AN R = T THEOREM FOR IMAGINARY QUADRATIC FIELDS

TOBIAS BERGER^{1} AND KRZYSZTOF KLOSIN^{2}

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

1. Introduction

Let K be a number field and ρ : Gal(\overline{K}/K) → GL_{2}(\overline{Q}_{p}) a continuous irreducible representation. It has been a subject of much interest and effort lately to determine which such Galois representations are modular, i.e., which such ρ’s have their L-function equal to an L-function of an automorphic representation of GL_{2}(A_{K}). Since the ground-breaking work of Wiles there has been a lot of progress in answering this question [Wil95, TW95, BCDT01, SW97, Fuj99, SW99, SW01, Tay02, Kis07].

This article is a continuation of our efforts to prove the modularity of continuous representations of Gal(\overline{K}/K) for K an imaginary quadratic field. This case is substantially different from other situations for which modularity has been proven so far since the associated symmetric space is a hyperbolic 3-manifold, so in particular tools from algebraic geometry are not available.

In a previous paper [BK09] we proved modularity of irreducible Galois representations of an imaginary quadratic K (in the minimal case) when the residual representation ρ_{0} (i.e., the composite of ρ with the map \overline{Z}_{p} ↠ \overline{F}_{p} after choosing an integral lattice in the space of ρ) is reducible under an additional condition that a certain L-value associated to the determinant of ρ has small p-adic valuation. As we showed in [BK09] this condition implies that the universal deformation ring of ρ_{0} is a discrete valuation ring. This article removes this condition.

2 TOBIAS BERGER^{1} AND KRZYSZTOF KLOSIN^{2}

We achieve this by developing a new commutative algebra criterion to prove “R = T ” theorems, applicable also in other situations (see Remark 1.2 and the discussion at the end of Section 6.4). For this fix a reducible, non-semisimple residual twodimensional Galois representation and study its minimal ordinary deformations with corresponding universal deformation ring R and universal deformation ρ_{R}. Following [BC06, Cal06] we define the ideal of reducibility I_{re} to be the smallest ideal I in R such that tr(ρ_{R}) mod I is the sum of two characters.

Date: 22 November 2009.

2000 Mathematics Subject Classification. 11F80, 11F55.

Key words and phrases. Galois representations, automorphic forms, deformations, ideal of reducibility, Eisenstein ideal, imaginary quadratic.

Assuming that there is only one (up to isomorphism) non-split extension of the two characters (in either order) which are the constituents of the semisimplification of the residual representation [BC06, Cal06] prove that I_{re} is principal. In this case our new commutative algebra criterion allows us to prove that R is isomorphic to T, a Hecke algebra, provided that R/I_{re} is a cyclic Artinian module and its order is no greater than the order of T/J for J an Eisenstein ideal of T (see Proposition 6.9 and Theorem 6.10). Lower bounds for the order of T/J have been proven in many cases starting with Mazur and Wiles [MW84]. For our case of imaginary quadratic fields we refer to [Ber09] which bounds val_{p}(#T/J) from below by the p-valuation (say n) of a Hecke L-value. Since we can rule out non-trivial reducible infinitesimal deformations by the uniqueness of our residual representation the cyclicity of R/I_{re} and an upper bound for its p-order (which again equals n) can be obtained using the Main Conjecture for Hecke characters of imaginary quadratic fields, proven by Rubin.

There are several consequences of the R = T theorem for both the Hecke algebra and the universal deformation ring. On the one hand the principality of the reducibility ideal implies that the Eisenstein ideal is also principal. On the other hand using our knowledge of the structure of the universal deformation ring we conclude that the Hecke algebra is a complete intersection (Corollary 6.14). Finally, the lower bound on the size of the Hecke congruence module translates under the isomorphism R ≅ T into a statement about cyclicity of a certain Galois group and the existence of certain reducible deformations (Corollary 6.16).

To give a more precise account, let c be the non-trivial automorphism of F, and let p > 3 be a prime split in the extension F/Q. Fix embeddings F ↪ \overline{Q} ↪ \overline{Q}_{p} ↪ C. Let F_{Σ} be the maximal extension of F unramified outside a finite set of places Σ and put G_{Σ} = Gal(F_{Σ}/F). Suppose F is a finite field of characteristic p and that χ_{0} : G_{Σ} → F^{×} is anticyclotomic (in general a weaker assumption of Σ-admissibility is enough - see Section 3) character ramified at the places dividing p. Suppose also that ρ_{0} : G_{Σ} → GL_{2}(F) is a continuous representation of the form

(1.1)  ρ_{0} = (\begin{matrix}1 & ∗ \\ 0 & χ_{0} \end{matrix})

and having scalar centralizer. In [BK09] we proved that under certain conditions on χ_{0} and Σ the residual representation ρ_{0} is unique up to isomorphism. We recall the relevant details in Section 3.

Following Mazur [Maz97] we study ordinary deformations of ρ_{0}. Let O be the ring of integers in a finite extension of Q_{p}. An O-deformation of ρ_{0} is a local complete Noetherian O-algebra A with residue field F and maximal ideal 𝔪_{A} together with a strict equivalence class of continuous representations ρ : G_{Σ} → GL_{2}(A) satisfying ρ_{0} = ρ mod 𝔪_{A}. An ordinary deformation is a deformation that satisfies

ρ|_{D_{𝔮}} ≅ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix})

AN R = T THEOREM 3

for 𝔮 | p, where χ_{i}|_{I_{𝔮}} = ϵ^{k_{i}} with integers k_{1} ≥ k_{2} depending on 𝔮 and ϵ is the p-adic cyclotomic character. Here D_{𝔮} and I_{𝔮} denote the decomposition group and the inertia group of 𝔮 | p.

A part of our approach rests on studying possible reducible deformations of ρ_{0} and showing that there are indeed not “too many” of those. As we have already shown in [BK09] the uniqueness of ρ_{0} itself implies that there are no non-trivial reducible infinitesimal deformations of ρ_{0}. In the current paper we go further and carefully study reducible deformations of ρ_{0} to other Artinian local rings.

To exhibit modular irreducible deformations we apply the cohomological congruences of [Ber09] and the Galois representations constructed by Taylor et al. using the strengthening of Taylor’s result in [BH07]. We also make use of a result of Urban [Urb05] who proves that ρ_{π}|_{D_{𝔮}} is ordinary at 𝔮 | p if π is ordinary at 𝔮. This together with the non-existence of a non-trivial reducible infinitesimal deformation implies the existence of an O-algebra surjection

(1.2)  R ↠ T,

where R is the universal Σ-minimal deformation ring (cf. Definition 4.8) and T is a Hecke algebra acting on cuspidal automorphic forms of GL_{2}(A_{F}) of weight 2 and fixed level.

Remark 1.1. As we remarked already in [BK09] the approach of [SW97] (where an analogous problem is studied for representations of Gal(\overline{Q}/Q)) breaks down in the imaginary quadratic case because of the non-existence of an ordinary reducible characteristic 0 deformation (cf. [BK09], Corollary 5.22). In [BK09] we assumed that the p-valuation of a Hecke L-value was small such that by the Main Conjecture there were even no reducible deformations to any Artinian rings larger than the residue field, which implied that the ideal of reducibility I_{re} of the universal deformation ring R was maximal. This simplified the proof of “R = T ”, because it follows from the results of Bellaiche-Chenevier [BC06] and Calegari [Cal06] that if R surjects onto a characteristic 0 ring (in our case T) and the ideal I_{re} is maximal, then R is a discrete valuation ring.

In this paper we exploit that in all cases there is, in fact, a surjection R/I_{re} ↠ T/J by the definitions of the Eisenstein ideal J and the ideal of reducibility I_{re}. By finding bounds on the orders of both sides we show that this map must be an isomorphism. Indeed, if we write n for the ϖ-valuation^{1} of a normalised Hecke L-value (for a Hecke character that gives rise to χ_{0} in (1.1)), then the order of the congruence module T/J is bounded from below by #O/ϖ^{n} by the full result of [Ber09]. On the deformation side we first bound the dimension of R in Section 5.1 by using the filtration of adρ_{0} by 1-dimensional pieces, similar to a calculation in [SW97]. Strengthening an argument of [BK09] and using a result of Urban we show in Section 5.2 that the Main Conjecture of Iwasawa Theory implies that there are no reducible ordinary deformations to O/ϖ^{m} for m > n. Together with the non-existence of non-trivial reducible infinitesimal deformations this implies that R/I_{re} = O/ϖ^{m} for m ≤ n (see Proposition 6.5). We therefore have that the surjection R/I_{re} ↠ T/J is an isomorphism, from which we deduce (using our commutative algebra criterion - Proposition 6.9) that R ≅ T in Theorem 6.10. using the principality of I_{re}.

1Here ϖ denotes a uniformizer of O.

Remark 1.2. As suggested to us by Frank Calegari our commutative algebra criterion allows one to replace the condition p∥B_{2,ω^{k−2}} from [Cal06] Theorem 2.3 by p | B_{2,ω^{k−2}}, which proves an R = T result for certain residually reducible representations of G_{Q}. More precisely, if T denotes the (classical) cuspidal Hecke algebra of weight 2 and level Γ_{1}(p) generated by T_{ℓ} for ℓ ≠ p, write J the maximal ideal of T containing T_{ℓ} − 1 − ℓ^{k−1} for all ℓ ≠ p. Under the assumption that the χ^{1−k}-eigenspace (where χ : G_{Q} → Z_{p}^{×} denotes the p-adic cyclotomic character) of the class group of Q(ζ_{p}) is cyclic, Calegari considers deformations of the unique non-split representation

\overline{ρ} = (\begin{matrix}1 & ∗ \\ 0 & χ^{k−1} \end{matrix}) mod p

with \overline{ρ} unramified over Q(ζ_{p}). He requires that the deformations ρ be unramified outside p, of determinant χω^{k−2}, where ω is the Teichmüller lift of χ mod p, and such that the restriction of ρ to Gal(\overline{Q}_{p}/L) arises as a generic fiber of a finite flat group scheme over the ring of integers of L. Here L stands for the completion at p of the maximal real subfield of Q(ζ_{p}). If one denotes by R the corresponding universal deformation ring, the arguments of our Sections 5.2 and 6.1 can be adapted, using [Cal06] Lemmas 4.2 and 4.5 and the Main conjecture proven by Mazur and Wiles [MW84], to prove the cyclicity and an upper bound for R/I_{re} in terms of the pvaluation of the relevant Bernoulli number. The lower bound by the same number on the size of the quotient of T_{J} by the Eisenstein ideal generated by T_{ℓ}−1−ℓ^{k−1} for all ℓ ≠ p is stated, for example, as Proposition 5.1 in [SW97]. By the commutative algebra criterion (Proposition 6.9) we can therefore conclude R = T_{J}.

The authors would like to thank Frank Calegari, Gaëtan Chenevier, Haruzo Hida and Ravi Ramakrishna for helpful discussions and comments.

2. Notation and terminology

2.1. Galois groups. Let F be an imaginary quadratic extension of Q of discriminant d_{F} ≠ 3, 4 and p > 3 a rational prime which splits in F. We fix once and for all a prime 𝔭 of F lying over p and write \overline{𝔭} for the prime of F such that (p) = pO_{F} = 𝔭\overline{𝔭}. Let Cl_{F} denote the class group of F. We assume that p ∤ # Cl_{F} and that any prime q | d_{F} satisfies q ≢ ±1 (mod p).

For a field K write G_{K} for the Galois group Gal(\overline{K}/K). If K is a finite extension of Q_{ℓ} for some rational prime ℓ, we write O_{K} (respectively ϖ_{K}, and F_{K}) for the ring of integers of K (respectively for a uniformizer of K, and O_{K}/ϖ_{K}O_{K}).

If K ⊃ F is a number field, O_{K} will denote its ring of integers. If 𝔮 is a place of K, we write K_{𝔮} for the completion of K with respect to the absolute value | · |_{𝔮} determined by 𝔮 and set O_{K,𝔮} = O_{K_{𝔮}} (if 𝔮 is archimedean, we set O_{K,𝔮} = K_{𝔮}). We also write ϖ_{𝔮} for a uniformizer of K_{𝔮}, 𝔓_{v} for the maximal ideal of O_{K,𝔮}, and k_{𝔮} for its residue field.

Fix once and for all compatible embeddings i_{𝔮} : \overline{F} ↪ \overline{F}_{𝔮} and \overline{F}_{𝔮} ↪ C, for every prime 𝔮 of F, so we will often regard elements of \overline{F}_{𝔮} as complex numbers without explicitly mentioning it. If w is a place of K ⊂ \overline{F}, it determines a place v of F, and we always regard K_{w} as a subfield of \overline{F}_{v} as determined by the embedding i_{v}. This also allows us to identify G_{K_{w}} with the decomposition group D\overline{v} ⊂ G_{K} of a place \overline{v} of \overline{F}. We will denote that decomposition group by D_{𝔮}. Abusing notation somewhat we will denote the image of D_{𝔮} in any quotient of G_{K} also by D_{𝔮}. We write I_{𝔮} ