} ,τ} \overline{ℚ}_{p} as the scalar τ(ϖ)^{⟨μ,ν_{τ} ⟩}. Since ν_{τ} ≤ λ_{τ} (as λ_{τ} is the highest weight of M_{λ_{τ}}) and μ ∈ X_{∗}(T_{n})_{−}, we see that ⟨μ, ν_{τ} ⟩ ≥ ⟨μ, λ_{τ} ⟩, so τ(ϖ)^{−⟨μ,λ_{τ} ⟩}μ(ϖ) stabilises M_{λ_{τ}} (𝒪_{F_{w}}) ⊗_{𝒪_{Fw} ,τ} \overline{ℤ}_{p} and part (i) follows.

To establish part (ii), since S ↪ S̃ ⊗ \overline{𝔽}_{p} is GL_{n}(F_{w})-equivariant, it suffices to show that (∏_{τ:F_{w}→\overline{ℚ}_{p}} τ(ϖ)^{−⟨μ,λ_{τ} ⟩})μ(ϖ) stabilises F_{a} ⊂ W_{λ} ⊗

\overline{𝔽}_{p}. Note that the induced action of τ(ϖ)^{−⟨μ,λ_{τ} ⟩}μ(ϖ) on the reduction M_{λ_{τ}} (𝒪_{F_{w}}) ⊗_{𝒪_{Fw} ,τ} \overline{𝔽}_{p} is the linear projection onto the ν_{τ} -weight spaces with ⟨μ, ν_{τ} ⟩ = ⟨μ, λ_{τ} ⟩. Let σ : k_{w} → \overline{𝔽}_{p} denote the embedding induced by τ. Since N_{a_{σ}} has a weight space decomposition, as it is an algebraic subrepresentation of M_{λ_{τ}} _{/𝔽_{p}}, it follows that τ(ϖ)^{−⟨μ,λ_{τ} ⟩}μ(ϖ) stabilises the subspace N_{a_{σ}}(k_{w}) ⊗_{k_{w},σ} \overline{𝔽}_{p}. This implies that part (ii) holds.

By comparing formulae (2.2.1) and (4.3.2) we see that part (iii) holds provided \overline{T}_{μ,ϖ}(μ(ϖ)) ∈ End_{\overline{𝔽}_{p}}(F_{a}) is the linear projection onto the ν_{σ}-weight spaces of N_{a_{σ}}(k_{w}) ⊗_{k_{w},σ} \overline{𝔽}_{p} such that ⟨μ, ν_{σ}⟩ = ⟨μ, a_{σ}⟩ for all σ.

Fix an embedding σ_{0} : k_{w} → \overline{𝔽}_{p} and let σ_{i} := σ_{0}^{p^{i}} for any integer i. Let f := [k_{w} : 𝔽_{p}]. By Steinberg’s tensor product theorem (Corollary II.3.17 in [Jan03]) we have ⨂_{i=0}^{f−1} N_{a_{σ_{i}}}^{(i)} ≅ N_{A}, where A := ∑_{i=0}^{f−1} p^{i}a_{σ_{i}} and where the superscript (i) denotes the i-th Frobenius twist (i.e. the composition with the i-th power of the Frobenius endomorphism of GL_{n}_{/𝔽_{p}}). Therefore

F_{a} ≅ ⨂_{σ} (N_{a_{σ}}(k_{w}) ⊗_{k_{w},σ} \overline{𝔽}_{p}) ≅ N_{A}(k_{w}) ⊗_{k_{w},σ_{0}} \overline{𝔽}_{p}.

Recall from §2.2 that \overline{T}_{μ,ϖ}(μ(ϖ)) projects onto the subspace of N_{A} that is invariant by the k_{w}-points of the unipotent radical of the parabolic of GL_{n} defined by −μ ∈ X_{∗}(T_{n})_{+}. It consists of the ν-weight spaces of N_{A} satisfying ⟨μ, ν⟩ = ⟨μ, A⟩ by Lemma 2.3 in [Her11a]. We can write ν = ∑_{i=0}^{f−1} p^{i}ν_{σ_{i}} where ν_{σ_{i}} is a weight of N_{a_{σ_{i}}} and we see that ⟨μ, ν⟩ = ⟨μ, A⟩ if and only if ⟨μ, ν_{σ_{i}}⟩ = ⟨μ, a_{σ_{i}}⟩ for all i, which is what we wanted to prove. □

Corollary 4.4.3. The action of (∏_{τ} τ(ϖ)^{− ∑_{i=1}^{j} λ_{τ,n+1−i}})T_{j} on S̃(W_{λ}) induces the action of \overline{T}_{j} ∈ ℋ(F_{a}) on S(F_{a}) ⊂ S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}.

4.5. A criterion for the vanishing of Hecke operators at p. In this subsection we explain how one can deduce information about Hecke operators from information about the associated Galois representation via Axiom Ã2. In particular, we deduce that certain Hecke operators in characteristic p have all eigenvalues equal to 0 in the situations that we consider in this paper.

First we recall a variant of a lemma of Deligne and Serre ([DS74]).

Lemma 4.5.1. Let A be a commutative \overline{ℤ}_{p}-algebra acting on a finite-rank free \overline{ℤ}_{p}-module M. If 𝔫 is a maximal ideal of A such that (M ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})_{𝔫} ≠ 0, then there exists a homomorphism of \overline{ℤ}_{p}-algebras θ : A → \overline{ℚ}_{p} whose kernel is contained in 𝔫, such that the θ-eigenspace of M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} is nonzero.

Proof. Since (M ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})_{𝔫} ≠ 0, we see that M_{𝔫} ≠ 0, and hence that (M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫} ≠ 0. Now M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} is a finite-dimensional \overline{ℚ}_{p}-vector space, of which the localisation (M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫} is a subspace. (More precisely, it is the maximal subspace on which all of the elements a ∉ 𝔫 act invertibly.) In particular, (M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫} is again finite-dimensional.

If we let A′ denote the image of A⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p} in End_{\overline{ℚ}_{p}}((M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫}), then A′

is a finite-dimensional commutative \overline{ℚ}_{p}-algebra, and hence admits a surjection θ′ : A′ → \overline{ℚ}_{p}. Since A′ acts faithfully on (M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫}, the θ′-eigenspace of (M ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔫} is then nonzero. If we let θ : A → \overline{ℚ}_{p} denote the homomorphism obtained by composing θ′ with the natural map A → A′, then θ satisfies the conditions of the lemma. □

In the remainder of this subsection we will follow the analysis of Section 2.7 of [Ger09], which considers the ordinary situation. Let E be an algebraic extension of ℚ_{p}. Let w|p be a place of F, and let ρ : G_{F_{w}} → GL_{n}(E) be a crystalline representation. Assume that E contains the images of all embeddings F_{w} ↪ \overline{E}. Let D = D_{cris}(ρ) = (B_{cris} ⊗_{ℚ_{p}} ρ)^{G_{Fw}}. This is a weakly admissible filtered φ-module; if F_{w}^{0} is the maximal absolutely unramified subfield of F_{w}, and ϕ^{0} is the absolute Frobenius automorphism of F_{w}^{0}, then D is a finite free F_{w}^{0} ⊗_{ℚ_{p}} E-module of rank n with

∘ a ϕ^{0}-semilinear, E-linear automorphism φ, and

∘ a separated and exhaustive decreasing filtration (Fil^{i} D_{F_{w}})_{i∈ℤ} on the tensor product D_{F_{w}} := D ⊗_{F_{w}^{0}} F_{w} by F_{w} ⊗_{ℚ_{p}} E-submodules.

For each embedding σ : F_{w}^{0} ↪ E, we let D_{σ} := D ⊗_{F_{w}^{0}⊗E,σ⊗1} E, so that D = ∏_{σ:F_{w}^{0}↪E} D_{σ}. Similarly, for each embedding τ : F_{w} ↪ E, we let D_{F_{w},τ} := D_{F_{w}} ⊗_{F_{w}⊗E,τ⊗1} E, so that D_{F_{w}} = ∏_{τ:F_{w}↪E} D_{F_{w},τ}. The filtration on D_{F_{w}} is induced by filtrations on each D_{F_{w},τ}, so that we may write Fil^{i}(D_{F_{w}}) = ∏_{τ:F_{w}↪E} Fil^{i}(D_{F_{w},τ}). Then by definition HT_{τ} (ρ) is the multiset of integers in which i ∈ ℤ occurs dim_{E} gr^{i}(D_{F_{w},τ}) times.

The map φ^{[F_{w}^{0}:ℚ_{p}]} induces an isomorphism of E-vector spaces φ^{[F_{w}^{0}:ℚ_{p}]} : D_{σ} ⟶^{∼} D_{σ} for each σ. Assume now that E is a finite extension of ℚ_{p}. Let D′ be a φ-stable free F_{w}^{0} ⊗_{ℚ_{p}} E-submodule of D. We define the Hodge and Newton numbers of D′ in the usual way, by forgetting the E-structure, and thinking of D′ as a finite-dimensional F_{w}^{0}-vector space. Thus

t_{H}(D′) = ∑_{i∈ℤ}(dim_{F_{w}} gr^{i} D′_{F_{w}})i,

t_{N}(D′) = ∑_{α∈ℚ}(dim_{F_{w}^{0}} D′_{α})α,

where D′_{α} is the slope-α part of D′. Since D is weakly admissible, we have t_{N}(D′) ≥ t_{H}(D′), with equality if and only if there is a crystalline sub-E-representation ρ′ of ρ with D′ = D_{cris}(ρ′).

In the following, we let val_{p} denote the valuation of \overline{ℚ}_{p}, normalised such that val_{p}(p) = 1. We note that the next Proposition does not require the running assumption that the extension F_{w}/ℚ_{p} is unramified, so we phrase the statement and proof without this assumption.

Proposition 4.5.2. Let λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}, and assume that r̄|_{G_{Fw}} has a crystalline lift ρ : G_{F_{w}} → GL_{n}(\overline{ℚ}_{p}) such that for all τ : F_{w} → \overline{ℚ}_{p} we have HT_{τ} (ρ) = −(λ_{τ,1} +n−1, …, λ_{τ,n}), and that φ^{[F_{w}:ℚ_{p}]} acting on the \overline{ℚ}_{p}-vector space D_{cris}(ρ) has characteristic polynomial

(X^{n} + ⋯ + (−1)^{j}q_{w}^{j(j−1)/2}t_{j}X^{n−j} + ⋯ + (−1)^{n}q_{w}^{n(n−1)/2}t_{n})^{[F_{w}:ℚ_{p}]},

where q_{w} = #k_{w}.

If for some 1 ≤ j ≤ n − 1 we have

(4.5.3)  val_{p}(t_{j}) = \frac{1}{[F_{w} : F_{w}^{0}]} ∑_{τ:F_{w}↪\overline{ℚ}_{p}} ∑_{i=1}^{j} λ_{τ,n+1−i},

then ρ|_{G_{Fw}} is reducible.

Note that by the proof below (or in applications, by Corollary 4.4.3) we know that val_{p}(t_{j}) is always at least as big as the right-hand side.

Proof. By the usual Baire category argument, we may assume that ρ is valued in GL_{n}(E) for some finite extension E/ℚ_{p}. Assume that E has been chosen large enough that it contains the images of all embeddings F_{w} ↪ \overline{ℚ}_{p} and that it contains all the roots of X^{n} + ⋯ + (−1)^{j}q_{w}^{j(j−1)/2}t_{j}X^{n−j} + ⋯ + (−1)^{n}q_{w}^{n(n−1)/2}t_{n}. Assume that for some 1 ≤ j ≤ n − 1 we have

val_{p}(t_{j}) = \frac{1}{[F_{w} : F_{w}^{0}]} ∑_{τ:F_{w}↪\overline{ℚ}_{p}} ∑_{i=1}^{j} λ_{τ,n+1−i}.

Let D = D_{cris}(ρ), and let α_{1}, …, α_{n} ∈ E denote the roots of

X^{n} + ⋯ + (−1)^{j}q_{w}^{j(j−1)/2}t_{j}X^{n−j} + ⋯ + (−1)^{n}q_{w}^{n(n−1)/2}t_{n}

ordered so that val_{p}(α_{1}) ≤ ⋯ ≤ val_{p}(α_{n}). For each embedding σ : F_{w}^{0} ↪ \overline{ℚ}_{p} choose an E-subspace D′_{σ} of D_{σ} such that the eigenvalues of φ^{[F_{w}^{0}:ℚ_{p}]} on it are α_{1}, …, α_{j}. Let D′ = ∏_{σ} D′_{σ}, so that D′ is a free F_{w}^{0} ⊗_{ℚ_{p}} E-submodule of D of rank j. We can ensure that D′ is φ-stable by first picking D′_{σ_{0}} for some fixed embedding σ_{0} and then defining D′_{σ∘(ϕ^{0})^{−i}} := φ^{i}(D′_{σ_{0}}) for all i. Then we have

t_{N}(D′) = \frac{dim_{ℚ_{p}} E}{dim_{ℚ_{p}} F_{w}^{0}} ∑_{i=1}^{j} val_{p}(α_{i})

≤ \frac{dim_{ℚ_{p}} E}{dim_{ℚ_{p}} F_{w}^{0}} (\frac{j(j − 1)}{2}[F_{w}^{0} : ℚ_{p}] + val_{p}(t_{j})).

