Draft: March 23, 2006
A REPORT ON THE AIM WORKING GROUP ON MOD p
LOCAL LANGLANDS
MATTHEW EMERTON

1. Introduction
This is a report on one of the working groups at the AIM workshop “p-adicrepresentations, modularity and beyond”. The goal of the working group was tomake some progress on understanding the (conjectural) mod p local Langlands
correspondence for GF when F is a (proper) finite extension of Qp . The group’s
discussion took place on the afternoon of Wednesday, 22nd of February.
The members of the group were motivated by the following ideas:(a) The theory of the mod p correspondence for GQp is now fairly well established,
and gives one confidence that a similar theory should exist when Qp is replaced bysome finite extension F .
(b) Local-global compatibility considerations suggest that any such correspondence should be related to Serre-type conjectures for global Galois representations,
say of GK for some number field K having F as a completion at some prime
above p. More precisely, the structure of the local correspondence should be related to the weight part of such a Serre-type conjecture. When K is a totally real
field unramified above p, such a conjecture on the weights has been recently been
made by Buzzard, Diamond, and Jarvis. This conjecture is furthermore phrased
in representation-theoretic terms, making a comparison with the conjectural mod
p local Langlands correspondence particularly natural.
The main participants in the working group were Breuil, Buzzard, Diamond,Emerton, Gee, Kisin, Schein, Schneider, and Taylor.
I would like to thank Fred Diamond for his helpful remarks on an earlier draftof this report.
2. The role of local-global considerations
2.1. The Buzzard-Diamond-Jarvis Serre-type conjecture on the weights.
Let K be a totally real field in which p is inert, and fix a quaternion algebra
D over K unramified at one real prime, and unramified at p. Fix some levelstructure away from p, and let Xr denote the Shimura curve with this given level1structure away from p, and of level pr at p. Write H 1 := lim
Hét(Xr/K , Fp ). This−→ris a GK × GL2 (F )-representation. Fix a two-dimensional continuous absolutely
irreducible representation ρ of GK over Fp which appears in the mod p cohomologyof the Shimura curve at level pN . Define M(ρ) := HomGK (ρ, H 1 ); this is a GL2 (F )representation, and the evaluation map gives a natural GK × GL2 (F )-equivariantembeddingρ ⊗Fp M(ρ) ,→ H 1 ,12

MATTHEW EMERTON

whose image is exactly the ρ-isotypical component of H 1 . Since ρ is absolutely
irreducible, one finds that for any congruence subgroup Γ(pr ), one hasr∼

1(Xr , Fp )),M (ρ)Γ(p ) −→ HomGK (ρ, Hétor equivalently,r∼1
ρ ⊗Fp M (ρ)Γ(p ) −→ ρ-isotypical part of Hét
(Xr , Fp ).

Taking r = 1, we find in particular that M(ρ)Γ(p) is a representation of GL2 (k).
The Serre-type conjecture of Buzzard, Diamond, and Jarvis (which is a theoremof Gee in many cases) predicts exactly which irreducible representations of GL 2 (k)over Fp appear as subrepresentations of M (ρ)Γ(p) . The list of V depends only on
ρ|GF (in fact, only on the restriction to the inertia subgroup IF of GF ). If ρ|GF issemi-simple, then the number of V that can appear is roughly 2f , where | k |= pf .
If ρ|GF is non semi-simple, then the V that can appear is a certain subset of thecorresponding set of weights attached to (ρ|GF ) ss .
Recall that for any representation V of GL2 (k) over Fp (which we can also thinkof a GL2 (OF ) representation, via the reduction map), there is an adjointness∼GL (F )HomGL2 (k) (V, M(ρ)Γ(p) ) −→ HomGL2 (F ) (c−IndGL22 (OF )Z V, M(ρ)).
(Here c−Ind means compact induction, and Z denotes the centre of GL2 (F ), whichis acting via some appropriate central character.) Thus the B-D-J conjecture
equally well predicts which possible maps
(1)

GL (F )

c−IndGL22 (OF )Z V → M(ρ)

can exist (for V an irreducible GL2 (k)-representation).
Actually, one can make a slightly stronger conjecture: A consideration of how
Hecke operators act shows that a map of the form (1) should factor through a map
GL (F )

c−IndGL22 (OF )Z V /(T − a) → M(ρ),
where T is the usual Hecke operator on the compact induction, a is some elementof Fp , equal to zero always in the case when ρ|GF is irreducible, and either zero orelse related to the eigenvalue of Frobenius on an unramified line in some twist of
ρ|GF , in the case when ρ|GF is reducible.
2.2. Local-global compatibility. Suppose that the level structure away from p
is chosen to precisely match with the prime-to-p Artin conductor of ρ. Then one
might optimistically hope that the GL2 (F )-representation M(ρ) depends only on
ρ|GF , and is the representation attached to ρ|GF via the conjectural mod p local
Langlands correspondence for GF . In fact, since one doesn’t (yet) have an a priori
characterization of this correspondence, this can possibly be taken as a defining
property of the correspondence (assuming that the hope is borne out).
Thus one can try to get information about the correspondence by getting information about the M(ρ). One in turn can try to get information about these via theGL (F )B-D-J conjecture, which restricts the possible maps c−IndGL22 (OF )Z V → M(ρ).

WORKING GROUP REPORT

33. Calculations
3.1. Preliminaries. One expects, based on experience from the case K = Q,
F = Qp , that if ρ|GF is not semisimple, then M(ρ) will be a non-trivial extensionof GL2 (F )-representations, whose semi-simplification will, however, match with thesemi-simplification of ρ|GF . (This is related to the fact that in this case the list of
Serre weights is smaller than in the semi-simple case.)
Thus, so as to not have to think about extensions, and in order to get themaximum number of Serre weights (and hence to derive as much information aboutM(ρ) as we can from the B-D-J conjecture) we will suppose that ρ|GF is semisimple.
Since from now on we will be focusing on ρ|GF we will write simply ρ to denote
this local Galois representation. Since we expect M(ρ) (where ρ is momentarily
still referring to the global ρ) to be related to ρ (which now, and from now on,refers to the restriction of our global ρ to GF ) via a local correspondence, we will
denote it by π(ρ).
To fix ideas, we will assume that F = Qp2 . Then ρ|IF (assuming it is semi-simple)
can have one of the following two forms (up to twist):
r +1+p(r1 +1)

ω4 00
p2 (r +1+p(r1 +1))
ω4 00

r +1+p(r1 +1)

ω2 000
1(when ρ|GF is irreducible)(when ρ|GF is reducible)

In each case, 0 ≤ r0 , r1 ≤ p − 1.
We now prepare to give the list of Serre weights (according to B-D-J) in eachcase (provided we are not in certain “boundary cases”). We adopt the convention
2that Symr0 stands for the representation of GL2 (k) on Symr Fp , and that Symr1
stands for the twist of this representation by the Frobenius automorphism (actingon GL2 (k), induced by Frobenius on k). We will similarly write det0 and det1 forthe determinant, and for its twist by Frobenius.
The following tables record each Serre weight, together with the associated T eigenvalue. (A caveat: in the B-D-J paper, they work with p-torsion in Jacobians
rather than étale cohomology with Fp coefficients, and take invariants after tensoring with V , rather than taking equivariant homs from V . Thus to convert
from B-D-J to the cohomological conventions used above, one should twist by acyclotomic character, and replace V by V ∗ . From now on we follow the B-D-J
conventions, and so write down the weights as they appear in B-D-J.)
Serre weights in the irreducible case:
Symr00 ⊗ Symr11(T = 0)p−2−r0
detr00 +1 det−1⊗ Symr11 +11 Sym01
detr11 +1 Symr00 −1 ⊗ Symp−2−r
110
detr00 detr11 +1 Symp−1−r⊗ Symp−3−r
10(T = 0)(T = 0)(T = 0).

4MATTHEW EMERTON

Serre weights in the reducible case:
Symr00 ⊗ Symr11(T = a1 6= 0)

0
1
detr00 +1 detr11 +1 Symp−3−r⊗ Symp−3−r01(T = a2 6= 0)

1
detp−1
detr11 Symr00 +1 ⊗ Symp−2−r
10(T = 0)0
Symp−2−r⊗ Symr11 +1 .
detr00 detp−110(T = 0).

In summary, for each of these V and a (and only these V and a), we get aGL (F )corresponding map c−IndGL22 (OF )Z V /(T − a) → π(ρ).
3.2. Some results of Paskunas. To save typing, from now on we write simply
GL (F )
c−Ind V to denote c−IndGL22 (OF )Z V.
Paskunas has shown (maybe excluding some boundary cases for r0 , r1 ) that there
is an embedding(2)1c−Ind detr11 +1 Symr00 −1 ⊗ Symp−2−r1Mc−Ind detr00 +1 Sym0p−2−r0 ⊗ Symr11 −1,→ c−Ind Symr00 ⊗ Symr11 /T.

