AN R = T THEOREM FOR IMAGINARY QUADRATIC FIELDSTOBIAS BERGER1 AND KRZYSZTOF KLOSIN2
Abstract. We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic eld assuming the uniquenessof the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply thecriterion one needs to show that the quotient of R by its ideal of reducibility
is cyclic Artinian of order no greater than the order of the congruence module T=J , where J is an Eisenstein ideal in the local Hecke algebra T. The
inequality is proven by applying the Main conjecture of Iwasawa Theory for
Hecke characters and using a result of [Ber09]. This strengthens our previous result [BK09] to include the cases of an arbitrary p-adic valuation of the
L-value, in particular, cases where R is not a discrete valuation ring. As aconsequence we show that the Eisenstein ideal is principal and that T is acomplete intersection.

1. Introduction
Let K be a number eld and  : Gal(K=K ) ! GL2 (Qp ) a continuous irreducible
representation. It has been a subject of much interest and e ort lately to determine which such Galois representations are modular, i.e., which such 's have theirL-function equal to an L-function of an automorphic representation of GL2 (AK ).
Since the ground-breaking work of Wiles there has been a lot of progress in answering this question [Wil95, TW95, BCDT01, SW97, Fuj99, SW99, SW01, Tay02,
Kis07].
This article is a continuation of our e orts to prove the modularity of continuous
representations of Gal(K=K ) for K an imaginary quadratic eld. This case issubstantially di erent from other situations for which modularity has been proven so
far since the associated symmetric space is a hyperbolic 3-manifold, so in particular
tools from algebraic geometry are not available.
In a previous paper [BK09] we proved modularity of irreducible Galois representations of an imaginary quadratic K (in the minimal case) when the residual
representation 0 (i.e., the composite of  with the map Zp “ Fp after choosing an
integral lattice in the space of ) is reducible under an additional condition that acertain L-value associated to the determinant of  has small p-adic valuation. As
we showed in [BK09] this condition implies that the universal deformation ring of
0 is a discrete valuation ring. This article removes this condition.
We achieve this by developing a new commutative algebra criterion to prove \R =T " theorems, applicable also in other situations (see Remark 1.2 and the discussion
Date : 22 November 2009.
2000 Mathematics Subject Classi cation. 11F80, 11F55.
Key words and phrases. Galois representations, automorphic forms, deformations, ideal of
reducibility, Eisenstein ideal, imaginary quadratic.
1

2

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

at the end of Section 6.4). For this x a reducible, non-semisimple residual twodimensional Galois representation and study its minimal ordinary deformations
with corresponding universal deformation ring R and universal deformation R .
Following [BC06, Cal06] we de ne the ideal of reducibility Ire to be the smallest
ideal I in R such that tr(R ) mod I is the sum of two characters.
Assuming that there is only one (up to isomorphism) non-split extension of thetwo characters (in either order) which are the constituents of the semisimpli cationof the residual representation [BC06, Cal06] prove that Ire is principal. In this caseour new commutative algebra criterion allows us to prove that R is isomorphic toT, a Hecke algebra, provided that R=Ire is a cyclic Artinian module and its order is
no greater than the order of T=J for J an Eisenstein ideal of T (see Proposition 6.9
and Theorem 6.10). Lower bounds for the order of T=J have been proven in manycases starting with Mazur and Wiles [MW84]. For our case of imaginary quadraticelds we refer to [Ber09] which bounds valp (#T=J ) from below by the p-valuation(say n) of a Hecke L-value. Since we can rule out non-trivial reducible in nitesimal
deformations by the uniqueness of our residual representation the cyclicity of R=Ire
and an upper bound for its p-order (which again equals n) can be obtained using
the Main Conjecture for Hecke characters of imaginary quadratic elds, proven by
Rubin.
There are several consequences of the R = T theorem for both the Hecke algebra and the universal deformation ring. On the one hand the principality of the
reducibility ideal implies that the Eisenstein ideal is also principal. On the other
hand using our knowledge of the structure of the universal deformation ring we conclude that the Hecke algebra is a complete intersection (Corollary 6.14). Finally,
the lower bound on the size of the Hecke congruence module translates under theisomorphism R = T into a statement about cyclicity of a certain Galois group andthe existence of certain reducible deformations (Corollary 6.16).
To give a more precise account, let c be the non-trivial automorphism of F , andlet p > 3 be a prime split in the extension F=Q. Fix embeddings F ,! Q ,! Qp ,!
C. Let F be the maximal extension of F unrami ed outside a nite set of places and put G = Gal(F =F ). Suppose F is a nite eld of characteristic p and that
0 : G ! F is anticyclotomic (in general a weaker assumption of -admissibility
is enough - see Section 3) character rami ed at the places dividing p. Suppose alsothat 0 : G ! GL2 (F) is a continuous representation of the form
(1.1)



0 = 10 

0and having scalar centralizer. In [BK09] we proved that under certain conditions
on 0 and  the residual representation 0 is unique up to isomorphism. We recall
the relevant details in Section 3.
Following Mazur [Maz97] we study ordinary deformations of 0 . Let O be thering of integers in a nite extension of Qp . An O-deformation of 0 is a local complete Noetherian O-algebra A with residue eld F and maximal ideal mA together
with a strict equivalence class of continuous representations  : G ! GL2 (A) satisfying 0 =  mod mA . An ordinary deformation is a deformation that satis esjD = 1 
q0 2

AN R = T THEOREM

3

for q j p, where i jI = ki with integers k1  k2 depending on q and  is thep-adic cyclotomic character. Here Dq and Iq denote the decomposition group andthe inertia group of q j p.
A part of our approach rests on studying possible reducible deformations of 0and showing that there are indeed not \too many" of those. As we have alreadyshown in [BK09] the uniqueness of 0 itself implies that there are no non-trivial
reducible in nitesimal deformations of 0 . In the current paper we go further andcarefully study reducible deformations of 0 to other Artinian local rings.
To exhibit modular irreducible deformations we apply the cohomological congruences of [Ber09] and the Galois representations constructed by Taylor et al. using
the strengthening of Taylor's result in [BH07]. We also make use of a result of
Urban [Urb05] who proves that  jD is ordinary at q j p if  is ordinary at q. This
together with the non-existence of a non-trivial reducible in nitesimal deformation
implies the existence of an O-algebra surjection(1.2)
R “ T;
where R is the universal -minimal deformation ring (cf. De nition 4.8) and T isa Hecke algebra acting on cuspidal automorphic forms of GL2 (AF ) of weight 2 andxed level.
Remark 1.1. As we remarked already in [BK09] the approach of [SW97] (wherean analogous problem is studied for representations of Gal(Q=Q)) breaks down in
the imaginary quadratic case because of the non-existence of an ordinary reduciblecharacteristic 0 deformation (cf. [BK09], Corollary 5.22). In [BK09] we assumed
that the p-valuation of a Hecke L-value was small such that by the Main Conjecture there were even no reducible deformations to any Artinian rings larger than
the residue eld, which implied that the ideal of reducibility Ire of the universal
deformation ring R was maximal. This simpli ed the proof of \R = T ", because it
follows from the results of Bellaiche-Chenevier [BC06] and Calegari [Cal06] that if
R surjects onto a characteristic 0 ring (in our case T) and the ideal Ire is maximal,
then R is a discrete valuation ring.
In this paper we exploit that in all cases there is, in fact, a surjection R=Ire “
T=J by the de nitions of the Eisenstein ideal J and the ideal of reducibility Ire.
By nding bounds on the orders of both sides we show that this map must be an
isomorphism. Indeed, if we write n for the $-valuation1 of a normalised Hecke
L-value (for a Hecke character that gives rise to 0 in (1.1)), then the order of thecongruence module T=J is bounded from below by #O=$n by the full result of[Ber09]. On the deformation side we rst bound the dimension of R in Section
5.1 by using the ltration of ad0 by 1-dimensional pieces, similar to a calculationin [SW97]. Strengthening an argument of [BK09] and using a result of Urban
we show in Section 5.2 that the Main Conjecture of Iwasawa Theory implies that
there are no reducible ordinary deformations to O=$m for m > n. Together with
the non-existence of non-trivial reducible in nitesimal deformations this implies
that R=Ire = O=$m for m  n (see Proposition 6.5). We therefore have thatthe surjection R=Ire “ T=J is an isomorphism, from which we deduce (using ourcommutative algebra criterion - Proposition 6.9) that R = T in Theorem 6.10.
using the principality of Ire .
qq

1Here $ denotes a uniformizer of O.

