c Cambridge University Press
Journal of the Inst. of Math. Jussieu (2009) 8(4), 669–692

doi:10.1017/S1474748009000036
Printed in the United Kingdom

669

A DEFORMATION PROBLEM FOR GALOIS REPRESENTATIONSOVER IMAGINARY QUADRATIC FIELDSTOBIAS BERGER1 AND KRZYSZTOF KLOSIN2
1

University of Cambridge, Department of Pure Mathematics andMathematical Statistics, Centre for Mathematical Sciences,
Cambridge CB3 0WB, United Kingdom (tberger@cantab.net)
2
Cornell University, Department of Mathematics, 310 Malott Hall,
Ithaca, NY 14853-4201, USA (klosin@math.cornell.edu)(Received 29 February 2008; accepted 11 May 2008)

Abstract We prove the modularity of minimally ramiﬁed ordinary residually reducible p-adic Galois
representations of an imaginary quadratic ﬁeld F under certain assumptions. We ﬁrst exhibit conditions
under which the residual representation is unique up to isomorphism. Then we prove the existence of
deformations arising from cuspforms on GL2 (AF ) via the Galois representations constructed by Taylor
et al. We establish a suﬃcient condition (in terms of the non-existence of certain ﬁeld extensions which
in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be adiscrete valuation ring and in that case we prove an R = T theorem. We also study reducible deformations
and show that no minimal characteristic 0 reducible deformation exists.
Keywords: Galois deformations; automorphic forms; imaginary quadratic ﬁelds; modularity
AMS 2000 Mathematics subject classiﬁcation: Primary 11F80; 11F55

1. Introduction
Starting with the work of Wiles [35,44] there has been a lot of progress in recent years on
modularity results for two-dimensional p-adic Galois representations of totally real ﬁelds(see, for example, [6, 14, 19, 29–31, 34]). The goal of this paper is to prove such a resultfor imaginary quadratic ﬁelds, a case that requires new techniques since the associatedsymmetric space√ has no√complex structure.
Let F = Q( −1), Q( −3) be an imaginary quadratic ﬁeld of discriminant dF . Undercertain assumptions we prove an ‘R = T ’ theorem for residually reducible twodimensional representations of the absolute Galois group of F . We pin down conditions(similar to [29], where an analogous problem is treated for representations of Gal(Q̄/Q))
that determine our residual representation up to isomorphism and then study its minimal ordinary deformations. Modular deformations are constructed using the congruences
involving Eisenstein cohomology classes of [3] and the result of Taylor on associating
Galois representations to certain cuspidal automorphic representations over imaginary

670

T. Berger and K. Klosin

quadratic ﬁelds (using the improvements of [5]). The approach of [29] to prove the isomorphism between universal deformation ring and Hecke algebra fails in our case becauseof the non-existence of an ordinary reducible characteristic 0 deformation. This failure,
however, allows under an additional assumption to show (using the method of [1]) that
the Eisenstein deformation ring is a discrete valuation ring. As in [7] it is then easy todeduce an ‘R = T ’ theorem.
To give a more precise account, let c be the non-trivial automorphism of F , and letp > 3 be a prime split in the extension F/Q. Fix embeddings F → Q̄ → Q̄p → C. Let
FΣ be the maximal extension of F unramiﬁed outside a ﬁnite set of places Σ and put
GΣ = Gal(FΣ /F ). Suppose F is a ﬁnite ﬁeld of characteristic p and that χ0 : GΣ → F ×
is an anticyclotomic character ramiﬁed at the places dividing p. Suppose also that ρ0 :GΣ → GL2 (F ) is a continuous representation of the form


1 ∗
ρ0 =
0 χ0
and having scalar centralizer. Under certain conditions on χ0 and Σ we show that ρ0 is
unique up to isomorphism (see § 3) and we ﬁx a particular choice. This setup is similar
to that of [29]. Note that, as explained in Remark 4.6, under our conditions ρ0 does notarise as twist by a character of the restriction of a representation of Gal(Q̄/Q).
Following Mazur [22] we study ordinary deformations of ρ0 . Let O be a local complete Noetherian ring with residue ﬁeld F . An O-deformation of ρ0 is a local complete
Noetherian O-algebra A with residue ﬁeld F and maximal ideal mA together with astrict equivalence class of continuous representations ρ : GΣ → GL2 (A) satisfying ρ0 = ρ
mod mA . An ordinary deformation is a deformation that satisﬁes


∗
χ
1
ρ|Dv ∼=
0 χ2
for v | p, where χi |Iv = ki with integers k1  k2 depending on v and  is the p-adic
cyclotomic character. Here Dv and Iv denote the decomposition group and the inertia
group of v | p, corresponding to F → Q̄p or the conjugate embedding, respectively.
To exhibit modular deformations we apply the cohomological congruences of [3] andthe Galois representations constructed by Taylor et al . using a strengthening of Taylor’s
result in [5]. We also make use of a result of Urban [40] who proves that ρπ |Dv is ordinary
at v | p if π is ordinary at v. We show that these results imply that there is an O-algebra
surjection
R  T,(1.1)
where R is the universal Σ-minimal deformation ring (cf. Deﬁnition 5.1) and T is a Hecke
algebra acting on cuspidal automorphic forms of GL2 (AF ) of weight 2 and ﬁxed level.
As in [7] we can deduce that the surjection (1.1) is, in fact, an isomorphism if R is adiscrete valuation ring (see Theorem 5.7). Using the method of [1] we prove in Proposition 5.8 that the latter reduces to the non-existence of reducible Σ-minimal deformations
to GL2 (O/2 O) (where  denotes a uniformizer of O). We then show (Theorem 5.12)

A deformation problem

671

that this last property can often be deduced from a condition on the L-value at 1 of a
Hecke character of inﬁnity type z/z̄ which is related to χ0 . Finally, we combine theseresults in Theorem 5.16 to prove the modularity of certain residually reducible Σ-minimalGΣ -representations. For an explicit numerical example where we can verify the conditionsof Theorem 5.16 see Example 5.19.
To demonstrate our modularity result we give here the following special case.
Theorem 1.1. Assume # ClF = 1, that p does not divide the class number of the rayclass ﬁeld of F of conductor p, and that any prime q | dF satisﬁes q ≡ ±1 (mod p). Let p
be the prime of F over (p) corresponding to the embedding F → Q̄ → Q̄p that we haveﬁxed. Let τ be the unramiﬁed Hecke character of inﬁnity type τ∞ (z) = z/z̄ and let τp :GΣ → Zp× be the associated p-adic Galois character. Assume that valp (Lint (1, τ )) = 1.
(For deﬁnitions see § 2.)
Let ρ : GΣ → GL2 (Q̄p ) be a continuous irreducible representation that is ordinary at
all places v | p. Suppose ρ̄ss ∼= 1 ⊕ τ̄p . If the following conditions are satisﬁed:(1) Σ ⊃ {v | pdF },(2) if v ∈ Σ, v  p, then τ̄p (Frobv ) = (#kv )±1 as elements of Fp ,(3) det(ρ) = τp ,(4) ρ is Σ-minimal,
then ρ is isomorphic to the Galois representation associated to a cuspform of GL2 (AF )of weight 2, twisted by the p-adic Galois character associated to a Hecke character ofinﬁnity type z.
We also study the existence of reducible deformations (see § 5.5). In contrast to thesituation in [29] there exists no reducible Σ-minimal O-deformation in our case, only anearly ordinary (in the sense of Tilouine [37]) reducible deformation which is, however,not de Rham at one of the places above p. This means that the method of [29] to prove
R = T via the numerical criterion of Wiles and Lenstra [20, 44] cannot be implemented
despite having all the ingredients on the Hecke side (i.e. a lower bound on the congruence
module measuring congruences between cuspforms and Eisenstein series).
The assumption on χ0 being anticyclotomic is not necessary for the methods of thispaper or the congruence result we use (see [3, Theorem 13]) but is necessary for constructing the modular deformations, and is related to a condition on the central character in
Taylor’s result on associating Galois representation to cuspforms. The restrictions in Definition 3.2 on the places contained in Σ and on the class group of the splitting ﬁeld of χ0
are similar to those of [29] and are essential for the uniqueness of ρ0 . Our methods do notallow to go beyond the Σ-minimal case (to achieve that in the Q-case, [29] uses Proposition 1 of [35], but its analogue fails for imaginary quadratic ﬁelds) or treat residually
irreducible Galois representations. To complement our study of the absolute deformation problem of a residually reducible Galois representation the reader is referred to theanalysis of the nearly ordinary relative deformation problem in [9].

672

T. Berger and K. Klosin

2. Notation and terminology
2.1. Galois groups
Let F be an imaginary quadratic extension of Q of discriminant dF = 3, 4 and p > 3 arational prime which splits in F . Fix a prime p of F lying over (p) and denote the otherprime of F over (p) by p̄. Let ClF denote the class group of F . We assume that p  # ClF
and that any prime q | dF satisﬁes q ≡ ±1 (mod p).
For a ﬁeld K write GK for the Galois group Gal(K̄/K). If K is a ﬁnite extension ofQ for some rational prime , we write OK (respectively K , and FK ) for the ring ofintegers of K (respectively for a uniformizer of K, and OK /K OK ).
If K ⊃ F is a number ﬁeld, OK will denote its ring of integers. If v is a place of K, we
write Kv for the completion of K with respect to the absolute value | · |v determined byv and set OK,v = OKv (if v is archimedean, we set OK,v = Kv ). We also write v for a
uniformizer of Kv , Pv for the maximal ideal of OK,v , and kv for its residue ﬁeld.
Fix once and for all compatible embeddings iv : F̄ → F̄ v and F̄ v → C, for every prime
v of F , so we will often regard elements of F̄ v as complex numbers without explicitly
mentioning it. If w is a place of K ⊂ F̄ , it determines a place v of F , and we always
regard Kw as a subﬁeld of F̄ v as determined by the embedding iv . This also allows us to
identify GKw with the decomposition group Dv̄ ⊂ GK of a place v̄ of F̄ . We will denotethat decomposition group by Dv . Abusing notation somewhat we will denote the imageof Dv in any quotient of GK also by Dv . We write Iv ⊂ Dv for the inertia group.
Let Σ be a ﬁnite set of places of K. Then KΣ will denote the maximal Galois extensionof K unramiﬁed outside the primes in Σ. We also write GΣ for GFΣ . Moreover, for apositive integer n, denote by µn the group of nth roots of unity.
2.2. Hecke characters
For a number ﬁeld K, denote by AK the ring of adeles of K and set A = AQ . By a
Hecke character of K we mean a continuous homomorphism×
λ : K × \ A×
K →C .

For a place v of K write λ(v) for the restriction of λ to Kv and λ(∞) for the restrictionof λ to v|∞ Kv . The latter will be called the inﬁnity type of λ. We also usually writeλ(v ) to mean λ(v) (v ). Given λ there exists a unique ideal fλ of K largest with respect×
to the following property: λ(v) (x) = 1 for every ﬁnite place v of K and x ∈ OK,v
suchthat x − 1 ∈ fλ OK,v . The ideal fλ is called the conductor of λ. If K = F , there is only
one archimedean place, which we will simply denote by ∞. For a Hecke character λ of F ,one has λ(∞) (z) = z m z̄ n with m, n ∈ R. If m, n ∈ Z, we say that λ is of type (A0 ). We
always assume that our Hecke characters are of type (A0 ). Write L(s, λ) for the Hecke
L-function of λ.
Let λ be a Hecke character of inﬁnity type z a (z/z̄)b with conductor prime to p. Assumea, b ∈ Z and a > 0 and b  0. Put

b
2π
alg
−a−2b
√
L (0, λ) := ΩΓ (a + b)L(0, λ),dF

A deformation problem

673

where Ω is a complex period. In most cases, this normalization is integral, i.e. lies in the
integer ring of a ﬁnite extension of Fp . See [4, Theorem 3] for the exact statement. PutLalg (0, λ) if valp (Lalg (0, λ))  0,int
L (0, λ) =1
otherwise.
For z ∈ C we write z̄ for the complex conjugate of z. The action of complex conjugation
×extends to an automorphism of A×
F and we will write x̄ for the image of x ∈ AF underthat automorphism.
For a Hecke character λ of F , we denote by λc the conjugate Hecke character of F
deﬁned by λc (x) = λ(x̄).
2.3. Galois representations
For a ﬁeld K and a topological ring R, by a Galois representation we mean a continuous
homomorphism ρ : GK → GLn (R). If n = 1 we usually refer to ρ as a Galois character.
We write K(ρ) for the ﬁxed ﬁeld of ker ρ and call it the splitting ﬁeld of ρ. If ρ is a Galois
character and M is an R-module, we denote by M (ρ) the R[GK ]-module M with the
GK -action given by ρ. If K is a number ﬁeld and v is a ﬁnite prime of K with inertia
group Iv we say that ρ is unramiﬁed at v if ρ|Iv = 1. If Σ is a ﬁnite set of places of K suchthat ρ is unramiﬁed at all v ∈ Σ, then ρ can be regarded as a representation of GKΣ .
Let E be a ﬁnite extension of Qp . Every Galois representation ρ : GK → GLn (E) can
be conjugated (by an element M ∈ GLn (E)) to a representation ρM : GK → GLn (OE ).
We denote by ρ̄M : GK → GLn (FE ) its reduction modulo E OE . It is sometimes calleda residual representation of ρ. The isomorphism class of its semisimpliﬁcation ρ̄ss
M isindependent of the choice of M and we simply write ρ̄ss .
Let  : GF → Zp× denote the p-adic cyclotomic character. For any subgroup G ⊂ GF
we will also write  for |G . Our convention is that the Hodge–Tate weight of  at p is 1.
Let λ be a Hecke character of F of type (A0 ) and Σλ = {v | pfλ }. We deﬁne (followingWeil) a p-adic Galois character
λp : GΣλ → F̄p×
associated to λ by the following rule. For a ﬁnite place v  pfλ of F , put λp (Frobv ) =ip (i−1∞ (λ(v ))) where Frobv denotes the arithmetic Frobenius at v. It takes values in the
integer ring of a ﬁnite extension of Fp .
Deﬁnition 2.1. For a topological ring R we call a Galois representation ρ : GΣ →GL2 (R) ordinary if


∗
χ
1
ρ|Dv ∼=
0 χ2
for v | p, where χi |Iv = ki with integers k1  k2 depending on v.
2.4. Automorphic representations of AF and their Galois representations
Set G = ResF/Q GL2 . For Kf an open compact subgroup of G(Af ), denote by S2 (Kf )the space of cuspidal automorphic forms of G(A) of weight 2, right-invariant under Kf

674

T. Berger and K. Klosin(for more details see § 3.1 of [38]). For ψ a ﬁnite order Hecke character write S2 (Kf , ψ)
 Kf
πffor the forms with central character ψ. This is isomorphic as a G(Af )-module tofor automorphic representations π of a certain inﬁnity type (see Theorem 2.2 below) withK
central character ψ. Here πf denotes the restriction of π to GL2 (Af ) and πf f stands for
the Kf -invariants.
For g ∈ G(Af ) we have the usual Hecke action of [Kf gKf ] on S2 (Kf ) and S2 (Kf , ψ).
For primes v such that the vth component of Kf is GL2 (OF,v ) we deﬁnev
Tv = Kf
Kf .
1
Combining the work of Taylor, Harris and Soudry with results of Friedberg–Hoﬀsteinand Laumon or Weissauer, one can show the following (see [5] for general case of cuspforms of weight k).
Theorem 2.2 (Berger and Harcos [5, Theorem 1.1]). Given a cuspidal automorphic
representation π of GL2 (AF ) with π∞ isomorphic to the principal series representation
corresponding to




|t2 |
t1
t1 ∗
→
t2
|t1 |
t2
and cyclotomic central character ψ (i.e. ψ c = ψ), let Σπ denote the set consisting of theplaces of F lying above p, the primes where π or π c is ramiﬁed, and the primes ramiﬁedin F/Q.
Then there exists a ﬁnite extension E of Fp and a Galois representationρπ : GΣπ → GL2 (E)such that if v ∈ Σπ , then ρπ is unramiﬁed at v and the characteristic polynomial ofρπ (Frobv ) is x2 −av (π)x+ψ(v )(#kv ), where av (π) is the Hecke eigenvalue corresponding
to Tv . Moreover, ρπ is absolutely irreducible.
Remark 2.3. Taylor has some additional technical assumption in [33] and only showed
the equality of the Hecke and Frobenius polynomial outside a set of places of zero density.
Conjecture 3.2 in [8] describes a conjectural extension of Taylor’s theorem.
Urban studied in [39] the case of ordinary automorphic representations π, and together
with results in [40] on the Galois representations attached to ordinary Siegel modular
forms proved the following theorem.
Theorem 2.4 (Urban [40, Corollary 2]). Let v be a prime of F lying over p. If π isunramiﬁed at v and ordinary at v, i.e. |av (π)|v = 1, then the Galois representation ρπ isordinary at v. Moreover,

Ψ1 ∗
∼
,
ρπ |Dv =Ψ2
where Ψ2 |Iv = 1 and Ψ1 |Iv = det ρπ |Iv = .

