ON DEFORMATION RINGS OF RESIDUALLY REDUCIBLEGALOIS REPRESENTATIONS AND R = T THEOREMSTOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Abstract. We study the crystalline universal deformation ring R (and its
ideal of reducibility I) of a mod p Galois representation ρ0 of dimension nwhose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents ρ1 and ρ2 . Under some assumptions on
Selmer groups associated with ρ1 and ρ2 we show that R/I is cyclic and often
finite. Using ideas and results of (but somewhat different assumptions from)
Bellaı̈che and Chenevier we prove that I is principal for essentially self-dual
representations and deduce statements about the structure of R. Using a newcommutative algebra criterion we show that given enough information on the
Hecke side one gets an R = T -theorem. We then apply the technique to modularity problems for 2-dimensional representations over an imaginary quadratic
field and a 4-dimensional representation over Q.

1. Introduction
Let F be a number field, Σ a finite set of primes of F and GΣ the Galois groupof the maximal extension of F unramified outside Σ. Let E be a finite extensionof Qp with ring of integers O and residue field F. Let ρ0 : GΣ → GLn (F) be anon-semi-simple continuous representation of the Galois group GΣ with coefficientsin F. Suppose that ρ0 has the formρ∗
ρ0 = 10 ρ2for two absolutely irreducible continuous representations ρi : GΣ → GLni (F) withn1 + n2 = n. The goal of this article is to study the crystalline universal deformation ring RΣ of ρ0 and in favorable cases show that it is isomorphic to a Hecke
algebra TΣ associated to automorphic forms on some algebraic group. Our approach relies on studying the ideal of reducibility I ⊂ RΣ as defined by Bellaı̈che
and Chenevier and the quotient RΣ /I. Roughly speaking the latter “captures” thereducible deformations, while the former captures the irreducible ones. As a first
result we prove that under some self-duality condition imposed on the deformations
the ideal I is principal (section 2). In contrast to [BC09] we do not assume thatthe trace of our universal deformation is “generically irreducible”. As a result wecannot affirm that I is generated by a non-zero divisor, but this is not needed forour main results. We can, however, still show that I is generated by a non-zerodivisor under a certain finiteness assumption (section 3), which is only used forred
results concerning RΣ, the quotient of RΣ by its nilradical.
Date: 10 February 2011.
2000 Mathematics Subject Classification. 11F80, 11F55.
Key words and phrases. Galois deformations, automorphic forms.
1

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TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

We then study the quotient RΣ /I. Under the following two major assumptions:• that the crystalline universal deformation rings of ρ1 and ρ2 are discrete
valuation rings (= O);• that the Selmer group HΣ1 (F, HomF (ρ2 , ρ1 )) is one-dimensional (see section
4 for the definition of HΣ1 ),we prove that the O-algebra structure map O → RΣ /I is surjective (section 6.3).
Combining this with the principality of I we show (section 6.4) that
• RΣ is a quotient of O[[X]],red• the reduced universal deformation ring RΣ
is a complete intersection.
The above properties give us enough control on the ring RΣ to formulate some numerical conditions (which if satisfied) imply an R = T theorem in this n-dimensional
context (section 8). In fact, our method can be summarized as follows. Suppose that we have an O-algebra surjection φ : RΣ  TΣ which induces a map
φ : RΣ /I  TΣ /φ(I). The surjection O  RΣ /I often factors through an isomorphism O/$m ∼= RΣ /I. In fact the size of RΣ /I is bounded from above by the size ofa certain Selmer group, HΣ1 (F, HomO (ρ̃2 , ρ̃1 ) ⊗ E/O), where ρ̃i denotes the uniquelift of ρi to GLni (O). Thus assuming #HΣ1 (F, HomO (ρ̃2 , ρ̃1 ) ⊗ E/O) ≤ #TΣ /φ(I)we conclude that φ is an isomorphism. We then apply a new commutative algebra
criterion (section 7) which uses principality of I as an input to allow us to conclude
that φ itself must have been an isomorphism.
One way to achieve the inequality #HΣ1 (F, HomO (ρ̃2 , ρ̃1 )⊗E/O) ≤ #TΣ /φ(I) isto relate both sides to the same L-value (these are the numerical conditions referred
to above). Many results bounding the right-hand side from below by the relevant
L-value are available in the literature (see sections 9 and 10 for examples of such
results). A corresponding upper-bound on #HΣ1 (F, HomO (ρ̃2 , ρ̃1 ) ⊗ E/O) can be
deduced from the relevant case of the Bloch-Kato conjecture. See Theorems 8.5
and 8.6, where the numerical conditions are stated precisely. In particular it isalso possible to apply our method if RΣ /I is infinite. In this case our commutative
algebra criterion is an alternative to the criterion of Wiles and Lenstra.
Let us now make some remarks about relations of our approach to other modularity results. As is perhaps obvious to the informed reader, it is different fromthe Taylor-Wiles method. Also, our residual representations are not “big” in thesense of Clozel, Harris and Taylor [CHT08]. There is some connection between
our setup and that of Skinner and Wiles [SW97], who studied residually reducible
2-dimesional representations of GQ , but the main arguments are different. A prototype of this method has already been employed by the authors to prove an R = T theorem for two-dimensional residually reducible Galois representations over animaginary quadratic field [BK09, BK11]. However, the assumptions of [BK11] aredifferent and the proofs reflect the “abelian” context of that article, and mostly
could not be generalized to the current setup. In particular the principality of theideal of reducibility in that context was a simple consequence of a certain uniqueness condition imposed on ρ0 . In the “non-abelian” setup the analogous uniquenesscondition is almost never satisfied. Finally let us note, that unlike recent higher
dimensional modularity results of Taylor et al. (e.g. [Tay08, Ger10, BLGGT10])
which prove Rred = T theorems, our method implies that RΣ is reduced.
In order to study crystalline deformations we establish certain functoriality properties of the Selmer groups HΣ1 (F, HomO (ρ̃i , ρ̃j ) ⊗ E/O) for i, j ∈ {1, 2}. In particular we need to know that these Selmer groups behave well with respect to taking

ON DEFORMATION RINGS

3

fixed-order torsion elements. On the other hand our numerical criteria strongly suggest that the bounds imposed to control the order of these Selmer groups should begiven by L-values. Conjecturally, it is the Bloch-Kato Selmer groups whose orders
are controlled by these L-values, hence we also relate compare these to our Selmer
groups HΣ1 (F, HomO (ρ̃i , ρ̃j ) ⊗ E/O). This is all done in section 4.
Let us now discuss the “numerical conditions” in more detail. They fall into
two categories depending on whether one has a reducible lift to characteristic zeroor not. We will focus here on the case when no such lift exists. (We refer thereader to Theorems 8.5 and 8.6 for the precise statement and to the discussion
following the theorems.) One of them is a lower bound on the order of the quotientTΣ /φ(I) = TΣ /J, where T is a certain “non-Eisenstein” Hecke algebra and J is
the corresponding “Eisenstein ideal”. Such quotients (and lower bounds on them)
have been studied by many authors, for example [SW99], [Ber09] (where J is indeed
the Eisenstein ideal) and [Bro07], [Klo09] (where J is the CAP ideal - see [Klo09]
for a precise definition), or [AK10] (where J is the “Yoshida ideal”). In general
this quotient measures congruences between automorphic forms with irreducible
Galois representation and a fixed automorphic form “lifted” from a proper Levi
subgroup (Eisenstein series, Saito-Kurowawa lift, Maass lift, Yoshida lift). Eachof these results bound this module from below by a certain L-value, which in fact
is the L-value which conjecturally gives the order of the Bloch-Kato Selmer group
Hf1 (F, HomO (ρ̃2 , ρ̃1 ) ⊗ E/O). The second numerical condition is the upper boundon the order of a related Selmer group HΣ1 (F, HomO (ρ̃2 , ρ̃1 ) ⊗ E/O) by the same
number. This condition thus seems to require (the $-part of) the Bloch-Katoconjecture for HomO (ρ̃2 , ρ̃1 ) and is currently out of reach in most cases when ρ1and ρ2 are not characters. So, our R = T result (Theorem 8.5) should be viewed
as a statement asserting that under certain assumptions on the Hecke side (the $part of) the Bloch-Kato conjecture for HomO (ρ̃2 , ρ̃1 ) (which in principle controls
extensions of ρ˜2 by ρ̃1 hence reducible deformations of ρ0 ) implies an R = T -theorem(which asserts modularity of both the reducible and the irreducible deformations of
ρ0 ). The fact that we can deduce modularity of all deformations from a statement
about just reducible deformations is a consequence of the principality of the idealof reducibility (whose size roughly speaking controls the irreducible deformations)and the commutative algebra criterion.
In the last two sections of the article we study two examples in which some (orall) of the conditions can be checked. The first example is still of an abelian natureand is in a sense a “crystalline” complement to our previous two articles [BK09]
and [BK11], where we studied ordinary deformations. The second example is muchless special and is in fact a prototypical higher-dimensional problem to which wehope our result may be applied. In this example we study certain irreducible fourdimensional crystalline deformations of a representation of the formρ (k/2 − 1) ∗,
ρ0 = f0
ρgwhere ρf and ρg are reductions of the Galois representations attached to two elliptic cusp forms f and g of weights 2 and k=even respectively. Using results of[AK10] and [BDSP10] which under some assumptions provide one of the numericalconditions (a lower bound on #TΣ /J) we prove that the Bloch-Kato conjecture inthis context implies that every such deformation of ρ0 is modular (i.e., comes froma Siegel modular form). For a precise statement see Theorem 10.3.

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TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Our method of proving principality of the ideal of reducibility (section 2) owes a
lot to the ideas of Bellaı̈che and Chenevier and the authors benefited greatly fromreading their book [BC09]. The authors would also like to thank the Mathematical Institute in Oberwolfach and the Max-Planck-Institut in Bonn, where part ofthis work was carried out for their hospitality. The first author would like to thankQueens’ College, Cambridge, and the second author would like to thank the Department of Mathematics at Université Paris 13, where part of this work was carriedout. We are also grateful to Joël Bellaı̈che, Gaëtan Chenevier, Neil Dummigan,
Matthew Emerton and Jacques Tilouine for helpful comments and conversations.
2. Principality of Reducibility ideals
Let A be a Noetherian henselian local (commutative) ring with maximal idealmA and residue field F and let R be an A-algebra. Let ρ : R → Mn (A) be amorphism of A-algebras and put T = tr ρ : R → A. We assume n! is invertible in
A and


τ1 ∗
ρ=
mod mA0 τ2
is a non-semisimple extension of τ2 by τ1 for two non-isomorphic absolutely irreducible representations τi of dimension ni .
Definition 2.1 ([BC09] Definition 1.5.2). The ideal of reducibility of T is thesmallest ideal I of A such that tr (ρ) mod I is the sum of two pseudocharacters
T1 , T2 such that Ti = tr τi mod mA . We will denote it by IT . (For a definition of apseudocharacter see e.g. [loc.cit], section 1.2.)
Definition 2.2 ([BC09] Section 1.2.4). The kernel of a pseudocharacter T : R → A
is the two-sided ideal of R defined by
ker T = {x ∈ R : ∀y ∈ R, T (xy) = 0}.
Remark 2.3. Note that T is an A-module homomorphism. If KT denotes thekernel of T as an A-module map, then clearly KT ⊃ ker T . This inclusion is in
general strict, and in fact it is often the case that ker T = ker ρ (see section 3).
Definition 2.4 ([BC09] Definition 1.3.1). Let S be an A-algebra. Then S is a
generalized matrix algebra (GMA) of type (n1 , n2 ) if S is equipped with a data ofidempotents E = {ei , ψi , i = 1, 2} with
(1) a pair of orthogonal idempotents e1 , e2 of sum 1,(2) for each i, an A-algebra isomorphism ψi : ei Sei → Mni (A),P2such that the trace T : S → A, defined by T (x) := i=1 tr (ψi (ei xei )), satisfies
T (xy) = T (yx) for all x, y ∈ S.
By [BC09] Example 1.2.4 and Section 1.2.5 we know that both Rρ := R/ ker ρ
and RT := R/ ker T are Cayley-Hamilton quotients (cf. Definition 1.2.3 in [BC09])of (R, T ) and the quotient map
ϕ : Rρ  RT
is an A-algebra morphism with kernel ker TRρ (so a two-sided ideal).
By [BC09] Lemma 1.4.3 we can now find suitable data of idempotents to give
both Rρ and RT the structure of a GMA:ON DEFORMATION RINGS