4

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Remark 1.2. As suggested to us by Frank Calegari our commutative algebra
criterion allows one to replace the condition pkB2;!k 2 from [Cal06] Theorem 2.3
by p j B2;!k 2 , which proves an R = T result for certain residually reducible
representations of GQ . More precisely, if T denotes the (classical) cuspidal Hecke
algebra of weight 2 and level 1 (p) generated by T` for ` 6= p, write J the maximal
ideal of T containing T` 1 `k 1 for all ` 6= p. Under the assumption that the
1 k -eigenspace (where  : GQ ! Zp denotes the p-adic cyclotomic character) of
the class group of Q(p ) is cyclic, Calegari considers deformations of the unique
non-split representation
 = 10 k 1 mod p
with  unrami ed over Q(p ). He requires that the deformations  be unrami ed
outside p, of determinant !k 2 , where ! is the Teichmuller lift of  mod p, andsuch that the restriction of  to Gal(Qp =L) arises as a generic ber of a nite atgroup scheme over the ring of integers of L. Here L stands for the completion at p of
the maximal real sub eld of Q(p ). If one denotes by R the corresponding universal
deformation ring, the arguments of our Sections 5.2 and 6.1 can be adapted, using[Cal06] Lemmas 4.2 and 4.5 and the Main conjecture proven by Mazur and Wiles[MW84], to prove the cyclicity and an upper bound for R=Ire in terms of the pvaluation of the relevant Bernoulli number. The lower bound by the same number
on the size of the quotient of TJ by the Eisenstein ideal generated by T` 1 `k 1 forall ` 6= p is stated, for example, as Proposition 5.1 in [SW97]. By the commutative
algebra criterion (Proposition 6.9) we can therefore conclude R = TJ .
The authors would like to thank Frank Calegari, Gaetan Chenevier, Haruzo Hida
and Ravi Ramakrishna for helpful discussions and comments.
2. Notation and terminology
2.1. Galois groups. Let F be an imaginary quadratic extension of Q of discriminant dF 6= 3; 4 and p > 3 a rational prime which splits in F . We x once and for alla prime p of F lying over p and write p for the prime of F such that (p) = pOF = pp.
Let ClF denote the class group of F . We assume that p - # ClF and that any primeq j dF satis es q 6 1 (mod p):
For a eld K write GK for the Galois group Gal(K=K ). If K is a nite extensionof Q` for some rational prime `, we write OK (respectively $K , and FK ) for the
ring of integers of K (respectively for a uniformizer of K , and OK =$K OK ).
If K  F is a number eld, OK will denote its ring of integers. If q is a placeof K , we write Kq for the completion of K with respect to the absolute value j  jq
determined by q and set OK;q = OK (if q is archimedean, we set OK;q = Kq ). We
also write $q for a uniformizer of Kq, Pv for the maximal ideal of OK;q , and kq
for its residue eld.
Fix once and for all compatible embeddings iq : F ,! F q and F q ,! C, for every
prime q of F , so we will often regard elements of F q as complex numbers without
explicitly mentioning it. If w is a place of K  F , it determines a place v of F ,and we always regard Kw as a sub eld of F v as determined by the embedding iv .
This also allows us to identify GKw with the decomposition group Dv  GK of aplace v of F . We will denote that decomposition group by Dq . Abusing notationq

AN R = T THEOREM

5

somewhat we will denote the image of Dq in any quotient of GK also by Dq . We
write Iq  Dq for the inertia group.
Let  be a nite set of places of K . Then K will denote the maximal Galois
extension of K unrami ed outside the primes in . We also write G for GF .
Moreover, for a positive integer n, denote by n the group of nth roots of unity.
2.2. Hecke characters. For a number eld 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 : K  n AK ! C :q of K write (q) for the restriction of  to Kq and (1) for the restriction
For a placeQof  to qj1 Kq . The latter will be called the in nity type of . We also usually
write ($q ) to mean (q) ($q ). Given  there exists a unique ideal f of K largest
with respect to the following property: (q) (x) = 1 for every nite place q of K such that x 1 2 f OK;q . The ideal f is called the conductor of .
and x 2 OK;q
If K = F , there is only one archimedean place, which we will simply denote by 1.
For a Hecke character  of F , one has (1) (z ) = z mz n with m; n 2 R. If m; n 2 Z,
we say that  is of type (A0 ). We always assume that our Hecke characters are oftype (A0 ). Write L(s; ) for the Hecke L-function of.
Let  be a Hecke character of in nity type z a zz b with conductor prime to p.
Assume a; b 2 Z and a > 0 and b  0. Put
b

2

alg
a
2
b
p
(a + b)  L(0; );L (0; ) :=dF

where is a complex period. In most cases, this normalization is integral, i.e., liesin the integer ring of a nite extension of Fp . See [Ber08] Theorem 3 for the exact
statement. Put
(
alg
alg
int
L (0; ) = L (0; ) if valp (L (0; ))  0
1otherwise:
For z 2 C we write z for the complex conjugate of z . The action of complex
conjugation extends to an automorphism of AF and we will write x for the imageof x 2 AF under that automorphism.
For a Hecke character  of F , we denote by c the Hecke character of F de nedby c (x) = (x).
2.3. Galois representations. For a eld K and a topological ring R, by a Galois
representation we mean a continuous homomorphism  : GK ! GLn (R). If n = 1
we usually refer to  as a Galois character. We write K () for the xed eld of ker and call it the splitting eld of . If  is a Galois character and M is an R-module,
we denote by M () the R[GK ]-module M with the GK -action given by . If K
is a number eld and q is a nite prime of K with inertia group Iq we say that
 is unrami ed at q if jI = 1. If  is a nite set of places of K such that  isunrami ed at all v 62 , then  can be regarded as a representation of GK .
Let E be a nite extension of Qp . Every Galois representation  : GK ! GLn (E )can be conjugated (by an element M 2 GLn (E )) to a representation M : GK !
GLn (OE ). We denote by M : GK ! GLn (FE ) its reduction modulo $E OE . It
is sometimes called a residual representation of . The isomorphism class of its
semisimpli cation ssM is independent of the choice of M and we simply write ss .
q

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

6

Let  : GF ! Zp denote the p-adic cyclotomic character. For any subgroup
G  GF we will also write  for jG. Our convention is that the Hodge-Tate weightof  at p is 1.
Let  be a Hecke character of F of type (A0 ) and  = fq j pf g. We de ne(following Weil) a p-adic Galois characterp : G ! F p

associated to  by the following rule: For a nite place q - pf of F , put p (Frobq ) =ip(i11 (($q ))) where Frobq denotes the arithmetic Frobenius at q. It takes values
in the integer ring of a nite extension of Fp .
De nition 2.1. For a topological ring R we call a Galois representation  : G !
GL2 (R) ordinary if




1

jD = 0 2ki
for q j p, where i jI =  with integers k1  k2 depending on q.
q

q2.4. 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 (for more details see Section 3.1 of [Urb95]). For a nite order Hecke
character write S2 (Kf ; ) forLthe forms with central character . This is isomorphic as a G(Af )-module to fKf for automorphic representations  of a certain
in nity type (see Theorem 2.2 below) with central character . Here f denotes
the restriction of  to GL2 (Af ) and fKf stands for the Kf -invariants.
For g 2 G(Af ) we have the usual Hecke action of [Kf gKf ] on S2 (Kf ) andS2 (Kf ; ). For primes q such that the vth component of Kf is GL2 (OF;v ) wede ne Tq = [Kf $q 1 Kf ].
Combining the work of Taylor, Harris, and Soudry with results of FriedbergHo stein and Laumon/Weissauer, one can show the following (see [BH07] for general case of cuspforms of weight k):
Theorem 2.2 ([BH07] Theorem 1.1). Given a cuspidal automorphic representation  of GL2 (AF ) with 1 isomorphic to the principal series representation
corresponding to






t1
t1  !
t2 7 jt1 j

jt2 j

t2
c
and cyclotomic central character (i.e., = ), let  denote the set consistingof the places of F lying above p, the primes where  or c is rami ed, and theprimes rami ed in F=Q.
Then there exists a nite extension E of Fp and a Galois representation
 : G ! GL2 (E )such that if q 62  , then  is unrami ed at q and the characteristic polynomialof  (Frobq ) is x2 aq ()x + ($q )(#kq ); where aq () is the Hecke eigenvalue
corresponding to Tq . Moreover,  is absolutely irreducible.