A deformation problem

675

Deﬁnition 2.5. Let E be a ﬁnite extension of Fp and ρ : GΣ → GL2 (E) a Galois
representation for a ﬁnite set of places Σ. We say that ρ is modular if there exists a
cuspidal automorphic representation π as in Theorem 2.2, such that ρ ∼= ρπ (possibly
after enlarging E).
From now on we ﬁx a ﬁnite extension E of Fp which we assume to be suﬃciently
large (see § 4.2 and Remark 4.5, where this condition is made more precise). To simplify
notation we put O := OE , F = FE and  = E .
3. Uniqueness of a certain residual Galois representation
Let Σ be a ﬁnite set of ﬁnite primes of F containing the primes lying over p. In thissection we study residual Galois representations ρ0 : GΣ → GL2 (F ) of the form

1 ∗
ρ0 =
χ0
having scalar centralizer for a certain class of characters χ0 (cf. Deﬁnition 3.2). We
show that for a ﬁxed χ0 there exists at most one such representation up to isomorphism(Corollary 3.7). The existence of such a representation follows from the global Euler
characteristic formula [24, Theorem 5.1], which implies
dimF (H 1 (GΣ , F (χ−10 )))  1.
In § 4 we show that there exists an ordinary one provided that valp (Lint (0, φ)) > 0 for acertain Hecke character φ of F such that the reduction of φp  is χ0 . Alternatively, one
could invoke the generalizations of Kummer’s criterion to imaginary quadratic ﬁelds (see,for example, [11, 17, 21, 45]).
Deﬁnition 3.1. Let R be a commutative ring, J ⊂ R an ideal, M a free R-module andN a submodule of M . We will say that N is saturated with respect to J if
N = {m ∈ M | mJ ⊂ N }.
Let χ0 : GΣ → F × be a Galois character. Let Sp be the set of primes of F (χ0 ) lyingover p. Write Mχ0 for v∈Sp (1 + Pv ) and Tχ0 for its torsion submodule. The quotientMχ0 /Tχ0 is a free Zp -module of ﬁnite rank. Let Ēχ0 be the closure in Mχ0 /Tχ0 of theimage of Eχ0 , the group of units of the ring of integers of F (χ0 ) which are congruent to 1
modulo every prime in Sp .
We will now restrict ourselves to studying a certain class of characters χ0 , which we willcall Σ-admissible. The deﬁnition of Σ-admissibility (Deﬁnition 3.2) pins down conditions
under which we are able to prove that the space Ext1GΣ (1, χ0 ), where 1 denotes the trivial
character, is one dimensional, i.e. that ρ0 is unique up to isomorphism (Corollary 3.7).
Deﬁnition 3.2. We say that χ0 is Σ-admissible if all of the following conditions are
satisﬁed.

676

T. Berger and K. Klosin(1) χ0 is ramiﬁed at p.
(2) If v ∈ Σ, then either χ0 is ramiﬁed at v or χ0 (Frobv ) = (#kv )±1 (as elementsof F ).
(3) χ0 is anticyclotomic, i.e. χ0 (cσc) = χ0 (σ)−1 for every σ ∈ GΣ and c the generatorof Gal(F/Q).
(4) The Zp -submodule Ēχ0 ⊂ Mχ0 /Tχ0 is saturated with respect to the ideal pZp .
(5) The χ−1
0 -eigenspace of the p-part of ClF (χ0 ) is trivial.
Remark 3.3. Note that Conditions (1) and (3) of Deﬁnition 3.2 imply that χ0 is also
ramiﬁed at p̄. Moreover, observe that χ0 is Σ-admissible if and only if χ−1
is. Indeed,0
Conditions (1)–(4) in Deﬁnition 3.2 are invariant under taking the inverse. Moreover,since χ0 is anticyclotomic, the extension F (χ0 )/Q is Galois, hence the χ−1
0 -eigenspace ofthe p-part of ClF (χ0 ) vanishes if and only if the χ0 -eigenspace does.
Remark 3.4. Condition (4) in Deﬁnition 3.2 implies that every  ∈ Ēχ0 which is a pth
power of an element of Mχ0 /Tχ0 must be a pth power of an element of Ēχ0 . In particular
the mapĒχ0 ⊗Zp F̄p → (Mχ0 /Tχ0 ) ⊗Zp F̄pis injective. One way to check Condition (4) in practice is to compute the p-part C of theclass group of F (χ0 )(µp ) as a Gal(F (χ0 )(µp )/F (χ0 ))-module. In particular, if ω denotes
the character of Gal(F (χ0 )(µp )/F (χ0 )) giving the action on µp , then Kummer theory
implies that Condition (4) is satisﬁed if the ω-part of C is trivial.
Letρ0 : GΣ → GL2 (F )be a Galois representation satisfying both of the following two conditions
1 ∗(Red) ρ0 =for a Σ-admissible character χ0 ;
χ0(Sc) ρ0 has scalar centralizer.
We have the following tower of ﬁelds: F ⊂ F (χ0 ) ⊂ F (ρ0 ). Note that p does notdivide [F (χ0 ) : F ]. Moreover, F (ρ0 )/F (χ0 ) is an abelian extension of exponent p,
hence Gal(F (ρ0 )/F (χ0 )) can be regarded as an Fp -vector space V0 on which the group
G := Gal(F (χ0 )/F ) operates Fp -linearly by conjugation and thus deﬁnes a representation
r0 : G → GLFp (V0 ),which is isomorphic to the irreducible Fp -representation associated with χ0−1 .

A deformation problem

677

Let L denote the maximal abelian extension of F (χ0 ) unramiﬁed outside the set Σ
and such that p annihilates Gal(L/F (χ0 )). Then, as before, V := Gal(L/F (χ0 )) is an
Fp -vector space endowed with an Fp -linear action of G, and one has
V ⊗Fp F̄p ∼=V ϕ,ϕ∈Hom(G,F̄p× )where for a Zp [G]-module N and an F̄ p -valued character ϕ of G, we write
N ϕ = {n ∈ N ⊗Zp F̄p | σn = ϕ(σ)n for every σ ∈ G}.

(3.1)

−1

Note that V0 ⊗Fp F̄p is a direct summand of V χ0 .
−1

Theorem 3.5. If χ0 is Σ-admissible, then dimF̄p V χ0 = 1.
Proof . Let L0 be the maximal abelian extension of F (χ0 ) of exponent p unramiﬁed
outside the set Σ and such that G acts on Gal(L0 /F (χ0 )) via the irreducible Fp -representation associated with χ−1
0 . It is enough to show that
dimF̄p (Gal(L0 /F (χ0 )) ⊗ F̄p )  1.
Condition (2) of Deﬁnition 3.2 ensures that L0 /F (χ0 ) is unramiﬁed outside the set{p, p̄}. Hence it is enough to study the extensions L/F (χ0 ) and L0 /F (χ0 ) with Σ ={p, p̄}. For p0 ∈ {p, p̄} let Sp0 be the set of primes of F (χ0 ) lying over p0 . Then Sp :=Sp ∪ Sp̄ . Write M (respectively T , Ē) for Mχ0 (respectively Tχ0 , Ēχ0 ). By Condition (5)of Deﬁnition 3.2 and class ﬁeld theory (see, for example, Corollary 13.6 in [42]) one has
Gal(L/F (χ0 )) ∼= (M/Ē) ⊗ Fp . Hence Gal(L0 /F (χ0 )) is a quotient of (M/Ē) ⊗ Fp . On
the other hand, using Condition (3) of Deﬁnition 3.2 one can show that Gal(L0 /F (χ0 ))is a quotient of (M/T ) ⊗ Fp . This follows from the fact that T is a product of the groupsµp ; thus χ0 being anticyclotomic by Condition (3) of Deﬁnition 3.2 cannot occur in T .
We will now study both (M/T ) ⊗ F̄ p and (M/Ē) ⊗ F̄ p , beginning with the former one.
Let G∨ := Hom(G, F̄ ×
p ). Since G is abelian, (M/T ) ⊗ F̄ p decomposes into a direct
sum of F̄ p [G]-modules(M/T ) ⊗ F̄p =(M/T )ψ ,ψ∈G∨with (M/T )ψ deﬁned as in (3.1). Note that we can reﬁne this by writing
Mp0 /Tp0 ,M/T =p0 ∈{p,p̄}

where Mp0 =G-stable.

v∈Sp0 (1 + Pv ) and Tp0 is the torsion subgroup of Mp0 . Each Mp0 /Tp0 is

Lemma 3.6. Let p0 ∈ {p, p̄}. For every ψ ∈ G∨ , we havedimF̄p (Mp0 /Tp0 )ψ = 1.

