A GEOMETRIC PERSPECTIVE ON THE BREUIL–MÉZARD CONJECTURE

MATTHEW EMERTON AND TOBY GEE

Abstract. Let p > 2 be prime. We state and prove (under mild hypotheses on the residual representation) a geometric refinement of the Breuil–Mézard conjecture for 2-dimensional mod p representations of the absolute Galois group of ℚ_{p}. We also state a conjectural generalisation to n-dimensional representations of the absolute Galois group of an arbitrary finite extension of ℚ_{p}, and give a conditional proof of this conjecture, subject to a certain R = 𝕋-type theorem together with a strong version of the weight part of Serre’s conjecture for rank n unitary groups. We deduce an unconditional result in the case of two-dimensional potentially Barsotti–Tate representations.

1. Introduction

Our aim in this paper is to revisit the Breuil–Mézard conjecture [BM02] from a geometric point of view. Let us explain what we mean by this. First recall that the Breuil–Mézard conjecture posits a formula (in terms of certain representationtheoretic data) for the Hilbert–Samuel multiplicity of the characteristic p fibre of certain local ℤ_{p}-algebras, namely those whose characteristic zero fibres parameterize two-dimensional potentially semistable liftings of some fixed continuous twodimensional Galois representation r̄ : G_{ℚ_{p}} → GL_{2}(𝔽), where 𝔽 is a finite field of characteristic p (the so-called potentially semistable deformation rings constructed in [Kis08]). One way in which a local ring can have multiplicity is if its Spec has more than one component: if its Spec is the union of n irreducible components, each with multiplicities μ_{i} (i = 1, …, n), then the multiplicity of the entire ring will be ∑_{i} μ_{i}. Our goal is to both explain and refine the Breuil–Mézard conjecture in these terms, by identifying the irreducible components of the various rings involved, in representation-theoretic terms, as well as to determine their multiplicities.

To be somewhat more precise, after recalling some background material in Section 2, in Section 3 we consider the case of two-dimensional representations of G_{ℚ_{p}}, as introduced above. In this case the Breuil–Mézard conjecture is a theorem of Kisin [Kis09a] (under very mild assumptions on r̄), and we are able to strengthen Kisin’s result so as to prove our geometric refinement of the conjecture. (We give a more detailed description of our results in this case in Subsection 1.1 below.) The possibility of such an extension is strongly suggested by the recent paper [BM12], and our results may be viewed as a sharpening of the results of ibid. (see Remark 1.1.9 below). In Section 4 we propose an extension of the Breuil–Mézard conjecture, and of our geometric refinement thereof, to the case of n-dimensional representations of G_{K}, for any finite extension K of ℚ_{p} and any positive integer n. Finally, in Section 5, we explain how the arguments of Section 3 may be extended to the case of n-dimensional representations, so as to prove an equivalence between the Breuil–Mézard conjecture (extended to the n-dimensional case) and its geometric refinement, under the assumption of a suitable R = 𝕋-type theorem, together with a strong form of the weight part of Serre’s conjecture for rank n unitary groups. In the case of two-dimensional potentially Barsotti–Tate representations, we deduce an unconditional geometric refinement of the results of [GK12].

The first author was supported in part by NSF grant DMS-1003339, and the second author by NSF grant DMS-0841491.

In an appendix we establish a technical result that allows us to realize representations of local Galois groups as restrictions of automorphic representations of global Galois groups.

1.1. Summary of our results in the case of two-dimensional representations of G_{ℚ_{p}}. We now explain in more detail our geometric refinement of the original Breuil–Mézard conjecture. To this end, we fix a finite extension E of ℚ_{p}, with ring of integers O, residue field 𝔽, and uniformizer π. As above, we also fix a continuous representation r̄: G_{ℚ_{p}} → GL_{2}(𝔽), and we let R^{□}(r̄) denote the universal lifting ring of r̄ over O. If m, n are integers with n ≥ 0 and τ is an inertial type defined over E, then we may consider the subset of Spec R^{□}(r̄)[1/p] consisting of those closed points that correspond to lifts of r̄ to characteristic zero which are potentially semistable with Hodge–Tate weights (m, m + n + 1) and inertial type τ. (We adopt the convention that the cyclotomic character has Hodge–Tate weight 1, though we caution the reader that this convention does not remain in force for the entire paper; see Section 1.3 for the precise conventions we will follow.) In [Kis08], Kisin proves that there is a reduced closed subscheme Spec R^{□}(m, n, τ, r̄) of Spec R^{□}(r̄) such that this subset is precisely the set of closed points of Spec R^{□}(m, n, τ, r̄)[1/p].^{1} The Breuil–Mézard conjecture addresses the problem of describing the characteristic p fibre of Spec R^{□}(m, n, τ, r̄), i.e. the closed subscheme Spec R^{□}(m, n, τ, r̄)/π of Spec R^{□}(r̄)/π. More precisely, the conjecture as originally stated in [BM02] gives a conjectural formula for the Hilbert–Samuel multiplicity of this local scheme. This conjecture was proved (under very mild assumptions on r̄) in [Kis09a]. In this paper we will prove a more precise statement, namely we will identify the underlying cycle of Spec R^{□}(m, n, τ, r̄)/π; that is, we will describe the irreducible components of this scheme, and the multiplicity with which each component appears. To explain this more carefully, suppose first that X is any Noetherian scheme. If Z is a closed subscheme of X, and 𝔭 is any point of X, then we may define the (Hilbert–Samuel) multiplicity e(Z, 𝔭) of Z at 𝔭 to be the Hilbert–Samuel multiplicity of the stalk O_{Z,𝔭}. Suppose now that Z is equidimensional of dimension d. If 𝔞 is a point of X of dimension d (i.e. whose closure \overline{\{𝔞\}} is of dimension d), then the stalk O_{Z,𝔞} is either zero (if 𝔞 ∉ Z) or an Artinian local ring (if 𝔞 ∈ Z, i.e. if 𝔞 is a generic point of Z, or, equivalently, if \overline{\{𝔞\}} is an irreducible component of Z), and the multiplicity e(Z, 𝔞) is simply the length of O_{Z,𝔞} as a module over itself, a quantity which can be interpreted geometrically as the multiplicity with which the component \overline{\{𝔞\}} appears in Z. Since Z contains only finitely many generic points, the formal sum Z(Z) := ∑_{𝔞} e(Z, 𝔞)𝔞 is well-defined as a d-dimensional cycle on X, and we refer to it as the cycle associated to Z. If 𝔭 is any point of X, then one has the formula

1In fact, in [Kis08] Kisin also fixes the determinants of the lifts that he considers, but we will suppress this technical point for now.

(1.1.1)  e(Z, 𝔭) = ∑_{𝔞} e(Z, 𝔞)e((\overline{\{𝔞\}}), 𝔭)

(where again the sum ranges over points 𝔞 of dimension d), allowing us to compute the multiplicity of Z at any point in terms of its associated cycle. We are interested in the case when X := Spec R^{□}(r̄)/π (an 8-dimensional Noetherian local scheme) and Z := Spec R^{□}(m, n, τ, r̄)/π for some m, n, τ. It is a theorem of [Kis08] that each of these closed subschemes Z is equidimensional of dimension 5, and so we may define the associated cycles Z(Spec R^{□}(m, n, τ, r̄)/π). Using this construction, we may in particular define a certain cycle on Spec R^{□}(r̄)/π attached to each Serre weight of r̄.

1.1.2. Definition. If σ is a Serre weight of r̄, write σ = σ_{m,n} := det^{m} ⊗Sym^{n}𝔽^{2} for integers m, n such that 0 ≤ n ≤ p−1, and define C_{m,n} := Z(Spec R^{□}(m, n, 𝟙, r̄)/π). (To avoid ambiguity, one could insist that m is chosen so that 0 ≤ m ≤ p − 2. However, the subscheme Spec R^{□}(m, n, 𝟙, r̄)/π is in fact independent of the particular choice of m used to describe σ.)

The following proposition describes these cycles quite explicitly. We will say that C_{m,n} consists of a single component if it has a single irreducible component, which is also reduced, and that it consists of two components if it consists of two reduced and irreducible components.

1.1.3. Proposition. Assume that the Breuil–Mézard Conjecture (i.e. Conjecture 3.1.1 below) holds for r̄.

(1) If r̄ is irreducible and σ_{m,n} is a Serre weight of r̄, then C_{m,n} consists of a single component, which has multiplicity one at the closed point of Spec R^{□}(r̄)/π.

(2) If r̄ is reducible and σ_{m,n} is a Serre weight of r̄ such that n < p − 2, or n = p − 2 and r̄ is a non-split extension of distinct characters, then C_{m,n} consists of a single component, which has multiplicity one at the closed point of Spec R^{□}(r̄)/π.

(3) If r̄ is reducible and σ_{m,n} is a Serre weight of r̄ with n = p − 1, so that

r̄|_{I_{p}} ∼ (\begin{matrix}ω^{m+1} & ∗ \\ 0 & ω^{m} \end{matrix}), then if ∗ is peu ramifiée and r̄ itself is a twist of

an extension of the trivial character by the mod p cyclotomic character, then C_{m,n} is a sum of two components, each having multiplicity one at the closed point of Spec R^{□}(r̄)/π. Otherwise C_{m,n} is a single component, having multiplicity one at the closed point of Spec R^{□}(r̄)/π.

(4) If σ_{m,n} is a Serre weight of r̄ with n = p − 2 and r̄ is split and pdistinguished, then C_{m,n} is a sum of two components, each having multiplicity one at the closed point of Spec R^{□}(r̄)/π.

(5) If σ_{m,n} is a Serre weight of r̄ with n = p − 2 and r̄ has scalar semisimplifiction, then C_{m,n} consists of a single component.

(6) If σ_{m,n} and σ_{m′,n′} are distinct Serre weights of r̄, then C_{m,n} and C_{m′,n′} have disjoint support, except if m ≡ m′ (mod p − 1), n = 0 and n′ = p − 1 (possibly after interchanging σ_{m,n} and σ_{m′,n′}), in which case C_{m,n} is equal to a component of C_{m′,n′}.

We remark that by the results of [Kis09a], the hypothesis of the preceeding proposition holds for most r̄.

We make one more definition before stating our main theorem.

1.1.4. Definition. Given integers a, b with b ≥ 0 and an inertial type τ (assumed to be defined over E), let σ(τ) denote the representation of GL_{2}(ℤ_{p}) over E associated to τ via Henniart’s inertial local Langlands correspondence, write σ(a, b, τ) := (det^{a} ⊗Sym^{b}E^{2}) ⊗_{E} σ(τ), and let \overline{σ(a, b, τ)}^{ss} denote the semi-simplification of the reduction mod π of (any) GL_{2}(ℤ_{p})-invariant O-lattice in σ(a, b, τ). (The representation so obtained is well-defined independent of the choice of invariant lattice.)

We may now state our geometric refinement of the Breuil–Mézard conjecture.

1.1.5. Theorem. Suppose that

r̄ ≁ (\begin{matrix}ωχ & ∗ \\ 0 & χ \end{matrix})

for any character χ. Fix integers m, n with n ≥ 0 and an inertial type τ, and for each Serre weight σ_{m,n} of r̄, let a_{m,n} denote the multiplicity with which σ_{m,n} appears as a constituent of \overline{σ(a, b, τ)}^{ss}. Then we have the following equality of cycles:

Z(Spec R^{□}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}C_{m,n}.

1.1.6. Remark. The usual form of the Breuil–Mézard conjecture, as stated in [BM02] and proved in [Kis09a], can be recovered from this result by applying the formula (1.1.1), and using the explicit description of the cycles C_{m,n} provided by Proposition 1.1.3. (Note that while Proposition 1.1.3 as stated assumes that the Breuil–Mézard conjecture holds for r̄, our proof of Proposition 1.1.3 will actually use Theorem 1.1.5 for r̄ as input, and so this argument is not circular.)

1.1.7. Remark. The hypothesis on r̄ in Theorem 1.1.5 is slightly weaker than that made in the analogous result in [Kis09a]. This is due to our use of potential modularity theorems to realise local representations globally, which is more flexible than the construction of [Kis09a] using CM forms. We remark that Paškūnas ([Paš12]) has reproved Kisin’s results (and our generalisation of them) by purely local means under a similarly weakened hypothesis on r̄.

1.1.8. Remark. Theorem 1.1.5 is proved via a refinement of the global argument made in [Kis09a], and uses the local arguments of [Kis09a] (using the p-adic Langlands correspondence) as an input. In particular, it does not give a new proof of the usual form of the Breuil–Mézard conjecture.

1.1.9. Remark. In the recent sequel [BM12] to their paper [BM02], Breuil and Mézard have constructed, for generic r̄, a correspondence between the irreducible components of Spec R^{□}(a, b, τ, r̄)/π and the Serre weights. We show (in Subsection 3.4) that this coincides with the correspondence σ_{m,n} ↦ C_{m,n}. (Note that when r̄ is generic in the sense of [BM12], each cycle C_{m,n} consists of a single component.) Thus our results may be reviewed as a refinement of those of [BM12]. We also note that in [BM12], the authors proceed by refining the local arguments of [Kis09a], while (as already noted) in this note we proceed by refining the global arguments of ibid. Thus the approaches of [BM12] and of the present note may be regarded as being somewhat complementary to one another.

1.1.10. Remark. In the paper [GK12], similar techniques to those of this paper are used to prove the Breuil–Mézard conjecture for two-dimensional potentially Barsotti–Tate representations of G_{K}. The key additional ingredients which are available in that case, but not in general, are the automorphy lifting theorems for potentially Barsotti–Tate representations proved in [Kis09b] and [Gee06]. The arguments of the present paper are in large part based on those of [GK12], which in turn relies on the strategy outlined in [Kis10]; in particular, our implementation of the patching argument for unitary groups is simply the natural adaptation of the arguments of [GK12] to higher rank unitary groups. Theorem 5.5.4 below gives a geometric refinement of some of the results of [GK12].

1.2. Acknowledgments. The debt that the arguments of this paper owe to the work of Mark Kisin will be obvious to the reader; in particular, several of our main arguments are closely based on arguments from [Kis09a], [Kis10] and [GK12]. We are also grateful to him, as well as to Christophe Breuil and Kevin Buzzard, for helpful comments on an earlier draft of this paper. We would like to thank Florian Herzig for helpful conversations about the representation theory of GL_{n}(𝔽_{q}), and Tom Barnet-Lamb, David Geraghty, Robert Guralnick, Florian Herzig, and Jack Shotton for helpful conversations about the material in Appendix A. We would like to thank the anonymous referee for a careful reading, and many helpful comments and suggestions.

1.3. Notation and Conventions. Throughout this paper, p will denote an odd prime.

If K is a field, then we let G_{K} denote its absolute Galois group. If K is furthermore a finite extension of ℚ_{p} for some p, then we write I_{K} for the inertia subgroup of G_{K}. If F is a number field and v is a finite place of F then we let Frob_{v} denote a geometric Frobenius element of G_{F_{v}}.

We let ε denote the p-adic cyclotomic character, and let \overline{ε} = ω the mod p cyclotomic character. We denote by ω̃ the Teichmüller lift of ω. We let ω_{2} denote a choice of a fundamental character of I_{ℚ_{p}} of niveau 2.

If K is a p-adic field, if ρ is a continuous de Rham representation of G_{K} over \overline{ℚ}_{p}, and if τ : K ↪ \overline{ℚ}_{p}, then we will write HT_{τ} (ρ) for the multiset of Hodge–Tate numbers 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 multiset 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 use this convention throughout the paper, except in Section 3, where we will use the opposite convention that ε has Hodge–Tate weight 1. We apologise for this, but it seems to us to be the best way to make what we write compatible with the existing literature.

Let K be a finite extension of ℚ_{p}, and let rec denote the local Langlands correspondence from isomorphism classes of irreducible smooth representations of GL_{n}(K) over ℂ to isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of W_{K} defined in [HT01]. Fix an isomorphism ı : \overline{ℚ}_{p} → ℂ. We define the local Langlands correspondence rec_{p} over \overline{ℚ}_{p} by ı ∘ rec_{p} = rec ∘ ı. This depends only on ı^{−1}(\sqrt{p}). We let Art_{K} : K^{×} → W_{K}^{ab} be the isomorphism provided by local class field theory, which we normalise so that uniformisers correspond to geometric Frobenius elements. We will write 𝟙 for the trivial n-dimensional representation of some group, the precise group and choice of n always being clear from the context.

When discussing deformations of Galois representations, we will use the terms “framed deformation” (which originated in [Kis09b]) and “lifting” (which originated in [CHT08]) interchangeably.

We write all matrix transposes on the left; so ^{t}A is the transpose of A.

2. Background on multiplicities and cycles

In this preliminary section we provide details on the notions of multiplicities and cycles that we outlined in the introduction.