AN R = T THEOREM

7

Urban studied in [Urb98] the case of ordinary automorphic representations , andtogether with results in [Urb05] on the Galois representations attached to ordinary
Siegel modular forms showed:
Theorem 2.3 (Corollary 2 of [Urb05]). Let q be a prime of F lying over p. If  isunrami ed at q and ordinary at q, i.e., jaq ()jq = 1, then the Galois representation
 is ordinary at q. Moreover jD 
=
q



1



 ;
2

where 2 jI = 1 and 1 jI = det  jI = :We conjecture that the assumption on the central character, inherent to themethod of proof of Theorem 2.2, is not necessary (see also [CD06] Conjecture 3.2).
Conjecture 2.4. Let  be a cuspidal automorphic representation as in Theorem
2.2 but without the assumption on the central character. Then the representation
 exists as above and the conclusion of Theorem 2.3 remains valid for it.
De nition 2.5. Let E be a nite extension of Fp and  : G ! GL2(E ) a Galois
representation for a nite set of places . We say that  is modular if there exists
a cuspidal automorphic representation  as in Theorem 2.2, such that  = (possibly after enlarging E ).
From now on we x a nite extension E of Fp which we assume to be suciently
large (see Section 4.2 and Remark 4.5, where this condition is made more precise).
To simplify notation we put O := OE , F = FE and $ = $E .
qqq

3. Uniqueness of a residual Galois representation
Let p = pp as before be a split prime. Let  be a nite set of nite primes of F
containing p. In this paper we will be studying deformations of a certain class ofresidual Galois representations which we now describe.
From now on let
0 : G ! GL2 (F)be a Galois representation
satisfying
both of the following conditions:(Red): 0 = 1 0 for a Galois character 0 : G ! F;(Sc): 0 has scalar centralizer.
De nition 3.1. We will say that a Galois character 0 : G ! F is -admissibleif(1) 0 is rami ed at p and at p;(2) if q 2 , then either 0 is rami ed at q or 0 (Frobq ) 6= (#kv )1 (as elementsof F);(3) dimF (H 1 (G ; F(0 ))) = dimF (H 1 (G ; F(0 1 ))) = 1:
Remark 3.2. It follows from the global Euler characteristic formula ([Mil06] Theorem 5.1), that the cohomology spaces in condition (3) of De nition 3.1 are at least
one-dimensional.
An important consequence of -admissibility of 0 is the uniqueness of 0 up toisomorphism. We record this fact as a proposition.

8

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Proposition 3.3. Suppose 0 : G ! GL2(F) is a Galois representation satisfying
condition (Red) and (Sc) for a -admissible character 0 . Then 0 
= 0 .
Proof. As 0 and 0 correspond to non-zero elements in ExtFp [G] (1; 0 ) = H 1 (G ; Fp (0 1 )),˜
the proposition follows from condition (3) in De nition 3.1.
We will be interested in studying deformations of a representation 0 satisfying
conditions (Red) and (Sc) for a -admissible character 0 .
Let us shortly explain the signi cance of these assumptions to the deformation
problem we study in the coming sections. Condition (Sc) is necessary to ensurethat the deformation functor studied in the following sections is representable.
The assumption that 0 is -admissible is crucial for our method. On the one
hand Proposition 3.3 guarantees that modular Galois representations which weconstruct (and which are de ned only up to semisimpli cation) are deformations of
0 . On the other hand, -admissibility of 0 is essential for ensuring that the idealof reducibility of the universal deformation is principal, which is crucial for ourcommutative algebra criterion (Proposition 6.9) that we use to prove modularity of-minimal deformations of 0 (Theorem 6.10).
We will now formulate some conditions (see Theorem 3.5 below) that ensure admissibility of a Galois character (for details see [BK09], Section 3). At the sametime we want to emphasize that there is no reason why every -admissible charactershould satisfy all of these conditions, however, it is in the context of Theorem 3.5
that we were able to construct examples of -admissible characters (see Remark 3.7
below). Also, it is only under condition (3) of Theorem 3.5 that one can relate (bytwisting, see [Ber09] Lemma 8) to cusp forms with cyclotomic character for which
one knows how to attach Galois representations to automorphic forms on GL2 (AF )by Theorem 2.2. Conjecturally, however, this condition should be unnecessary (seeConjecture 2.4).
De nition 3.4. Let R be a commutative ring, J  R an ideal, M a free R-moduleand N a submodule of M . We will say that N is saturated with respect to J if
N = fm 2 M j mJ  N g:
Let 0 : G ! F be a Galois
Q character. Let Sp be the set of primes of F (0 )lying over p. Write M0 for q2Sp (1 + Pv ) and T0 for its torsion submodule.
The quotient M0 =T0 is a free Zp -module of nite rank. Let E 0 be the closurein M0 =T0 of the image of E0 , the group of units of the ring of integers of F (0 )which are congruent to 1 modulo every prime in Sp .
In [BK09] the authors proved the following theorem (see Theorem 3.5 and theproof of Corollary 3.6 in [loc. cit.]).
Theorem 3.5. Let 0 : G ! F be a Galois character. Assume that 0 satis es
all of the following conditions:(1) 0 is rami ed at p;(2) if q 2 , then either 0 is rami ed at q or 0 (Frobq ) 6= (#kq )1 (aselements of F);(3) 0 is anticyclotomic, i.e., 0 (cc) = 0 () 1 for every  2 G and c thegenerator of Gal(F=Q);(4) the Zp -submodule E 0  M0 =T0 is saturated with respect to the ideal pZp ,(5) The 0 1 -eigenspace of the p-part of ClF (0 ) is trivial.

AN R = T THEOREM

9Then 0 is -admissible.
Remark 3.6. Note that conditions (1) and (3) of Theorem 3.5 imply that 0 is
also rami ed at p. Observe that 0 satis es conditions (1)-(5) of Theorem 3.5 if andonly if so does 0 1 . Indeed, conditions (1)-(4) in Theorem 3.5 are invariant undertaking the inverse. Moreover, since 0 is anticyclotomic, the extension F (0 )=Q is
Galois, hence the 0 1 -eigenspace of the p-part of ClF (0 ) vanishes if and only if the0 -eigenspace does.
Remark 3.7. Condition (4) in Theorem 3.5 implies that every  2 E 0 which is apth power of an element of M0 =T0 must be a pth power of an element of E 0 . In
particular the map
E 0 Zp Fp ! (M0 =T0 ) Zp Fp
is injective. One way to check condition (4) in practice is to compute the p-part Cof the class group of F (0 )(p ) as a Gal(F (0 )(p )=F (0 ))-module. In particular,
if ! denotes the character of Gal(F (0 )(p )=F (0 )) giving the action on p , thenKummer theory implies that Condition (4) is satis ed if the !-part of C is trivial.
In [BK09], Section 6, the authors exhibited an example of a Galois representation
whose mod $-reduction satis ed conditions (Red) and (Sc) for a Galois charactersatisfying conditions (1)-(5) of Theorem 3.5.

