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 groupof Qp . We also state a conjectural generalisation to n-dimensional representations of the absolute Galois group of an arbitrary finite extension of Qp , andgive 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 ageometric point of view. Let us explain what we mean by this. First recall thatthe Breuil–Mézard conjecture posits a formula (in terms of certain representationtheoretic data) for the Hilbert–Samuel multiplicity of the characteristic p fibre ofcertain local Zp -algebras, namely those whose characteristic zero fibres parameterize two-dimensional potentially semistable liftings of some fixed continuous twodimensional Galois representation r̄ : GQp → GL2 (F), where F is a finite field of
characteristic p (the so-called potentially semistable deformation rings constructedin [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, eachwith multiplicities µi (i = 1, . . . , n), then the multiplicity of the entire ring will bePi µi . Our goal is to both explain and refine the Breuil–Mézard conjecture in theseterms, by identifying the irreducible components of the various rings involved, inrepresentation-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 GQp ,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 amore 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 GK , for any finite extension K of Qp and any positive integer n. Finally,
The first author was supported in part by NSF grant DMS-1003339, and the second author by
NSF grant DMS-0841491.
1

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MATTHEW EMERTON AND TOBY GEE

in Section 5, we explain how the arguments of Section 3 may be extended to thecase of n-dimensional representations, so as to prove an equivalence between the
Breuil–Mézard conjecture (extended to the n-dimensional case) and its geometricrefinement, under the assumption of a suitable R = T-type theorem, together witha 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].
In an appendix we establish a technical result that allows us to realize representations of local Galois groups as restrictions of automorphic representations ofglobal Galois groups.
1.1. Summary of our results in the case of two-dimensional representations of GQp . We now explain in more detail our geometric refinement of theoriginal Breuil–Mézard conjecture. To this end, we fix a finite extension E of Qp ,with ring of integers O, residue field F, and uniformizer π. As above, we also fix acontinuous representation r̄ : GQp → GL2 (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 potentiallysemistable 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 wecaution 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 ofSpec R (r̄)/π. More precisely, the conjecture as originally stated in [BM02] gives aconjectural formula for the Hilbert–Samuel multiplicity of this local scheme. Thisconjecture 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 ofthis 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
OZ,p . Suppose now that Z is equidimensional of dimension d. If a is a point of
X of dimension d (i.e. whose closure {a} is of dimension d), then the stalk OZ,a iseither zero (if a 6∈ Z) or an Artinian local ring (if a ∈ Z, i.e. if a is a generic pointof Z, or, equivalently, if {a} is an irreducible component of Z), and the multiplicity e(Z, a) is simply the length of OZ,a as a module over itself, a quantity which
can be interpreted geometrically as the multiplicity with which the component {a}appears in
PZ. 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

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

THE BREUIL–MÉZARD CONJECTURE

3

it as the cycle associated to Z. If p is any point of X , then one has the formulaX (1.1.1)e(Z, p) =e(Z, a)e {a} , pa(where again the sum ranges over points a of dimension d), allowing us to computethe 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] thateach of these closed subschemes Z is equidimensional of dimension 5, and so wemay 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 Serreweight of r̄.
1.1.2. Definition. If σ is a Serre weight of r̄, write σ = σm,n := detm ⊗Symn F2 for
integers m, n such that 0 ≤ n ≤ p−1, and define Cm,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
Cm,n consists of a single component if it has a single irreducible component, whichis 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 σm,n is a Serre weight of r̄, then Cm,n consistsof a single component, which has multiplicity one at the closed point ofSpec R (r̄)/π.
(2) If r̄ is reducible and σm,n is a Serre weight of r̄ such that n < p − 2, orn = p − 2 and r̄ is a non-split extension of distinct characters, then Cm,nconsists of a single component, which has multiplicity one at the closed pointof Spec R (r̄)/π.
(3) If r̄ is reducible
 m+1 andσm,n is a Serre weight of r̄ with n = p − 1, so thatω∗, then if ∗ is peu ramifiée and r̄ itself is a twist ofr̄|Ip ∼
0ωm
an extension of the trivial character by the mod p cyclotomic character,then Cm,n is a sum of two components, each having multiplicity one at theclosed point of Spec R (r̄)/π. Otherwise Cm,n is a single component, having
multiplicity one at the closed point of Spec R (r̄)/π.
(4) If σm,n is a Serre weight of r̄ with n = p − 2 and r̄ is split and pdistinguished, then Cm,n is a sum of two components, each having multiplicity one at the closed point of Spec R (r̄)/π.
(5) If σm,n is a Serre weight of r̄ with n = p − 2 and r̄ has scalar semisimplifiction, then Cm,n consists of a single component.
(6) If σm,n and σm0 ,n0 are distinct Serre weights of r̄, then Cm,n and Cm0 ,n0have disjoint support, except if m ≡ m0 (mod p − 1), n = 0 and n0 = p − 1(possibly after interchanging σm,n and σm0 ,n0 ), in which case Cm,n is equal
to a component of Cm0 ,n0 .

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MATTHEW EMERTON AND TOBY GEE

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 GL2 (Zp ) over E associated to τ via Henniart’s inertial local Langlands correspondence, write σ(a, b, τ ) :=ss
(deta ⊗Symb E 2 ) ⊗E σ(τ ), and let σ(a, b, τ ) denote the semi-simplification of the
reduction mod π of (any) GL2 (Zp )-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 6∼

ωχ
0


∗
χ

for any character χ. Fix integers m, n with n ≥ 0 and an inertial type τ , and foreach Serre weight σm,n of r̄, let am,n denote the multiplicity with which σm,n appearsss
as a constituent of σ(a, b, τ ) . Then we have the following equality of cycles:
 X
am,n Cm,n .
Z Spec R (a, b, τ, r̄)/π =m,n1.1.6. 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 theformula (1.1.1), and using the explicit description of the cycles Cm,n provided by
Proposition 1.1.3. (Note that while Proposition 1.1.3 as stated assumes that theBreuil–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 thanthat 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 andMézard 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 σm,n 7→ Cm,n . (Note thatwhen r̄ is generic in the sense of [BM12], each cycle Cm,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.

THE BREUIL–MÉZARD CONJECTURE

51.1.10. 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 GK . The key additional ingredients which areavailable 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 inturn relies on the strategy outlined in [Kis10]; in particular, our implementation of
the patching argument for unitary groups is simply the natural adaptation of thearguments of [GK12] to higher rank unitary groups. Theorem 5.5.4 below gives ageometric refinement of some of the results of [GK12].
1.2. Acknowledgments. The debt that the arguments of this paper owe to thework of Mark Kisin will be obvious to the reader; in particular, several of our mainarguments 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 FlorianHerzig for helpful conversations about the representation theory of GLn (Fq ), andTom Barnet-Lamb, David Geraghty, Robert Guralnick, Florian Herzig, and Jack
Shotton for helpful conversations about the material in Appendix A. We would liketo 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 GK denote its absolute Galois group. If K is furthermore a finite extension of Qp for some p, then we write IK for the inertia subgroupof GK . If F is a number field and v is a finite place of F then we let Frobv denotea geometric Frobenius element of GFv .
We let ε denote the p-adic cyclotomic character, and let ε = ω the mod p
cyclotomic character. We denote by ωe the Teichmüller lift of ω. We let ω2 denotea choice of a fundamental character of IQp of niveau 2.
If K is a p-adic field, if ρ is a continuous de Rham representation of GK overQp , and if τ : K ,→ Qp , then we will write HTτ (ρ) for the multiset of Hodge–Tate
numbers of ρ with respect to τ . By definition, if W is a de Rham representationof GK over Qp and if τ : K ,→ Qp then the multiset HTτ (W ) contains i with
b
GK
K(i))
. Thus for example HT (ε) = {−1}. We will
multiplicity dim (W ⊗
Qp

τ,K

τ

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 theexisting literature.
Let K be a finite extension of Qp , and let rec denote the local Langlands
correspondence from isomorphism classes of irreducible smooth representationsof GLn (K) over C to isomorphism classes of n-dimensional Frobenius semisimple Weil–Deligne representations of WK defined in [HT01]. Fix an isomorphismı : Qp → C. We define the local Langlands correspondence recp over Qp by√abı ◦ recp = rec ◦ ı. This depends only on ı−1 ( p). We let ArtK : K × → WK
be the isomorphism provided by local class field theory, which we normalise so thatuniformisers correspond to geometric Frobenius elements. We will write 1 for the

6

MATTHEW EMERTON AND TOBY GEE

trivial n-dimensional representation of some group, the precise group and choice ofn 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 originatedin [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 andcycles that we outlined in the introduction.
2.1. Hilbert–Samuel multiplicities. Recall that if A is a Noetherian local ringwith maximal ideal m of dimension d, and M is a finite A-module, then there isA
polynomial PM(X) of degree at most d (the Hilbert–Samuel polynomial of M ),A(n) is equal
uniquely determined by the requirement that for n  0, the value PM
n+1
to the length of M/m
M as an A-module.
2.1.1. Definition. The Hilbert–Samuel multiplicity e(M, A) is defined to be d!
A(X). We write e(A) for e(A, A).
times the coefficient of X d in PM
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 denotethe scheme-theoretic support of M (i.e. Z is the closed subscheme of X cut
out by the annihilator ideal I ⊂ OX of M). For any point x ∈ X , we writee(M, x) to denote the Hilbert–Samuel multiplicity e(Mx , OZ,x ).
(2) If Z is a closed subscheme of X , then we write e(Z, x) := e(OZ , x) for allx ∈ X. (Note that Z coincides with the scheme-theoretic support of OZ ,and so by definition this is equal to the multiplicity e(OZ,x ) of the local
ring OZ,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 Mx = 0, then e(M, x) = 0.
(2) If x is a generic point of the scheme-theoretic support Z of M (so that
OZ,x is an Artinian ring), then e(M, x) is simply the length of Mx as an
OZ,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 {x} is of dimension d.
(2) A d-dimensional cycle on X is a formal finite Z-linear combination of pointsof X of dimension d.
(3) We write X ≥ 0 if X is in fact a Z≥0 -linear combination of points of X ofdimension d, and we write X ≥ Y if X − Y ≥ 0.
X
2.2.4. Definition. If Z =nx x is a d-dimensional cycle on X , then for any
dim(x)=dpoint y ∈ X , we define the multiplicity e(Z, y) via the formulaX
e(Z, y) :=nx e({x}, y).
dim(x)=dTHE BREUIL–MÉZARD CONJECTURE

7(Here {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 Zd (M) associated to M as follows:Xe(M, x)x,
Zd (M) :=dim(x)=d

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 pointof 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 inthe sum defining Zd (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, thenby Remark 2.2.2, the multiplicity e(M, x) is simply the length of Mx as
an OZ,x -module.)(2) If the support of M is finite-dimensional of dimension d, then we write
simply Z(M) := Zd (M).
(3) If Z is a closed subset of X , then we write Zd (Z) := Zd (OZ ), 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 Zd (Z) = 0.
2.2.7. Lemma. If 0 → M0 → M → M00 → 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 M0 and M00 are of dimension ≤ d), thenZd (M) = Zd (M0 ) + Zd (M00 ).
Proof. Let Z (resp. Z 0 and Z 00 ) denote the support of M (resp. of M0 and M00 ), sothat Z 0 , Z 00 ⊆ Z. As already noted in the statement of Definition 2.2.5, if x ∈ X isof dimension d, then e(M, x) (resp. e(M0 , x), resp. e(M00 , x)) is simply the lengthof Mx as an OZ,x -module (resp. the length of M0x as an OZ 0 ,x -module, resp. thelength of M00x as an OZ 00 ,x -module). Since each of OZ 0 ,x and OZ 00 ,x is a quotientof OZ,x , the claimed additivity of cycles follows from the additivity of lengths inexact 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.