Replacing r0 , r1 by p − 1 − r0 , p − 1 − r1 , and twisting appropriately, we similarly
obtain an embedding(3)1c−Ind detr11 +1 Symr00 −1 ⊗ Symp−2−r
1

Mc−Ind detr00 +1 Sym0p−2−r0 ⊗ Symr11 −101,→ c−Ind detr00 detr11 Symp−1−r⊗ Symp−1−r/T.
01Paskunas has furthermore shown that the cokernels of (2) and (3) are iso∼morphic. Finally, he has constructed a certain irreducible quotient πr0 ,r1 (−→detr00 detr11 ⊗πp−1−r0 ,p−1−r1 ) of these cokernels.
Currently, no other irreducible quotient of c−Ind Symr00 ⊗ Symr11 /T seems tohave been described in the literature. (But see the Postscript below.)
3.3. Some deductions: the reducible case.
Let’s consider the reducible case first. Then considering the first two Serreweights, we find maps
c−Ind Symr00 ⊗ Symr11 /(T − a1 ) → π(ρ)
and0c−Ind detr00 +1 detr11 +1 Symp−3−r⊗ Sym1p−3−r1 /(T − a2 ) → π(ρ),
0where a1 , a2 6= 0.
If we consider the third and fourth Serre weights, they require T = 0, and alsomatch under the Paskunas intertwining. Thus it is natural to guess that they are
simultaneously explained by a map
detp−1
detr11 ⊗πr0 +1,p−2−r1 → π(ρ).
0WORKING GROUP REPORT

5The upshot is that for reducible ρ, we guess that?

π(ρ) = c−Ind Symr00 ⊗ Symr11 /(T − a1 )M
01c−Ind detr00 +1 detr11 +1 Symp−3−r⊗ Symp−3−r/(T − a2 )01M
detp−1
detr11 πr0 +1,p−2−r1 .
03.4. Some deductions: the irreducible case. In this case, we find that theweights that appear for ρ are not preserved under the Paskunas intertwiners. Thusfor any weight V , say the first – namely Symr00 ⊗Symr11 – whatever the corresponding
mapc−Ind Symr00 ⊗ Symr11 → π(ρ)
is, it must not factor through the cokernel of (2) (since otherwise, using the Paskunas intertwiner, it would also factor through the cokernel of (3), which would give
an impossible weight).
Thus it must be non-zero on the source of (2). Now again a consideration ofweights shows that it is trivial on the second summand in this source, and so must
be non-trivial on the first summand – which fortunately does correspond to anallowed weight! It must furthermore factor through the T = 0 quotient of thissummand. Again, it cannot factor through the Paskunas cokernel, and so thisprocess iterates.
During this iteration, we find each weight of ρ appearing, and so the first weight
actually “explains” all of the others (in the sense that it forces them to appear).
So it seems that π(ρ) is a certain quotient of c−Ind Symr00 ⊗Symr11 whose precise
structure we don’t yet understand.
3.5. Further computations. There are some boundary cases of the B-D-J conjecture (e.g. r0 = r1 = 0) when we get additional intertwiners in the compact
induction, for example∼c−Ind 1 ⊗ 1/T −→ c−Ind Symp−1⊗ Symp−1,01which one finds are compatible with the weight predictions of B-D-J. This gives
further evidence for the consistency/compatibility of the B-D-J conjectures andmod p Langlands.
3.6. Postscript. Following the discussion on Feb 22, further discussions took place,
involving Breuil, Buzzard, Diamond, Emerton, Gee, and Taylor (in various permutations). These discussions led to a much firmer understanding of the case Q p2 ,as well as to significant progress towards understanding the case of Q pf for general f . It now seems very clear that the structure of the c−Ind Sym~r /T and theirirreducible quotients is intimately related to – and to a surprisingly large extent
controlled by – the B-D-J weight conjecture.
One natural question that was raised was whether the weights appear in M(ρ) Γ(p)with multiplicity one (i.e. whether HomGL2 (k) (V, M(ρ)Γ(p) ) is one-dimensional), atleast generically. It seems to be accepted by the experts that this should be so, but
it would be worthwhile to have precise statements and arguments.
If one grants this multiplicity one statement in the case when F = Qp2 for someglobal ρ for which ρ|GF is irreducible (which is not of exceptional weight from the6MATTHEW EMERTON

point of view of the B-D-J conjecture) then the discussion of Subsection 3.4 implies
that for any Serre weight V of ρ, the image of the corresponding map
GL (F )

c−IndGL22 (OF )Z V /T → M(ρ)
is independent of V , and is irreducible, but is not one of the irreducible representations constructed by Paskunas. It would naturally be of great interest to construct
such a quotient directly.
Mathematics Department, Northwestern University, 2033 Sheridan Rd., Evanston,IL 60208
E-mail address: emerton@math.northwestern.edu