4. Deformations of 0
In this section we will x a residual representation 0 arising as the reductionof modular Galois representations  : G ! GL2 (O) and de ne a deformation
problem.
4.1. Eisenstein congruences. Let 1 ; 2 be two Hecke characters with in nity
types (11) (z ) = z and (21) (z ) = z 1. Put = 1 2 .
Denote by S the nite set of places where both i are rami ed, but  is unrami ed. Write Mi for the conductor of i . For an ideal N in OF and a nite place q
of F put Nq = NOF;q . We de ne

K 1(Nq ) =and
Now put
(4.1)







a b 2 GL (O ) j a 1; c  0 mod N ;
q
2 F;q
c d

U 1 (Nq ) = fk 2 GL2 (OF;q ) j det(k)  1 mod Nq g:
Kf :=

Y

2

q S

U 1 (M1;q )Y

2=

K 1 ((M1 M2 )q )  G(Af ):q S

From now on, let  be a nite set of places of F containingS := fq j M1 Mc1 M2 Mc2 g [ fq j pdF g:
We denote by T() the O-subalgebra of EndO (S2 (Kf ; )) generated by the
Hecke operators Tq for all places q 62 . Following [Tay88] (p. 107) we de ne idempotents ep and ep , commuting with each other and with T() acting on S2 (Kf ; ).
They are characterized by the property that any element h 2 X := epep S2 (Kf ; )which is an eigenvector for Tp and Tp satis es jap (h)jp = jap (h)jp = 1, where ap (h)(resp. ap (h)) is the Tp-eigenvalue (resp. Tp-eigenvalue) corresponding to h. Let

10

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Tord() denote the quotient algebra of T() obtained by restricting the Hecke
operators to X .
Let J ()  T() be the ideal generated byfTq 1 ($q )  #kq 2 ($q) j q 62 g:
De nition 4.1. Denote by m() a maximal ideal of Tord () containing the imageof J (). We set T := Tord ()m() . Moreover, set J := J ()T . We refer to J
as the Eisenstein ideal of T .
Theorem 4.2 ([Ber05], Theorem 6.3, [Ber09] Theorem 14). Let  be an unrami ed

Hecke character of in nity type (1) (z ) = z 2 . There exist Hecke characters 1 ; 2with 1 =2 =  such that their conductors are divisible only by rami ed primes orinert primes not congruent to 1 mod p and such that#(T =J )  #(O=(Lint (0; ))):
Proof. [Ber09] Theorem 14 states this inequality for the Hecke algebra T(). However, the Eisenstein cohomology class used in the proof of [Ber09] Theorem 14 isordinary because by [Ber08] Lemma 9 its Tp-eigenvalue (resp. Tp -eigenvalue) is thep-adic unit p1 (p) + 2 (p) (resp. p1 (p) + 2 (p)). Therefore one can prove thestatement for the ordinary cuspidal Hecke algebra.
˜
Remark 4.3. If  is unrami ed then p  is anticyclotomic (see [Ber09] Lemma
1). The condition on the conductor of the auxiliary character 1 together with ourassumption on the discriminant of F therefore ensure that for 0 = p  conditions(1)-(3) of Theorem 3.5 are automatically satis ed for  = S .
The assumption on the rami cation of  can be relaxed. For example, Proposition 16 and Theorem 28 of [Ber08] and Proposition 9 and Lemma 11 of [Ber09]
imply the following:
Theorem 4.4. Let 1, 2 be as at the start of this section with p - #(OF =M1M2 ).
Assume both M1 and M2 are coprime to (p) and the conductor of  := 1 =2 and;)divisible only by primes split in F=Q. Suppose LL(0(0;) 2 O. If the torsion part ofHc2 (SKf ; Zp ) is trivial, where
SKf = G(Q)nG(A)=Kf U (2)Cthen#(T =J )  #(O=(Lint (0; 1 =2 ))):
Remark 4.5. In fact, by replacing Zp by the appropriate coecient system, theresult is true for characters 1 ; 2 of in nity type zz m and z m 1 , respectively,
for m  0. For Theorems 4.2 and 4.4, the eld E needs to contain the values of thenite parts of 1 and 2 as well as Lint(0; 1 =2 ).
We will from now on assume that we are either in the situation of Theorem 4.2
or 4.4 and x the characters 1 ; 2 and  = 1 =2 , with corresponding conditions
on the set  and de nitions of Kf , T , and J . We also assume from now on thatvalp (Lint(0; )) > 0, however our main result (Theorem 6.13) remains vacuously
true when Lint(0; ) is a p-adic unit (see Remark 5.10). Put 0 = p  and assume
that 0 is -admissible. If we are in the situation of Theorem 4.4 then suppose
also that M1 and M2 are not divisible by any primes q such that #kq  1 mod p.
(This last assumption is only used in the proof of Theorem 4.9.)

AN R = T THEOREM

4.2. De nition of 0 . Write

S2 (Kf ; )m() =M 2 

11fKf

for a nite set  of ordinary cuspidal automorphic representations with central
character , such that fKf 6= 0. The set  is non-empty by Theorem 4.2 under
our assumption that valp (Lint(0; )) > 0.
Let  2  : Let  : G ! GL2 (E ) be the Galois representation attached to 
by Conjecture 2.4 (This is another point where we assume that E is large enough).
If we assume that c =  then the condition on the central character in Theorem
2.2 can be satis ed (after possibly twisting with a nite character), see [Ber09]
Lemma 8, so we do not need to refer to Conjecture 2.4 in this case.
The representation  is unrami ed at all q 2= S , and satis es
tr  (Frobq ) = aq ()anddet  (Frobq ) = ($q )  #(kq ):Q
By de nition, T injects into 2 EndO (Kf ). Since Tq acts on  Q
by multiplication by aq () 2 O the Hecke algebra T embeds, in fact, into B = 2 O(in particular T has no Z-torsion).
Observe that (tr  ())2 2 T  B for all  2 G . This follows from theChebotarev Density Theorem and the continuity of  (note that T is a niteO-algebra).
Fix  2  for the rest of this subsection. De ne 0 :=  2;1p . Then 0satis estr 0 (Frob q)  1 + (p )(Frob q) (mod $) for q 2= S ;anddet 0 =  2;2p   = p :
By choosing a suitable lattice  one can ensure that 0 has image inside GL2 (O).
The Chebotarev Density Theorem and the Brauer-Nesbitt Theorem imply that(0 )ss = 1  p :
By Theorem 2.2 0 is irreducible, and by Theorem 2.3 0 is ordinary so a standard argument (see e.g. Proposition 2.1 in [Rib76]) shows the lattice  may bechosen in such a way that 0 is not semi-simple and


1

0
(4.2)
 =


p 

and is ordinary. We put(4.3)
0 := 0 :Note that the isomorphism class of 0 is in fact independent of the choice of  2 
by Proposition 3.3.
Remark 4.6. The representation 0 satis es conditions (Red) and (Sc) of Section
3. The fact that 0 is ordinary combined with (4.2) implies that