8

MATTHEW EMERTON AND TOBY GEE

2.2.9. Lemma. Let X be a Noetherian scheme of finite dimension d, and let f ∈OX (X ) be regular (i.e. a non-zero divisor in each stalk of OX ). If M is an f -torsion
free coherent sheaf on X which is supported in dimension d − 1, then M/f M 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/f M) is of dimension ≤ d − 2, as claimed.

2.2.10. Lemma. Let X be a Noetherian integral scheme of finite dimension d, andlet f ∈ OX (X ) be non-zero. If M is an f -torsion free coherent
 sheaf on X whichis generically free of rank one, then Zd−1 (M/f M) = Z V (f ) .
Proof. Let x be the generic point of X , and ix : Spec κ(x) → X the canonical map.
By assumption Mx is one-dimensional over κ(x), and so we may find a morphismof quasi-coherent sheaves M → (ix )∗ κ(x) whose kernel M0 is torsion. The image
M00 of this morphism is a coherent subsheaf of (ix )∗ κ(x), and hence is containedin an invertible sheaf L. Let L0 denote the cokernel of the inclusion M00 ,→ L.
Consider first the exact sequence
0 → M0 → M → M00 → 0.
All the terms in this sequence are f -torsion free, and so
0 → M0 /f M0 → M/f M → M00 /f M00 → 0
is again exact. Lemma 2.2.9 shows that Zd−1 (M0 /f M0 ) = 0, and Lemma 2.2.7
then implies that
(2.2.11)Zd−1 (M/f M) = Zd−1 (M00 /f M00 ).

Next consider the exact sequence 0 → L0 [f ] → L0 → L0 → L0 /f L0 → 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)Zd−1 (L0 [f ]) = Zd−1 (L0 /f L0 ).

Finally, consider the exact sequence 0 → M00 → L → L0 → 0. The first two
terms in this sequence are f -torsion free, and so it induces an exact sequence
0 → L0 [f ] → M00 /f M00 → L/f L → L0 /f L0 → 0. Again applying Lemma 2.2.7,
together with (2.2.12), we find that Zd−1 (M00 /f M00 ) = Zd−1 (L/f L). Combining
this with (2.2.11), together
 with the fact that L is an invertible sheaf, we find that
Zd−1 (M/f M) = Z V (f ) , as required.

2.2.13. Proposition. Let X be a Noetherian scheme of finite dimension d, andf ∈ OX (X ) be regular (i.e. a non-zero divisor in each stalk
P of OX ). If M is an
f -torsion free coherent sheaf on X , and if Zd (M) =dim(x)=d nx x, where, as
indicated, x runs over the d-dimensional points of X , then the support of M/f M
has dimension ≤ d − 1, andX

Zd−1 (M/f M) =nx Zd−1 {x} ∩ V (f ) .
dim(x)=dP
Proof. We argue by induction on the quantity n :=dim x=d nx , with the casewhen 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 nx > 0. We may then find a
non-zero surjection Mx → κ(x) (since by definition nx is the length of Mx ), which
induces a non-zero map of quasi-coherent sheaves M → (ix )∗ κ(x) (where ix is theTHE BREUIL–MÉZARD CONJECTURE

9canonical map Spec κ(x) → X ). Let M0 denote the kernel of this map, and M00 its
image, so that the sequence 0 → M0 → M → M00 → 0 is exact.
The sheaf M0 is f -torsion free (since it is a subsheaf of M) while the sheaf
M00 is f -torsion free, supported on {x}, and generically free of rank one over thiscomponent (being a non-zero coherent subsheaf of (ix )∗ κ(x)). Thus we obtain ashort exact sequence
0 → M0 /f M0 → M/f M → M00 /f M00 → 0,and Lemmas 2.2.7 and 2.2.10 show that
Zd−1 (M/f M) = Zd−1 (M0 /f M0 ) + Zd−1 (M00 /f M00 )= Zd−1 (M0 /f M0 ) + Zd−1 {x} ∩ V (f ) .
The proposition follows by induction.



We close this section with a result about the product of cycles. In fact, we willneed 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 ofthis 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
b k B/q is a quotient of A⊗b k B of dimension d + e. Henceof dimension e, then A/p⊗b k B/q is a closed subscheme of Spec A⊗
b k B of dimension d + e, and weSpec A/p⊗write
b k B/q),Z(Spec A/p) ×k Z(Spec B/q) := Z(Spec A/p⊗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 onb k B.
Spec A⊗
If M and N are finitely generated A- and B-modules respectively, giving rise tocoherent sheaves M and N on Spec A and Spec B respectively, then the completedb k B, which web k N gives rise to a coherent sheaf on Spec A⊗
tensor product M ⊗b .
denote by MN
2.2.14. Lemma. In the context of the preceding discussion, if the support of Mband N are of dimensions d and e respectively, then the support of MN
is ofdimension d + e, andb ) = Zd (M) ×k Ze (N ).
Zd+e (MNb ,Proof. This is standard; we sketch the proof. Restricting to the support of MN
we may assume that M (resp. N ) is a faithful A-module (resp. B-module), sothat A has dimension d and B has dimension e. As in (for example) the proofb k B are precisely
of [BLGHT11, Lemma 3.3(5)], the distinct minimal primes of A⊗b k B)+q(A⊗b k B), where p is a minimal prime of A and q is a minimal prime
the p(A⊗of B. We are thus reduced to checking that for each such p, q, we haveb k N, (A/p)⊗b k (B/q)) = e(M, (A/p))e(N, (B/q)),e(M ⊗which follows from Lech’s lemma [Mat89, Thm. 14.12] exactly as in the proof of
Proposition 1.3.8 of [Kis09a].
10

MATTHEW EMERTON AND TOBY GEE

3. The geometric Breuil–Mézard Conjecture for two-dimensional
representations of GQp
In this section we explain the Breuil–Mézard conjecture in its original setting,that of two-dimensional representations of GQp , and state our geometric version.
We then recall some of the details of the proof of the original formulation of theconjecture from [Kis09a], and show that the proof may be extended to prove thegeometric 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]. Fixa prime p > 2 and a finite extension E of Qp (our coefficient field), with ring ofintegers O, residue field F, and uniformiser π. We assume that #F > 5, so that
PSL2 (F) is a simple group.
Let r̄ : GQp → GL2 (F) be a continuous representation, and let τ : IQp →GL2 (E) be an inertial type, i.e. a representation with open kernel which extends
to WQp . Fix integers a, b with b ≥ 0 and a de Rham character ψ : GQp → O×,ψ(a, b, τ, r̄) be the framedsuch that ψε = det r̄. We let R,ψ (a, b, τ, r̄) and Rcr
deformation O-algebras which are universal for framed deformations of r̄ whichhave 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 σcr (τ ) denote the finite-dimensional irreducible E-representations of GL2 (Zp ) corresponding to τ via Henniart’s inertial
local Langlands correspondence, we set σ(a, b, τ ) = (deta ⊗Symb E 2 ) ⊗E σ(τ ) andσ cr (a, b, τ ) = (deta ⊗Symb E 2 ) ⊗E σ cr (τ ), and we let La,b,τ (respectively Lcr
a,b,τ ) bea GL2 (Zp )-stable O-lattice in σ(a, b, τ ) (respectively σ cr (a, b, τ )). Write σm,n for
the representation detm ⊗Symn F2 of GL2 (Fp ), 0 ≤ m ≤ p − 2, 0 ≤ n ≤ p − 1, sothat we may write∼am,n(La,b,τ ⊗O F)ss −→ ⊕m,n σm,n
,
and
acr
m,n
ss ∼
(Lcr
a,b,τ ⊗O F) −→ ⊕m,n σm,n ,for some integers am,n , acrm,n .
We can now state the Breuil–Mézard conjecture [BM02].
3.1.1. Conjecture. There are integers µm,n (r̄) depending only on m, n, and r̄,such that for any a, b, τ with det τ = ε1−2a−b ψ|IQp ,Xe(R,ψ (a, b, τ, r̄)/π) =am,n µm,n (r̄),
m,n

and
,ψ
e(Rcr
(a, b, τ, r̄)/π) =X

acrm,n µm,n (r̄).

m,n3.1.2. 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. Notethat if the conjecture is true for all a, b, τ , then it is easy to see that µm,n (r̄) = 0

THE BREUIL–MÉZARD CONJECTURE

11

unless det r̄|IQp = ω 2m+n+1 , and that if this holds then we must have µm,n (r̄) =
,ψ
e
e(Rcr
(m,
e n, 1, r̄)/π), where m
e is chosen so that ψ|IQp = ε2m+n+1. So all the valuesµm,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 asωχ ∗sumptions that r̄ 6∼
for any χ, and that if r̄ has scalar semisimplification
0 χthen 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 Cm,ndepending only on m, n, and r̄, such that for any a, b, τ with det τ = ε−2a−b−1 ψ|IQp ,X
Z(R,ψ (a, b, τ, r̄)/π) =am,n Cm,n ,
m,n

and
,ψ
Z(Rcr
(a, b, τ, r̄)/π) =X

acrm,n Cm,n .