678

T. Berger and K. Klosin

Proof of Lemma 3.6. Note that to decompose (Mp0 /Tp0 ) ⊗ F̄ p it is enough to decompose v∈Sp Pv ⊗ F̄p , since (1+Pv )/(torsion) ∼= Pv as Zp [Dv ]-modules. It is not diﬃcult
0
to see thatPv ⊗ F̄p ∼F̄p (φ),
=
φ∈G∨

v∈Sp0

where F̄p (φ) denotes the one-dimensional F̄p -vector space on which G acts via φ. The
lemma follows easily.
We are now ready to complete the proof of Theorem 3.5. Recall that the tensor product
Gal(L0 /F (χ0 )) ⊗ F̄ p is both a quotient of (M/T ) ⊗ F̄ p and of (M/Ē) ⊗ F̄ p . Since(M/T ) ⊗ F̄ p = (Mp /Tp ) ⊗ F̄ p × (Mp̄ /Tp̄ ) ⊗ F̄ p , Lemma 3.6 implies that(F̄p (ψ) ⊕ F̄p (ψ)).

(M/T ) ⊗ F̄p =ψ∈G∨On the other hand, one hasĒ ⊗ F̄p =F̄p (ψ),ψ∈G∨ \{1}where 1 denotes the trivial character. Now, Condition (4) of Deﬁnition 3.2 ensures thatthe mapĒ ⊗ F̄p → (M/T ) ⊗ F̄pis an injection (see Remark 3.4). So, Gal(L0 /F (χ0 )) ⊗ F̄ p is a quotient of((M/T ) ⊗ F̄p )/(Ē ⊗ F̄p ) ∼= F̄p (1) ⊕ F̄p (1) ⊕F̄p (ψ).
ψ∈G∨ \{1}Since χ0 = 1, we have dimF̄p (Gal(L0 /F (χ0 )) ⊗ F̄p )  1, which is what we wanted toshow.
Corollary 3.7. Suppose ρ : GΣ → GL2 (F ) is a Galois representation satisfying conditions (Red) and (Sc). Then ρ ∼= ρ0 .
Proof . As ρ0 and ρ correspond to elements in ExtF̄p [GΣ ] (1, χ0 ) ∼= H 1 (GΣ , F̄p (χ−1
0 )),
−1
1
it is enough to show that H (GΣ , F̄p (χ0 )) is one dimensional as an F̄ p -vector space.
Using the inﬂation-restriction sequence we see that H 1 (GΣ , F̄p (χ−10 )) is isomorphic to
χ−1∼
0 , F̄ (χ−1 )).
HomG (ker χ0 , F̄p (χ−1p0 )) = HomF̄p [G] (V0−1
Since V χ0 ∼= F̄p (χ−10 ) by Theorem 3.5, the corollary follows.



4. Modular forms and Galois representations
In this section we exhibit irreducible ordinary Galois representations that are residually
reducible and arise from weight 2 cuspforms.

A deformation problem

679

4.1. Eisenstein congruences
Let φ1 , φ2 be two Hecke characters with inﬁnity types φ1 (z) = z and φ2 (z) = z −1 .
Put γ = φ1 φ2 .
Denote by S the ﬁnite set of places where both φi are ramiﬁed, but φ = φ1 /φ2 isunramiﬁed. Write Mi for the conductor of φi . For an ideal N in OF and a ﬁnite place v
of F put Nv = NOF,v . We deﬁneab∈ GL2 (OF,v ) a − 1, c ≡ 0 mod Nv ,K 1 (Nv ) =
c d
(∞)

(∞)

and
U 1 (Nv ) = {k ∈ GL2 (OF,v ) | det(k) ≡ 1 mod Nv }.
Now put
K 1 ((M1 M2 )v ) ⊂ G(Af ).

U 1 (M1,v )

Kf :=
v∈S

(4.1)

v ∈S
/

From now on, let Σ be a ﬁnite set of places of F containing
Sφ := {v | M1 Mc1 M2 Mc2 } ∪ {v | pdF }.
We denote by T (Σ) the O-subalgebra of EndO (S2 (Kf , γ)) generated by the Hecke
operators Tv for all places v ∈ Σ. Following [32, p. 107] we deﬁne idempotents ep and ep̄ ,commuting with each other and with T (Σ) acting on S2 (Kf , γ). They are characterized
by the property that any element h ∈ X := ep ep̄ S2 (Kf , γ) which is an eigenvector
for Tp and Tp̄ satisﬁes |ap (h)|p = |ap̄ (h)|p = 1, where ap (h) (respectively ap̄ (h)) is the
Tp -eigenvalue (respectively Tp̄ -eigenvalue) corresponding to h. Let T ord (Σ) denote thequotient algebra of T (Σ) obtained by restricting the Hecke operators to X.
Let J(Σ) ⊂ T (Σ) be the ideal generated by{Tv − φ1 (v )#kv − φ2 (v ) | v ∈ Σ}.
Deﬁnition 4.1. Denote by m(Σ) the maximal ideal of T ord (Σ) containing the image ofJ(Σ). We set TΣ := T ord (Σ)m(Σ) . Moreover, set JΣ := J(Σ)TΣ . We refer to JΣ as theEisenstein ideal of TΣ .
Theorem 4.2 (Berger [2, Theorem 6.3], [3, Theorem 14]). Let φ be an unramiﬁed
Hecke character of inﬁnity type φ(∞) (z) = z 2 . There exist Hecke characters φ1 , φ2 with
φ1 /φ2 = φ such that their conductors are divisible only by ramiﬁed primes or inert primes
not congruent to ±1 mod p and such that#(TΣ /JΣ )  #(O/(Lint (0, φ))).
Proof . Theorem 14 in [3] states this inequality for the Hecke algebra T (Σ). However,the Eisenstein cohomology class used in the proof of Theorem 14 in [3] is ordinary
because by [4, Lemma 9] its Tp -eigenvalue (respectively Tp̄ -eigenvalue) is the p-adic unit
pφ1 (p) + φ2 (p) (respectively pφ1 (p̄) + φ2 (p̄)). Therefore, one can prove the statement forthe ordinary cuspidal Hecke algebra.
680

T. Berger and K. Klosin

Remark 4.3. If φ is unramiﬁed then φp  is anticyclotomic (see [3, Lemma 1]). The
condition on the conductor of the auxiliary character φ1 together with our assumption on
the discriminant of F therefore ensure that for χ0 = φp , Condition (2) of Σ-admissibility
is automatically satisﬁed for all primes v ∈ Sφ .
The assumption on the ramiﬁcation of φ can be relaxed. For example, Proposition 16
and Theorem 28 of [4] and Proposition 9 and Lemma 11 of [3] imply the following
theorem.
Theorem 4.4. Let φ1 , φ2 be as at the start of this section. Assume both M1 and M2 are
coprime to (p) and divisible only by primes split in F/Q and that p  #(OF /M1 M2 )× .
Suppose (φ1 /φ2 )c = φ1 /φ2 . If the torsion part of Hc2 (SKf , Zp ) is trivial, where
SKf = G(Q)\G(A)/Kf U (2)C ×
then#(TΣ /JΣ )  #(O/(Lint (0, φ1 /φ2 ))).
Remark 4.5. In fact, by replacing Zp by the appropriate coeﬃcient system, the result
is true for characters φ1 , φ2 of inﬁnity type z z̄ −m and z −m−1 , respectively, for m  0.
For Theorems 4.2 and 4.4, the ﬁeld E needs to contain the values of the ﬁnite parts of
φ1 and φ2 as well as Lint (0, φ1 /φ2 ).
We will from now on assume that we are either in the situation of Theorem 4.2 orTheorem 4.4 and ﬁx the characters φ1 , φ2 and φ = φ1 /φ2 , with corresponding conditions
on the set Σ and deﬁnitions of Kf , TΣ and JΣ . We also assume from now on thatvalp (Lint (0, φ)) > 0. Put χ0 = φp  and assume that χ0 is Σ-admissible. If we are in thesituation of Theorem 4.4, then suppose also that M1 and M2 are not divisible by any
primes v such that #kv ≡ 1 mod p. (This last assumption is only used in the proof of
Theorem 5.2.)
4.2. Residually reducible Galois representations
Write
K

πf f

S2 (Kf , γ)m(Σ) =π∈ΠΣ

for a ﬁnite set ΠΣ of ordinary cuspidal automorphic representations with central characK
ter γ, such that πf f = 0. The set ΠΣ is non-empty by Theorem 4.2 under our assumptionthat valp (Lint (0, φ)) > 0.
Let π ∈ ΠΣ . Let ρπ : GΣ → GL2 (E) be the Galois representation attached to π by
Theorem 2.2. (This is another point where we assume that E is large enough.) The condition on the central character in Theorem 2.2 can be satisﬁed (after possibly twisting witha ﬁnite character) under our assumptions on φ (see [3, Lemma 8]). The representation/ Sφ , and satisﬁes
ρπ is unramiﬁed at all v ∈tr ρπ (Frobv ) = av (π)

A deformation problem

681

and
det ρπ (Frobv ) = γ(v )#kv .
By deﬁnition, TΣ injects into π∈ΠΣ EndO (π Kf ). Since Tv acts on π by multiplication
by av (π) ∈ O the Hecke algebra TΣ embeds, in fact, into B = π∈ΠΣ O.
Observe that (tr ρπ (σ))π∈ΠΣ ∈ TΣ ⊂ B for all σ ∈ GΣ . This follows from the Chebotarev density theorem and the continuity of ρπ (note that TΣ is a ﬁnite O-algebra).