Since for each embedding τ : F_{w} ↪ E the elements of HT_{τ} (ρ) are λ_{τ,n} < λ_{τ,n−1} + 1 < ⋯ < λ_{τ,1} + n − 1, we also have that

t_{H}(D′) ≥ \frac{dim_{ℚ_{p}} E}{dim_{ℚ_{p}} F_{w}} ∑_{τ:F_{w}↪\overline{ℚ}_{p}} ∑_{i=1}^{j}(λ_{τ,n+1−i} + i − 1)

= \frac{dim_{ℚ_{p}} E}{dim_{ℚ_{p}} F_{w}^{0}} [F_{w} : F_{w}^{0}]^{−1}(\frac{j(j − 1)}{2}[F_{w} : ℚ_{p}] + [F_{w} : F_{w}^{0}] val_{p}(t_{j}))

≥ t_{N}(D′).

Since D is weakly admissible we have t_{N}(D′) ≥ t_{H}(D′), so we have t_{H}(D′) = t_{N}(D′), so that ρ is reducible. □

Corollary 4.5.4. Assume that r̄|_{G_{Fw}} is irreducible, and assume that Axioms Ã1 and Ã2 of Subsection 4.3 hold. Then Axiom A1 of Subsection 4.2 holds.

Proof. Let V = F_{a} be a weight of GL_{n}(k_{w}) such that S(V)_{𝔪} ≠ 0, and let λ ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})} be the lift of a. As S(F_{a}) ↪ S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}, Axiom Ã1 implies that S(F_{a}) is finite-dimensional. For 1 ≤ j ≤ n − 1 let us write T̃_{j} := (∏_{τ} τ(ϖ)^{− ∑_{i=1}^{j} λ_{τ,n+1−i}})T_{j}. Suppose that α_{j} ∈ \overline{𝔽}_{p} for 1 ≤ j ≤ n − 1 are eigenvalues of the T̃_{j} on (S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})_{𝔪}. By Corollary 4.4.3 it suffices

to show that α_{j} = 0 for all j. We apply Lemma 4.5.1 with M = S̃(W_{λ}), A = 𝕋[T̃_{1}, …, T̃_{n−1}], and the maximal ideal 𝔫 that is the kernel of \overline{θ} : A → 𝕋/𝔪 = \overline{𝔽}_{p} sending T̃_{j} to α_{j}. The θ-eigenspace provided by that lemma lies in (S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔪} (as ker θ ⊂ ker \overline{θ}) and θ(T̃_{j}) ∈ \overline{ℤ}_{p} lifts α_{j} for all j. If

we had α_{j} ≠ 0 for some j, then the eigenvalue of T̃_{j} would be a unit, so the eigenvalue of T_{j} would satisfy equation (4.5.3). Hence it would follow from Axioms Ã1, Ã2 and Proposition 4.5.2 that there is a lift ρ of r̄|_{G_{Fw}} such that ρ is reducible. Since r̄|_{G_{Fw}} is irreducible by assumption, this is impossible. The result follows. □

Theorem 4.5.5. Assume that r̄|_{G_{Fw}} is irreducible. If Axioms Ã1–Ã3 of Subsection 4.3 are satisfied by S̃, and if S is an \overline{𝔽}_{p}-subspace of S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} which is stable under GL_{n}(F_{w}) and 𝕋 and such that GL_{n}(F_{w}) acts smoothly on it, then S also satisfies Axioms A1–A2 of Subsection 4.2.

Proof. Corollary 4.5.4 shows that Ã1 and Ã2 together imply A1. If F_{w} = ℚ_{p}, and a = (a_{1}, …, a_{n}) is restricted, let λ ∈ ℤ_{+}^{n} denote the lift of a. From the embedding (4.4.1) we see that (S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})_{𝔪} ≠ 0, hence (S̃(W_{λ}) ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔪} ≠ 0. From Ã1 and Ã2 we deduce that the first part of A2 holds. For the second and third parts of A2 we argue similarly, letting a GL_{n}(k_{w})-stable lattice R_{0} in the cuspidal (resp. principal series) representation R such that R_{0} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} contains F_{a} as a subrepresentation play the role of W_{λ} and using axioms Ã1 and Ã3. □

5. Weight elimination

We preserve all the notation of the preceding section, and assume that Axiom A2 holds. We will however specialise to the case when n = 3 and our fixed place w|p of F is split, that is, F_{w} = ℚ_{p}. The Serre-type conjecture in [Her09] lets us describe a set W_{w}^{?}(r̄) of weights in which r̄ should be modular, and which is conjectured to coincide with the set of all “regular” weights of r̄. Our goal in this section is to prove “weight elimination”, i.e. that if r̄ is modular of some sufficiently generic weight, then this weight does in fact lie in the set W_{w}^{?}(r̄).

The construction of W_{w}^{?}(r̄) is recalled in Subsection 5.1, while elimination itself is proved in Subsection 5.2.

5.1. The definition of W_{w}^{?}(r̄). The set W_{w}^{?}(r̄) has a purely local definition depending only on r̄|_{G_{Fw}}, and we will sometimes refer to it as W_{w}^{?}(r̄|_{G_{Fw}}) in order to emphasise the local nature of the definition. In Section 6 of [Her09] we naturally associated to a tame Galois representation \overline{ρ} : G_{ℚ_{p}} → GL_{3}(\overline{𝔽}_{p}) a representation V (\overline{ρ}) of GL_{3}(𝔽_{p}) over \overline{ℚ}_{p}. (It is a Deligne–Lusztig representation up to sign and only depends on \overline{ρ}|_{I_{ℚp}}.) We also defined a certain operator ℛ on the set of irreducible \overline{𝔽}_{p}-representations of GL_{3}(𝔽_{p}). Let JH(\overline{V (\overline{ρ})}) denote the set of Jordan–Hölder constituents of the reduction mod p of a lattice inside V (\overline{ρ}). We let

W_{w}^{?}(r̄|_{G_{Fw}}) = ℛ(JH(\overline{V (r̄|_{G_{Fw}})})).

5.2. Elimination. Before we get to our main result on weight elimination we need some preliminary lemmas. We recall from §6.4 in [Her09] the shorthand notation for inertial Galois representations τ : I_{ℚ_{p}} → GL_{3}(\overline{𝔽}_{p}) that can be extended to G_{ℚ_{p}}, at least in the special case we need. Suppose that ξ ∈ S_{3} is a permutation of the set {1, 2, 3} of order 3 and that μ = (μ_{1}, μ_{2}, μ_{3}) ∈ ℤ^{3}. Then we define τ(ξ, μ) := ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where

ψ = ω_{3}^{∑_{i=0}^{2} μ_{ξ^{i}(1)}p^{i}}.

(Recall that the tame fundamental character ω_{3} was defined in Subsection 3.3.)

Lemma 5.2.1. Suppose that 0 ≤ x − y ≤ p − 3 and 0 ≤ y − z ≤ p − 3 and that \overline{ρ} : G_{F_{w}} → GL_{3}(\overline{𝔽}_{p}) is irreducible. Then F (x, y, z) ∈ W_{w}^{?}(\overline{ρ}) if and only if

\overline{ρ}|_{I_{Fw}} ≅ τ(ξ, (x + 2, y + 1, z)), for some ξ ∈ S_{3} of order 3,

or x − z > p − 2 and

\overline{ρ}|_{I_{Fw}} ≅ τ(ξ, (z + p, y + 1, x − p + 2)), for some ξ ∈ S_{3} of order 3.

Proof. This follows from Proposition 7.4 of [Her09]. □

We need a lemma to help us distinguish niveau 3 inertial Galois representations of the form τ(ξ, (a, b, c)) with ξ ∈ S_{3} of order 3.

Lemma 5.2.2. Suppose n ∈ ℤ. Then n is not divisible by p^{2} + p + 1 if and only if one of the following happens:

(i) n = x + py + p^{2}z with x > y ≥ z and x − z ≤ p.

(ii) n = p^{2}x + py + z with x ≥ y > z and x − z ≤ p.

The two cases are disjoint. In either case, (x, y, z) is uniquely determined by n.

Proof. Without loss of generality, we can assume that 0 ≤ n < p^{2} + p + 1. (Adding p^{2} + p + 1 to n corresponds to adding 1 to each of x, y, z.)

If n ≠ 0, in base p + 1 we have n − 1 = α(p + 1) + β with 0 ≤ α ≤ p − 1, 0 ≤ β ≤ p. If α+β ≤ p−1, we take (x, y, z) = (α+β +1, α, 0). If α+β ≥ p, we take (x, y, z) = (1, α + 2 − p, α + β + 1 − 2p). This establishes the “only if” direction.

Now note that 0 ≤ n < p^{2}+p+1 is equivalent to z = 0 if n is as in (i) (resp. to x = 1 if n is as in (ii)). Each of these cases happens \binom{p+1}{2} = (p^{2} + p)/2 times. This implies the remaining claims. □

Lemma 5.2.3. Suppose a > b > c, a − c ≤ p, suppose that x > y > z, x − z ≤ p, and suppose that a + b + c = x + y + z. If

τ(ξ, (a, b, c)) ≅ τ(ξ′, (x, y, z))

for some ξ, ξ′ ∈ S_{3} of order 3, then ξ = ξ′ and (a, b, c) = (x, y, z).

Proof. Note that τ(ξ, (a, b, c))^{∨} ≅ τ(ξ^{−1}, (−c, −b, −a)). Thus, by dualising if necessary, we may assume that ξ = (1 2 3). It follows that one of p^{2}x+py+z, x + py + p^{2}z has to coincide with one of

a + pb + p^{2}c,
b + pc + p^{2}a ≡ (b + 1) + pc + p^{2}(a − p),
c + pa + p^{2}b ≡ (c + p) + p(a − 1) + p^{2}b

modulo p^{3} − 1. Our conditions on (a, b, c) and (x, y, z) imply that the first of these five expressions is as in case (ii) and the others are as in case (i) of Lemma 5.2.2. As those two cases are disjoint, we see that ξ′ = (1 2 3). Lemma 5.2.2 shows that (x, y, z) has to agree with one of (a, b, c), (b+1, c, a− p), (c + p, a − 1, b) modulo (p − 1, p − 1, p − 1)ℤ. Since a + b + c = x + y + z it follows that (a, b, c) = (x, y, z). □

Lemma 5.2.4. If (x, y, z) and (a, b, c) in ℤ_{+}^{3} are restricted weights, then F (x, y, z) ≅ F (a, b, c) if and only if x − y = a − b, y − z = b − c, and x + y + z ≡ a + b + c (mod 3(p − 1)).

Proof. This follows immediately from the discussion in §4.1.2. □

We can now state and prove our main result on weight elimination. (Recall from Definition 4.2.2 that if r̄: G_{F} → GL_{3}(\overline{𝔽}_{p}) is continuous and irreducible, then we let W_{w}(r̄) denote the set of weights V for which r̄ is modular of weight V. This set depends on the axiomatic setup of Section 4.)

Theorem 5.2.5. Suppose that r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}), and let w|p be a place of F such that r̄|_{G_{Fw}} is irreducible and such that Axiom A2 is satisfied. Continue to assume that F_{w} = ℚ_{p}. Suppose that the weight F (x, y, z) is such that either

(5.2.6)  x − z < p − 3

or that

(5.2.7)  x − y < p − 5, y − z < p − 5, and x − z > p + 1.

Then F (x, y, z) ∈ W_{w}(r̄) implies F (x, y, z) ∈ W_{w}^{?}(r̄|_{G_{Fw}}).

Proof. Suppose that F (x, y, z) ∈ W_{w}(r̄).

We first suppose that (5.2.7) holds. Let a = y, b = x − (p − 1), c = z. By Lemma 2.1.3 and Frobenius reciprocity we see that F (x, y, z) is a constituent of the reduction mod p of the principal series Ind_{B_{3}(k_{w})}^{GL_{3}(k_{w})}(χ^{a} ⊗ χ^{b} ⊗ χ^{c}), where B_{3} ⊂ GL_{3} is the Borel subgroup of upper-triangular matrices and χ : k_{w}^{×} = 𝔽_{p}^{×} → \overline{ℚ}_{p}^{×} is the Teichmüller lift. (Note that the principal series is unchanged, up to semisimplification, when the characters are permuted.) By Axiom A2 there is a lift of r̄|_{G_{Fw}} to a potentially semistable representation ρ : G_{F_{w}} → GL_{3}(\overline{ℚ}_{p}) with Hodge–Tate weights −2, −1, 0 and WD(ρ)|_{I_{Fw}} ≅ ω^{a} ⊕ ω^{b} ⊕ ω^{c}. By Theorem 3.3.13, we have r̄|_{I_{Fw}} ≅ ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where either

(5.2.8)  ψ = ω_{3}^{(y+a_{0})+p(z+a_{2})+p^{2}(x−p+1+a_{1})}

or

(5.2.9)  ψ = ω_{3}^{(y+2−a_{2})+p(x−p+3−a_{1})+p^{2}(z+2−a_{0})}

with (a_{0}, a_{1}, a_{2}) ∈ {(1, 1, 1), (1, 2, 0), (2, 1, 0)}. (Here we use that x − y < p − 3, y − z < p − 3, x − z > p + 1.)