2.1. Hilbert–Samuel multiplicities. Recall that if A is a Noetherian local ring with maximal ideal 𝔪 of dimension d, and M is a finite A-module, then there is polynomial P_{M}^{A}(X) of degree at most d (the Hilbert–Samuel polynomial of M), uniquely determined by the requirement that for n ≫ 0, the value P_{M}^{A}(n) is equal to the length of M/𝔪^{n+1}M as an A-module.

2.1.1. Definition. The Hilbert–Samuel multiplicity e(M, A) is defined to be d! times the coefficient of X^{d} in P_{M}^{A}(X). We write e(A) for e(A, A).

Note in particular that if A is Artinian, then e(M, A) is simply the length of M as an A-module.

2.2. Cycles. Let X be a Noetherian scheme.

2.2.1. Definition. (1) Let M be a coherent sheaf on X, and write Z to denote the scheme-theoretic support of M (i.e. Z is the closed subscheme of X cut out by the annihilator ideal I ⊂ O_{X} of M). For any point x ∈ X, we write e(M, x) to denote the Hilbert–Samuel multiplicity e(M_{x}, O_{Z,x}).

(2) If Z is a closed subscheme of X, then we write e(Z, x) := e(O_{Z}, x) for all x ∈ X. (Note that Z coincides with the scheme-theoretic support of O_{Z}, and so by definition this is equal to the multiplicity e(O_{Z,x}) of the local ring O_{Z,x}.)

2.2.2. Remark. In some situations the multiplicity e(M, x) is particularly simple to describe.

(1) If x does not lie in the support of M, i.e. if M_{x} = 0, then e(M, x) = 0.

(2) If x is a generic point of the scheme-theoretic support Z of M (so that O_{Z,x} is an Artinian ring), then e(M, x) is simply the length of M_{x} as an O_{Z,x}-module.

2.2.3. Definition. (1) We say that a point x ∈ X is of dimension d, and write

dim(x) = d, if its closure \overline{\{x\}} is of dimension d.

(2) A d-dimensional cycle on X is a formal finite ℤ-linear combination of points of X of dimension d.

(3) We write X ≥ 0 if X is in fact a ℤ_{≥0}-linear combination of points of X of dimension d, and we write X ≥ Y if X − Y ≥ 0.

2.2.4. Definition. If Z = ∑_{dim(x)=d} n_{x}x is a d-dimensional cycle on X, then for any

point y ∈ X, we define the multiplicity e(Z, y) via the formula

e(Z, y) := ∑_{dim(x)=d} n_{x}e(\overline{\{x\}}, y).

(Here \overline{\{x\}} denotes the closure of the point x.)

2.2.5. Definition. (1) If d ≥ 0 is a non-negative integer, and M is a coherent sheaf on X whose support has dimension ≤ d, then we define the ddimensional cycle Z_{d}(M) associated to M as follows:

Z_{d}(M) := ∑_{dim(x)=d} e(M, x)x,

where, as indicated, the sum ranges over all points of X of dimension d. (Our assumption on the dimension of the support of M ensures that any point of dimension d lying in the support M is necessarily a generic point of that support, and hence that there are only finitely many such points lying in the support of M. Thus all but finitely many terms appearing in the sum defining Z_{d}(M) vanish, and so this sum is in fact well-defined. Note also that, if we let Z denote the scheme-theoretic support of M, then by Remark 2.2.2, the multiplicity e(M, x) is simply the length of M_{x} as an O_{Z,x}-module.)

(2) If the support of M is finite-dimensional of dimension d, then we write simply Z(M) := Z_{d}(M).

(3) If Z is a closed subset of X, then we write Z_{d}(Z) := Z_{d}(O_{Z}), and denote this simply by Z(Z) if Z is finite-dimensional of dimension d.

2.2.6. Remark. If Z is equidimensional of some finite dimension d, then Z(Z) encodes the irreducible components of Z, together with the multiplicity with which each component appears in Z. If Z has dimension less than d, then Z_{d}(Z) = 0.

2.2.7. Lemma. If 0 → M′ → M → M′′ → 0 is an exact sequence of coherent sheaves on X, such that the support of M is of dimension ≤ d (or equivalently, such that the supports of each of M′ and M′′ are of dimension ≤ d), then

Z_{d}(M) = Z_{d}(M′) + Z_{d}(M′′).

Proof. Let Z (resp. Z′ and Z′′) denote the support of M (resp. of M′ and M′′), so that Z′, Z′′ ⊆ Z. As already noted in the statement of Definition 2.2.5, if x ∈ X is of dimension d, then e(M, x) (resp. e(M′, x), resp. e(M′′, x)) is simply the length of M_{x} as an O_{Z,x}-module (resp. the length of M′_{x} as an O_{Z′,x}-module, resp. the length of M′′_{x} as an O_{Z′′,x}-module). Since each of O_{Z′,x} and O_{Z′′,x} is a quotient of O_{Z,x}, the claimed additivity of cycles follows from the additivity of lengths in exact sequences. □

The following lemma records formula (1.1.1), which was stated in the introduction.

2.2.8. Lemma. If M is a coherent sheaf on X with support of dimension d, and if Z(M) is the cycle associated to M, then for any point y ∈ X, we have the formula

e(M, y) = e(Z(M), y).

Proof. This follows immediately from Theorem 14.7 of [Mat89]. □

We now establish a cycle-theoretic version of [Kis09a, Prop. 1.3.4], whose proof follows the same lines as the proof of that result.

2.2.9. Lemma. Let X be a Noetherian scheme of finite dimension d, and let f ∈ O_{X} (X) be regular (i.e. a non-zero divisor in each stalk of O_{X}). If M is an f-torsion free coherent sheaf on X which is supported in dimension d − 1, then M/fM is supported in dimension d − 2.

Proof. Since M is f-torsion free, no generic point of Supp(M) is contained in V (f). Thus Supp(M/fM) is of dimension ≤ d − 2, as claimed. □

2.2.10. Lemma. Let X be a Noetherian integral scheme of finite dimension d, and let f ∈ O_{X} (X) be non-zero. If M is an f-torsion free coherent sheaf on X which is generically free of rank one, then Z_{d−1}(M/fM) = Z(V (f)).

Proof. Let x be the generic point of X, and i_{x} : Spec κ(x) → X the canonical map. By assumption M_{x} is one-dimensional over κ(x), and so we may find a morphism of quasi-coherent sheaves M → (i_{x})_{∗}κ(x) whose kernel M′ is torsion. The image M′′ of this morphism is a coherent subsheaf of (i_{x})_{∗}κ(x), and hence is contained in an invertible sheaf L. Let L′ denote the cokernel of the inclusion M′′ ↪ L.

Consider first the exact sequence

0 → M′ → M → M′′ → 0.

All the terms in this sequence are f-torsion free, and so

0 → M′/fM′ → M/fM → M′′/fM′′ → 0

is again exact. Lemma 2.2.9 shows that Z_{d−1}(M′/fM′) = 0, and Lemma 2.2.7 then implies that

(2.2.11)  Z_{d−1}(M/fM) = Z_{d−1}(M′′/fM′′).

Next consider the exact sequence 0 → L′[f] → L′ → L′ → L′/fL′ → 0. Since all the terms in this exact sequence are torsion, and so supported in dimension d − 1, we see from Lemma 2.2.7 that

(2.2.12)  Z_{d−1}(L′[f]) = Z_{d−1}(L′/fL′).

Finally, consider the exact sequence 0 → M′′ → L → L′ → 0. The first two terms in this sequence are f-torsion free, and so it induces an exact sequence 0 → L′[f] → M′′/fM′′ → L/fL → L′/fL′ → 0. Again applying Lemma 2.2.7, together with (2.2.12), we find that Z_{d−1}(M′′/fM′′) = Z_{d−1}(L/fL). Combining this with (2.2.11), together with the fact that L is an invertible sheaf, we find that Z_{d−1}(M/fM) = Z(V (f)), as required. □

2.2.13. Proposition. Let X be a Noetherian scheme of finite dimension d, and f ∈ O_{X} (X) be regular (i.e. a non-zero divisor in each stalk of O_{X}). If M is an f-torsion free coherent sheaf on X, and if Z_{d}(M) = ∑_{dim(x)=d} n_{x}x, where, as indicated, x runs over the d-dimensional points of X, then the support of M/fM has dimension ≤ d − 1, and

Z_{d−1}(M/fM) = ∑_{dim(x)=d} n_{x}Z_{d−1}(\overline{\{x\}} ∩ V (f)).

Proof. We argue by induction on the quantity n := ∑_{dim x=d} n_{x}, with the case when n = 0 being handled by Lemma 2.2.9. Suppose now that n is an arbitrary positive integer, and choose x of dimension d such that n_{x} > 0. We may then find a non-zero surjection M_{x} → κ(x) (since by definition n_{x} is the length of M_{x}), which induces a non-zero map of quasi-coherent sheaves M → (i_{x})_{∗}κ(x) (where i_{x} is the canonical map Spec κ(x) → X). Let M′ denote the kernel of this map, and M′′ its image, so that the sequence 0 → M′ → M → M′′ → 0 is exact.

The sheaf M′ is f-torsion free (since it is a subsheaf of M) while the sheaf M′′ is f-torsion free, supported on \overline{\{x\}}, and generically free of rank one over this component (being a non-zero coherent subsheaf of (i_{x})_{∗}κ(x)). Thus we obtain a short exact sequence

0 → M′/fM′ → M/fM → M′′/fM′′ → 0,

and Lemmas 2.2.7 and 2.2.10 show that

Z_{d−1}(M/fM) = Z_{d−1}(M′/fM′) + Z_{d−1}(M′′/fM′′)

= Z_{d−1}(M′/fM′) + Z_{d−1}(\overline{\{x\}} ∩ V (f)).

The proposition follows by induction. □

We close this section with a result about the product of cycles. In fact, we will need to apply such a result in the context of a completed tensor product of complete Noetherian local k-algebras, for some field k (which is fixed for the remainder of this discussion), and so we restrict our attention to that particular context.

Note that if A and B are complete Noetherian local k-algebras, and if 𝔭 and 𝔮 are primes of A and B respectively, such that A/𝔭 is of dimension d and B/𝔮 is of dimension e, then A/𝔭ˆ⊗_{k}B/𝔮 is a quotient of Aˆ⊗_{k}B of dimension d + e. Hence Spec A/𝔭ˆ⊗_{k}B/𝔮 is a closed subscheme of Spec Aˆ⊗_{k}B of dimension d + e, and we write

Z(Spec A/𝔭) ×_{k} Z(Spec B/𝔮) := Z(Spec A/𝔭ˆ⊗_{k}B/𝔮),

and then extend this by linearity to a bilinear product from d-dimensional cycles on Spec A and e-dimensional cycles on Spec B to (d + e)-dimensional cycles on Spec Aˆ⊗_{k}B.

If M and N are finitely generated A- and B-modules respectively, giving rise to coherent sheaves M and N on Spec A and Spec B respectively, then the completed tensor product M ˆ⊗_{k}N gives rise to a coherent sheaf on Spec Aˆ⊗_{k}B, which we denote by Mˆ⊠N.

2.2.14. Lemma. In the context of the preceding discussion, if the support of M and N are of dimensions d and e respectively, then the support of Mˆ⊠N is of dimension d + e, and

Z_{d+e}(Mˆ⊠N) = Z_{d}(M) ×_{k} Z_{e}(N).

Proof. This is standard; we sketch the proof. Restricting to the support of Mˆ⊠N, we may assume that M (resp. N) is a faithful A-module (resp. B-module), so that A has dimension d and B has dimension e. As in (for example) the proof of [BLGHT11, Lemma 3.3(5)], the distinct minimal primes of Aˆ⊗_{k}B are precisely the 𝔭(Aˆ⊗_{k}B)+𝔮(Aˆ⊗_{k}B), where 𝔭 is a minimal prime of A and 𝔮 is a minimal prime of B. We are thus reduced to checking that for each such 𝔭, 𝔮, we have

e(M ˆ⊗_{k}N, (A/𝔭)ˆ⊗_{k}(B/𝔮)) = e(M, (A/𝔭))e(N, (B/𝔮)),

which follows from Lech’s lemma [Mat89, Thm. 14.12] exactly as in the proof of Proposition 1.3.8 of [Kis09a]. □

3. The geometric Breuil–Mézard Conjecture for two-dimensional representations of G_{ℚ_{p}}

In this section we explain the Breuil–Mézard conjecture in its original setting, that of two-dimensional representations of G_{ℚ_{p}}, and state our geometric version. We then recall some of the details of the proof of the original formulation of the conjecture from [Kis09a], and show that the proof may be extended to prove the geometric version. The one difference from Kisin’s notation and that of the present paper is that we prefer not to fix one of the Hodge–Tate weights of our Galois representations to be 0; this makes no essential difference to any of the arguments, but the additional flexibility that this notation gives us is convenient in the exposition. We remind the reader that in this section, our conventions for Hodge–Tate weights are that ε has Hodge–Tate weight 1.

3.1. The conjecture. We begin by recalling some notation from [Kis09a]. Fix a prime p > 2 and a finite extension E of ℚ_{p} (our coefficient field), with ring of integers O, residue field 𝔽, and uniformiser π. We assume that #𝔽 > 5, so that PSL_{2}(𝔽) is a simple group.

Let r̄ : G_{ℚ_{p}} → GL_{2}(𝔽) be a continuous representation, and let τ : I_{ℚ_{p}} → GL_{2}(E) be an inertial type, i.e. a representation with open kernel which extends to W_{ℚ_{p}}. Fix integers a, b with b ≥ 0 and a de Rham character ψ : G_{ℚ_{p}} → O^{×} such that \overline{ψε} = det r̄. We let R^{□,ψ}(a, b, τ, r̄) and R_{cr}^{□,ψ}(a, b, τ, r̄) be the framed deformation O-algebras which are universal for framed deformations of r̄ which have determinant ψε, and are potentially semistable (respectively potentially crystalline) with Hodge–Tate weights (a, a + b + 1) and inertial type τ. As in Section 1.1.2 of [Kis09a], we let σ(τ) and σ_{cr}(τ) denote the finite-dimensional irreducible E-representations of GL_{2}(ℤ_{p}) corresponding to τ via Henniart’s inertial local Langlands correspondence, we set σ(a, b, τ) = (det^{a} ⊗Sym^{b}E^{2}) ⊗_{E} σ(τ) and σ^{cr}(a, b, τ) = (det^{a} ⊗Sym^{b}E^{2}) ⊗_{E} σ^{cr}(τ), and we let L_{a,b,τ} (respectively L_{a,b,τ}^{cr} ) be a GL_{2}(ℤ_{p})-stable O-lattice in σ(a, b, τ) (respectively σ^{cr}(a, b, τ)). Write σ_{m,n} for the representation det^{m} ⊗Sym^{n}𝔽^{2} of GL_{2}(𝔽_{p}), 0 ≤ m ≤ p − 2, 0 ≤ n ≤ p − 1, so that we may write

(L_{a,b,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{m,n}σ_{m,n}^{a_{m,n}},

and

(L_{a,b,τ}^{cr} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{m,n}σ_{m,n}^{a_{m,n}^{cr}},

for some integers a_{m,n}, a_{m,n}^{cr}.

We can now state the Breuil–Mézard conjecture [BM02].

3.1.1. Conjecture. There are integers μ_{m,n}(r̄) depending only on m, n, and r̄, such that for any a, b, τ with det τ = ε^{1−2a−b}ψ|_{I_{ℚp}},

e(R^{□,ψ}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}μ_{m,n}(r̄),

and

e(R_{cr}^{□,ψ}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}^{cr}μ_{m,n}(r̄).

3.1.2. Remark. By a straightforward twisting argument, the truth of the conjecture is independent of the choice of ψ, so we may assume that ψ is crystalline. Note that if the conjecture is true for all a, b, τ, then it is easy to see that μ_{m,n}(r̄) = 0 unless det r̄|_{I_{ℚp}} = ω^{2m+n+1}, and that if this holds then we must have μ_{m,n}(r̄) = e(R_{cr}^{□,ψ}(m̃, n, 𝟙, r̄)/π), where m̃ is chosen so that ψ|_{I_{ℚp}} = ε^{2m̃+n+1}. So all the values μ_{m,n}(r̄) are determined by the crystalline deformation rings in low weight.

3.1.3. Remark. Conjecture 3.1.1 was proved in [Kis09a] under the additional as-

sumptions that r̄ ≁ (\begin{matrix}ωχ & ∗ \\ 0 & χ \end{matrix}) for any χ, and that if r̄ has scalar semisimplification

then r̄ is scalar.

We may now state our geometric version of the Breuil–Mézard conjecture.