Fix π ∈ ΠΣ for the rest of this subsection. Deﬁne ρπ := ρπ ⊗ φ−12,p . Then ρπ satisﬁes
tr ρπ (Frob v) ≡ 1 + (φp )(Frob v) (mod )andfor v ∈/ Sφ ,det ρπ = γφ−22,p  = φp .

By choosing a suitable lattice Λ one can ensure that ρπ has image inside GL2 (O). The
Chebotarev density theorem and the Brauer–Nesbitt theorem imply that(ρ̄π )ss ∼= 1 ⊕ φ̄p ¯.
By Theorem 2.2 ρπ is irreducible, so a standard argument (see, for example, Proposition 2.1 in [27]) shows the lattice Λ may be chosen in such a way that ρ̄π is not semisimple
and

1 ∗

ρ̄π =
.
(4.2)
φ̄p ¯Hence ρ̄π satisﬁes conditions (Red) and (Sc) of § 3. By Theorem 2.4, ρπ is ordinary which
combined with (4.2) implies that

1

.
(4.3)
ρ̄π |Dp̄ ∼=(φ̄p ¯)|Dp̄
We putρ0 := ρ̄π .

(4.4)

Remark 4.6. It follows from the proof of Theorem 3.5 that since ρ0 splits when restricted
to Dp̄ (see (4.3)), the extension F (ρ0 )/F (χ0 ) is totally ramiﬁed at p and split at p̄, so,
in particular, there exists τ ∈ Ip such that
1 1
ρ0 (τ ) =.
1
This implies that no twist of ρ0 by a character is invariant under c ∈ Gal(F/Q) and sono character twists of ρ0 and the deformations of ρ0 considered in the following sections
arise from base change.
Furthermore, the ordinary modular deformations of ρ0 in § 5.2 cannot be induced from acharacter of a quadratic extension of F because such representations split when restricted
to the decomposition groups Dv for v | p. This follows from Urban’s result (Theorem 2.3)
and the restriction of these characteristic 0 representations being semisimple on an open
subgroup of each of the decomposition groups.

682

T. Berger and K. Klosin

5. Deformations of ρ0
Let Σ, φ, χ0 and ρ0 be as in § 4. Recall that we have assumed that χ0 is Σ-admissible
and have shown in § 4.2 that ρ0 satisﬁes conditions (Red) and (Sc) of § 3. Hence byCorollary 3.7, ρ0 is unique up to isomorphism. By Remark 4.6 the extension F (ρ0 )/F (χ0 )is ramiﬁed at p but splits at p̄. In this section we study deformations of ρ0 .
5.1. Deﬁnitions
Denote the category of local complete Noetherian O-algebras with residue ﬁeld F by
LCN(E). An O-deformation of ρ0 is a pair consisting of A ∈ LCN(E) and a strict equivalence class of continuous representations ρ : GΣ → GL2 (A) such that ρ0 = ρ (mod mA ),where mA is the maximal ideal of A. As is customary we will denote a deformation bya single member of its strict equivalence class. Note that the Hodge–Tate weights of φp 
are −1 at p and +1 at p̄.
Following [29] we make the following deﬁnition.
Deﬁnition 5.1. We say that an O-deformation ρ : GΣ → GL2 (A) of ρ0 is Σ-minimal
if ρ is ordinary,det ρ = φp ,and for all primes v ∈ Σ such that #kv ≡ 1 (mod p) one has1∼
ρ|Iv =.
φp |Iv
Note that by our assumption on the conductor of φ, we in fact have φp |Iv = 1 for v as
above. Also the ordinarity condition means in this case that


1
∗
 ∗
and ρ|Ip̄ ∼.
ρ|Ip ∼==
−1
1
Since ρ0 has a scalar centralizer and Σ-minimality is a deformation condition in thesense of [22], there exists a universal deformation ring which we will denote by RΣ,O ∈LCN(E), and a universal Σ-minimal O-deformation ρΣ,O : GΣ → GL2 (RΣ,O ) such thatfor every A ∈ LCN(E) there is a one-to-one correspondence between the set of O-algebra
maps RΣ,O → A (inducing identity on F ) and the set of Σ-minimal deformations ρ :GΣ → GL2 (A) of ρ0 .
5.2. Irreducible modular deformations of ρ0
The arguments from § 4.2 together with the uniqueness of ρ0 (Corollary 3.7) can now
be reinterpreted as the following theorem.
Theorem 5.2. For any π ∈ ΠΣ there is an O-algebra homomorphism rπ : RΣ,O  O
inducing ρπ .

A deformation problem

683

Proof . The only property left to be checked is Σ-minimality. This is clear since ρπ isunramiﬁed away from Sφ , and no v ∈ Sφ satisﬁes #kv ≡ 1 (mod p) by construction (ifwe are in the case of Theorem 4.2) or assumption (in the case of Theorem 4.4).

Remark 5.3. The assumption on the conductors of φ1 , φ2 made at the end of § 4.1 could
be relaxed if local–global compatibility was known for the Galois representations constructed by Taylor. For a discussion of the Langlands conjecture for imaginary quadraticﬁelds see [8, Conjecture 3.2].
Proposition 5.4. There does not exist any non-trivial upper-triangular Σ-minimal
deformation of ρ0 to GL2 (F [x]/x2 ).
Proof . Let ρ : GΣ → GL2 (F [x]/x2 ) be an upper-triangular Σ-minimal deformation.
Then ρ has the form
1 + xα∗χ0 + xβ
for α : GΣ → F + a group homomorphism (here F + denotes the additive group of F )and β : GΣ → F a function.
By ordinarity of ρ we have det ρ = χ0 , which forces β = −αχ0 . Let v be a prime of F
and consider the restriction of α to Iv . If v ∈ Σ, v  p and #kv ≡ 1 mod p, one must have(by local class ﬁeld theory) that α(Iv ) = 0. If v ∈ Σ and #kv ≡ 1 mod p (respectivelyv = p), then Σ-minimality (respectively ordinarity at p) implies that α(Iv ) = 0. Thus α
can only be ramiﬁed at p̄. However, since ρ is ordinary at p̄, ρ|Ip̄ can be conjugated to arepresentation of the form

1
.
∗ χ0
This, together with the fact that χ0 is ramiﬁed at p̄ (see the remark after Deﬁnition 3.2)
easily implies that α must be unramiﬁed at p̄. Since p  # ClF , we must have α = 0.
Hence ρ is of the form
1 ∗
χ0and for G = ker(χ0 ) ⊂ GΣ we have
ρ|G =


1 b0 + xb1
1

for b0 , b1 : G → F + group homomorphisms. Note that F (ρ)/F (χ0 ) is thus an abelian
extension unramiﬁed outside Σ which is annihilated by p. Moreover, Gal(F (χ0 )/F ) acts−1
on Gal(F (ρ)/F (χ0 )) via χ−1. Hence Gal(F (ρ)/F (χ0 )) ⊗Fp F̄ p is a quotient of V χ0 with0−1
V χ0 as in § 3. By an argument analogous to that in the proof of Corollary 3.7, it follows
from Theorem 3.5 that ρ must be the trivial deformation.
Proposition 5.5. The universal deformation ring RΣ,O is generated as an O-algebraby traces.

684

T. Berger and K. Klosin

Proof . We follow the argument of [7, Lemma 4.2]. If suﬃces to show that any non-trivial
deformation of ρ0 to GL2 (F [x]/x2 ) is generated by traces. Let ρ be such a deformation.
Observe that for σ ∈ Gal(Q̄/F (χ0 )) the element ρ(σ) can be written as
1 + xa(σ) b0 (σ) + xb1 (σ),xc(σ)
1 + xd(σ)
so det(ρ)(σ) − tr(ρ)(σ) = −1 − xb0 (σ)c(σ). Since c is non-trivial by Proposition 5.4, theChebotarev density theorem implies there exists a σ such that xb0 (σ)c(σ) = 0. Since
det(ρ)(σ) = 1, it follows that the traces of ρ generate F [x]/x2 .
Lemma 5.6. The image of the map RΣ,O → π∈ΠΣ O given by x → (rπ (x))π is TΣ .
Proof . The O-algebra RΣ,O is generated by the set {tr ρΣ,O (Frobv ) | v ∈ Σ}. Forv ∈ Σ, we haverπ (tr ρΣ,O (Frobv )) = φ2,p (Frobv )−1 av (π).
Hence the image of the map in the lemma is the closure of the O-subalgebra of π∈ΠΣ O
generated by the set {φ2,p (Frobv )−1 Tv | v ∈ Σ} which is the same as the closure of the
O-subalgebra of π∈ΠΣ O generated by the set {Tv | v ∈ Σ} which in turn is TΣ .
By Lemma 5.6 we obtain a surjective O-algebra homomorphism r : RΣ,O  TΣ . Asin [7] we can now deduce the following theorem.
Theorem 5.7. If RΣ,O is a discrete valuation ring and if
valp (Lint (0, φ)) > 0,then the map r : RΣ,O → TΣ deﬁned above is an isomorphism.
Proof . This follows easily because the Hecke algebra is torsion free since by deﬁnition
it acts faithfully on a vector space of characteristic 0.

5.3. When is RΣ,O a discrete valuation ring?
Set Ψ := φp  and write Ψ2 for Ψ (mod 2 ).
Proposition 5.8. Assume that ρ0 does not admit any Σ-minimal upper-triangular deformation to GL2 (O/2 O). Then RΣ,O is a discrete valuation ring.
Remark 5.9. The condition on the non-existence of a Σ-minimal upper-triangular deformation of ρ0 to GL2 (O/2 O) follows from the following condition on the character φ(or, which is the same, on the splitting ﬁeld F (Ψ2 ) of Ψ2 ): there does not exist an abelianp-extension L of F (Ψ2 ), unramiﬁed outside p such that Gal(L/F (Ψ2 )) is isomorphic to a
Z[Gal(F (Ψ2 )/F )]-submodule of (O/2 O)(Ψ2−1 ) on which Gal(F (Ψ2 )/F ) operates faithfully. Indeed, as in the proof of Proposition 5.4, the condition of Σ-minimality forces any
such deformation to be of the form

1 ∗
0 Ψ2
with ∗ corresponding to an extension of F (Ψ2 ) unramiﬁed away from p.

A deformation problem

685

Proof of Proposition 5.8. We brieﬂy recall some general facts about Eisenstein representations from § 3 of [7] and § 2 of [1]. Let (A, m, k) be a local p-adically complete ring.
Let G be a topological group and consider a continuous representation ρ : G → GL2 (A)such that tr(ρ) mod m is the sum of two distinct characters τi : G → k × , i = 1, 2.
Deﬁnition 5.10. The ideal of reducibility of A is the smallest ideal I of A such thattr(ρ) mod I is the sum of two characters.
Lemma 5.11 (Bellaı̈che and Chenevier [1, Corollaire 2], Calegari [7, Lemma3.4]). Suppose A is Noetherian, that the ideal of reducibility is maximal, and that
dimk Ext1cts,k[G] (τ1 , τ2 ) = dimk Ext1cts,k[G] (τ2 , τ1 ) = 1.
If A admits a surjective map to a ring of characteristic 0, then A is a discrete valuationring.
We apply this lemma for G = GΣ , A = RΣ,O , τ1 = 1 and τ2 = χ0 . Σ-admissibilityof χ0 (which implies Σ-admissibility of its inverse by Remark 3.3) guarantees that thedimension condition in Lemma 5.11 is satisﬁed. Moreover, since RΣ,O → TΣ is surjective
and TΣ is a ring of characteristic zero, we infer that RΣ,O is a discrete valuation ring
whenever the ideal of reducibility I of RΣ,O is maximal. This is the case if and only ifthere does not exist a surjection RΣ,O /I  F [x]/x2 or RΣ,O /I  O/2 O, or, by theuniversality of RΣ,O if ρ0 does not admit any non-trivial Σ-minimal deformations of ρ0
to GL2 (F [x]/x2 ) or GL2 (O/2 O) that are upper-triangular. The latter cannot occur by
assumption and the former by Proposition 5.4.
Note that Gal(F (Ψ )/F ) ∼= Γ × ∆ with Γ ∼= Zp and ∆ a ﬁnite group.
Theorem 5.12. Assume p  #∆. If#(O/Lint (0, φ)) = p[O:Zp ] ,then ρ0 does not admit any Σ-minimal upper-triangular deformation to GL2 (O/2 O).
In particular, RΣ,O is a discrete valuation ring.
Remark 5.13. Let O be the ring of integers in any ﬁnite extension of Qp containing Lint (0, φ). Note that the L-value condition in Theorem 5.12 is equivalent to#(O /Lint (0, φ)) = p[O :Zp ] .
Proof . Put χ̃0 = Ψ |∆ . Write X∞ for Gal(M (F (Ψ ))/F (Ψ )) with M (F (Ψ )) the maximal abelian pro-p-extension of F (Ψ ) unramiﬁed away from the primes lying over p and−1(X∞ ⊗ O)χ̃0 the χ̃−1
0 -part of X∞ ⊗ O. Moreover, write M (F (Ψ2 ))Ψ for the maximalabelian pro-p-extension of F (Ψ2 ) unramiﬁed away from p on which Gal(F (Ψ2 )/F ) acts
via Ψ −1 . We will use the following two lemmas.
Lemma 5.14. We have−1

−1#((X∞ ⊗ O)χ̃0 /(γ − Ψ −1 (γ))(X∞ ⊗ O)χ̃0 )  #(O/Lint (0, φ)).

686

T. Berger and K. Klosin

Lemma 5.15. We have
−1

−1

−1

#(Gal(M (F (Ψ2 ))Ψ /F (Ψ2 ))⊗O)χ̃0  #((X∞ ⊗O)χ̃0 /(γ−Ψ −1 (γ))(X∞ ⊗O)χ̃0 ). (5.1)
We ﬁrst show how Theorem 5.12 follows from these lemmas. Suppose that L as inRemark 5.9 existed. Then one would have L ⊂ M (F (Ψ2 ))Ψ . One also has F (Ψ2 )F (ρ0 ) ⊂M (F (Ψ2 ))Ψ , hence−1#(Gal(F (Ψ2 )F (ρ0 )/F (Ψ2 )) ⊗ O)χ̃0  #(Gal(M (F (Ψ2 ))Ψ /F (Ψ2 )) ⊗ O)χ˜0

−1

,

(5.2)

but F (Ψ2 )F (ρ0 ) = L, because Gal(F (Ψ2 )/F ) does not act faithfully on the group
Gal(F (Ψ2 )F (ρ0 )/F (Ψ2 )). It is easy to see that the quantity on the left-hand side of (5.2)
is p[O:Zp ] . Hence, if the conditions of Theorem 5.12 are satisﬁed, the inequalities inLemmas 5.14 and 5.15 become equalities and this easily implies that F (Ψ2 )F (ρ0 ) =
M (F (Ψ2 ))Ψ . Thus L cannot exist.
Proof of Lemma 5.14. For any Galois character τ : GF → O× put Aτ = E/O(τ )and set G := Gal(F (Ψ )/F ). It follows from Proposition 5.21 (see § 5.5) that the module
HomG (X∞ , AΨ −1 ) is ﬁnite. By [15, Propositions 2.2(i) and 2.3],
HomG (X∞ , AΨ −1 ) ∼= SstrAΨ −1 (F ),1where SstrAΨ −1 (F ) ⊂ H (GF , AΨ −1 ) denotes the strict Selmer group deﬁned by Greenberg(see [15, § 1] for a deﬁnition). Note that the class number restriction in [15] is not required
for these results.
It is clear that
str
∼ str
Sstr
AΨ −1 (F ) = SA(Ψ −1 )c (F ) = SAΨ (F ).

The duality result of [16, Theorem 2] implies an isomorphism
∼ str
Sstr
AΨ (F ) = SAΨ −1  (F )if both Selmer groups are ﬁnite. By the observation at the beginning of the proof weknow that SstrAΨ (F ) is ﬁnite. For the Selmer group of the dual character the arguments ofthe proof of Proposition 2.2 of [15] imply that
Sstr
AΨ −1  (F ) → HomG (X∞ , AΨ −1 ).
By applying the main conjecture of Iwasawa theory, Wiles [44, p. 532] proves that
# HomG (X∞ , AΨ −1 )  #(O/Lint (0, φ)).
(For similar results towards the Bloch–Kato conjecture see also [15] who treats imaginary
quadratic ﬁelds of class number one but Hecke characters of general inﬁnity types.)
Finally, it is easy to see that−1

−1

# HomG (X∞ , AΨ −1 ) = #(X∞ ⊗ O)χ̃0 /(γ − Ψ −1 (γ))(X∞ ⊗ O)χ̃0 .


A deformation problem

687

Proof of Lemma 5.15. The restriction provides a surjective O-linear homomorphism−1−1(X∞ ⊗ O)χ̃0  (Gal(M (F (Ψ2 ))/F (Ψ2 )) ⊗ O)χ̃0 .
Since Gal(F (Ψ2 )/F ) acts on Gal(M (F (Ψ2 ))Ψ /F (Ψ2 )) via Ψ −1 the composite
−1

−1

−1

(X∞ ⊗ O)χ̃0  (Gal(M (F (Ψ2 ))/F (Ψ2 )) ⊗ O)χ̃0  (Gal(M (F (Ψ2 ))Ψ /F (Ψ2 )) ⊗ O)χ̃0(5.3)
clearly factors through−1−1(X∞ ⊗ O)χ̃0 /(γ − Ψ −1 (γ))(X∞ ⊗ O)χ̃0 .