Next we use that F (x, y, z) is a constituent of the reduction of two cuspidal representations. Choose a ℚ_{p}-linear embedding of F_{w,3} into \overline{ℚ}_{p} and let χ_{3} : k_{w,3}^{×} → F_{w,3}^{×} → \overline{ℚ}_{p}^{×} denote the Teichmüller lift. By formula (7.8) in [Her09] we see that F (x, y, z) is a constituent of the reduction mod p of the cuspidal representation R_{𝕋}^{θ} in the notation of §2.4, where θ = χ_{3}^{a+pb+p^{2}c} and a = y+1, b = x − (p − 1), c = z − 1. (Note that all highest weights in that formula are restricted, due to our bounds on (x, y, z).) By Axiom A2 there is a lift of r̄|_{G_{Fw}} to a potentially semistable representation ρ : G_{F_{w}} → GL_{3}(\overline{ℚ}_{p}) with Hodge–Tate weights −2, −1, 0 and WD(ρ)|_{I_{Fw}} ≅ ω_{3}^{a+pb+p^{2}c} ⊕ ω_{3}^{b+pc+p^{2}a} ⊕ ω_{3}^{c+pa+p^{2}b}. By Theorem 3.3.13, we have r̄|_{I_{Fw}} ≅ ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where either

(5.2.10)  ψ = ω_{3}^{(y+1+a_{0})+p(z−1+a_{2})+p^{2}(x−p+1+a_{1})}

with (a_{0}, a_{1}, a_{2}) ∈ {(0, 2, 1), (1, 1, 1), (1, 2, 0)} or

(5.2.11)  ψ = ω_{3}^{(y+1+a_{0})+p(x−p+1+a_{2})+p^{2}(z−1+a_{1})}

with a_{0}, a_{1}, a_{2} ∈ [0, 2] with a_{1} +a_{2} +a_{3} = 3. (Here we use that x−y < p−2, y − z < p − 5, x − z > p.)

Similarly we claim that F (x, y, z) is a constituent of the reduction mod p of the cuspidal representation R_{𝕋}^{θ} where θ = χ_{3}^{c+pb+p^{2}a} and a = x + 1, b = z + p − 1, c = y − 1. Equivalently, F (x, y, z)^{∨} ≅ F (−z, −y, −x) is a constituent of the reduction of (R_{𝕋}^{θ})^{∨} ≅ R_{𝕋}^{θ^{−1}}. (For the last isomorphism see p. 136 in [DL76].) This is again true by formula (7.8) in [Her09]. By Axiom A2 there is a lift of r̄|_{G_{Fw}} to a potentially semistable representation ρ : G_{F_{w}} → GL_{3}(\overline{ℚ}_{p}) with Hodge–Tate weights −2, −1, 0 and WD(ρ)|_{I_{Fw}} ≅ ω_{3}^{c+pb+p^{2}a} ⊕ ω_{3}^{b+pa+p^{2}c} ⊕ ω_{3}^{a+pc+p^{2}b}. By Theorem 3.3.13, we have r̄|_{I_{Fw}} ≅ ψ ⊕ ψ^{p} ⊕ ψ^{p^{2}}, where either

(5.2.12)  ψ = ω_{3}^{(y+2−a_{0})+p(z+1−a_{2})+p^{2}(x−p+4−a_{1})}

with a_{0}, a_{1}, a_{2} ∈ [0, 2] with a_{1} + a_{2} + a_{3} = 3, or

(5.2.13)  ψ = ω_{3}^{(y+2−a_{0})+p(x−p+3−a_{2})+p^{2}(z+2−a_{1})}

with (a_{0}, a_{1}, a_{2}) ∈ {(0, 2, 1), (1, 1, 1), (1, 2, 0)}. (Here we use that x − y < p − 5, y − z < p − 2, x − z > p.)

Lemma 5.2.3 allows us to distinguish between the various inertial Galois representations. (Note that whenever Theorem 3.3.13 applies, it shows that r̄|_{I_{Fw}} ≅ τ(ξ, (a + b_{0}, b + b_{1}, c + b_{2})) for some ξ ∈ S_{3} of order 3 and some b_{i} ∈ [0, 2] with ∑ b_{i} = 3. We have a+b_{0} > b+b_{1} > c+b_{2} and (a+b_{0})−(c+b_{2}) ≤ p due to the assumptions placed on (a, b, c). Moreover, whenever we applied that theorem, we had a+b+c = x+y +z +p−1.) In particular, we see that the possibilities for r̄|_{I_{Fw}} described in (5.2.8), (5.2.10), (5.2.12) are distinct from the possibilities described in (5.2.9), (5.2.11), (5.2.13).

The above results thus show that either

(5.2.14)  r̄|_{I_{Fw}} ≅ τ((1 3 2), (y + b_{0}, x − p + 1 + b_{1}, z + b_{2}))

with (b_{0}, b_{1}, b_{2}) ∈ {(1, 2, 0), (2, 1, 0)} (use (5.2.8) and (5.2.10)), or

(5.2.15)  r̄|_{I_{Fw}} ≅ τ((1 2 3), (y + b_{0}, x − p + 1 + b_{1}, z + b_{2}))

with (b_{0}, b_{1}, b_{2}) ∈ {(1, 1, 1), (2, 1, 0)} (use (5.2.9) and (5.2.13)).

On the other hand, Lemma 5.2.1 shows that F (x, y, z) ∈ W_{w}^{?}(r̄|_{G_{Fw}}) is equivalent to r̄|_{I_{Fw}} being isomorphic to τ(ξ, (x+2, y+1, z)) or τ(ξ, (z+p, y+ 1, x − p + 2)) for some ξ ∈ S_{3} of order 3, but it is not hard to see that this is equivalent to (5.2.14) or (5.2.15). It follows that F (x, y, z) ∈ W_{w}^{?}(r̄|_{G_{Fw}}).

Now suppose that (5.2.6) holds. By Axiom A2 there is a lift of r̄|_{G_{Fw}} to a crystalline representation ρ : G_{F_{w}} → GL_{3}(\overline{ℚ}_{p}) with Hodge–Tate weights −(x + 2), −(y + 1), −z. Since r̄|_{G_{Fw}} is irreducible and (x + 2) − z ≤ p − 2, a standard calculation in Fontaine–Laffaille theory (cf. Theorem 1.2’ of [Sch08c]) shows that r̄|_{I_{Fw}} ≅ τ(ξ, (x+2, y+1, z)) for some ξ ∈ S_{3} of order 3 and thus F (x, y, z) ∈ W_{w}^{?}(r̄|_{G_{Fw}}) by Lemma 5.2.1. (Alternatively we could proceed as in the previous case, using for example the two principal series representations whose reduction contains F (x, y, z). But then we would need to assume that x − y > 2, y − z > 2, and x − z < p − 5.) □

6. Weight cycling for GL_{3}

In this section we further develop the general weight cycling formalism of Subsection 2.3 in the particular case of the group GL_{3}(ℚ_{p}), and then use it to prove our main theorem. More precisely, in Subsection 6.1 we prove Proposition 6.1.3, which makes Corollary 2.3.3 completely explicit in this context, while in Subsection 6.2, we prove Theorem 6.2.3, which is the main theorem of the paper in the axiomatic setting of Section 4; as explained in the introduction, it establishes that if r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}) is modular of some strongly generic weight, then the set W_{w}^{?}(r̄) of conjectured weights consists precisely of those generic weights for which r̄ is modular. (See Definition 6.2.2 for the definition of generic and strongly generic weights.)

6.1. Local results. We start with a basic but important lemma on parabolically induced representations for the group GL_{3}(𝔽_{p}). We introduce the following notation for standard parabolic subgroups of GL_{3}: let P_{1} :=

P_{(0,0,−1)} = ( \begin{matrix}∗ & ∗ & ∗ \\ ∗ & ∗ & ∗ \\ & & ∗ \end{matrix} ) and P_{2} := P_{(0,−1,−1)} = ( \begin{matrix}∗ & ∗ & ∗ \\ & ∗ & ∗ \\ & ∗ & ∗ \end{matrix} ).

Suppose that V is an irreducible \overline{𝔽}_{p}-representation of GL_{3}(𝔽_{p}) We say that V lies in the lower alcove if V ≅ F (x, y, z) with x − y ≥ 0, y − z ≥ 0, and x − z < p − 2. We say that V lies in the closure of the lower alcove if the same conditions hold, except that x − z ≤ p − 2. We say that V lies in the upper alcove if V ≅ F (x, y, z) with x − y < p − 1, y − z < p − 1, and x − z > p − 2. (These notions do not depend on the choice of (x, y, z) by Lemma 5.2.4.)

Lemma 6.1.1. Suppose that (a, b, c) ∈ ℤ^{3} with a − b > 0, b − c > 0, and a − c < p − 1. Then the induced representation Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b, c)) is of length three with irreducible constituents given by

F (b, c, a − p + 1), F (b + p − 1, a, c), F (a, b, c).

The induced representation Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (c, b − p + 1)) is of length six with irreducible constituents given by

F (c + p − 1, b, a − p + 1), F (c + p − 1, a, b), F (c + p − 2, a, b + 1),

F (a − 1, b, c + 1), F (b − 1, c, a − p + 2), F (a, c, b − p + 1).

These nine irreducible representations of GL_{3}(𝔽_{p}) are non-isomorphic, by Lemma 5.2.4. (It helps to note that the second, the fourth, and the ninth weight lie in the upper alcove, and the others lie in the closure of the lower alcove.)

Proof. Recall that we denote by B_{n} ⊂ GL_{n} the Borel subgroup of uppertriangular matrices and by T_{n} ⊂ B_{n} the diagonal maximal torus. We have a short exact sequence

0 → F (c, b − p + 1) → Ind_{B_{2}(𝔽_{p})}^{GL_{2}(𝔽_{p})} (F (b) ⊗ F (c)) → F (b, c) → 0.

(We get the two maps using Frobenius reciprocity together with Lemma 2.3 in [Her11a]. For dimension reasons exactness follows, since the representations at the ends are non-isomorphic.) Tensoring with F (a) and parabolically inducing to GL_{3}(𝔽_{p}) we see that the principal series representation

Π := Ind_{B_{3}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b) ⊗ F (c))

is an extension between the two induced representations in the statement of this lemma. But Π = \overline{R_{1}(a, b, c)} in the Grothendieck group, for example by Lemma 4.7 in [Her09] (note that a factor (−1)^{n−r} is missing on the left-hand side). Here, R_{ξ}(μ) for ξ ∈ S_{3} and μ ∈ X^{∗}(T_{3}) denotes a Deligne–Lusztig representation of GL_{3}(𝔽_{p}) over \overline{ℚ}_{p}. We can thus determine the irreducible constituents of Π from the formula in the proof of Proposition 7.4 in [Her09]. Decomposing dual Weyl modules using Proposition 3.18 in [Her09], we obtain the nine irreducible representations listed above. Since they are distinct, for any irreducible constituent V of Π, the principal series Π has a unique quotient with socle V. We compute the socle of Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a)⊗F (b, c)) using Lemma 2.3 in [Her11a]: if V = F (x, y, z) is in the socle, then by Frobenius reciprocity we have a GL_{1}(𝔽_{p}) × GL_{2}(𝔽_{p})- linear map F (z) ⊗ F (x, y) → F (a) ⊗ F (b, c), so by our bounds on (a, b, c) we see that V ≅ F (b, c, a − p + 1). Thus Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b, c)) is the unique quotient of Π with socle F (b, c, a − p + 1).

Let α_{2} be the simple root (0, 1, −1) ∈ X^{∗}(T_{3}), whose corresponding simple reflection is (2 3) under the natural identification of the Weyl group with S_{3}. Jantzen [Jan84] considers a natural intertwining map

\overline{T}_{α_{2}} : Ind_{B_{3}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b) ⊗ F (c)) → Ind_{B_{3}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (c) ⊗ F (b))

and gives in Satz 4.1 the following formula for its image, in the Grothendieck group of GL_{3}(𝔽_{p})-representations over \overline{𝔽}_{p}:

im(\overline{T}_{α_{2}}) = \frac{1}{2} ∑_{0≤l≤b−c}(\overline{R_{1}(a, b − l, c + l)} − \overline{R_{(2 3)}(a, b − l, c + l)}).

Note that Jantzen computes the restriction of this representation to SL_{3}(𝔽_{p}) in §5.2 of [Jan84], but it is not completely straightforward to deduce the result for GL_{3}(𝔽_{p}) due to the fact that the restriction of Π to SL_{3}(𝔽_{p}) may have repeated Jordan–Hölder factors. Instead, we redo his calculation for GL_{3}(𝔽_{p}), using his result on the reduction mod p of Deligne–Lusztig representations. From the proof of Proposition 7.4 in [Her09] we find that for all (i, j, k) ∈ ℤ^{3}, \overline{R_{1}(i, j, k)} − \overline{R_{(2 3)}(i, j, k)} equals

(W (j + p − 1, i, k) − W (j + p − 2, i, k + 1))
+(W (i, j, k) − W (i, j − 1, k + 1))
+(W (j, k, i − p + 1) − W (j − 1, k + 1, i − p + 1))
+(W (k + p − 1, i, j) − W (k + p − 2, i, j + 1)),

where W (μ) denotes the dual Weyl module of highest weight μ. Thus the sum over l above is telescoping and we obtain eight terms initially. But two of them are dual Weyl modules with non-dominant highest weights, which we can make dominant using formula (3.5) in [Her09]. After simplifying we find that im(\overline{T}_{α_{2}}) equals

W (b + p − 1, a, c) + W (a, b, c) + W (b, c, a − p + 1) − W (c + p − 2, a, b + 1).