1
(4.4)
0 jD =(p )jD :
p

p

12

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Note that in fact (4.4) together with condition (2) in De nition 3.1 implies that theextension F (0 )=F (0 ) is unrami ed away from p.
Remark 4.7. In Remark 4.6 of [BK09] we pointed out that the extension F (0)=F (0)under the conditions of Theorem 3.5 is totally rami ed at p, so, in particular, there
exists  2 Ip such that
0 ( ) = 1 11 :This implies that no twist of 0 by a character is invariant under c 2 Gal(F=Q)
and so no character twists of 0 arise from base change. However, we would liketo withdraw our additional statement in Remark 4.6 of [BK09] (and corresponding
remark in Section 4.6 of [Ber09]) that this implies that (twists of) deformations of 0cannot arise from base change. The G -invariant lattice in E 2 that we consider neednot be invariant under complex conjugation. In fact, Hida proved in [Hid82] theexistence of deformations of a twist of 0 for anticyclotomic characters 0 startingfrom congruences of CM forms over Q.
4.3. Deformation problem. Recall that (p) = pp is split. Let , , 0 and 0 be
as above. Recall that we have assumed that 0 is -admissible and have shown in
Section 4.2 that 0 satis es conditions (Red) and (Sc) of Section 3. Hence by Proposition 3.3, 0 is unique up to isomorphism. By (4.4) the extension F (0 )=F (0 )splits at p. In this section we study deformations of 0 .
Denote the category of local complete Noetherian O-algebras with residue eld
F by LCN(E ). An O-deformation of 0 is a pair consisting of A 2 LCN(E ) and astrict equivalence class of continuous representations  : G ! GL2 (A) such that0 =  (mod mA ), where mA is the maximal ideal of A. As is customary we will
denote a deformation by a single member of its strict equivalence class. Note thatthe Hodge-Tate weights of p  are 1 at p and +1 at p.
Following [SW97] we make the following de nition:
De nition 4.8. We say that an O-deformation  : G ! GL2 (A) of 0 is minimal if  is ordinary,det  = p ;and for all primes q 2  such that #kq  1 (mod p) one has


1

j =
:
Iq

p j Iq

Note that by our assumption on the conductor of , we in fact have p jI = 1 forq as above. Also the ordinarity condition means in this case that


1


j =
q

Ip

and

 1





jI 
=  1 :p

Note that 0 satis es all the conditions in De nition 4.8. Since 0 has a scalar
centralizer and -minimality is a deformation condition in the sense of [Maz97],
there exists a universal deformation ring which we will denote by R 2 LCN(E ),and a universal -minimal O-deformation ;O : G ! GL2 (R ) such that for every A 2 LCN(E ) there is a one-to-one correspondence between the set of O-algebraAN R = T THEOREM

13

maps R ! A (inducing identity on F) and the set of -minimal deformations
 : G ! GL2 (A) of 0 .
The arguments from Section 4.2 together with the uniqueness of 0 (Proposition
3.3 - recall that we are all the time assuming that 0 is -admissible) can now be

reinterpreted as:
Theorem 4.9. For any  2  there is an O-algebra homomorphism r : R “ O
inducing 0 .
Proof. The only property left to be checked is -minimality. This is clear since is unrami ed away from S , and no q 2 S satis es #kq  1 (mod p) byconstruction (if we are in the case of Theorem 4.2) or assumption (in the case of˜
Theorem 4.4).
The next three propositions were proved in [BK09] (see Propositions 5.4, 5.5 andLemma 5.6 in [loc. cit.]).
Proposition 4.10. There does not exist any non-trivial upper-triangular -minimal
deformation of 0 to GL2 (F[x]=x2 ).
Proposition 4.11. The universal deformation ring R is generated as an Oalgebra by traces.
Proposition 4.12. The image of the map R ! Q2 O given by x 7! (r (x))
is T .
Corollary 4.13. There exists a surjective O-algebra homomorphism r : R “ T .
˜
Proof. This is a direct consequence of Proposition 4.12.
5. Galois cohomology calculations
In this section we bound the Krull dimension of the universal deformation ringR via its tangent space. We conclude that R is a quotient of O[[X ]], which will
later allow us to prove that R is a complete intersection (see Corollary 6.14). We
also study reducible deformations of 0 to O=$n .
5.1. The Krull dimension of R is less than 3. Let 0 be as in Section 4 andlet U be the representation space for 0 . Then U is a two-dimensional F-vector
space with a G -stable ltration
0  U1  U ;where U1 is the 1-dimensional F-vector space on which G acts trivially. The
quotient U2 = U =U1 is a 1-dimensional F-vector space on which G acts via 0 .
Let W = ad0 = HomF (U ; U ) be the adjoint representation, and let
W Sel = ff 2 W : f (U )  U1 ; f (U1 ) = 0g:
Recall that 0 splits when restricted to Ip (cf. (4.4)). De ne the Ip -submodule
W Sel;t = ff 2 W : f (U )  U2 ; f (U2 ) = 0g:
For q 2  let
8
H 1 (Ip ; W =W Sel) if q = p;>><
H 1 (Ip ; W =W Sel;t) if q = p;
Hq = > H
1 (Iq ; W )if #kq  1 mod p;>:
0otherwise.

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

14

De ne the following Selmer group:MH1 (F; W ) = ker(H 1 (G ; W ) ! Hq ):2

q

Proposition 5.1.

dimF H1 (F; W )  1:
Proof. We proceed in a similar manner to the proof of Proposition 3.1 in [SW97]
Section 3. Let 1   comprise those primes in  such that #kq  1 mod p,
together with p and p. Let W1 = HomF (U2 ; U ), and let W2 = HomF (U1 ; U ). Thereis a commutative diagram of G -modules
// W1
// W
// W2
// 0
0
// F
// W =W Sel
// W2
// 0
0
having exact rows. Similarly, we have a commutative diagram of Ip -modules

0

››

››

››

// W1

// W

// W2

// 0

// W1
// W =W Sel;t
// F
// 0
0
having exact rows. (The third vertical arrow is induced by a splitting of the exact sequence for W2 as Ip -modules below.) Together these induce the following
commutative diagram of cohomology groups:// H 1 (G ; W1 )// H 1 (G ; W )// H 1 (G ; W2 ) ;
0
››

››

0
where

››
// X

››

L

//

21 Hq››qM

X = H 1 (Ip ; F) 21 nfpg››// Y

H 1 (Iq ; W1 )q

and

Y = H 1 (Ip ; F) M

21 nfpg

H 1 (Iq ; W2 ):q

Note that the rows in the diagram are exact. Indeed, the only thing to check is theexactness on the left, which can be veri ed easily by observing that the H 0 -terms
form a short exact sequence in each of the cases. Hence by the snake lemma there
is an exact sequence
0 ! ker( ) ! ker( ) = H1 (F; W ) ! ker( ):We claim now that ker( ) = 0 so that ker( ) ,! ker( ). First observe that W1 ts
into the short exact sequencef
F ! 0:
0 ! W Sel ! W1 !

AN R = T THEOREM

15

Since W1 
= 0 0 1, it is clear that W1G = 0 and hence the associated long exact
cohomology sequence yields the exact sequence1) 1
0 ! F ! H 1 (G ; W Sel) ! H 1 (G ; W1 ) H (f!
H (G ; F):
1
Since W Sel = F(0 ), it follows from the -admissibility of 0 that the second
arrow is surjective. Hence H 1 (f ) is injective.

Lemma 5.2.

ker( ) ,! ker(H 1 (G ; F) !

M

21

H 1 (Iq ; F))) = 0:q

Proof. Consider the following diagram

H 1 (G ; W1 )

H 1 (f ) // 1
H (G ; F)
:=q21 resq

f0 L// q2 H 1 (Iq ; F)X1
01
where f is the identity on H (Ip ; F) and is induced by f on the rest of the compo››››

nents. By de nitions of the maps involved it is clear that the diagram is commutative. Since H 1 (f ) is injective as observed above, if c 2 ker , its image must lie inker , hence the inclusion of the kernels in the statement of the lemma follows. Now
consider c 2 ker , i.e., c is unrami ed at all q 2 1 . But since F is a p-power orderabelian group, we see that c is also unrami ed at all q 62 1 . Hence it is unrami ed
everywhere. Since H 1 (G ; F) = Hom(G ; F) and by our assumption p - #ClF it is˜clear that c = 0.
Similarly, W2 ts into the short exact sequence
0 ! F ! W2 ! Hom(U1 ; U2 ) = F(0 ) ! 0:
The associated long exact cohomology sequence yields the commutative diagram// H 1 (G ; W2 )// H 1 (G ; F(0 ))H 1 (G ; F) ﬄ
L

21

q

f1
H 1 (Iq ; F) ﬄ 
››

››
// Y
//

L

21 nfpg

qf2
H 1 (Iq ; F(0 ))››having exact rows. As above we see that ker(f1 ) = 0 and therefore
ker( ) ,! ker(f2 )  H 1 (G ; F(0 )):Since the character 0 is -admissible we have that dimF H 1 (G ; F(0 )) = 1. This˜
concludes the proof of Proposition 5.1.
Proposition 5.3. We haveH1 (F; ad0 0 ) = Hom(mR =(m2R + $R ); F) = HomO alg(R ; F[x]=x2 ):Proof. See for example [dS97], Theorem 15 which together with Lemma 5.4 yields
the Proposition.
˜
Lemma 5.4. Hp is the cohomological condition corresponding to ordinarity at p of
an in nitesimal deformation, i.e., a deformation to GL2 (F[X ]=X 2).

