MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE
EISENSTEIN IDEALTOBIAS BERGER AND KRZYSZTOF KLOSIN

Abstract. For a totally real field F , a finite extension F of Fp and a Galois
character χ : GF → F× unramified away from a finite set of places Σ ⊃ {p | p}
1 (F, χ−1 ). In [BK15] it wasconsider the Bloch-Kato Selmer group H := HΣ
proved that the number d of isomorphism classes of (non-semisimple, reducible)
residual representations ρ giving rise to lines in H which are modular by someρf (also unramified outside Σ) satisfies d ≥ n := dimF H. This was proved
under the assumption that the order of a congruence module is greater than orequal to that of a divisible Selmer group. We show here that if in addition therelevant local Eisenstein ideal J is non-principal, then d > n. When F = Q
we prove the desired bounds on the congruence module and the Selmer group.
We also formulate a congruence condition implying the non-principality of J
that can be checked in practice, allowing us to furnish examples where d > n.

1. Introduction
Let p be an odd prime and let Σ be a finite set of primes of Q containing p where
each prime ` ∈ Σ, ` 6= p satisfies ` 6≡ 1 (mod p). Write GΣ for the absolute Galois
group of the maximal Galois extension of Q unramified outside of Σ. Let E be a
finite extension of Qp with integer ring O, uniformizer $ and O/$O = F. Letχ : GΣ → F× be a character. Consider a non-split extension of GΣ -modules
0 → F → ρ → F(χ) → 0.
In this paper we are interested in the modularity of ρ in the following sense: Fix
a positive integer N divisible only by the primes in Σ − {p}. We will say that ρis modular (of level N ) if there exists a newform f (of level N ) giving rise to a(irreducible) Galois representation ρf : GΣ → GL2 (E) and a GΣ -stable O-latticein the space of ρf such that with respect to this lattice the mod $ reduction ρf of
ρf is isomorphic to ρ (as representations).
This is a very strong notion of modularity for two reasons:(1) we require that ρf ∼= ρ rather than simply tr ρf = tr ρ and(2) we do not allow ρf to be ramified at primes outside of Σ.
The requirement (2) stands in contrast with the work of Hamblen and Ramakrishna [HR08] who prove modularity of such ρ by ρf in the sense of (1), but allow
for additional ramification of ρf . More specifically, they show the existence of a
The first author’s research was supported by the EPSRC Grant EP/R006563/1. The second
author was supported by the Young Investigator Grant #H98230-16-1-0129 from the National
Security Agency, a Collaboration for Mathematicians Grant #578231 from the Simons Foundation and by a PSC-CUNY award jointly funded by the Professional Staff Congress and the City
University of New York.
1

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TOBIAS BERGER AND KRZYSZTOF KLOSIN

characteristic zero lift ρ : GΣ0 → GL2 (O) of ρ for some set Σ0 ⊃ Σ and then use
the modularity theorem of Skinner and Wiles [SW99] to conclude modularity of ρ.
To the best of our knowledge the question of modularity of ρ in our strong sense
has never been studied despite being rather natural. (In the semi-simple reducible
case such an analysis was carried out by Billerey and Menares in [BM18] using a
different method.) While we are not able to prove that all ρ as above are modular inthis sense, this is perhaps not to be expected. In particular not all such extensions
will in general be modular if we fix the level N as there are only finitely many formsof fixed level (we also fix the weight by imposing a condition on the determinant).
So, in particular enlarging F (which increases the number of isomorphism classesof ρ) will produce non-modular extensions. This prompts an intriguing question:
given N how many of the extensions ρ are modular of level N ? In this article wegive a lower bound on this number when ρ is in the image of the Fontaine-Laffaillefunctor as we now explain. While we limit most of our discussion here for simplicity
to the case of Q, we prove some of our results for a general totally real field F (seebelow).
Any isomorphism class ρ in the category of representations gives rise to a linein the residual Bloch-Kato Selmer group HΣ1 (Q, χ−1 ) (where we do not impose any
conditions on primes in Σ other than p). We showed in [BK15] that under someassumptions the group HΣ1 (Q, χ−1 ) has a basis consisting of modular extensions,
i.e., that at least n := dim HΣ1 (Q, χ−1 ) such isomorphism classes of ρ are modular.
Improving this bound (which is the main goal of this paper) is a tougher problem
and we show it is related to the structure of the Eisenstein ideal J of the (local)
cuspidal Hecke algebra T. We obtain the most satisfactory answer for F = Q. In
this case we show that if J is not principal and the Selmer group HΣ1 (Q, χ) (“forextensions in the opposite order” of characters to the one in ρ) is one-dimensional,
then the number of modular isomorphism classes of the representations ρ is strictly
larger than n (under some restrictions on Σ and χ) - cf. Corollary 5.9.
One of the immediate consequences of our results is that if J is not principal
then dim HΣ1 (Q, χ−1 ) > 1 (since in a one-dimensional Selmer group there is only
one line!). We note that Wake and Wang-Erickson [WWE18] give a cohomologicallower bound on the number of generators of the Eisenstein ideal for modular forms
of weight 2 and trivial nebentypus. A side effect of our result (but one that applies
to the case of k > 2 or k = 2 and non-trivial nebentypus, so not the case studied
in [WWE18]) is that it provides a condition in the converse direction, i.e., J notprincipal implies dim HΣ1 > 1.
In the process of proving Corollary 5.9 (i.e., when F = Q) we establish a lower
bound on the congruence module T/J by a certain Bernoulli number with correction
factors. Previous results of this kind include Theorem 5.1 in [SW97], which applies
in the case of k = 2 and non-trivial nebentypus and an analogous result of Mazur[Maz77], Proposition II.9.7 (for k = 2, prime level and trivial nebentypus). We alsoestablish a corresponding upper bound on the relevant Bloch-Kato Selmer groupwhich together with the T/J-bound are key for the existence of a modular basis ofHΣ1 (Q, χ−1 ). We also prove bounds on other Selmer groups that allow one to checkwhen dimF HΣ1 (Q, χ) = 1 and dimF (Q, χ−1 ) > 1 (the case when our theorem isinteresting).

MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

3

For a general F we obtain a similar result. However the existence of corresponding bounds on T/J and the Selmer group, while expected to hold, is not yet
known.
Let us discuss the organization of the paper. In section 2 we establish basic notation and facts regarding Selmer groups and Fontaine-Laffaille representations. In
section 3 we study the relevant Hecke algebra T along with its quotients Tτ corresponding to newforms whose Galois representations reduce to different isomorphismclasses of (reducible) residual representations τ . We also define the Eisenstein ideal
J and prove a preliminary result guaranteeing the existence of more than n modular
Galois extensions (Proposition 3.9). In section 4 we introduce and study the idealsof reducibility of the Galois representations ρτ : GΣ → GL2 (Tτ ) (whose existence
we prove) showing their principality under the assumption that dimF HΣ1 (Q, χ) = 1.
This allows us to strengthen Proposition 3.9 to Theorem 4.8. In section 5 we
strengthen Theorem 4.8 further in the case F = Q by proving an equality between
the orders of T/J and the relevant divisible Selmer group. In section 6 we establish
bounds on certain Selmer groups allowing us (among other things) to verify thecondition dimF HΣ1 (Q, χ) = 1 for two examples which we discuss in section 7.
We would like to thank David Spencer for informing us about [BM18] and[Spe18]. We are also grateful to Neil Dummigan and Carl Wang-Erickson for helpfulcomments and to Jordi Guàrdia and Haluk Sengün for help with the calculations
for the Examples in section 7. We are also greatly indebted to the anonymousreferee whose careful reading of the manuscript spared us from a few errors andhelped us considerably improve the exposition.
2. Setup
Let F be a totally real 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. Assume that if q ∈ Σ, then N q 6≡ 1 (mod 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 slight abuse of notation). Let E be a (sufficiently large)
finite extension of Qp with ring of integers O and residue field F. We fix a choiceof a uniformizer $. We will write  for the p-adic cyclotomic character,  for its
mod p reduction, and ω for the Teichmüller lift of . For a local ring A we write
mA for its maximal ideal.
2.1. Fontaine-Laffaille representations. Let n be any positive integer. Supposer : GΣ → GLn (F)is a continuous homomorphism.
We recall from [CHT08] p. 35 the definition of a Fontaine-Laffaille representation: Let p | p and A be a local complete Noetherian Zp -algebra with residue field
F. A representation ρ : Dp → GLn (A) is Fontaine-Laffaille if for each Artinian
quotient A0 of A, ρ ⊗ A0 lies in the essential image of the Fontaine-Laffaille functor
G (for its definition see e.g. [BK13] Section 5.2.1). We also call a continuous finitedimensional GΣ -representation V over Qp Fontaine-Laffaille if, for all primes p | p,
it is crystalline and Fil0 D = D and Filp−1 D = (0) for the filtered vector space

4

TOBIAS BERGER AND KRZYSZTOF KLOSIN

D = (Bcrys ⊗Qp V )Dp defined by Fontaine (for details see again [BK13] Section
5.2.1).
For j ∈ {1, 2} let τj : GΣ → GLnj (F) be an absolutely irreducible continuous
representation. Assume that τ1 ∼6 τ2 . Consider the set of isomorphism classes of=n-dimensional residual Fontaine-Laffaille representations of the form:


τ1 ∗
(2.1)
τ=
: GΣ → GLn (F),τ2which are non-semi-simple (n = n1 + n2 ).
2.2. Selmer groups. For a p-adic GΣ -module M (finitely generated or cofinitely
generated over O - for precise definitions cf. [BK13], section 5) we define the Selmer
1group HΣ1 (F, M ) to be the subgroup of Hcont(FΣ , M ) consisting of cohomology
classes which are crystalline in the sense of Bloch-Kato at all primes p of F dividing
p, i.e.
YHΣ1 (F, M ) = ker(H 1 (GΣ , M ) →(H 1 (Fp , M )/Hf1 (Fp , M )).
p|p

For GΣ -modules M occurring as O-lattices T in E-vector spaces V or as divisible modules V /T the crystalline conditions Hf1 (Fp , M ) are as defined by BlochKato in [BK90] (cf. also section 1 in [Rub00]). For GK -modules M of finite cardinality we use Fontaine-Laffaille theory to define the local condition: If K denotes an unramified extension of Qp then if M is in the essential image of the
Fontaine-Laffaille functor G we define Hf1 (K, M ) as the image of Ext1MF O (1FD , D)
in H 1 (K, M ) ∼= Ext1O[GK ] (1, M ), where MF O is the category of filtered Dieudonnémodules, G(D) = M and 1FD is the unit filtered Dieudonné module defined in
Lemma 4.4 of [BK90]. Note that we place no restrictions at the primes in Σ that
do not lie over p. For more details cf. [loc.cit.].
3. The rings Tτ
Proposition 3.1. Suppose ρ : GΣ → GLn (E) is irreducible and satisfies
ρss ∼
(3.1)
= τ1 ⊕ τ2 ,where ρss denotes the semi-simplification of any residual representation of ρ. Then
there exists a lattice inside E n so that with respect to that lattice the mod $ reduction ρ of ρ has the form


τ1 ∗
ρ=
0 τ2
and is non-semi-simple.
Proof. This argument goes back to Ribet and in this form is a special case of[Urb01], Theorem 1.1, where the ring B in [loc.cit.] is the discrete valuation ringO.
For τ as in (2.1) let Φτ,E be the set of isomorphism classes of Fontaine-Laffailleat p | p Galois representations ρ : GΣ → GLn (E) such that there exists a GΣ stable lattice L in the space of ρ so that the mod $-reduction of ρL equals τ . The
following is a higher-dimensional analogue of Lemma 2.13(ii) from [SW99]:
Proposition 3.2 ([BK15], Proposition 3.2). One has Φτ,E ∩ Φτ 0 ,E = ∅ if τ ∼6 τ 0.
=

MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

5

For the rest of this section set n = 2, τ1 = 1 and τ2 = χ = ψk−1 , where ψis unramified at p and k is an integer such that 2 ≤ k ≤ p − 1. Write ψ̃ for theTeichmüller lift of ψ and set χ̃ = ψ̃k−1 .
Let N be an ideal of OF divisible only by primes in Σ which do not lie over p. We
consider the space Sk (N, ψ̃) of cuspidal Hilbert modular forms (over the field F )of parallel weight k ≥ 2, level Γ0 (N) and character ψ̃. Let T0 be the O-subalgebraof EndC Sk (N, ψ̃) generated by the Hecke operators Tq for all q 6∈ Σ. Set J 0 to be
the ideal of T0 generated by the set {Tq − (1 + ψ̃(q)(N q)k−1 ) | q 6∈ Σ}. Let m bea maximal ideal of T0 containing J 0 and set T to be the completion of T0 at theideal m.
Definition 3.3. We will call J := J 0 T the (local) Eisenstein ideal (associated toψ̃).
We refer to the surjective O-algebra homomorphisms λ : T  O as Hecke
eigensystems. For each such λ we denote by τ̃λ : GΣ → GL2 (E) the corresponding(irreducible) Galois representation. Using Proposition 3.1 we see that there exists
a lattice in E 2 with respect to which τ̃λ is valued in GL2 (O) such that its mod $reduction τ̃ λ is non-semisimple. Proposition 3.2 guarantees that the isomorphismclass of τ̃ λ is independent of the choice of such a lattice. In view of this we will
simply write τλ for the non-semi-simple residual Galois representation attached to
λ (well-defined up to isomorphism). We write Tτ for the image of the canonical
map
Y
T→
O,
λ:τλ ∼=τi.e., the quotient of T corresponding to all Hecke eigensystems whose associatedresidual non-semisimple Galois representations are isomorphic to τ . If no τλ isisomorphic to τ we set Tτ = 0. We will denote by Jτ the image of J in Tτ .
Remark 3.4. It is clear that T and Tτ are finitely generated O-modules. Furthermore, #T/J < ∞ as otherwise, as we show below, there would exist a surjective
O-algebra map φ : T → O factoring through T/J. Such a map would correspond to a cuspidal eigenform f , i.e., φ = φf , where φf sends Hecke operators
to their eigenvalues corresponding to f , with J ⊂ ker φ. So, the eigenvalue of Tq
for f would equal 1 + ψ̃(q)(N q)k−1 , but this (using Tchebotarev Density Theorem
and the Brauer-Nesbitt Theorem) would contradict the irreducibility of the Galois
representation τ̃φ attached to f .
Let us now show that if #T/J = ∞, then such a φ would indeed exist. So,
suppose #T/J = ∞. Then T/J = Os × T as an O-module with T finite and s > 0.
Hence T/J is not of finite length as an O-module, and it is easy to see that it isalso not of finite length as a module over itself. Since T is Noetherian, it follows
that there is a prime ideal p of T/J which is not maximal (cf. Theorem 2.14 in[Eis95]), hence T/(J + p) is an infinite domain (as all finite domains are fields).
This implies that the structure map O → T/(J + p) is injective (as T is a finitely
generated O-module), and so the domain T/(J + p) is finite over O, thus we may
assume it equals O as O is assumed to be sufficiently large. Hence the canonicalmap T/J  T/(J + p) = O gives us the O-algebra surjection.
Note that isomorphism classesof Fontaine-Laffaille residual representations τ :
1 ∗GΣ → GL2 (F) such that τ =are in one-to-one correspondence with lines inχ6

TOBIAS BERGER AND KRZYSZTOF KLOSIN

HΣ1 (F, χ−1 ). Since 2 ≤ k < p the representations τ̃λ (and τλ ) are Fontaine-Laffailleat primes lying over p.

1 ∗Definition 3.5. We will say that (an isomorphism class of) τ =: GΣ →
χ
GL2 (F) is modular if there exists λ : T → O such that τλ ∼= τ (in other words, if
Tτ 6= 0).
Remark 3.6. Note that the requirement in Definition 3.5 is stronger than the usual
definition of modularity which simply asks that tr τ = tr τλ for τ̃λ : GΣ → GL2 (E).
Theorem 3.7 (Corollary 4.8 in [BK15]). Suppose that #HΣ1 (F, χ̃−1 ⊗ E/O) ≤#T/J. Then there exists a basis B of HΣ1 (F, χ−1 ) such that each τ ∈ B is modular.
Proof. Let us only explain why Assumption 2.4 in [BK15] used in Corollary 4.8
therein is satisfied. For this it is enough to show that there are no non-trivial
infinitesimal deformations of 1, respectively χ. This can be proved exactly as [BK13]
Proposition 9.5 since p - # ClF and N q 6≡ 1 mod p for all q ∈ Σ.
Remark 3.8. The assumption that #HΣ1 (F, χ̃−1 ⊗ E/O) ≤ #T/J is used in the
proof of Corollary 4.8 in [BK15]. The left-hand side of the inequality encodes certain
crystalline GΣ -extensions of torsion O-modules while the right-hand side encodes
corresponding modular extensions (arising from Eisenstein congruences). Henceit can be viewed as in some sense ensuring an abundance of reducible modular
deformations of appropriate type. Roughly speaking, the Selmer group on the left
hand side should be bounded by a certain L-value by virtue of the relevant caseof the Bloch-Kato Conjecture. Then the inequality in the assumption reflects thebelief that Eisenstein congruences should be controlled by the same L-value. In
section 5 we will prove that these inequalities are often satisfied when F = Q.
Let T denote the set of isomorphism classes of residual Galois representationsof the form (2.1) such that τ1 = 1 and τ2 = χ. Let Tmod be the subset of T
consisting of isomorphism classes which are modular. Note that by Proposition 3.1
each element of Tmod can be identified with a line in HΣ1 (F, χ−1 ) and Theorem 3.7
gives a sufficient condition for the existence of at least dimF HΣ1 (F, χ−1 )-many suchlines. These lines span the Selmer group, but a natural question to ask is if one
could strengthen the conditions of Theorem 3.7 to guarantee the existence of evenmore modular lines. This is achieved by Proposition 3.9 which is the first main
result of this paper.
As the proof of the proposition uses some results from [BK15], let us briefly explain the nature of these results. For every modular line arising from τ the quotientTτ /Jτ 6= 0 and it contributes to the quotient T/J which “records” Eisenstein congruences arising from all modular τ as above. While in general these ‘contributions’
do not exactly add up to the order of T/J, in the extreme case when there are just1−1enough modular lines to
Q span HΣ (F, χ ) (i.e., the lines are linearly independent)
one can show that # τ Tτ /Jτ ≤ #T/J (Proposition 5.1 in [BK15]). For this
one uses the fact that these lines contribute to “linearly independent” lines in thecorresponding divisible Selmer group HΣ1 (F, χ̃−1 ⊗ E/O), whose order is boundedfrom above by #T/J by our assumption. On the other hand a commutative algebra result proved in [BKK14] shows that under someQ fairly mild hypotheses on a
local complete O-subalgebra of full rankQT ⊂ A = i Ai , Ai = Oni together withprojections ϕi : A  Ai one has that # i ϕi (T )/ϕi (J) ≥ #T /J provided ϕi (J) isMODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

7

principal for every i with equality if and only if J is principal as well (this is statedin Proposition 5.4 in [BK15]). Combining these two results allows us to prove thefollowing proposition.
Proposition 3.9. Suppose that #HΣ1 (F, χ̃−1 ⊗ E/O) ≤ #T/J. If Jτ is principal
for every τ ∈ Tmod but J is not principal, then the set Tmod of modular isomorphismclasses has cardinality strictly greater than dimF HΣ1 (F, χ−1 ).
Proof. Let us first note that by Remark 3.4 we have that T is finitely generated
as an O-module and #T/J < ∞, hence the results of [BK15] and [BKK14] apply.
By Theorem 3.7 we know that there exists a modular basis B of HΣ1 (F, χ−1 ), so
in particular #Tmod ≥ dimF HΣ1 (F, χ−1 ). Suppose that in fact equality holds. By
Proposition 5.1 in [BK15] we have that
Y
#T/J ≥ #Tτ /Jτ .
τ ∈Tmod

Since any modular extension gives rise to an element of Tmod , we see that anyother modular basis of HΣ1 (F, χ−1 ) must be obtained from B by scaling its elements,
i.e., B is ‘projectively unique’ in the terminologyof [BK15]. Then by Proposition
Q
5.4 in [BK15] we get that #T/J = # τ ∈Tmod Tτ /Jτ . This however implies that J
is principal by Corollary 2.7 of [BKK14] - note that principality of Jτ is necessary
for the application of the corollary (cf. p. 73 of [BKK14]).

For future use we note that the opposite inequality #HΣ1 (F, χ̃−1 ⊗E/O) ≥ #T/J
always holds:
Proposition 3.10. One has#HΣ1 (F, χ̃−1 ⊗ E/O) ≥ #T/J.
Proof. This is proved by applying Urban’s lattice construction, as explained in the
proof of [BK15] Lemma 4.4 (we do not need the assumptions 2.5 and 4.2 there as
we just want an inequality of orders).
In the next section we show that if one assumes one-dimensionality of the “opposite” Selmer group HΣ1 (F, χ) then principality of each Jτ follows.
4. Ideal of reducibility and its principality
Let G be a group and A be a complete Noetherian local O-algebra (with residue
field F) which is reduced. Set R = A[G]. Let τ1 , τ2 : G → GLni (F) be two absolutely irreducible representations with τ1 6∼= τ2 . Set n := n1 + n2 and assume thatn! is invertible in A. Let T be a (residually multiplicity free) pseudo-representation
T : R → A of dimension n. Following [BC09] we define the ideal of reducibility of T
to be the smallest ideal I of A such that T = T1 +T2 mod I, where T1 , T2 are pseudorepresentations with the property that Ti = tr τi mod mA . Let ρ : R → Mn (A) bean A-algebra homomorphism. Suppose that the mod mA reduction ρ : R → Mn (F)of ρ has the form


τ1 ∗
ρ=
τ2
and is non-semi-simple. We define the ideal of reducibility of ρ to be the ideal of
reducibility of the pseudo-representation tr ρ.

8

TOBIAS BERGER AND KRZYSZTOF KLOSIN

Write F :=QFrac(A), the total ring of fractions of A, which is a finite prods
uct of fields i=1 Ai (cf. e.g., [BC09], section 1.7). Fix Sij ⊂ Ext1F[G] (τi , τj )one-dimensional subspaces for (i, j) ∈ {(1, 2), (2, 1)}. Assume that the pseudorepresentation tr ρi : R → Ai is absolutely irreducible for every i = 1, 2, . . . , s.
Moreover, assume that ρ : R → Mn (F) which factors through F[G] → Mn (F) gives
rise to a non-trivial element in S21 .
Proposition 4.1 ([BC09], Proposition 1.7.4). One has
dimF Ext1(R/ ker ρ)/mA (R/ ker ρ) (τ2 , τ1 ) = 1.
Proof. Let us only note that Proposition 1.7.4 in [BC09] concerns ker T instead ofker ρ. However, it follows from Proposition 1.6.4 of [BC09] along with our assumption on absolute irreducibility of tr ρi that ker ρ = ker T .

The goal of this section is to give a sufficient condition guaranteeing that I isprincipal. Before we begin let us briefly explain the method. If the dimension ofExt1(R/ ker ρ)/mA (R/ ker ρ) (τ1 , τ2 ) (“opposite direction”) is also one, I would be principal by Proposition 1.7.5 of [BC09]. To prove this we use Urban’s construction
to obtain an A-module T ⊕ A together with a G-action which modulo mA gives anon-split extension in the “opposite direction”. If T = A, then this extension isa reduction of a representation of G into GL2 (A) and Proposition 1.7.4 in [BC09]
gives us the desired one-dimensionality. In the proof of Theorem 4.2 we formulate
a condition that allows us to conclude that T /mA T = F and essentially deducefrom this that T = A by Nakayama’s Lemma.
From now on assume that A is finite over O. We will later apply this for A = Tτ
for which this assumption is satisfied (cf. Remark 3.4). Then by Theorem 1.1 in[Urb01] there exists an A-lattice L in F n and an A-lattice T in F such that
(4.1)

0 → τ2 ⊗A T /mA T → L ⊗A F → τ1 ⊗A F → 0.