3.1.4. Conjecture. For each 0 ≤ m ≤ p − 2, 0 ≤ n ≤ p − 1, there is a cycle C_{m,n} depending only on m, n, and r̄, such that for any a, b, τ with det τ = ε^{−2a−b−1}ψ|_{I_{ℚp}},

Z(R^{□,ψ}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}C_{m,n},

and

Z(R_{cr}^{□,ψ}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}^{cr}C_{m,n}.

3.1.5. Remark. Note again that if the conjecture is true for all choices of a, b, τ then with the assumptions and notation of Remark 3.1.2 we must have C_{m,n} = Z(R_{cr}^{□,ψ}(m̃, n, 𝟙, r̄)/π).

The following is our main result towards Conjecture 3.1.4.

3.1.6. Theorem. If r̄ ≁ (\begin{matrix}ωχ & ∗ \\ 0 & χ \end{matrix}) for any χ, then Conjecture 3.1.4 holds for r̄.

3.2. The proof of Theorem 3.1.6 via patching. In this subsection we present the proof of Theorem 3.1.6. To this end, we fix a continuous representation r̄ : G_{ℚ_{p}} → GL_{2}(𝔽). We begin by realising this representation as the restriction of a global Galois representation, by a similar (but simpler) argument to that of Appendix A of [GK12].

3.2.1. Proposition. There is a totally real field F and a continuous irreducible representation \overline{ρ} : G_{F} → GL_{2}(𝔽) such that

(1) p splits completely in F;

(2) \overline{ρ} is totally odd;

(3) \overline{ρ}(G_{F}) = GL_{2}(𝔽);

(4) if v ∤ p is a place of F then \overline{ρ}|_{G_{Fv}} is unramified;

(5) if v|p is a place of F then \overline{ρ}|_{G_{Fv}} ≅ r̄;

(6) [F : ℚ] is even;

(7) \overline{ρ} is modular.

Proof. By Proposition 3.2 of [Cal12], we may find F and \overline{ρ} satisfying all but the last two conditions. By Proposition 8.2.1 of [Sno09], there is a finite Galois extension F′/F in which all places above p split completely such that \overline{ρ}|_{G_{F′}} is modular. If we make a further quadratic extension (linearly disjoint from \overline{F}^{ker \overline{ρ}} over F, and in which the primes above p split completely), if necessary, to ensure that [F′ : ℚ] is even, and replace F with F′, then the result follows. □

For the remainder of the subsection, we follow the arguments of [Kis09a, §2.2] very closely, and we do our best to conform to the notation used there.

We choose a finite set S of finite places of F, containing all the places v|p and at least one other place. Using [Kis09a, Lem. 2.2.1], we can and do choose S so that conditions (1)–(4) of [Kis09a, §2.2] hold.

We denote by D the quaternion algebra with centre F which is ramified at all infinite places of F and unramified at all finite places (so the set Σ considered in [Kis09a, §2.1.1] is empty). We fix a maximal order O_{D} of D, for each finite place v we choose an isomorphism (O_{D})_{v} ⟶^{∼} M_{2}(O_{F_{v}}), and we define U = ∏_{v} U_{v} ⊂ (D ⊗_{F} 𝔸_{F}^{∞})^{×} to be the following compact open subgroup of ∏_{v}(O_{D})_{v}^{×}: if v|p or v ∉ S then U_{v} = (O_{D})_{v}^{×}, while if v ∈ S but v ∤ p, then U_{v} consists of the matrices which are upper-triangular and unipotent modulo ϖ_{v}, where ϖ_{v} is a uniformiser of F_{v}. The subgroup U is sufficiently small in the sense of [Kis09a, §2.1.1].

We write Σ_{p} for the set of places v|p of F (since Σ is empty, this is consistent with [Kis09a]). For each v|p, we let R_{v}^{□,ψ}(\overline{ρ}|_{G_{Fv}}) denote the universal framed deformation O-algebra for \overline{ρ}|_{G_{Fv}} with determinant ψε, and we define

R_{∞} := ˆ⊗_{v|p,O}R_{v}^{□,ψ}(\overline{ρ}|_{G_{Fv}})[[x_{1}, …, x_{g}]],

where x_{1}, …, x_{g} are formal variables, and the integer g is chosen as in [Kis09a, Prop. 2.2.4].

For each place v|p of F, we choose integers a_{v}, b_{v} with b_{v} ≥ 0, together with an inertial type τ_{v}, and let ∗ be either cr or nothing (the same choice of ∗ being made for all v|p). We assume that det τ_{v} = ε^{1−2a_{v}−b_{v}}ψ|_{I_{ℚp}} for each v. We write^{2}

R̄_{∞} := ˆ⊗_{v|p,O}R_{∗}^{□,ψ}(a_{v}, b_{v}, τ_{v}, \overline{ρ}|_{G_{Fv}})[[x_{1}, …, x_{g}]].

If we write W_{σ} := ⊗_{v|p}L_{a_{v},b_{v},τ_{v}}^{∗}, then W_{σ} is a finite free O-module with an action of ∏_{v|p} GL_{2}(O_{F_{v}}), and the quotient W_{σ}/πW_{σ} is then a representation of ∏_{v|p} GL_{2}(O_{F_{v}}) over 𝔽. Fix a Jordan–Hölder filtration

0 = L_{0} ⊂ ⋯ ⊂ L_{s} = W_{σ}/πW_{σ}

of W_{σ}/πW_{σ} by ∏_{v|p} GL_{2}(O_{F_{v}})-subrepresentations. If we write σ_{i} := L_{i}/L_{i−1}, then σ_{i} ≅ ⊗_{v|p} det^{m_{v,i}} ⊗Sym^{n_{v,i}}𝔽^{2} for some uniquely determined integers m_{v,i} ∈ {0, …, p − 2} and n_{v,i} ∈ {0, …, p − 1}.

The patching construction of [Kis09a, § 2.2.5] then gives, for some integer denoted by h + j in [Kis09a], and formal variables y_{1}, …, y_{h+j},

• an (R̄_{∞}, O[[y_{1}, …, y_{h+j}]])-bimodule M_{∞}, finite free over O[[y_{1}, …, y_{h+j}]], and

• a filtration of M_{∞}/πM_{∞} by (R̄_{∞}/π, 𝔽[[y_{1}, …, y_{h+j}]])-bimodules, say

0 = M_{∞}^{0} ⊂ M_{∞}^{1} ⊂ ⋯ ⊂ M_{∞}^{s} = M_{∞}/πM_{∞},

such that each M_{∞}^{i}/M_{∞}^{i−1} is a finite free 𝔽[[y_{1}, …, y_{h+j}]]-module, and such that

• the isomorphism class of M_{∞}^{i}/M_{∞}^{i−1} as an (R_{∞}, 𝔽[[y_{1}, …, y_{h+j}]])-bimodule depends only on the isomorphism class of σ_{i} as a ∏_{v|p} GL_{2}(O_{F_{v}})-module, and not on the choices of a_{v}, b_{v} and τ_{v}. (It is immediate from the finiteness argument used in patching that this can be achieved for any finite collection of tuples (a_{v}, b_{v}, τ_{v}), and since there are only countably many tuples, a diagonalization argument allows us to assume independence for all tuples.)

2The ring R̄_{∞} depends upon the particular choices of a_{v}, b_{v}, τ_{v} and ∗, although (following [Kis09a]) we do not indicate this in the notation.

For any Serre weight σ_{m,n}, we now write ψ_{cr} for a crystalline character lifting ω^{−1} det r̄, and we define (in the notation of Remark 3.1.2)

μ_{m,n}(r̄) := e(R_{cr}^{□,ψ_{cr}}(m̃, n, 𝟙, r̄)/π)

and

C_{m,n} := Z(R_{cr}^{□,ψ_{cr}}(m̃, n, 𝟙, r̄)/π).

Proof of Theorem 3.1.6. Fix a, b, τ and ∗, and set a_{v} = a, b_{v} = b, τ_{v} = τ for each v|p. Lemma 2.2.11 of [Kis09a] (together with its proof) shows that e(R̄_{∞}/π) ≥ e(M_{∞}/πM_{∞}, R̄_{∞}/π), and the following conditions are equivalent.

(1) M_{∞} is a faithful R̄_{∞}-module.

(2) M_{∞} is a faithful R̄_{∞}-module which has rank 1 at all generic points of R̄_{∞}.^{3} (3) e(R̄_{∞}/π) = e(M_{∞}/πM_{∞}, R̄_{∞}/π).

(4) e(R̄_{∞}/π) ≤ e(M_{∞}/πM_{∞}, R̄_{∞}/π). (Note that the proof of [Kis09a, Lem. 2.2.11], as written, literally applies only when a = 0, but in fact it goes through unchanged for any value of a.) Furthermore, the argument of Corollary 2.2.17 of [Kis09a] (which uses the p-adic Langlands correspondence for GL_{2}(ℚ_{p}), together with the weight part of Serre’s conjecture) establishes the inequality e(R̄_{∞}/π) ≤ e(M_{∞}/πM_{∞}, R̄_{∞}/π), so that in fact all of the conditions (1)–(4) actually hold. (This is where our hypothesis regarding r̄ is used.)

Let R̄_{∞}^{i} := (ˆ⊗_{v|p}R_{cr}^{□,ψ_{cr}}(m̃_{v,i}, n_{v,i}, 𝟙, \overline{ρ}|_{G_{Fv}})/π)[[x_{1}, …, x_{g}]], regarded as a quotient of R_{∞}. Since the isomorphism class of M_{∞}^{i}/M_{∞}^{i−1} as an R_{∞}-module depends only on the isomorphism class of σ_{i}, one sees as in the proof of Lemma 2.2.13 of [Kis09a] that the action of R_{∞} on M_{∞}^{i}/M_{∞}^{i−1} factors through R̄_{∞}^{i}. (In brief: it is enough to check for each place v_{0}|p that the action of R_{v_{0}}^{□,ψ}/π factors through R_{cr}^{□,ψ_{cr}}(m̃_{v_{0},i}, n_{v_{0},i}, 𝟙, \overline{ρ}|_{G_{Fv_{0}}} )/π. In order to do this, one applies the construction with a_{v} = m̃_{v_{0},i}, b_{v} = n_{v_{0},i}, τ_{v_{0}} an appropriate scalar type, and the other τ_{v} chosen so that σ_{m_{v,i},n_{v,i}} is a Jordan–Hölder factor of L_{a_{v},b_{v},τ_{v}} ⊗_{O} 𝔽 for v ≠ v_{0}.) Furthermore, the R_{∞}-module M_{∞}^{i}/M_{∞}^{i−1} is supported on all of Spec R̄_{∞}^{i}, by the results of Section 4.6 of [Gee11] (cf. the final paragraph of the proof of [Kis09a, Prop. 2.2.15]). Finally, we note that R̄_{∞}^{i} is generically reduced (see the proof of [Kis09a, Prop. 2.2.15]).

Lemma 2.2.14 shows that

(3.2.2)  Z(R̄_{∞}/π) = (∏_{v|p} Z(R_{∗}^{□,ψ}(a, b, τ, r̄)/π)) × Z(Spec 𝔽[[x_{1}, …, x_{g}]]),

while, identifying each of the modules M_{∞}/πM_{∞}, M_{∞}^{i}/M_{∞}^{i−1}, etc., with the corresponding sheaves on Spec R_{∞}/π that they give rise to, we compute that

(3.2.3) Z(M_{∞}/πM_{∞}) = ∑_{i} Z(M_{∞}^{i}/M_{∞}^{i−1})

≥ ∑_{i} Z(Spec R̄_{∞}^{i}/π) = (∏_{v|p} ∑_{m,n} a_{m,n}^{∗}C_{m,n}) × Z(Spec 𝔽[[x_{1}, …, x_{g}]]),

3We remark that there is a typo in this part of the statement of [Kis09a, Lem. 2.2.11]; R_{∞} is written there, rather than R̄_{∞}.

the first equality following by Lemma 2.2.7, the inequality following from the fact, noted above, that the support of M_{∞}^{i}/M_{∞}^{i−1} coincides with the generically reduced closed subscheme Spec R̄_{∞}^{i} of Spec R_{∞}/π, and the second equality following from another application of Lemma 2.2.14. Also, since M_{∞} is π-torsion free, and generically free of rank one over each component of Spec R̄_{∞} (condition (2) above), Proposition 2.2.13 shows that

(3.2.4)  Z(R̄_{∞}/π) = Z(M_{∞}/πM_{∞}).

Putting these computations together, we find that

(∏_{v|p} Z(R_{∗}^{□,ψ}(a, b, τ, r̄)/π)) × Z(Spec 𝔽[[x_{1}, …, x_{g}]]) =^{(3.2.2)} Z(R̄_{∞}/π)

=^{(3.2.4)} Z(M_{∞}/πM_{∞}, R̄_{∞}/π) ≥^{(3.2.3)} (∏_{v|p} ∑_{m,n} a_{m,n}^{∗}C_{m,n}) × Z(Spec 𝔽[[x_{1}, …, x_{g}]]).

However, if we apply Lemma 2.2.8 (with y the closed point of Spec R̄_{∞}/π) to pass to the corresponding multiplicities, then this inequality on cycles gives a corresponding inequality on multiplicities, which is in fact an equality, by Lemma 2.3.1 of [Kis09a]. Thus this inequality of cycles is an equality (note that any non-zero cycle must have non-zero multiplicity at the unique closed point of Spec R̄_{∞}/π), and we deduce that

∏_{v|p} Z(R_{∗}^{□,ψ}(a, b, τ, r̄)/π) = ∏_{v|p} ∑_{m,n} a_{m,n}^{∗}C_{m,n},

and thus that

Z(R_{∗}^{□,ψ}(a, b, τ, r̄)/π) = ∑_{m,n} a_{m,n}^{∗}C_{m,n},

as required. □

3.2.5. Remark. Even if r̄ ∼ (\begin{matrix}ωχ & ∗ \\ 0 & χ \end{matrix}) for some χ, it is presumably possible to

use the above arguments to show that our geometric Breuil–Mézard conjecture is equivalent to the usual one, and that both are equivalent to the equivalent conditions of Lemma 2.2.11 of [Kis09a]. We have not done so, because parity issues with Hilbert modular forms make the argument rather longer than one would wish, and in any case we prove a similar statement in far greater generality in Section 5 (see Theorem 5.5.2, which together with Lemma 4.3.1 shows the equivalence of our geometric conjecture with the usual one). (Note that since p > 2 all the Serre weights occurring in the reduction mod p of L_{a,b,τ} have the same parity, so it is possible to circumvent parity problems by twisting, but the details are a little unpleasant to write out.)

3.3. Analysis of components. For a given r̄, there may be several different Serre weights σ_{m,n} for which μ_{m,n}(r̄) ≠ 0. We now examine the different cycles C_{m,n}.

We write W (r̄) for the set of σ_{m,n} for which μ_{m,n}(r̄) ≠ 0 (that is: the set of Serre weights for r̄). The only cases where W (r̄) contains more than one element are as follows (throughout it is understood that we always impose the conditions that 0 ≤ m ≤ p − 2 and 0 ≤ n ≤ p − 1):

• If r̄ is irreducible, say

r̄|_{I_{ℚp}} ∼ ω^{m} ⊗ (\begin{matrix}ω_{2}^{n+1} & 0 \\ 0 & ω_{2}^{p(n+1)} \end{matrix}) ,

then W (r̄) = {σ_{m,n}, σ_{m+n,p−1−n}}.

• If r̄ is reducible but indecomposable, with

r̄|_{I_{ℚp}} ∼ (\begin{matrix}ω^{m+1} & ∗ \\ 0 & ω^{m} \end{matrix})

a peu ramifiée extension, then W (r̄) = {σ_{m,0}, σ_{m,p−1}}.

• If r̄ is reducible and decomposable, with

r̄|_{I_{ℚp}} ∼ (\begin{matrix}ω^{m+n+1} & 0 \\ 0 & ω^{m} \end{matrix})

such that n ≤ p − 3, then – if 0 < n < p − 3, we have W (r̄) = {σ_{m,n}, σ_{m+n+1,p−3−n}};

– if n = p − 3 and p > 3, we have W (r̄) = {σ_{m,p−3}, σ_{m−1,0}, σ_{m−1,p−1}};

– if p = 3 and n = 0, we have W (r̄) = {σ_{m,0}, σ_{m,2}, σ_{m+1,0}, σ_{m+1,2}}.

The relationships between the cycles C_{m,n} are as follows. Note in particular that

by Theorem 3.1.6, if r̄ ≁ (\begin{matrix}ωχ & ∗ \\ 0 & χ \end{matrix}) for any χ, then the assumption in the following

Proposition is automatic.