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

16

Proof. Let  be an in nitesimal deformation of 0 with det() = = 0 . Write
(g)0 (g) 1 = 1 + c (g):Then c de nes a 1-cocycle with values in ad0 0 . For g 2 Ip , we have for a suitablebasis (with respect to which 0 splits and  is lowertriangular)


0
0
c (g) 2  0 = W Sel;t;so the image of [c ] in Hp is zero, as desired. Conversely, if this condition is satis ed
˜then jI is lower-triangular , hence so is jD and  is ordinary at p.
Corollary 5.5. We have R = O[[X ]]=I for an ideal I .
Proof. This is an application of Nakayama, see e.g. [dS97] Theorem 15 or [Til96],
Lemma 5.1. Note that we have H1 (F; ad 0 ) = H1 (F; ad0 0 ) since p - # ClF by˜
our assumption.
5.2. No reducible deformation beyond p-valuation of L-value. Set := p and write r for mod $r with r > 0. Note that Gal(F ( )=F ) =   with
= Zp and  a nite abelian group. Assume p - #. Set ~0 := j.
For a nite abelian extension K of F write M (K ) for the maximal abelian prop-extension of K unrami ed away from the primes lying over p. For a character' : Gal(K=F ) ! O write M (K )' for the maximal abelian pro-p-extension of
K unrami ed away from the primes lying over p such that Gal(K=F ) acts on
Gal(M (K )' =K ) via ' 1 .
Lemma 5.6. Any -minimal uppertriangular deformation r of 0 to O=$r musthave the form


1

:
r =
p

p

r

Proof. We will prove this by induction on r. It is true for r = 1, so assume thisholds for r = k. So, we can write

k b
1
+
$
 =
;
k+1

d

where : G ! F is a group homomorphism (the group operation on F being
addition). Arguing as in the proof of Proposition 5.4 of [BK09] we see that can
only be rami ed at p and p. Ordinarity at p (respectively at p) forces k+1 jI(respectively k+1 jI ) to have a free O=$k+1 -submodule (respectively quotient)on which Ip (respectively Ip ) acts trivially. This together with the fact that 0
is rami ed at p and at p easily imply that jI = jI = 0. Finally, using theassumption that p - # ClF we conclude that = 0. Since det k+1 = k+1 , we get
d = k+1 .
˜
Write e for the rami cation index of O over Zp . Note that the exponent of O=$r
is pdr=ee , where for a real number , d e denotes the smallest integer n  . By aslight abuse of terminology we will say that a deformation  of 0 is upper-triangular
if there exists a member of the strict equivalence of  that is upper-triangular.
Lemma 5.7. Suppose there exists a -minimal uppertriangular deformation r of
0 to GL2 (O=$r ). Then there exists a surjective map of groups
Gal(M (F ( r )) r =F ( r )) “ Z=pdr=ee :
p

p

p

p

AN R = T THEOREM

17

Proof. Consider the b-entry of r as a function b : G ! O=$r . Suppose thatthe restriction bjGal(F (r )=F ( r )) 2 $O=$r O. This means that the mod $ reduction of b is zero after we split r , so that F (0 )  F ( r ), but this is impossible as F (0 )=F is non-abelian. So, we get that bjGal(F (r )=F ( r )) 62 $O=$r O.
Hence Gal(F (r )=F ( r )) has a cyclic quotient of order pdr=ee . Since F (r ) ˜
M (F ( r ) r ), we are done.
Write Lint(0; ) = u$n, where u is a unit. One has valp ($) = 1=e. We also havefor any x 2 O, that valp (x) = 1e val$ (x).
Remark 5.8. Note that we have the following sequence of equalities:(5.1) #O=Lint (0; ) = #O=$n = pn[O:Zp ]e= pn[F:Fp] =
int

int

= pval$ (L (0;))[F:Fp] = pvalp (L (0;))[O:Zp] :Proposition 5.9. Let n be as above. There is no upper-triangular -minimal
deformation n+1 : G ! GL2 (O=$n+1 ).
Proof. Suppose n+1 exists. By Lemma 5.7 there exists an elementx 2 Gal(M (F ( n+1 )) n+1 =F ( n+1 ))
d
(
n
+1)
=e
e
of order p. Note that since p - #, the character ~0 is the Teichmuller liftof a character with values in F . Let F0 be the smallest sub eld Fp  F0  Fsuch that the image of (the reduction of) ~0 is contained in F0 . Let O0 be thecorresponding extension of Zp , i.e., Zp  O0  O. Note that we can assume
without loss of generality that O0 is unrami ed over Zp . By Lemma 5.6, n+1 :G ! GL2 (O=$n+1 ) has the form


1
b
(

)
n+1 () =:n+1 ( )Let y 2  be a generator (note that  is cyclic as a group isomorphic to a subgroup of F0 ). Then it is easy to see that ~0 1 (yk )b(x) are all elements of orderpd(n+1)=ee , and they generate a subgroup H of b(Gal(F (n+1 )=F ( n+1 )) of order(pd(n+1)=ee )[F0 :Fp] = (pd(n+1)=ee )[O0 :Zp] - the last equality is true because O0 =Zp is
unrami ed. (This subgroup still has exponent only pd(n+1)=ee .)Now we have
11
1
(H Zp O)~0 = (H Zp O0 O0 O)~0 = (H Zp O0 )~0 O0 O;
and hence(5.2)
1
1#(H Zp O)~0 = (#(H Zp O0 )~0 )[O:O0]  (#H )[O:O0 ] = pd(n+1)=ee[O:Zp ] =pd(n+1)=eee[F:Fp ]  p(n+1)[F:Fp] = #O=$n+1 :But the group H can be regarded as a subgroup of Gal(F (n+1 )=F ( n+1 )). We
have F (n+1 )  M (F ( n+1 )) , hence we arrive at a contradiction by Theorem
5.11 below.
˜
Remark 5.10. Note that the proof of Proposition 5.9 is valid also for n = 0. Henceit shows that in the case when Lint(0; ) is a p-adic unit, 0 as in Section 3 is not-minimal.

