MODULARITY LIFTING THEOREMS - NOTES FOR POSTECH ...

MODULARITY LIFTING THEOREMS - NOTES FOR
POSTECH WINTER SCHOOL ON SERRE’S CONJECTURE
TOBY GEE
1. The local Langlands correspondence for GL2
Let K be a finite extension of Qp, with ring of integers OK, uniformiser
πK and residue field OK/πK = k. Let valK : K × ↠ Z, and let | · |K =
(#k) − valK(α) . Let IK be the inertia subgroup of GK, the absolute Galois
group of K, and let WK be the Weil group of K. By definition, WK is the
subgroup of GK consisting of elements which map to an integral power of
the Frobenius in Gk (the absolute Galois group of k). The group WK is
a topological group, but its topology is not the subspace topology of GK;
rather, the topology is determined by demanding that IK is open, and has
its usual topology. Let FrobK ∈ GK/IK ∼ = Gk be the geometric Frobenius
element.
Let Knr = K IK
be the maximal unramified extension of K, and let
Ktame = ∪ (n,p)=1K nr (π 1/n
K ) be the maximal tamely ramified extension. Then
PK := Gal(K/Ktame ) is the unique Sylow pro-p subgroup of IK. Let
ζ = (ζN) be a compatible system of primitive roots of unity (i.e. ζa ab = ζb).
Then we have a character
tζ : IK/PK ∼ −→ �
Zl,
defined by
σ(π 1/n
K )
π 1/n
K
l�=p
= ζ (tζ(σ) (mod n))
n
.
If σ ∈ WK, then tζ(στσ −1 ) = ɛ(σ)tζ(τ), where ɛ is the cyclotomic character.
We let tζ,l be the composite of tζ and the projection to Zl.
Local class field theory is summarised in the following statement.
Theorem 1.1. Let W ab
K denote the group WK/[WK, WK]. Then there are
unique isomorphisms ArtK : K × → W ab
K such that
(i) if K ′ /K is a finite extension, then ArtK ′ = ArtK ◦N K ′ /K, and
These notes are based on a lecture course given by Richard Taylor at Harvard University
in 2009.
1

2 TOBY GEE
(ii) we have a commutative square
K × ArtK ��
valK
����
Z
W ab
K
����
��
Frob Z K
where the bottom arrow is the isomorphism sending a ↦→ Frob a K .
The irreducible representations of the group W ab
K are just the characters
of WK, and local class field theory gives a simple description of them, as
representations of K × = GL1(K). The local Langlands correspondence for
GLn is a kind of n-dimensional generalisation of this, giving a description
of the n-dimensional representations of WK in terms of the representation
theory of GLn(K).
Let L be a field of characteristic 0.
Definition 1.2. A representation of WK over L is a representation (on a
finite-dimensional L-vector space) which is continuous if L has the discrete
topology; that is, it is a representation with open kernel.
A Weil-Deligne representation of WK on a finite-dimensional L-vector
space V is a pair (r, N) consisting of a representation r : WK → GL(V ),
and an endomorphism N ∈ End(V ) such that for all σ ∈ WK,
r(σ)Nr(σ) −1 = (#k) −vK(σ) N,
where vK : WK → Z is determined by σ|K
nr = FrobvK(σ) K .
Remark 1.3. (i) Since IK is compact and open in WK, if r is a representation
of WK then r(IK) is finite.
(ii) N is necessarily nilpotent.
Exercise 1.4. (i) Show that if (r, V ) is a representation of WK and
m ≥ 1 then the following defines a Weil-Deligne representation
Sp m(r) with underlying vector space V m : we let WK act via
r| Art −1
K |m−1
K
⊕ r| Art−1 K |m−2
K ⊕ · · · ⊕ r,
and let N be the natural isomorphism from r| Art −1
K |i−1
K
for each i < m − 1, and be 0 on r| Art −1
K |m−1
K .
to r| Art−1
K |i K
(ii) Show that every Weil-Deligne representation is isomorphic to a direct
sum of representations Sp mi (ri).
(iii) Show that if (r, V, N) is a Weil-Deligne representation of WK, and
K ′ /K is a finite extension, then (r|G K ′ , V, N) is a Weil-Deligne
representation of WK ′.
(iv) Suppose that r is a representation of WK. Show that if σ ∈ WK
then for some positive integer n, r(σ n ) is in the centre of r(WK).
(v) Assume further that σ /∈ IK. Show that for any τ ∈ WK there exists
n ∈ Z and m > 0 such that r(σ n ) = r(τ m ).

MODULARITY LIFTING THEOREMS 3
(vi) Show that for a representation r of WK, the following conditions
are equivalent:
(a) r is semisimple.
(b) r(σ) is semisimple for all σ ∈ WK.
(c) r(σ) is semisimple for some σ /∈ IK.
(vii) Let (r, N) be a Weil-Deligne representation of WK. Set ˜r(σ) =
r(σ) ss . Prove that (˜r, N) is also a Weil-Deligne representation of
WK.
Definition 1.5. We say that a Weil-Deligne representation (r, N) is Frobeniussemsimple
if r is semisimple. With notation as above, we say that (˜r, N) is
the Frobenius semisimplification of (r, N).
Definition 1.6. If L is an algebraic extension of Ql for some l, then we say
that an element A ∈ GLn(L) is bounded if it has determinant in O ×
L , and
characteristic polynomial in OL[X].
Exercise 1.7. A is bounded if and only if it stabilises an OL-lattice in L n .
Definition 1.8. Let L be an algebraic extension of Ql for some l. Then we
say that r is bounded if r(σ) is bounded for all σ ∈ WK.
Exercise 1.9. Show r is bounded if and only if r(σ) is bounded for some
σ /∈ IK.
Proposition 1.10. (Grothendieck’s monodromy theorem) Suppose that l �=
p, that K/Qp is finite, and that V is a finite-dimensional L-vector space,
with L an algebraic extension of Qp. Fix φ ∈ WK a lift of FrobK and a
compatible system (ζN) of primitive roots of unity. If ρ : GK → GL(V )
is a continuous representation then there is a finite extension K ′ /K and a
uniquely determined nilpotent N ∈ End(V ) such that for all σ ∈ IK ′,
ρ(σ) = exp(Ntζ,l(σ)).
For all σ ∈ WK, we have ρ(σ)Nρ(σ) −1 = #k −v(σ) N. In fact, we have an
equivalence of categories WD = WDζ,φ from the category of continuous representations
of GK on finite-dimensional L-vector spaces to the category of
bounded Weil-Deligne representations on finite-dimensional L-vector spaces,
taking
ρ ↦→ (V, r, N), r(τ) := ρ(τ) exp(−tζ,l(φ −vK(τ) τ)N).
The functors WDζ ′ ,φ ′ and WDζ,φ are naturally isomorphic.
Remark 1.11. Note that since N is nilpotent, the exponential here is just a
polynomial - there are no convergence issues!
We sketch the proof as a series of exercises.
Exercise 1.12. (i) Show that there exists a GK-stable OL-lattice Λ ⊂
V . Show that the if GK ′ is the kernel of the induced map GK →
Aut(Λ/lΛ), then K ′ /K is a finite extension, and ρ(GK ′) is pro-l.
Show that ρ|IK ′ factors through tζ,l : IK ′ → Zl.