By Proposition 3.18 in [Her09] this equals

F (b + p − 1, a, c) + F (a, b, c) + F (b, c, a − p + 1).

Finally, im(\overline{T}_{α_{2}}) is isomorphic to Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b, c)), since both are quotients of Π with socle F (b, c, a−p+1). (Note that even the codomain of \overline{T}_{α_{2}} has this socle.) This completes the proof of the first part of the lemma. The second part follows from our computation of the union of the constituents above. □

Remark 6.1.2. With some more work it is possible to compute the complete submodule structure of Ind_{B_{3}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (a) ⊗ F (b) ⊗ F (c)). This uses also Jantzen’s 3-step filtration with semisimple graded pieces of this representation [Jan84].

Now we go back to the setting of Section 2, in the special case of the algebraic group G = GL_{3} with F taken to be ℚ_{p}. (We could equally well assume that F/ℚ_{p} is totally ramified, but we will not need it.) We have K = GL_{3}(ℤ_{p}). Suppose that V is a weight of the group GL_{3}(𝔽_{p}), i.e. an irreducible \overline{𝔽}_{p}-representation of GL_{3}(𝔽_{p}). Recall that we defined Hecke operators \overline{T}_{1} = \overline{T}_{(0,0,1),ϖ} and \overline{T}_{2} = \overline{T}_{(0,1,1),ϖ} in ℋ_{G}(V) in Subsection 4.2.

Proposition 6.1.3. Suppose that V = F (x, y, z), where (x, y, z) is a restricted weight and that π is a smooth GL_{3}(ℚ_{p})-representation over \overline{𝔽}_{p}.

(i) Suppose x − y > 0, y − z > 0, and x − z < p − 1. If \overline{T}_{1} fails to be injective on (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then (V′ ⊗_{\overline{𝔽}_{p}} π)^{K} ≠ 0 where V′ is one of

F (z + p − 1, x, y), F (x, z, y − p + 1).

If \overline{T}_{2} fails to be injective on (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then (V′ ⊗_{\overline{𝔽}_{p}} π)^{K} ≠ 0 where V′ is one of

F (y, z, x − p + 1), F (y + p − 1, x, z).

(ii) Suppose x − y < p − 1, y − z < p − 1, and x − z > p − 1. If \overline{T}_{1} fails to be injective on (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then (V′ ⊗_{\overline{𝔽}_{p}} π)^{K} ≠ 0 where V′ is one of

F (x, z + p − 1, y), F (x − 1, z + p − 1, y + 1),

F (y − 1, x − p + 1, z + 1), F (z + p − 2, y, x − p + 2),

F (z + 2p − 2, x, y).

If \overline{T}_{2} fails to be injective on (V ⊗_{\overline{𝔽}_{p}} π)^{K}, then (V′ ⊗_{\overline{𝔽}_{p}} π)^{K} ≠ 0 where V′ is one of

F (y, x − p + 1, z), F (y − 1, x − p + 1, z + 1),

F (x − 1, z + p − 1, y + 1), F (z + p − 2, y, x − p + 2),

F (y, z, x − 2p + 2).

In each case, the last weight lies in the upper alcove and the other weights lie in the closure of the lower alcove.

Proof. We apply Corollary 2.3.3 with λ = (0, 0, 1), respectively λ = (0, 1, 1). In these two cases it is clear that K ∩^{λ(ϖ)}K contains ker(K → GL_{3}(𝔽_{p})), so by Lemma 2.1.4 we see that π|_{K} contains an irreducible constituent of

ker(Ind_{P_{−λ}(𝔽_{p})}^{GL_{3}(𝔽_{p})} V^{N_{−λ}(𝔽_{p})} → V).

By Lemma 2.3 in [Her11a] the induced representation equals

I_{1} := Ind_{P_{1}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (x, y) ⊗ F (z)), resp. I_{2} := Ind_{P_{2}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (x) ⊗ F (y, z)).

To compute the constituents of I_{2}, we apply Lemma 6.1.1 with a = x, b = y, c = z if x − z < p − 1 and with a = x, b = z + p − 1, c = y if x − z > p − 1. This completes the case when \overline{T}_{2} fails to be injective.

We will compute the constituents of I_{1} from those of I_{2}. The outer automorphism ζ : g ↦ ( \begin{matrix}& & 1 \\ & 1 \\ 1 \end{matrix} ) ·^{t}g^{−1} · ( \begin{matrix}& & 1 \\ & 1 \\ 1 \end{matrix} ) of GL_{3}(𝔽_{p}) maps every irreducible

\overline{𝔽}_{p}-representation F (x, y, z) to its dual F (−z, −y, −x). (Consider formal characters on the level of algebraic group representations.) Thus, when we apply ζ to I_{2} we obtain Ind_{P_{1}(𝔽_{p})}^{GL_{3}(𝔽_{p})} (F (−z, −y)⊗F (−x)) and its constituents are dual to the ones of I_{2}. It remains to relabel (−z, −y, −x) as (x, y, z) to obtain I_{1}. Concretely, we obtain the constituents of I_{1} from those of I_{2} by first dualising and then relabelling (−z, −y, −x) as (x, y, z). It is now easy to verify that we get the required constituents, so this completes the case when \overline{T}_{1} fails to be injective. □

6.2. The main theorem. We preserve all the notation of Section 4, and assume that Axioms A1 and A2 holds. As in Section 5 we assume furthermore that n = 3 and that F_{w} = ℚ_{p}. In this subsection we prove our main theorem in the axiomatic setting. However, before doing so, we have to make precise the notion of a weight being (strongly) generic. To this end we introduce the following definitions.

Definition 6.2.1. Suppose that δ ∈ ℤ_{≥0}. We say that an irreducible \overline{𝔽}_{p}- representation of GL_{3}(𝔽_{p}) is δ-generic if it isomorphic to F (x, y, z) for some (x, y, z) ∈ ℤ_{+}^{3} such that

−1 + δ < x − y < p − 1 − δ,
−1 + δ < y − z < p − 1 − δ,
|x − z − (p − 2)| > δ.

Note that by Lemma 5.2.4 this definition does not depend on the choice of (x, y, z).

Definition 6.2.2. We say that a weight V is generic (resp. strongly generic) if V is 4-generic (resp. 6-generic).

If r̄ : G_{F} → GL_{3}(\overline{𝔽}_{3}) is continuous and irreducible, then we recall that W_{w}(r̄) denotes the set of weights V for which r̄ is modular of weight V, and we let W_{gen,w}(r̄) := {V ∈ W_{w}(r̄) : V is generic}. With this, we are finally ready to state and prove our main theorem.

Theorem 6.2.3. Suppose that r̄: G_{F} → GL_{3}(\overline{𝔽}_{p}) is a continuous representation, and let w|p be a place of F such that r̄|_{G_{Fw}} is irreducible and such that Axioms A1 and A2 are satisfied. Continue to assume that F_{w} = ℚ_{p}. If r̄ is modular of some strongly generic weight, then W_{gen,w}(r̄) = W_{w}^{?}(r̄).

Proof. By Theorem 5.2.5 we know that W_{gen,w}(r̄) ⊂ W_{w}^{?}(r̄). To establish the reverse inclusion we will show that all weights in W_{w}^{?}(r̄) are generic and that W_{w}^{?}(r̄) ⊂ W_{w}(r̄).

Step 1: We show that either (a) r̄|_{I_{Fw}} or (b) r̄|_{I_{Fw}}^{∨} ⊗ \overline{ε}^{2} is isomorphic to τ((1 2 3), (a + 2, b + 1, c)) for some (a, b, c) ∈ ℤ^{3} such that a − b > 5, b − c > 4, a − c < p − 7. By assumption we can pick a 6-generic weight F (x, y, z) ∈ W_{w}(r̄). Thus F (x, y, z) ∈ W_{w}^{?}(r̄) and we get from Lemma 5.2.1 that either r̄|_{I_{Fw}} ≅ τ(ξ, (x + 2, y + 1, z)), or x − z > p − 2 and r̄|_{I_{Fw}} ≅ τ(ξ, (z + p, y + 1, x − p + 2)), where ξ ∈ S_{3} is of order 3. If x − z < p − 2 and ξ = (1 2 3) then conclusion (a) holds with (a, b, c) = (x, y, z). If x − z > p − 2 and r̄|_{I_{Fw}} is isomorphic to τ((1 2 3), (z + p, y + 1, x − p + 2)) or to τ((1 3 2), (x + 2, y + 1, z)) ≅ τ((1 2 3), (z + p + 1, y, x − p + 2)), then it is easy to check that conclusion (a) holds (for the obvious choices of (a, b, c)). For the three remaining cases, note that either r̄|_{G_{Iw}}^{∨} ⊗ \overline{ε}^{2} ≅ τ((1 3)ξ(1 3), (−z + 2, −y + 1, −x)) or x − z > p − 2 and r̄|_{G_{Iw}}^{∨} ⊗ \overline{ε}^{2} ≅ τ((1 3)ξ(1 3), (−x + p, −y + 1, −z − p + 2)). Compared to the first three cases, (x, y, z) is interchanged with (−z, −y, −x) and ξ with (1 3)ξ(1 3). We thus find that conclusion (b) holds.

From now on until the last step we suppose that conclusion (a) holds in Step 1.

Step 2: We analyse the set W_{w}^{?}(r̄|_{G_{Fw}}). Using Lemma 5.2.1 (see Lemma 7.6 in [Her09] for details) we see that it consists of the “obvious weights in the lower alcove”,

F (a, b, c), F (c + p − 2, a, b + 1), F (b, c − 1, a − p + 2),

the “obvious weights in the upper alcove”,

F (c + p − 2, b + 1, a − p + 1), F (b + p − 1, a + 1, c − 1), F (a, c, b − p + 1),

and the “shadow weights in the upper alcove”,

F (c + p − 2, b, a − p + 2), F (b + p − 1, a, c), F (a, c − 1, b − p + 2).

(In general we say that a weight F (x, y, z) that lies in the upper alcove is the shadow of the weight F (z + p − 2, y, x − p + 2) in the lower alcove. The two highest weights are related by a simple reflection in the affine Weyl group.) Note that the map θ : (a, b, c) ↦ (c + p − 2, a, b + 1) preserves τ((1 2 3), (a + 2, b + 1, c)) and the bounds a − b > 5, b − c > 4, a − c < p − 7. In particular, it induces a permutation of W_{w}^{?}(r̄|_{G_{Fw}}). It cyclically permutes each of the three kinds of weights above. (Note that it preserves these weights, though not the particular choice of highest weights used in the list above. See Lemma 5.2.4.)

We now verify that all nine weights above are 4-generic. For F (a, b, c) we have

a − b = (a − b − 5) + 5 > 3,
b − c = (b − c − 4) + 4 > 3,
a − c = p − 7 − (p − 7 − a + c) < p − 6,

and this also covers its shadow F (c + p − 2, b, a − p + 2). For F (c + p − 2, b + 1, a − p + 1) we have

(c + p − 2) − (b + 1) = p − 7 − (b − c − 4) < p − 5,
(b + 1) − (a − p + 1) = p − 5 − (a − b − 5) < p − 5,
(c + p − 2) − (a − p + 1) = (p − 7 − a + c) + p + 4 > p + 2.

We have written the inequalities in such a way that they immediately carry over for the other six weights using θ. (Note that θ permutes {a − b − 5, b − c − 4, p − 7 − a + c}.)

In the next three steps we consider an arbitrary element V ∈ W_{w}(r̄) and analyse the effect of weight cycling.

Step 3: Suppose that V is an obvious weight in the lower alcove. Using θ, we may assume without loss of generality that V ≅ F (a, b, c). By Lemma 4.2.3 we have an ℋ_{G}(V)-equivariant isomorphism

S(V)_{𝔪} ≅ (V ⊗_{\overline{𝔽}_{p}} S_{𝔪})^{GL_{3}(𝒪_{Fw})} .

Axiom A1 implies that both \overline{T}_{1} and \overline{T}_{2} in ℋ_{G}(V) act nilpotently on the finite-dimensional vector space S(V)_{𝔪}. We can thus apply weight cycling for both Hecke operators.

In the case of \overline{T}_{1} Proposition 6.1.3 shows that S(V′)_{𝔪} ≠ 0 where V′ is isomorphic to either F (c + p − 1, a, b) or F (a, c, b − p + 1). Suppose the first possibility holds. Then F (c + p − 1, a, b) ∈ W_{w}(r̄) ⊂ W_{w}^{?}(r̄|_{G_{Fw}}) by Theorem 5.2.5. Since (c+p−1)−b < p−2 it would have to equal one of the obvious weights in the lower alcove. But Lemma 5.2.4 implies that this is not the case, so F (c + p − 1, a, b) ∉ W_{w}(r̄). Thus F (a, c, b − p + 1) ∈ W_{w}(r̄).