18

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2This implies that when n = 0 Theorem 6.13 remains vacuously true, since condition (5) in that theorem can never be satis ed.
Theorem 5.11. One has
1#(Gal(M (F ( n+1 )) =F ( n+1)) O)~0  #O=Lint(0; ) = #(O=$n ):Proof. One estimates the order of the left hand side by relating it to the Selmer
group of a Hecke character. Using standard methods one can bound the order ofthe latter from above by the L-value applying the main conjecture proven by Rubin.
For details see Lemma 5.14 and 5.15 of [BK09], where this is done for n = 1. The
˜
proofs of both of these lemmas generalize easily to arbitrary n.
6. R = TIn this section we will prove the main result which asserts that the surjection in
Corollary 4.13 is an isomorphism. As a consequence we obtain a result on modularity of Galois representations and deduce some properties of the Hecke algebra
and cyclicity of a certain Galois group. As in Section 5.2 in what follows we assume
that p - #. As before, throughout this section we assume that 0 is -admissible.
6.1. The ideal of reducibility. We brie y recall some general facts about Eisenstein representations from Section 3 of [Cal06] and Section 2 of [BC06]: Let (R; mR ; F)be a local p-adically complete ring. Let G be a topological group and consider acontinuous representation  : G ! GL2 (R) such that tr () mod mR is the sum of
two distinct characters i : G ! F ; i = 1; 2. Moreover, assume thatdimF Ext1cts;F[G](1 ; 2 ) = dimF Ext1cts;F[G] (2 ; 1 ) = 1:
De nition 6.1. The ideal of reducibility of R is the smallest ideal I of R such thattr () mod I is the sum of two characters. We will denote the ideal of reducibilityof R by Ire .
Proposition 6.2. The ideal Ire is principal.
Proof. See [Cal06], Proof of Lemma 3.4. This uses in a crucial way condition (3)of -admissibility of 0 (see De nition 3.1).
˜
Theorem 6.3 (Urban). Let (R; mR; F) be a local Artinian ring. Let 1 , 2, and be three representations of a topological group G with coecients in R (with having image in GLm (R)). Assume the following are true:  and 1  2 have the same characteristic polynomials;
 The mod mR -reductions 1 and 2 of 1 and 2 respectively are absolutely
irreducible and non-isomorphic; The mod mR -reduction  of  is indecomposable and the subrepresentationof  is isomorphic to 1 .
Then there exists g 2 GLm (R) such that



(
h
)

1
1
(h) = g
2 (h) g
for all h 2 G.
Proof. This is Theorem 1 in [Urb99].
˜AN R = T THEOREM

19

Corollary 6.4. Let R 2 LCN(E ) and suppose  : G ! GL2(R) is a -minimal
deformation of 0 . Let I  R be an ideal such that R=I 2 LCN(E ) and is an Artinring. Then I contains the ideal of reducibility of R if and only if  mod I is anupper-triangular deformation of 0 to GL2 (R=I ).
Proof. If  mod I is isomorphic to an upper-triangular deformation of 0 to GL2 (R=I ),then clearly tr  mod I is a sum of two characters (which are distinct, since theymust reduce to 0 and 1 modulo mR ), so I contains the ideal of reducibility. The
converse is an easy consequence of Theorem 6.3.
˜
int
Proposition 6.5. Assume n = val$ (L (0; )) > 0. Then one has R =Ire =O=$r where 0 < r  n.
Proof. Write S for R =Ire . Then S is a local complete ring. Moreover, by Corollary 5.5 we have that S is a quotient of O[[X ]], and hence R =$R (and thus
S=$S ) is a quotient of F[[X ]]. But F[[X ]] is a dvr, so S=$S = F[[X ]]=X m forsome m 2 Z+ [ f1g. (By F[[X ]]=X 1 we mean F[[X ]].) We will rst show thatm = 1. Suppose m 6= 1. Then HomO alg(R ; S=$S ) contains at least two elements - the map R “ F ,! S=$S and the surjection R “ S=$S . These
two elements give rise to two distinct elements in HomO alg(R ; F[X ]=X 2), thetrivial one and the surjection R “ S=$S “ F[X ]=X 2. By the de nition of R
there is a one-to-one correspondence between the deformations to F[X ]=X 2 and elements of HomO alg (R ; F[X ]=X 2). The trivial element corresponds to the trivial
deformation to F[X ]=X 2, i.e., with image contained in GL2 (F), which is clearly
upper-triangular. However, the deformation corresponding to the surjection mustalso be upper-triangular by Corollary 6.4 since ker(R “ S=$S “ F[X ]=X 2) contains Ire and F[X ]=X 2 is Artinian. But we know by Proposition 4.10 that 0 does
not admit any non-trivial -minimal upper-triangular deformations to F[X ]=X 2.
Hence we must have m = 1.
Thus by the complete version of Nakayama's Lemma ([Eis95], Exercise 7.2) weknow that S is generated (as a O-module) by one element. So S = O=$r withr 2 Z+ [ f1g. Finally we must have 0 < r  n, r 6= 1 since by Corollary 6.4 if
r > n or r = 1, then there would be an upper-triangular -minimal deformationof 0 to O=$n+1 , which is impossible by Proposition 5.9.
˜
Remark 6.6. Note that the fact that S is a quotient of O[[X ]] is actually notnecessary for the proof of Proposition 6.5. Indeed, it suces to know that S (andhence S=$S ) is topologically nitely generated as an O-algebra. It is an easy factthat then S=$S = F if and only if S=$S has no quotient isomorphic to F[X ]=X 2.
Corollary 6.7. Let n = val$ (Lint(0; )) > 0. Suppose there exists  2  suchthat tr 0 (Frobq ) = 1 + (p )(Frobq ) mod $n for all q 62 . Write r;n for thecomposite R “ O “ O=$n (the rst arrow being r ). Then Ire = ker r;n .
Proof. By the de nition of the ideal of reducibility we have Ire  ker r;n . But, by
Proposition 6.5, R =Ire = O=$r for 0 < r  n, so we have O=$r = R =Ire “R = ker r;n = O=$n and the corollary follows.
˜
6.2. A commutative algebra criterion and R=T theorem. Let R and S
denote complete local Noetherian O-algebras with residue eld F. Suppose that S
is nitely generated as a module over Zp and has no Z-torsion. Write mR and mS
for the maximal ideals of R and S respectively.

20

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Lemma 6.8. Let x 2 S be such that #(S=xS ) < 1. Then multiplication by x is
injective.
Proof. Consider the exact sequencex S ! S=xS ! 0:
0 ! ker x ! S !
Since S is nitely generated as a module over Zp we can tensor the exact sequencewith Qp and conclude that since S=xS Qp = 0, we must also have ker x Qp = 0.
Since ker x is a nitely generated Zp -module it must be a nite group. Hence if
a 2 ker x, pm a = 0 for some m 2 Z0 . Thus a is a torsion element, which implies
˜
a = 0 by our assumption on S .
Proposition 6.9. Suppose there exists a surjective O-algebra map  : R “ S
inducing identity on the residue elds and an element  2 R such that the bottom
map in the following commutative diagram
R



// S

// S=( )S
R=R
is an isomorphism. If R=R = O=$r for some positive integer r, then  is an››››

isomorphism.
Proof. Write x = (). We have (k ) = xk . First, we are going to show that
R=k R = S=xk S for every positive integer k. Indeed, the map R=k R ! S=(k )Sis surjective because R ! S is. So, it remains to prove injectivity. The mapk 1 R=k R ! k R=k+1 R given by multiplication by  is clearly surjective, so we
get#(R=R)  #(R=2 R)  : : : :Moreover, we have the short exact sequence
0 ! k 1 R=k R ! R=k R ! R=k 1 R ! 0;
so #(R=k R) = #(R=k 1 R)#(k 1 R=k R). Combining this equation with theprevious sequence of inequalities, we get by induction on k that #(R=k R) (#(R=R))k . On the other hand, the maps

x
x 2 3 x
x k 1 k
S=xS “xS=x2 S “x S=x S “ : : : “x S=x Sare all injective because multiplication by x on S is injective by Lemma 6.8. Thus

we get#(S=xk S ) = #(S=xS )k = #(R=R)k  #(R=k R):
This proves injectivity of the map R=k R “ S=xk S .
Hence
lim R=k R = lim S=xk S:

kk
So, it suces to show that R = lim R= R and S = lim S=xk S . The rst followskkfrom Lemma 7.14 on page 197 in [Eis95] since for every power of mR , say mkR
there is a power s of  such that ()s  mkR (indeed, take s = k) and for every
power of (), say ()k there is a power of msR of mR such that msR  ()k (indeed,kAN R = T THEOREM

21

since mR = (; $), we get mrR  (), so s = kr works). The situation for S is
analogous.
˜
We will now state some consequences of Proposition 6.9.
Theorem 6.10. The map r : R “ T in Corollary 4.13 is an isomorphism.
Proof. Write J for the G -representation induced by R ! T =J . By de nitionof J and the Chebotarev density theorem we see that tr J = 1 + p , hencer 1 (J )  Ire and hence J  r(Ire ). By Proposition 6.2 the ideal Ire is principal
and we x a generator  of Ire . Thus the surjectionR =R “ T =r()T “ T =J ;must be an isomorphism since R =R = O=$r with r  n by Proposition 6.5
and #(T =J )  #O=$n by Theorems 4.2 and 4.4. Here n = val$ (Lint (0; )).
Hence r induces the commutative diagram in Proposition 6.9 with R = R , S = T(note that T is Z-torsion-free and nitely generated as a module over Zp sinceby de nition it acts faithfully on a nite dimensional vector space of characteristic
˜zero).
int
Corollary 6.11. Let n = val$ (L (0; )). Suppose there exists  2  such thattr 0 (Frobq ) = 1 + (p )(Frobq ) mod $n for all q 62 . Then R = T = O, i.e.,R = T is a discrete valuation ring.
Proof. By Corollary 6.7 we have r 1 ($n O) = Ire . So, we see that r induces thecommutative diagram in Proposition 6.9 with R = R , S = O,  a generator of˜
Ire .
Remark 6.12. Note that Corollary 6.11 implies that if n = val$ (Lint(0; )) > 0
and there exists an automorphic representation  2  such that  has Hecke
eigenvalues congruent to the Hecke eigenvalues of an Eisenstein series mod $n ,then  = fg.
6.3. Modularity theorem. In this section we state a modularity theorem which
is a consequence of the results of the previous sections. To make its statement
self-contained, we explicitly include all the assumptions we have made so far.
Theorem 6.13. Let F be an imaginary quadratic eld and p > 3 a rational prime
which splits in F . Fix a prime p of F over p. Assume that p - # ClF and that any
prime q j discF satis es q 6 1 (mod p): Let 1 , 2 be Hecke characters of F withsplit conductors and of in nity type z and z 1 respectively such that  := 1 =2
is unrami ed. Assume that the conductor M1 of 1 is coprime to (p) and thatp - #(OF =M1 ) . Write Gal(F (p )=F ) =   with = Zp and  a nitegroup. Assume p - #:Let  : G ! GL2 (E ) be a continuous irreducible representation that is ordinary
at all places q j p. Suppose ss = 1  2 with 1 = 1;p , 2 = 2;p . Set
0 := 1 2 1 . If all of the following conditions are satis ed:(1)   fq j pdF M1 Mc1 g,(2) p - #(3) 0 is -admissible (cf. De nition 3.1),(4) det() = 1 2 ,(5)  2;1p is -minimal,
then  is modular in the sense of De nition 2.5.

22

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

6.4. More consequences of our result. We will now state some corollaries of
the R = T theorem proved in the previous section. As before, we assume that 0
is -admissible. We will also discuss how our approach can be used to partially
recover the results of Skinner and Wiles [SW99].
Corollary 6.14. The Eisenstein ideal J is a principal ideal and the Hecke algebraT is a complete intersection (and hence Gorenstein).
Proof. The rst statement follows immediately from Theorem 6.10 and the factthat the ideal of reducibility Ire (which is principal - see Proposition 6.2) is mapped
exactly onto the Eisenstein ideal J (see proof of Theorem 6.10). The fact that T
is Gorenstein follows from Proposition 6.4 in [Bas63]. Note that the proposition
is applicable since the maximal ideal of T is generated by $ and a generator of
the ideal of reducibility (see Proof of Proposition 6.9) and p is clearly a non-zerodivisor. Finally since R = T = O[[X ]]=I with the ideal I of codimension 1, theGorenstein condition is equivalent with principality of I ([Eis95], Corollary 21.20).
Hence T is a complete intersection.
˜
Remark 6.15. The properties of the Hecke algebra and the Eisenstein ideal statedin Corollary 6.14 were proved for their counterparts over Q by Mazur ([Maz77],
Theorem 11, and Chapter 2, Section 14). See also a discussion in [CE05] (page 99
and Corollary 3.17) where these properties, like here, are derived as a consequenceof an R = T theorem.
Corollary 6.16. Let the notation be as in Section 5. There exists a -minimal
upper-triangular deformation of 0 to GL2 (O=$n ). Moreover, the O-module
1
Gal(M (F ( n )) =F ( n )) O)~0
is isomorphic to O=$n .
Proof. The rst statement follows from the isomorphisms R =Ire == T =J 
O=$n combined with Theorem 6.3. The existence of the upper-triangular deformation together with Lemma 5.7 provides an element of Gal(M (F ( n )) n =F ( n ))of order pdn=ee . As in the proof of Theorem 5.11 we see that this element generates
an O-submodule of1
Gal(M (F ( n )) =F ( n )) O)~0of order #O=$n . Hence the second statement follows from Theorem 5.11.
˜
Remark 6.17. While the upper bounds on the order of the Galois group in Corollary 6.16 are predicted by the Main Conjecture of Iwasawa Theory, their exact Omodule structure is in general a mystery. Our results show that the upper bounds
are in fact optimal and provide a rather de nitive answer to structure question,
however only in the case of a -admissible character, where one assumes at theoutset the cyclicity of the group of extensions (see De nition 3.1).
Finally, we will discuss to what extent our method provides an alternative tothe approach of Skinner and Wiles in [SW99] for proving an R = T theorem for
2-dimensional, p-adic, residually reducible, ordinary representations of GQ withunique non-semisimple reduction which are unrami ed outside nitely many primes.
First note that the assumptions of [SW99] are weaker than ours, because they only
imply that the group of extensions of 1 by the character as in De nition 3.1 is onedimensional, but not so if one reverses the order of 1 and the character. However,AN R = T THEOREM

23

even with that weaker assumption in place our Proposition 4.10 carries over (withthe same proof) showing that there are no upper-triangular -minimal deformations of the residual representation 0 of [SW99] to GL2 (F[X ]=X 2). Using this we
proceed as in the proof of our Proposition 6.5 to show that the universal deformamintion ring (denoted in [SW99] by Rmin= O=$k;O ) has the property that R;O =Ire 
for some k 2 Z+ [ f1g, where Ire denotes the ideal of reducibility. Note that
by Remark 6.6 it is not necessary to know that Rmin;O is a quotient of O[[X ]].
Since [SW99] in fact do exhibit a -minimal upper-triangular deformation Eis
;O tominGL2 (O), we must have k = 1, i.e., R;O =Ire = O. In particular, Ire is the kernel
Eisof the map Rmin;O “ O correspoding to ;O , which in [SW99] is denoted by I .
It follows therefore that the surjection of Rmin;O onto the (full, not just cuspidal)
Hecke algebra (that in [SW99] is denoted by Tmin;O ) descends to an isomorphism
min
Eis
Eis

T
=I
,
where
I
denotestheEisenstein
ideal (see also PropoRmin
=I
=
;O
;O re
sition 3.12 in [CE05] where an analogous statement is proved in the case when
the residual representation is reducible and semi-simple). In particular our method
gives an alternative proof to the statement that every reducible -minimal deformation of 0 is modular. However, we cannot apply our commutative algebra criterionmin(Proposition 6.9) to conclude that Rmin;O = T;O . First, as mentioned above, theassumptions of [SW99] are weaker than our assumption of -admissibility, so do
not imply that the ideal of reducibility is principal, and secondly, Rmin;O =Ire is notnite. In [SW99] and [CE05] the R = T statement is proved using the numericalcriterion of Wiles and Lenstra which requires one to compare the size of Ire =Ire2
with the size of the quotient of the cuspidal Hecke algebra by I Eis .
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1 University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge, CB3 0WB, United Kingdom.
2 Universit
e Paris 13, Departement de Mathematiques, Institut Galilee, 99, avenue
Jean-Baptiste Clement, 93430 - Villetaneuse, France