m,n3.1.5. 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 Cm,n =
,ψ
Z(Rcr
(m,
e n, 1, r̄)/π).
The following is our main result towards Conjecture 3.1.4.
ωχ ∗3.1.6. Theorem. If r̄ 6∼
for any χ, then Conjecture 3.1.4 holds for r̄.
0 χ
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 :GQp → GL2 (F). We begin by realising this representation as the restriction ofa global Galois representation, by a similar (but simpler) argument to that ofAppendix A of [GK12].
3.2.1. Proposition. There is a totally real field F and a continuous irreducible
representation ρ : GF → GL2 (F) such that(1) p splits completely in F ;(2) ρ is totally odd;(3) ρ(GF ) = GL2 (F);(4) if v - p is a place of F then ρ|GFv is unramified;(5) if v|p is a place of F then ρ|GFv ∼= r̄;(6) [F : Q] is even;(7) ρ is modular.
Proof. By Proposition 3.2 of [Cal12], we may find F and ρ satisfying all but the lasttwo conditions. By Proposition 8.2.1 of [Sno09], there is a finite Galois extension
F 0 /F in which all places above p split completely such that ρ|GF 0 is modular. If
ker ρ

we make a further quadratic extension (linearly disjoint from F
over F , and inwhich the primes above p split completely), if necessary, to ensure that [F 0 : Q] is
even, and replace F with F 0 , then the result follows.
12

MATTHEW EMERTON AND TOBY GEE

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 thatconditions (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 OD of D, for each finite placeQ∼v we choose an isomorphism (OD )v −→ M2 (OFv ), and we define
U = v Uv ⊂Q×
(D ⊗F A∞to be the following compact open subgroup of v (OD )×v : if v|p orF )v∈/ S then Uv = (OD )×
,
while
if
v
∈
S
but
v
p,
then
U
consistsofthematricesvv
which are upper-triangular and unipotent modulo $v , where $v is a uniformiserof Fv . The subgroup U is sufficiently small in the sense of [Kis09a, §2.1.1].
We write Σp for the set of places v|p of F (since Σ is empty, this is consistent with
[Kis09a]). For each v|p, we let Rv,ψ (ρ|GFv ) denote the universal framed deformation
O-algebra for ρ|GFv with determinant ψε, and we defineb v|p,O Rv,ψ (ρ|GF )[[x1 , . . . , xg ]],R∞ := ⊗v
where x1 , . . . , xg 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 av , bv with bv ≥ 0, together with aninertial type τv , and let ∗ be either cr or nothing (the same choice of ∗ being madefor all v|p). We assume that det τv = ε1−2av −bv ψ|IQp for each v. We write2
b v|p,O R∗,ψ (av , bv , τv , ρ|GF )[[x1 , . . . , xg ]].
R̄∞ := ⊗v
If we write Wσ := ⊗v|p L∗av ,bv ,τv , then Wσ is a finite free O-module with anQaction of v|p GL2 (OFv ), and the quotient Wσ /πWσ is then a representation ofQ
v|p GL2 (OFv ) over F. Fix a Jordan–Hölder filtration
0 = L0 ⊂ · · · ⊂ Ls = Wσ /πWσ
Q

of Wσ /πWσ by v|p GL2 (OFv )-subrepresentations. If we write σi := Li /Li−1 ,then σi ∼= ⊗v|p detmv,i ⊗Symnv,i F2 for some uniquely determined integers mv,i ∈{0, . . . , p − 2} and nv,i ∈ {0, . . . , p − 1}.
The patching construction of [Kis09a, § 2.2.5] then gives, for some integer denotedby h + j in [Kis09a], and formal variables y1 , . . . , yh+j ,• an (R̄∞ , O[[y1 , . . . , yh+j ]])-bimodule M∞ , finite free over O[[y1 , . . . , yh+j ]],and• a filtration of M∞ /πM∞ by (R̄∞ /π, F[[y1 , . . . , yh+j ]])-bimodules, say
01s
0 = M∞⊂ M∞⊂ · · · ⊂ M∞
= M∞ /πM∞ ,ii−1such that each M∞/M∞
is a finite free F[[y1 , . . . , yh+j ]]-module, and such
that
i
i−1
• the isomorphism class of M∞/M∞
as an (R∞ , F[[yQ1 , . . . , yh+j ]])-bimodule
depends only on the isomorphism class of σi as a v|p GL2 (OFv )-module,
and not on the choices of av , bv and τv . (It is immediate from the finitenessargument used in patching that this can be achieved for any finite collection
2The ring R̄∞ depends upon the particular choices of av , bv , τv and ∗, although (following[Kis09a]) we do not indicate this in the notation.

THE BREUIL–MÉZARD CONJECTURE

13

of tuples (av , bv , τv ), and since there are only countably many tuples, adiagonalization argument allows us to assume independence for all tuples.)For any Serre weight σm,n , we now write ψcr for a crystalline character liftingω −1 det r̄, and we define (in the notation of Remark 3.1.2)
,ψcrµm,n (r̄) := e(Rcr(m,
e n, 1, r̄)/π)

and
,ψcr
Cm,n := Z(Rcr(m,e n, 1, r̄)/π).

Proof of Theorem 3.1.6. Fix a, b, τ and ∗, and set av = a, bv = b, τv = τ for eachv|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 GL2 (Qp ), 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 isused.)
i
,ψcr
b v|p Rcr
Let R̄∞:= ⊗(me v,i , nv,i , 1, ρ|GFv )/π [[x1 , . . . , xg ]], regarded as a quoi
i−1tient of R∞ . Since the isomorphism class of M∞/M∞
as an R∞ -module depends
only on the isomorphism class of σi , one sees as in the proof of Lemma 2.2.13 ofii−1
i[Kis09a] that the action of R∞ on M∞/M∞factors through R̄∞. (In brief: it
,ψ
is enough to check for each place v0 |p that the action of Rv0 /π factors through
,ψcr
Rcr
(m
e v0 ,i , nv0 ,i , 1, ρ|GFv )/π. In order to do this, one applies the construction
0
with av = me v0 ,i , bv = nv0 ,i , τv0 an appropriate scalar type, and the other τv
chosen so that σmv,i ,nv,i is a Jordan–Hölder factor of Lav ,bv ,τv ⊗O F for v 6= v0 .)ii−1iFurthermore, the R∞ -module M∞/M∞
is supported on all of Spec R̄∞, by theresults of Section 4.6 of [Gee11] (cf. the final paragraph of the proof of [Kis09a,i
Prop. 2.2.15]). Finally, we note that R̄∞is generically reduced (see the proofof [Kis09a, Prop. 2.2.15]).
Lemma 2.2.14 shows thatY
Z(R∗,ψ (a, b, τ, r̄)/π) × Z(Spec F[[x1 , . . . , xg ]]),
(3.2.2)Z(R̄∞ /π) =
v|p
i
i−1
while, identifying each of the modules M∞ /πM∞ , M∞/M∞
, etc., with the corresponding sheaves on Spec R∞ /π that they give rise to, we compute thatXii−1
(3.2.3) Z(M∞ /πM∞ ) =
Z(M∞
/M∞
)
i

≥

X
i

i
Z(Spec R̄∞
/π) =

Y X


a∗m,n Cm,n × Z(Spec F[[x1 , . . . , xg ]]),

v|p m,n3We remark that there is a typo in this part of the statement of [Kis09a, Lem. 2.2.11]; R∞ iswritten there, rather than R̄∞ .

14

MATTHEW EMERTON AND TOBY GEE

the first equality following by Lemma 2.2.7, the inequality following from the fact,ii−1
noted above, that the support of M∞/M∞
coincides with the generically reducediclosed subscheme Spec R̄∞ of Spec R∞ /π, and the second equality following fromanother 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Y
(3.2.2)
Z(R∗,ψ (a, b, τ, r̄)/π) × Z(Spec F[[x1 , . . . , xg ]]) = Z(R̄∞ /π)v|p(3.2.4)=

Z(M∞ /πM∞ , R̄∞ /π)(3.2.3) Y X≥

a∗m,n Cm,n × Z(Spec F[[x1 , . . . , xg ]]).

v|p m,nHowever, if we apply Lemma 2.2.8 (with y the closed point of Spec R̄∞ /π) to pass tothe 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 thatYYX
a∗m,n Cm,n ,Z(R∗,ψ (a, b, τ, r̄)/π) =v|p m,nv|pand thus that
Z(R∗,ψ (a, b, τ, r̄)/π) =X

a∗m,n Cm,n ,m,n

as required.






ωχ ∗
for some χ, it is presumably possible to0 χ
use the above arguments to show that our geometric Breuil–Mézard conjecture isequivalent 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, andin any case we prove a similar statement in far greater generality in Section 5 (seeTheorem 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 La,b,τ have the same parity, so it is possible tocircumvent parity problems by twisting, but the details are a little unpleasant towrite out.)
3.2.5. Remark. Even if r̄ ∼3.3. Analysis of components. For a given r̄, there may be several different Serreweights σm,n for which µm,n (r̄) 6= 0. We now examine the different cycles Cm,n .
We write W (r̄) for the set of σm,n for which µm,n (r̄) 6= 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):

THE BREUIL–MÉZARD CONJECTURE

15• If r̄ is irreducible, sayr̄|IQp ∼ ω m ⊗

 n+1
ω2
0

0



p(n+1)

ω2,then W (r̄) = {σm,n , σm+n,p−1−n }.
• If r̄ is reducible but indecomposable, with
 m+1

ω
∗
r̄|IQp ∼0ωma peu ramifiée extension, then W (r̄) = {σm,0 , σm,p−1 }.
• If r̄ is reducible and decomposable, with m+n+1ω0r̄|IQp ∼0ωmsuch that n ≤ p − 3, then– if 0 < n < p − 3, we have W (r̄) = {σm,n , σm+n+1,p−3−n };– if n = p − 3 and p > 3, we have W (r̄) = {σm,p−3 , σm−1,0 , σm−1,p−1 };– if p = 3 and n = 0, we have W (r̄) = {σm,0 , σm,2 , σm+1,0 , σm+1,2 }.
The relationships between
 Cm,n are as follows. Note in particular that the cyclesωχ ∗for any χ, then the assumption in the following
by Theorem 3.1.6, if r̄ 6∼
0 χ
Proposition is automatic.
3.3.1. Proposition. Assume that Conjecture 3.1.4 holds for r̄. Then the cycles
Cm,n have disjoint support, except for the cycles Cm,0 and Cm,p−1 when both are
non-zero. In this latter case there is an equality Cm,0 = Cm,p−1 , except if r̄ is a
twist of a (possibly split) peu ramifiée extension of the trivial character by the modp cyclotomic character, in which case Cm,p−1 is the sum of two irreducible cycles,
one of which is Cm,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 ourgeometric formulation of the Breuil–Mezard conjecture. We begin by noting that it
follows from Corollary 1.7.14 of [Kis09a]that the cycles
 Cm,n are irreducible, except
ω m+1∗
in the cases that n = p − 1 and r̄|IQp ∼is peu ramifiée, or n = p − 2
0
ωm


µ1 0
and r̄ ∼ ω m ⊗for distinct unramified characters µ1 and µ2 . Thus to0 µ2
prove the claimed disjointness of cycles, it is enough to prove that the cycles are
not equal.
We first consider the cases where Cm,0 and Cm,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 σp−1 , so we conclude that
C0,p−1 = Z(R,ψcr (e0, 0, 1, r̄)/π),,ψcr eC0,0 = Z(Rcr(0, 0, 1, r̄)/π).