The case of \overline{T}_{2} is very similar. We eliminate F (b, c, a − p + 1) and deduce that F (b + p − 1, a, c) ∈ W_{w}(r̄).

Step 4: Suppose that V is an obvious weight in the upper alcove. Using θ, we may assume without loss of generality that V ≅ F (a, c, b − p + 1). We apply weight cycling with \overline{T}_{2} and deduce from Proposition 6.1.3 that S(V′)_{𝔪} ≠ 0 where V′ is isomorphic to one of F (c + p − 1, a, b), F (c + p − 2, a, b+1), F (a−1, b, c+1), F (b−1, c, a−p+2), and F (c+p−1, b, a−p+1). As in the previous step we can eliminate all these possibilities except for the second. Let us give some detail for the last weight, the only one that lies in the upper alcove. We have

(c + p − 1) − b = p − 5 − (b − c − 4) < p − 5,
b − (a − p + 1) = p − 6 − (a − b − 5) < p − 5,
(c + p − 1) − (a − p + 1) = (p − 7 − a + c) + p + 5 > p + 1,

so we may indeed apply Theorem 5.2.5 to conclude that F (c + p − 1, b, a − p + 1) ∈ W_{w}^{?}(r̄|_{G_{Fw}}), a contradiction. Thus F (c + p − 2, a, b + 1) ∈ W_{w}(r̄).

Step 5: Suppose that V is a shadow weight in the upper alcove. Using θ, we may assume without loss of generality that V ≅ F (b + p − 1, a, c). We apply weight cycling with \overline{T}_{1} and deduce that S(V′)_{𝔪} ≠ 0 where V′ is isomorphic to one of F (b, c, a − p + 1), F (b − 1, c, a − p + 2), F (a − 1, b, c + 1), F (c+p−2, a, b+1), and F (c+p−1, b, a−p+1). In the previous two steps we eliminated all these possibilities except for the third. So F (c+p−2, a, b+1) ∈ W_{w}(r̄).

Step 6: We now put the previous three steps together and take into account the symmetry θ. Schematically the cycling process does the following.

\xymatrix{ & obvious/upper \ar[dr]^{_{\overline{T}_{2}}} \\ obvious/lower \ar[ur]^{_{\overline{T}_{1}}} \ar[dr]_{_{\overline{T}_{2}}} &  & obvious/lower \\  & shadow/upper \ar[ur]_{_{\overline{T}_{1}}}}

It is clear that it passes through all nine weights in W_{w}^{?}(r̄|_{G_{Fw}}). (The weights on the left and on the right are related by θ.) This completes the proof of the theorem, in case conclusion (a) in Step 1 holds.

Step 7: Finally, we deal with the case when conclusion (b) holds in Step 1. The situation is dual to the one we considered: since W_{w}^{?}(r̄|_{G_{Fw}})^{∨} = W_{w}^{?}(r̄|_{G_{Fw}}^{∨} ⊗ \overline{ε}^{2}) by Proposition 6.23 in [Her09], the weights in Step 2 become dualised. A weight V lies in the same alcove as its dual V^{∨} and if one of them is generic, so is the other. The argument in the proof of Proposition 6.1.3 shows that for i = 1, 2, the implied weights for V if \overline{T}_{i} fails to be injective are dual to the implied weights for V^{∨} if \overline{T}_{3−i} fails to be injective. (By “implied weights” we mean the weights V′ listed in Proposition 6.1.3 for a given weight V and Hecke operator \overline{T}_{i}.) □

Remark 6.2.4. In Steps 4 and 5 we only applied weight cycling with one of the two Hecke operators. In either situation it is not hard to see that, if we apply weight cycling with the other Hecke operator we can only eliminate two out of the five possibilities.

7. Global applications

In this section we show that the axioms of Section 4 are satisfied for certain spaces of automorphic forms on definite unitary groups, and we establish our main theorems for these groups. Subsection 7.1 introduces notation and terminology related to unitary groups and automorphic forms on them. Subsection 7.2 presents various results relating Galois representations, automorphic forms, and Serre weights; we signal in particular Definition 7.3.5, in which we give precise meaning to the notion of a global residual Galois representation being modular of a given Serre weight. Subsection 7.4 verifies that the axioms of Section 4 are satisfied, and the main theorems are deduced in Subsection 7.5. Finally, Subsection 7.6 uses automorphic induction of Hecke characters to produce specific examples to which our main theorems apply.

7.1. Automorphic forms on unitary groups. Throughout this subsection, and those that follow, we fix an imaginary CM field F with maximal totally real subfield F^{+}, and let c denote the non-trivial element of Gal(F/F^{+}). We assume that every place of F^{+} above p splits in F. (Note that in our main theorems, we will further assume that p splits completely in F.) We now proceed to introduce the various objects that will be required for our global theorems.

7.1.1. Unitary groups. We fix a positive integer n (which will eventually be specialised to equal 3), and let G be a reductive group over F^{+} which is an outer form of GL_{n}, which splits over F, and which has the property that for every infinite place v of F^{+}, G(F_{v}^{+}) is compact. (We could relax the assumption that G splits over F to G becoming an inner form over F that splits at all places dividing p, but we refrain from doing so in order to keep notation simple.)

It will be convenient to fix an integral model for G. We may find a nonzero N ∈ 𝒪_{F^{+}} that is prime to p such that G admits a reductive (in particular, smooth) model 𝒢 over 𝒪_{F^{+}}[1/N], which furthermore admits an isomorphism ι : 𝒢_{/𝒪_{F} [1/N]} ⟶^{∼} (GL_{n})_{/𝒪_{F} [1/N]} . (To see that such a model exists, we may argue as follows. Since G splits over F, there is an involution # of the second kind on M := M_{n}(F) such that for any F^{+}-algebra A, we have

G(A) = {g ∈ (M ⊗_{F^{+}} A)^{×}|gg^{#} = 1}.

Choose any 𝒪_{F} -order M_{1} in M, and let M_{0} := M_{1} ∩ M_{1}^{#}, so that M_{0} is a #-stable 𝒪_{F} -order in M. Then we have a model G_{0} over 𝒪_{F^{+}} for G defined by

G_{0}(A) = {g ∈ (M_{0} ⊗_{𝒪_{F^{+}}} A)^{×}|gg^{#} = 1}

for any 𝒪_{F^{+}}-algebra A. This model is connected reductive outside the finite set of places at which M_{0} is not maximal, so by localising we obtain a reductive model over some 𝒪_{F^{+}}[1/N].

In order to see that we may assume that p ∤ N, recall that we are free to modify an order at any finite set of places, so as places above p split in F/F^{+}, we can assume that M_{0} is maximal at each place dividing p. In order to obtain the isomorphism ι, it is enough to show that over some 𝒪_{F} [1/N] (with p ∤ N), M_{0} becomes isomorphic to M_{n}(𝒪_{F}). By weak approximation, we can choose g ∈ GL_{n}(F) so that at each place w|p of F, we have the equality g_{w}(M_{0} ⊗ 𝒪_{F_{w}})g_{w}^{−1} = M_{n}(𝒪_{F_{w}}). Then gM_{0}g^{−1} becomes equal to M_{n}(𝒪_{F}) after localising at some N that is prime to p.)

If v is any finite place of F^{+} which splits in F, and if w is a place of F lying above v, then ι gives rise to an isomorphism

ι_{w} : G(F_{v}^{+}) ⟶^{∼} GL_{n}(F_{w}),

which restricts to an isomorphism 𝒢(𝒪_{F_{v}^{+}}) ⟶^{∼} GL_{n}(𝒪_{F_{w}}) if v does not divide N. If w^{c} is the other prime of F lying over v, then c induces an isomorphism c : GL_{n}(F_{w}) ⟶^{∼} GL_{n}(F_{w^{c}}), and our assumption that G is an outer form of GL_{n} ensures that c ∘ ι_{w} is conjugate (by an element of GL_{n}(𝒪_{F_{w^{c}}}), if v ∤ N) to the inverse transpose of ι_{w^{c}}.

7.1.2. Modular forms and Hecke operators away from p. If W is an \overline{ℤ}_{p}- module with an action of 𝒢(𝒪_{F^{+},p}) (where 𝒪_{F^{+},p} := ∏_{v|p} 𝒪_{F_{v}^{+}}), and if U is a compact open subgroup of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}) ⊂ G(𝔸_{F^{+}}^{∞}), then we let S(U, W) denote the space of modular forms on G of level U with coefficients W, i.e. the space of functions

f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → W

such that f(gu) = u_{p}^{−1}f(g) for all u ∈ U. (Note that when W = W_{λ} with W_{λ} as in §7.1.4, these give examples of spaces of algebraic modular forms in the sense of [Gro99].)

For any U as above, we may write G(𝔸_{F^{+}}^{∞}) = ∐_{i} G(F^{+})t_{i}U for some finite set {t_{i}}. Then there is an isomorphism

S(U, W) → ⨁_{i} W^{U∩t_{i}^{−1}G(F^{+})t_{i}}

given by f ↦ (f(t_{i}))_{i}. We say that U is sufficiently small if for some finite place v of F^{+} the projection of U to G(F_{v}^{+}) contains no element of finite order other than the identity. If U is sufficiently small, then for each i as above we have U ∩ t_{i}^{−1}G(F^{+})t_{i} = {1}, so for any W as above and any \overline{ℤ}_{p}-algebra A, we have a natural isomorphism

(7.1.3)  S(U, W) ⊗_{\overline{ℤ}_{p}} A ⟶^{∼} S(U, W ⊗_{\overline{ℤ}_{p}} A).

If U is any compact open subgroup as above, then we will say that U is unramified at a place v of F^{+} which splits in F and does not divide N if U = 𝒢(𝒪_{F_{v}^{+}})U^{v} for a compact open subgroup U^{v} of G(𝔸_{F^{+}}^{∞,v}). (Recall that 𝒢(𝒪_{F_{v}^{+}}) ≅ GL_{n}(𝒪_{F_{w}}) via ι_{w}, for either of the places w of F lying over v.) We will say that U is unramified at p if it is unramified at all places v|p. We note that U is unramified at all but finitely many places of F^{+} that split in F.

Given U, we let 𝒫_{U} denote the set of finite places of F which lie above split places of F^{+} at which U is unramified and which do not divide pN, and we let 𝒫 ⊂ 𝒫_{U} denote a subset with finite complement. We then define 𝕋^{𝒫} as in Subsection 4.2. The algebra 𝕋^{𝒫} acts on S(U, W) (for any 𝒢(𝒪_{F^{+},p})- module W) via the Hecke operators

T_{w}^{(j)} := ι_{w}^{−1} [GL_{n}(𝒪_{F_{w}}) (\begin{matrix}ϖ_{w}1_{j} & 0 \\ 0 & 1_{n−j} \end{matrix}) GL_{n}(𝒪_{F_{w}})]

for w ∈ 𝒫 and ϖ_{w} a uniformiser in 𝒪_{F_{w}}. Note that T_{w^{c}}^{(j)} = T_{w}^{(n−j)}(T_{w}^{(n)})^{−1} on S(U, W).

7.1.4. Modular forms of weight λ. Let (ℤ_{+}^{n})_{0}^{Hom(F,\overline{ℚ}_{p})} be the elements λ of (ℤ_{+}^{n})^{Hom(F,\overline{ℚ}_{p})} with

λ_{σ,i} + λ_{σc,n+1−i} = 0

for all σ ∈ Hom(F, \overline{ℚ}_{p}) and 1 ≤ i ≤ n. For the remainder of this subsection

fix λ ∈ (ℤ_{+}^{n})_{0}^{Hom(F,\overline{ℚ}_{p})}. For any place w|p of F let λ_{w} denote the projection of λ to (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}. In §4.1.1 we defined a finite free \overline{ℤ}_{p}-module W_{λ_{w}} with an action of GL_{n}(𝒪_{F_{w}}). We give this an action of 𝒢(𝒪_{F_{v}^{+}}) via ι_{w}. This depends only on v := w|_{F^{+}}, as ι_{w^{c}} is conjugate to the inverse-transpose of c ∘ ι_{w} (consider the formal characters of the M_{λ_{τ}}), so we will also denote it by W_{λ_{v}}. Let W_{λ} be the finite free \overline{ℤ}_{p}-module with an action of 𝒢(𝒪_{F^{+},p}) defined by

W_{λ} := ⨂_{v|p} W_{λ_{v}}.

If A is a \overline{ℤ}_{p}-module and U is a compact open subgroup of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}), then we let

S_{λ}(U, A) := S(U, W_{λ} ⊗_{\overline{ℤ}_{p}} A).

From (7.1.3), we see that if U is sufficiently small, then there is a natural isomorphism

(7.1.5)  S_{λ}(U, \overline{ℤ}_{p}) ⊗_{\overline{ℤ}_{p}} A ⟶^{∼} S_{λ}(U, A).