3.3.1. Proposition. Assume that Conjecture 3.1.4 holds for r̄. Then the cycles C_{m,n} have disjoint support, except for the cycles C_{m,0} and C_{m,p−1} when both are non-zero. In this latter case there is an equality C_{m,0} = C_{m,p−1}, except if r̄ is a twist of a (possibly split) peu ramifiée extension of the trivial character by the mod p cyclotomic character, in which case C_{m,p−1} is the sum of two irreducible cycles, one of which is C_{m,0}.

Proof. It is presumably possible to establish this in most cases via direct computations with Fontaine-Laffaille theory; however, we take the opportunity to use our geometric formulation of the Breuil–Mezard conjecture. We begin by noting that it follows from Corollary 1.7.14 of [Kis09a] that the cycles C_{m,n} are irreducible, except

in the cases that n = p − 1 and r̄|_{I_{ℚp}} ∼ (\begin{matrix}ω^{m+1} & ∗ \\ 0 & ω^{m} \end{matrix}) is peu ramifiée, or n = p − 2

and r̄ ∼ ω^{m} ⊗ (\begin{matrix}μ_{1} & 0 \\ 0 & μ_{2} \end{matrix}) for distinct unramified characters μ_{1} and μ_{2}. Thus to

prove the claimed disjointness of cycles, it is enough to prove that the cycles are not equal.

We first consider the cases where C_{m,0} and C_{m,p−1} are nonzero. For notational simplicity, we make a twist by ω^{−m}, and thus further assume that m = 0. Consider the trivial inertial type 𝟙. Examining Henniart’s appendix to [BM02], we see that σ(𝟙) = St, the Steinberg representation, while σ^{cr}(𝟙) = 𝟙, the trivial representation. The reduction mod p of St is just σ_{p−1}, so we conclude that

C_{0,p−1} = Z(R^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄)/π),
C_{0,0} = Z(R_{cr}^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄)/π).

By definition, R_{cr}^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄) is a quotient of R^{□,ψ}(0, 0, 𝟙, r̄), and as they have the same dimension, we see that Spec R_{cr}^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄) is a union of irreducible components of Spec R^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄).

It is easy to see that the two rings are actually equal unless r̄ is a twist of a peu ramifié extension of 1 by ω, because the only non-crystalline semistable crystalline representations with Hodge–Tate weights 0 and 1 are unramified twists of extensions of 1 by ε, and so it suffices to check, in the case when r̄ is a twist of a peu ramifié extension of 1 by ω, that Spec R^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄)/π) has two distinct components. In the case that the extension is non-split, the framed deformation ring is formally smooth over the deformation ring, and the relevant rings are computed in Theorem 5.3.1(i) of [BM02]; in particular, they do verify that the Specs of their reductions mod π contain two distinct components.

We now explain another way to see that Spec R^{□,ψ_{cr}}(˜0, 0, 𝟙, r̄)/π) consists of two distinct components, which works equally well in the case when r̄ is split. Namely, a two-dimensional crystalline representation with Hodge–Tate weights 0 and 1 with reducible reduction is necessarily an extension of characters, whose restrictions to I_{p} are trivial and cyclotomic respectively, and it is uniquely determined by its associated pseudo-representation. The cycle C_{0,0} is thus directly seen to be irreducible, and it has non-trivial image in the associated pseudo-deformation space. On the other hand, as we already observed, a genuinely semi-stable two-dimensional deformation of r̄ with Hodge–Tate weights 0 and 1 is necessarily a twist of an extension of the trivial character by the cyclotomic character, with the possible twist being uniquely determined (since we have fixed the determinant to be ψε). Such extensions are determined by their L-invariant, and so one can give an explicit description of the Zariski closure of the space of genuinely semi-stable deformations, and show that this closure, as well as its reduction mod π, is irreducible. Moreover, its reduction mod π does not coincide with C_{0,0}, since its image in the associated pseudo-deformation space is simply the closed point. Thus we have shown that C_{0,p−1} is the sum of two distinct irreducible components.

Consider next the case that r̄|_{I_{ℚp}} ∼ ω^{m} ⊗ (\begin{matrix}ω_{2}^{n+1} & 0 \\ 0 & ω_{2}^{p(n+1)} \end{matrix}) with 0 < n < p − 1.

We need to show that C_{m,n} ≠ C_{m+n,p−1−n}. Consider the inertial type ω^{m+n} ⊕ ω^{m}, and choose ψ so that ψ|_{I_{ℚp}} = εω^{2m+n}. Then by for example Lemmas 3.1.1 and 4.2.4 of [CDT99], the semisimplification of the reduction modulo p of σ(ω^{m+n}⊕ω^{m}) (the representation of GL_{2}(ℤ_{p}) associated to ω^{m+n}⊕ω^{m} by Henniart’s inertial local Langlands correspondence, which is the inflation to GL_{2}(ℤ_{p}) of a principal series representation of GL_{2}(𝔽_{p})) has Jordan–Hölder factors σ_{m,n} and σ_{m+n,p−1−n}, so that we have

Z(R^{□,ψ}(0, 1, ω^{m+n} ⊕ ω^{m}, r̄)/π) = C_{m,n} + C_{m+n,p−1−n}.

By Theorem 6.22 of [Sav05] (noting that the framed deformation ring is formally smooth over the deformation ring, since r̄ is irreducible), we have R^{□,ψ}(0, 1, ω^{m+n}⊕ ω^{m}, r̄)/π ≅ 𝔽[[U, V, W, X, Y]]/(XY), so C_{m,n} ≠ C_{m+n,p−1−n}, as required.

It remains to treat the cases where r̄ ∼ \overline{χ}_{1}⊕\overline{χ}_{2} is a direct sum of two distinct characters. In this case, one knows that all of the relevant lifting rings R_{cr}^{□,ψ_{cr}}(m̃, n, 𝟙, r̄) are in fact ordinary lifting rings (cf. Corollary 1.7.14 and Remark 1.7.16 of [Kis09a]). It thus suffices to note that the cycles that we have to prove are distinct correspond respectively to liftings which contain a submodule lifting \overline{χ}_{1} or which contain a submodule lifting \overline{χ}_{2}, and since \overline{χ}_{1} ≠ \overline{χ}_{2}, it is easy to see that the cycles are distinct. □

Proof of Proposition 1.1.3. The assertions about the number of components of C_{m,n} follow from Proposition 3.3.1 above together with Corollary 1.7.14 of [Kis09a], and the disjointness of the supports of the C_{m,n} follows from Proposition 3.3.1. □

3.4. Comparison with the results of [BM12]. In [BM12], Breuil and Mézard construct, for generic r̄, a correspondence between the irreducible components of Spec R^{□,ψ}(a, b, τ, r̄)/π and the Serre weights for r̄ which appear with positive multiplicity in the mod p reduction of (det^{a} ⊗Sym^{b}E^{2}) ⊗_{E} σ(τ). The definition of the correspondence is given in [BM12, Thm. 1.5]: namely, if 𝔞 is a generic point of Spec R^{□,ψ}(a, b, τ, r̄)/π, then the associated Serre weight is the GL_{2}(ℤ_{p})-socle of a certain GL_{2}(ℚ_{p})-representation obtained from the universal Galois representation into GL_{2}(R^{□,ψ}(a, b, τ, r̄)/(π, 𝔞)) via the p-adic local Langlands correspondence.

According to our geometric formulation of the Breuil–Mézard conjecture, the component \overline{\{𝔞\}} of Spec R^{□,ψ}(a, b, τ, r̄)/π is equal to Spec R_{cr}^{□,ψ_{cr}}(m̃, n, 𝟙, r̄)/π), for some Serre weight σ_{m,n} of r̄. Since the p-adic local Langlands correspondence is functorial, we find that the GL_{2}(ℤ_{p})-socle of the GL_{2}(ℚ_{p})-representation attached to 𝔞, thought of as a generic point of Spec R^{□,ψ}(a, b, τ, r̄)/π, is the same as the GL_{2}(ℤ_{p})-socle of the GL_{2}(ℚ_{p})-representation attached to 𝔞, thought of as the generic point of Spec R_{cr}^{□,ψ_{cr}}(m̃, n, 𝟙, r̄)/π). But in this latter case, the GL_{2}(ℤ_{p})- socle in question is equal to σ_{m,n}, as follows from the fact that the correspondence of [BM12] is compatible with the original conjecture of [BM02]. This shows that the correspondence of [BM12] is precisely the correspondence C_{m,n} ↦ σ_{m,n}.

4. The Breuil–Mézard Conjecture for GL_{n}

4.1. The numerical conjecture. We begin by formulating a natural generalisation of the Breuil–Mézard conjecture for n-dimensional representations. We now fix the notation we will use for the rest of the paper, which differs in some respects from that of Section 3, but is closer to that used in the literature on automorphy lifting theorems for unitary groups. We remind the reader that for the rest of the paper, our convention on Hodge–Tate weights is that the Hodge–Tate weight of ε is −1. Let K/ℚ_{p} be a finite extension with ring of integers O_{K} and residue field k, let E/ℚ_{p} be a finite extension with ring of integers O, uniformiser π and residue field 𝔽, and let r̄: G_{K} → GL_{n}(𝔽) be a continuous representation. Assume that E is sufficiently large, and in particular that E contains the images of all embeddings K ↪ \overline{ℚ}_{p}. Let ℤ_{+}^{n} denote the set of tuples (λ_{1}, …, λ_{n}) of integers with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{n}. For any λ ∈ ℤ_{+}^{n}, view λ as a dominant character of the algebraic group GL_{n/O} in the usual way, and let M′_{λ} be the algebraic O_{K}-representation of GL_{n} given by

M′_{λ} := Ind_{B_{n}}^{GL_{n}}(w_{0}λ)_{/O_{K}}

where B_{n} is the Borel subgroup of upper-triangular matrices of GL_{n}, and w_{0} is the longest element of the Weyl group (see [Jan03] for more details of these notions, and note that M′_{λ} has highest weight λ). Write M_{λ} for the O_{K}-representation of GL_{n}(O_{K}) obtained by evaluating M′_{λ} on O_{K}. For any λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,E)} we write L_{λ} for the O-representation of GL_{n}(O_{K}) defined by

⊗_{τ:K↪E}M_{λ_{τ}} ⊗_{O_{K},τ} O.

Given any a ∈ ℤ_{+}^{n} with p − 1 ≥ a_{i} − a_{i+1} for all 1 ≤ i ≤ n − 1, we define the k-representation P_{a} of GL_{n}(k) to be the representation obtained by evaluating Ind_{B_{n}}^{GL_{n}}(w_{0}a)_{k} on k, and let N_{a} be the irreducible sub-k-representation of P_{a} generated by the highest weight vector (that this is indeed irreducible follows for example from II.2.8(1) of [Jan03] and the appendix to [Her09]). We say that an element a ∈ (ℤ_{+}^{n})^{Hom(k,𝔽)} is a Serre weight if

• for each σ ∈ Hom(k, 𝔽) and each 1 ≤ i ≤ n − 1 we have

p − 1 ≥ a_{σ,i} − a_{σ,i+1},

• and for each σ we have 0 ≤ a_{σ,n} ≤ p − 1, and not all a_{σ,n} = p − 1. If a ∈ (ℤ_{+}^{n})^{Hom(k,𝔽)} is a Serre weight then we define an irreducible 𝔽-representation F_{a} of GL_{n}(k) by

F_{a} := ⊗_{τ∈Hom(k,𝔽)}N_{a_{τ}} ⊗_{k,τ} 𝔽.

The representations F_{a} are absolutely irreducible and pairwise non-isomorphic, and every irreducible 𝔽-representation of GL_{n}(k) is of the form F_{a} for some a (see for example the appendix to [Her09]). We say that an element λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,E)} is a lift of a Serre weight a ∈ (ℤ_{+}^{n})^{Hom(k,𝔽)} if for each σ ∈ Hom(k, 𝔽) there is an element τ ∈ Hom_{ℚ_{p}}(K, E) lifting σ such that λ_{τ} = a_{σ}, and for all other τ′ ∈ Hom_{ℚ_{p}}(K, E) lifting σ we have λ_{τ′} = 0. We have a partial ordering ≤ on Serre weights, where b ≤ a if and only if a − b is a sum of (positive) simple roots.

4.1.1. Lemma. If λ is a lift of a then L_{λ} ⊗_{O} 𝔽 has socle F_{a}, and every other Jordan–Hölder factor of L_{λ} ⊗_{O} 𝔽 is of the form F_{b} with b < a.

Proof. This follows from sections 5.8 and 5.9 of [Hum06] (noting that the orderings ≤ and ≤_{ℚ} coincide for GL_{n}). □

Let τ : I_{K} → GL_{n}(E) be a representation with open kernel which extends to W_{K}, and take λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,E)}. Let r̄ : G_{K} → GL_{n}(𝔽) be a continuous representation. If E′/E is a finite extension, we say that a potentially crystalline representation ρ : G_{K} → GL_{n}(E′) has Hodge type λ if for each τ : K ↪ E,

HT_{τ} (ρ) = {λ_{τ,1} + n − 1, λ_{τ,2} + n − 2, …, λ_{τ,n}}.

We say that ρ has inertial type τ if the restriction to I_{K} of the Weil–Deligne representation associated to ρ is equivalent to τ.

4.1.2. Proposition. For each λ, τ there is a unique quotient R_{r̄,λ,τ}^{□} of the universal lifting O-algebra R□ ; r̄ for r̄ with the following properties.

(1) R_{r̄,λ,τ}^{□} is reduced and p-torsion free, and R_{r̄,λ,τ}^{□} [1/p] is formally smooth and equidimensional of dimension n^{2} + [K : ℚ_{p}]n(n − 1)/2.

(2) If E′/E is a finite extension, then an O-algebra homomorphism R□ ; r̄ → E′ factors through R_{r̄,λ,τ}^{□} if and only if the corresponding representation G_{K} → GL_{n}(E′) is potentially crystalline of Hodge type λ and inertial type τ.

(3) R_{r̄,λ,τ}^{□} /π is equidimensional.

Proof. All but the final point are proved in [Kis08], and the final statement is a straightforward consequence of the first (cf. the proof of Lemme 2.1 of [BM12]). □

If τ is trivial we will write R_{r̄,λ}^{□} for R_{r̄,λ,τ}^{□}. The Breuil–Mézard conjecture predicts the Hilbert–Samuel multiplicity e(R_{r̄,λ,τ}^{□} /π). In order to state the conjecture, it is first necessary to make a conjecture on the existence of an inertial local Langlands correspondence for GL_{n}. The following is a folklore conjecture.

4.1.3. Conjecture. If τ is an inertial type, then there is a finite-dimensional smooth irreducible \overline{ℚ}_{p}-representation σ(τ) of GL_{n}(O_{K}) such that if τ̃ is any Frobeniussemisimple Weil–Deligne representation of W_{K} over \overline{ℚ}_{p}, then the restriction of (rec_{p}^{−1}(τ̃)^{∨}) to GL_{n}(O_{K}) contains (an isomorphic copy of) σ(τ) as a subrepresentation if and only if τ̃|_{I_{K}} ∼ τ and N = 0 on τ̃. If p > n then σ(τ) is unique up to isomorphism.

4.1.4. Remark. A very similar conjecture is formulated in [Con10], which also proves some partial results in the case n = 3. We have formulated this conjecture only for representations with N = 0, as we will only formulate our generalisations of the Breuil–Mézard conjecture for potentially crystalline representations, in order to avoid complications in the global arguments of Section 5. We expect that a “semistable” version of the conjecture will also be valid, where one removes the conclusion that N = 0 on τ, but adds the requirement that rec_{p}^{−1}(τ̃) ⊗ | det |^{(n−1)/2} be generic.

Conjecture 4.1.3 is proved in the case n = 2 by Henniart in the appendix to [BM02]. It has been proved for any n for supercuspidal representations by Paškūnas ([Paš05]), but to the best of our knowledge it is open in general. However, the important point for us is the existence of σ(τ), rather than its uniqueness, and this is known in general. The following is a special case of Proposition 6.5.3 of [BC09] (see also [Con12]).

4.1.5. Theorem. If τ is an inertial type, then there is a finite-dimensional smooth irreducible \overline{ℚ}_{p}-representation σ(τ) of GL_{n}(O_{K}) such that if τ̃ is any pure Frobeniussemisimple Weil–Deligne representation of W_{K} over \overline{ℚ}_{p}, then the restriction of (rec_{p}^{−1}(τ̃) ⊗ | det |^{(n−1)/2}) to GL_{n}(O_{K}) contains (an isomorphic copy of) σ(τ) as a subrepresentation if and only if τ̃|_{I_{K}} ∼ τ and N = 0 on τ̃.

Enlarging E if necessary, we may assume that σ(τ) is defined over E. Since it is a finite-dimensional representation of the compact group GL_{n}(O_{K}), it contains a GL_{n}(O_{K})-stable O-lattice L_{τ}. Set L_{λ,τ} := L_{τ} ⊗_{O} L_{λ}, a finite free O-module with an action of GL_{n}(O_{K}). Then we may write

(L_{λ,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{a}F_{a}^{n_{a}},

where the sum runs over the Serre weights a ∈ (ℤ_{+}^{n})^{Hom(k,𝔽)}, and the n_{a} are nonnegative integers. Then the generalised Breuil–Mézard conjecture is the following.

4.1.6. Conjecture. There exist integers μ_{a}(r̄) depending only on r̄ and a such that e(R_{r̄,λ,τ}^{□} /π) = ∑_{a} n_{a}μ_{a}(r̄).

4.1.7. Remark. (1) Assuming the conjecture, the integers μ_{a}(r̄) may be computed recursively as follows. Let λ_{a} be a lift of a. If a is in the lowest alcove, then L_{λ} ⊗_{O} 𝔽 = F_{a}, and we have μ_{a}(r̄) = e(R_{r̄,λ_{a}}^{□}/π). In general, by Lemma 4.1.1 we see that we may compute μ_{a}(r̄) given e(R_{r̄,λ_{a}}^{□}/π) and the values μ_{b}(r̄), b < a.

(2) The reader might have expected the sum on the right hand side to only be over weights a which are predicted Serre weights for r̄. However, according to the philosophy explained in the introduction to [GK12], the predicted Serre weights a for r̄ should be precisely the a for which μ_{a}(r̄) ≠ 0.

(3) The reader might wonder why we have formulated our n-dimensional analogues of the Breuil–Mézard conjectures for liftings without fixed determinant, when the conjecture is more usually stated as in Section 3 for lifts with fixed determinant. The reason is that in the global arguments we will make in Section 5 using unitary groups, it is the lifting rings without fixed determinant that arise naturally. However, the conjectures with and without fixed determinant are actually equivalent, at least as long as p > n, as follows from Lemma 4.3.1 below.

4.2. The geometric conjecture. We may now state our geometric conjecture, which is entirely analogous to Conjecture 3.1.4.

4.2.1. Conjecture. For each Serre weight a there is a cycle C_{a} depending only on r̄ and a such that

Z(R_{r̄,λ,τ}^{□} /π) = ∑_{a} n_{a}C_{a}.

4.2.2. Remark. Again, if one assumes the conjecture then one can inductively compute the cycles C_{a} in terms of the cycles Z(R_{r̄,λ_{a}}^{□}/π).

4.3. Twisting by characters. We now establish a lemma which implies in particular that the Breuil–Mézard conjecture for liftings without a fixed determinant (as we have formulated it above) is equivalent to the analogous conjecture for liftings with a fixed determinant (at least if p > n). Note that, for a given inertial type τ and Hodge type λ, there is a character ψ_{λ,τ} : I_{K} → O^{×} such that any lift ρ of r̄ of Hodge type λ and inertial type τ necessarily has det ρ|_{I_{K}} = ψ_{λ,τ}. Let ψ : G_{K} → O^{×} be a character such that \overline{ψ} = det r̄ and ψ|_{I_{K}} = ψ_{λ,τ}. We let R_{r̄,λ,τ}^{□,ψ} denote the quotient of R_{r̄,λ,τ}^{□} corresponding to lifts with determinant ψ. In the case n = 2 and K = ℚ_{p} this is [BM02, 2.2.2.9].

4.3.1. Lemma. Suppose that p > n. Then R_{r̄,λ,τ}^{□} ≅ R_{r̄,λ,τ}^{□,ψ} [[X]].

Proof. Consider a finite extension E′/E and a point x : R_{r̄,λ,τ}^{□} → E′ with corresponding representation ρ′. Let θ = ψ^{−1} det ρ′, so that θ is an unramified character with trivial reduction. Since p > n, we see that θ has an O_{E′}-valued n-th root. From this observation is it easy to see that if ρ^{ψ} : G_{K} → GL_{n}(R_{r̄,λ,τ}^{□,ψ}) is the universal lifting of determinant ψ, and μ_{x} is the unramified character taking a Frobenius element to x, then ρ^{ψ} ⊗ μ_{1+X} ∘ det : G_{K} → GL_{n}(R_{r̄,λ,τ}^{□,ψ} [[X]]) is the universal lifting of arbitrary determinant, as required. □

5. Global patching arguments

Our goal in this section is to employ the Taylor–Wiles–Kisin patching method so as to generalize, to the extent possible, the results of Section 3 to the n-dimensional context. These arguments are essentially the natural n-dimensional generalisation of the arguments of Section 4 of [GK12]. We closely follow the approaches of [Kis09a] and [Tho12] (which in turn follows [BLGG11] and [CHT08]). In particular, in the actual implementation of the patching method we follow [Tho12] very closely, although, because our ultimate interests are local, we will sometimes make stronger global assumptions in order to simplify the arguments; these stronger assumptions can always be achieved in our applications. Before getting to the patching argument itself, we include a number of preliminary subsections in which we briefly recall the necessary background material on automorphic forms and Galois representations, referring the reader to [Tho12] for more details.

5.1. Basic set-up. We put ourselves in the setting of Section 4, so that K/ℚ_{p} is a finite extension, and r̄ : G_{K} → GL_{n}(𝔽) is a continuous representation. We also assume from now on that p > 2.

As in Section 3, we begin by globalising r̄. Since the standard global context in which to study higher-dimensional Galois representations is that of automorphic forms on unitary groups, we briefly recall the various concepts that are required to discuss Galois representations in that setting.

To begin with, we recall from [CHT08] that G_{n} denotes the group scheme over ℤ defined to be the semidirect product of GL_{n} × GL_{1} by the group {1, j}, which acts on GL_{n} × GL_{1} by

j(g, μ)j^{−1} = (μ ·^{t}g^{−1}, μ).

We have a homomorphism ν : G_{n} → GL_{1}, sending (g, μ) to μ and j to −1. We refer the reader to Section 2.1 of [CHT08] for a thorough discussion of G_{n}, and of the relationship between G_{n}-valued representations and essentially conjugate self-dual GL_{n}-valued representations.

5.1.1. Terminology. To ease notation, we adopt the following convention with regard to Galois representations with values in G_{n}(\overline{𝔽}_{p}): if F is an imaginary CM field with maximal totally real subfield F^{+}, and \overline{ρ} : G_{F^{+}} → G_{n}(\overline{𝔽}_{p}) is a continuous representation with \overline{ρ}(G_{F}) ⊂ GL_{n}(\overline{𝔽}_{p}) × GL_{1}(\overline{𝔽}_{p}), then we write \overline{ρ}|_{G_{F}} for the restriction of \overline{ρ} to G_{F}, regarded as a representation G_{F} → GL_{n}(\overline{𝔽}_{p}), and similarly for \overline{ρ}(G_{F (ζ_{p})}) and \overline{ρ}|_{G_{Fṽ}} (for places ṽ of F).

Recall that the notion of an adequate subgroup of GL_{n}(\overline{𝔽}_{p}) is defined in [Tho12]. We will not need the details of the definition; some examples of representations whose image is adequate will be constructed in Appendix A.

We now state our basic hypothesis related to the globalization of r̄. Namely, we assume that there is an imaginary CM field F with maximal totally real subfield F^{+}, together with a continuous representation \overline{ρ} : G_{F^{+}} → G_{n}(\overline{𝔽}_{p}), such that

• F/F^{+} is unramified at all finite places,

• [F^{+} : ℚ] is divisible by 4,

• every place v|p of F^{+} splits in F,

• \overline{ρ} is automorphic in the sense of Definition 5.3.1 below; in particular, ν ∘\overline{ρ} = \overline{ε}^{1−n}δ_{F/F^{+}}^{n}, where δ_{F/F^{+}} is the quadratic character corresponding to F/F^{+},

• \overline{ρ}^{−1}(GL_{n}(\overline{𝔽}_{p}) × GL_{1}(\overline{𝔽}_{p})) = G_{F},

• \overline{ρ} is unramified at primes v ∤ p,

• \overline{ρ}(G_{F (ζ_{p})}) is adequate, so that in particular \overline{ρ}|_{G_{F}} is irreducible,

• \overline{F}^{ker ad \overline{ρ}|_{G_{F}}} does not contain F (ζ_{p}), and

• for each place v|p of F^{+}, there is a place ṽ of F lying over v such that

Fṽ ≅ K and \overline{ρ}|_{G_{Fṽ}} is isomorphic to r̄. We say that such an (F and) \overline{ρ} is a suitable globalization of r̄.

5.1.2. Remark. It will not always be the case that such a representation \overline{ρ} exists, because the assumption that \overline{ρ}(G_{F (ζ_{p})}) is adequate implies that p ∤ n. On the other hand, if we assume that p ∤ n and that Conjecture A.3 holds for r̄, then by Corollary A.7 there is a suitable globalization of r̄.

We briefly explain the motivation for the various conditions that we require. The first three ensure the existence of a convenient unitary group on which to work, with the property that it is isomorphic to GL_{n} at places dividing p. The final condition ensures that \overline{ρ} can be used to study r̄. The remaining conditions are imposed in order to use patching constructions of [Tho12]; some of them are imposed in order to simplify these constructions. (When considering the last three conditions, the reader should recall our terminological convention of (5.1.1).)

5.2. Unitary groups and algebraic automorphic forms. There is a reductive algebraic group G/F^{+} with the following properties (cf. Section 6 of [Tho12]):

• G is an outer form of GL_{n}, with G_{/F} ≅ GL_{n/F}.

• If v is a finite place of F^{+}, G is quasi-split at v.

• If v is an infinite place of F^{+}, then G(F_{v}^{+}) ≅ U_{n}(ℝ). As in section 3.3 of [CHT08] we may define a model for G over O_{F^{+}}. If v is a place of F^{+} which splits as ww^{c} over F, then we have an isomorphism

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

Let E/ℚ_{p} be a finite extension with ring of integers O and residue field 𝔽, which we assume is chosen to be sufficiently large that \overline{ρ} is valued in G_{n}(𝔽). From now on we will often regard \overline{ρ}|_{G_{F}} as being valued in GL_{n}(𝔽), and we write V_{𝔽} for the underlying 𝔽-vector space of \overline{ρ}|_{G_{F}}.

Let S_{p} denote the set of places of F^{+} lying over p, and for each v ∈ S_{p} fix a place ṽ of F lying over v. Let S̃_{p} denote the set of places ṽ for v ∈ S_{p}. Write F_{p}^{+} := F ⊗_{ℚ} ℚ_{p}, O_{F_{p}^{+}} := O_{F} ⊗_{ℤ} ℤ_{p}. Let W be a finite O-module with an action of G(O_{F_{p}^{+}}), and let U ⊂ G(𝔸_{F^{+}}^{∞}) be a compact open subgroup with the property that for each u ∈ U, if u_{p} denotes the projection of u to G(F_{p}^{+}), then u_{p} ∈ G(O_{F_{p}^{+}}). We say that U is sufficiently small if for all t ∈ G(𝔸_{F^{+}}^{∞}), t^{−1}G(F^{+})t ∩ U does not contain an element of order p. Let S(U, W) denote the space of algebraic modular forms on G of level U and weight W, i.e. the space of functions

f : G(F^{+})\G(𝔸_{F^{+}}^{∞}) → W

with f(gu) = u_{p}^{−1}f(g) for all u ∈ U. If U is sufficiently small, then the functor W ↦ S(U, W) is exact.

For each v ∈ S_{p} choose an inertial type τ_{v} : I_{F_{v}} → GL_{n}(E) and a weight λ_{v} ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (Fṽ,E)}, and let L_{λ_{v},τ_{v}} be the O-representation of GL_{n}(O_{Fṽ}) defined in Section 4. Write L_{λ,τ} for the tensor product of the L_{λ_{v},τ_{v}}, regarded as a representation of G(O_{F^{+},p}) by letting G(O_{F^{+},p}) act on L_{λ_{v},τ_{v}} via ιṽ, and for any O-algebra A we write S_{λ,τ} (U, A) for S(U, L_{λ,τ} ⊗_{O} A).

5.3. Hecke algebras and Galois representations. This assumption made above that \overline{F}^{ker ad \overline{ρ}|_{G_{F}}} does not contain F (ζ_{p}) means that we can and do choose a finite place v_{1} ∉ S_{p} of F^{+} which splits over F such that v_{1} does not split completely in F (ζ_{p}), and ad \overline{ρ}|_{G_{F}} (Frob_{v_{1}}) = 1. (We make this last assumption in order to simplify the deformation theory at v_{1}; in particular these assumptions will imply that the local unrestricted lifting ring at v_{1} is smooth, and that all liftings are unramified.)

Let U = ∏_{v} U_{v} be a compact open subgroup of G(𝔸_{F^{+}}^{∞}) with U_{v} a compact open subgroup of G(F_{v}^{+}) such that:

• U_{v} = G(O_{F_{v}^{+}}) for all v which split in F other than v_{1};

• U_{v_{1}} is the preimage of the upper triangular matrices under

G(O_{F_{v_{1}}^{+}} ) → G(k_{v_{1}}) ⟶^{∼}_{ι_{w_{1}}} GL_{n}(k_{v_{1}})

where w_{1} is a place of F over v_{1};

• U_{v} is a hyperspecial maximal compact subgroup of G(F_{v}^{+}) if v is inert in

F. Then U is sufficiently small (by the choice of U_{v_{1}}). Let T = S_{p} ∪ {v_{1}}. We let 𝕋^{T,univ} be the commutative O-polynomial algebra generated by formal variables T_{w}^{(j)} for all 1 ≤ j ≤ n, w a place of F lying over a place v of F^{+} which splits in F and is not contained in T. The algebra 𝕋^{T,univ} acts on S_{λ,τ} (U, O) 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 ∉ T and ϖ_{w} a uniformiser in O_{F_{w}}. We denote by 𝕋_{λ,τ}^{T} (U, O) the image of T^{T,univ} in End_{O}(S_{λ,τ} (U, O)).

5.3.1. Definition. We say that a maximal ideal 𝔪 of 𝕋^{T,univ} with residue field of characteristic p is automorphic if for some (λ, τ) as above we have S_{λ,τ} (U, O)_{𝔪} ≠ 0. We say that a representation \overline{ρ} : G_{F^{+}} → G_{n}(\overline{𝔽}_{p}) is automorphic if there is an automorphic maximal ideal 𝔪 of 𝕋^{T,univ} such that if v ∉ T is a place of F^{+} which splits as v = ww^{c} in F, then \overline{ρ}|_{G_{F}} (Frob_{w}) has characteristic polynomial equal to the image of X^{n} + ⋯ + (−1)^{j}(Nw)^{j(j−1)/2}T_{w}^{(j)}X^{n−j} + … (−1)^{n}(Nw)^{n(n−1)/2}T_{w}^{(n)}.

In the following we will make a number of arguments that are vacuous unless S_{λ,τ} (U, O)_{𝔪} ≠ 0 for the particular (λ, τ) under consideration, but for technical reasons we do not make this assumption. Since \overline{ρ} is a suitable globalization of r̄ in the sense of Section 5, it is automorphic by assumption, and there is a maximal ideal 𝔪 of 𝕋^{T,univ} associated to \overline{ρ} as in Definition 5.3.1. Let G_{F^{+},T} := Gal(F (T)/F^{+}), where F (T) is the maximal extension of F unramified outside of T and infinity.

5.3.2. Proposition. There is a unique continuous lift

ρ_{𝔪} : G_{F^{+},T} → G_{n}(𝕋_{λ,τ}^{T} (U, O)_{𝔪})

of \overline{ρ}, which satisfies (1) ρ_{𝔪}^{−1}((GL_{n} × GL_{1})(𝕋_{λ,τ}^{T} (U, O)_{𝔪})) = G_{F}.

(2) ν ∘ ρ_{𝔪} = ε^{1−n}δ_{F/F^{+}}^{n}.

(3) ρ_{𝔪} is unramified outside T. If v ∉ T splits as ww^{c} in F then ρ_{𝔪}(Frob_{w}) has characteristic polynomial

X^{n} + ⋯ + (−1)^{j}(Nw)^{j(j−1)/2}T_{w}^{(j)}X^{n−j} + … (−1)^{n}(Nw)^{n(n−1)/2}T_{w}^{(n)}.

(4) For each place v ∈ S_{p}, and each O-algebra homomorphism

x : 𝕋_{λ,τ}^{T} (U, O)_{𝔪} → E′,

where E′/E is a finite extension, the representation x∘ρ_{𝔪}|_{G_{Fṽ}} is potentially crystalline of Hodge type λ_{v} and inertial type τ_{v}.