16

MATTHEW EMERTON AND TOBY GEE,ψcr eBy definition, Rcr(0, 0, 1, r̄) is a quotient of R,ψ (0, 0, 1, r̄), and as they have,ψcr ethe same dimension, we see that Spec Rcr(0, 0, 1, r̄) is a union of irreducible
components of Spec R,ψcr (e0, 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 extensionsof 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,ψcr (e0, 0, 1, r̄)/π) has two distinct components. In
the case that the extension is non-split, the framed deformation ring is formallysmooth 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,ψcr (e0, 0, 1, r̄)/π) consists of two
distinct components, which works equally well in the case when r̄ is split. Namely, atwo-dimensional crystalline representation with Hodge–Tate weights 0 and 1 with
reducible reduction is necessarily an extension of characters, whose restrictions
to Ip are trivial and cyclotomic respectively, and it is uniquely determined by
its associated pseudo-representation. The cycle C0,0 is thus directly seen to beirreducible, 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 C0,0 , since its image in the associatedpseudo-deformation space is simply the closed point. Thus we have shown that
C0,p−1 is the sum of two distinct irreducible components.
 n+1

ω2
0
with 0 < n < p − 1.
Consider next the case that r̄|IQp ∼ ω m ⊗p(n+1)
0
ω2
We need to show that Cm,n 6= Cm+n,p−1−n . Consider the inertial type ω
e m+n ⊕ ω
em,
2m+n
ω
. Then by for example Lemmas 3.1.1 andand choose ψ so that ψ|IQp = εe4.2.4 of [CDT99], the semisimplification of the reduction modulo p of σ(eω m+n ⊕ ω
em)
m+n
m
e
⊕ω
e by Henniart’s inertial local(the representation of GL2 (Zp ) associated to ωLanglands correspondence, which is the inflation to GL2 (Zp ) of a principal series
representation of GL2 (Fp )) has Jordan–Hölder factors σm,n and σm+n,p−1−n , sothat we haveZ(R,ψ (0, 1, ω
e m+n ⊕ ωe m , r̄)/π) = Cm,n + Cm+n,p−1−n .
By Theorem 6.22 of [Sav05] (noting that the framed deformation ring is formallysmooth over the deformation ring, since r̄ is irreducible), we have R,ψ (0, 1, ω
e m+n ⊕m∼ω
e , r̄)/π = F[[U, V, W, X, Y ]]/(XY ), so Cm,n 6= Cm+n,p−1−n , as required.
It remains to treat the cases where r̄ ∼ χ1 ⊕χ2 is a direct sum of two distinct char,ψcr
acters. In this case, one knows that all of the relevant lifting rings Rcr(m,e 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 correspondrespectively to liftings which contain a submodule lifting χ1 or which contain a

THE BREUIL–MÉZARD CONJECTURE

17

submodule lifting χ2 , and since χ1 6= χ2 , it is easy to see that the cycles are distinct.
Proof of Proposition 1.1.3. The assertions about the number of components of Cm,nfollow from Proposition 3.3.1 above together with Corollary 1.7.14 of [Kis09a], andthe disjointness of the supports of the Cm,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 ofSpec R,ψ (a, b, τ, r̄)/π and the Serre weights for r̄ which appear with positive multiplicity in the mod p reduction of (deta ⊗Symb E 2 ) ⊗E σ(τ ). The definition of thecorrespondence is given in [BM12, Thm. 1.5]: namely, if a is a generic point ofSpec R,ψ (a, b, τ, r̄)/π, then the associated Serre weight is the GL2 (Zp )-socle of acertain GL2 (Qp )-representation obtained from the universal Galois representation
into GL2 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
,ψcr
(m,
e n, 1, r̄)/π), forcomponent {a} of Spec R,ψ (a, b, τ, r̄)/π is equal to Spec Rcr
some Serre weight σm,n of r̄. Since the p-adic local Langlands correspondence
is functorial, we find that the GL2 (Zp )-socle of the GL2 (Qp )-representation attached to a, thought of as a generic point of Spec R,ψ (a, b, τ, r̄)/π, is the same as
the GL2 (Zp )-socle of the GL2 (Qp )-representation attached to a, thought of as the,ψcrgeneric point of Spec Rcr(m,e n, 1, r̄)/π). But in this latter case, the GL2 (Zp )socle in question is equal to σm,n , as follows from the fact that the correspondenceof [BM12] is compatible with the original conjecture of [BM02]. This shows thatthe correspondence of [BM12] is precisely the correspondence Cm,n 7→ σm,n .
4. The Breuil–Mézard Conjecture for GLn
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 fromthat 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/Qp be a finite extension with ring of integers OK and residue field k, let E/Qp
be a finite extension with ring of integers O, uniformiser π and residue field F, andlet r̄ : GK → GLn (F) be a continuous representation. Assume that E is sufficiently
large, and in particular that E contains the images of all embeddings K ,→ Qp .
Let Zn+ denote the set of tuples (λ1 , . . . , λn ) of integers with λ1 ≥ λ2 ≥ · · · ≥ λn .
For any λ ∈ Zn+ , view λ as a dominant character of the algebraic group GLn/O in
the usual way, and let Mλ0 be the algebraic OK -representation of GLn given bynMλ0 := IndGL
Bn (w0 λ)/OK

where Bn is the Borel subgroup of upper-triangular matrices of GLn , and w0 is thelongest element of the Weyl group (see [Jan03] for more details of these notions,and note that Mλ0 has highest weight λ). Write Mλ for the OK -representation ofGLn (OK ) obtained by evaluating Mλ0 on OK . For any λ ∈ (Zn+ )HomQp (K,E) we write
Lλ for the O-representation of GLn (OK ) defined by⊗τ :K,→E Mλτ ⊗OK ,τ O.

18

MATTHEW EMERTON AND TOBY GEEGiven any a ∈ Zn+ with p − 1 ≥ ai − ai+1 for all 1 ≤ i ≤ n − 1, we define
the k-representation Pa of GLn (k) to be the representation obtained by evaluatn
ing IndGL
Bn (w0 a)k on k, and let Na be the irreducible sub-k-representation of Pa
generated by the highest weight vector (that this is indeed irreducible follows forexample from II.2.8(1) of [Jan03] and the appendix to [Her09]). We say that anelement a ∈ (Zn+ )Hom(k,F) is a Serre weight if• for each σ ∈ Hom(k, F) and each 1 ≤ i ≤ n − 1 we havep − 1 ≥ aσ,i − aσ,i+1 ,• and for each σ we have 0 ≤ aσ,n ≤ p − 1, and not all aσ,n = p − 1.
If a ∈ (Zn+ )Hom(k,F) is a Serre weight then we define an irreducible F-representation
Fa of GLn (k) by
Fa := ⊗τ ∈Hom(k,F) Naτ ⊗k,τ F.
The representations Fa are absolutely irreducible and pairwise non-isomorphic, andevery irreducible F-representation of GLn (k) is of the form Fa for some a (see forexample the appendix to [Her09]). We say that an element λ ∈ (Zn+ )HomQp (K,E) is alift of a Serre weight a ∈ (Zn+ )Hom(k,F) if for each σ ∈ Hom(k, F) there is an elementτ ∈ HomQp (K, E) lifting σ such that λτ = aσ , and for all other τ 0 ∈ HomQp (K, E)
lifting σ we have λτ 0 = 0. We have a partial ordering ≤ on Serre weights, whereb ≤ 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 Lλ ⊗O F has socle Fa , and every otherJordan–Hölder factor of Lλ ⊗O F is of the form Fb with b < a.
Proof. This follows from sections 5.8 and 5.9 of [Hum06] (noting that the orderings≤ and ≤Q coincide for GLn ).
Let τ : IK → GLn (E) be a representation with open kernel which extends
to WK , and take λ ∈ (Zn+ )HomQp (K,E) . Let r̄ : GK → GLn (F) be a continuous
representation. If E 0 /E is a finite extension, we say that a potentially crystalline
representation ρ : GK → GLn (E 0 ) has Hodge type λ if for each τ : K ,→ E,
HTτ (ρ) = {λτ,1 + n − 1, λτ,2 + n − 2, . . . , λτ,n }.
We say that ρ has inertial type τ if the restriction to IK 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 Rr̄,λ,τ
of the universal
lifting O-algebra Rr̄ for r̄ with the following properties.
(1) Rr̄,λ,τ
is reduced and p-torsion free, and Rr̄,λ,τ
[1/p] is formally smooth andequidimensional of dimension n2 + [K : Qp ]n(n − 1)/2.
(2) If E 0 /E is a finite extension, then an O-algebra homomorphism Rr̄ → E 0factors through Rr̄,λ,τ
if and only if the corresponding representation GK →0
GLn (E ) is potentially crystalline of Hodge type λ and inertial type τ .
(3) Rr̄,λ,τ/π is equidimensional.