4 TOBY GEE
(ii) Choose σ ∈ IK ′ such that tζ,l(σ) generates tζ,l(IK ′). By considering
the action of conjugation by φ, show that the eigenvalues of ρ(σ)
are all l-power roots of unity. Hence show that one may make a
further finite extension K ′′ /K ′ such that the elements of ρ(IK ′′) are
all unipotent.
(iii) Deduce the existence of a unique nilpotent N ∈ End(V ) such that
for all σ ∈ IK ′′, ρ(σ) = exp(Ntζ,l(σ)). [Hint: use the logarithm map
(why are there no convergence issues?).]
(iv) Complete the proof of the proposition, by showing that (r, N) is a
Weil-Deligne representation. Where does the condition that r is
bounded come in?
One advantage of Weil-Deligne representations over Galois representations
is that there are no subtle topological issues: the topology on the
Weil-Deligne representation is the discrete topology. This allows one to
describe representations in a way that is “independent of L”.
Weil-Deligne representations are the objects on the “Galois” side of the
local Langlands correspondence. We now describe the objects on the “automorphic”
side. These will be representations (π, V ) of GLn(K) on (usually
infinite-dimensional) C-vector spaces.
Definition 1.13. We say that (π, V ) is smooth if for any vector v ∈ V , the
stabiliser of v in GLn(K) is open. We say that (π, V ) is admissible if it is
smooth, and for any open subgroup K ⊂ V , V K is finite-dimensional.
For example, a smooth one-dimensional representation of K × is the same
thing as a continuous character.
Fact 1.14. (i) If π is smooth and irreducible then it is admissible.
(ii) Schur’s lemma is true, and in particular if π is admissible and irreducible
then it has a central character χπ : K × → C × .
In general these representations are classified in terms of the (super)cuspidal
representations. We won’t need the details of this classification, and accordingly
we won’t define the cuspidal representations.
Let B be the subgroup of GL2 consisting of upper-triangular matrices.
Define δ : B → K × by
δ
�� ��
a ∗
= ad
0 d
−1 .
Given two characters χ1, χ2 : K × → C × , we may view χ1 ⊗ χ2 as a representation
of B by
�
a
χ1 ⊗ χ2 :
0
�
∗
↦→ χ1(a)χ2(d).
d
Then we define a representation χ1 × χ2 of GL2(K) by
χ1 × χ2 = n-Ind GL2(K)
B (χ1 ⊗ χ2)
:= {φ : GL2(K) → C|φ(hg) = (χ1 ⊗ χ2)(h)|δ(h)| 1/2
K φ(g) for all h ∈ B, g ∈ GL2(K)}

MODULARITY LIFTING THEOREMS 5
where GL2(K) acts by (gφ)(g ′ ) = φ(g ′ g), and we only allow smooth φ,
i.e. functions for which there is an open subgroup U of GL2(K) such that
φ(gu) = φ(g) for all g ∈ GL2(K), u ∈ U.
The representation χ1 × χ2 has length at most 2, but is not always irreducible.
It is always the case that χ1 × χ2 and χ2 × χ1 have the same
Jordan-Hölder factors. If χ1 × χ2 is irreducible then we say that it is a
principal series representation.
Fact 1.15. (i) χ1 × χ2 is irreducible unless χ1/χ2 = | det | ±1 .
(ii) χ×χ| det | has a one-dimensional irreducible subrepresentation, and
the corresponding quotient is irreducible. We denote this quotient
by Sp 2(χ).
We will let χ1 ⊞ χ2 denote the unique irreducible quotient of χ1 × χ2.
These, and one-dimensional characters, are all the non-cuspidal irreducible
admissible representations of GL2(K). We say that an irreducible smooth
representation π of GL2(K) is discrete series if it is of the form Sp 2(χ) or
is cuspidal.
The local Langlands correspondence provides a unique family of bijections
recK from the set of irreducible smooth representations of GLn(K) to the set
of n-dimensional Frobenius semisimple Weil-Deligne representations of WK
over C, satisfying a list of properties. In order to be uniquely determined, one
needs to formulate the correspondence for all n at once, and the properties
are expressed in terms of L- and ɛ-factors, neither of which we have defined.
Accordingly, we will not make a complete statement of the local Langlands
correspondence, but will rather state the properties of the correspondence
that we will need to use. It is also possible to define the correspondence in
global terms, as we will see later, and indeed at present the only proof of
the correspondence is global.
Fact 1.16. We now list some properties of recK for n = 1, 2.
(i) If n = 1 then recK(π) = π ◦ Art −1
K .
(ii) If χ is a smooth character, recK(π ⊗(χ◦det)) = recK(π)⊗recK(χ).
(iii) recK(Sp2(χ)) = Sp2(recK(χ)). (iv) If χ1/χ2 �= | det | ±1 , then recK(χ1 ⊞ χ2) = recK(χ1) ⊕ recK(χ2).
(v) recK(π) is unramified (i.e. N = 0 and the restriction to IK is trivial)
if and only if π = χ1 ⊞χ2 with χ1/χ2 �= | det | ±1 both unramified
characters (i.e. trivial on O ×
K ), or π = χ ◦ det for some unramified
character χ. These conditions are equivalent to πGL2(OK) �= 0, in
which case it is one-dimensional.
(vi) π is discrete series if and only if recK(π) is indecomposable.
1.1. Hecke operators. Let φ be a compactly supported C-valued function
on GL2(OK)\ GL2(K)/ GL2(OK). Concretely, these are functions which
vanish outside of a finite number of double cosets GL2(OK)g GL2(OK). The
set of such functions is in fact a ring, with the multiplication being given by
convolution. To be precise, we fix µ the (left and right) Haar measure on

6 TOBY GEE
GL2(K) such that µ(GL2(OK)) = 1, and we define
�
(φ1 ∗ φ2)(x) = φ1(g)φ2(g −1 x)dµx.
GL2(K)
Of course, this integral is really just a finite sum. One can check without too
much difficulty that the ring H of these Hecke operators is just C[Tp, S ±1
p ],
where Tp is the characteristic function of
� �
πK 0
GL2(OK) GL2(OK)
0 1
and Sp is the characteristic function of
� �
πK 0
GL2(OK)
GL2(OK).
0 πK
The algebra H acts on an irreducible admissible GL2(K)-representation
π. Given φ ∈ H, we obtain a linear map π(φ) : π → πGL2(OK) , by
�
π(φ)(v) = φ(g)π(g)vdµg.
GL2(K)
In particular, if π is unramified then π(φ) acts via a scalar on the onedimensional
C-vector space π GL2(OK) . We will now compute this scalar
explicitly.
Exercise 1.17. (i) Show that we have decompositions
�
πK
GL2(OK)
0
�
�
0
πK
GL2(OK) =
πK
0
�
0
GL2(OK),
πK
GL2(OK)
and
� �
πK 0
GL2(OK) =
0 1
⎛
⎝
�
α∈OK (mod πK)
⎞
� �
πK α
GL2(OK) ⎠
0 1
� � �
1 0
GL2(OK).
0 πK
(ii) Suppose that π = (χ|·|)◦det with χ unramified. Show that πGL2(OK) =
π, and that Sp acts via χ(πK) 2 (#k) −1 , and that Tp acts via (#k1/2 +
#k−1/2 )χ(πK).
(iii) Suppose that χ1, χ2 are unramified characters and that χ1 �= χ2|·| ±1
K .
Let π = χ1 ⊕ χ2. Using the Iwasawa decomposition GL2(K) =
B(K) GL2(OK), check that πGL2(OK) �� is �� one-dimensional, and is
a b
spanned by a function φ0 with φ0
= χ1(a)χ2(d). Show
0 d
that π(Sp) acts on πGL2(OK) via (χ1χ2)(πK), and that π(Tp) acts
via #k1/2 (χ1(πK) + χ2(πK)).

MODULARITY LIFTING THEOREMS 7
2. Automorphic representations and Galois representations
Let F be a totally real field, and let D/F be a central simple algebra of
dimension 4. Letting S(D) be the set of places v of F at which D is ramified,
i.e. for which D ⊗F Fv is a division algebra (equivalently, is not isomorphic
to M2(Fv)), it is known that S(D) classifies D up to isomorphism, and
that S(D) can be any finite set of places of D of even cardinality (so for
example S(D) is empty if D = M2(F )). We will now define some spaces of
automorphic forms on D × .
For each v|∞ fix kv ≥ 2 and ηv ∈ Z such that kv + 2ηv − 1 = w is
independent of v. These will be the weights of our modular forms. Let GD
be the algebraic group over Q such that for any R-algebra Q, GD(R) =
(D ⊗Q R) × . For v|∞, we define a compact mod centre subgroup of GD(Fv)
as follows: if v ∈ S(D) we let Uv = GD(Fv) ∼ = H × , and if v /∈ S(D), so
that GD(Fv) ∼ = GL2(R), we take Uv = R × � �
a b
SO(2). If γ = ∈ GL2(R)
c d
and z ∈ C − R, we let j(γ, z) = cz + d. One checks easily that j(γδ, z) =
j(γ, δz)j(δ, z).
We now define a representation (τv, Wv) of Uv over C for each v|∞. If
v ∈ S(D), we have GD(Fv) ↩→ GL2(Fv) ∼ = GL2(C) which acts on C 2 , and
we let (τv, Wv) be the representation
(Sym kv−2 C 2 ) ⊗ (∧ 2 C 2 ) ηv .
If v /∈ S(D), then we have Uv ∼ = R × SO(2), and we take Wτ = C, with
τv(γ) = j(γ, i) kv (det γ) ηv−1 .
We write U∞ = �
v|∞ Uv, W∞ = ⊗v|∞Wv, τ∞ = ⊗v|∞τv. Let A = AQ be
the adeles of Q, and let A∞ be the finite adeles. We then define SD,k,η to
be the space of functions φ : GD(Q)\GD(A) → W∞ which satisfy
(i) φ(gu∞) = τ∞(u∞) −1 φ(g) for all u∞ ∈ U∞ and g ∈ GD(A).
(ii) There is a nonempty open subset U ∞ ⊂ GD(A ∞ ) such that φ(gu) =
φ(g) for all u ∈ U ∞ , g ∈ GD(A).
(iii) Let S∞ denote the infinite places of F . If g ∈ GD(A ∞ ) then the
function
defined by
(C − R) S∞−S(D) → W∞
h∞(i, . . . , i) ↦→ ⊗v∈S∞τv(hv)φ(ghv)
is holomorphic. [Note that this function is well-defined by the first
condition, as U∞ is the stabiliser (i, . . . , i).]
(iv) If S(D) = ∅ then for all g ∈ GD(A) = GL2(AF ), then we have
� � �
1 x
φ( g)dx = 0.
0 1
F \AF

8 TOBY GEE
If in addition we have F = Q, then we furthermore demand that for
all g ∈ GD(A ∞ ), h∞ ∈ GL2(R) + the function φ(gh∞)|ℑ(h∞i)| k/2
is bounded on C − R.
There is a natural action of GD(A ∞ ) on SD,k,η by right-translation, i.e.
(gφ)(x) := φ(xg).
Exercise 2.1. While this definition may at first sight appear rather mysterious,
it is just a generalisation of the familiar spaces of cuspidal modular
forms. For example, take F = Q, S(D) = ∅, k∞ = k, and η∞ = 0. Define
U1(N) = {g ∈ GL2( ˆ � �
∗ ∗
Z)|g ≡ (mod N)}.
0 1
(i) Let GL2(Q) + be the subgroup of GL2(Q) consisting of matrices with
positive determinant. Show that the intersection of GL2(Q) + and
U1(N) inside GL2(A∞ � �
) is Γ1(N), the matrices in SL2(Z) congruent
1 ∗
to (mod N). [Hint: what is
0 1
ˆ Z × ∩ Q × ?]
(ii) Use the facts that GL2(A) = GL2(Q)U1(N) GL2(R) + [which follows
from strong approximation for SL2 and the fact that det U1(N) =
ˆZ × ] and that A × = Q × Z ˆ× ×
R >0 to show that S U1(N)
D,k,0 can naturally be
identified with a space of functions
satisfying
φ : Γ1(N)\ GL2(R) + → C
φ(gu∞) = j(u∞, i) −k φ(g)
for all g ∈ GL2(R) + , u∞ ∈ R × >0 SO(2).
(iii) Show that the stabiliser of i ∈ in GL2(R) + is R × >0 SO(2). Hence deduce
a natural isomorphism between S U1(N)
D,k,0 and Sk(Γ1(N)), which
takes a function φ as above to the function (gi ↦→ j(g, i) kφ(g), g ∈ GL2(R) + .
The case that S∞ ⊂ S(D) is particularly simple; then if U ⊂ GD(A∞ ) is
open, SU D,2,0 is just the set of C-valued functions on
GD(Q)/GD(A)/GD(R)U,
which is a finite set.
We will now examine the action of Hecke operators on these spaces.
Choose an order OD ⊂ D (that is, a Z-subalgebra of D which is finitely
generated as a Z-module and for which OD ⊗Z Q ∼ −→ D). For example, if
D = M2(F ), one may take OD = M2(OF ).
For all but finite many finite places v of F we can choose an isomorphism
Dv ∼ = M2(Fv) such that this isomorphism induces an isomorphism OD ⊗OF
∼
OFv −→ M2(OFv). Then GD(A∞ �
) is the subset of elements g = (gv) ∈
v∤∞ GD(Fv) such that gv ∈ GL2(OFv) for almost all v.

MODULARITY LIFTING THEOREMS 9
We now wish to describe certain irreducible representations of GD(A) in
terms of irreducible representations of the GL2(Fv). More generally, we have
the following construction. Let I be an indexing set and let Vi be a C-vector
space. Suppose that we are given 0 �= ei ∈ Vi for almost all i. Then we
define the restricted tensor product
⊗ ′ {ei} Vi := lim
−→
J⊂I
⊗i∈JVi,
where the sets J contain all the places for which ei is not defined, and where
the transition maps for the colimit are given by “tensoring with the ei”. It
can be checked that ⊗ ′ {ei} Vi ∼ = ⊗ ′ {fi} Vi if for almost all i, ei and fi span the
same line.
Definition 2.2. We call a representation (π, V ) of GD(A ∞ ) admissible if
(i) for any v ∈ V , the stabiliser of v is open, and
(ii) for any U ⊂ GD(A ∞ ) an open subgroup, dimC V U < ∞.
Fact 2.3. If πv is an irreducible smooth (so admissible) representation of
GL2(OFv )
GD(Fv) with πv �= 0 for almost all v, then ⊗ ′ πv := ⊗ ′ �
π GL2 (OF,v )
�πv
v
is an irreducible admissible representation of GD(A ∞ ), and any irreducible
representation of GD(A ∞ ) arises in this way for unique πv.
We have a global Hecke algebra, which decomposes as a restricted product
of the local Hecke algebras in the following way. For each finite place v of F
we choose Uv ⊂ GD(Fv) a compact open subgroup, such that Uv = GL2(OFv)
for almost all v. Let µv be a Haar measure on GD(Fv), chosen such that
for almost all v we have µv(GL2(OFv)) = 1. Then there is a unique Haar
measure µ on GD(A∞ ) such that for any Uv as above, if we set U = �
v Uv ⊂
GD(A∞ ), then µ(U) = �
v µv(Uv). Then there is a decomposition
Cc(U\GD(A ∞ )/U)µ ∼ = ⊗ ′ {1Uv µv}Cc(Uv\GD(Fv)/Uv)µv,
and the actions of these Hecke algebras are compatible with the decomposition
π = ⊗ ′ πv.
Fact 2.4. SD,k,η is a semisimple admissible representation of GD(A ∞ ).
Definition 2.5. The irreducible constituents of SD,k,η are called the cuspidal
automorphic representations of GD(A ∞ ) of weight (k, η).
Fact 2.6. (Strong multiplicity one for GL2) Suppose that S(D) = ∅. Then
every irreducible consituent of SD,k,η has multiplicity one. In fact if π (respectively
π ′ ) is a cuspidal automorphic representation of weight (k, η) (respectively
(k ′ , η ′ )) such that πv ∼ = π ′ v for almost all v then k = k ′ , η = η ′ ,
and π = π ′ .
Fact 2.7. (The theory of newforms) If n is an ideal of OF , write
U1(n) = {g ∈ GL2( ÔF
� �
∗ ∗
)|g ≡ (mod n)}.
0 1

10 TOBY GEE
If π is a cuspidal automorphic representation of GD(A ∞ ) then there is a
unique ideal n such that π U1(n) is one-dimensional, and π U1(m) �= 0 if and
only if n|m. We call n the conductor of π.
Analogous to the theory of admissible representations of GL2(K), K/Qp
finite that we sketched above, there is a theory of admissible representations
of D × , D a nonsplit quaternion algebra over K. Since D × /K × is
compact, any irreducible smooth representation of D × is finite-dimensional.
There is a bijection JL from the irreducible smooth representations of D ×
to the discrete series representations of GL2(K), determined by a character
identity.
Fact 2.8. (The global Jacquet-Langlands correspondence)
(i) The only finite-dimensional cuspidal automorphic representations of
GD(A∞ ) occur if S(D) ⊃ S∞ and kv = 2 for all v ∈ S∞, in which
case there are 1-dimensional representations, which factor through
the reduced determinant.
(ii) There is a bijection JL from the infinite-dimensional cuspidal automorphic
representations of GD(A∞ ) of weight (k, η) to the cuspidal
automorphic representations π of GL2(A∞ F ) of weight (k, η)
such that πv is discrete series for all finite places v ∈ S(D). Furthermore
if v /∈ S(D) then JL(π)v = πv, and if v ∈ S(D) then
JL(π)v = JL(πv).
Fact 2.9. (The existence of Galois representations associated to cuspidal
modular forms) Let π be an irreducible admissible representation of GL2(AF ).
Then there is a CM field Lπ and for each finite place λ of Lπ a continuous
irreducible Galois representation
such that
rλ(π) : GF → GL2(Lπ,λ)
(i) if πv is unramified and v does not divide the residue characteristic of
λ, then rλ(π)|GFv is unramified, and the characteristic polynomial
of Frobv is X2 − tvX + (#kv)sv, where tv and sv are the eigen-
GL2(OFv )
values of Tv and Sv respectively on πv . [Note that by the
Chebotarev density theorem, this already characterises rλ(π) up to
isomorphism.]
(ii) More generally, WD(rλ(π)|GFv )F −ss ∼ = recFv(πv ⊗ | det | −1/2 ).
(iii) If v divides the residue characteristic of λ then rλ(π)|GFv is de Rham
with τ-Hodge-Tate weights ητ , ητ + kτ − 1, where τ : F ↩→ Lπ ⊂ C
is an embedding lying over v. If πv is unramified then rλ(π)|GFv is
crystalline.
(iv) If cv is a complex conjugation, then det rλ(π)(cv) = −1.
Fact 2.10. (Base change) Let E/F be a cyclic extension of totally real fields
of prime degree. Let Gal(E/F ) = 〈σ〉 and let Gal(E/F ) ∨ = 〈δ E/F 〉. Let π be

MODULARITY LIFTING THEOREMS 11
a cuspidal automorphic representation of GL2(A∞ F ) of weight (k, η). Then
there is a cuspidal automorphic representation BCE/F (π) of GL2(AE) of
weight (BCE/F (k), BCE/F (η)) such that
(i) for all finite places v of E, recEv(BCE/F (π)v) = (recF (π v|F v|F ))|WEv .
In particular, rλ(BCE/F (π)) ∼ = rλ(π)|GE .
(ii) BCE/F (k)v = kv|F , BCE/F (η)v = ηv|F .
(iii) BCE/F (π) ∼ = BCE/F (π ′ ) if and only if π ∼ = π ′ ⊗ (δi E/F ◦ ArtF ◦ det)
for some i.
(iv) A cuspidal automorphic representation π of GL2(A∞ E ) is in the image
of BCE/F if and only if π ◦ σ ∼ = π.
Definition 2.11. We say that r : GF → GL2(Q p) is modular (of weight (k, η))
if it is isomorphic to i(rλ(π)) for some cuspidal automorphic representation
π (of weight (k, η)) and some i : Lπ ↩→ Q p lying over λ.
Proposition 2.12. Suppose that r : GF → GL2(Qp) is a continuous representation,
and that E/F is a solvable Galois extension of totally real fields.
Then r|GE is modular if and only if r is modular.
Exercise 2.13. Prove the above proposition as follows.
(i) Use induction to reduce to the case that E/F is cyclic of prime
degree.
(ii) Suppose that r|GE is modular, say ∼= r|GE i(rλ(π)). Use strong
multiplicity one to show that π ◦ σ ∼ = π. Deduce that there is an
automorphic representation π ′ such that BCE/F (π ′ ) = π.
(iii) Use Schur’s lemma to deduce that there is a character χ of GF such
that r ∼ = i(rλ(π ′ )) ⊗ χ. Conclude that r is modular.
We can make use of this result to make considerable simplifications in our
proofs of modularity lifting theorems. It is frequently employed in conjunction
with the following fact from class field theory.
Fact 2.14. Let K be a number field, and let S be a finite set of places of
K. For each v ∈ S, let Lv be a finite Galois extension of Kv. Then there
is a finite solvable Galois extension M/K such that for each place w of M
above a place v ∈ S there is an isomorphism Lv ∼ = Mw of Kv-algebras.
Note that we are allowed to have infinite places in S, so that if K is totally
real we may choose to make L totally real by an appropriate choice of the
Lv.
3. Modularity lifting theorems
Assume that p > 2 is prime, let F/Q be totally real, and let ρ : GF →
GL2(Q p) be a continuous irreducible representation. We will now be a little
slack with our notation for Galois representations attached to automorphic
representations, being content to write rp(π) for some i(rλ(π)) as above. We

12 TOBY GEE
want to prove that ρ is modular. For this to be true, we will have to assume
some conditions. Firstly:
◦ ρ is unramified at almost all primes, and is de Rham at places
dividing p, with distinct Hodge-Tate weights.
◦ ρ is odd, i.e. det ρ(cv) = −1 for any complex conjugation cv.
These conditions are conjecturally enough. However, we can’t prove a theorem
anything like as strong as this. There are a number of assumptions
that we need to make that appear to be intrinsic to the method.
◦ ρ is modular, say ρ ∼ = ¯rp(π).
◦ ρ|G F (ζp) is irreducible.
◦ π has the same weight as ρ, in a sense that we will make precise
below.
There is another assumption that we will make that can be considerably
relaxed, but not as yet removed.
◦ If v|p is a place of F , then Fv/Qp is unramified, and the Hodge-Tate
weights of ρ are all {0, k − 1} with 0 ≤ k ≤ p − 1. Furthermore πv
is unramified for all v|p, and π has weight (k, 0).
We begin by making a number of simplifications. Firstly, we use Grothendieck’s
monodromy theorem, local-global compatibility, and solvable base change,
which together show that we may assume that:
◦ At every finite place v ∤ p of F , ρ|GFv is either unramified, or has
unipotent ramification.
◦ Each πv is either unramified or is Sp2(χ) with χ unramified.
Less obviously, but explained in Ken Ribet’s talk, we can assume that
◦ Each πv is unramified.
[Sketch: base change so that if v is a place with πv Steinberg, than Nv ≡ 1
(mod p). Then produce a congruence between the Steinberg and a ramified
principal series, and then make another base change to replace this with an
unramified principal series.] For the method we’re going to use, we will also
make a further base change and assume that:
◦ If v ∤ p is a place at which ρ|GFv is ramified, then Nv ≡ 1 (mod p),
and ρ(GFv) = 1.
We can finally also assume that det ρ = det rp(π); the ratio of these characters
is everywhere unramified and residually trivial, thus has finite p-power
order, and in particular is necessarily trivial on each complex conjugation,
so can be trivialised by base change.
By a standard argument using the Baire category theorem and the fact
that GF is compact, we may assume that ρ : GF → GL2(K), K/Qp finite.
Again using the compactness of GF , after conjugation we may assume that
ρ : GF → GL2(O), where O is the ring of integers of K, with maximal ideal λ
and residue field O/λ = k. We will sometimes assume that K is sufficiently

MODULARITY LIFTING THEOREMS 13
large (so that various polynomials have roots in K, for example), but the
reader can check that we only ever require a finite extension.
For simplicity of exposition, we will now restrict to a special case. This
is purely to make our notation simpler; the arguments that we make go
through in exactly the same way in the general case. We will restrict to the
case F = Q, and will assume that all of the above assumptions apply, and
that ρ is only ramified at one prime other than p. Thus we want to prove
the following theorem.
Theorem 3.1. Let p > 2 be prime and let ρ : GQ → GL2(O) be irreducible.
Suppose that
◦ ρ|G is irreducible.
F (ζp)
◦ ρ is unramified except for at p and some prime l.
◦ ρ|GQ has unipotent ramification, l ≡ 1 (mod p), and ρ(GQl ) = 1.
l
◦ There is a cuspidal automorphic representation π of GL2(A) of
weight (k, 0), with πv unramified for all primes v and 2 ≤ k ≤ p−1,
such that ρ ∼ = ¯rp(π), and det ρ = det rp(π).
is crystalline with Hodge-Tate weights (0, k − 1).
◦ r|GQp
Then ρ is modular.
Write χ = det ρ from now on.
3.1. Local deformation theory. We will need to consider several lifting
rings for ρ|GQ (which is the trivial 2-dimensional representation of GQl l ,
with l ≡ 1 (mod p)). Write rloc l : GQl → GL2(Rloc l ) for the universal lifting
with determinant χ. Choose φ ∈ GQl a lift of the Frobenius, and σ ∈
IQl a generator of tame inertia, so that φσφ−1 = σl . Note that rloc l is
determined by the data of the two matrices rloc l (φ), rloc l (σ) satisfying this
relation; the deformation theory is genuinely explicit in this case! With some
(non-trivial) messing around with pairs of two by two matrices, one can prove
the following facts (or for a more sophisticated approach, which establishes
the analogous results for n-dimensional representations, see [Tay08]).
Definition 3.2. (i) Let Pnr be the minimal ideal of R loc
r loc
l (σ) = 12.
l
modulo which
(ii) For any root of unity ζ, we let Pnr be the minimal ideal of R loc
l
modulo which r loc
l (σ) has characteristic polynomial (X − ζ)(X −
ζ−1 ).
(iii) Let Pmult be the minimal ideal of Rloc modulo which rloc
l (σ) has
l
characteristic polynomial (X−1) 2 , and l(tr rloc l (φ)) 2 = (1+l) 2 det rloc l (φ).
[The motivation for the definition of Pmult is that we are attempting to
describe the unipotent liftings, and if you assume that rloc � �
1 1
l (σ) = ,
0 1
this is the relation forced on rloc l (φ). All but one Qp-point of Rloc l /Pmult will
have N �= 0, in fact.]

14 TOBY GEE
Fact 3.3. The minimal primes of Rloc l are precisely √ Pnr, √ �
Pmult, and the
Pζ for ζ �= 1. We have √ P1 = √ Pnr ∩ √ Pmult.
Write Rl,1, Rl,ζ, Rl,nr, Rl,mult for the corresponding quotients of R loc
l .
Fact 3.4. (i) Rl,ζ[1/p] is formally smooth and geometrically irreducible.
(ii) Rl,nr is formally smooth over O (and thus geometrically irreducible).
(iii) Rl,mult[1/p] is formally smooth and geometrically irreducible.
(iv) Spec Rl,1 = Spec Rl,nr∪Spec Rl,mult and Spec Rl,1/λ = Spec Rl,nr/λ∪
Spec Rl,mult/λ are both a union of two irreducible components.
We will make use of these facts later, and also of the trivial but crucial
fact that Rl,1/λ = Rl,ζ/λ.
3.2. Global deformations. Let ? denote one of 1, ζ, nr, mult. Let R be a
complete local Noetherian O-algebra with residue field k. By a deformation
to R of ρ of type ? we mean a ker(GL2(R) → GL2(k))-conjugacy class of
liftings of ρ to r : GQ → GL2(R) such that:
(i) r unramified outside {l, p}, and det r = χ.
(ii) The map Rloc l → R determined by ρ|GQ factors through Rl,?.
l
(iii) The map Rloc p → R determined by ρ|GQp factors through Rp,k, the
universal lifting ring coming from Fontaine-Laffaille theory.
A universal deformation of type ? exists, which we denote ρuniv ? : GQ →
GL2(Runiv ? ). Note that ρ, the deformation that we wish to prove is modular,
is a deformation of type 1.
A framed deformation of type ? is a ker(GL2(R) → GL2(k))-conjugacy
class of (r, {rq}q=l,p, {αq}q=l,p) where r lifts ρ and is as above, the rq are
= αqrqα −1
q .
lifts of r|GQq , and the αq ∈ ker(GL2(R) → GL2(k)) satisfy r|GQq
Here conjugacy by β replaces r by βrβ −1 , leaves rq alone, and replaces αq
by βαq.
There is a universal framed deformation over R� ? , and there are natural
maps Rloc q → R� ? , determined by the rq. Let Rloc ? = Rp,k ˆ⊗Rl,?, which has
relative dimension 7 over O. We also have a map Runiv ? → R� ? , and a non-
canonical isomorphism Runiv ? [[A1, . . . , A7]] ∼ = R� ? , given by choosing matrices
αq.
A standard calculation, together with the Poitou-Tate sequence, shows
that R� ? is topologically generated over Rloc
? by
dim H 1 l,p,⊥ (GQ, ad 0 ρ(1))
elements, where “H1 l,p,⊥ ” denotes the cohomology classes that are unramified
outside l and p. We denote this quantity by u.
Lemma 3.5. For all N ≥ 1 there is a finite set QN of primes such that
#QN = u, and for all q ∈ QN,
(i) q ≡ 1 (mod p N ),
(ii) ρ(Frobq) has distinct eigenvalues,

MODULARITY LIFTING THEOREMS 15
and H 1 l,p,⊥,QN (GQ, (ad 0 ρ)(1)) = (0) (this is defined in the same way as above,
except that we demand that our classes are trivial at places in QN).
This may be proved by the Cebotarev density theorem.
We need to understand the deformations that we’re permitting at places
in QN.
Lemma 3.6. Suppose that (q − 1) is exactly divisible by pm , and that
ρ(Frobq) has distinct eigenvalues. Then the universal lifting ring for lifts of
ρ|GQq with determinant χ|GQq is isomorphic to O[[x, y, B, u]]/((1+u)pm −1),
with the universal lift being given by
� �−1 � � � �
1 y α + B 0 1 y
φ ↦→
,
x 1 0 χ(φ)/(α + B) x 1
where α is a lift to O of one of the eigenvalues of ρ(φ), and
�
1
σ ↦→
x
�−1 �
y 1 + u
1 0
0
(1 + u) −1
� �
1
x
�
y
.
1
Exercise 3.7. Prove this lemma as follows.
� �
α 0
(i) Suppose that ρ(φ) = . Let ρ : GQq 0 β
→ GL2(A) be a lift
of ρ. By Hensel’s lemma, there are a, b ∈ mA such that ρ(φ) has
characteristic polynomial (X − (α + a))(X − (β + b)). Show that
there are x, y ∈ mA such that
� � � �
1
1
ρ(φ) = (α + a)
x
x
and
ρ(φ)
� � � �
y
y
= (β + b)
1
1
(ii) Since ρ|GQq is unramified, ρ(σ) = 1, so we may write
�
1
x
�−1 �
y 1
ρ(σ)
1 x
� �
y 1 + u
=
1 w
�
v
1 + z
with u, v, w, z ∈ mA. Use the commutation relation between ρ(φ)
and ρ(σ) to show that v = w = 0.
(iii) Use the fact that χ|GQq is unramified to show that 1+z = (1+u)−1 .
(iv) Show that (1 + u) q = 1 + u, and deduce that (1 + u) q−1 = 1.
(v) Deduce that (1 + u) pm = 1.
(vi) Complete the proof of the lemma.
The rather vague moral to take away from this calculation is that this
local lifting ring is not smooth, but that if we were able to pass to a limit
with m we would obtain a smooth ring. This in effect is what we will do.

16 TOBY GEE
Define R� as above, except that we no longer require the deformations
?,QN
to be unramified at q ∈ QN. R� is still generated by u elements over
?,QN
Rloc ? , by another duality calculation.
�
Write ∆QN = q∈QN Sylp((Z/qZ) × ). For simplicity of notation, let’s suppose
that this is isomorphic to (Z/pNZ) u (again, the general case works in
the same way, but the notation is slightly more complicated). By the above,
∼= χα ⊕ χβ, and we have a homomor-
runiv |GQq is reducible, say runiv
?,QN ?,QN |GQq
phism Sylp((Z/qZ) × ) → (Runiv ?,QN )× induced by local class field theory and
χα|IQq . Taking the product over q ∈ QN, we have a map ∆QN → (Runiv
?,QN )× .
So R� ?,QN is a module over O[∆QN ], and if we let aN denote the augmentation
ideal of O[∆QN ], then R� ?,QN /aN = Runiv ? .
We now have to introduce modular forms into the picture. At this point
we will be rather brief, as to give the full details would take a prohibitive
amount of space. This is also the one place where the theory is better
developed over a totally real field other than Q; indeed, the optimal choice
of field is a field F with [F : Q] even. We then choose a division algebra
D with S(D) = ∞. We now define spaces of “algebraic modular forms” for
GD(A∞ ) ∼ = GL2(A∞ F ). These are an integral p-adic avatar of the spaces of
modular forms considered above, and we can use them to study congruences
between modular forms, and in particular to relate the global deformation
rings we are considering to Hecke algebras.
Let U = �
v Uv ⊂ GL2(A∞ F ) be a compact open subgroup, with Uv =
GL2(OFv) for all v|p. Fix χ0 : (A∞ F )× /F × → O × a continuous character.
Let Λk denote the representation of GL2(OF,l) given by
⊗τ:F ↩→K Sym k−2 O 2 .
If A is a finite O-module, we let Sk,χ0 (U, A) denote the space of functions
satisfying
φ : D × \ GL2(A ∞ F ) → Λ ⊗O A
φ(guz) = χ0(z)u −1
p φ(g)
for all g ∈ GL2(A∞ F ), u ∈ U, z ∈ (A∞ F )× . We will sometimes write S(U, A)
for Sk,χ0 (U, A). Note that S(U, O) is a finite free O-module.
We have an action of Hecke operators on S(U, A); if we let S be the set
of places v of F which either divide p or for which Uv �= GL2(OFv), then
we have an action of the Hecke operators Tv, Sv at all places v /∈ S, via
the usual formulas we found above. Let TU denote the O-subalgebra of
EndO(Sk,χ0 (U, O)) generated by these operators; this is a commutative Oalgebra,
and is finite and free as an O-module. It is not so hard to relate
these spaces to the spaces of modular forms we considered previously.
∼
Fact 3.8. Fix an isomorphism ı : Qp −→ C. Then the ı-linear ring homomorphisms
θ : TU → C are in natural bijection with the cuspidal automorphic
representations of GD(A∞ F ) of weight (k, 0) such that πv is unramified

MODULARITY LIFTING THEOREMS 17
for all v /∈ S and for all v|p, and such that the central character of π is
z ↦→ ı((Nzp) k−2χ0(z ∞ ))(Nz∞) 2−k . The correspondence is given by π ↦→ θ,
where for example θ(Tv) is the eigenvalue of Tv on ı−1 � �
GL2(OFv )
π .
We wish to construct a Galois representation ρU : GF → GL2(TU). This
should be compatible with the above bijection in the sense that if π corresponds
to θ : TU → C as above, then the composite GF → GL2(TU) →
GL2(Q p) (given by ı −1 θ) is just rp(π) up to isomorphism.
This representation is constructed by “patching together” the representations
rp(π). We will not give the details, noting only that the key input
is the following result of Carayol.
Fact 3.9. Let R be a complete local Noetherian O-algebra with residue field
k, and let ρ1, ρ2 : Γ → GLn(R) be continuous representations, where Γ is a
topologically finitely generated profinite group. Assume that ρ 1 = ρ 2 and is
absolutely irreducible. Then:
(i) if a ∈ GLn(R) and aρ1a −1 = ρ1 then a ∈ R × .
(ii) if tr ρ1 = tr ρ2 then there exists a ∈ ker(GLn(R) → GLn(k)) such
that aρ1a −1 = ρ2.
(iii) if S is a closed subring of R and is also a complete local Noetherian
O-algebra with residue field k, with mS = mR ∩ S, and tr(ρ(GK)) ⊂
S, then there is an a ∈ ker(GLn(R) → GLn(k)) such that aρa −1 :
GK → GLn(S).
The proof of this is elementary but somewhat involved; one reduces to
the Artinian case and argues by induction on the length of R, reducing to
(in the third case) R = k[ɛ]/(ɛ2 ), S = k, where one argues directly (note
though that even this case is not trivial).
We will now return to the main argument, and will abuse the truth by
pretending that Q is a totally real field of even degree, allowing us to use
the above spaces and notation [of course, in reality we would make a base
change to put ourselves in that situation]. Now choose χ0 to correspond
to the central character of our given π. For each finite set of places Q
not containing l or p, let UQ = � UQ,v, where UQ,v = GL2(OF,v) unless
either v = l �or v �∈
Q. If v = l we take the subset U1(v) of matrices
1 ∗
congruent to (mod v), and if v ∈ Q we take the subset of matrices
0 1
� �
a b
∈ U0(v) (the matrices which are upper-triangular modulo v) such
c d
that a/d ↦→ 1 ∈ ∆v, the maximal p-power quotient of k × v . Define U ′ Q in the
same way, but using U0(v) at v = l. If ζ is a p-th root of unity (possibly equal
to 1), we define SQ,ζ(U, O) to be the subspace of Sk,χ0 (UQ, O) consisting of
(ul)u−1 p φ(g) for all u ∈ U ′ Q , where
functions φ which satisfy φ(gu) = ψ −1
ζ
v

18 TOBY GEE
ψζ : U0(l) → O × � �
a b
is the composite of the character ↦→ a/d (mod l)
c d
with the character corresponding under local class field theory to σ ↦→ ζ.
Now let m∅ denote the maximal ideal of the Hecke algebra T1,U (which
∅
acts on S1,∅(U, O)) corresponding to the Hecke eigenvalues of π mod p, and
write S1,∅ := S(U∅, O)m , T ∅ 1,∅ := TU∅,m . Let ζ be a primitive p-th root
∅
of unity, and define Sζ,∅ and Tζ,∅ in the same way. Note that we have
Sζ,∅/λ = S1,∅/λ. In the same way as above, we find that there are Galois
representations ρmod 1,∅ : GQ → GL2(T1,∅) and ρmod ζ,∅ : GQ → GL2(Tζ,∅) lifting
ρ.
Furthermore, it follows from local-global compatibility that there are natural
maps Runiv 1 → T1,∅ and Runiv ζ → Tζ,∅. These maps are actually surjective
(consider the images of the characteristic polynomials of the Frobenius
elements). So to complete the proof it suffices to show that
(as we would then have that ker(R univ
1
SuppRuniv(S1,∅) = Spec R
1
univ
1
→ T1,∅) is nilpotent (see the material
on nearly faithful modules below), so any O-point of Runiv 1 is a O-point of
T1,∅ and is thus modular).
It turns out to be easier to prove the corresponding statement for Runiv ζ ;
this is the “minimal” case. Before 2006, the statement for Runiv 1 was deduced
from this one by Ihara’s lemma, which is unknown in dimension greater than
two, and was a significant obstacle to extending the theory. The argument
that we give here is Richard Taylor’s version of the Taylor-Wiles-Kisin argument,
which avoids the use of Ihara’s lemma.
Now, for each Q = QN as above, we consider a Hecke algebra TUQ acting
on S(UQ, O). It is generated by the operators Tv, Sv for all places v /∈ Q,
v ∤ l, p, and by the Hecke operators
� � � �
πv 0
Uv = UQ,v UQ,v
0 1
for v ∈ Q, where πv is a uniformiser for OFv (as the notation suggests, Uv is
independent of the choice of uniformiser). There is a maximal ideal mQ of
TUQ , which is induced by m∅ and the requirement that it also contains Uv−αv
for each v ∈ Q. We write S1,Q = SQ,1(U, O)mQ , T1,Q = (TUQ )mQ , and define
Sζ,Q, Tζ,Q similarly. Again, we have a natural surjection Runiv 1,Q → T1,Q, and
thus we can view S1,Q as an Runiv 1,Q -module.
Fact 3.10. S1,Q is finite and free over O[∆Q], and the quotient S1,Q/aQ ∼ =
S 1,∅.
[This is not so hard to prove; one uses local-global compatibility to express
the action of ∆Q in terms of explicit Hecke operators, and reduces to a
concrete statement. It is worth noting however that this statement is far