Proof. This may be proved in the same way as Proposition 3.4.4 of [CHT08], making use of Corollaire 5.3 of [Lab09], Theorem 1.1 of [BLGGT12], and Theorem 4.1.5. (Note that the purity assumption in Theorem 4.1.5 holds by Theorem 1.2 of [Car12], which builds on the results of [Shi11] and [CH11].) □

We note that our assumptions on v_{1} imply that S_{λ,τ} (U, O)_{𝔪} ⊗_{ℤ_{p}} ℚ_{p} is free over 𝕋_{λ,τ}^{T} (U, O)_{𝔪}[1/p] of rank n! (where we adopt the convention that this remark is true if both 𝕋_{λ,τ}^{T} (U, O)_{𝔪}[1/p] and S_{λ,τ} (U, O)_{𝔪} ⊗_{ℤ_{p}} ℚ_{p} are zero). (This just comes from the fact that the Iwahori invariants of an unramified principal series representation of GL_{n} have dimension n!.)

5.4. Deformations to G_{n}. Let S be a set of places of F^{+} which split in F, with S_{p} ⊆ S. As in [CHT08], we will write F (S) for the maximal extension of F unramified outside S and infinity, and from now on we will write G_{F^{+},S} for Gal(F (S)/F^{+}). Regard \overline{ρ} as a representation of G_{F^{+},S}. We will freely make use of the terminology (of liftings, framed liftings etc) of Section 2 of [CHT08].

Choose a place ṽ_{1} of F above v_{1}. Let T̃ denote the set of places ṽ, v ∈ T. For each v ∈ T, we let R□ ; ṽ denote the reduced and p-torsion free quotient of the universal O-lifting ring of \overline{ρ}|_{G_{Fṽ}}. For each v ∈ S_{p}, write R□,λ_{v},τ_{v} ; ṽ for R_{\overline{ρ}|_{G_{Fṽ}} ,λ_{v},τ_{v}}^{□}.

Consider (in the terminology of [CHT08]) the deformation problem

S := (F/F^{+}, T, T̃, O, \overline{ρ}, ε^{1−n}δ_{F/F^{+}}^{n}, \{R□ ; ṽ_{1}\} ∪ \{R□,λ_{v},τ_{v} ; ṽ\}_{v∈S_{p}}).

There is a corresponding universal deformation ρ_{S}^{univ} : G_{F^{+},T} → G_{n}(R_{S}^{univ}) of \overline{ρ}. In addition, there is a universal T -framed deformation ring R_{S}^{□_{T}} in the sense of Definition 1.2.1 of [CHT08], which parameterises deformations of \overline{ρ} of type S together with particular local liftings for each ṽ ∈ T̃. The lifting of Proposition 5.3.2 and the universal property of ρ_{S}^{univ} gives an O-homomorphism

R_{S}^{univ} ↠ 𝕋_{λ,τ}^{T} (U, O)_{𝔪},

which is surjective by property (3) of ρ_{𝔪}.

5.5. Patching. We now follow the proof of Theorem 6.8 of [Tho12]. We wish to consider auxiliary sets of primes in order to apply the Taylor–Wiles–Kisin patching method. Since \overline{ρ}(G_{F (ζ_{p})}) is adequate by assumption, by Proposition 4.4 of [Tho12] we see that we can (and do) choose an integer q ≥ [F^{+} : ℚ]n(n − 1)/2 and for each N ≥ 1 a tuple (Q_{N}, Q̃_{N}, \{\overline{ψ}ṽ\}_{v∈Q_{N}}) such that

• Q_{N} is a finite set of finite places of F^{+} of cardinality q which is disjoint from T and consists of places which split in F;

• Q̃_{N} consists of a single place ṽ of F above each place v of Q_{N};

• Nv ≡ 1 mod p^{N} for v ∈ Q_{N};

• for each v ∈ Q_{N}, \overline{ρ}|_{G_{Fṽ}} ≅ s̄ṽ ⊕ \overline{ψ}ṽ where \overline{ψ}ṽ is an eigenspace of Frobenius corresponding to an eigenvalue α_{v}, on which Frobenius acts semisimply. Write d_{N}^{v} = dim \overline{ψ}ṽ.

For each v ∈ Q_{N}, let R\overline{ψ}ṽ ; ṽ denote the quotient of R□ ; ṽ corresponding to lifts r : G_{Fṽ} → GL_{n}(A) which are ker(GL_{n}(A) → GL_{n}(k))-conjugate to a lift of the form s ⊕ ψ, where s is an unramified lift of s̄ṽ and ψ is a lift of \overline{ψ}ṽ for which the image of inertia is contained in the scalar matrices. We let S_{Q_{N}} denote the deformation problem

S_{Q_{N}} := (F/F^{+}, T ∪ Q_{N}, T̃ ∪ Q̃_{N}, O, \overline{ρ}, ε^{1−n}δ_{F/F^{+}}^{n},

\{R□ ; ṽ_{1}\} ∪ \{R□,λ_{v},τ_{v} ; ṽ\}_{v∈S_{p}} ∪ \{R\overline{ψ}ṽ ; ṽ\}_{v∈Q_{N}}).

We let R_{S_{Q_{N}}}^{univ} denote the corresponding universal deformation ring, and we let R_{S_{Q_{N}}}^{□_{T}} denote the corresponding universal T -framed deformation ring. We define

R^{loc} := (ˆ⊗_{v∈S_{p}}R□,λ_{v},τ_{v} ; ṽ) ˆ⊗R□ ; ṽ_{1}

where all completed tensor products are taken over O. By Proposition 4.4 of [Tho12], we may also assume that

• the ring R_{S_{Q_{N}}}^{□_{T}} can be topologically generated over R^{loc} by q−[F^{+} : ℚ]n(n−

1)/2 elements. Let U_{0}(Q_{N}) = ∏_{v} U_{0}(Q_{N})_{v} and U_{1}(Q_{N}) = ∏_{v} U_{1}(Q_{N})_{v} be the compact open subgroups of G(𝔸_{F^{+}}^{∞}) defined by (for i = 0, 1) U_{i}(Q_{N})_{v} = U_{v} if v ∉ Q_{N}, and

U_{0}(Q_{N})_{v} = ι−1 ; ṽ𝔭ṽ ; N, U_{1}(Q_{N})_{v} = ι−1 ; ṽ𝔭ṽ ; N,1 if v ∈ Q_{N}, where 𝔭ṽ ; N and 𝔭ṽ ; N,1 are the

parahoric subgroups defined in [Tho12], corresponding to the partition n = (n − dṽ ; N) + dṽ ; N. We have natural maps

𝕋_{λ,τ}^{T ∪Q_{N}} (U_{1}(Q_{N}), O) ↠ 𝕋_{λ,τ}^{T ∪Q_{N}} (U_{0}(Q_{N}), O) ↠ 𝕋_{λ,τ}^{T ∪Q_{N}} (U, O) ↪ 𝕋_{λ,τ}^{T} (U, O).

Thus 𝔪 determines maximal ideals of the first three algebras in this sequence which we denote by 𝔪_{Q_{N}} for the first two and 𝔪 for the third. Note also that 𝕋_{λ,τ}^{T ∪Q_{N}} (U, O)_{𝔪} = 𝕋_{λ,τ}^{T} (U, O)_{𝔪} by the proof of Corollary 3.4.5 of [CHT08]. For each v ∈ Q_{N} choose an element ϕṽ ∈ G_{Fṽ} lifting the geometric Frobenius element of G_{Fṽ}/I_{Fṽ} and let ϖṽ ∈ O_{Fṽ} be the uniformiser with Art_{Fṽ} ϖṽ = ϕṽ|_{F ab ; ṽ}. As in Proposition 5.9 of [Tho12] there are commuting projection operators pr_{ϖṽ} ∈ End_{O}(S_{λ,τ} (U_{i}(Q_{N}), O)_{𝔪_{Q_{N}}}). Write M = S_{λ,τ} (U, O)_{𝔪}, and for i = 0, 1 we write

M_{i,Q_{N}} = ( ∏_{v∈Q_{N}} pr_{ϖṽ} ) S_{λ,τ} (U_{i}(Q_{N}), O)_{𝔪_{Q_{N}}} .

Let 𝕋_{i,Q_{N}} denote the image of 𝕋_{λ,τ}^{T ∪Q_{N}} (U_{i}(Q_{N}), O) in End_{O}(M_{i,Q_{N}}), let Δ_{Q_{N}} = U_{0}(Q_{N})/U_{1}(Q_{N}), and let 𝔞_{Q_{N}} denote the kernel of the augmentation map

O[Δ_{Q_{N}}] → O.

For i = 0, 1 and α ∈ Fṽ of non-negative valuation, we have the Hecke operator

V_{α} = ι−1 ; ṽ ([𝔭ṽ ; N,1 (\begin{matrix}1_{n−dṽ ; N} & 0 \\ 0 & A_{α} \end{matrix}) 𝔭ṽ ; N,1]) ,

where A_{α} = diag(α, 1, …, 1). Exactly as in the proof of Theorem 6.8 of [Tho12], we have:

(1) The map

( ∏_{v∈Q_{N}} pr_{ϖṽ}) : M → M_{i,Q_{N}}

is an isomorphism.

(2) M_{1,Q_{N}} is free over O[Δ_{Q_{N}}] with

M_{1,Q_{N}} /𝔞_{Q_{N}} ⟶^{∼} M_{0,Q_{N}} .

(3) For each v ∈ Q_{N}, there is a character with open kernel Vṽ : F × ; ṽ → 𝕋_{1,Q_{N}}^{×} so that

(a) for each α ∈ Fṽ of non-negative valuation, V_{α} = Vṽ(α) on M_{1,Q_{N}};

(b) (ρ_{𝔪_{Q_{N}}} ⊗_{𝕋_{λ,τ}^{T ∪Q_{N}} (U_{i}(Q_{N}),O)_{𝔪_{Q_{N}}}} 𝕋_{1,Q_{N}})|_{W_{Fṽ}} ≅ s⊕ψ with s an unramified

lift of s̄ṽ and ψ a lift of \overline{ψ}ṽ with I_{Fṽ} acting on ψ via the scalar (Vṽ ∘

Art_{Fṽ}^{−1}).

The above shows, in particular, that the lift ρ_{𝔪_{Q_{N}}} ⊗ 𝕋_{1,Q_{N}} of \overline{ρ} is of type S_{Q_{N}} and gives rise to a surjection R_{S_{Q_{N}}}^{univ} ↠ 𝕋_{1,Q_{N}}. Thinking of Δ_{Q_{N}} as the product of the inertia subgroups in the maximal abelian p-power order quotient of ∏_{v∈Q_{N}} G_{Fṽ}, we obtain a homomorphism Δ_{Q_{N}} → (R_{S_{Q_{N}}}^{univ} )^{×} by considering ψ as above in some basis. We thus have homomorphisms

O[Δ_{Q_{N}}] → R_{S_{Q_{N}}}^{univ} → R_{S_{Q_{N}}}^{□_{T}}

and natural isomorphisms R_{S_{Q_{N}}}^{univ} /𝔞_{Q_{N}} ≅ R_{S}^{univ} and R_{S_{Q_{N}}}^{□_{T}} /𝔞_{Q_{N}} ≅ R_{S}^{□_{T}}, and the surjection R_{S_{Q_{N}}}^{univ} ↠ 𝕋_{1,Q_{N}} is a homomorphism of O[Δ_{Q_{N}}]-algebras.

Fix a filtration by 𝔽-subspaces

0 = L_{0} ⊂ L_{1} ⊂ ⋯ ⊂ L_{s} = L_{λ,τ} /πL_{λ,τ}

such that each L_{i} is G(O_{F^{+},p})-stable, and for each i = 0, 1, …, s − 1, the quotient σ_{i} := L_{i+1}/L_{i} is absolutely irreducible. This induces a filtration

0 = M^{0} ⊂ M^{1} ⊂ ⋯ ⊂ M^{s} = M/πM.

We may now patch just as in the proof of [Tho12, Thm. 6.8], keeping track of filtrations as in [Kis09a, § (2.2.9)]. (See also section 4.3 of [GK12] in the case n = 2.) More precisely, we patch together the R^{loc}[Δ_{Q_{N}}]-modules M_{1,Q_{N}} ⊗_{R_{S_{Q_{N}}}^{univ}} R_{S_{Q_{N}}}^{□_{T}} via

Lemma 6.10 of [Tho12]. Having made this construction, if we set

R_{∞} := (ˆ⊗_{v∈S_{p},O}R□ ; ṽ) ˆ⊗_{O}R□ ; ṽ_{1}[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}]],

R̄_{∞} := (ˆ⊗_{v∈S_{p},O}R□,λ_{v},τ_{v} ; ṽ) ˆ⊗_{O}R□ ; ṽ_{1}[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}]]

= R^{loc}[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}]],

and

S_{∞} := O[[z_{1}, …, z_{n^{2}#T} , y_{1}, …, y_{q}]],

for formal variables x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, y_{1}, …, y_{q} and z_{1}, …, z_{n^{2}#T}, and if we let 𝔞 denote the kernel of the augmentation map S_{∞} → O, then we see that there exists:

• An S_{∞}-module M_{∞} which is simultaneously an R̄_{∞}-module such that the image of R̄_{∞} in End(M_{∞}) is an S_{∞}-algebra.

• A filtration by R̄_{∞}-modules

0 = M_{∞}^{0} ⊂ M_{∞}^{1} ⊂ ⋯ ⊂ M_{∞}^{s} = M_{∞}/πM_{∞}

whose graded pieces are finite free S_{∞}-modules.

• A surjection of R^{loc}-algebras R̄_{∞} → R_{S}^{univ}.

• An isomorphism of R̄_{∞}-modules

M_{∞}/𝔞M_{∞} ⟶^{∼} M,

which identifies M_{∞}^{i}/𝔞M_{∞}^{i} with M^{i}, and such that the induced R̄_{∞}-module structure on M coincides with the R_{S}^{univ}-module structure on M, via the surjection R̄_{∞} → R_{S}^{univ} mentioned in the previous point.

We furthermore claim that we can make the above construction so that, for each value of i = 1, 2, … s, the (R_{∞}, S_{∞})-bimodule structure on M_{∞}^{i}/M_{∞}^{i−1} (arising from its (R̄_{∞}, S_{∞})-bimodule structure, and the surjection R_{∞} → R̄_{∞}) and the isomorphism M_{∞}^{i}/(𝔞M_{∞}^{i}, M_{∞}^{i−1}) ⟶^{∼} M^{i}/M^{i−1} depends only on (U, 𝔪 and) the isomorphism class of L_{i}/L_{i−1} as a G(O_{F^{+},p})-representation, but not on (λ, τ). For any finite collection of pairs (λ, τ) this follows by the same finiteness argument used during patching. Since the set of (λ, τ) is countable, the claim follows from a diagonalization argument.

It will be convenient in the following to refer to a tuple (a_{v})_{ṽ∈S̃_{p}} as a Serre weight, where each a_{v} is a Serre weight for GL_{n}(kṽ), where kṽ is the residue field of Fṽ. We write F_{a} = ⊗_{ṽ∈S̃_{p}}F_{a_{v}}, a representation of G(O_{F^{+},p}). For a a Serre weight, we denote by M_{∞}^{a} the R_{∞}/π-module constructed above (using the construction for all (λ, τ)) when L_{i}/L_{i−1} ⟶^{∼} F_{a}, and we set

μ′_{a}(\overline{ρ}) = (n!)^{−1}e(M_{∞}^{a}, R_{∞}/π)

and

Z′_{a}(\overline{ρ}) = (n!)^{−1}Z(M_{∞}^{a}).

(Note that a priori this is only a cycle with ℚ-coefficients rather than ℤ-coefficients, but in the situations in which our theorems apply, it will follow that it is in fact a cycle with ℤ-coefficients.)

For each v|p we write

(L_{λ_{v},τ_{v}} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ∑_{a_{v}} F_{a_{v}}^{n_{av}} ,

so that

(L_{λ,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ∑_{a} F_{a}^{n_{a}},

where n_{a} = ∏_{v} n_{a_{v}}.

The following technical, but crucial, lemma is a slight refinement (and generalisation to the n-dimensional setting) of Lemma 4.3.8 of [GK12].

5.5.1. Lemma. For each a, μ′_{a}(\overline{ρ}) is a non-negative integer. Moreover, for any fixed (λ, τ) the following conditions are equivalent:

(1) The support of M ⊗_{ℤ_{p}}ℚ_{p} meets every irreducible component of Spec R^{loc}[1/p].

(2) M_{∞} ⊗_{ℤ_{p}} ℚ_{p} is a faithfully flat R̄_{∞}[1/p]-module which is everywhere locally free of rank n!.

(3) R_{S}^{univ} is a finite O-algebra and M ⊗_{ℤ_{p}} ℚ_{p} is a faithful R_{S}^{univ}[1/p]-module.