5

Lemma 2.5. There exist data of idempotents E T = {eTi , ψiT , i = 1, 2} for RT andE ρ = {eρi , ψiρ , i = 1, 2} for Rρ such that for † = ρ, T
(1) T (e†i ) = ni ,(2) ϕ(eρi ) = eTi ,(3) T (ei xei ) = tr τi (x) mod mA ,(4) If i 6= j, T (ei xej yei ) ∈ mA for any x, y ∈ R,(5) ψiρ ◦ ϕ = ψiT ,(6) ψiρ lifts τi |ei Rρ ei : ei Rρ ei → Mni (F) such that for all x ∈ ei Rρ ei , T (x) =tr ψiρ (x).
These data of idempotents define A-submodules A†i,j of R† for † = T, ρ such that
there are canonical isomorphisms of A-algebras"#
Mn1 (A†1,1 )
Mn1 ,n2 (A†1,2 )† ∼R =Mn2 ,n1 (A†2,1 )
Mn2 (A†2,2 )and ϕ(Aρi,j ) = ATi,j .
Remark 2.6. If R is endowed with an anti-automorphism τ (see below) thenLemma 1.8.3 of [BC09] ensures that the idempotents ei as in Lemma 2.5 can bechosen so that τ (e†i ) = e†i .
Proof. We lift the idempotents of R/ ker T in [BC09] Lemma 1.4.3 and 1.8.3 compatibly to Rρ and RT , i.e. such that ϕ(eρi ) = eTi (by first lifting them to RT andρ
Tthen further to Rρ ). We also choose the ψi,jand ψi,j
in Lemma 1.4.3 compatibly soρρTTthat we can also pick Ei ∈ ei R ei and Ei ∈ ei R ei (as in [BC09] Notation 1.3.3)
with ϕ(Eiρ ) = EiT .
Define A-submodules A∗i,j = Ei∗ R∗ Ej∗ ⊂ R∗ for ∗ = ρ, T . Note that this is how[BC09] Proposition 1.4.4(i) defines ATi,j (i.e. via [BC09] Lemma 1.4.3). By theabove we then haveϕ(Aρi,j ) = ATi,j .
Proposition 2.7. One has IT = T (AT1,2 AT2,1 ).
Proof. This follows from [BC09], Proposition 1.5.1.

Proposition 2.8. The A-module AT1,2 is an A-module generated over A by one
element.
Proof. By [BC09] Lemma 1.3.7 one can conjugate by an invertible matrix withvalues in A (we use here that, since A is local, every finite type projective A-module
is free) to get ρ adapted to E in the sense of [BC09] Definition 1.3.6.
Now by [BC09] Proposition 1.3.8 we know that ρ(R) is the standard GMA attached to some ideals Aρ1,2 , Aρ2,1 of A. Put Aρ1,1 = Aρ2,2 = A. The definition ofadaptedness to the data of idempotents E means concretely that for every r ∈ Ra1,1 (r) a1,2 (r)ρ(r) =
a2,1 (r) a2,2 (r)with ai,j (r) ∈ Mni ,nj (Aρi,j ) and a1,1 (r) ≡ τ1 (r) mod mA and a2,2 (r) ≡ τ2 (r)mod mA . Now since ρ ⊗ F must still be a non-split extension of τ2 by τ1 we

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

6

deduce that for the image Aρ1,2 of the ideal Aρ1,2 in A/mA we have Aρ1,2 6= 0, henceAρ1,2 = A.
By the arguments in the proof of Proposition 1.3.8 we see that we obtain theideals Aρi,j of A from Aρi,j via A-linear maps fi,j (for definition see [loc. cit]), i.e.,Aρi,j = fi,j (Aρi,j ). The maps fi,j are injective since ρ is on Rρ , hence we concludeρthat Aρ1,2 ∼= A. By Lemma 2.5 we have ϕ(A1,2 ) = AT1,2 . Hence AT1,2 is generated
over A by one element.
∼
To show that IT is principal using Proposition 2.8, we will now show that AT1,2 =T
A2,1 as A-modules under an additional assumption on the existence of an involution
on R.
Let τ : R → R be an anti-automorphism (i.e., τ (xy) = τ (y)τ (x)) of A-algebras
such that τ 2 = id. For an A-algebra B, and an A-algebra homomorphism ρ : R →Mn (B) put ρ⊥ = t (ρ ◦ τ ).
Example 2.9. Here are some examples of anti-automorphisms if R = A[G] forsuitable Galois groups G:(1) τ : g 7→ g −1 corresponding to ρ⊥ = ρ∗ (contragredient);(2) τ : g 7→ cg −1 c for c an order 2 element in G corresponding to ρ⊥ = (ρc )∗ ;(3) τ : g 7→ χ−1 (g)g −1 for a character χ : G → O× corresponding to ρ⊥ =ρ∗ ⊗ χ−1 .
Assume in addition that
(2.1)

T ◦ τ = T and tr τi ◦ τ = tr τi ,(i = 1, 2).

Remark 2.10. By [BC09] p.47, if ρ is a semisimple representation, valued in afield, then T being invariant under τ is equivalent to ρ⊥ ∼= ρ.
Theorem 2.11. If τ as in (2.1) exists, then AT1,2 ∼= AT2,1 and IT is a principal
ideal of A.
Proof. The first assertion follows from [BC09], Lemma 1.8.5(ii) (here we use thatker T is stable under involution τ which is not true for ker ρ in general). By Proposition 2.7 we have IT = T (AT1,2 AT2,1 ). Let gi,j be a generator of ATi,j , i.e., write
ATi,j = gi,j A. Then IT = T (g1,2 Ag2,1 A) = AT (g1,2 g2,1 ) ⊂ A.
Remark 2.12. If τ as in (2.1) does not exist, but τ1 and τ2 are characters which
satisfy
dimF H 1 (G, Hom(τ1 , τ2 )) = dimF H 1 (G, Hom(τ2 , τ1 )) = 1
then IT is principal by a result of Bellaı̈che-Chenevier and Calegari (see for example[Cal06], Proof of Lemma 3.4).
3. ker ρ = ker T
Let ρ : R → Mn (A) and T : R → A be as in the previous section. The goal ofthis section is to prove Proposition 3.1. If one replaces the assumption that A/IT
be finite with the assumption that T ⊗ Ks is irreducible for every s, then this isproved in [BC09], Proposition 1.6.4. In this section we assume that A isQreduced,
write K for its total fraction ring, which is a finite product of fields K = s∈S Ks .
Proposition 3.1. Assume that A is reduced, infinite but #A/IT < ∞. Thenker ρ = ker T .

ON DEFORMATION RINGS

7

Proof. We clearly have ker ρ ⊂ ker T . Put S := R/ ker ρ ∼= ρ(R) ⊂ Mn (A). Write
T 0 for the pseudocharacter on S induced by T . We will show using a sequence oflemmas that (S/ ker T 0 ) ⊗ K ∼= Mn (K). This implies that S ⊗ K ∼= Mn (K) and0
therefore that ker T ⊗ K = 0. Note that ker T 0 injects into ker T 0 ⊗ K because theother three maps in the following (commutative) diagram
SO

/ S⊗K
O

ker T 0/ ker T 0 ⊗ K

are injective. So ker T 0 = 0 which finishes the proof of the proposition.
Q
Q
To show (S/ ker T 0 )⊗A K ∼= Mn (K) we first note that A ,→ s As ⊂ s Ks = K,
where the products are over all minimal primes ps and As = A/ps .
Lemma 3.2. As is infinite for all s.
Proof. If As is finite then As is a field because it is a domain. Hence ps equals theunique maximal ideal of A, so ps is the only minimal prime ideal, hence A ⊂ As isa finite field, contradicting our assumption.

Lemma 3.3. We have A/IT ⊗A K = 0 and hence IT ⊗A K = K.
Proof. By flatness of tensoring with K it suffices to show that A/IT ⊗A K = 0.
Denote by φs : A  As . Then A/IT  As /φs (IT ) and the latter must be finite, so
by Lemma 3.2 φs (IT ) 6= 0. This implies that As /φs (IT ) ⊗As Ks = 0. Now observe
that
Y
Y
Y
A/IT ⊗A K = A/IT ⊗A
Ks =A/IT ⊗A Ks =As /φs (IT ) ⊗As Ks = 0.
s

s

s


By [BC09], Proposition 1.4.4(ii) we have
Mn1 (A)
Mn1 ,n2 (A1,2 )S/ ker T 0 ∼⊂ Mn (K),=
Mn2 ,n1 (A2,1 )
Mn2 (A)for some fractional ideals (in the sense of [BC09], p.27) A1,2 , A2,1 ⊂ K.
Lemma 3.4. We have A1,2 ⊗ K = A2,1 ⊗ K = K.
Q
Q
Proof. Let I be any K-submodule of K = s∈S Ks . Then I = s∈T ⊂S Ks . By
the definition of a fractional ideal ([BC09] p.27) there exists fi,j ∈ A such thatfi,j Ai,j ⊂ A so
now that
Q we have Ai,j ⊗ K ⊂ K by flatness of ⊗A K. Assume
Q
A
⊗
K
=
K
⊂
K.
This
implies
that
A
A
⊗
K
=
0
1,2
A
s
2,1
1,2
s∈T ⊂Ss∈T ⊂T Ks ⊂
Q
∼
K
since
it
is
a
K-submodule.
Since
A
A
with
A
as
in
Lemma 2.5
=
s
i,j
i,j
i,j
s∈T ⊂Sby [BC09], Theorem 1.4.4(ii), we have the following surjective map of A-modules
T0
A2,1 ⊗A A1,2 ∼= A1,2 ⊗A A2,1  A1,2 A2,1 −→ IT (the last map is a surjection by
Proposition 2.7), so A1,2 ⊗A2,1 ⊗K surjects onto IT ⊗K which equals K by Lemma
3.3, hence we must have T 0 = T = S.
This finishes the proof of the proposition.



Corollary 3.5. Assume that A is reduced, inifinite but #A/IT < ∞. Also assume
that τ as in (2.1) exists. Then IT is principal and generated by a non-zero-divisor.

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TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Proof. Principality of IT follows from Theorem 2.11. Arguing as in the proofs of
Propositions 1.7.4 and 1.7.5 in [BC09], one sees that IT is generated by f1,2 f2,1
(with fi,j as in the proof of Lemma 3.4), which is a non-zero divisor.

4. Functoriality of short crystalline representations
In Theorem 6.4 we want to relate residual Selmer groups to Bloch-Kato Selmer
groups. In this section we define these and collect some results of Fontaine-Laffailleand Bloch-Kato on short crystalline representations and deduce a functoriality of
the Selmer groups with respect to short exact sequences of finite Galois modules.
Our exposition is influenced by that of [DFG04] Section 2.1 and [Wes00].
4.1. Notation for Galois cohomology. For any field F , we write GF for its
absolute Galois group Gal(F /F ) (for some implicit fixed choice of algebraic closure
F ). If F is a field and M is a topological abelian group with an action of GF , wealways assume that this action is continuous with respect to the profinite topology
on GF and the given topology on M . If L/K is an extension of fields and M isa topological Gal(L/K)-module, then we write H i (L/K, M ) for the cohomology
group H i (Gal(L/K), M ), computed with continous cochains. If L is a separable
algebraic closure of K then we just write H i (K, M ).
4.2. Local cohomology groups. Fix a prime p and let O be the ring of integers
in a finite extension of Qp and uniformizer $. For a prime ` let K be a finite
extension of Q` . Let M be an O-module with an O-linear action of GK . We callM a p-adic GK -module over O if one of the following holds:(1) M is finitely generated, i.e. a finitely generated Zp -module and the GK action is continuous for the p-adic topology on M ;(2) M is discrete, i.e. a torsion Zp -module of finite corank (i.e. M is isomorphic
as a Zp -module to (Qp /Zp )r ⊕ M 0 for some r ≥ 0 and some Zp -module M 0
of finite order) and the GK -action on M is continuous for the discrete
topology on M ;(3) M is a finite-dimensional Qp vector space and the GK -action is continuous
for the p-adic topology on M .
M is both finitely generated and discrete if and only if it is of finite cardinality.
Definition 4.1. A local finite-singular structure on M consists of a choice of Osubmodule N (K, M ) ⊆ H 1 (K, M ).
4.2.1. ` = p. Consider first ` = p. Assume that K is unramified over Qp . We will
be using the crystalline local finite-singular structure, defined in the following.
Let T ⊆ V be a GK -stable Zp -lattice and put W = V /T . For n ≥ 1, put
Wn = {x ∈ W : $n x = 0} ∼= T /$n T.
Following Bloch and Kato we define N (K, V ) = Hf1 (K, V ) = ker(H 1 (K, V ) →H 1 (K, Bcrys ⊗ V )), denote by Hf1 (K, T ) its pullback via the natural map T ,→ V
and let N (K, W ) = Hf1 (K, W ) = im(Hf1 (K, V ) → H 1 (K, W )).
For finitely generated p-adic GK -modules we recall the theory of FontaineLaffaille [FL82], following the exposition in [CHT08] Section 2.4.1. Let MF O(“Dieudonné modules”) denote the category of finitely generated O-modules M
together with a decreasing filtration Fili M by O-submodules which are O-direct
summands with Fil0 M = M and Filp−1 M = (0) and Frobenius linear maps

ON DEFORMATION RINGS

9

P i i
Φi : Fili M → M with Φi |Fili+1 M = pΦi+1 andΦ Fil M = M . They define
an exact, fully faithful covariant functor G of O-linear categories from MF O (intheir notation Gṽ and MF O,ṽ ) to the category of finitely generated O-modules
with continuous action by GK . Its essential image is closed under taking subquotients and contains quotients of lattices in short crystalline representations defined
as follows: We call V a continuous finite-dimensional GK -representation over Qp
short crystalline if, for all places v | p, Fil0 D = D and Filp−1 D = (0) for thefiltered vector space D = (Bcrys ⊗Qp V )Gv defined by Fontaine. Note that thisdiffers slightly from the definition in Section 1.1.2 of [DFG04] and follows instead
the more restrictive setting of [CHT08] Section 2.4.1.
For any p-adic GK -module M of finite cardinality in the essential image of G we
define Hf1 (K, M ) as the image of Ext1MF O (1FD , D) in H 1 (K, M ) ∼= Ext1O[GK ] (1, M ),where G(D) = M and 1FD is the unit filtered Dieudonné module defined in Lemma
4.4 of [BK90].
Remark 4.2. Note that we define Hf1 (K, W ) and Hf1 (K, Wn ) in two different
ways (using the Bloch-Kato definition for the first group and the G-functor for thelatter). However, it is in fact true that the isomorphism W = lim Wn induces an−→nisomorphism Hf1 (K, W ) = lim Hf1 (K, Wn ) (cf. Proposition 2.2 in [DFG04]).
−→n

Lemma 4.3. Letji

0 → T 0 → T → T 00 → 0
be an exact sequence of finite p-adic GK -modules in the essential image of G. Then
there is an exact sequence of O-modules
0 → H 0 (K, T 0 ) → H 0 (K, T ) → H 0 (K, T 00 ) → Hf1 (K, T 0 ) → Hf1 (K, T ) → Hf1 (K, T 00 ) → 0.
Proof. Let D∗ be elements of MF O such that G(D∗ ) = T ∗ . This follows from thefunctoriality of the Ext-functor and Ext0 (1, D) = H 0 (K, G(D)) and Ext2 (1, D) = 0
for any Dieudonné module D.

By Lemma 4.3 we have the following commutative diagram with exact rows:0/ H 0 (K, T 00 )/j∗ H 0 (K, T )0/ H 0 (K, T 00 )/j∗ H 0 (K, T )=/ H 1 (K, T 0 )/ H 1 (K, T )f⊆/ H 1 (K, T 0 )/ H 1 (K, T 00 )

f

f

⊆

i∗


/ H 1 (K, T )

⊆

j∗


/ H 1 (K, T 00 )This implies
Hf1 (K, T 00 ) = j∗ Hf1 (K, T )and1
Hf1 (K, T 0 ) = i−1∗ Hf (K, T ),by comparing the first row with the exact sequence
1
1
1
0 → H 0 (K, T 00 )/j∗ H 0 (K, T ) → i−1∗ Hf (K, T ) → Hf (K, T ) → j∗ Hf (K, T ) → 0

of [Wes00] Lemma I.3.1.
In the terminology of [Wes00] this says that the local finite-singular crystalline
structures on T 0 and T 00 are the induced structures giving the crystalline finitesingular structure on T .

/0

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

10Corollary 4.4. Let W and Wn be as above. Then we have an exact sequence ofO-modules
0 → H 0 (K, W )/$n → Hf1 (K, Wn ) → Hf1 (K, W )[$n ] → 0.
Proof. We apply Lemma 4.3 to the exact sequence·$ n

0 → Wn → Wm −−→ Wm−n → 0
for m ≥ n. This implies the exactness of
0 → H 0 (K, Wm−n )/$n H 0 (K, Wm ) → Hf1 (K, Wn ) → Hf1 (K, Wm )[$n ] → 0.
By taking lim we get a short exact sequence−m→
0 → H 0 (K, W )/$n → Hf1 (K, Wn ) → (lim Hf1 (K, Wm ))[$n ] → 0,−m→so we conclude by Remark 4.2.



4.2.2. ` 6= p. For primes ` 6= p we define the unramified local finite-singular structure on any p-adic GK -module M over O as
1
N (K, M ) = Hur(K, M ) = ker(H 1 (K, M ) → H 1 (Kur , M )),where Kur is the maximal unramified extension of K.
ji
For an exact sequence 0 → M 0 → M → M 00 → 0 of unramified p-adic GK modules over O [Wes00] Lemma I.2.1 shows that this structure on M induces theunramified structures on M 0 and M 00 , i.e.
11
Hur(K, M 00 ) = j∗ Hur(K, M )

and
(4.1)

1
1
Hur
(K, M 0 ) = i−1∗ Hur (K, M ).

Let V be a continuous finite-dimensional GK -representation over Qp and T ⊆ V
be a GK -stable Zp -lattice and put W = V /T . Bloch-Kato then define the following
finite-singular structures on V , T and W :1
Hf1 (K, V ) = Hur(K, V ),i

Hf1 (K, T ) = i−1 Hf1 (K, V ) for T ,→ Vand
Hf1 (K, W ) = im(Hf1 (K, V ) → H 1 (K, W )).
1
By [Rub00] Lemma 1.3.5 we have Hf1 (K, W ) = Hur(K, W )div . Following [Rub00]
1
Definition 1.3.4 we define Hf (K, Wn ) just as the inverse image of Hf1 (K, W ) underthe map H 1 (K, Wn ) → H 1 (K, W ). Call this the minimally ramified structure.
For the minimally ramified structure it follows (see e.g. [Rub00] Corollary 1.3.10)
that lim Hf1 (K, Wm ) = Hf1 (K, W ). Note that by [Rub00] Lemma 1.3.5(iv) the−m→minimally ramified structure agrees with the unramified structure (i.e. Hf1 (K, W ) =1
1
Hur(K, W ) and Hf1 (K, Wn ) = Hur(K, Wn )) if W is unramified.

ON DEFORMATION RINGS

11

4.3. Global Selmer groups. Let F be a number field and let Σ be a fixed finite
set of finite places of F containing the places Σp lying over p. Assume that p
is unramified in F/Q. For every place v we fix embeddings of F ,→ F v . We
write FΣ for the maximal (Galois) extension of F unramified outside Σ and all thearchimedean places and set GΣ = Gal(FΣ /F ).
We use the terminology of p-adic finitely generated (or discrete) GΣ -modules
similar to the corresponding local notions.
For any p-adic GΣ -module M we defined the crystalline local finite-singularstructure Hf1 (Fv , M ) for v | p.
Definition 4.5. We define the Selmer group HΣ1 (F, M ) of M as the kernel of themapY
H 1 (FΣ , M ) →H 1 (Fv , M )/Hf1 (Fv , M ).
v∈ΣpNote that this Selmer group does not impose any conditions at places in Σ\Σp .
Let V be a continuous finite-dimensional representation of GΣ over Qp which isshort crystalline. Let T ⊆ V be a GΣ -stable lattice and put W = V /T and Wn as
before.
For v - p let Hf1 (Fv , M ) denote the minimally ramified structure on M = W, Wn ,as defined above. We will also require the definition of the Bloch-Kato Selmer
group, which has more restrictive local conditions:(4.2)

Hf1 (F, W ) = ker(H 1 (FΣ , W ) →Y

H 1 (Fv , W )/Hf1 (Fv , W ),v∈Σwhere Hf1 (Fv , W ) = 0 for v | ∞.
This Bloch-Kato Selmer group is conjecturally related to special L-values. The
two groups HΣ1 (F, W ) and Hf1 (F, W ) coincide if the latter also has no local conditions at v ∈ Σ\Σp , i.e. when Hf1 (Kv , W ) = H 1 (Kv , W ). The following Lemma will
be useful to identify such situations:
Put V ∗ = HomO (V, E(1)), T ∗ = HomO (T, O(1)) and W ∗ = V ∗ /T ∗ . We define
the v-Euler factor(4.3)

Pv (V ∗ , X) = det(1 − XFrobv |(V ∗ )Iv ).

Lemma 4.6. HΣ1 (F, W ) = Hf1 (F, W ) if for all places v ∈ Σ, v - p we have
(1) Pv (V ∗ , 1) ∈ O∗
(2) Tam0v (T ∗ ) = 1.
Here the Tamagawa factor Tam0v (T ∗ ) equals #H 1 (Fv , T ∗ )tor × |Pv (V ∗ , 1)|p (see[Fon92], Section 11.5).
Proof. Consider a finite place v ∈ Σ. By [Rub00] Proposition 1.4.3 (i) we see that
H 1 (Fv , W )/Hf1 (Fv , W ) is isomorphic to Hf1 (Fv , T ∗ ).
Since the Euler factor Pv (V ∗ , 1) 6= 0 we have that H 0 (Fv , V ∗ ) = 0 = Hf1 (Fv , V ∗ )and so Hf1 (Fv , T ∗ ) = H 1 (Fv , T ∗ )tor (see Fontaine, Asterisque 206, 1992, Section
11.5).
To conclude the lemma we note that Hf1 (F, W ) has additional local conditionsat infinity compared to the definition of HΣ1 (F, W ). However, for an archimedean
place v we get that H 1 (R, W ) = 0 since Gal(C/R) has order 2 and W is pro-p,
and our assumption that p > 2.


12

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Remark 4.7. We remark that the triviality of H 0 (Fv , V ∗ ) and Hf1 (Fv , V ∗ ) imply
via the long exact sequence associated to 0 → T → V → W → 0 that
Hf1 (Fv , T ∗ ) ∼= H 0 (Fv , W ∗ ).
In H 0 (Fv , W ∗ ) one has a subgroup ((V ∗ )Iv /(T ∗ )Iv )Frobv =1 , which has order |Pv (V ∗ , 1)|−1$
In fact, the long exact Iv -cohomology sequence
0 → (T ∗ )Iv → (V ∗ )Iv → (W ∗ )Iv → H 1 (Iv , T ∗ ) → H 1 (Iv , V ∗ )vtells us that the index of ((V ∗ )Iv /(T ∗ )Iv )Frobv =1 in H 0 (Fv , W ∗ ) is given by #(H 1 (Iv , T ∗ )Gtor ).
0∗
By Proposition 4.2.2 in [FPR94] we know that the latter equals Tamv (T ). Thisimplies that Tam0v (T ∗ ) is trivial if W Iv is divisible.