MODULARITY LIFTING THEOREMS 19
from formal in general, and e.g. would fail if we were working with modular
forms over an imaginary quadratic field.]
Define S � 1,Q := SQ ⊗ R univ
1,Q R� 1,Q , an R� 1,Q -module. Similarly, define T� 1,Q =
T1,Q ⊗ R univ
1,Q R� 1,Q = T1,Q[[A1, . . . , A7]]. For each N, choose O[[T1, . . . , Tu]] ↠
O[∆QN ] (which we can do because #QN = u). This gives us a map
O[[T1, . . . , Tu, A1, . . . , A7]] → R� 1,QN . Since R� is topologically generated
1,QN
over R loc
1
by u elements, we can also choose a surjection Rloc
1 [[X1, . . . , Xu]] →
R � 1,QN .
We now wish to “take the limit as N → ∞”. It’s not immediately
clear that we can do this, because the sets of primes QN are unrelated
for different N. However, this isn’t a problem; the trick is to consider each
S � 1,QM /(pN , A N 1 , . . . , AN 7 ) as a module for each (O/pN )[(Z/p N ) u ][A1, . . . , A7]/(A N 1 , . . . , AN 7 ),
M ≥ N. For fixed N there are only finitely many isomorphism classes of
such modules, and this allows us to choose a sequence of pairs (M, N) for
which the resulting modules can be patched together.
In fact, we also need to patch the actions of R loc
1 [[X1, . . . , Xu]], and we
also want to do the same thing simultaneously for the ζ-objects. Having
carried this out, we find ourselves in the following situation.
◦ S � 1,∞ and S� ζ,∞ are finite free O[[T1, . . . , Tu, A1, . . . , A7]]-modules,
and are also Rloc 1 [[X1, . . . , Xu]] (respectively Rloc ζ [[X1, . . . , Xu]])-modules.
In fact, the action of O[[T1, . . . , Tu, A1, . . . , A7]] on S� 1,∞ (respectively
on S� ζ,∞ ) may be factored through Rloc 1 [[X1, . . . , Xu]] (respectively
Rloc ζ [[X1, . . . , Xu]]). [For this last point, note that it suffices to
observe that the image of O[[T1, . . . , Tu, A1, . . . , A7]] in EndO(S� ?,∞ )
is contained in the image of Rloc ? [[X1, . . . , Xu]], which follows from
the corresponding result at each finite level.]
◦ S� 1,∞ /λ and S� ζ,∞λ are isomorphic as modules for k[[T1, . . . , Tu, A1, . . . , A7]]
and for (Rloc 1 /λ)[[X1, . . . , Xu]] = (Rloc ζ /λ)[[X1, . . . , Xu]].
◦ S� 1,∞ /(T1, . . . , Tu, A1, . . . , A7) ∼ = S1,∅ as an Runiv 1 -module, and
S� ζ,∞ /(T1, . . . , Tu, A1, . . . , A7) ∼ = Sζ,∅ as an Runiv ζ -module.
We now employ some commutative algebra.
Definition 3.11. Let A be a Noetherian ring, and let M be a finitely generated
A-module. Say that M is nearly faithful over A if any of the following
equivalent conditions hold:
(i) Supp M = Spec A.
(ii) Supp M contains all the minimal primes of Spec A.
(iii) AnnA M is nilpotent.
This terminology was introduced by Richard Taylor. We will need 3
general results about near faithfulness.
Lemma 3.12. (i) If M is a nearly faithful A-module and I is an ideal
of A, then M/IM is a nearly faithful A/I-module.

20 TOBY GEE
(ii) Suppose that A is a catenary local O-algebra, that each irreducible
component of Spec A has the same dimension, and that each generic
point of Spec A has characteristic 0. Suppose that every prime of A
minimal over λA contains a unique minimal prime of A. Let M be
a finitely generated O-torsion free A-module. If M/λM is a nearly
faithful A/λA-module, then M is a nearly faithful A-module.
(iii) Suppose that A is a Noetherian local ring, that Spec A is irreducible,
and that M is a finitely generated A-module. Suppose that the mAdepth
of M is greater than or equal to the Krull dimension of A.
Then M is a nearly faithful A-module.
Proof. (i) We need to show that (M/IM)P �= 0 for every prime I ⊂ P .
If not, then we have MP /IP MP = 0, and by Nakayama’s lemma we
conclude that MP = 0, a contradiction.
(ii) This requires slightly more work, but is still elementary. See Lemma
2.2(2) of [Tay08].
(iii) This uses some homological algebra - see Lemma 2.3 of [Tay08].
�
We now apply these results as follows. Consider firstly S � ζ,∞
as a mod-
ule for R loc
ζ [[X1, . . . , Xu]]. We employ the third criterion above with A =
Rloc ζ [[X1, . . . , Xu]]. Since Spec Rloc ζ,l
is irreducible, we see that Spec Rloc
ζ [[X1, . . . , Xu]]
is irreducible, so that it is enough to check that the mA-depth of S� ζ,∞ is at
least dim A. But A has the same dimension as O[[T1, . . . , Tu, A1, . . . , A7]],
considered as an
so it is enough to prove the corresponding assertion for S� ζ,∞
O[[T1, . . . , Tu, A1, . . . , A7]]-module. But S� ζ,∞ is a finite free O[[T1, . . . , Tu, A1, . . . , A7]]module,
so this is immediate, and we conclude that S� ζ,∞ is a nearly faithful
Rloc ζ [[X1, . . . , Xu]]-module.
From the third criterion, we see that S� ζ,∞ /λ is a nearly faithful (Rloc
ζ λ)[[X1, . . . , Xu]]module,
so that S� 1,∞ /λ is a nearly faithful (Rloc 1 λ)[[X1, . . . , Xu]]-module.
We now use the second criterion to conclude that S� 1,∞ is a nearly faithful
Rloc 1 [[X1, . . . , Xu]]-module (note that any complete local noetherian ring is
catenary (even excellent)). Finally, using the first criterion again we see that
S� 1,∞ /(T1, . . . , A1, . . . , A7) ∼ = S1,∅ is a nearly faithful Runiv 1 -module.
As above, this means that ker(Runiv 1 → T1,∅) is nilpotent, so any O-point
of Runiv 1 is a O-point of T1,∅ and is thus modular. In particular ρ is modular,
as required.
References
[Tay08] Richard Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois
representations. II, Pub. Math. IHES 108 (2008), 183–239.
Department of Mathematics, Northwestern University, 2033 Sheridan Road,
Evanston, IL 60208-2730, USA
E-mail address: gee@math.northwestern.edu