(4) e(R̄_{∞}/π) = (n!)^{−1} ∑_{a} n_{a}e(M_{∞}^{a}, R_{∞}/π) = ∑_{a} n_{a}μ′_{a}(\overline{ρ}).

(5) Z(R̄_{∞}/π) = (n!)^{−1} ∑_{a} n_{a}Z(M_{∞}^{a}) = ∑_{a} n_{a}Z′_{a}(\overline{ρ}).

Proof. By Proposition 4.1.2 and our assumptions on the primes in T, R̄_{∞}[1/p] is formally smooth, and equidimensional of dimension q + n^{2}#T = dim S_{∞}[1/p]. Furthermore, the morphism Spec R̄_{∞}[1/p] → Spec R^{loc}[1/p], corresponding to the R^{loc}[1/p]-algebra structure on R̄_{∞}[1/p], induces a bijection on irreducible components; below we denote this bijection by Z ↦ Z′.

Since M_{∞} is finite free over S_{∞}, and the image of R̄_{∞} in End(M_{∞}) is an S_{∞}- algebra, M_{∞} ⊗_{ℤ_{p}} ℚ_{p} has depth at least q+n^{2}#T at every maximal ideal of R̄_{∞}[1/p] in its support; so by the Auslander–Buchsbaum formula, it has depth exactly q + n^{2}#T at every maximal ideal of R̄_{∞}[1/p] in its support, and M_{∞} ⊗_{ℤ_{p}} ℚ_{p} has projective dimension 0 over R̄_{∞}[1/p]. Since M_{∞} ⊗_{ℤ_{p}} ℚ_{p} is finite over S_{∞}[1/p], we see that it is finite flat over R̄_{∞}[1/p], and so its support is a union of irreducible components of Spec R̄_{∞}[1/p].

If Z ⊂ Spec R̄_{∞}[1/p] is an irreducible component in the support of M_{∞} ⊗_{ℤ_{p}} ℚ_{p}, then Z is finite over Spec S_{∞}[1/p] and of the same dimension, because M_{∞} ⊗_{ℤ_{p}} ℚ_{p} is finite free over S_{∞}[1/p]. Hence the map Z → Spec S_{∞}[1/p] is surjective. In particular, the fibre of Z over the closed point 𝔞 of Spec S_{∞}[1/p] is non-zero, and hence gives a point in the support of M_{∞}/𝔞M_{∞} = M lying in the component Z′ of Spec R^{loc}[1/p] corresponding to Z. As M has rank n! over any point of R_{S}^{univ}[1/p] in its support, it follows that M_{∞} ⊗_{ℤ_{p}} ℚ_{p} is in fact locally free of rank n! over all of Z.

Thus we see that M_{∞}⊗_{ℤ_{p}}ℚ_{p} is faithful over R̄_{∞}[1/p], or equivalently, has support consisting of the union of all the irreducible components of R̄_{∞}[1/p], if and only if the support of M meets each irreducible component of Spec R^{loc}[1/p], and that, in this case, it is furthermore locally free of rank n!. This shows that (1) and (2) are equivalent.

Using Proposition 1.3.4 of [Kis09a] we also see that each μ′_{a}(\overline{ρ}) is a non-negative integer, and that

e(R̄_{∞}/π) ≥ (n!)^{−1}e(M_{∞}/πM_{∞}, R̄_{∞}/π) = (n!)^{−1} ∑_{a} n_{a}e(M_{∞}^{a}, R_{∞}/π)

with equality if and only if M_{∞} is a faithful R̄_{∞}-module (in which case, as already observed, it is necessarily locally free of rank n!). Thus (2) and (4) are equivalent.

If M_{∞} is a faithful R̄_{∞}-module, then R̄_{∞}, being a subring of the ring of S_{∞}- endomorphisms of the finite S_{∞}-module M_{∞}, is finite over S_{∞}, and so R_{S}^{univ}, which is a quotient of R̄_{∞}/𝔞, is a finite O-module. This shows that (2) implies (3). Now assume (3), so that R_{S}^{univ} is a finite O-algebra; we see by Proposition 1.5.1 of [BLGGT13] that the image of

Spec R_{S}^{univ} → Spec R^{loc}

meets every component of Spec R^{loc}[1/p]. Hence, since M ⊗_{ℤ_{p}} ℚ_{p} is a faithful R_{S}^{univ}- module by assumption, we see that (1) holds. Thus (1)–(4) are all equivalent.

It remains to show that (5) is equivalent to the other conditions. From Lemma 2.2.8, we see that (5) implies (4). Now assume that (2) holds. Then M_{∞} is πtorsion free and generically free of rank n! over each component of Spec R̄_{∞}, so by Proposition 2.2.13 we have

Z(R̄_{∞}/π) = (n!)^{−1}Z(M_{∞}/πM_{∞}).

Identifying each of the modules M_{∞}/πM_{∞}, M_{∞}^{i}/M_{∞}^{i−1}, etc., with the corresponding sheaves on Spec R_{∞}/π that they give rise to, we see from Lemma 2.2.7 that we have

Z(M_{∞}/πM_{∞}) = ∑_{i} Z(M_{∞}^{i}/M_{∞}^{i−1})

= ∑_{a} n_{a}Z(M_{∞}^{a}).

The result follows. □

For each Serre weight a ∈ (ℤ_{+}^{n})^{Hom(k,𝔽)}, let λ_{a} be a fixed lift of a. As in Remark 4.1.7(1), we may inductively define unique integers μ_{a}(r̄) such that for any Serre weight b, if we write

(L_{λ_{b},𝟙} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{a}F_{a}^{m_{a,b}},

then

e(R_{r̄,λ_{b},𝟙}^{□}/π) = ∑_{a} m_{a,b}μ_{a}(r̄).

(In Remark 4.1.7(1) we were assuming the Breuil–Mézard conjecture, but the inductive process defining the μ_{a}(r̄) does not rely on any cases of the conjecture.) Similarly, we may inductively define unique cycles C_{a} such that

Z(R_{r̄,λ_{b},𝟙}^{□}/π) = ∑_{a} m_{a,b}C_{a}.

We now prove the main result of this section, a conditional equivalence between the geometric formulation of the Breuil–Mézard conjecture, the numerical formulation, and a global statement. For the convenience of the reader, we recall the various assumptions of this section in the statement of the theorem.

5.5.2. Theorem. Let p > 2 be prime, let K/ℚ_{p} be a finite extension, and let r̄: G_{K} → GL_{n}(𝔽) be a continuous representation, with 𝔽 a finite extension of 𝔽_{p}. Assume that there is a suitable globalization \overline{ρ} of r̄ as in Section 5.1 (for example, by Corollary A.7 this is guaranteed if p ∤ n and Conjecture A.3 holds for r̄).

Suppose that the equivalent conditions of Lemma 5.5.1 hold whenever each λ_{v} = λ_{a_{v}} for some Serre weight a_{v} and each τ_{v} = 𝟙.

Then, if λ ∈ (ℤ_{+}^{n})^{Hom_{ℚp} (K,E)} and τ is an inertial type for G_{K}, and if we write

(L_{λ,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{a}F_{a}^{n_{a}},

the following conditions are equivalent:

(1) The equivalent conditions of Lemma 5.5.1 hold when λ_{v} = λ and τ_{v} = τ for each v|p.

(2) e(R_{r̄,λ,τ}^{□} /π) = ∑_{a} n_{a}μ_{a}(r̄).

(3) Z(R_{r̄,λ,τ}^{□} /π) = ∑_{a} n_{a}C_{a}.

Proof. Consider conditions (4) and (5) of Lemma 5.5.1 in the case that each λ_{v} = λ_{b_{v}} for some Serre weight b_{v} and each τ_{v} = 𝟙. Note that in this case we have (L_{λ,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{a}F_{a}^{m_{a,b}} where m_{a,b} = ∏_{v} m_{a_{v},b_{v}}. Throughout this proof we will write \overline{ρ}_{v} for \overline{ρ}|_{G_{Fṽ}} (which is isomorphic to r̄).

As remarked above, the conditions on v_{1} ensure that R□ ; ṽ_{1} is formally smooth over O, say R□ ; ṽ_{1} ≅ O[[t_{1}, …, t_{n^{2}}]]. Our assumption that the equivalent conditions of Lemma 5.5.1 hold implies that

∑_{a} m_{a,b}μ′_{a}(\overline{ρ}) = e(R̄_{∞}/π)

= ∏_{v|p} e(R_{\overline{ρ}_{v},λ_{bv} ,𝟙}^{□}/π)

= ∏_{v|p} ∑_{a_{v}} m_{a_{v},b_{v}}μ_{a_{v}}(r̄)

= ∑_{a} m_{a,b} ∏_{v|p} μ_{a_{v}}(r̄),

where the second equality holds by Proposition 1.3.8 of [Kis09a]. Similarly

∑_{a} m_{a,b}Z′_{a}(\overline{ρ}) = Z(R̄_{∞}/π)

= ∏_{v|p} Z(R_{\overline{ρ}_{v},λ_{bv} ,𝟙}^{□}/π) × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]])

= ∏_{v|p} ∑_{a_{v}} m_{a_{v},b_{v}}C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]])

= ∑_{a} m_{a,b} ∏_{v|p} C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]]),

where we have made use of Lemma 2.2.14. If we define a partial order on the tuples (a_{v})_{v|p} of Serre weights by writing (a_{v})_{v|p} ≥ (b_{v})_{v|p} to mean that each a_{v} ≥ b_{v}, then an easy induction using Lemma 4.1.1 shows that we must in fact have that for each Serre weight a,

μ′_{a}(\overline{ρ}) = ∏_{v|p} μ_{a_{v}}(r̄)

and

Z′_{a}(\overline{ρ}) = ∏_{v|p} C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]]).

We now consider the conditions of Lemma 5.5.1 in the case that each λ_{v} = λ and each τ_{v} = τ. By definition (and Lemma 2.2.14), we have

Z(R̄_{∞}/π) = ∏_{v|p} Z(R_{\overline{ρ}_{v},λ,τ}^{□} /π) × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]]).

Using the relation above, we have

∑_{a} n_{a}Z′_{a}(\overline{ρ}) = ∑_{a} ∏_{v|p} n_{a_{v}}C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]])

= ∏_{v|p} ∑_{a_{v}} n_{a_{v}}C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]]).

Now, condition (5) of Lemma 5.5.1 states that

Z(R̄_{∞}/π) = ∑_{a} n_{a}Z′_{a}(\overline{ρ}),

which is equivalent to

∏_{v|p} Z(R_{\overline{ρ}_{v},λ,τ}^{□} /π) × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]])

= ∏_{v|p} ∑_{a_{v}} n_{a_{v}}C_{a_{v}} × Z(𝔽[[x_{1}, …, x_{q−[F^{+}:ℚ]n(n−1)/2}, t_{1}, …, t_{n^{2}}]])

and thus to

Z(R_{r̄,λ,τ}^{□} /π) = ∑_{a} n_{a}C_{a},

giving the equivalence of (1) and (3). The equivalence of (1) and (2) may be proved by a formally identical argument. □

5.5.3. Remark. We regard the assumption of Theorem 5.5.2 (that the conditions of Lemma 5.5.1 hold for all Serre weights) as a strong form of the weight part of Serre’s conjecture, saying that each set of components of the local crystalline deformation rings in low weight is realised by a global automorphic Galois representation. Under this assumption, Theorem 5.5.2 may be regarded as saying that instances of the Breuil–Mézard conjecture (and its geometric refinement) are equivalent to certain R = 𝕋 theorems.

One case in which Theorem 5.5.2 has unconditional consequences is the case that n = 2 and λ = 0, where the automorphy lifting theorems of [Kis09b] and [Gee06] can be applied. We now give such an application, where we make use of the results of [GK12] to give a particularly clean statement. Note that when n = 2, the inductive definition of the cycles C_{a} above is trivial, and we simply set C_{a} := Z(R_{r̄,λ_{a},𝟙}^{□}/π).

Suppose that K/ℚ_{p} is absolutely unramified, and that n = 2; then we say that a Serre weight a is a predicted Serre weight for r̄ if it is one of the weights predicted in [BDJ10]. We say that a is regular if we have 0 ≤ a_{σ,1} − a_{σ,2} < p − 1 for each σ ∈ Hom(k, 𝔽).

5.5.4. Theorem. Let p > 2 be prime, let K/ℚ_{p} be an absolutely unramified finite extension, and let r̄: G_{K} → GL_{2}(𝔽) be a continuous representation, with 𝔽 a finite extension of 𝔽_{p}. Suppose that every predicted Serre weight for r̄ is regular. Then, for any inertial type τ for G_{K}, if we write (L_{0,τ} ⊗_{O} 𝔽)^{ss} ⟶^{∼} ⊕_{a}F_{a}^{n_{a}}, we have Z(R_{r̄,0,τ}^{□} /π) = ∑_{a} n_{a}C_{a}.

Proof. This will follow from Theorem 5.5.2 once we have checked that all of the hypotheses hold. Since n = 2 and p > 2, we have p ∤ n and thus a suitable globalization will exist provided that Conjecture A.3 holds for r̄; but this follows from the proof of Theorem A.1.2 of [GK12] (which shows that r̄ has a potentially Barsotti–Tate lift) and Lemma 4.4.1 of op.cit. (which shows that any potentially Barsotti–Tate representation is potentially diagonalizable).

It remains to check that the equivalent conditions of Lemma 5.5.1 hold whenever λ_{v} = 0 and τ_{v} = τ (for an arbitrary τ) for all v|p, and whenever τ_{v} is trivial for all v and each λ_{v} corresponds to a Serre weight. In the first (respectively second) case, this follows from Lemma 4.4.1 (respectively Lemma 4.4.2) of [GK12], exactly as in the proof of Corollary 4.4.3 of [GK12]. □

5.5.5. Remark. In particular, Theorem 5.5.4 confirms Conjecture 1.4 of [BM12] in the case that each k_{τ} = 2, and extends Théorème 1.5 of ibid. to arbitrary types.

Appendix A. Realising local representations globally

In this appendix we realise local representations globally, using potential automorphy theorems. In the case n = 2, analogous results were proved by similar techniques in Appendix A of [GK12]. We would like to thank Robert Guralnick and Florian Herzig for the proof of the following fact.

A.1. Lemma. Suppose that p > 2 and p ∤ n. Let (GL_{n}.2)(𝔽_{p^{m}}) denote the subgroup of GL_{2n}(𝔽_{p^{m}}) generated by block diagonal matrices of the form (g,^{t}g^{−1}) and a matrix J with J(g,^{t}g^{−1})J^{−1} = (^{t}g^{−1}, g). Then for m sufficiently large, both (GL_{n}.2)(𝔽_{p^{m}}) ⊂ GL_{2n}(\overline{𝔽}_{p}) and GL_{n}(𝔽_{p^{m}}) ⊂ GL_{n}(\overline{𝔽}_{p}) are adequate.

Proof. In the case of GL_{n}(𝔽_{p^{m}}), this is immediate from Theorem 1.2 and Lemma 1.4 of [Gur12b]. For (GL_{n}.2)(𝔽_{p^{m}}), since we are assuming that (p, 2n) = 1, it follows from Theorem 1.5 of [Gur12a] (which shows the corresponding result for the underlying algebraic group GL_{n}.2) and the proof of Theorem 1.2 of [Gur12b] that the result will be true provided that Ext_{(GL_{n}.2)(𝔽_{p^{m}})}^{1}(V, V) = 0 for m sufficiently large, where V = 𝔽_{p^{m}}^{2n} is the space on which (GL_{n}.2)(𝔽_{p^{m}}) acts.

The result then follows easily from inflation-restriction, the corresponding result for GL_{n}(𝔽_{p^{m}}) (which is proved in the course of the proof of Theorem 1.2 of [Gur12b]), and the fact that the standard representation of GL_{n}(𝔽_{p^{m}}) has a different central character to its dual provided that p^{m} > 3. □

A.2. Proposition. Let K/ℚ_{p} be a finite extension, and let r̄: G_{K} → GL_{n}(\overline{𝔽}_{p}) be a continuous representation, with p ∤ n. Then there is a CM field L with maximal totally real subfield L^{+}, and a continuous irreducible representation \overline{ρ} : G_{L^{+}} → G_{n}(\overline{𝔽}_{p}), such that

• each place v|p of L^{+} splits in L;

• for each place v|p of L^{+}, L_{v}^{+} ≅ K and there is a place ṽ of L lying over v

such that \overline{ρ}|_{G_{Lṽ}} ≅ r̄;

• ν ∘ \overline{ρ} = \overline{ε}^{1−n}δ_{L/L^{+}}^{n}, where δ_{L/L^{+}} is the quadratic character corresponding

to L/L^{+};
• \overline{ρ}^{−1}(GL_{n}(\overline{𝔽}_{p}) × GL_{1}(\overline{𝔽}_{p})) = G_{L};

• \overline{ρ}(G_{L(ζ_{p})}) = GL_{n}(𝔽_{p^{m}}) for some sufficiently large m as in Lemma A.1 (so

in particular, \overline{ρ}(G_{L(ζ_{p})}) is adequate);
• \overline{L}^{ker \overline{ρ}} does not contain L(ζ_{p});
• if v ∤ p is a finite place of L^{+}, then \overline{ρ}|_{G_{L^{+}_{v}}} is unramified.