Proposition 4.8. If H 0 (FΣ , W ) = 0 then we haveHΣ1 (F, Wn ) ∼= HΣ1 (F, W )[$n ].
Proof. We note that the local finite-singular structures on Wn are induced fromthose on W under the natural inclusion Wn ,→ W (by (4.1) for v - p or by Corollary
4.4 and the discussion preceding it for v | p). Using this, one shows by a diagramchase (see proof of [Wes00] Lemma II.3.1) that the exact sequence
×$ n

0 → Wn → W → W → 0
gives rise to an exact sequence
0 → H 0 (FΣ , W )/$n → HΣ1 (F, Wn ) → HΣ1 (F, W )[$n ] → 0.
To conclude this section, we define the notion of a crystalline representation,
following [CHT08] p. 35. Let v | p and A be a complete Noetherian Zp -algebra. A
representation ρ : GFv → GLn (A) is crystalline if for each Artinian quotient A0 of
A, ρ ⊗ A0 lies in the essential image of G.
5. Setup for universal deformation ring
5.1. Main assumptions. Let F be a number field and p > 2 a prime with p # ClF and p unramified in F/Q. Let Σ be a finite set of finite places of F containing
all the places lying over p. Let GΣ denote the Galois group Gal(FΣ /F ), where FΣ
is the maximal extension of F unramified outside Σ. For every prime q of F we fix
compatible embeddings F ,→ F q ,→ C and write Dq and Iq for the corresponding
decomposition and inertia subgroups of GF (and also their images in GΣ by a slightabuse of notation). Let E be a (sufficiently large) finite extension of Qp with ringof integers O and residue field F. We fix a choice of a uniformizer $. Consider thefollowing n-dimensional residual representation:


ρ1 ∗
ρ0 =
: GΣ → GLn (F).
ρ2
We assume that ρ1 and ρ2 are absolutely irreducible and non-isomorphic (of dimensions n1 , n2 respectively with n1 + n2 = n) and that ρ0 is non-semisimple. From
now on assume p - n!. Furthermore, we assume that ρ0 is crystalline at the primesof F lying over p.

ON DEFORMATION RINGS

13

For i = 1, 2 let Ri,Σ denote the universal deformation ring (so in particulara local complete Noetherian O-algebra with residue field F) classifying all GΣ deformations of ρi that are crystalline at the primes dividing p. So, in particular
we do not impose on our lifts any conditions at primes in Σ \ Σp .
Assumption 5.1. In what follows we make the following assumptions:(1) dimF HΣ1 (F, HomF (ρ2 , ρ1 )) = 1.
(2) R1,Σ = R2,Σ = O. Set ρ̃i , i = 1, 2 to be the unique deformations of ρi to
GLni (O).
Note that Assumptions 5.1 put certain restrictions on the ramification propertiesof the representations ρi . Set Vi,j := HomO (ρ̃i , ρ̃j ) ⊗ E/O for i, j ∈ {1, 2}. Fix aGΣ -stable O-lattice Ti,j in Vi,j and write Wi,j = Vi,j /Ti,j . Assumption 5.1(2) isequivalent to the following two assertions:• HΣ1 (F, Wi,i [$]) = 0 for i = 1, 2.
• There exists a crystalline lift of ρi to GLni (O).
So, apart from the existence of the lift, both conditions (1) and (2) can be viewed asconditions on some Selmer groups, more specifically HΣ1 (F, Wi,i [$]) and HΣ1 (F, W2,1 [$]) =HΣ1 (F, HomF (ρ2 , ρ1 )). When Σ consists only of the primes of F lying above p, thenHΣ1 (F, Wi,j ) = Hf1 (F, Wi,j ) and the size of the latter group is (conjecturally) controlled by an (appropriately normalized) L-value Li,j . By Proposition 4.8 we have
Hf1 (F, Wi,j [$]) = Hf1 (F, Wi,j )[$]. In particular if Σ = Σp and Li,i is a p-adic unitand L2,1 has $-adic valuation equal to 1, the conditions on the Selmer groups are
satisfied. (A weaker condition guaranteeing cyclicity of Hf1 (F, W2,1 ) would suffice,but cannot be read off from an L-value.) However, in the situations when Σ 6= Σp ,the Selmer groups HΣ1 are not necessarily the same as the Bloch-Kato Selmer groups
Hf1 . For all the applications that we have in mind the following assumption on theset Σ allows us to control the orders of Selmer groups involved in the arguments:
Assume that for all places v ∈ Σ, v - p and all pairs (i, j) ∈ {(1, 1), (2, 2), (2, 1)}
we have(1) Pv ((Vi,j )∗ , 1) ∈ O∗
(2) Tam0v ((Ti,j )∗ ) = 1.
By Lemma 4.6 we then know that we have HΣ1 (F, Wi,j ) = Hf1 (F, Wi,j ), so inthis case the L-value conditions discussed above suffice. Also note that in thecase i = j, Wi,i = ad ρ̃i = ad0 ρ̃i ⊕ F, so the condition reduces to a condition on
HΣ1 (F, ad0 ρ̃i ⊗E/O) as long as we assume that Σ does not contain any prime v with#kv ≡ 1 mod p because then the condition p - # ClF ensures that HΣ1 (F, F) = 0.
5.2. Definitions. From now on we assume that the representations ρ1 and ρ2 aswell as the set Σ satisfy Assumption 5.1 and that ρ0 is crystalline. Denote thecategory 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 equivalenceclass of continuous representations ρ : GΣ → GLn (A) such that ρ0 = ρ (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.
Definition 5.2. We say that an O-deformation ρ : GΣ → GLn (A) of ρ0 is crystalline if ρ|Dq is crystalline at the primes q lying over p.
Lemma 5.3. The representation ρ0 has scalar centralizer.

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

14
Proof. Let

A
C


B
∈ GLn (F) lie in the centralizer of ρ0 , i.e.,
D


 


A B ρ1 f
ρ1 f
A B
=
,
C D 0 ρ2
0 ρ2 C D

where all the matrices are assumed to have appropriate sizes. Then Cρ1 = ρ2 C,
hence C = 0, because ρ1 ∼6 ρ2 . This forces A (resp. D) to lie in the centralizer of ρ1=(resp. ρ2 ), hence A and D are scalar matrices (equal to, say, a and d respectively)
by Schur’s lemma, since ρ1 and ρ2 are absolutely irreducible. Now, since ρ0 is not
split, there exists g ∈ GΣ such that ρ1 (g) = In1 and ρ2 (g) = In2 (identity matrices),but f (g) 6= 0. Then the identity
af + Bρ2 = ρ1 B + f d
implies that a = d, hence it reduces to Bρ2 = ρ1 B, which implies that B = 0 sinceρ1 6∼= ρ2 .
Since ρ0 has a scalar centralizer and crystallinity is a deformation condition in
the sense of [Maz97], there exists a universal deformation ring which we will denote
00)∈ LCN(E), and a universal crystalline O-deformation ρ0Σ : GΣ → GLn (RΣby RΣ
such that for every A ∈ LCN(E) there is a one-to-one correspondence between the0
set of O-algebra maps RΣ
→ A (inducing identity on F) and the set of crystalline
deformations ρ : GΣ → GLn (A) of ρ0 .
Suppose that there exists an anti-automorphism τ as in (2.1).
Definition 5.4. For A ∈ LCN(E) we call a crystalline deformation ρ : GΣ →GLn (A) τ -self-dual or simply self-dual if τ is clear from the context if
tr ρ = tr ρ ◦ τ.
Proposition 5.5. The functor assigning to an object A ∈ LCN(E) the set of strict
equivalence classes of self-dual crystalline deformations to GLn (A) is representable0by the ideal generated by {tr ρΣ (g) − tr ρΣ (τ (g)) | g ∈ GΣ }.
by the quotient of RΣ
We will denote this quotient by RΣ and will write ρΣ for the corresponding universal
deformation.
red
for the quotient of RΣ by its nilradical and ρred
We write RΣΣ for the correred. We
sponding (universal) deformation, i.e., the composite of ρΣ with RΣ  RΣ00will also write Ire ⊂ RΣ for the ideal of reducibility of tr ρΣ and Ire ⊂ RΣ for theredideal of reducibility of tr ρ0Σ , and finally Ire
for the ideal of reducibility of tr ρred
Σ .
The results of Section 1 tell us:
red
red
Proposition 5.6. The ideal of reducibility Ire ⊂ RΣ (resp. Ire⊂ RΣ) of tr ρΣ(resp. tr ρred)isprincipal.
Σ

6. Upper-triangular deformations of ρ0
In this section we study deformations of ρ0 to complete local rings whose trace
splits as a sum of two pseudocharacters.

ON DEFORMATION RINGS

15

6.1. No infinitesimal upper-triangular deformations.
Definition 6.1. We will say that a crystalline deformation is upper-triangular ifsome member of its strict equivalence class has the form
A1 (g) B(g)ρ(g) =
for all g ∈ GΣ0
A2 (g)
with Ai (g) an ni × ni -matrix.
Proposition 6.2. Under Assumption 5.1 (1) and (2) there does not exist any
non-trivial upper-triangular crystalline deformation of ρ0 to GLn (F[x]/x2 ).
 0

ρ1 ∗
0
Proof. Let ρ =be such a deformation. By Assumption 5.1 (2), we have
ρ02
0that ρi is strictly equivalent to ρi for i = 1, 2. By conjugating it by an upperblock-diagonal matrix with entries in F and identity matrices in the blocks on the
diagonal we may assume that ρ0i = ρi . Assume ∗ = f + xg. In the basis
 
 
 
 
1
0
x
0
,
,
,
,
0
1
0
x
the representation ρ0 has the following form

ρ1 f

ρ2
ρ0 = g ρ1Hence it has a subquotient isomorphic toρτ := 1



.
f
ρ2


g
.
ρ2

Note that τ as a subquotient of a crystalline representation is still crystalline, thus
g gives rise to an element in HΣ1 (F, HomF (ρ2 , ρ1 )). If g is the trivial class, thenwe get ρ0 ∼= ρ0 as claimed, so assume that g is non-trivial. Then we must have
A B ] ∈ GL (F) such thatτ ∼= ρ0 by Assumption 5.1(1). Hence there exists Y := [ C2
DY ρ0 = τ Y . Using the fact that ρ1 , ρ2 are irreducible and non-isomorphic an easy
calculation shows that a = A, d = D must be scalars, C = 0 and thatg = d−1 (af + Bρ2 − ρ1 B).
Set


Z=

1


−d−1 Bx∈ GL2 (F[x]/x2 ).
1 + ad x

Then one checks easily thatZρ0 = ρ0 Z,
hence we are done.



6.2. Study of upper-triangular deformations to cyclic O-modules. The following lemma is immediate.
Lemma 6.3. Assume Assumption 5.1 (2). Let R ∈ LCN(E). Then (up to strict
equivalence) any crystalline uppertriangular deformation ρ of ρ0 to R must havethe form


ρ1,R
∗
ρ=
,
ρ2,R

16

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

where ρi,R stands for the composite of ρ̃i with the O-algebra structure map O → R.
Proof. This follows immediately from Assumption 5.1 (2).

Put W = HomO (ρ̃2 , ρ̃1 ) ⊗ E/O and Wn = {x ∈ W : $n x = 0}.
Theorem 6.4. Suppose there exists a positive integer m such that#HΣ1 (F, W ) ≤ #O/$m .
Then ρ0 does not admit any upper-triangular crystalline deformations to GLn (O/$m+1 ).
Proof. Let ρm+1 be such a block-uppertriangular deformation. By Lemma 6.3 ρm+1
must have the form
ρ̃1 mod $m+1b
ρm+1 =.
ρ̃2 mod $m+1
Since ρm+1 is crystalline it gives rise to an element E in HΣ1 (F, Wm+1 ). We claimthat E ∈/ HΣ1 (F, Wm+1 )[$m ]. Consider the following diagram:(Wm+1 /W1 )GΣ/ H 1 (FΣ , W1 )

H 1 (FΣ , Wm+1 )$m(
H 1 (FΣ , Wm+1 )The vertical sequence is induced from the exact sequence 0 → W1 → Wm+1 →$m

Wm+1 /W1 → 0, the horizontal from 0 → Wm → Wm+1 → W1 → 0.
Note that Wn ∼= T /$n by x 7→ $n x. This isomorphism is GΣ -equivariant since
the action is O-linear. This implies that W2 /W1 ∼= T /$T ∼= W1 as GΣ -modules.
By our assumption that ρ1 and ρ2 are irreducible and non-isomorphic we know that
Hom(ρ2 , ρ1 )GΣ = 0, so we get
W1GΣ = (W2 /W1 )GΣ = 0.
Note that (Wm+1 /W1 )GΣ = 0 follows from (W2 /W1 )GΣ = 0 since Wm+1 surjects
onto W2 under multiplication by $m−1 . Therefore, if $m E = 0 then E would haveto lie in the kernel of the horizontal map. This map corresponds, however, under
the isomorphism of Wk ∼= T /$k T , to the morphism
H 1 (FΣ , T /$m+1 T )

mod $

→

H 1 (FΣ , T /$T ).

Hence the image of E under the horizontal map corresponds to the non-split extension given by ρ0 . This proves the claim.
By the structure theorem of finitely generated modules over the PID O, the module HΣ1 (F, Wm+1 ) must be isomorphic to a direct sum of modules of the form O/$r .
Since E ∈/ HΣ1 (F, Wm+1 )[$m ], the module HΣ1 (F, Wm+1 ) must have a submodule
isomorphic to O/$m+1 . We claim that W1GΣ = 0 also implies H 0 (FΣ , W ) = 0.
For this consider a ∈ W GΣ . If a 6= 0, then there exists n such that $n a = 0
but $n−1 a 6= 0. Since the GΣ -action is O-linear, a$n−1 lies in W1G = 0, so
a = 0, which proves the claim. By the claim and Proposition 4.8, HΣ1 (F, Wm+1 ) =ON DEFORMATION RINGS

17

HΣ1 (F, W )[$m+1 ]. By our assumption on the bound on #HΣ1 (F, W ) this contradicts
the existence of ρm+1 .
Remark 6.5. The existence of an m as in Theorem 6.4 follows essentially from(the $-part of) the Bloch-Kato conjecture for the module HomO (ρ̃2 , ρ̃1 ) and its
value should equal the $-adic valuation of a special L-value associated with thismodule divided by an appropriate period. See also section 5.1 to see how one candeal with primes v ∈ Σ \ Σp .
6.3. Cyclicity of RΣ /Ire .
Theorem 6.6. Let R be a local complete Noetherian O-algebra with residue field F.
If T : R[GΣ ] → R is a pseudocharacter such that T is the trace of a d-dimensional
absolutely irreducible representation, then there exists a unique (up to isomorphism)
representation ρT : GΣ → GLd (R) such that tr ρT = T .
Proof. This is Theorem 2.18 in [Hid00].

Theorem 6.7. Let (R, mR , F) be a local Artinian (or complete Hausdorff ) ring.
Let σ1 , σ2 , and σ be three representations of a topological group G with coefficientsin R (with σ having image in GLn (R)). Assume the following are true:• σ and σ1 ⊕ σ2 have the same characteristic polynomials;• The mod mR -reductions σ 1 and σ 2 of σ1 and σ2 respectively are absolutely
irreducible and non-isomorphic;• The mod mR -reduction σ of σ is indecomposable and the subrepresentationof σ is isomorphic to σ 1 .
Then there exists g ∈ GLn (R) such that
σ1 (h)∗
σ(h) = gg −1
σ2 (h)
for all h ∈ G.
Proof. This is Theorem 1 in [Urb99].

00Corollary 6.8. Let I ⊂ RΣ
be an ideal such that RΣ
/I ∈ LCN(E) and is an Artin0ring. Then I contains the ideal of reducibility of RΣ if and only if ρ0Σ mod I is an0/I).
upper-triangular deformation of ρ0 to GLn (RΣProof. If ρ0Σ mod I is isomorphic to an upper-triangular deformation of ρ0 to0/I), then clearly tr ρ0Σ mod I is the sum of two traces reducing to tr ρ1 +
GLn (RΣtr ρ2 , so I contains the ideal of reducibility. We will now prove the converse. Suppose I contains the ideal of reducibility. Then by definition tr ρ0Σ = T1 + T2 mod
I for two pseudocharacters T1 , T2 such that T i = tr ρi . Since ρi are absolutely
00irreducible it follows from Theorem 6.6 that there exist ρTi : RΣ
/I[GΣ ] → RΣ/Isuch that Ti = tr ρTi mod I. By [BC09], section 1.2.3 and the fact that p - n! onehastr ρ0Σ

(mod I) = tr ρT1 + tr ρT2 = tr (ρT1 ⊕ ρT2 ) =⇒ χρ0Σ mod I = χρT1 ⊕ρT2 ,where χ stands for the characteristic polynomial. By the Brauer-Nesbitt Theorem(or Theorem 6.6 for R = F) we conclude that ρTi ∼= ρi , so we can apply Theorem
6.7 to get that ρ0Σ mod I is isomorphic to a block-upper-triangular representation,0 ×
0
say σ. Using the fact that the map (RΣ) → (RΣ/I)× is surjective we see that

18

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN20
we can further conjugate σ (over RΣ/I) to a block-upper-triangular deformation ofρ0 .


Lemma 6.9. If R is a local complete Noetherian O-algebra then it is a quotient ofO[[X1 , X2 , . . . , Xs ]].
Proof. This is Theorem 7.16a,b of [Eis95].

Proposition 6.10. Assume Assumption 5.1 (1), (2). Then the structure map
00
O → RΣ,O/Ire
is surjective.
Before we prove the proposition we will show that it implies the corresponding
redstatement for Ire and Ire.
Lemma 6.11. Let// B ,

ϕ

A



F
be a commutative diagram of commutative A-algebras. Define TB via the commutative diagram
T
/A .
A[G]
ϕ


TB
B[G]

ϕ


/B

Then ϕ induces a surjection
A/IT  B/ITB .
Proof. It is enough to show that ϕ(IT ) ⊂ ITB . Indeed, assuming this, ϕ induces
a well-defined map A/IT → B/ITB , which must be a surjection since ϕ is. Since
A/ϕ−1 (ITB ) ∼= B/ITB , we see that T modulo ϕ−1 (ITB ) is a sum of pseudocharacters, hence ϕ−1 (ITB ) ⊃ IT . Since ϕ is a surjection it follows that ITB ⊃ ϕ(IT ). 
Corollary 6.12. Assume Assumption 5.1 (1), (2). Then the structure maps O →redredRΣ /Ire and O → RΣ/Ire
are surjective.
Proof. This follows immediately from Proposition 6.10 and Lemma 6.11 where A =0redRΣ, B = RΣ or B = RΣ.
00
Proof of Proposition 6.10. Write S for RΣ
/Ire. Then S is a local complete ring.
Moreover, by Lemma 6.9 we have that S is a quotient of O[[X1 , . . . , Xs ]], and hence
0
0
RΣ
/$RΣ
(and thus S/$S) is a quotient of F[[X1 , . . . , Xs ]]. We first claim that infact S/$S = F. Indeed, assume otherwise, i.e., that S/$S = F[[X1 , . . . , Xs ]]/J
and s > 0, then S/$S admits a surjection, say φ onto F[X]/X 2 , i.e., there are00
at least two distinct elements of HomO−alg (RΣ, F[X]/X 2 ) - the map RΣ F ,→
2
0
2
F[X]/X and the surjection RΣ  F[X]/X arising from φ. By the definition
0of RΣ
there is a one-to-one correspondence between the deformations to F[X]/X 2
0and elements of HomO−alg (RΣ, F[X]/X 2 ). The trivial element corresponds to thetrivial deformation to F[X]/X 2 , i.e., with image contained in GL2 (F), which isclearly upper-triangular. However, the deformation corresponding to the surjection
0
must also be upper-triangular by Corollary 6.8 since ker(RΣ S/$S  F[X]/X 2 )02contains Ire and F[X]/X is Artinian. But we know by Proposition 6.2 that ρ0 does

ON DEFORMATION RINGS

19

not admit any non-trivial crystalline upper-triangular deformations to F[X]/X 2 .
Hence we arrive at a contradiction. So, it must be the case that S/$S = F.
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.
0
Proposition 6.13. The ring RΣ
is topologically generated as an O-algebra by theset

S := {tr ρ0Σ (Frobv ) | v 6∈ Σ}.
tr
Proof. Let RΣ
be the closed (and hence complete by [Mat89] Theorem 8.1) O0
trsubalgebra of RΣ
generated by the set S. Let I0tr be the smallest closed ideal of RΣ
containing the set

T := {tr ρ0Σ (Frobv ) − tr ρ̃1 (Frobv ) − tr ρ̃2 (Frobv ) | v 6∈ Σ}.
Note that tr ρ0Σ (Frobv ) − tr ρ̃1 (Frobv ) − tr ρ̃2 (Frobv ) ≡ 0 (mod $) for v 6∈ Σ, so
tr
00I0tr 6= RΣ
. Also note that I0 := I0tr RΣ
is the smallest closed ideal of RΣ
containing
0T . We will now show that I0 = Ire . Indeed, by the Chebotarev density theorem we000. On the other hand since RΣ
/Ire
get tr ρ0Σ = tr ρ̃1 + tr ρ̃2 (mod I0 ), hence I0 ⊃ Ire0
is complete Hausdorff, we can apply Corollary 6.8 to the ideal Ire to conclude that0
) is an upper-triangular deformation of ρ0 and thus by Lemma 6.3 weρ0Σ (mod Ire00.
). It follows that I0 ⊂ Ire
must have tr ρ0Σ = tr ρ̃1 + tr ρ̃2 (mod Iretr/I0tr isNote that since tr ρ̃i is O-valued, the O-algebra structure map O → RΣtrtrtr
surjective, hence RΣ
/(I0tr + $RΣ
) = F. Thus in particular mtr := I0tr + $RΣ
is thetrmaximal ideal of RΣ
. Moreover, the containment
tr
0
RΣ
,→ RΣ

(6.1)
gives rise to an O-algebra map
(6.2)

tr
0
RΣ
/I0tr → RΣ
/I0 ,

0
0
which must be surjective since the object on the right equals RΣ
/Ire
by the above0
0
argument and RΣ /Ire is generated by 1 as an O-algebra by Proposition 6.10. Thismap descends to
tr
0
0
0
tr
0
0
tr
F = RΣ
/mtr → RΣ/mtr RΣ
= RΣ
/(I0 + $RΣ
) = RΣ
/(Ire+ $RΣ
) = F,

which is an isomorphism since (6.2) was surjective. Note that the maps (6.1) andtrtris complete (which means complete(6.2) are in fact RΣ-algebra maps and since RΣ
trwith respect to m ) we can apply the complete version of Nakayama’s lemma to
tr0
conclude that RΣ= RΣ.
Proposition 6.14. Assume Assumption 5.1 and #HΣ1 (F, HomO (ρ̃2 , ρ̃1 )) ≤ #O/$m .
00Then RΣ/Ire= O/$s for some 0 < s ≤ m. The same conclusion is true for RΣ /Ireredred
and for RΣ /Ire.
00
Proof. By Proposition 6.10 we have that RΣ
/Ire= O/$s for some s ∈ Z+ ∪ {∞}.
But we must have 0 < r ≤ m, since by Corollary 6.8 if r > m or r = ∞, thenthere would be an upper-triangular crystalline deformation of ρ0 to O/$m+1 , which
is impossible by Theorem 6.4. The last assertion of the Proposition follows from
Lemma 6.11.
20

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

6.4. Some consequences of the principality of Ire . Below we list some consequences of principality of Ire in our context.
Lemma 6.15. If R is a local complete Noetherian O-algebra and there exists r ∈ Rsuch that the structure map O → R/rR is surjective, then R is a quotient of O[[X]].
Proof. Since R/(r, $) = F, the ideal (r, $) ⊂ R is maximal. Hence by Theorem
7.16 in [Eis95] there exists an O-algebra map Φ : O[[X, Y ]]  R sending X to r
and Y to $. But this map factors through Ψ : O[[X, Y ]]  O[[X]] sending X to
X and Y to $ (indeed, ker Ψ = (Y − $)O[[X, Y ]] ⊂ ker Φ).

red
red
red
Proposition 6.16. If RΣ/Ire
6= O, then RΣ
is Gorenstein.
red
red
Proof. First note that by Proposition 6.10 our assumption implies that RΣ/Irered
is finite. Thus by Corollary 3.5 the ideal Ire is generated by a non-zero divisor.
red
Hence in particular the maximal ideal of RΣ
contains a non-zerodivisor. Thus wered
can apply [Bas63], Proposition 6.4 to conclude that RΣ
is Gorenstein.

red
red
red
Proposition 6.17. If RΣ/Ire
6= O, then RΣ
is a complete intersection.
red
Proof. By Lemma 6.15 we know that RΣ= O[[X]]/J. Note that codim(J) =red
reddim RΣ which because Ire is principal equals (cf. e.g. [AM69], Corollary 11.18)
redred+ 1 = 1 since RΣ /Ire is finite. It follows from [Eis95], Corollary 21.20/Ire
dim RΣ
red
that RΣ is a complete intersection.