Proof. All but the final point are proved in [Kis08], and the final statement is astraightforward consequence of the first (cf. the proof of Lemme 2.1 of [BM12]). If τ is trivial we will write Rr̄,λfor Rr̄,λ,τ. The Breuil–Mézard conjecture predictsthe Hilbert–Samuel multiplicity e(Rr̄,λ,τ /π). In order to state the conjecture, it isTHE BREUIL–MÉZARD CONJECTURE

19

first necessary to make a conjecture on the existence of an inertial local Langlands
correspondence for GLn . The following is a folklore conjecture.
4.1.3. Conjecture. If τ is an inertial type, then there is a finite-dimensional smooth
irreducible Qp -representation σ(τ ) of GLn (OK ) such that if τe is any Frobeniussemisimple Weil–Deligne representation of WK over Qp , then the restriction of(rec−1τ )∨ ) to GLn (OK ) contains (an isomorphic copy of) σ(τ ) as a subrepresenp (etation if and only if τe|IK ∼ τ and N = 0 on τe. If p > n then σ(τ ) is unique up toisomorphism.
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 generalisationsof 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 theconclusion that N = 0 on τ , but adds the requirement that rec−1τ ) ⊗ | det |(n−1)/2p (ebe 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, theimportant point for us is the existence of σ(τ ), rather than its uniqueness, and thisis 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 Qp -representation σ(τ ) of GLn (OK ) such that if τe is any pure Frobeniussemisimple Weil–Deligne representation of WK over Qp , then the restriction of(rec−1τ ) ⊗ | det |(n−1)/2 ) to GLn (OK ) contains (an isomorphic copy of) σ(τ ) as ap (esubrepresentation if and only if τe|IK ∼ τ and N = 0 on τe.
Enlarging E if necessary, we may assume that σ(τ ) is defined over E. Since it isa finite-dimensional representation of the compact group GLn (OK ), it contains a
GLn (OK )-stable O-lattice Lτ . Set Lλ,τ := Lτ ⊗O Lλ , a finite free O-module withan action of GLn (OK ). Then we may write∼(Lλ,τ ⊗O F)ss −→ ⊕a Fana ,where the sum runs over the Serre weights a ∈ (Zn+ )Hom(k,F) , and the na are nonnegative integers. Then the generalised Breuil–Mézard conjecture is the following.
4.1.6. Conjecture.
There exist integers µa (r̄) depending only on r̄ and a such thatP
e(Rr̄,λ,τ/π) = a na µa (r̄).
4.1.7. Remark.
(1) Assuming the conjecture, the integers µa (r̄) may be computed recursively as follows. Let λa be a lift of a. If a is in the lowest

alcove, then Lλ ⊗O F = Fa , and we have µa (r̄) = e(Rr̄,λ/π). In general,aby Lemma 4.1.1 we see that we may compute µa (r̄) given e(Rr̄,λ/π) andathe values µb (r̄), b < a.
(2) The reader might have expected the sum on the right hand side to only beover weights a which are predicted Serre weights for r̄. However, according

20

MATTHEW EMERTON AND TOBY GEE

to the philosophy explained in the introduction to [GK12], the predicted
Serre weights a for r̄ should be precisely the a for which µa (r̄) 6= 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 Ca depending only onr̄ and a such thatX
na Ca .
Z(Rr̄,λ,τ/π) =a4.2.2. Remark. Again, if one assumes the conjecture then one can inductivelycompute the cycles Ca in terms of the cycles Z(Rr̄,λ/π).
a4.3. 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 ψλ,τ : IK → O× such that any lift
ρ of r̄ of Hodge type λ and inertial type τ necessarily has det ρ|IK = ψλ,τ . Let,ψψ : GK → O× be a character such that ψ = det r̄ and ψ|IK = ψλ,τ . We let Rr̄,λ,τ
denote the quotient of Rr̄,λ,τ
corresponding to lifts with determinant ψ. In the casen = 2 and K = Qp this is [BM02, 2.2.2.9].
,ψ
∼
4.3.1. Lemma. Suppose that p > n. Then Rr̄,λ,τ= Rr̄,λ,τ [[X]].
→ E 0 with correProof. Consider a finite extension E 0 /E and a point x : Rr̄,λ,τ0−10sponding representation ρ . Let θ = ψ det ρ , so that θ is an unramified characterwith trivial reduction. Since p > n, we see that θ has an OE 0 -valued n-th root.
,ψ) is the univerFrom this observation is it easy to see that if ρψ : GK → GLn (Rr̄,λ,τ
sal lifting of determinant ψ, and µx is the unramified character taking a Frobenius
,ψelement to x, then ρψ ⊗ µ1+X ◦ det : GK → GLn (Rr̄,λ,τ
[[X]]) is the universal liftingof 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 generalisationof 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

THE BREUIL–MÉZARD CONJECTURE

21

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 thenecessary 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/Qp isa finite extension, and r̄ : GK → GLn (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 Gn denotes the group scheme overZ defined to be the semidirect product of GLn × GL1 by the group {1, j}, whichacts on GLn × GL1 byj(g, µ)j −1 = (µ · t g −1 , µ).
We have a homomorphism ν : Gn → GL1 , sending (g, µ) to µ and j to −1. We refer
the reader to Section 2.1 of [CHT08] for a thorough discussion of Gn , and of therelationship between Gn -valued representations and essentially conjugate self-dual
GLn -valued representations.
5.1.1. Terminology. To ease notation, we adopt the following convention withregard to Galois representations with values in Gn (Fp ): if F is an imaginary CM
field with maximal totally real subfield F + , and ρ : GF + → Gn (Fp ) is a continuous
representation with ρ(GF ) ⊂ GLn (Fp ) × GL1 (Fp ), then we write ρ|GF for the
restriction of ρ to GF , regarded as a representation GF → GLn (Fp ), and similarlyfor ρ(GF (ζp ) ) and ρ|GFve (for places ve of F ).
Recall that the notion of an adequate subgroup of GLn (Fp ) 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 ρ : GF + → Gn (Fp ), 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 ,• ρ is automorphic in the sense of Definition 5.3.1 below; in particular, ν ◦ρ =n+
ε1−n δF/F,+ , where δF/F + is the quadratic character corresponding to F/F−1• ρ (GLn (Fp ) × GL1 (Fp )) = GF ,• ρ is unramified at primes v - p,
• ρ(GF (ζp ) ) is adequate, so that in particular ρ|GF is irreducible,
ker ad ρ|GF• F
does not contain F (ζp ), and• for each place v|p of F + , there is a place ve of F lying over v such thatFve ∼= K and ρ|GFve is isomorphic to r̄.

We say that such an (F and) ρ is a suitable globalization of r̄.
5.1.2. Remark. It will not always be the case that such a representation ρ exists,
because the assumption that ρ(GF (ζp ) ) is adequate implies that p - n. On the other

22

MATTHEW EMERTON AND TOBY GEE

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, withthe property that it is isomorphic to GLn at places dividing p. The final condition
ensures that ρ can be used to study r̄. The remaining conditions are imposed inorder to use patching constructions of [Tho12]; some of them are imposed in order
to simplify these constructions. (When considering the last three conditions, thereader 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 GLn , with G/F ∼= GLn/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(Fv+ ) ∼= Un (R).
As in section 3.3 of [CHT08] we may define a model for G over OF + . If v is a placeof F + which splits as wwc over F , then we have an isomorphism∼ιw : G(OFv+ ) −→ GLn (OFw ).
Let E/Qp be a finite extension with ring of integers O and residue field F, which
we assume is chosen to be sufficiently large that ρ is valued in Gn (F). From now
on we will often regard ρ|GF as being valued in GLn (F), and we write VF for theunderlying F-vector space of ρ|GF .
Let Sp denote the set of places of F + lying over p, and for each v ∈ Sp fix aplace ve of F lying over v. Let S̃p denote the set of places ṽ for v ∈ Sp . Write
Fp+ := F ⊗Q Qp , OFp+ := OF ⊗Z Zp . Let W be a finite O-module with an action ofG(OFp+ ), and let U ⊂ G(A∞F + ) be a compact open subgroup with the property thatfor each u ∈ U , if up denotes the projection of u to G(Fp+ ), then up ∈ G(OFp+ ).
−1
G(F + )t ∩ U does not
We say that U is sufficiently small if for all t ∈ G(A∞F + ), tcontain 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−1
p f (g) for all u ∈ U . If U is sufficiently small, then the functor
W 7→ S(U, W ) is exact.
For each v ∈ Sp choose an inertial type τv : IFv → GLn (E) and a weightλv ∈ (Zn+ )HomQp (Fve ,E) , and let Lλv ,τv be the O-representation of GLn (OFve ) defined in Section 4. Write Lλ,τ for the tensor product of the Lλv ,τv , regarded as arepresentation of G(OF + ,p ) by letting G(OF + ,p ) act on Lλv ,τv via ιve, and for any
O-algebra A we write Sλ,τ (U, A) for S(U, Lλ,τ ⊗O A).
5.3. Hecke algebras and Galois representations. This assumption made above
ker ad ρ|GF
does not contain F (ζp ) means that we can and do choose a finitethat F
place v1 ∈/ Sp of F + which splits over F such that v1 does not split completely inF (ζp ), and ad ρ|GF (Frobv1 ) = 1. (We make this last assumption in order to simplify
the deformation theory at v1 ; in particular these assumptions will imply that the
local unrestricted lifting ring at v1 is smooth, and that all liftings are unramified.)THE BREUIL–MÉZARD CONJECTURE

23

Q
Let U = v Uv be a compact open subgroup of G(A∞F + ) with Uv a compact open
subgroup of G(Fv+ ) such that:• Uv = G(OFv+ ) for all v which split in F other than v1 ;• Uv1 is the preimage of the upper triangular matrices under∼G(OFv+ ) → G(kv1 ) −→ GLn (kv1 )1

ιw1

where w1 is a place of F over v1 ;• Uv is a hyperspecial maximal compact subgroup of G(Fv+ ) if v is inert inF.
Then U is sufficiently small (by the choice of Uv1 ). Let T = Sp ∪ {v1 }. We letTT,univ be the commutative O-polynomial algebra generated by formal variables(j)
Tw 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 TT,univ acts on Sλ,τ (U, O) via the Hecke
operators




$w 1j
0
Tw(j) := ι−1
(O
)
(O
)
GL
GL
n
Fw
n
Fw
w
0
1n−j
for w 6∈ T and $w a uniformiser in OFw . We denote by TTλ,τ (U, O) the image of
T T,univ in EndO (Sλ,τ (U, O)).
5.3.1. Definition. We say that a maximal ideal m of TT,univ with residue field of
characteristic p is automorphic if for some (λ, τ ) as above we have Sλ,τ (U, O)m 6= 0.
We say that a representation ρ : GF + → Gn (Fp ) is automorphic if there is an
automorphic maximal ideal m of TT,univ such that if v ∈/ T is a place of F + whichcsplits as v = ww in F , then ρ|GF (Frobw ) has characteristic polynomial equal to(j)(n)
the image of X n + · · · + (−1)j (Nw)j(j−1)/2 Tw X n−j + . . . (−1)n (Nw)n(n−1)/2 Tw .
In the following we will make a number of arguments that are vacuous unlessSλ,τ (U, O)m 6= 0 for the particular (λ, τ ) under consideration, but for technicalreasons we do not make this assumption. Since ρ is a suitable globalization of r̄ inthe sense of Section 5, it is automorphic by assumption, and there is a maximal ideal
m of TT,univ associated to ρ as in Definition 5.3.1. Let GF + ,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
ρm : GF + ,T → Gn (TTλ,τ (U, O)m )of ρ, which satisfies
T(1) ρ−1m ((GLn × GL1 )(Tλ,τ (U, O)m )) = GF .
n(2) ν ◦ ρm = ε1−n δF/F+.
(3) ρm is unramified outside T . If v ∈/ T splits as wwc in F then ρm (Frobw )has characteristic polynomial
X n + · · · + (−1)j (Nw)j(j−1)/2 Tw(j) X n−j + . . . (−1)n (Nw)n(n−1)/2 Tw(n) .
(4) For each place v ∈ Sp , and each O-algebra homomorphism
x : TTλ,τ (U, O)m → E 0 ,where E 0 /E is a finite extension, the representation x◦ρm |GFve is potentiallycrystalline of Hodge type λv and inertial type τv .