Proof. Choose m such that r̄(G_{K}) ⊂ GL_{n}(𝔽_{p^{m}}) and m is sufficiently large as in Lemma A.1, and set 𝔽 = 𝔽_{p^{m}}. We now apply Proposition 3.2 of [Cal12]; in the notation of that result, we take G = G_{n}(𝔽), and we let E be any totally real field with the property that if v|p is a place of E, then E_{v} ≅ K. We let w ∤ p be a finite place of E, and we choose a character \overline{χ}_{w} : G_{E_{w}} → 𝔽^{×} such that \overline{χ}_{w}|_{I_{Fw}} surjects onto 𝔽^{×}. We take F = E(ζ_{p}), and we take S to be the set of places of E which either divide p or else are infinite, together with the place w. We take c_{v} = j ∈ G_{n}(𝔽) for each infinite place v of S, and for each place v|p of E we let H_{v}/E_{v} correspond to \overline{K}^{ker r̄}/K, and let ϕ_{v} : Gal(H_{v}/E_{v}) → G_{n}(𝔽) correspond to (r̄, \overline{ε}^{1−n}) : G_{K} → GL_{n}(𝔽) × GL_{1}(𝔽) ⊂ G_{n}(𝔽). We let H_{w} = \overline{E}_{w}^{ker \overline{χ}_{w}}, and let ϕ_{w} = (1, \overline{χ}_{w}) : Gal(H_{w}/E_{w}) → GL_{n}(𝔽) × GL_{1}(𝔽) ⊂ G_{n}(𝔽).

Writing M^{+}/E for the field denoted K in [Cal12], we see that M^{+} is a totally real field, and that we have a surjective representation \overline{ρ} : G_{M^{+}} → G_{n}(𝔽), with the properties that

• for each place v|p of M^{+}, M_{v}^{+} ≅ K and there is a place ṽ of L lying over v

such that \overline{ρ}|_{Mṽ} ≅ r̄,
• w splits completely in M^{+}, and if w′|w then ν ∘ \overline{ρ}|_{G_{M{+ \atop w′}}} = \overline{χ},

• for each place v|∞ of M^{+}, \overline{ρ}(c_{v}) = j, where c_{v} is a complex conjugation at

v,
• ν ∘ \overline{ρ}|_{G_{M^{+}_{v}}} = \overline{ε}^{1−n} if v|p,
• \overline{M^{+}}^{ker \overline{ρ}} does not contain M^{+}(ζ_{p}).

Define M/M^{+} by G_{M} = \overline{ρ}^{−1}(GL_{n}(𝔽) × GL_{1}(𝔽)). Since \overline{ρ}(c_{v}) = j for each complex conjugation c_{v}, we see that M is an imaginary CM field. Similarly, we see that each place of M^{+} lying over p or w splits in M.

Now consider the character \overline{ϕ} := (ν ∘\overline{ρ})\overline{ε}^{n−1}δ_{M/M^{+}}^{n}. By construction, we see that \overline{ϕ}(c_{v}) = 1 for each complex conjugation c_{v}, that \overline{ϕ}|_{G_{M^{+}_{v}}} = 1 for each place v|p, and

that \overline{ϕ}|_{I_{M{+ \atop w′}}} = \overline{χ}|_{I_{M{+ \atop w′}}} for each place w′|w. Replace M^{+} by N^{+} := \overline{M^{+}}^{ker \overline{ϕ}}, M by

N := N^{+}M, and \overline{ρ} by \overline{ρ}|_{G_{N^{+}}} . Then N^{+}/M^{+} is a totally real abelian extension in which all places dividing p split completely, and it is linearly disjoint from M^{+}(ζ_{p}) over M^{+} (by considering the ramification at places above w).

To complete the proof, it suffices to choose a totally real finite Galois extension L^{+}/N^{+} such that

• each place of N^{+} lying over p splits completely in L^{+},

• \overline{ρ}|_{G_{L^{+}_{v}}} is unramified for all finite places v ∤ p of L^{+}, and

• L^{+} is linearly disjoint from \overline{N^{+}}^{ker \overline{ρ}|_{G_{N^{+}}}} (ζ_{p}) over N^{+}.

The first two of these conditions concern the splitting behaviour of a finite number of places of N^{+}, so the existence of an extension satisfying all but the third condition is guaranteed by Lemma 2.2 of [Tay03]. In order to satisfy the third condition as well, for each of the (finitely many) Galois intermediate fields \overline{N^{+}}^{ker \overline{ρ}|_{G_{N^{+}}}} (ζ_{p})/L_{i}/N^{+} with the property that Gal(L_{i}/N^{+}) is simple one chooses a finite place of N^{+} which does not split completely in L_{i}, and arranges that it splits completely in L^{+} (cf. the proof of Proposition 6.2 of [BLGHT11]). □

In order to study the Breuil–Mézard conjecture globally, we need to be able to realise local mod p representations as part of global automorphic Galois representations. In order to do this, we will apply the above construction, and then show that the representation \overline{ρ} is potentially automorphic. This essentially follows from the results of [BLGGT13]. However, the main theorems of [BLGGT13] make no attempt to control the local behaviour at finite places of the extensions over which potential automorphy is proved, while it is crucial for us that we do not change the local fields at places dividing p. Fortunately, a simple trick using restriction of scalars that we learned from Richard Taylor allows us to deduce the potential automorphy results that we need from those of [BLGGT13]. We will need the following conjecture.

A.3. Conjecture. Let K/ℚ_{p} be a finite extension, and r̄ : G_{K} → GL_{n}(\overline{𝔽}_{p}) a continuous representation. Then r̄ has a potentially diagonalizable lift r : G_{K} → GL_{n}(\overline{ℚ}_{p}) which has regular Hodge–Tate weights.

A.4. Remark. If n = 2 then Conjecture A.3 is easily verified by a Galois cohomology calculation (for example, any potentially Barsotti–Tate lift is potentially diagonalizable, and the result is then immediate in the irreducible case, and in the reducible case is a special case of Lemma 6.1.6 of [BLGG12]). We anticipate that a similar argument works more generally.

A.5. Lemma. Let p ∤ 2n be prime. Let K/ℚ_{p} be a finite extension, and let r̄ : G_{K} → GL_{n}(\overline{𝔽}_{p}) be a continuous representation. Assume that Conjecture A.3 holds for r̄. Let \overline{ρ} : G_{L^{+}} → G_{n}(\overline{𝔽}_{p}) be the representation provided by Lemma A.2. Then there is a lift ρ : G_{L^{+}} → G_{n}(\overline{ℚ}_{p}) of \overline{ρ} such that

• ν ∘ ρ = ε^{1−n}δ_{L/L^{+}}^{n},

• ρ is unramified outside of places dividing p, and

• if w|p is a place of L, then ρ|_{G_{Fw}} is potentially diagonalizable with regular Hodge–Tate weights.

Proof. In the case that p > 2(n + 1), this follows at once from Theorem 4.3.1 of [BLGGT13], together with Conjecture A.3. As explained in the first paragraph of Theorem A.4.1 of [BLGG13], the only obstacle to extending Theorem 4.3.1 of [BLGGT13] to the case p ∤ 2n is the problem of relaxing Proposition 3.3.1 of [BLGGT13] to hold under this assumption, and in particular, it is necessary to know that the induction of a representation from a degree 2 subgroup is adequate. However, in our situation, this is guaranteed by Lemma A.1. □

At this point, we would like to apply Theorem 4.5.1 of [BLGGT13] to establish that for some CM extension F/L, ρ|_{G_{F}} is (in the terminology of [BLGGT13]) automorphic. However, for our applications we need the places of L above p to split completely in F, which is not guaranteed by Theorem 4.5.1 of [BLGGT13]. Accordingly, we need to give a slight modification of the proof of loc. cit.

A.6. Proposition. Maintain the notation and assumptions of Lemma A.5. Then there is a CM extension F/L which is linearly disjoint from L^{ker \overline{ρ}}(ζ_{p}) over L, such that each place of L lying over p splits completely in F, and ρ|_{G_{F}} is automorphic in the terminology of [BLGGT13].

Proof. The proof is a straightforward modification of the arguments of [BLGGT13], specifically Theorem 3.1.2 and Proposition 3.3.1 of op. cit. We sketch the details. We will freely use the notation and terminology of [BLGGT13]. As in the proof of Proposition 3.3.1 of [BLGGT13], we may choose a character ψ : G_{L} → \overline{ℚ}_{p}^{×} such that

• ψ is crystalline at all places above p.

• R := I(ρ|_{G_{L^{+}}} ⊗ (ψ, ε^{−n}δ_{L/L^{+}}^{n+1})) : G_{L^{+}} → GSp_{2n}(\overline{ℚ}_{p}) has multiplier ε^{1−2n}, and is crystalline with distinct Hodge–Tate weights at all places lying over p.

• R̄(G_{L^{+}(ζ_{p})}) is adequate. (This follows from the choice of 𝔽 = 𝔽_{p^{m}}, which was chosen large enough that the conclusion of Lemma A.1 holds.)

We now employ a slight variant of the proof of Theorem 3.1.2 of [BLGGT13]. We let the set I of loc. cit. consist of just the single element {1}, we set n_{1} = 2n, and we put l_{1} = p, r_{1} := R̄, F = F_{0} = L^{+}, F^{(avoid)} = L(ζ_{p}). We then apply the constructions of loc. cit., choosing in particular an auxiliary positive integer N, and constructing a geometrically irreducible scheme T̃ = T_{r̄_{1}×r̄′_{1}} over Spec L^{+}(ζ_{N})^{+}.

Choose a solvable extension (L′)^{+}/L^{+} of totally real fields such that

• (L′)^{+} is linearly disjoint from \overline{L}^{ker \overline{ρ}}(ζ_{p}) over L^{+}, and

• for each place v|p of (L′)^{+}(ζ_{N})^{+}, there is a point P_{v} ∈ T̃((L′)^{+}(ζ_{N})_{v}^{+}) with

v(t_{1}(P_{v})) < 0.

We now construct a geometrically irreducible scheme T̃′ over Spec(L′)^{+}(ζ_{N})^{+} in exactly the same way as T̃ is constructed over Spec L^{+}(ζ_{N})^{+}, and we then set T′ := Res_{(L′)^{+}(ζ_{N})^{+}/L^{+}} T̃′, a geometrically irreducible scheme over Spec L^{+}.

We can then apply Proposition 3.1.1 of [BLGGT13], to find a finite Galois extension F^{+}/L^{+} of totally real fields in which the places of L^{+} over p split completely, and a point of P ∈ T′(F^{+}) which satisfies the assumptions (relating to linear disjointness from a certain field, and on the v(T_{i}(P))) on the field F′ in the proof of Theorem 3.1.2 of [BLGGT13]. (We can assume that the places of L^{+} over p split completely in F^{+} by the assumption on the places of (L′)^{+}(ζ_{N})^{+} lying over p.)

Regarding P as an F^{+}(L′)^{+}(ζ_{N})^{+}-point of T̃′, it then follows exactly as in loc. cit. that there is a continuous lift R′ : G_{F^{+}(L′)^{+}(ζ_{N})^{+}} → GSp_{2n}(\overline{ℚ}_{p}) of the restriction \overline{ρ}|_{G_{F^{+}(L′)^{+}(ζ_{N})^{+}}} such that

• R′ is ordinary, and

• R′ is automorphic. Let M be a quadratic totally imaginary extension of F^{+}(L′)^{+}(ζ_{N})^{+} such that

• all places of F^{+}(L′)^{+}(ζ_{N})^{+} above p split in M, and

• M is linearly disjoint from \overline{L}^{ker \overline{ρ}}F^{+}(L′)^{+}(ζ_{p}) over F^{+}(L′)^{+}.

It follows from Theorem 4.2.1 of [BLGGT13] that R|_{G_{M}} is automorphic, and then from Lemma 2.1.1 of [BLGGT13] that ρ|_{G_{F^{+}(L′)^{+}(ζ_{N})^{+}L}} is automorphic. Since the extension F^{+}(L′)^{+}(ζ_{N})^{+}L/F^{+}L is solvable, it follows from Lemma 1.4 of [BLGHT11] that ρ|_{G_{F^{+}L}} is automorphic. Since the places of L^{+} over p split completely in F^{+}, the claim follows upon taking F = F^{+}L. □

A.7. Corollary. Suppose that p ∤ 2n, that K/ℚ_{p} is a finite extension, and let r̄: G_{K} → GL_{n}(\overline{𝔽}_{p}) be a continuous representation for which Conjecture A.3 holds. Then there is an imaginary CM field F and a continuous irreducible representation \overline{ρ} : G_{F^{+}} → G_{n}(\overline{𝔽}_{p}) such that

• each place v|p of F^{+} splits in F, and has F_{v}^{+} ≅ K,

• for each place v|p of F^{+}, there is a place ṽ of F lying over v with \overline{ρ}|_{G_{Fṽ}} isomorphic to r̄,

• \overline{ρ} is unramified outside of p,

• \overline{ρ}^{−1}(GL_{n}(\overline{𝔽}_{p}) × GL_{1}(\overline{𝔽}_{p})) = G_{F},

• \overline{ρ}(G_{F (ζ_{p})}) is adequate,

• \overline{F}^{ker \overline{ρ}} does not contain F (ζ_{p}),

• \overline{ρ} is automorphic in the sense of Definition 5.3.1, and in particular μ ∘ \overline{ρ} = \overline{ε}^{1−n}δ_{F/F^{+}}^{n}.

Proof. This follows from Proposition A.6 and the theory of base change between GL_{n} and unitary groups, cf. Proposition 2.2.7 of [Ger09]. □

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[Con10] William Conley, Inertial types and automorphic representations with prescribed ramification, Ph.D. thesis, UCLA, 2010.

[Con12] , In preparation, 2012.

[Gee06] Toby Gee, A modularity lifting theorem for weight two Hilbert modular forms, Math. Res. Lett. 13 (2006), no. 5-6, 805–811.

[Gee11] , Automorphic lifts of prescribed types, Mathematische Annalen 350 (2011), 107–144, 10.1007/s00208-010-0545-z.

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

[GK12] Toby Gee and Mark Kisin, The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, preprint, 2012.

[Gur12a] Robert Guralnick, Adequate subgroups II, Bulletin of Mathematical Sciences 2 (2012), no. 1, 193–203.

[Gur12b] Robert M. Guralnick, Adequacy of representations of finite groups of Lie type, Appendix A to Dieulefait’s paper “Automorphy of Symm^{5}(GL(2)) and base change”, 2012.

[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.

[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.

[Hum06] James E. Humphreys, Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, vol. 326, Cambridge University Press, Cambridge, 2006.

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

[Kis08] Mark Kisin, Potentially semi-stable deformation rings, J. Amer. Math. Soc. 21 (2008), no. 2, 513–546.

[Kis09a] , The Fontaine-Mazur conjecture for GL_{2}, J. Amer. Math. Soc. 22 (2009), no. 3, 641–690.

[Kis09b] , Moduli of finite flat group schemes, and modularity, Ann. of Math. (2) 170 (2009), no. 3, 1085–1180.

[Kis10] , The structure of potentially semi-stable deformation rings, Proceedings of the International Congress of Mathematicians, vol. II (2010), 294–311.

[Lab09] Jean-Pierre Labesse, Changement de base CM et séries discrètes, preprint, 2009.

[Mat89] Hideyuki Matsumura, Commutative ring theory, second ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid.

[Paš05] Vytautas Paškūnas, Unicity of types for supercuspidal representations of GL_{N}, Proc.

London Math. Soc. (3) 91 (2005), no. 3, 623–654.

[Paš12] , On the Breuil–Mézard conjecture, 2012.

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

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

[Sno09] Andrew Snowden, On two dimensional weight two odd representations of totally real fields, 2009, preprint.

[Tay03] Richard Taylor, On icosahedral Artin representations. II, Amer. J. Math. 125 (2003), no. 3, 549–566.

[Tho12] Jack Thorne, On the automorphy of l-adic Galois representations with small residual image, Journal of the Institute of Mathematics of Jussieu (2012), 1–66.

Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston, IL 60208

   E-mail address, Matthew Emerton: emerton@math.northwestern.edu
   E-mail address, Toby Gee: gee@math.northwestern.edu