We now recall the relationship between our spaces of (algebraic) modular forms and the space of (classical) automorphic forms on G. Write S_{λ}(\overline{ℚ}_{p}) for the direct limit of the spaces S_{λ}(U, \overline{ℚ}_{p}) over compact open subgroups U of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}) (with the transition maps being the inclusions S_{λ}(U, \overline{ℚ}_{p}) ⊂ S_{λ}(V, \overline{ℚ}_{p}) whenever V ⊂ U). Concretely, S_{λ}(\overline{ℚ}_{p}) is the set of functions

f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}

such that there is a compact open subgroup U of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}) with

f(gu) = u_{p}^{−1}f(g)

for all u ∈ U, g ∈ G(𝔸_{F^{+}}^{∞}). This space has a natural left action of G(𝔸_{F^{+}}^{∞}) via

(g · f)(h) := g_{p}f(hg).

Recall that we have fixed an isomorphism ı : \overline{ℚ}_{p} ⟶^{∼} ℂ. Fix a set of embeddings \{τ̃ : F → \overline{ℚ}_{p}\} such that Hom(F, \overline{ℚ}_{p}) = \{τ̃\} ⊔ \{τ̃ ∘ c\}. Let σ_{λ} denote the representation of G(F_{∞}^{+}) given by ⨂τ̃ M_{λτ̃}(ℂ), where G(F_{∞}^{+}) acts on M_{λτ̃}(ℂ) via the homomorphism G(F_{∞}^{+}) → G(ℂ) ⟶^{ι} GL_{n}(ℂ), where both maps are induced by ı ∘ τ̃ : F → ℂ. As usual, σ_{λ} does not depend on the choice of the τ̃ up to isomorphism. Let 𝒜 denote the space of automorphic forms on G(F^{+})\G(𝔸_{F^{+}}). As in the proof of Proposition 3.3.2 of [CHT08] one easily obtains the following lemma.

Lemma 7.1.6. There is an isomorphism of G(𝔸_{F^{+}}^{∞})-modules

S_{λ}(\overline{ℚ}_{p}) ⊗_{\overline{ℚ}_{p},ı} ℂ ⟶^{∼} Hom_{G(F_{∞}^{+})}(σ_{λ}^{∨}, 𝒜).

In particular, we note that S_{λ}(\overline{ℚ}_{p}) is a semisimple admissible G(𝔸_{F^{+}}^{∞})- module.

7.2. Galois representations attached to modular forms. In the following theorem we recall the results we will need regarding the existence of Galois representations attached to modular forms on the unitary group G. In the statement of the theorem, we let |.|^{(1−n)/2} : F^{×} → \overline{ℚ}_{p}^{×} denote the unique square root of |.|^{1−n} whose composite with ı : \overline{ℚ}_{p} ⟶^{∼} ℂ takes positive values. We will write rec_{w} for rec_{F_{w}}.

Theorem 7.2.1. If π is an irreducible subrepresentation of the G(𝔸_{F^{+}}^{∞})- representation S_{λ}(\overline{ℚ}_{p}), then there is a continuous semisimple representation

r_{π} : G_{F} → GL_{n}(\overline{ℚ}_{p})

such that (i) r_{π}^{c} ≅ r_{π}^{∨} ⊗ ε^{1−n}. (ii) The representation r_{π} is de Rham, and is crystalline if π has level

prime to p. If τ : F ↪ \overline{ℚ}_{p} then

HT_{τ} (r_{π}) = {λ_{τ,1} + n − 1, …, λ_{τ,n}}.

(iii) If v ∤ p is a place of F^{+} which splits as v = ww^{c} in F, then

WD(r_{π}|_{G_{Fw}})^{F−ss} ≅ rec_{w}((π_{v} ∘ ι_{w}^{−1}) ⊗ | · |^{(1−n)/2}).

(iv) If w|p is a place of F, write v = w|_{F^{+}}. Then

WD(r_{π}|_{G_{Fw}})^{ss} ≅ rec_{w}((π_{v} ∘ ι_{w}^{−1}) ⊗ | · |^{(1−n)/2})^{ss}.

Proof. This follows from Lemma 7.1.6, Theorem 6.1 and Remark 6.3 of [Whi11], and the main results of [Shi11], [CH09], [Car12] and [BLGGT12]. More specifically, under a mild hypothesis on the weight λ, a representation r_{π} satisfying (i), (ii) and (iii) is constructed in [Shi11], using the cohomology of unitary Shimura varieties. In [CH09] this construction is extended to general λ via congruences, and properties (i) and (ii) are checked, and (iii) is checked up to semisimplification. That (iii) holds without semisimplification is the main result of [Car12], and (iv) is the main result of [BLGGT12]. □

7.3. Serre weights. We now wish to define what it means for an irreducible representation r̄: G_{F} → GL_{n}(\overline{𝔽}_{p}) to be modular of some weight. The basic notion required for this is that of a Serre weight, which we now define.

Definition 7.3.1. A Serre weight is an isomorphism class of irreducible smooth \overline{𝔽}_{p}-representations of 𝒢(𝒪_{F^{+},p}).

Remark 7.3.2. We will engage in the same abuse of language with regard to Serre weights as the one that was signalled in the case of (general) weights in Remark 2.1.2.

The kernel of the natural map 𝒢(𝒪_{F^{+},p}) → ∏_{v|p} 𝒢(k_{v}), where k_{v} denotes the residue field of F_{v}^{+}, is a normal pro-p group, and so a Serre weight may equally well be regarded as an isomorphism class of irreducible representations of the product ∏_{v|p} 𝒢(k_{v}). If we fix a prime ṽ of F lying over v for each v lying over p, then the isomorphisms ιṽ induce an isomorphism

(7.3.3)  ∏_{v|p} 𝒢(k_{v}) ⟶^{∼} ∏_{v|p} GL_{n}(kṽ),

and hence an irreducible representation of this product is nothing but a tensor product of irreducible representations of the groups GL_{n}(kṽ). This relates the concept of Serre weight to the general notion of weight discussed in Subsections 2.1 and in §4.1.2. This allows us to give an “explicit” description of all Serre weights, as we now explain.

First note that if w|p, then c induces an isomorphism k_{w} ⟶^{∼} k_{w^{c}}, and hence a bijection Hom(k_{w^{c}}, \overline{𝔽}_{p}) ⟶^{∼} Hom(k_{w}, \overline{𝔽}_{p}) via σ ↦ σc. In this way

we obtain an action of c on (ℤ_{+}^{n})^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})}. We let (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} denote the subset of (ℤ_{+}^{n})^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} consisting of elements a such that for each w|p, if σ ∈ Hom(k_{w}, \overline{𝔽}_{p}) and 1 ≤ i ≤ n then

a_{σ,i} + a_{σc,n+1−i} = 0.

We say that an element a ∈ (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} is a restricted weight if, for each w|p, the projection a_{w} ∈ (ℤ_{+}^{n})^{Hom(k_{w},\overline{𝔽}_{p})} is restricted in the sense of §4.1.2; that is, if, for each w|p and each σ ∈ Hom(k_{w}, \overline{𝔽}_{p}), we have

a_{σ,i} − a_{σ,i+1} ≤ p − 1

for all 1 ≤ i ≤ n − 1.

Let a ∈ (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} be a restricted weight. For any place w|p of F we defined in §4.1.2 a finite free \overline{𝔽}_{p}-module F_{a_{w}} with an action of GL_{n}(k_{w}). We give this an action of 𝒢(k_{v}) via ι_{w}. Just as in the case of W_{λ} in §7.1.4, this depends only on v := w|_{F^{+}}, so we will also denote it by F_{a_{v}}.

We then define the Serre weight F_{a} to be the tensor product

F_{a} := ⨂_{v|p} F_{a_{v}},

thought of as an irreducible representation of ∏_{v|p} 𝒢(k_{v}), or equivalently, of 𝒢(𝒪_{F^{+},p}).

Lemma 7.3.4. Every Serre weight admits a representative of the form F_{a},

for some restricted weight a ∈ (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})}.

Proof. This follows from the isomorphism (7.3.3), together with the fact, noted in §4.1.2, that every irreducible representation of GL_{n}(k_{w}) is of the form F_{a_{w}} for some restricted weight a_{w} ∈ (ℤ_{+}^{n})^{Hom(k_{w},\overline{𝔽}_{p})}. □

By virtue of this lemma (and taking into account Remark 7.3.2), from now on we will typically denote a Serre weight by F_{a}, for some restricted

weight a ∈ (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})}.

If p is unramified in F, then we may introduce the notion of the lift of a restricted weight. Namely, if p is unramified in F, so that Hom(F, \overline{ℚ}_{p}) = ∐_{w|p} Hom(F_{w}, \overline{ℚ}_{p}) and Hom(𝒪_{F} , \overline{𝔽}_{p}) = ∐_{w|p} Hom(k_{w}, \overline{𝔽}_{p}) are in natural bi-

jection, then we say that a weight λ ∈ (ℤ_{+}^{n})_{0}^{Hom(F,\overline{ℚ}_{p})} is the lift of a restricted weight a if for each w|p, we have that λ_{w} is the lift of a_{w} in the sense of §4.1.2; i.e. if for each w|p and each τ ∈ Hom(F_{w}, \overline{ℚ}_{p}), we have λ_{τ} = a\overline{τ}.

Suppose now that r̄ : G_{F} → GL_{n}(\overline{𝔽}_{p}) is a continuous irreducible representation. If 𝒫 is a cofinite subset of the finite places of F which lie above split places of F^{+} and do not lie over p with the property that r̄ is unramified at all the places lying in 𝒫, then we may associate a maximal ideal 𝔪 of 𝕋^{𝒫} to r̄, as in Subsection 4.2. (We remark that, conversely, 𝔪 determines r̄ by the Chebotarev density theorem.)

Definition 7.3.5. If r̄: G_{F} → GL_{n}(\overline{𝔽}_{p}) is continuous and irreducible, then we say that r̄ modular of Serre weight F_{a}, if there is a compact open subset U of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}) which is unramified at p, a subset 𝒫 ⊂ 𝒫_{U} with finite complement such that r̄ is unramified at all the places lying in 𝒫, and such that, if 𝔪 is the maximal ideal in 𝕋^{𝒫} associated to r̄, then S(U, F_{a})_{𝔪} ≠ 0.

We say that r̄ is modular if it is modular of some Serre weight.

We write W (r̄) := {F_{a} | r̄ is modular of Serre weight F_{a}}, and refer to this as the set of Serre weights of r̄. (Thus r̄ is modular if and only if W (r̄) ≠ ∅.)

Remark 7.3.6. If, in the context of the preceding definition, we have that S(U, F_{a})_{𝔪} ≠ 0, then certainly S(U′, F_{a})_{𝔪} ≠ 0 for any compact open subgroup U′ of U. (Here we have assumed, without loss of generality, that 𝒫 ⊂ 𝒫_{U′}(⊂ 𝒫_{U}), so that 𝕋^{𝒫} acts on both spaces of modular forms.) Given U we can choose U′ ⊂ U that is sufficiently small and unramified at p. Thus, when working in the context of this definition, we may assume that the level U that appears is sufficiently small, and we will assume this from now on.

Remark 7.3.7. Note that if r̄: G_{F} → GL_{n}(\overline{𝔽}_{p}) is modular then r̄^{c} ≅ r̄^{∨}\overline{ε}^{n−1}. (Note that in Subsection 4.2 we defined the maximal ideal 𝔪 via the characteristic polynomials of r̄^{∨}(Frob_{v}), which is why we have \overline{ε}^{n−1} rather than \overline{ε}^{1−n} here.)

7.4. Verifying the characteristic zero axioms. We now apply the theory developed in the earlier sections of this paper to the specific situation of the unitary groups considered earlier in this section. To this end, we will define appropriate spaces S and S̃ which satisfy the axioms of Section 4. We maintain the notation established earlier in this section, and furthermore assume that p is unramified in F.

Assume that r̄ is modular of Serre weight F_{a}, in the sense of Defini-

tion 7.3.5, for some restricted weight a ∈ (ℤ_{+}^{n})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})}. Then, by definition, there is a compact open subgroup U of G(𝔸_{F^{+}}^{∞,p}) × 𝒢(𝒪_{F^{+},p}) that is unramified at p and a subset 𝒫 ⊂ 𝒫_{U} with finite complement such that S(U, F_{a})_{𝔪} ≠ 0, where 𝔪 is the maximal ideal of 𝕋^{𝒫} corresponding to r̄. Shrinking 𝒫 if necessary, by Remark 7.3.6 we may (and do) assume that U is sufficiently small.

We fix a place w|p of F, and assume that r̄|_{G_{Fw}} is irreducible. We write v = w|_{F^{+}}, and write U = U_{v} × U^{v}, where U^{v} ⊂ G(𝔸_{F^{+}}^{∞,v}) is compact open.

We now set up some further notation in preparation for defining the spaces S and S̃.

If W is any \overline{ℤ}_{p}-module with an action of 𝒢(𝒪_{F^{+},p}) such that 𝒢(𝒪_{F_{v}^{+}}) acts trivially, we define S(U^{v}, W) to be the space of functions

f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → W

such that f(gu) = u_{p}^{−1}f(g) for all g ∈ G(𝔸_{F^{+}}^{∞}) and all u ∈ U^{v}. It is an \overline{ℤ}_{p}- module with an action of G(F_{v}^{+}), given by (γ · f)(g) = f(gγ) for γ ∈ G(F_{v}^{+}) and g ∈ G(𝔸_{F^{+}}^{∞}), and a commuting action of 𝕋^{𝒫}. Let S^{sm}(U^{v}, W) be the largest smooth G(F_{v}^{+})-subrepresentation of S(U^{v}, W), i.e.