As in [Urb01] (see also [Klo09], p. 159-160) we get a cocycle c ∈ H 1 (G, Hom(τ1 , τ2 )⊗T /mA T ) and a map
ι : Hom(T /mA T , F) → Ext1F[G] (τ1 , τ2 ) = H 1 (G, Hom(τ1 , τ2 )),f 7→ (1 ⊗ f )(c),which is injective by Lemma 4.5 in [BK15].
Theorem 4.2. If the image of ι lies in S12 , then I is principal.
Proof. We have T /mA T = Fs for some s ∈ Z+ . Since S12 = F, the injectivity of ι
implies that s = 1. Hence (4.1) itself is an element of S12 . Moreover by a complete
version of Nakayama’s Lemma, T is generated by 1 element, say x ∈ T , as an
A-module. We claim that this implies that T = A. Indeed, consider the A-module
map φ : A  T given by r 7→ rx. We will show that this map is injective. Suppose
a is in the kernel. Then a annihilates T . However, by definition of T and the factthat A is reduced and hence embeds
into its ring of fractions F we can consider
Q
x and a as elements of F = i Ai , i.e., write them as a = (a1 , a2 , . . . , as ) andx = (x1 , x2 , . . . , xs ). We want to show that a = 0.
Let J be the set of i such that ai 6= 0. First note that if j ∈ J , then xA ⊗A Aj =0. Indeed, if j ∈ J , then since ax = 0, we must have xj = 0, so xα ⊗ 1 =xαa ⊗ 1/aj = 0 for all α ∈ A. Secondly note that if j 6∈ J , then xA ⊗A Aj is of

MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

dimension ≤ 1 as an Aj -vector space. Indeed, letwrite πj for the map A → Aj . ThenPk xαk ⊗ βk ∈ xA ⊗A Aj and!
Xxαk ⊗βk =kXx⊗πj (αk )βk = x⊗kXπj (αk )βk9!
= (x⊗1)·kXπj (αk )βk,khence indeed xA ⊗A Aj is spanned over Aj by x ⊗ 1.
Thus we getYYY
T ⊗A F = xA ⊗A
Ai =xA ⊗A Ai =xA ⊗A Ai
i

i

i6∈J

and each piece of
Qthe product is either 0 or Aj . Since T is a lattice we must have
T ⊗A F = F = i Ai , and this forces J = ∅.
Hence L ∼= An , so (4.1) is the reduction of a representation R → Mn (A). Thus
by [BC09], Proposition 1.7.4, we get thatdimF Ext1(R/ ker ρ)/mA (R/ ker ρ) (τ1 , τ2 ) = 1
and thus by [loc.cit.], Proposition 1.7.5 the ideal I is principal.

Lemma 4.3. Let τ ∈ Tmod . There exists a representation ρτ : GΣ → GL2 (Tτ )that reduces to τ modulo mTτ .
Proof. Consider the representationYρ0τ : GΣ →GL2 (O) ⊂ GL2 (Frac(Tτ ))λ:τλ ∼=τ

given by the representations τ̃λ . We now proceed as in the proof of Theorem 6.2
in [BK15] replacing Rτtr,0 there with Tτ . We only give a brief outline here as theargument is essentially identical. Using Theorem 4.1 in [BK15] we deduce theexistence of a Galois invariant lattice L in the representation space Frac(Tτ )2 of
ρ0τ and a Tτ -lattice Tτ ⊂ Frac(Tτ ) which fits into the exact sequence(4.2)

0 → Tτ /Iτ Tτ → L ⊗Tτ Tτ /Iτ → χ̃ ⊗O Tτ /Iτ → 0,

where Iτ is the ideal of reducibility of the pseudo-representation tr τ .
As in the proof of Theorem 6.2 in [BK15] one notes that L ∼= Tτ ⊕ Tτ as Tτ modules and then shows that Tτ /Iτ Tτ ⊗Tτ F ∼= F, so we get Tτ = Tτ as in the
proof of Theorem 4.2 above. Thus (4.2) gives rise to a representation ρτ as in thestatement of the Lemma.
Remark 4.4. We note that Lemma 4.3 does not imply that there is a representation of GΣ into GL2 (T). In the residually irreducible case this is in fact the case(cf. Lemma 3.27 in [DDT97]). Also if one assumes that τ is unique (i.e., that there
is only one isomorphism class of non-semisimple residual representations with semisimplification 1 ⊕ χ) this is also true and follows from the fact that in this case theuniversal deformation ring is generated by traces (cf. Corollary 3.2 in [SW97] andProposition 7.13 in [BK13]). However, in general (when several different τ s exist),
this need no longer be the case. Lemma 4.3 can be viewed as providing a substitute for the existence of a representation into GL2 (T) when one fixes a particular
residual representation τ . However, while Tτ is a quotient of T, in general there isno natural map Tτ → T.

10

TOBIAS BERGER AND KRZYSZTOF KLOSIN

Using Lemma 4.3 we can write Iτ for the ideal of reducibility of ρτ . Let us now
apply Theorem 4.2 to our situation with A = Tτ . Note that the cuspidality of Tτ
ensures that the assumption of absolute irreducibility of the generic components ofρτ is satisfied.
Lemma 4.5. One has Jτ = Iτ .
Proof. This can be proved like Lemma 2.9 in [BK15]: As tr ρτ ≡ 1 + ψ(q)N (q)k−1
mod Jτ for all ideals q 6∈ Σ, one clearly has Iτ ⊂ Jτ . To prove the opposite
containment it is enough to show that if tr ρτ ≡ T1 + T2 mod I 0 for some ideal
I 0 ⊂ Tτ and some characters T1 , T2 which reduce to 1 and ψk−1 respectively, thenit has to be the case that T1 = 1 and T2 = ψk−1 . This follows from the factthat Assumption 2.4 in [BK15] is satisfied in our situation (cf. Proof of Theorem
3.7).

Proposition 4.6. If dimF HΣ1 (F, χ) = 1 then Iτ is a principal ideal.
Proof. Because τ is an actual representation, Proposition 4.1 gives us thatdimF Ext1(Tτ [GΣ ]/ ker ρτ )/mτ (Tτ [GΣ ]/ ker ρτ ) (χ, 1) = 1.
We set A = Tτ , G = GΣ and S21 = HΣ1 (F, χ). The claim follows from Theorem
4.2 and Lemma 4.7 below.
Lemma 4.7. The image of ι : Hom(T /mT T , F) → H 1 (GΣ , χ) is contained inHΣ1 (F, χ).
Proof. This follows from Lemma 4.5 in [BK15], except that we do not need theassumptions 2.5 and 4.2 there, as we do not claim surjectivity of ι here.
Combined with Proposition 3.9 we obtain the following result.
Theorem 4.8. Suppose that #HΣ1 (F, χ̃−1 ⊗ E/O) ≤ #T/J. If dimF HΣ1 (F, χ) = 1
and the Eisenstein ideal J is not principal, then #Tmod > dimF HΣ1 (F, χ−1 ).
We end this section by stating a cohomological criterion guaranteeing the principality of the Eisenstein ideal.
Corollary 4.9. Suppose that #HΣ1 (F, χ̃−1 ⊗ E/O) ≤ #T/J. Suppose furthermorethat dimF HΣ1 (F, χ) = dimF HΣ1 (F, χ−1 ) = 1. Then J is principal.
Proof. In this case there is only one line in HΣ1 (F, χ−1 ) which is modular by Theorem 3.7, i.e., we must have #Tmod = 1. The claim now follows directly from
Theorem 4.8.

5. F = Q
In this section we take F = Q. As in section 3 we set τ1 = 1 and τ2 = χ whereχ is a character ramified at p. By class field theory we can write χ = ω k−1 ψ forsome k with 2 ≤ k ≤ p − 1 and a character ψ unramified at p. We write ψ̃ for theTeichmüller lift of ψ and χ̃ = ψ̃k−1 . The assumption that N q 6≡ 1 (mod p) for allq ∈ Σ is unnecessary for any of the results in this section.

MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

11

5.1. Proving #HΣ1 (Q, χ̃−1 ⊗ E/O) ≤ #T/J. For the convenience of the reader
let us recall our setup. We denote by T0 the Hecke algebra acting on the space ofcusp forms Sk (Γ0 (N)) (as before N ∈ Z+ is only divisible by primes in Σ − {p}),i.e., the O-subalgebra of EndC (Sk (Γ0 (N))) generated by T` for all ` - Np. Set J 0
to be the ideal of T0 generated by the operators T` − (1 + ψ̃(`)`k−1 ) for all ` 6∈ Σ.
Let m be the maximal ideal of T0 containing J 0 and write T for the completion of
T0 at m. Set J to be the image of J 0 in T.
PutY
η(ψ̃, k) := Bk (ψ̃) ·(1 − ψ̃(`)`k ),`∈Σ−{p}where Bk (ψ̃) is the kth Bernoulli number of ψ̃. Here we treat ψ̃ as a Dirichlet
character of Z/MZ rather than of Z/NZ, where M is the largest factor of N only
divisible by primes dividing the conductor of ψ̃ (in other words we do not setψ̃(`) = 0 if ` - cond(ψ̃)).
Remark 5.1. It is expected that #T/J ≥ #O/η(ψ̃, k) as long as k > 2 or k = 2
but ψ 6= 1. The case k = 2 and ψ = 1 is slightly different. For Σ = {p, `} with ` aprime different from p Mazur [Maz77] Proposition II.9.7 proved
`−1
valp (#T/J) = [O : Zp ] valp (num).
12This corresponds to η(1 (mod `), k) where we - different to our convention above- take ψ̃ = 1 as a Dirichlet character modulo `, i.e. put ψ̃(`) = 0. In the proof of
Proposition 5.2 below the case k = 2, ψ = 1 is excluded due to the different form of
the constant term of the Eisenstein series. See also [Oht14] and [Yoo16] who treat
a related Hecke algebra when k = 2, ψ = 1 and the level is composite.
We now prove that #T/J ≥ #O/η(ψ̃, k) under some conditions.
Proposition 5.2. Let k ≥ 2. If k = 2 assume that ψ 6= 1. Let N = cond(ψ̃),Σ = {p, `, q | N } for some prime ` - N p. Then there exists m > 0 such that
#T/J ≥ #O/η(ψ̃, k) for N = N `m .
Remark 5.3. We note that our proof in fact shows that #T̃/J˜ ≥ #O/η(ψ̃, k),where T̃ is the Hecke algebra including Tp , and J˜ has the additional generator
˜ We do not use the congruence
Tp − (1 + ψ̃(p)pk−1 ). Note that T/J  T̃/J.
˜
module T̃/J in this paper, but for other applications it might be of interest thatthe corresponding cusp forms congruent to the Eisenstein series are ordinary at p.
Let us also note that for Proposition 5.2 we allow for the primes dividing N to becongruent to 1 mod p.
Proof of Proposition 5.2. We partially adapt arguments from lectures notes by Skinner from 2002 which treat the case of weight k = 2 (making explicit Wiles’ argumentin the proof of the totally real Iwasawa Main Conjecture).
If η(ψ̃, k) ∈ O× then there is nothing to prove. So assume val$ (η(ψ̃, k)) > 0.
Let φ be a non-trivial Dirichlet character of conductor M such that φ(−1) = (−1)l
for l ≥ 1. Set
∞L(φ, 1 − l) X X
φ(d)dl−1  q n ∈ Ml (M, φ)
El (φ) =
+
2
n=1
d|n

12

TOBIAS BERGER AND KRZYSZTOF KLOSIN

to be the Eisenstein series of weight l whose constant term is L(φ, 1−l)/2 (cf. [Miy89],
Theorem 4.7.1).
Proposition 5.4 ([Oza17] Proposition 0.3). If l = 2 assume that φ 6= 1. The
constant term of El (φ) at the cusp [u : v] ∈ P1 (Q) equals φ(u)−1 L(φ, 1 − l)/2 if
M | v and zero otherwise.
By a generalisation of a result of Washington (see [Sun10] Theorem 4) we knowthat there exists an auxiliary character ϕ of conductor `m for some m > 0 (which
we fix from now on) with ϕ(−1) = (−1)k−1 such thatL(ψ̃ϕ, 0)L(ϕ−1 , 2 − k) ∈ O× .

(5.1)
Then we put

G := E1 (ψ̃ϕ) · Ek−1 (ϕ−1 ) ∈ Mk (N `m , ψ̃)and deduce that its constant terms are
(
−1
,2−k)
if N `m | v
ψ̃ −1 (u) L(ψ̃ϕ,0)L(ϕ
4
0
else.
In the following we will use G, which clearly has p-integral Fourier coefficientsand a constant term which is a p-unit, to prove a congruence of the following
Eisenstein series to a cusp form. Put(5.2)

Fm (z) := Ek (ψ̃)(`m−1 z) − ψ̃(`)`k Ek (ψ̃)(`m z).

Proposition 1.2 in [BM16] (generalized to k ≥ 2 (and ψ 6= 1 if k = 2) in [BM18]
Proposition 4) states that the constant term of Ek (ψ̃)(M z) at a cusp [u : v] is given
by(0
if N - vg
(5.3)

B
k
g
g
k,ψ̃
ψ̃( M
−ψ̃(u)−1 2k) otherwise,M
where g = gcd(v, M ).
We apply this result with M = `m−1 and M = `m to compute that the constantterm of Fm at the cusp [u : v] equals
Bk,ψ̃L(ψ̃, 1 − k)(1 − ψ̃(`)`k ) = ψ̃(u)−1(1 − ψ̃(`)`k )2k2
if N `m | v and zero otherwise.
This now allows us to get a bound on T/J: Define−ψ̃(u)−1

H = Fm −η(ψ̃, k)· G,
a0 (G)kwhere a0 (G) denotes the constant term of G at infinity (which is a p-unit - seeabove). Then the previous discussion shows that H ∈ Sk (N `m , ψ̃) with q-expansion
coefficients in O.
We can then define a surjective O-algebra homomorphism φ : T/J  O/η(ψ̃, k)such that Tq 7→ 1 + ψ̃(q)q k−1 for all primes q - N `p as follows:First note that H has a Fourier coefficient which is a p-unit. To see this, notea`m−1 (Fm ) = a1 (Ek (ψ̃)) = 1, soa`m−1 (H) = 1 −η(ψ̃, k)· a m−1 (G) ∈ O× ,a0 (G)k `MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

13

where an denotes the n-th Fourier coefficient of the respective modular form.
This allows us to extend H to an O-basis of Sk (`m N, O), say m0 = H, m1 , . . . mr .
Let t ∈ T. Then
r
X
tm0 =
λi (t)mi , for λi (t) ∈ O.
i=0

We can now define the (surjective) O-module homomorphism φ : T → O/η(ψ̃, k)by φ(t) = λ0 (t) (mod η(ψ̃, k)), and it is easy to check that this, in fact, is evena ring homomorphism, and that it factors through T/J since Tq − 1 − ψ̃(q)q k−1
annihilates Fm .
Remark 5.5. Dummigan-Fretwell [DF14], Billerey-Menares [BM18], and Spencer[Spe18] use similar linear combinations of Eisenstein series to prove mod p congruences using the Deligne-Serre lifting lemma. Note, however, that our Fm has
non-vanishing constant terms only for N `m | v, which makes it possible to remove
them by using the auxiliary G and prove the full expected T/J bound. By [BKK14]
Proposition 4.3 this gives a lower bound on the amount and depth of Eisensteincongruences:
For a Hecke eigensystem λ : T → O write mλ for the depth of its p-adic congruence with Ek (ψ̃), i.e., mλ is the largest integer s such that λ(T` ) ≡ 1 + ψ̃(`)`k−1
mod $s for every ` 6∈ Σ. Write e for the ramification index of O over Zp . Then
combining Proposition 5.2 with [BKK14] Proposition 4.3 (see also Remark 5.13)
we obtain[E : Qp ] X
mλ ≥ valp (#T/J) ≥ valp (#O/η(ψ̃, k)).
eλIt is possible to add more auxiliary primes to Σ in Proposition 5.2, i.e. proving
additional “congruences of local origin” (in the terminology of [DF14]). For our
examples in section 7 we note here the case of two auxiliary primes:
Proposition 5.6. Let k ≥ 2. If k = 2 assume that ψ 6= 1. Let N = cond(ψ̃),Σ = {p, `, r, q | N } for some primes `, r - N p. Assume there exists a character ϕ of
conductor `m1 rm2 for some m1 , m2 > 0 with ϕ(−1) = (−1)k−1 such thatL(ψ̃ϕ, 0)L(ϕ−1 , 2 − k) ∈ O× .

(5.4)

Then we have #T/J ≥ #O/η(ψ̃, k) for N = N `m1 rm2 .
Proof. We define G as in the proof of Proposition 5.2 using the given character
ϕ of conductor `m1 rm2 . Its constant term is as above, except that the condition
for non-vanishing is now N `m1 rm2 | v. The necessary modification in the proof of
Proposition 5.2 is to further stabilize the Eisenstein series Ek (ψ̃): We put
Fm1 ,m2 (z) := Fm1 (rm2 −1 z) − rk ψ̃(r)Fm1 (rm2 z).
In terms of Ek (ψ̃) this is given by
Ek (ψ̃)(`m1 −1 rm2 −1 z)−`k ψ̃(`)Ek (ψ̃)(`m1 rm2 −1 z)−rk ψ̃(r)Ek (ψ̃)(`m1 −1 rm2 z)+(`r)k ψ̃(`r)Ek (ψ̃)(`m1 rm2 z).
We apply again Proposition 1.2 in [BM16] to check that the constant term of
Fm1 ,m2 at a cusp [u : v] is non-zero only if N `m1 rm2 | v and in that case is givenψ̃,k)by −ψ̃(u)−1 η(2k.