7. A commutative algebra criterion
Let R and S denote complete local Noetherian O-algebras with residue field F.
Suppose that S is a finitely generated free O-module.
Theorem 7.1. Suppose there exists a surjective O-algebra map φ : R  S inducing
identity on the residue fields and π ∈ R such that the following diagram
(7.1)

/S

φ

R

R/πR


/ S/φ(π)Scommutes. Write φn for the map φn : R/π n R  S/φ(π)n S. Assume #φ(π)S/φ(π)2 S <∞. Suppopse φ1 : R/πR  S/φ(π)S is an isomorphism.
• If R/πR ∼= O/$r for some positive integer r, then φ is an isomorphism.
• If R/πR ∼= O and the induced map πR/π 2 R  φ(π)S/φ(π)2 S is an isomorphism, then φ is an isomorphism.
In the case when R/πR = O, Theorem 7.1 gives an alternative to the criterionof Wiles and Lenstra to prove R = T . Let us briefly recall this criterion. Suppose
we have the following commutative diagram of surjective O-algebra maps
(7.2)

R
πR

 
O

/S

φ

πS

and for A = R or S set ΦA := (ker πA )/(ker πA )2 and ηA = πA (AnnA ker πA ).

ON DEFORMATION RINGS

21

Theorem 7.2 (Wiles and Lenstra). #ΦR ≤ #O/ηS if and only if φ is an isomorphism of complete intersections.
Proposition 7.3. Suppose diagram (7.2) commutes. Suppose that ker πR is aprincipal ideal of R generated by some π ∈ R and suppose that #ΦR ≤ #O/ηS ,then φ is an isomorphism.
Proof. By our assumption we have
#πR/π 2 R = #ΦR ≤ #O/ηS .

(7.3)

On the other hand the right-hand-side of (7.3) is bounded from above by #ΦS (seee.g. formula (5.2.3) in [DDT97]). However, note that since φ is surjective it follows
that φ(ker πR ) = ker πS , hence ΦS = φ(π)S/φ(π)2 S. Hence we can apply Theorem
7.1 to conclude that φ is an ismorphism.
Proof of Theorem 7.1. Consider the following commutative diagram with exact rows.
(7.4)

0

/ πR/π n R/ R/π n R0
/ φ(π)S/φ(π)n Sα/ R/πR

β

/0

φ1


/ S/φ(π)n S

/ S/φ(π)S/0

We will show that R/π n R ∼= S/φ(π)n S for all n. By (7.4) and snake lemma it isenough to show that α is an isomorphism for all n > 1.
Set x = φ(π). Note that α is clearly surjective, because φ is. On the other
hand, the multiplication by π (resp. by x) induces surjective maps: π k−1 R/π k R 
π k R/π k+1 R (resp. xk−1 S/xk S  xk S/xk+1 S). So, arguing as in the proof of
Proposition 6.9 in [BK11] we have #(πR/π k R) ≤ #(πR/π 2 R)k−1 and #(xS/xk R) =#(xS/x2 S)k−1 because by Lemma 6.7 of [BK11] we get that the multiplication by
x is injective on xS (apply this lemma for xS instead of S - note that xS beinga submodule of a finitely generated torsion free O-module is also finitely generated and torsion-free). If R/πR ∼= xS/x2 S by assumption. If= O, then πR/π 2 R ∼r
R/πR = O/$ O for some positive integer r we deduce that πR/π 2 R ∼= xS/x2 S asin the proof of Proposition 6.9 in [loc.cit.] and (arguing inductively) in both cases
we finally obtain πR/π k R ∼= xS/xk S, which is what we wanted. So,
lim R/π n R ∼S/xn S.
= lim
←
−
←
−
n
n
Now, consider the following commutative diagram with exact rows
0

/R

ι /


/S

/ coker ι/0/ coker ι0

/0

∼ φ

φ

0

lim R/π n R←−nlim S/φ(π)n S
←
−
n

ι/ 0

where the maps ι and ι0 are injective because R (resp. S) are separated (withrespect to the maximal ideals hence with respect to any non-unit ideals). The firstvertical map is surjective and the second is an isomorphism, hence by snake lemma
the first vertical map is an isomorphism as well.
TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

22

8. R = T theorems
Fix a (semi-simple) p-adic Galois representation ρπ0 : GF → GLn (E) which
factors through GΣ and satisfies:
(8.1)
ρss ∼
= ρ1 ⊕ ρ2 .
π0

Proposition 8.1. If ρπ0 is irreducible then there exists a lattice L inside E n sothat with respect to that lattice the mod $ reduction ρπ0 of ρπ0 has the form


ρ1 ∗
ρπ0 =
0 ρ2
and is non-semi-simple.
Proof. This is a special case of [Urb01], Theorem 1.1, where the ring B in [loc.cit.]
is a discrete valuation ring = O.
Set
ρ0 := ρπ0 .
Let Π be the set of Galois representations ρπ : GΣ → GLn (E) (with ρπ semisimple but not necessarily irreducible) for which there exists a crystalline deformation ρ0π : GΣ → GLn (O) of ρ0 such that one has(ρ0 )ss ∼=/E ρπ .
πRemark 8.2. Our choice of notation is motivated by potential applications ofthese results. In applications ρπ0 will be the Galois representation attached tosome automorphic representation π0 and Π will be (in one-to-one correspondencewith) the subset of (L-packets of) automorphic representations π whose associatedGalois representation ρπ satisfies the above condition.
Proposition 8.3. Assume Assumption 5.1(1). If ρ : GΣ → GLn (E) is irreducible
and crystalline and ρss = ρss
0 , then ρ ∈ Π.
Proof. By Proposition 8.1 ρ is E-isomorphic to a representation ρ0 : GΣ → GLn (O)with ρ0 upper-triangular and non-semi-simple. Since ρ0 is crystalline its reduction
gives rise to a non-zero element inside HΣ1 (F, HomF (ρ2 , ρ1 )) and by Assumption
5.1(1) this group is one-dimensional.
Remark 8.4. In contrast to Proposition 8.3 if ρ is reducible (and by assumption
semi-simple) it is not always the case that ρ ∈ Π. For example Skinner and Wiles in[SW99] studied a minimal (ordinary) deformation problem for residually reducible
2-dimensional Galois representations. In [loc.cit] they assert the existence of an
upper-triangular Σ-minimal deformation ρ0 of ρ0 to GL2 (O) based on argumentsfrom Kummer theory. The semi-simplification of this deformation is the Galois
representation ρE2,ϕ associated to a certain Eisenstein series E2,ϕ (see page 10523
in [loc.cit.] for a definition of E2,ϕ ), hence we take ρ0E2,ϕ = ρ0 . The difficulty isin showing the existence of a representation whose semi-simplification agrees with
ρE2,ϕ , but which reduces to ρ0 hence is non-semi-simple (this is where the Kummer
theory is used). In contrast to the case considered in [SW99], the authors showedthat in the case of 2-dimensional Galois representations over an imaginary quadratic field F there is no upper-triangular Σ-minimal deformation of ρ0 to GL2 (O)([BK09], Corollary 5.22). So, in particular if one considers an Eisenstein series (say

ON DEFORMATION RINGS

23

E) over F then there is no representation ρ0E whose semi-simplification is isomorphic
to ρE and for which one has ρ0E = ρ0 and at the same time ρ0E is minimal.
Let Π be as above. Then one obtains an O-algebra map
Y
RΣ →
O.
ρπ ∈Π

We (suggestively) write TΣ for the image of this map and denote the resulting
surjective O-algebra map RΣ  TΣ by φ.
Theorem 8.5. Suppose the set Π is finite. Assume Assumption 5.1 (1) and (2).
Suppose there exists an anti-automorphism τ of RΣ [GΣ ] such that tr ρΣ ◦ τ = tr ρΣand tr ρi ◦ τ = tr ρi for i = 1, 2. In addition suppose that there exists a positive
integer m such that the following two “numerical” conditions are satisfied:(1) #HΣ1 (F, HomO (ρ̃2 , ρ̃1 ) ≤ #O/$m ,(2) #TΣ /φ(Ire ) ≥ #O/$m .
Then the map φ : RΣ  TΣ is an isomorphism.
Proof. This is just a summary of our arguments so far. The existence of τ guarantees principality of the ideal of reducibility Ire . Condition (1) in Theorem 8.5
implies (by Proposition 6.14) that #RΣ /Ire = O/$s for 0 < s ≤ m. This combinedwith condition (2) guarantees that φ descends to an isomorphism φ1 : RΣ /Ire →TΣ /φ(Ire ). Hence by Theorem 7.1(1) we conclude that φ is an isomorphism.

Theorem 8.6. Suppose the set Π is finite. Assume Assumption 5.1. Suppose
there exists an anti-automorphism τ of RΣ [GΣ ] such that tr ρΣ ◦ τ = tr ρΣ andtr ρi ◦ τ = tr ρi for i = 1, 2. In addition suppose that
(1) TΣ /φ(Ire ) = O,
(2) #Ire /(Ire )2 ≤ #(φ(Ire )TΣ )/(φ(Ire )TΣ )2 .
Then the map φ : RΣ  TΣ is an isomorphism.
Proof. This is proved analogously to Theorem 8.6 but uses Theorem 7.1(2). Notethat Corollary 6.12 combined with condition (1) of Theorem 8.6 yields RΣ /Ire =TΣ /φ(Ire ) = O, hence the map φ1 in Theorem 7.1 is an isomorphism.
Remark 8.7. In applying Theorems 8.5 and 8.6 in practice one identifies TΣ witha local complete Hecke algebra. Then condition (2) may be a consequence of a lower
bound on the order of TΣ /J, where J could be the relevant congruence ideal (e.g.,
Eisenstein ideal - see section 9 or Yoshida ideal - see section 10). See for example[BK09] and [BK11], where such a condition (which is a consequence of a result
proved in [Ber09]) is applied in the context of Theorem 8.5. On the other hand in[SW99] one shows that the condition needed to apply the criterion of Wiles andLenstra is satisfied and this implies (cf. Proposition 7.3 and its proof) that condition(2) in Theorem 8.6 is satisfied. On the other hand condition (1) of Theorem 8.5
seems to require (the $-part of) the Bloch-Kato conjecture for HomO (ρ̃2 , ρ̃1 ) andis in most cases when ρ1 and ρ2 are not characters currently out of reach. Hencein this case Theorem 8.5 should be viewed as a statement asserting that undercertain assumptions, (the $-part of) the Bloch-Kato conjecture for HomO (ρ̃2 , ρ̃1 )(which in principle controls extensions of ρ˜2 by ρ̃1 hence reducible deformations ofρ0 ) implies an R = T -theorem (which asserts modularity of both the reducible andthe irreducible deformations of ρ0 ).