5.4. Modularity theorem
In this section we state a modularity theorem which is a consequence of the results ofthe previous sections. To make its statement self-contained, we explicitly include all the
assumptions we have made so far.
Theorem 5.16. Let φ1 , φ2 be Hecke characters of F with split conductors and of
inﬁnity type z and z −1 respectively such that φ := φ1 /φ2 is unramiﬁed. Assume that the
conductor M1 of φ1 is coprime to (p) and that p  #(OF /M1 )× . Moreover, assume thatvalp (Lint (0, φ)) > 0.
Let ρ : GΣ → GL2 (E) be a continuous irreducible representation that is ordinary at
all places v | p. Suppose ρ̄ss ∼= χ1 ⊕ χ2 with χ1 = φ1,p , χ2 = φ2,p . Set χ0 := χ1 χ−1
2 . Ifall of the following conditions are satisﬁed:(1) Σ ⊃ {v | pdF M1 Mc1 },(2) the representation ρ̄ ⊗ χ−1
2 admits no upper-triangular Σ-minimal deformation toGL2 (O/2 O),(3) χ0 is Σ-admissible,(4) det(ρ) = φ1 φ2 ,(5) ρ ⊗ φ−12,p is Σ-minimal,
then ρ is modular in the sense of Deﬁnition 2.5.
Remark 5.17. Write Gal(F (Ψ )/F ) = Γ × ∆ with Γ ∼= Zp . If p  #∆ then by Theorem 5.12, Condition (2) in Theorem 5.16 can be replaced by #(O/Lint (0, φ)) = p[O:Zp ] .
Remark 5.18. Theorem 4.4 and Remark 4.5 show that the conditions for the conductor
and inﬁnity type of φ can be relaxed if one imposes a condition on the torsion freenessof a cohomology group.

688

T. Berger and K. Klosin

Example 5.19. We now turn to a numerical example in which we can verify the conditions of√ Theorem 5.16 (under an additional assumption which we discuss below). Let
F = Q( −51) and p = 5 (which splits in F ). Since the class number is 2, there are twounramiﬁed Hecke characters of inﬁnity type z 2 . For each of them the functional equation
relates the L-value at 0 to the L-value at 0 of a Hecke character of inﬁnity type z̄/z.
The latter one in turn is equal (by the Weil lifting; see, for example, [25, Theorem 4.8.2]
or [18, Theorem 12.5]) to the L-value at 1 of a weight 3 modular form of level 51 and2character the Kronecker symbol ( −51· ). Let φ be the Hecke character of inﬁnity type z
3corresponding to the modular form with q-expansion starting with q + 3q + . . . . Using
Magma [10] one calculates (see Remark 5.20) thatval5 (Lint (0, φ))  1.
Assuming that the 5-valuation is exactly 1 (see Remark 5.20 explaining the computational
issues involved) this is enough to satisfy Condition (2) of Theorem 5.16 (cf. Remarks 5.13
and 5.17). The character χ0 = φp  is Σ-admissible for appropriate sets Σ (i.e. they satisfy
Conditions (1), (3), (4) and (5) of Deﬁnition 3.2) because the ray class ﬁeld of conductor 5(a degree 16 extension over F ) has class number 3 (as calculated by Magma assuming theGeneralized Riemann Hypothesis). Here we use that the splitting ﬁeld F (χ0 ) is containedin the ray class ﬁeld of F of conductor 5.
Remark 5.20. In our calculation above we use an operation in Magma called LRatio
which calculates a rational normalization of the L-value of a modular form using modular
symbols. This calculation gives 5-valuation equal to 1. Because of the diﬀerent period usedby Magma we can only conﬁrm that this provides a lower bound on the 5-valuation of
Lint (0, φ) = L(0, φ)/Ω 2 , for Ω the Neron period of a suitable elliptic curve with complex
multiplication by F (see, for example, [13, p. 768]). This follows from the following
relations between the diﬀerent periods.
(1) The proof of Lemma 7.1 of [12] shows that the period used by Magma (RealVolume)is an integral multiple of the canonical period Ω(f )+ deﬁned by Vatsal [41] (up todivisors of N k! for the level N = 51 and weight k = 3 of the modular form).
(2) Vatsal [41] proves that one can ﬁnd a Dirichlet character χ such thatτ (χ̄)

L(1, f, χ)(−2πi)Ω(f )±(with χ(−1) = (−1)± ) is a 5-unit. Note that Vatsal’s condition that ρ̄f is absolutely
irreducible is satisﬁed in our case and Ω(f )− ∼ Ω(f )+ because f has complex
multiplication. Here we write ‘∼’ to indicate equivalence up to 5-unit. Because
2+
πL(1, f, χ) ∼ L(0, φ · resQ
F (χ̄)) this implies that π · Ω(f ) is a 5-integral multiple
2of Ω .
5.5. A reducible deformation of ρ0
Let Ψ = φp . Then χ0 = Ψ̄ . For a ﬁnite set of primes S of F , let LΨ (S) denote the maximal abelian pro-p extension of F (Ψ ) unramiﬁed outside S and such that Gal(F (Ψ )/F )acts on Gal(LΨ (S)/F (Ψ )) via Ψ −1 .

A deformation problem

689

Proposition 5.21. The group Gal(LΨ (Σ \ {p̄})/F (Ψ )) is a torsion Zp -module.
Proof . The Σ-admissibility of χ0 implies that the extension LΨ (Σ\{p̄})/F (Ψ ) is unramiﬁed away from the primes lying over p. Then the claim follows from the anticyclotomic
main conjecture of Iwasawa theory for imaginary quadratic ﬁelds (see [23, 28, 36]) afternoting that L(0, φ) = 0.

Corollary 5.22. There does not exist a Σ-minimal reducible deformation of ρ0 into
GL2 (A) if A is not a torsion O-algebra.
Proof . As in Proposition 5.4 such a deformation would have to be of the form

1 ∗
ρ=
.
Ψ

(5.4)

By ordinarity, one must also haveρ|Ip̄ ∼=

1
Ψ |Ip̄

,which implies that the upper shoulder ∗ in (5.4) corresponds to an extension L/F (Ψ )which is unramiﬁed away from primes lying over p. Since A is not a torsion Zp -module,
this would contradict Proposition 5.21.
Remark 5.23. In [29] Skinner and Wiles prove an R = T theorem for deformations ofa certain class of reducible (non-semisimple) residual representations of GQ of the form
[ 1 χ∗ ] for χ : GQ → F̄ ×
p a continuous character. They apply the numerical criterion ofWiles and Lenstra [20, 44] by ﬁrst relating the size of the relevant universal deformation
ring to a special value of the L-function of χ. They achieve this by studying the Galois
cohomology of ad ρ for a Σ-minimal reducible deformation ρ with values in a characteristic zero Zp -algebra O. Here Σ is a ﬁnite set of primes of Q satisfying similar conditions
to the ones we imposed on our sets Σ. Corollary 5.22 means that their method cannot
be applied in our case.
Even though no Σ-minimal characteristic zero deformations of ρ0 exist, we now showthat if one drops the ordinarity condition at p̄, it is possible to construct a reducible(non-ordinary) deformation of ρ0 into GL2 (O).
Proposition 5.24. There exists a unique deformation ρ : GΣ → GL2 (O) of ρ0 of the
form

1 ∗
∼
ρ=
.
Ψ
The extension F (ρ)/F (Ψ ) is unramiﬁed away from {p, p̄}.

690

T. Berger and K. Klosin

Proof . To show that a deformation of the desired form exists (and is unique up to strict
equivalence) it is enough to show that
H 1 (GΣ , O(Ψ −1 )) ∼
=O

(5.5)

as O-modules. Noting the equality of the Euler–Poincaré characteristics
(−1)i rkO (H i (GΣ , O(Ψ −1 ))) =(−1)i dimF (H i (GΣ , F (χ−1
0 )))
i

i

(see, for example, [26, Lemma A.1.8]), it follows from the global Euler characteristic
formula [24, Theorem 5.1] thatrkO (H 1 (GΣ , O(Ψ −1 )))  1.