14

TOBIAS BERGER AND KRZYSZTOF KLOSIN

Assume, for example, that N `m1 | v, but r - v. Then by (5.3) the constant termof Fm1 ,m2 is given byBk,ψ̃−ψ̃(u)−12kψ̃(r1−m2 )ψ̃(r−m2 )ψ̃(r−m2 )ψ̃(r1−m2 )
k
k
k
+
(`r)
−
`
ψ̃(`)
−
r
ψ̃(r)
ψ̃(`r)(rm2 −1 )k(rm2 −1 )k(rm2 )k(rm2 )k!
= 0.

If N `m1 ri kv for 1 ≤ i < m2 the term g = gcd(v, M ) in the constant term
formula for all M to be considered in (5.3) just has an extra ri , so all terms get
multiplied by ri ψ̃(ri ). By symmetry this also deals with the cases where N `j rm2 kv
for 0 ≤ j < m1 . The remaining cases N `i rj | v with i < m1 and j < m2 can betreated similarly.
Finally, if N `m1 rm2 | v then the constant term becomesBk,ψ̃ Bk,ψ̃−ψ̃(u)−1
1 − `k ψ̃(`) − rk ψ̃(r) + (`r)k ψ̃(`r) = −ψ̃(u)−1(1−ψ̃(`)`k )(1−ψ̃(r)rk ).
2k
2k
Proposition 5.7. One has #HΣ1 (Q, χ̃−1 ⊗ E/O) ≤ #O/η(ψ̃, k).
Proof. Consider the following diagram of fields with corresponding Galois groups:
L∞
X∞

Q∞ Q(ψ̃ω k−1 )Γ∆Q(ψ̃ω k−1 )Q∞
Γ=<γ>∼=Zp∆

Q
Here Q(ψ̃ω k−1 ) denotes the splitting field of ψ̃ω k−1 and L∞ is the maximal abelian
extension of Q∞ Q(ψ̃ω k−1 ) unramified everywhere.
We first prove that
(5.5)

1
#H{p}
(Q, χ̃−1 ⊗ E/O) ≤ #O/Bk (ψ̃).

This follows from the Main Conjecture of Iwasawa theory proven by Mazur-Wiles,
as we briefly explain for the convenience of the reader: For K = Q or Q∞ and ϕ acharacter of GK put
Y1
HGr(K, E/O(ϕ)) := ker(H 1 (K, E/O(ϕ)) →H 1 (Iv , E/O(ϕ))).
v

A result of Flach (see [Och00] Proposition 4.1(1)) tells us that11
H{p}(Q, E/O(ψ̃ −1 1−k )) ⊆ HGr(Q, E/O(ψ̃ −1 1−k )).

Let Ψ = ψ̃ −1 ω 1−k and X∞,Ψ be the Ψ-isotypical component of X∞ for the ac1tion of ∆. We have X∞,Ψ = Hom(HGr(Q∞ , E/O(Ψ)), E/O). Using the Γ-module
structure of X∞,Ψ from this we get
1
X∞,Ψ /(T − (κ1−k− 1)) = Hom(HGr(Q, E/O(Ψ(/ω)1−k )), E/O),
0MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

15

where κ0 = (/ω)(γ). Since both modules are finite and Ψ(/ω)1−k = ψ̃ −1 1−k we
get
1
#HGr
(Q, E/O(ψ̃ −1 1−k )) = #X∞,Ψ /(T − (κ1−k− 1)).
0
Since X∞,Ψ has no finite Λ := Zp [[Γ]]-submodules (see [MW84] Proposition 1 on
p. 193) one obtains#X∞,Ψ /(T − (κ1−k− 1)) ≤ #Λ/(gΨ , T − (κ1−k− 1)),
00
where gΨ ∈ Λ is the characteristic power series of X∞,Ψ . By the Main Conjecture(see [MW84] Theorem p. 214) we havegΨ (κs0 − 1) = Lp (ωΨ−1 , s),where the latter is the p-adic L-function with the following interpolation property(see [Was97] Theorem 5.11):
Bn (ψ̃), for n ≥ 1.
n
Setting n = k and observing that (1 − ψ̃(p)pk−1 ) ∈ O× we obtain (5.5).
A repeated application of Lemma 6.2 in the next section (by selecting s in thatlemma to be sufficiently large and taking n in that lemma to be k − 1) leads us now
to the bound by η(ψ̃, k) on HΣ1 (Q, χ̃−1 ⊗ E/O).

Lp (ωΨ−1 , 1 − n) = −(1 − ψ̃(p)pn−1 )From now on assume that ψ̃, Σ and T are as in Proposition 5.2 or Proposition
5.6. By combining Propositions 3.10, 5.2 (or 5.6) and 5.7 we obtain the following
corollary.
Corollary 5.8. We have
#T/J = #O/η(ψ̃, k) = #HΣ1 (Q, χ̃−1 ⊗ E/O).
Then in the case F = Q we obtain the following stronger versions of Theorem
4.8 and Corollary 4.9.
Corollary 5.9. If dimF HΣ1 (Q, χ) = 1 and the Eisenstein ideal J is not principal,
then #Tmod > dimF HΣ1 (Q, χ−1 ).

1 ∗Remark 5.10. Suppose we consider the set of extensions ρ =: GΣ0 →
χ
0
GL2 (F) with χ ramified at all primes in Σ ⊃ {p}. Then Corollary 5.9 can beviewed as asserting that more than dimF HΣ1 (Q, χ−1 ) of these extensions arise frommodular representations ρf which are ramified at no more than one additional prime(the prime ` in Proposition 5.2, i.e., Σ = Σ0 ∪ {`}) as long as J is not principal anddimF HΣ1 (Q, χ) = 1.
Corollary 5.11. Suppose that dimF HΣ1 (Q, χ) = dimF HΣ1 (Q, χ−1 ) = 1. Then J
is principal.
5.2. Congruence criterion. The assumption that the Eisenstein ideal is not principal may be difficult to check directly, so we will translate it here into a criterion
that relies on counting congruences. We still let ψ̃, Σ and T be as in Proposition
5.2 or as in Proposition 5.6.
We assume now that E contains all the values of Hecke eigensystems λ : T → Qp
congruent to 1 + ψ̃(`)`k−1 modulo a prime above p. For a Hecke eigensystem
λ : T → O write mλ for the depth of its p-adic congruence with Ek (ψ̃), i.e., mλ is

16

TOBIAS BERGER AND KRZYSZTOF KLOSIN

the largest integer s such that λ(T` ) ≡ 1 + ψ̃(`)`k−1 mod $s for every ` 6∈ Σ. Writee for the ramification index of O over Zp .
Theorem 5.12. Assume that dimF HΣ1 (Q, χ) = 1. If[E : Qp ] X
mλ > valp (#O/η(ψ̃, k))e
λor equivalently
1X
mλ > valp (η(ψ̃, k))
e

(5.6)

λ

then J is not principal and #Tmod > dimF HΣ1 (Q, χ−1 ).
Proof. Assume J is principal. Writing Tλ = O, Jλ = $mλ O, T = T and J as
before for the Eisenstein ideal, we can apply Corollary 2.7 in [BKK14] (again noteas in Proposition 3.9 that the principality of the Jλ s is necessary for the applicationof the corollary) to conclude that then!
Y
X
[E : Qp ] Xmλ .
valp (#T /J) = valp #Tλ /Jλ =valp (#O/$mλ ) =
e
λ

λ

λ

The left-hand side equals valp (#O/η(ψ̃, k)) by Corollary 5.8, hence the above sequence of equalities contradicts our assumption. So we conclude that J is notprincipal. On the other hand the same corollary tells us that valp (#T /J) =valp (#HΣ1 (Q, χ̃⊗E/O)), so we can apply Proposition 3.9 to conclude that #Tmod >
dimF HΣ1 (Q, χ−1 ).
Remark 5.13. We would like to take this opportunity to correct the inequality(4.1) in [BKK14] and inequalities in Examples 5.1, 5.3 and 5.4 therein. All of themshould have an extra factor of [E : Qp ] on the left-hand side. Proposition 3.1 andequality (1.1) of [BKK14] remain correct as stated.
6. Analysis of HΣ1 (Q, F(n))
In this section we prove bounds on certain Selmer groups. The assumption that` 6≡ 1 (mod p) for all ` | N is not needed for Proposition 6.1 and Lemma 6.2.
Proposition 6.1. For 2 ≤ k ≤ p − 1 and k even we havevalp (#HΣ1 (Q, F(1−k))) ≥ [F : Fp ] min{valp (B1,ωk−1 ), 1} +X

min{valp (1 − `k ), 1} .