24

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

Remark 8.8. If an anti-automorphism τ in Theorems 8.5 and 8.6 does not exist,
but instead one has
dimF H 1 (GΣ , HomF (ρ1 , ρ2 )) = dimF H 1 (GΣ , HomF (ρ2 , ρ1 )) = 1,
then the conclusions of Theorems 8.5 and 8.6 still hold by Remark 2.12.
9. 2-dimensional Galois representations of an imaginary quadratic
field - the crystalline caseIn this and in the next section we will describe how the method outlined in thepreceding sections can be applied in concrete situations. We begin with the casewhen F is an imaginary quadratic field, ρ1 and ρ2 are characters. This is similar tothe problem studied in [BK11], but covers the case of crystalline deformations (asopposed to ordinary minimal deformations considered in [loc.cit.]). Because of thissimilarity with [BK11], we will discuss only the aspects in which this case differsfrom the ordinary case and will refer the reader to [BK09] and [BK11] for mostdetails and definitions. In the next section we will study another case, this time
when the representations ρ1 and ρ2 are 2-dimensional and will consider an R = T
problem for residually reducible 4-dimensional Galois representations of GQ .
9.1. The setup. Let F be an imaginary quadratic extension of Q of discriminant
dF 6= 3, 4 and p > 3 a rational prime which is unramified in F . We fix once and for
all a prime p of F lying over (p). As before, we fix for every prime q embeddings
F ,→ F q ,→ C and write Dq and Iq for the corresponding decomposition and inertia
subgroups. We assume that p - # ClF and that any prime q | dF satisfies q 6≡ ±1(mod p).
Let Σ be a finite set of finite primes of F containing all the primes lying over p.
Let χ0 : GΣ → F× be a Galois character and


1 ∗
ρ0 =
: GΣ → GL2 (F)χ0be a non-semi-simple Galois representation.
9.2. Assumption 5.1. We will now describe sufficient conditions under which Assumption 5.1 is satisfied.
Q
Let Sp be the set of primes of F (χ0 ) lying over p. Write Mχ0 for q∈Sp (1 + Pv )and Tχ0 for its torsion submodule. The quotient Mχ0 /Tχ0 is a free Zp -module offinite rank. Let E χ0 be the closure in Mχ0 /Tχ0 of the image of Eχ0 , the group ofunits of the ring of integers of F (χ0 ) which are congruent to 1 modulo every primein Sp .
Definition 9.1. We say that χ0 : GΣ → F× is Σ-admissible if it satisfies all of thefollowing conditions:(1) χ0 is ramified at p;(2) if q ∈ Σ, then either χ0 is ramified at q or χ0 (Frobq ) 6= (#kq )±1 (as elementsof F);(3) if q ∈ Σ, then #kq 6≡ 1 (mod p);(4) χ0 is anticyclotomic, i.e., χ0 (cσc) = χ0 (σ)−1 for every σ ∈ GΣ and c thegenerator of Gal(F/Q);(5) the Zp -submodule E χ0 ⊂ Mχ0 /Tχ0 is saturated with respect to the ideal
pZp ,ON DEFORMATION RINGS

25(6) The χ−1
0 -eigenspace of the p-part of ClF (χ0 ) is trivial.
Remark 9.2. Note that χ0 is Σ-admissible if and only if χ−1
0 is (cf. Remark 3.3
in [BK09]).
9.2.1. Assumption 5.1(1). Set G = Gal(F (χ0 )/F ). Let L denote the maximalabelian extension of F (χ0 ) unramified outside the set Σ and such that p annihilates
Gal(L/F (χ0 )). Then V := Gal(L/F (χ0 )) is an Fp -vector space endowed with an
Fp -linear action of G, and one hasM
V ϕ,
V ⊗Fp Fp ∼=×ϕ∈Hom(G,Fp )where for a Zp [G]-module N and an Fp -valued character ϕ of G, we write
N ϕ = {n ∈ N ⊗Zp Fp | σn = ϕ(σ)n for every σ ∈ G}.

(9.1)

−1

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

Proposition 9.3. One has dimFp V χ0 = 1.
Proof. If p is a split prime this assertion has been proved in [BK09] (see Theorem3.5). For an inert p the proof is essentially the same, so let us just point out how
to reconcile some of the issues that arise in the inert case (for notation we refer
the reader to the proof of Thereom 3.5 in [loc.cit]). In particular as opposed to thesplit case, in the inert case one gets that for every ψ ∈ G∨ ,dimFp (M/T )ψ = 2.
For this one can argue as follows: Since the ramification index of p in F (χ0 ) is
no greater than p2 − 1, the p-adic logarithm gives a Dv -equivariant isomorphism∼,→ 1 + Pv
for every v | p. This L
followed by the injection 1 + Pp+2
Pp+2
= 1 + Pp+2
v
v
v
p+2
∼
yields an isomorphism of G-modules⊗ Qp = (M/T ) ⊗ Qp . It is not
v|p Pvdifficult to see that
Y
M
M
Pp+2
⊗ Qp ∼Qp (φ) ∼Qp (φ) ⊕ Qp (φ),
=
=
v
q∈Sp

φ∈Gal(F (χ0 )/Q)∨φ∈G∨

where Qp (φ) denotes the one-dimensional Qp -vector space on which G (or Gal(F (χ0 )/Q))
acts via φ. The claim follows easily from this. However, since we now only haveone prime of F lying over p, this still gives us (as in the split case) thatM((M/T ) ⊗ Fp )/(E ⊗ Fp ) ∼Fp (ψ).
= Fp (1) ⊕ Fp (1) ⊕ψ∈G∨ \{1}Since χ0 6= 1 we are done.



As in the proof of Corollary 3.7 in [BK09] Proposition 9.3 implies that the space
H 1 (GΣ , Fp (χ−10 )) is one-dimensional and hence we obtain the following corollary(note that ρ0 itself is crystalline, so the extension it gives rise to lies in the Selmer
group).
Corollary 9.4. The pair (1, χ0 ) for a Σ-admissible character χ0 satisfies Assumption 5.1(1).

26

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

9.2.2. Assumption 5.1(2). Write ρi for the character 1 or χ0 .
Proposition 9.5. There does not exist any non-trivial crystalline deformation of
ρi to GL1 (F[x]/x2 ).
Proof. Let ρ : GΣ → GL1 (F[x]/x2 ) be a crystalline deformation of ρi . Then since
ρ−1is also crystalline we can without loss of generality assume that ρ has the formi
ρ = 1 + xα for α : GΣ → F+ a group homomorphism (here F+ denotes the additive
group of F).
Let q be a prime of F and consider the restriction of α to Iq . If q ∈ Σ, q - p then#kv 6≡ 1 mod p by Definition 9.1(3), and thus one must have (by local class field
theory) that α(Iq ) = 0. Thus α can only be ramified at the primes lying over p.
The proposition thus follows easily from the following lemma and the assumptionthat p - # ClF .

Lemma 9.6. A p-power order crystalline character ψ : GΣ → (F[x]/x2 )× must be
unramified at primes lying above p.
Proof. Since a character as above can be thought of as a 2-dimensional representation ρ : GΣ → GL2 (F) of the form
1 0ρ(σ) =,
α 1
it is enough to show that for q lying over p a crystalline extension of the trivial
one-dimensional F-representation of GFq by itself must be unramified at primes
lying over p. However, such an extension is necessarily split by Remark 6.13, p.589
of [FL82].

Corollary 9.7. The pair (1, χ0 ) for a Σ-admissible character χ0 satisfies Assumption 5.1(2).
9.3. Bounding the Selmer group. From now on we will make a particular choiceof χ0 and Σ. Let φ1 , φ2 be two Hecke characters of infinity types z and z −1 respectively, and set φ = φ1 /φ2 . Let φp denote the p-adic Galois character corresponding
to φ. Set Ψ := φp  and χ0 = Ψ. Assume that Σ contains all the primes dividing
M1 M2 M1c M2c discF p, where Mi denotes the conductor of φi .
Let Lint (0, φ) be the special L-value attached to φ as in [BK09]. Write W for
HomO (Ψ, 1) ⊗ E/O.
Conjecture 9.8. #Hf1 (F, W ) ≤ #O/$m , where m = val$ (Lint (0, φ)).
Remark 9.9. Conjecture 9.8 can in many cases be deduced from the Main conjecture proven by Rubin [Rub91]. If φ−1 = ψ 2 for ψ a Hecke character associated to a
CM elliptic curve, then one can argue as follows. By Proposition 4.4.3 in [Dee99] andusing that Hf1 (F, W ) ∼= Hf1 (F, W c ), we have #Hf1 (F, W ) = #Hf1 (F, E/O(φ−1p )).
Thus we can use Corollary 4.3.4 in [Dee99] which together with the functional
equation satisfied by L(0, φ) implies the desired inequality.
Corollary 9.10. Assume that χ0 is Σ-admissible and that Conjecture 9.8 holds forφ. Then #HΣ1 (F, W ) ≤ #O/$m , where m = val$ (Lint (0, φ)).
Proof. Let v ∈ Σ\Σp . First note that since Ψ is O× -valued one must have W Iv = W
or W Iv = 0. So, in particular W Iv is divisible. Hence by Remark 4.7, we get that
Tam0v (T ∗ ) = 1, so by Lemma 4.6 it is enough to show that Pv (V ∗ , 1) ∈ O× . Let

ON DEFORMATION RINGS

27

Σun be the subset of Σ \ Σp consisting of those primes v for which χ0 is unramified.
If v 6∈ Σun , then this Euler factor is 1. Otherwise one has
Pv (V ∗ , 1)−1 = 1 − Ψ(Frobv ) ≡ 1 − χ0 (Frobv ) · #kv(mod $).

Because χ0 is Σ-admissible (cf. Definition 9.1(2)) we are done by Conjecture 9.8.
From now on assume that χ0 is Σ-admissible and that Conjecture 9.8 holds forφ. Let RΣ denote the crystalline universal deformation ring of ρ0 and Ire its idealof reducibility.
Corollary 9.11. One has RΣ /Ire = O/$s for some 0 < s ≤ m, with m as above.
Proof. This follows from Conjecture 9.8 and Proposition 6.14.



Remark 9.12. Note that this proof of Corollary 9.11 differs from (and is simplerthan) the proof of Theorem 5.12 in [BK09] in that we do not need to relate theSelmer groups to Galois groups. This is so because the proof of Theorem 6.4
interprets (which is perhaps more natural) upper-triangular deformations directly
as cohomology classes in the Selmer group.
9.4. Modularity of crystalline residually reducible 2-dimensional Galois
representations over F .
Remark 9.13. By Remark 9.2 and Corollary 9.4 one also has dimF H 1 (GΣ , Fp (χ0 )) =1, hence by Remark 2.12, the ideal of reducibility Ire ⊂ RΣ is principal.
From now on assume that φ is unramified or that we are in the situation ofTheorem 4.4 of [BK09]. Let TΣ denote the Hecke algebra defined in section 4 of
[BK09], except we do not restrict to the ordinary part. Conjecture 5 of [Ber09]
asserted that the Galois representation ρπ attached to an automorphic representation π over F is crystalline if π is unramified at p. This has now been provenin many cases by A. Jorza [Jor10]. When it is satisfied we obtain by universality
a canonical map ψ : RΣ  TΣ and the set Π in section 8 can be identified withthe set ΠΣ from section 4.2 of [BK09]. By Theorem 14 of [Ber09] condition (2) of
Theorem 8.5 is satisfied with m as in Conjecture 9.8. Hence gathering all this, we
can apply Theorem 8.5 (using Remark 9.13 instead of the existence of τ ) to deduce
the following modularity result.
Theorem 9.14. The map ψ : RΣ  TΣ is an isomorphism.
From this one easily has the following modularity theorem.
Theorem 9.15. Let F , p and Σ be as above. Let φ be an unramified Hecke character of infinity type z 2 and let χ0 = φp . Assume χ0 is Σ-minimal and that
Conjecture 9.8 holds φ. Let ρ : GΣ → GL2 (E) be an irreducible continuous Galois
representation and suppose that ρss ∼= 1 ⊕ χ0 . If ρ is crystalline at the primes of F
lying over p then (a twist of ) ρ is modular.
10. 4-dimensional Galois representations of Q - Yoshida lifts
In this section we apply our methods to study the deformation ring of a 4dimensional residually reducible Galois representation of GQ .