24

MATTHEW EMERTON AND TOBY GEE

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 v1 imply that Sλ,τ (U, O)m ⊗Zp Qp is free overTTλ,τ (U, O)m [1/p] of rank n! (where we adopt the convention that this remark is true
if both TTλ,τ (U, O)m [1/p] and Sλ,τ (U, O)m ⊗Zp Qp are zero). (This just comes fromthe fact that the Iwahori invariants of an unramified principal series representationof GLn have dimension n!.)5.4. Deformations to Gn . Let S be a set of places of F + which split in F , withSp ⊆ 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 GF + ,S for Gal(F (S)/F + ).
Regard ρ as a representation of GF + ,S . We will freely make use of the terminology(of liftings, framed liftings etc) of Section 2 of [CHT08].
Choose a place ve1 of F above v1 . Let Te denote the set of places ve, v ∈ T .
For each v ∈ T , we let Rve denote the reduced and p-torsion free quotient of theuniversal O-lifting ring of ρ|GFve . For each v ∈ Sp , write Rve,λv ,τv for Rρ|,λv ,τv .
G
Consider (in the terminology of [CHT08]) the deformation problem

Fve,λv ,τvn
S := (F/F + , T, Te, O, ρ, ε1−n δF/F}v∈Sp ).
+ , {Rve1 } ∪ {Rve

There is a corresponding universal deformation ρuniv: GF + ,T → Gn (RSuniv ) of ρ. In
S
addition, there is a universal T -framed deformation ring RST in the sense of Definition 1.2.1 of [CHT08], which parameterises deformations of ρ of type S together
with particular local liftings for each ve ∈ Te. The lifting of Proposition 5.3.2 andgives an O-homomorphism
the universal property of ρuniv
S
RSuniv  TTλ,τ (U, O)m ,which is surjective by property (3) of ρm .
5.5. Patching. We now follow the proof of Theorem 6.8 of [Tho12]. We wish toconsider auxiliary sets of primes in order to apply the Taylor–Wiles–Kisin patching
method. Since ρ(GF (ζp ) ) 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 eache N , {ψ ve}v∈Q ) such thatN ≥ 1 a tuple (QN , Q
N• QN 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 ;e N consists of a single place ve of F above each place v of QN ;• Q
• Nv ≡ 1 mod pN for v ∈ QN ;• for each v ∈ QN , ρ|GFve ∼= s̄ve ⊕ ψ ve where ψ ve is an eigenspace of Frobenius
corresponding to an eigenvalue αv , on which Frobenius acts semisimply.
Write dvN = dim ψ ve.
ψFor each v ∈ QN , let Rve ve denote the quotient of Rve corresponding to lifts r :GFve → GLn (A) which are ker(GLn (A) → GLn (k))-conjugate to a lift of the forms ⊕ ψ, where s is an unramified lift of s̄ve and ψ is a lift of ψ ve for which the image

THE BREUIL–MÉZARD CONJECTURE

25

of inertia is contained in the scalar matrices. We let SQN denote the deformation
probleme N , O, ρ, ε1−n δ n + ,SQN := (F/F + , T ∪ QN , Te ∪ Q
F/Fψ{Rve1 } ∪ {Rve,λv ,τv }v∈Sp ∪ {Rve ve }v∈QN ).
We let RSuniv
denote the corresponding universal deformation ring, and we let RSQT
QN
N
denote the corresponding universal T -framed deformation ring. We defineb v∈Sp Rve,λv ,τv ⊗Rb ve
Rloc := ⊗1
where all completed tensor products are taken over O. By Proposition 4.4 of[Tho12], we may also assume that• the ring RSQT can be topologically generated over Rloc by q−[F + : Q]n(n−N1)/2 elements.
Q
Q
Let U0 (QN ) = v U0 (QN )v and U1 (QN ) = v U1 (QN )v be the compact open
/ QN , andsubgroups of G(A∞F + ) defined by (for i = 0, 1) Ui (QN )v = Uv if v ∈
−1 v
v
e
v
e
e
v
e
and
p
if
v
∈
Q
,
where
p
,
U
(Q
)
=
ι
p
U0 (QN )v = ι−1
p
N
1
N v
N,1 are the
N
N,1
N
v
e
v
e
parahoric subgroups defined in [Tho12], corresponding to the partition n = (n −eedvN
) + dvN. We have natural maps
∪QN
∪QN
∪QN
TTλ,τ
(U1 (QN ), O)  TTλ,τ(U0 (QN ), O)  TTλ,τ
(U, O) ,→ TTλ,τ (U, O).

Thus m determines maximal ideals of the first three algebras in this sequencewhich we denote by mQN for the first two and m for the third. Note also that
∪QN
TTλ,τ
(U, O)m = TTλ,τ (U, O)m by the proof of Corollary 3.4.5 of [CHT08]. For
each v ∈ QN choose an element φve ∈ GFve lifting the geometric Frobenius elementof GFve /IFve and let $ve ∈ OFve be the uniformiser with ArtFve $ve = φve|Fveab . As
in Proposition 5.9 of [Tho12] there are commuting projection operators pr$ve ∈EndO (Sλ,τ (Ui (QN ), O)mQN ). Write M = Sλ,τ (U, O)m , and for i = 0, 1 we writeY
Mi,QN = 
pr$ve  Sλ,τ (Ui (QN ), O)mQN .
v∈QN∪QN
Let Ti,QN denote the image of TTλ,τ
(Ui (QN ), O) in EndO (Mi,QN ), let ∆QN =U0 (QN )/U1 (QN ), and let aQN denote the kernel of the augmentation map

O[∆QN ] → O.
For i = 0, 1 and α ∈ Fve of non-negative valuation, we have the Hecke operator




1n−dvNe
0
−1
v
e
v
e
Vα = ιvepN,1p,0
Aα N,1
where Aα = diag(α, 1, . . . , 1). Exactly as in the proof of Theorem 6.8 of [Tho12],
we have:(1) The mapY
pr$ve : M → Mi,QNv∈QN

is an isomorphism.

26

MATTHEW EMERTON AND TOBY GEE(2) M1,QN is free over O[∆QN ] with∼

M1,QN /aQN −→ M0,QN .
(3) For each v ∈ QN , there is a character with open kernel Vve : Fve× → T×
1,QN
so that(a) for each α ∈ Fve of non-negative valuation, Vα = Vve(α) on M1,QN ;(b) (ρmQN ⊗TT ∪QN (U (Q ),O)
T1,QN )|WFve ∼= s ⊕ ψ with s an unramified
λ,τ

i

N

mQ

N

lift of s̄ve and ψ a lift of ψ ve with IFve acting on ψ via the scalar (Vve ◦Art−1
Fve ).
The above shows, in particular, that the lift ρmQN ⊗ T1,QN of ρ is of type SQN andgives rise to a surjection RSuniv T1,QN . Thinking of ∆QN as the product of the
QN
Qinertia subgroups in the maximal abelian p-power order quotient of v∈QN GFve ,)× by considering ψ as above in some
we obtain a homomorphism ∆QN → (RSuniv
QN
basis. We thus have homomorphisms
O[∆QN ] → RSuniv→ RSQT
Q
N

N/aQN ∼and natural isomorphisms RSuniv= RST , and the= RSuniv and RSQT /aQN ∼QN
Nsurjection RSuniv T1,QN is a homomorphism of O[∆QN ]-algebras.
Q
N

Fix a filtration by F-subspaces
0 = L0 ⊂ L1 ⊂ · · · ⊂ Ls = Lλ,τ /πLλ,τsuch that each Li is G(OF + ,p )-stable, and for each i = 0, 1, . . . , s − 1, the quotient
σi := Li+1 /Li 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 offiltrations as in [Kis09a, § (2.2.9)]. (See also section 4.3 of [GK12] in the case n = 2.)
More precisely, we patch together the Rloc [∆QN ]-modules M1,QN ⊗RSuniv RSQT via
QN

N

Lemma 6.10 of [Tho12]. Having made this construction, if we set

b v∈Sp ,O Rve ⊗b O Rve [[x1 , . . . , xq−[F + :Q]n(n−1)/2 ]],R∞ := ⊗1b v∈Sp ,O Rve,λv ,τv ⊗b O Rve [[x1 , . . . , xq−[F + :Q]n(n−1)/2 ]]R̄∞ := ⊗1= Rloc [[x1 , . . . , xq−[F + :Q]n(n−1)/2 ]],andS∞ := O[[z1 , . . . , zn2 #T , y1 , . . . , yq ]],for formal variables x1 , . . . , xq−[F + :Q]n(n−1)/2 , y1 , . . . , yq and z1 , . . . , zn2 #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 theimage of R̄∞ in End(M∞ ) is an S∞ -algebra.
• A filtration by R̄∞ -modules
01s
0 = M∞⊂ M∞⊂ · · · ⊂ M∞
= M∞ /πM∞

whose graded pieces are finite free S∞ -modules.
• A surjection of Rloc -algebras R̄∞ → RSuniv .