`∈Σ−{p}

Proof. By Fontaine-Laffaille theory (see e.g.
Fontaine-Laffaille Dp extension[Bre01] Proposition 9.1.2(i)) any

0 → F → ρ → F(k − 1) → 0
1
is split on Ip , so Hf1 (Qp , F(1−k)) = Hur(Qp , F(1−k)) := (ker(H 1 (Qp , F(1−k)) →1
H (Ip , F(1 − k))). We therefore have
!
Y
1
H{p}
(Q, F(1 − k)) = ker H 1 (Q, F(1 − k)) →H 1 (I` , F(1 − k)) .
`MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

17

As in section 2 of [Ski06] we can argue that restriction to GQ(µp ) gives
1
H{p}(Q, F(1 − k)) = HomGal(Q(µp )/Q) (CQ(µp ) , F(1 − k)),where CQ(µp ) denotes the class group of Q(µp ).
The p-primary part of CQ(µp ) on which the action of Gal(Q(µp )/Q) is via ω 1−k
has order given by the p-primary part of L(0, ω k−1 ) = −B1,ωk−1 by [MW84] The1
orem 2 p. 216 (see also [Ski06] Theorem 2.1.3). This shows that #H{p}(Q, F(1 −1−kωk)) ≥ (#Fp /B1,ωk−1 )[F:Fp ] (equality holds if CQ(µis cyclic).
p)
The proposition now follows from Lemma 6.2 below applied with n = k − 1.

Lemma 6.2. Let n be an integer and set m := valp (ψ̃(`)`n+1 − 1) for ψ̃ a Dirichlet
character unramified away from Σ − {p}. Assume m < ∞ and let s ≥ me be an
integer, where e is the ramification index of O over Zp . Set W = E/O(ψ̃ −1 −n )and Ws = W [$s ]. Suppose ` ∈ Σ − {p} and let Σ0 ⊂ Σ with ` 6∈ Σ0 . Then one has#HΣ1 0 ∪{`} (Q, Ws ) ≤ (#O/pm O)#HΣ1 0 (Q, Ws ).
Proof. First assume that W is ramified at `. Since ψ̃ is the Teichmüller lift of ψ itmust then be the case W1 is also ramified at `, so W I` = 0. We then use [BK13]
Lemma 5.6 to conclude thatHΣ1 0 ∪{`} (Q, Ws ) = HΣ1 0 (Q, Ws ).
From now on assume that W is unramified at `. By [Rub00], Theorem 1.7.3 wehave an exact sequence
0 → HΣ1 0 (Q, Ws ) → HΣ1 0 ∪{`} (Q, Ws ) →H 1 (Q` , Ws ).
1 (Q , W )
Hur
`
s

1
1
Lemma 1.3.8(ii) in [Rub00] tells us that Hur(Q` , Ws ) = Hf1 (Q` , Ws ), where Hur(Q` , Ws ) :=1
1
ker(H (Q` , Ws ) → H (I` , Ws )). We also get(6.1)

H 1 (I` , Ws ) = Hom(I`tame , Ws ) = Hom(Zp (1), Ws ) = Ws (−1).

Here I`tame denotes the image of I` in the tame quotient of GQ` . This gives an
H 1 (Q` ,Ws )upper bound of (#F)s = #Ws on the order of the quotient H. To prove1
ur (Q` ,Ws )1
the claim it is enough to show that the image of the map H (Q` , Ws ) → H 1 (I` , Ws )has order not greater than #O/pm O. To do so consider the inflation-restriction
sequence (where we set G := Gal(Qur` /Q` )):H 1 (G, Ws ) → H 1 (Q` , Ws ) → H 1 (I` , Ws )G → H 2 (G, Ws ).
The last group in the above sequence is zero since G ∼= Ẑ and Ẑ has cohomologicaldimension one. This means that the image of the restriction map H 1 (Q` , Ws ) →H 1 (I` , Ws ) equals H 1 (I` , Ws )G . Let us show that the latter module has order≤ #O/pm O. Indeed,(6.2) H 1 (I` , Ws )G = HomG (I` , Ws ) = HomG (I`tame , Ws )= HomG (Zp (1), $−s O/O(ψ̃ −1 −n )) = HomG (Zp , $−s O/O(ψ̃ −1 −n−1 )).
So, φ ∈ H 1 (I` , Ws ) lies in H 1 (I` , Ws )G = HomG (Zp , $−s O/O(ψ̃ −1 −n−1 )) if andonly if φ(x) = g · φ(g −1 · x) = g · φ(x) = ψ̃ −1 −n−1 (g)φ(x) for every x ∈ I` and18

TOBIAS BERGER AND KRZYSZTOF KLOSIN

every g ∈ G, i.e., if and only if(6.3)(ψ̃ −1 −n−1 (g) − 1)φ(x) ∈ O

for every x ∈ I` , g ∈ G.

Since Frob` topologically generates G, we see that (6.3) holds if and only if it holds
for every x ∈ I` and for g = Frob` . So condition (6.3) becomes(6.4)(1 − ψ̃ −1 (`)`−n−1 )φ(x) ∈ O

for every x ∈ I` .

Since valp (1−ψ̃ −1 (`)`−n−1 ) = valp (ψ̃(`)`n+1 −1) = m, we get that φ(x) ∈ p−m O/O,
as claimed.
When s = 1 and ψ = 1 we prove a slightly stronger result.
Lemma 6.3. Let n be an integer. Suppose ` ∈ Σ − {p} and m := min{valp (`n+1 −1), 1}. Let Σ0 ⊂ Σ with ` 6∈ Σ0 . Write q = #F. Then one has(6.5)#HΣ1 0 ∪{`} (Q, F(−n)) ≤ q m #HΣ1 0 (Q, F(−n))

with equality when m = 0.
Proof. If m = 0 the inequality (6.5) follows directly from Lemma 6.2, while theopposite inequality is clear. Now assume m = 1. As before, set W = E/O(−n)
and Ws = W [$s ]. Then [Rub00], Theorem 1.7.3 gives us again an exact sequence(6.6)

0 → HΣ1 0 (Q, W1 ) → HΣ1 0 ∪{`} (Q, W1 ) →H 1 (Q` , W1 ),
1 (Q , W )Hur`1and as in the proof of Lemma 6.2 we see that the order of the module on the rightis bounded by q. This yields (6.5).
For HΣ1 (Q, F(k − 1)) on the other hand it is in general not possible to relate to1(Qp , F(k − 1)) (but see [Rub00]
pieces of class groups, as Hf1 (Qp , F(k − 1)) 6= Hur
Proposition 1.6.4(ii) for k = 1).
Proposition 6.4. One has(6.7)

valp (#HΣ1 (Q, F(k − 1))) ≤ valp (#H 1 (Gal(Q{p} /Q), F(k − 1)))
X
+ [F : Fp ]min{valp (1 − `k−2 ), 1}.
`∈Σ−{p}

Proof. Let us first assume that one has(6.8)

1
valp (#HΣ1 (Q, F(k − 1))) ≤ valp (#H{p}(Q, F(k − 1)))
X
+ [F : Fp ]min{valp (1 − `k−2 ), 1}.
`∈Σ−{p}1
The Selmer group H{p}(Q, F(k − 1)) is certainly no larger than the Selmer groupwhere all the classes are unramified away from p and we impose no condition atp. This last Selmer group is isomorphic to H 1 (Gal(Q{p} /Q), F(k − 1)). Here Q{p}stands for the maximal algebraic extension of Q unramified away from p. This gives
us the claim of the Proposition. Hence it remains to prove (6.8), but this followsby (a possibly repeated application of) Lemma 6.3 where we set n = 1 − k and notethat valp (`k−2 − 1) = valp (`2−k − 1).
We will use the following proposition with r = k − 1.

MODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

19

Proposition 6.5. Suppose r ∈ Z, with r 6≡ 1 mod (p−1) and that the r -eigenspaceof the p-part C of the class group of Q(µp ) is trivial. Then dimF H 1 (Gal(Q{p} /Q), F(r)) ≤1.
Proof. Using the inflation-restriction sequence it is enough to show thatdimF HomGal(Q{p} /Q) ((ker r )ab , F(r)) ≤ 1.
By Class Field Theory this reduces the problem to studying the units for the splitting field of χ0 := r as a G := Gal(Q(χ0 )/Q)-module as we now explain. In fact,
a similar analysis has been carried in section 3 of [BK09] for imaginary quadratic
fields. The current situation is simpler, so we will only sketch the argument here
and refer the reader to [BK09] for details. Let P be the unique prime of Q(χ0 )over p. Write OP for the valuation ring of the completion of Q(χ0 ) at P. Let T××
be the torsion subgroup of the group of units OP. Then OP/T is a free Zp -moduleof rank d := [Q(χ0 ) : Q]. Since the r -eigenspace of C is trivial, by Proposition
13.6 in [Was97] we see that any element of HomGal(Q{p} /Q) ((ker χ0 )ab , F(r)) gives
×rise to a G-equivariant homomorphism from OP
to F(r). As T ∼= µp and so G
×acts on T by  we see that such a homomorphism will factor through OP/T as×r 6≡ 1 mod (p − 1). In fact it will also clearly factor through (OP /T ) ⊗ F. Using
×
OP
/T ∼
F-vector= 1 + POP ∼= POP , it is enough to decompose the d-dimensionalL
space POP ⊗ F as a G-module. One easily sees that POP ⊗ F = φ∈G∨ F(φ),
where G∨ denotes the group of homomorphisms φ : G → F× and F(φ) is the
one-dimensional F-vector space on which G acts via φ. So, the eigenspace of i in
POP ⊗ F is at most one-dimensional for each i.

7. Example
We end with some examples, where the conditions of Theorem 5.12 are satisfied.
7.1. S6 (77). Let p = 19, k = 6, Σ = {7, 11, 19}, and consider χ = ω k−1 (i.e.
ψ = 1). Let E be a sufficiently large finite extension of Qp (to be specified later)
and F its residue field as before. Since p - (1 − 74 )(1 − 114 ) we have by Lemma 6.3
1
that HΣ1 (Q, χ) = H{p}(Q, χ). By Propositions 6.4 and 6.5 (note that k − 1 = 5 6≡ 1
mod 18) we know that the latter is at most 1-dimensional since the relevant pieceof the class group of Q(µp ) is trivial by Herbrand’s theorem as p - B14 .
As already discussed in [DF14] Example 5.6 there exist a cuspform f of weight
6 and level 7 congruent to an Eisenstein series mod p or more precisely, thataq (f ) ≡ 1 + q k−1 modulo a prime above p for primes q 6= 7 (there is also such acongruence for an eigenform g ∈ S6 (11)), so by Ribet’s lattice construction
 we knowχ ∗that there exists a non-trivial crystalline (Fontaine-Laffaille) extension, so
0 1
dimF HΣ1 (Q, χ) = 1.
While our arguments below (together with Theorem 5.12) imply in particularthat dimF HΣ1 (Q, χ−1 ) ≥ 2 (so the question of the number of modular extensions
becomes relevant) we note that this also follows from Proposition 6.1 since p |(1 − 76 ) and p | (1 − 116 ).
Since val19 (η(1 (mod 77), 6)) = val19 (B6 (1 − 76 )(1 − 116 )) = 2 and one can checkusing SAGE [The18] that there exists a character of conductor 77 satisfying (5.4)

20

TOBIAS BERGER AND KRZYSZTOF KLOSIN(so m1 = m2 = 1 in the statement of Proposition 5.6) we are in the position to
apply Theorem 5.12 as long as we can show (5.6) holds.
Note that Proposition 5.6 does not guarantee the existence of a newform of level
77 congruent to an Eisenstein series (the bound on T/J could be achieved by theoldforms coming from level 7 and 11). See [BM16] Conjecture 3.2 and Remark 4.3
for related discussions.
MAGMA calculations show that S6 (Γ0 (77)) has 5 Galois conjugacy classes of
newforms. One of these, let us call it h, has a coefficient field Kh of degree 4 over
Q and its Hecke eigenvalues aq (h) ≡ 1 + q 5 mod P for some small primes q and P
one of the 3 primes above p in Kh with ramification index 1 and inertial degree 1.
In S6 (77) we therefore have two oldforms arising from f ∈ S6 (7) and g ∈ S6 (11),and the newform h. If we consider the compositum of their coefficient fields wehave the following situation:
Kf Kg Kh

Kf

Kg
2

3

Kh
4

Q
In Kf Kg Kh the ideal (19) factors as (P1 · · · P9 )2 . Using MAGMA one checks
that for one of the primes above 19 (P4 in MAGMA’s numbering, which has inertial
degree 1) one has aq (f ) ≡ 1 + q 5 mod P4 , aq (f ) 6≡ 1 + q 5 mod P24 , aq (g) ≡ 1 + q 5
mod P24 , and aq (h) ≡ 1 + q 5 mod P24 . (The difference in exponents is explainedby the fact that p ramifies only in Kf , not in Kg or Kh .)We check these congruences hold for all primes q ≤ 3696 except for q = 7, 11. By
the argument from [DF14] Example 5.6 (using twists by the quadratic character ofconductor 77) this establishes the congruence for all eigenvalues an for n coprime
6
1
to 77, as the Sturm bound for S6 (772 ) equals 12772 (1 + 17 )(1 + 11
) = 3696.
Already for F the completion of Kf Kg Kh at P4 we therefore have a surplus of
Eisenstein congruences, sinceX
11/emλ = (1 + 2 + 2) > 2 = val19 (η(1 (mod 77), 6)).
2This implies such a surplus over an extension of F that contains the eigenvalues ofall eigenforms in S6 (77), i.e., over a field which is sufficiently large for the applicationof Theorem 5.12 - this is our sufficiently large extension E.
It is not a priori clear that the representations associated to these three cuspforms
are not all isomorphic modulo p. But since the assumptions of Theorem 5.12
are satisfied, we can deduce the existence of more than dimF HΣ1 (Q, χ−1 ) = 2
modular lines in HΣ1 (Q, χ−1 ) and we have also proved that the Eisenstein ideal is
not principal.
7.2. S6 (51). The fact that the surplus of congruences was visible already over thefield F in the previous example depended on the ramification behaviour of p in
the three coefficient fields. In general however, one may have to go up all the way
to a field which contains the eigenvalues of all the congruent eigenforms
P to get adefinitive answer whether there indeed is a surplus. Otherwise our sum
mλ mayMODULARITY OF RESIDUAL GALOIS EXTENSIONS AND THE EISENSTEIN IDEAL

21

be missing some congruent forms which do not give rise to Hecke eigensystems
T → O when O is too small. Let us illustate this by the following example.
Let p = 13, k = 6, and Σ = {3, 13, 17}. Again these primes are chosen such thatpk(1 − 36 ), pk(1 − 176 ) and p - (1 − 34 )(1 − 174 ). Here we have cuspidal eigenforms
congruent to Eisenstein series (away from 3 and 17) f ∈ S6 (3) and g ∈ S6 (17),both with rational eigenvalues, and a newform h ∈ S6 (51) with coefficient field Khof degree 4 over Q also congruent to the Eisenstein series.
In Kh one has that (13) = P1 P2 for two prime ideals P1 , P2 , of inertial degree
1 and 3, respectively. Using MAGMA one checks thataq (f ) ≡ aq (g) ≡ aq (h) ≡ 1 + q 5

mod P1 ,(for all primes q 6= 3, 17 up to the Sturm bound 1836 for S6 (512 )) so there is asurplus of congruencesXmλ ≥ 1 + 1 + 1 = 3 > 2 = val13 (η(1 (mod 51), 6)).
But h is not congruent to 1 + q 5 modulo P2 , so for this prime we do not see asurplus of congruences, as one checks that aq (f ), aq (g) 6≡ 1 + q 5 mod P22 , soXmλ = 1 + 1 = 2 = val13 (η(1 (mod 51), 6)).
But over the Galois closure of Kh of degree 24 for any of the 8 primes above p
there is a conjugate of h which is congruent to the Eisenstein series modulo thatprime.
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