28

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

10.1. Setup. Let Sn (N ) denote the space of (elliptic) cusp forms of weight n andlevel N . Assume that p > k ≥ 4 is even and that N is a square-free positive
integer with p - N . We will also assume that all primes l | N satisfy l 6≡ 1 (modp). Let f ∈ S2 (N ) and g ∈ Sk (N ) be two eigenforms whose residual (mod p)
Galois representations are absolutely irreducible and mutually non-isomorphic. Fora positive integer n write SnS (N ) for the space of Siegel modular forms φ which are
cuspidal and satisfy
A B
det(CZ+D)−n φ((AZ+B)(CZ+D)−1 ) = φ(Z) for∈ Sp4 (Z); C ≡ 0 (mod N ).
C D
Here Z is in the Siegel upper-half space.
Theorem 10.1 (Yoshida). There exists a C-linear map
S(N )Y : S2 (N ) ⊗ Sk (N ) → Sk/2+1

such that
Lspin (s, Y (f ⊗ g)) = L(s − k/2 + 1, f )L(s, g)
up to the Euler factors at the primes dividing N . In particular the lift Y (f ⊗ g) isa Hecke eigenform for primes away from N .
Let Σ denote the finite set of finite places of Q consisting of p and the primes
dividing N . For a Siegel cuspidal eigenform φ (away from Σ) denote by ρφ : GΣ →GL4 (E) the Galois representation attached to φ by Weissauer [Wei05] and Laumon[Lau05]. The representations are crystalline at p by [FC90] Théorème VI.6.2. It
follows from Theorem 10.1 thatρY (f ⊗g) ∼= ρf (k/2 − 1) ⊕ ρg ,where ρf and ρg denote the Galois representations attached to f and g by Eichler, Shimura and Deligne. Note that because the determinants of the two twodimensional summands match, the image of ρY (f ⊗g) is contained (possibly after
conjugating) in GSp4 (O) and not just in GL4 (O). Let S nY denote the orthogonal
S
complement (under the standard Petersson inner product on Sk/2+1(N )) of the imf,gnY
age of the map Y and let S ⊂ S
denote the subspace spanned by eigenforms
φ whose Galois representation satisfy the following two conditions:• ρφ is irreducible;• The semisimplification of the reduction mod $ (with respect to some latticein E 4 ) of ρφ is isomorphic to ρf (k/2 − 1) ⊕ ρg .
S
Let TS denote the O-subalgebra of EndO (Sk/2+1(N )) generated by the local Heckef,galgebras away from Σ, and let TΣ = T be the image of TS inside EndO (S f,g ).
Then (if non-zero) Tf,g is a local, complete Noetherian O-algebra with residue field
F which is finitely generated as a module over O. Let Ann(Y (f ⊗ g)) ⊂ TS denotethe annihilator of Y (f ⊗g). It is a prime ideal and one has TS / Ann(Y (f ⊗g)) ∼= O.
Let If,g = ψ(Ann(Y (f ⊗ g))), where ψ : TS  Tf,g is the projection map. It is an
ideal.

Conjecture 10.2. Suppose m = val$ (LN,alg (1 + k/2, f × g)) > 0. Then
#Tf,g /If,g ≥ #O/$m .
Here LN,alg (1 + k/2, f × g) denotes appropriately normalized special value of the
convolution L-function of f and g.

ON DEFORMATION RINGS

29

In a recent preprint Agarwal and the second author have proved this conjecture
in many cases (cf. [AK10], Theorem 6.5 and Corollary 6.10) under some additional
assumptions (among them that f and g are ordinary). See also [BDSP10] for asimilar result. As a consequence of this conjecture we get that Tf,g 6= 0 whenever
the L-value is not a unit. Also, the conjecture implies that the space S f,g 6= 0. Let
F ∈ S f,g 6= 0 be an eigenform. Then its Galois representation ρF : GΣ → GL4 (E)is irreducible, but the semi-simplification of its reduction mod $ has the form4∼ρssF = ρf (k/2 − 1) ⊕ ρg . Using Proposition 8.1 we fix a lattice L ⊂ E such thatwith respect to that lattice ρF is non-semi-simple and has the formρf (k/2 − 1) ∗
ρF =
.
0
ρg
Set
ρ0 := ρF .
10.2. Assumption 5.1. In what follows we impose Assumption 5.1. Let us briefly
discuss some sufficient conditions under which Assumption 5.1 is satisfied. The
Selmer group in Part (1) is equal to Hf1 (Q, HomO (ρ̃2 , ρ̃1 )) ⊗ F as long as we assume
the conditions of Lemma 4.6. The condition on the Tamagawa factor is satisfied if
W Iv is divisible, where W = HomO (ρ̃2 , ρ̃1 )) ⊗ E/O. This is proven in [BDSP10]
Lemma 3.2(i) under the additional assumption that there does not exist a newformh ∈ S2 (N ), h 6= f which is congruent (away from Σ) to f (mod $) and similarly
there does not exist a newform h ∈ Sk (N ), h 6= g which is congruent (away fromΣ) to g (mod $). In what follows we assume that the local L-factors in Lemma 4.6
are p-adic units, and hence a necessary and sufficient condition for the Assumption
5.1(1) to be satisfied is that the Bloch-Kato Selmer group be cyclic (which, assuming
the relation to an L-value predicted by the Bloch-Kato conjecture, is guaranteed
for example when val$ (LN,alg (1 + k/2, f × g)) = 1).
On the other hand one can also formulate some sufficient conditions under which
Assumption 5.1(2) is satisfied. We will only discuss the case of ρ2 = ρg , the case of
ρ1 being similar. Suppose that ρ : GΣ → GL2 (O) is another crystalline lift of ρ2 . In
particular ρ is semi-stable at p, hence the Fontaine-Mazur conjecture predicts thatit should be modular. This conjecture is true in many cases. In particular it is true
when ρ is unramified outside finitely primes, ramified at p and (short)p crystalline,with ρ|GQ(p) absolutely irreducible and modular (here Q(p) = Q( (−1)(p−1)/2 p))
by a Theorem of Diamond, Flach and Guo ([DFG04], Theorem 0.3). In our case
ρ is ramified at p because det ρ is, so if we assume in addition that ρg |GQ(p) is
absolutely irreducible, we can conclude that there exists a modular form h suchthat ρ ∼= ρh . Since we assume (in accordance with Assumption 5.1 - see discussion
following that assumption) that Σ does not contain any primes congruent to 1 modp, we have HΣ1 (F, ad0 ρ̃i ⊗ E/O) = HΣ1 (F, ad ρ̃i ⊗ E/O) as explained in section 5.1.
Hence we must have det ρh = det ρg , so h is necessarily of weight k. Since ourdeformations are unramified outside Σ (and crystalline at p), the level of the form
h can only be divisible by the primes dividing N . In this case Assumption 5.1(2) isequivalent to an assertion that there does not exist a newform h ∈ S2 (N 2 ), h 6= f
which is congruent (away from Σ) to f (mod $) and similarly there does not exist
a newform h ∈ Sk (N 2 ), h 6= g which is congruent (away from Σ) to g (mod $).
Indeed, it follows from a result of Livne ([Liv89], Theorem 0.2) that under ourassumptions concerning the primes in Σ, the form f (resp. g) cannot be congruent

30

TOBIAS BERGER1 AND KRZYSZTOF KLOSIN2

to a form of level divisible by l3 for a prime l | N (note that N is square-free by
assumption). Alternatively one can use Theorem 1.5 in [Jar99] which works for all
totally real fields. So, Assumption 5.1(2) follows from just a slight strengtheningof the congruence conditions already imposed to satisfy Assumption 5.1(1).
Alternatively, the Selmer group HΣ1 (Q, ad ρi ), i = 1, 2 could be related to asymmetric-square L-value using the Bloch-Kato conjecture and Lemma 4.6 together
with Remark 4.7. For the divisibility of (W ∗ )Iv for W = ad0 ρf ⊗E/O we can argue
as follows, as explained to one of us by Neil Dummigan: Assume again that f is
not congruent (away from Σ) to another newform modulo $. Then with respect tosome basis x, y, both ρf and ρf send a generator of the p-part of the tame inertia
1 1group at v to the matrix. It follows that the Iv -fixed parts of both Sym2 ρf
0 1
and Sym2 ρf are two-dimensional, spanned by x2 and xy − yx. Hence the Iv -fixedpart of Sym2 ρf ⊗ E/O is divisible. We observe that Sym2 ρf differs from W ∗ justby a Tate twist.
10.3. Deformations. From now on we assume that Assumptions 5.1(1) and (2)hold, Σ = {l | N } ∪ {p}. Consider an anti-automorphism τ : GΣ → GΣ given byτ (g) = (g)k−1 g −1 (see Example 2.9(3)). Note that tr ρi ◦ τ = tr ρi for i = 1, 2. By
Remark 1 of [Wei05] we also know that for any Siegel modular form φ of parallel
weight k/2 + 1, the Galois representation ρφ (in particular, also ρ0 ) is essentiallyself-dual with respect to τ as defined above, i.e. that ρ∗φ ∼= ρφ 1−k .
We study deformations ρ of ρ0 such that• ρ is crystalline at p;• tr ρ ◦ τ = tr ρ.
This deformation problem is represented by a universal couple (RΣ , ρΣ ). By Proposition 5.6 the ideal of reducibility Ire of RΣ is principal. Moreover since RΣ is generated by traces (Proposition 6.13), we get an O-algebra surjection φ : RΣ  Tf,g .
(Note that even though the Hecke operators are involved in all the coefficients of thecharacteristic polynomial of the Frobenius elements, all of them can be expressedby the trace.) The ($-part of the) Bloch-Kato conjecture (together with Lemma
4.6 - see the discussion above) predicts that
(10.1)#HΣ1 (Q, HomO (ρg , ρf (k/2 − 1))) ≤ #O/$mwith m as above. At the moment this conjecture is beyond our reach. Moreover,
it is not clear that the periods used to define the algebraic L-value involved in theMain Conjecture and the one defining LN,alg above coincide (something we haveassumed when writing (10.1)). If we assume (10.1) then Proposition 6.14 implies
that RΣ /Ire ∼= O/$s for s ≤ m. So the induced map RΣ /Ire  Tf,g /φ(Ire ) =f,g
T /If,g is an isomorphism. Thus by Theorem 8.5, we get that φ is an isomorphism.
In particular we have proved the following theorem:
Theorem 10.3. Let f , g and Ψ be as above and assume that Assumption 5.1 issatisfied and that equation (10.1) as well as Conjecture 10.2 hold. Let ρ : GΣ →GL4 (E) be an irreducible Galois representation and suppose that
ρss = ρf (k/2 − 1) ⊕ ρg .
S
Moreover assume that ρ is crystalline. Then there exists F 0 ∈ Sk/2+1(N ) such that∼ρ = ρF 0 ,ON DEFORMATION RINGS

31

i.e., ρ is modular.

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1 University of Sheffield, School of Mathematics and Statistics, Hicks Building,Hounsfield Road, Sheffield S3 7RH, United Kingdom.
2 Department of Mathematics, Queens College, City University of New York, 65-30
Kissena Blvd, Flushing, NY 11367, USA