S^{sm}(U^{v}, W) = \underrightarrow{lim}_{U′_{v} ≤ U_{v} open} S(U^{v}, W)^{U′_{v}}.

If W is a finite free \overline{ℤ}_{p}-module with an action of 𝒢(𝒪_{F^{+},p}) such that 𝒢(𝒪_{F_{v}^{+}}) acts trivially, we define S^{lalg}(U^{v}, W) to be the space of functions

f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → W

such that f(gu) = u_{p}^{−1}f(g) for all g ∈ G(𝔸_{F^{+}}^{∞}) and all u ∈ U^{v} and such that for all g ∈ G(𝔸_{F^{+}}^{∞}) and all η ∈ W^{∨} the map G(F_{v}^{+}) → \overline{ℤ}_{p}, γ ↦ ⟨f(gγ), η⟩ is locally algebraic. Note that S^{lalg}(U^{v}, W) has natural, commuting actions of G(F_{v}^{+}) and 𝕋^{𝒫}.

We let S = S^{sm}(U^{v}, ⊗_{v′≠v}F_{a_{v′}}) and S̃ = S^{lalg}(U^{v}, ⊗_{v′≠v}W_{λ_{v′}}), where

λ ∈ (ℤ_{+}^{n})_{0}^{Hom(F,\overline{ℚ}_{p})} denotes the lift of a. Note that these spaces do not depend on a_{w} and λ_{w}. We identify G(F_{v}^{+}) with GL_{n}(F_{w}) via ι_{w}. Then S and S̃ have commuting actions of 𝕋 and GL_{n}(F_{w}), and the action of GL_{n}(F_{w}) on S is smooth.

Lemma 7.4.1. Recall that we are assuming that U is sufficiently small. We have a natural isomorphism

S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ⟶^{∼} S^{sm}(U^{v}, ⊗_{v′≠v}(W_{λ_{v′}} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}))

that is compatible with the 𝕋- and GL_{n}(F_{w})-actions.

Proof. Note first that G(F^{+})\G(𝔸_{F^{+}}^{∞}) is compact, as G is anisotropic at infinity. This implies in particular that the above map is well defined. The above map is surjective, since any element of the right-hand side takes only finitely many values so that we can lift it to a smooth element of S̃. (Here we use that U is sufficiently small.)

Let f ∈ S̃ be such that f ⊗ 1 is in the kernel of the map. Let e_{1}^{∨}, …, e_{d}^{∨} denote a basis of (⊗_{v′≠v}W_{λ_{v′}})^{∨}. By compactness of G(F^{+})\G(𝔸_{F^{+}}^{∞}) and by definition of S̃ we can find a compact open subgroup U′_{v} ⊂ GL_{n}(𝒪_{F_{w}}) with the following property: writing G(𝔸_{F^{+}}^{∞}) = ∐_{i=1}^{r} G(F^{+})g_{i}U′_{v}U^{v} there are algebraic functions f_{ij} : U′_{v} → \overline{ℤ}_{p} such that ⟨f(g_{i}u′_{v}), e_{j}^{∨}⟩ = f_{ij}(u′_{v}) for all i and all u′_{v} ∈ U′_{v}. Since Res_{F_{w}/ℚ_{p}} GL_{n} is defined over ℚ_{p}, there is a finite extension K of ℚ_{p} with uniformiser ϖ_{K} such that all f_{ij} take values in 𝒪_{K}. Since f ⊗ 1 is in the kernel, the f_{ij} take values in ϖ_{K}𝒪_{K}. It follows that ϖ_{K}^{−1}f ∈ S̃, so f ⊗ 1 = 0. □

By the lemma we get an embedding S ↪ S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} that is compatible with the 𝕋- and GL_{n}(F_{w})-actions.

Lemma 7.4.2. If Ṽ is a free finite-rank \overline{ℤ}_{p}-module equipped with a locally algebraic action of U_{v} = GL_{n}(𝒪_{F_{w}}) and M is a finite free \overline{ℤ}_{p}-module with an action of ∏_{v′≠v} 𝒢(𝒪_{F_{v}^{+}}), then there is a natural isomorphism of 𝕋-modules

(Ṽ ⊗_{\overline{ℤ}_{p}} S^{lalg}(U^{v}, M))^{GL_{n}(𝒪_{Fw})} ⟶^{∼} S(U, Ṽ ⊗_{\overline{ℤ}_{p}} M).

Proof. Note that we have a 𝕋- and GL_{n}(F_{w})-equivariant map η : Ṽ ⊗ S(U^{v}, M) → S(U^{v}, Ṽ ⊗M) given by x⊗f_{0} ↦ f(g) := x⊗f_{0}(g). It is an isomorphism since Ṽ is finite free as \overline{ℤ}_{p}-module. By restricting to locally algebraic functions on the left-hand side and by passing to GL_{n}(𝒪_{F_{w}})-invariants, we get a map η′ as in the statement of the lemma that is moreover injective.

Pick a basis e_{1}, …, e_{d} of Ṽ. Given any element f ∈ S(U, Ṽ ⊗_{\overline{ℤ}_{p}} M), write η^{−1}(f) = ∑ e_{i} ⊗ f_{i} for unique f_{i} ∈ S(U^{v}, M). To show that η′ is surjective, we need to prove that f_{i} ∈ S^{lalg}(U^{v}, M) for all i. But a simple computation, using the fact that f is GL_{n}(𝒪_{F_{w}})-invariant, shows that for fixed g ∈ G(𝔸_{F^{+}}^{∞}) and m^{∨} ∈ M^{∨}, the function γ ↦ ⟨f_{i}(gγ), m^{∨}⟩ on GL_{n}(𝒪_{F_{w}}) is a linear combination of matrix coefficients of Ṽ, hence locally algebraic. □

A similar but easier argument establishes the following lemma.

Lemma 7.4.3. If V is a free finite-rank \overline{𝔽}_{p}-module equipped with a smooth action of U_{v} = GL_{n}(𝒪_{F_{w}}) and M is an \overline{𝔽}_{p}-module with an action of ∏_{v′≠v} 𝒢(𝒪_{F_{v}^{+}}), then there is a natural isomorphism of 𝕋-modules

(V ⊗_{\overline{𝔽}_{p}} S^{sm}(U^{v}, M))^{GL_{n}(𝒪_{Fw})} ⟶^{∼} S(U, V ⊗_{\overline{𝔽}_{p}} M).

Proposition 7.4.4. In the present setting, Axioms Ã1–Ã3 hold.

Proof. To verify Axiom Ã1, first note that S̃(Ṽ) is finite free by Lemma 7.4.2 (as U is sufficiently small). It remains to show that

(Ṽ ⊗_{\overline{ℤ}_{p}} S̃)^{GL_{n}(𝒪_{Fw})} ⊗_{\overline{ℤ}_{p}} A ⟶^{∼} (Ṽ ⊗_{\overline{ℤ}_{p}} S̃ ⊗_{\overline{ℤ}_{p}} A)^{GL_{n}(𝒪_{Fw})}

when A = \overline{𝔽}_{p} or A = \overline{ℚ}_{p}. In the first case we can rewrite the left-hand side using Lemma 7.4.2, (7.1.3), Lemma 7.4.3, and Lemma 7.4.1 (in this order):

(Ṽ ⊗_{\overline{ℤ}_{p}} S̃)^{GL_{n}(𝒪_{Fw})} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}

≅ S(U, Ṽ ⊗_{\overline{ℤ}_{p}} ⨂_{v′≠v} W_{λ_{v′}}) ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}

≅ S(U, Ṽ ⊗_{\overline{ℤ}_{p}} ⨂_{v′≠v} W_{λ_{v′}} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})

≅ ((Ṽ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p}) ⊗_{\overline{𝔽}_{p}} S^{sm}(U^{v}, ⊗_{v′≠v}(W_{λ_{v′}} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})))^{GL_{n}(𝒪_{Fw})}

≅ (Ṽ ⊗_{\overline{ℤ}_{p}} S̃ ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p})^{GL_{n}(𝒪_{Fw})} .

In the second case note that the map is injective because \overline{ℚ}_{p} is \overline{ℤ}_{p}-flat. Any element of the right-hand side can be written as m ⊗ λ, where m ∈ Ṽ ⊗_{\overline{ℤ}_{p}} S̃

and λ ∈ \overline{ℚ}_{p}^{×}. Since Ṽ ⊗_{\overline{ℤ}_{p}} S̃ is \overline{ℤ}_{p}-torsion free it follows that m is GL_{n}(𝒪_{F_{w}})- stable, as required.

To verify Axiom Ã2, let λ_{w} ∈ (ℤ_{+}^{n})^{Hom(F_{w},\overline{ℚ}_{p})}. Our definition of S̃ only depended on λ away from w, w^{c}, so we are free to modify λ at w and w^{c} so that it agrees with the given λ_{w} at w. By Lemma 7.4.2 and Axiom Ã1 we have S̃(W_{λ_{w}} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) = S(U, W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}), so by assumption S_{λ}(U, \overline{ℚ}_{p})_{𝔪} ≠ 0. It follows from Lemma 4.5.1, as well as Theorem 7.2.1(iii) together with Corollary 3.1.2 of [CHT08] and the Chebotarev density theorem that there is an irreducible G(𝔸_{F^{+}}^{∞})-subrepresentation π of S_{λ}(\overline{ℚ}_{p}) with r̄_{π} ≅ r̄^{∨}. By Theorem 7.2.1, r_{π}|_{G_{Fw}} is crystalline with HT_{τ} (r_{π}|_{G_{Fw}}) = (λ_{τ,1} + n − 1, …, λ_{τ,n}) for all τ : F_{w} → \overline{ℚ}_{p}, so r_{π}^{∨}|_{G_{Fw}} provides the required lift. The claim about the characteristic polynomial of φ^{[F_{w}:ℚ_{p}]} follows from Theorem 7.2.1(iv) together with Corollary 3.1.2 of [CHT08].

It remains to verify Axiom Ã3. Suppose that R is a representation of GL_{n}(k_{w}) as in Axiom Ã3. As our definition of S̃ only depended on λ away from w, w^{c}, we are free to modify λ at w and w^{c} so that λ_{w} = 0 at w. By choosing a GL_{n}(k_{w})-stable lattice R_{0} in R and applying Axiom Ã1 and Lemma 7.4.2 to the smooth representation Ṽ = R_{0} we have S̃(R) ≅ S(U, R ⊗_{\overline{ℚ}_{p}} (W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})). Therefore

S̃(R)_{𝔪} ≅ S(U, R ⊗_{\overline{ℚ}_{p}} (W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}))_{𝔪}
≅ S(U′, R ⊗_{\overline{ℚ}_{p}} (W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}))_{𝔪}^{U/U′}
≅ Hom_{U_{v}} (R^{∨}, S(U′, W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔪}),

where U′ is the kernel of the map U → U_{v} ⟶^{ι_{w}} GL_{n}(𝒪_{F_{w}}) → GL_{n}(k_{w}). (In the second line, note that u ∈ U acts on a function f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → R ⊗_{\overline{ℚ}_{p}} (W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p}) by (u · f)(g) = u_{p}f(gu).) As in the preceding paragraph

(the verification of Axiom Ã2) there is thus an irreducible subrepresentation π of S_{λ}(\overline{ℚ}_{p}) such that π^{U′} ∩ S(U′, W_{λ} ⊗_{\overline{ℤ}_{p}} \overline{ℚ}_{p})_{𝔪} ≠ 0 and π|_{𝒢(𝒪_{F^{+}_{v}})} contains

R^{∨}. In particular, r̄_{π} ≅ r̄^{∨}. Since λ_{w} = 0, it follows from Theorem 7.2.1 and Proposition 2.4.1 that r_{π}^{∨}|_{G_{Fw}} provides the required lift. □

7.5. The main theorem for definite unitary groups. From now on we assume that n = 3 and that p splits in F.

The set W^{?}(r̄) of predicted Serre weights for r̄ is defined in terms of the set of predicted weights W^{?}(r̄|_{G_{Fw}}) for the representations r̄|_{G_{Fw}} for w|p a place of F (see Subsection 5.1). Namely, if v is a place of F^{+} lying over p, if w is a place of F lying over v, if V_{v} is an irreducible representation of 𝒢(k_{v}) over \overline{𝔽}_{p}, and if w is a place of F lying over v, then we introduce the following condition:

(7.5.1)  V_{v} ∘ ι_{w}^{−1} ∈ W_{w}^{?}(r̄|_{G_{Fw}}).

Definition 7.5.2. We define the set of predicted Serre weights for r̄ to be

W^{?}(r̄) := {V = ⊗_{v|p}V_{v} : V_{v} satisfies condition (7.5.1) for all w|v|p}.

Remark 7.5.3. In Definition 7.5.2, it suffices to fix a place ṽ over each place v|p of F^{+}, and then to impose the condition (7.5.1) just at these chosen places ṽ. This follows from the facts that r̄^{c} ≅ r̄^{∨} ⊗ \overline{ϵ}^{2}, that W_{w}^{?}(r̄|_{G_{Fw}})^{∨} = W_{w}^{?}(r̄|_{G_{Fw}}^{∨} ⊗ \overline{ε}^{2}) (by Proposition 6.23 in [Her09]) and that ι_{w^{c}} is conjugate to the inverse transpose of c ∘ ι_{w}. Indeed, taken together, these imply that imposing condition (7.5.1) at ṽ is equivalent to imposing it at v^{c}.