(5.6)

On the other hand, uniqueness of ρ0 (Corollary 3.7) implies thatdimF (H 1 (GΣ , O(Ψ −1 ))/)  dimF (H 1 (GΣ , F (Ψ −1 ))) = 1.
This, together with (5.6) and Nakayama’s lemma gives (5.5). Finally, note that Σ-admissibility of χ0 forces F (ρ)/F (χ0 ) to be unramiﬁed away from {p, p̄}.
Remark 5.25. The representation ρ in Proposition 5.24 is not ordinary. Indeed, if itwere ordinary the representation ρ|Dp̄ would have an unramiﬁed quotient. Since it clearly
has an unramiﬁed submodule, it would be split and thus the upper shoulder ∗ would
correspond to a non-Zp -torsion extension of F (Ψ ) unramiﬁed away from p, which does
not exist by Proposition 5.21. On the other hand, ρ is nearly ordinary in the sense of
Tilouine (see, for example, Deﬁnition 3.1 of [43]) with respect to the upper-triangular
Borels at p and p̄. Since one has
1∗
ρ|Ip̄ ∼,=
|Ip̄
the representation ρ is, however, not de Rham.
Acknowledgements. The authors would like to thank Adebisi Agboola, Trevor
Arnold, Frank Calegari, John Coates, Matthew Emerton, Ralph Greenberg, Chris Skinner and Jacques Tilouine for helpful discussions and comments. We would also like tothank the anonymous referee for his/her careful reading and comments: in particular, forsuggesting improvements to the proofs of Theorem 3.5 and Proposition 5.24.
References
1.
2.
3.

J. Bellaı̈che and G. Chenevier, Lissité de la courbe de Hecke de GL2 aux points
Eisenstein critiques, J. Inst. Math. Jussieu 5 (2006), 333–349.
T. Berger, An Eisenstein ideal for imaginary quadratic ﬁelds, Thesis, University ofMichigan, Ann Arbor (2005).
T. Berger, An Eisenstein ideal for imaginary quadratic ﬁelds and the Bloch–Kato conjecture for Hecke characters, Compositio Math., in press.

A deformation problem
4.
5.
6.
7.
8.
9.
10.
11.

12.

13.
14.
15.
16.
17.
18.
19.
20.

21.

22.
23.
24.
25.
26.

27.

691

T. Berger, Denominators of Eisenstein cohomology classes for GL2 over imaginary
quadratic ﬁelds, Manuscr. Math. 125(4) (2008), 427–470.
T. Berger and G. Harcos, l-adic representations associated to modular forms overimaginary quadratic ﬁelds, Int. Math. Res. Not. 23 (2007), article ID rnm113.
C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic
curves over Q: wild 3-adic exercises, J. Am. Math. Soc. 14 (2001), 843–939 (electronic).
F. Calegari, Eisenstein deformation rings, Compositio Math. 142 (2006), 63–83.
F. Calegari and N. M. Dunfield, Automorphic forms and rational homology 3spheres, Geom. Topol. 10 (2006), 295–329 (electronic).
F. Calegari and B. Mazur, Nearly ordinary Galois deformations over arbitrary number
ﬁelds, J. Inst. Math. Jussieu 8 (2009), 99–177.
W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I, The user
language, J. Symbolic Comput. 24(3–4) (1997), 235–265.
J. Coates and A. Wiles, Kummer’s criterion for Hurwitz numbers, in Algebraic Number
Theory, Kyoto Int. Symposium, Research Institute for Mathematical Sciences, Universityof Kyoto, Kyoto, 1976, pp. 9–23 (Japan Society for the Promotion of Science, Tokyo,
1977).
N. Dummigan, W. Stein and M. Watkins, Constructing elements in Shafarevich–Tategroups of modular motives, in Number theory and algebraic geometry, London Mathematical Society Lecture Notes, Volume 303, pp. 91–118 (Cambridge University Press, 2003).
T. Finis, Divisibility of anticyclotomic L-functions and theta functions with complex
multiplication, Annals Math. (2) 163 (2006), 767–807.
K. Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint(1999).
L. Guo, General Selmer groups and critical values of Hecke L-functions, Math. Annalen297(2) (1993), 221–233.
L. Guo, On a generalization of Tate dualities with application to Iwasawa theory, Compositio Math. 85(2) (1993), 125–161.
H. Hida, Kummer’s criterion for the special values of Hecke L-functions of imaginary
quadratic ﬁelds and congruences among cusp forms, Invent. Math. 66(3) (1982), 415–459.
H. Iwaniec, Topics in classical automorphic forms, Graduate Studies in Mathematics,
Volume 17 (American Mathematical Society, Providence, RI, 1997).
M. Kisin, The Fontaine–Mazur conjecture for GL2 , preprint (2007).
H. W. Lenstra Jr, Complete intersections and Gorenstein rings, in Elliptic Curves,
Modular Forms, and Fermat’s Last Theorem, Hong Kong, 1993, Series in Number Theory,
Volume I, pp. 99–109 (International Press, Cambridge, MA, 1995).
´ Lozano-Robledo, Bernoulli numbers, Hurwitz numbers, p-adic L-functions andA.
Kummer’s criterion, Rev. R. Acad. Cienc. Exactas Fı́s. Nat. Ser. A Mat. 101(1) (2007),
1–32.
B. Mazur, An introduction to the deformation theory of Galois representations, in Modular Forms and Fermat’s Last Theorem, Boston, MA, 1995, pp. 243–311 (Springer, 1997).
B. Mazur and J. Tilouine, Représentations galoisiennes, diﬀérentielles de Kähler et‘conjectures principales’, Publ. Math. IHES 71 (1990), 65–103.
J. S. Milne, Arithmetic duality theorems, 2nd edn (BookSurge, Charleston, SC, 2006).
T. Miyake, Modular forms (transl. from Japanese by Y. Maeda) (Springer, 1989).
B. Perrin-Riou, p-adic L-functions and p-adic representations (transl. from the 1995
French original by L. Schneps and revised by the author), Société Mathématique deFrance/American Mathematical Society Texts and Monographs, Volume 3 (American
Mathematical Society, Providence, RI, 2000).
K. A. Ribet, A modular construction of unramiﬁed p-extensions of Q(µp ), Invent. Math.
34(3) (1976), 151–162.

692
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.

39.
40.

41.
42.
43.
44.
45.

T. Berger and K. Klosin
K. Rubin, The ‘main conjectures’ of Iwasawa theory for imaginary quadratic ﬁelds, Invent.
Math. 103(1) (1991), 25–68.
C. M. Skinner and A. J. Wiles, Ordinary representations and modular forms, Proc.
Natl Acad. Sci. USA 94 (1997), 10 520–10 527.
C. M. Skinner and A. J. Wiles, Residually reducible representations and modular
forms, Publ. Math. IHES 89 (1999), 5–126.
C. M. Skinner and A. J. Wiles, Nearly ordinary deformations of irreducible residual
representations, Annales Fac. Sci. Toulouse Math. 10(1) (2001), 185–215.
R. Taylor, On congruences between modular forms, Thesis, Princeton University (1988).
R. Taylor, l-adic representations associated to modular forms over imaginary quadraticﬁelds, II, Invent. Math. 116 (1994), 619–643.
R. Taylor, Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu 1(2002), 125–143.
R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Annals
Math. (2) 141 (1995), 553–572.
J. Tilouine, Sur la conjecture principale anticyclotomique, Duke Math. J. 59 (1989),
629–673.
J. Tilouine, Deformations of Galois representations and Hecke algebras (Mehta ResearchInstitute of Mathematics and Mathematical Physics, Allahabad, 1996).
E. Urban, Formes automorphes cuspidales pour GL2 sur un corps quadratique imaginaire: valeurs spéciales de fonctions L et congruences, Compositio Math. 99(3) (1995),
283–324.
E. Urban, Module de congruences pour GL(2) d’un corps imaginaire quadratique etthéorie d’Iwasawa d’un corps CM biquadratique, Duke Math. J. 92(1) (1998), 179–220.
E. Urban, Sur les représentations p-adiques associées aux représentations cuspidales de
GSp4/Q , in Formes automorphes, II, Le cas du groupe GSp(4), pp. 151–176, Astérisque,No. 302 (Société Mathématique de France, Paris, 2005).
V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397–
419.
L. C. Washington, Introduction to cyclotomic ﬁelds, 2nd edn, Graduate Texts in Mathematics, Volume 83 (Springer, 1997).
T. Weston, Iwasawa invariants of Galois deformations, Manuscr. Math. 118(2) (2005),
161–180.
A. Wiles, Modular elliptic curves and Fermat’s last theorem, Annals Math. (2) 141(1995), 443–551.
R. I. Yager, A Kummer criterion for imaginary quadratic ﬁelds, Compositio Math. 47(1)(1982), 31–42.