THE BREUIL–MÉZARD CONJECTURE

27• An isomorphism of R̄∞ -modules∼

M∞ /aM∞ −→ M,iiwhich identifies M∞/aM∞with M i , and such that the induced R̄∞ -module
structure on M coincides with the RSuniv -module structure on M , via thesurjection R̄∞ → RSuniv mentioned in the previous point.
We furthermore claim that we can make the above construction so that, for eachi
i−1
value of i = 1, 2, . . . s, the (R∞ , S∞ )-bimodule structure on M∞/M∞(arisingfrom its (R̄∞ , S∞ )-bimodule structure, and the surjection R∞ → R̄∞ ) and the
∼
i
i
i−1
isomorphism M∞
/(aM∞
, M∞
) −→ M i /M i−1 depends only on (U, m and) the
isomorphism class of Li /Li−1 as a G(OF + ,p )-representation, but not on (λ, τ ). For
any finite collection of pairs (λ, τ ) this follows by the same finiteness argumentused during patching. Since the set of (λ, τ ) is countable, the claim follows from adiagonalization argument.
It will be convenient in the following to refer to a tuple (av )ve∈Sep as a Serreweight, where each av is a Serre weight for GLn (kve), where kve is the residue field ofFve. We write Fa = ⊗ve∈Sep Fav , a representation of G(OF + ,p ). For a a Serre weight,awe denote by M∞
the R∞ /π-module constructed above (using the construction for∼all (λ, τ )) when Li /Li−1 −→ Fa , and we seta, R∞ /π)µ0a (ρ) = (n!)−1 e(M∞anda
Za0 (ρ) = (n!)−1 Z(M∞).
(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 acycle with Z-coefficients.)
For each v|p we write
X n∼Favav ,
(Lλv ,τv ⊗O F)ss −→av

so that∼(Lλ,τ ⊗O F)ss −→X

Fana ,

aQ
where na = v nav .
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, µ0a (ρ) is a non-negative integer. Moreover, for any
fixed (λ, τ ) the following conditions are equivalent:(1) The support of M ⊗Zp Qp meets every irreducible component of Spec Rloc [1/p].
(2) M∞ ⊗Zp Qp is a faithfully flat R̄∞ [1/p]-module which is everywhere locally
free of rank n!.
(3) RSuniv is a finite O-algebra and M ⊗Zp Qp is a faithful RSuniv [1/p]-module.
PPa(4) e(R̄∞ /π) = (n!)−1 Pa na e(M∞, R∞ /π)= a na µ0a (ρ).
Pa(5) Z(R̄∞ /π) = (n!)−1 a na Z(M∞) = a na Za0 (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 + n2 #T = dim S∞ [1/p].
Furthermore, the morphism Spec R̄∞ [1/p] → Spec Rloc [1/p], corresponding to the

28

MATTHEW EMERTON AND TOBY GEE

Rloc [1/p]-algebra structure on R̄∞ [1/p], induces a bijection on irreducible components; below we denote this bijection by Z 7→ Z 0 .
Since M∞ is finite free over S∞ , and the image of R̄∞ in End(M∞ ) is an S∞ algebra, M∞ ⊗Zp Qp has depth at least q +n2 #T at every maximal ideal of R̄∞ [1/p]in its support; so by the Auslander–Buchsbaum formula, it has depth exactly q +n2 #T at every maximal ideal of R̄∞ [1/p] in its support, and M∞ ⊗Zp Qp has
projective dimension 0 over R̄∞ [1/p]. Since M∞ ⊗Zp Qp 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∞ ⊗Zp Qp ,then Z is finite over Spec S∞ [1/p] and of the same dimension, because M∞ ⊗Zp Qp
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, andhence gives a point in the support of M∞ /aM∞ = M lying in the component Z 0 ofSpec Rloc [1/p] corresponding to Z. As M has rank n! over any point of RSuniv [1/p]in its support, it follows that M∞ ⊗Zp Qp is in fact locally free of rank n! over allof Z.
Thus we see that M∞ ⊗Zp Qp is faithful over R̄∞ [1/p], or equivalently, has supportconsisting 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 Rloc [1/p], and that, in
this case, it is furthermore locally free of rank n!. This shows that (1) and (2) areequivalent.
Using Proposition 1.3.4 of [Kis09a] we also see that each µ0a (ρ) is a non-negative
integer, and thate(R̄∞ /π) ≥ (n!)−1 e(M∞ /πM∞ , R̄∞ /π) = (n!)−1Xana e(M∞, R∞ /π)awith equality if and only if M∞ is a faithful R̄∞ -module (in which case, as alreadyobserved, 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 RSuniv , which
is a quotient of R̄∞ /a, is a finite O-module. This shows that (2) implies (3). Now
assume (3), so that RSuniv is a finite O-algebra; we see by Proposition 1.5.1 of[BLGGT13] that the image ofSpec RSuniv → Spec Rloc
meets every component of Spec Rloc [1/p]. Hence, since M ⊗Zp Qp is a faithful RSuniv 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 haveZ(R̄∞ /π) = (n!)−1 Z(M∞ /πM∞ ).

THE BREUIL–MÉZARD CONJECTURE

29

i
i−1
Identifying each of the modules M∞ /πM∞ , M∞/M∞
, etc., with the corresponding
sheaves on Spec R∞ /π that they give rise to, we see from Lemma 2.2.7 that we haveXii−1
Z(M∞ /πM∞ ) =
Z(M∞
/M∞
)
i

=

X

a
na Z(M∞).

a

The result follows.

For each Serre weight a ∈ (Zn+ )Hom(k,F) , let λa be a fixed lift of a. As in Remark
4.1.7(1), we may inductively define unique integers µa (r̄) such that for any Serreweight b, if we write∼m(Lλb ,1 ⊗O F)ss −→ ⊕a Fa a,b ,then
e(Rr̄,λ
/π) =
b ,1

X

ma,b µa (r̄).

a(In Remark 4.1.7(1) we were assuming the Breuil–Mézard conjecture, but the inductive process defining the µa (r̄) does not rely on any cases of the conjecture.)
Similarly, we may inductively define unique cycles Ca such thatXma,b Ca .
Z(Rr̄,λ
/π)
=
,1
b
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 thevarious assumptions of this section in the statement of the theorem.
5.5.2. Theorem. Let p > 2 be prime, let K/Qp be a finite extension, and letr̄ : GK → GLn (F) be a continuous representation, with F a finite extension of Fp .
Assume that there is a suitable globalization 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 λv =λav for some Serre weight av and each τv = 1.
Then, if λ ∈ (Zn+ )HomQp (K,E) and τ is an inertial type for GK , and if we write∼(Lλ,τ ⊗O F)ss −→ ⊕a Fana ,the following conditions are equivalent:(1) The equivalent conditions of Lemma 5.5.1 hold when λv = λ and τv = τ foreach v|p.
P(2) e(Rr̄,λ,τ/π) = a na µa (r̄).
P(3) Z(Rr̄,λ,τ/π) = a na Ca .
Proof. Consider conditions (4) and (5) of Lemma 5.5.1 in the case that each λv =λbv for some Serre weight bv and each τv = 1. Note that in this case we haveQ∼m(Lλ,τ ⊗O F)ss −→ ⊕a Fa a,b where ma,b = v mav ,bv . Throughout this proof wewill write ρv for ρ|GFve (which is isomorphic to r̄).
As remarked above, the conditions on v1 ensure that Rve1 is formally smoothover O, say Rve1 ∼= O[[t1 , . . . , tn2 ]]. Our assumption that the equivalent conditions

30

MATTHEW EMERTON AND TOBY GEEof Lemma 5.5.1 hold implies thatX
ma,b µ0a (ρ) = e(R̄∞ /π)
a

=

Y

=

YX

e(Rρv ,λbv ,1 /π)v|pmav ,bv µav (r̄)

v|p av

=

X

ma,b

a

Y

µav (r̄),v|pwhere the second equality holds by Proposition 1.3.8 of [Kis09a]. SimilarlyXma,b Za0 (ρ) = Z(R̄∞ /π)
a

=

Y

=

YX

Z(Rρv ,λbv ,1 /π) × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])v|pmav ,bv Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])v|p av

=

X

ma,b

a

Y

Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]),

v|pwhere we have made use of Lemma 2.2.14. If we define a partial order on the tuples(av )v|p of Serre weights by writing (av )v|p ≥ (bv )v|p to mean that each av ≥ bv ,then an easy induction using Lemma 4.1.1 shows that we must in fact have thatfor each Serre weight a,Yµ0a (ρ) =µav (r̄)
v|p

and
Y

Za0 (ρ) =Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]).

v|p

We now consider the conditions of Lemma 5.5.1 in the case that each λv = λand each τv = τ . By definition (and Lemma 2.2.14), we haveY
Z(R̄∞ /π) =Z(Rρv ,λ,τ /π) × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]).
v|p

Using the relation above, we have
XY
X
na Za0 (ρ) =nav Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])
a

a

=

v|p

YX

nav Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]]).

v|p av

Now, condition (5) of Lemma 5.5.1 states that
X
Z(R̄∞ /π) =na Za0 (ρ),
aTHE BREUIL–MÉZARD CONJECTURE

31

which is equivalent toY
Z(Rρv ,λ,τ /π) × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])
v|p

=

YX

nav Cav × Z(F[[x1 , . . . , xq−[F + :Q]n(n−1)/2 , t1 , . . . , tn2 ]])v|p avand thus to
Z(Rr̄,λ,τ/π) =Xna Ca ,

agiving the equivalence of (1) and (3). The equivalence of (1) and (2) may be provedby a formally identical argument.

5.5.3. Remark. We regard the assumption of Theorem 5.5.2 (that the conditions ofLemma 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 theBreuil–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 casethat 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 useof the results of [GK12] to give a particularly clean statement. Note that whenn = 2, the inductive definition of the cycles Ca above is trivial, and we simply set/π).
Ca := Z(Rr̄,λa ,1Suppose that K/Qp 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 predictedin [BDJ10]. We say that a is regular if we have 0 ≤ aσ,1 − aσ,2 < p − 1 for eachσ ∈ Hom(k, F).
5.5.4. Theorem. Let p > 2 be prime, let K/Qp be an absolutely unramified finite
extension, and let r̄ : GK → GL2 (F) be a continuous representation, with F a finite
extension of Fp . Suppose that every predicted Serre weight for r̄ is regular. Then,∼for any inertial type τ for GK , if we write (L0,τ ⊗O F)ss −→ ⊕a Fana , we haveP
Z(Rr̄,0,τ /π) = a na Ca .
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 suitableglobalization 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 wheneverλv = 0 and τv = τ (for an arbitrary τ ) for all v|p, and whenever τv is trivial for
all v and each λv 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] inthe case that each kτ = 2, and extends Théorème 1.5 of ibid. to arbitrary types.

32

MATTHEW EMERTON AND TOBY GEEAppendix 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 (GLn .2)(Fpm ) denote the subgroup of GL2n (Fpm ) generated by block diagonal matrices of the form (g, t g −1 ) anda matrix J with J(g, t g −1 )J −1 = (t g −1 , g). Then for m sufficiently large, both(GLn .2)(Fpm ) ⊂ GL2n (Fp ) and GLn (Fpm ) ⊂ GLn (Fp ) are adequate.
Proof. In the case of GLn (Fpm ), this is immediate from Theorem 1.2 and Lemma
1.4 of [Gur12b]. For (GLn .2)(Fpm ), since we are assuming that (p, 2n) = 1, it
follows from Theorem 1.5 of [Gur12a] (which shows the corresponding result for theunderlying algebraic group GLn .2) and the proof of Theorem 1.2 of [Gur12b] thatthe result will be true provided that Ext1(GLn .2)(Fpm ) (V, V ) = 0 for m sufficiently
large, where V = F2n
pm is the space on which (GLn .2)(Fpm ) acts.
The result then follows easily from inflation-restriction, the corresponding result for GLn (Fpm ) (which is proved in the course of the proof of Theorem 1.2
of [Gur12b]), and the fact that the standard representation of GLn (Fpm ) has adifferent central character to its dual provided that pm > 3.

A.2. Proposition. Let K/Qp be a finite extension, and let r̄ : GK → GLn (Fp ) bea continuous representation, with p - n. Then there is a CM field L with maximal
totally real subfield L+ , and a continuous irreducible representation ρ : GL+ →Gn (Fp ), such that• each place v|p of L+ splits in L;∼• for each place v|p of L+ , L+e of L lying over v
v = K and there is a place v∼
such that ρ|GLve = r̄;n• ν ◦ ρ = ε1−n δL/L
+ , where δL/L+ is the quadratic character corresponding+
to L/L ;• ρ−1 (GLn (Fp ) × GL1 (Fp )) = GL ;• ρ(GL(ζp ) ) = GLn (Fpm ) for some sufficiently large m as in Lemma A.1 (soin particular, ρ(GL(ζp ) ) is adequate);ker ρdoes not contain L(ζp );• L• if v - p is a finite place of L+ , then ρ|GL+ is unramified.
v

Proof. Choose m such that r̄(GK ) ⊂ GLn (F ) and m is sufficiently large as inLemma A.1, and set F = Fpm . We now apply Proposition 3.2 of [Cal12]; in thenotation of that result, we take G = Gn (F), and we let E be any totally real fieldwith the property that if v|p is a place of E, then Ev ∼= K. We let w - p be a
finite place of E, and we choose a character χw : GEw → F× such that χw |IFw
surjects onto F× . We take F = E(ζp ), 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 takecv = j ∈ Gn (F) for each infinite place v of S, and for each place v|p of E we
ker r̄
let Hv /Ev correspond to K
/K, and let φv : Gal(Hv /Ev ) → Gn (F) correspond
ker χ
1−n
) : GK → GLn (F) × GL1 (F) ⊂ Gn (F). We let Hw = E w w , and letto (r̄, ε
φw = (1, χw ) : Gal(Hw /Ew ) → GLn (F) × GL1 (F) ⊂ Gn (F).
pm

THE BREUIL–MÉZARD CONJECTURE

33

Writing M + /E for the field denoted K in [Cal12], we see that M + is a totallyreal field, and that we have a surjective representation ρ : GM + → Gn (F), with theproperties that• for each place v|p of M + , Mv+ ∼= K and there is a place ve of L lying over v
such that ρ|Mve ∼= r̄,• w splits completely in M + , and if w0 |w then ν ◦ ρ|GM + = χ,
w0

• for each place v|∞ of M + , ρ(cv ) = j, where cv is a complex conjugation atv,• ν ◦ ρ|GM + = ε1−n if v|p,
v

• M+

ker ρ

does not contain M + (ζp ).

Define M/M + by GM = ρ−1 (GLn (F) × GL1 (F)). Since ρ(cv ) = j for each complex
conjugation cv , we see that M is an imaginary CM field. Similarly, we see thateach place of M + lying over p or w splits in M .
nNow consider the character φ := (ν ◦ρ)εn−1 δM/M
+ . By construction, we see thatφ(cv ) = 1 for each complex conjugation cv , that φ|GM + = 1 for each place v|p, andv

that φ|IM + = χ|IM + for each place w0 |w. Replace M + by N + := M +
w0

ker φ

, M byw0

N := N + M , and ρ by ρ|GN + . Then N + /M + is a totally real abelian extension inwhich all places dividing p split completely, and it is linearly disjoint from M + (ζp )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+ ,• ρ|GL+ is unramified for all finite places v - p of L+ , andv• L+ is linearly disjoint from N +ker ρ|GN+(ζp ) over N + .

The first two of these conditions concern the splitting behaviour of a finite number ofplaces 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,ker ρ|GN+
for each of the (finitely many) Galois intermediate fields N +(ζp )/Li /N ++with the property that Gal(Li /N ) is simple one chooses a finite place of N + which
does not split completely in Li , 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 torealise local mod p representations as part of global automorphic Galois representations. In order to do this, we will apply the above construction, and then showthat the representation ρ is potentially automorphic. This essentially follows fromthe results of [BLGGT13]. However, the main theorems of [BLGGT13] make noattempt 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 ofscalars 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.

34

MATTHEW EMERTON AND TOBY GEE

A.3. Conjecture. Let K/Qp be a finite extension, and r̄ : GK → GLn (Fp ) acontinuous representation. Then r̄ has a potentially diagonalizable lift r : GK →GLn (Qp ) 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 potentiallydiagonalizable, 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 thata similar argument works more generally.
A.5. Lemma. Let p - 2n be prime. Let K/Qp be a finite extension, and let r̄ :GK → GLn (Fp ) be a continuous representation. Assume that Conjecture A.3 holds
for r̄. Let ρ : GL+ → Gn (Fp ) be the representation provided by Lemma A.2. Then
there is a lift ρ : GL+ → Gn (Qp ) of ρ such thatn• ν ◦ ρ = ε1−n δL/L+,
• ρ is unramified outside of places dividing p, and• if w|p is a place of L, then ρ|GFw 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 toknow 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 establishthat for some CM extension F/L, ρ|GF 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 Lker ρ (ζp ) over L, suchthat each place of L lying over p splits completely in F , and ρ|GF 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 : GL → Qp suchthat
• ψ is crystalline at all places above p.
n+1
1−2n• R := I(ρ|GL+ ⊗ (ψ, ε−n δL/L,+ )) : GL+ → GSp2n (Qp ) has multiplier εand is crystalline with distinct Hodge–Tate weights at all places lying over p.
• R̄(GL+ (ζp ) ) is adequate. (This follows from the choice of F = Fpm , 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 n1 = 2n,

THE BREUIL–MÉZARD CONJECTURE

35

and we put l1 = p, r1 := R̄, F = F0 = L+ , F (avoid) = L(ζp ). We then apply theconstructions of loc. cit., choosing in particular an auxiliary positive integer N , andconstructing a geometrically irreducible scheme Te = Tr̄1 ×r̄10 over Spec L+ (ζN )+ .
Choose a solvable extension (L0 )+ /L+ of totally real fields such thatker ρ(ζp ) over L+ , and• (L0 )+ is linearly disjoint from L
0 ++• for each place v|p of (L ) (ζN ) , there is a point Pv ∈ Te((L0 )+ (ζN )+v ) withv(t1 (Pv )) < 0.
We now construct a geometrically irreducible scheme Te0 over Spec(L0 )+ (ζN )+in exactly the same way as Te is constructed over Spec L+ (ζN )+ , and we then set
T 0 := Res(L0 )+ (ζN )+ /L+ Te0 , 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 0 (F + ) which satisfies the assumptions (relating to linear disjointness from a certain field, and on the v(Ti (P ))) on the field F 0 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 (L0 )+ (ζN )+ lying over p.)
Regarding P as an F + (L0 )+ (ζN )+ -point of Te0 , it then follows exactly as in loc.
cit. that there is a continuous lift R0 : GF + (L0 )+ (ζN )+ → GSp2n (Qp ) of the restriction
ρ|GF + (L0 )+ (ζ )+ such thatN

• R0 is ordinary, and• R0 is automorphic.
Let M be a quadratic totally imaginary extension of F + (L0 )+ (ζN )+ such that• all places of F + (L0 )+ (ζN )+ above p split in M , andker ρ +F (L0 )+ (ζp ) over F + (L0 )+ .
• M is linearly disjoint from L
It follows from Theorem 4.2.1 of [BLGGT13] that R|GM is automorphic, and thenfrom Lemma 2.1.1 of [BLGGT13] that ρ|GF + (L0 )+ (ζ )+ L is automorphic. SinceN
the extension F + (L0 )+ (ζN )+ L/F + L is solvable, it follows from Lemma 1.4 of[BLGHT11] that ρ|GF + 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/Qp is a finite extension, and letr̄ : GK → GLn (Fp ) be a continuous representation for which Conjecture A.3 holds.
Then there is an imaginary CM field F and a continuous irreducible representation
ρ : GF + → Gn (Fp ) such that• each place v|p of F + splits in F , and has Fv+ ∼= K,• for each place v|p of F + , there is a place ve of F lying over v with ρ|GFve
isomorphic to r̄,• ρ is unramified outside of p,• ρ−1 (GLn (Fp ) × GL1 (Fp )) = GF ,• ρ(GF (ζp ) ) is adequate,
ker ρ

• F
does not contain F (ζp ),• ρ is automorphic in the sense of Definition 5.3.1, and in particular µ ◦ ρ =n
ε1−n δF/F+.
Proof. This follows from Proposition A.6 and the theory of base change between
GLn and unitary groups, cf. Proposition 2.2.7 of [Ger09].
36

MATTHEW EMERTON AND TOBY GEE

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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