Definition 7.5.4. We say that a Serre weight V = ⊗_{v|p}V_{v} is generic (resp. strongly generic) if V_{v} ∘ι_{w}^{−1} is a generic (resp. strongly generic) weight in the sense of Definition 6.2.2 for all places w|p of F. We let W_{gen}(r̄) denote the set of generic Serre weights in W (r̄).

Theorem 7.5.5. Let F be an imaginary CM field in which the prime p splits completely, and suppose that r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}) is a continuous representation. For all places w|p of F suppose that r̄|_{G_{Fw}} is irreducible. Then W_{gen}(r̄) ⊂ W^{?}(r̄).

Proof. Fix a Serre weight V = ⊗_{v|p}V_{v} ∈ W_{gen}(r̄). Since the definition of W^{?}(r̄) is local, we need only prove that if w|p is a place of F with w|_{F^{+}} = v, then V_{v} ∘ ι_{w}^{−1} ∈ W_{w}^{?}(r̄|_{G_{Fw}}). We set up the axiomatic framework as in Subsection 7.4 using the place w and the Serre weight V. Then we see by Lemma 7.4.3 that V_{v} ∘ ι_{w}^{−1} ∈ W_{w}(r̄). We can now conclude by Proposition 7.4.4 and Theorems 4.5.5, 5.2.5. □

Theorem 7.5.6. Let F be an imaginary CM field in which the prime p splits completely, and suppose that r̄ : G_{F} → GL_{3}(\overline{𝔽}_{p}) is a continuous representation. For all places w|p of F suppose that r̄|_{G_{Fw}} is irreducible. If r̄ is modular of some strongly generic Serre weight, then W_{gen}(r̄) = W^{?}(r̄).

Proof. By Theorem 7.5.5 we have W_{gen}(r̄) ⊂ W^{?}(r̄). By assumption there is some V = ⊗_{v|p}V_{v} ∈ W_{gen}(r̄) such that r̄ is modular of Serre weight V. Fix some place w|p of F. Arguing with each such w in turn, one over each place v|p of F^{+}, and iterating, we see that it suffices to show that if V′_{v}∘ι_{w}^{−1} ∈ W_{w}^{?}(r̄|_{G_{Fw}}) then (⊗_{v′≠v}V_{v′})⊗V′_{v} ∈ W (r̄). Setting up the axiomatic framework as in Subsection 7.4 using the place w and the Serre weight V, we see that this follows from Proposition 7.4.4 and Theorems 4.5.5, 6.2.3. □

7.6. Automorphic induction. In this brief section we show that one can construct many examples in which our main theorem (Theorem 6.2.3) applies. Firstly, choose a CM field F with maximal totally real subfield F^{+} such that F/F^{+} is unramified at all finite places and such that p splits completely in F. (For example, choose any CM field F in which p splits completely, and use Lemma 4.1.2 of [CHT08] to find a Galois totally real field K^{+} in which p splits completely such that the extension F K^{+}/F^{+}K^{+} is everywhere unramified. The condition that F/F^{+} is unramified at all finite places is used below in order to apply Theorem 5.4 of [Lab11].) Fix a

weight λ ∈ (ℤ_{+}^{3})_{0}^{Hom(F,\overline{ℚ}_{p})}.

By Theorem 6 of chapter 10 of [AT09] (and its proof, which shows that we may assume that p is unramified in the extension), we may choose a cyclic totally real extension L^{+}/F^{+} of degree 3 in which every place of F^{+} lying over p is inert. Let L = F L^{+}. By Lemma 4.1.5 of [CHT08], we may choose a crystalline character ψ : G_{L} → \overline{ℚ}_{p}^{×} such that

∘ ψψ^{c} = ε^{−2},

∘ if w|p is a place of F and τ : F_{w} ↪ \overline{ℚ}_{p}, then
⋃ HTτ̃(ψ|_{G_{Lw}}) = {λ_{τ,1} + 2, λ_{τ,2} + 1, λ_{τ,3}},

where τ̃ runs through all embeddings L_{w} ↪ \overline{ℚ}_{p} such that τ̃|_{F_{w}} = τ.

Assume furthermore that for each τ we have λ_{τ,1} − λ_{τ,3} ≤ p − 2. Set r_{ψ} := Ind_{G_{L}}^{G_{F}} ψ. Then for each place w|p of F the representation r̄_{ψ}^{∨}|_{G_{Fw}} is irreducible (use the bounds on the λ_{τ,i} and the assumption that each place of F^{+} lying over p is inert in L^{+}), so in particular the global representation r̄_{ψ}^{∨} is irreducible. Then by Theorems 4.2 and 5.1 of [AC89], together with Theorem 5.4 of [Lab11] and Lemma 7.1.6 above, it follows

that r̄_{ψ}^{∨} is modular of Serre weight F_{a}, where a ∈ (ℤ_{+}^{3})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} is the unique restricted weight of which λ is the lift. (Note that W_{a} ⊗_{\overline{ℤ}_{p}} \overline{𝔽}_{p} ≅ F_{a} due to our bounds on the λ_{τ,i}, for example by Proposition 3.18 in [Her09]. Note also that there is a unitary group satisfying our assumptions which is furthermore quasisplit at all finite places, cf. Section 2.1 of [Ger09].)

Thus if we choose any restricted weight a ∈ (ℤ_{+}^{3})_{0}^{∐_{w|p} Hom(k_{w},\overline{𝔽}_{p})} such that F_{a_{w}} is strongly generic and in the lower alcove for each w|p, the representation r̄_{ψ}^{∨} constructed above (with λ the lift of a) satisfies the assumptions of Theorem 6.2.3. (Note that the condition that λ_{τ,1} −λ_{τ,3} ≤ p−2 is automatic from the assumption that all F_{a_{w}} are strongly generic.)

References

[AC89] James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989.

[ADP02] Avner Ash, Darrin Doud, and David Pollack, Galois representations with conjectural connections to arithmetic cohomology, Duke Math. J. 112 (2002), no. 3, 521–579.

[AS00] Avner Ash and Warren Sinnott, An analogue of Serre’s conjecture for Galois representations and Hecke eigenclasses in the mod p cohomology of GL(n, Z), Duke Math. J. 105 (2000), no. 1, 1–24.

[AT09] Emil Artin and John Tate, Class field theory, AMS Chelsea Publishing, Providence, RI, 2009, Reprinted with corrections from the 1967 original.

[BDJ10] Kevin Buzzard, Fred Diamond, and Frazer Jarvis, On Serre’s conjecture for mod ℓ Galois representations over totally real fields, Duke Math. J. 155 (2010), no. 1, 105–161.

[BH10] Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence, III: the general case, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 497–553.

[BLGG12] Thomas Barnet-Lamb, Toby Gee, and David Geraghty, Serre weights for U(n), in preparation, 2012.

[BLGGT12] Thomas Barnet-Lamb, Toby Gee, David Geraghty, and Richard Taylor, Local-global compatibility for l = p, II, to appear in Ann. Sci. École Norm. Sup., 2012.

[BM02] Christophe Breuil and Ariane Mézard, Multiplicités modulaires et représentations de GL_{2}(Z_{p}) et de Gal(\overline{Q}_{p}/Q_{p}) en l = p, Duke Math. J. 115 (2002), no. 2, 205–310, With an appendix by Guy Henniart.

[Bre97] Christophe Breuil, Représentations p-adiques semi-stables et transversalité de Griffiths, Math. Ann. 307 (1997), no. 2, 191–224.

[Bre98] , Construction de représentations p-adiques semi-stables, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 3, 281–327.

[Bre99] , Représentations semi-stables et modules fortement divisibles, Invent. Math. 136 (1999), no. 1, 89–122.

[Bre02] , Integral p-adic Hodge theory, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 51–80.

[Car05] Xavier Caruso, Conjecture de l’inertie modérée de Serre, Ph.D. thesis, Univ.

Paris 13, available at http://tel.archives-ouvertes.fr/tel-00011202,

2005.

[Car06] , Représentations semi-stables de torsion dans le cas er < p − 1, J. Reine Angew. Math. 594 (2006), 35–92.

[Car11] , 𝔽_{p}-représentations semi-stables, to appear in Ann. Inst. Fourier, 2011.

[Car12] Ana Caraiani, Local-global compatibility and the action of monodromy on nearby cycles, to appear in Duke Math. J., 2012.

[CDT99] Brian Conrad, Fred Diamond, and Richard Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, J. Amer. Math. Soc. 12 (1999), no. 2, 521–567.

[CH09] Gaëtan Chenevier and Michael Harris, Construction of automorphic Galois representations, II, preprint, 2009.

[CHT08] Laurent Clozel, Michael Harris, and Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations, Publ. Math. Inst. Hautes Études Sci. (2008), no. 108, 1–181, With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras.

[CL09] Xavier Caruso and Tong Liu, Quasi-semi-stable representations, Bull. Soc. Math. France 137 (2009), no. 2, 185–223.

[DL76] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161.

[Dou07] Darrin Doud, Supersingular Galois representations and a generalization of a conjecture of Serre, Experiment. Math. 16 (2007), no. 1, 119–128.

[DS74] Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975).

[Edi92] Bas Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), no. 3, 563–594.

[Gee11a] Toby Gee, Automorphic lifts of prescribed types, Math. Ann. 350 (2011), no. 1, 107–144.

[Gee11b] , On the weights of mod p Hilbert modular forms, Invent. Math. 184 (2011), no. 1, 1–46.

[Ger09] David Geraghty, Modularity lifting theorems for ordinary Galois representations, preprint, 2009.

[GG12] Toby Gee and David Geraghty, Companion forms for unitary and symplectic groups, Duke Math. J. 161 (2012), no. 2, 247–303.

[Gro99] Benedict H. Gross, Algebraic modular forms, Israel J. Math. 113 (1999), 61–93.

[GS11a] Toby Gee and David Savitt, Serre weights for mod p Hilbert modular forms: the totally ramified case, J. Reine Angew. Math. 660 (2011), 1–26.

[GS11b] , Serre weights for quaternion algebras, Compos. Math. 147 (2011), no. 4, 1059–1086.

[Her09] Florian Herzig, The weight in a Serre-type conjecture for tame n-dimensional Galois representations, Duke Math. J. 149 (2009), no. 1, 37–116.

[Her11a] , The classification of irreducible admissible mod p representations of a p-adic GL_{n}, Invent. Math. 186 (2011), no. 2, 373–434.

[Her11b] , A Satake isomorphism in characteristic p, Compos. Math. 147 (2011), no. 1, 263–283.

[Hid81] Haruzo Hida, On congruence divisors of cusp forms as factors of the special values of their zeta functions, Invent. Math. 64 (1981), no. 2, 221–262.

[HT01] Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich.

[Jan84] Jens Carsten Jantzen, Filtrierungen der Darstellungen in der Hauptserie endlicher Chevalley-Gruppen, Proc. London Math. Soc. (3) 49 (1984), no. 3, 445–482.

[Jan03] , Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003.

[Kud94] Stephen S. Kudla, The local Langlands correspondence: the non-Archimedean case, Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 365–391.

[Lab11] J.-P. Labesse, Changement de base CM et séries discrètes, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, Int. Press, Somerville, MA, 2011, pp. 429–470.

[Liu08] Tong Liu, On lattices in semi-stable representations: a proof of a conjecture of Breuil, Compos. Math. 144 (2008), no. 1, 61–88.

[Rib83] Kenneth A. Ribet, Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1.

[Roc98] Alan Roche, Types and Hecke algebras for principal series representations of split reductive p-adic groups, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 3, 361–413.

[Sav05] David Savitt, On a conjecture of Conrad, Diamond, and Taylor, Duke Math. J. 128 (2005), no. 1, 141–197.

[Sav08] , Breuil modules for Raynaud schemes, J. Number Theory 128 (2008), no. 11, 2939–2950.

[Sch08a] Michael M. Schein, On modular weights of Galois representations, Math. Res. Lett. 15 (2008), no. 3, 537–542.

[Sch08b] , Weights in Serre’s conjecture for Hilbert modular forms: the ramified case, Israel J. Math. 166 (2008), 369–391.

[Sch08c] , Weights in Serre’s conjecture for GL_{n} via the Bernstein-Gelfand-Gelfand complex, J. Number Theory 128 (2008), no. 10, 2808–2822.

[Sch08d] , Weights of Galois representations associated to Hilbert modular forms, J. Reine Angew. Math. 622 (2008), 57–94.

[Shi11] Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741.

[Spr98] T. A. Springer, Linear algebraic groups, second ed., Progress in Mathematics, vol. 9, Birkhäuser Boston Inc., Boston, MA, 1998.

[Tay06] Richard Taylor, On the meromorphic continuation of degree two L-functions, Doc. Math. (2006), no. Extra Vol., 729–779 (electronic).

[Whi11] Paul-James White, Tempered automorphic representations of the unitary group, preprint, 2011.

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

E-mail address: emerton@math.northwestern.edu

Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA

E-mail address: gee@math.northwestern.edu

Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA

E-mail address: herzig@math.ias.edu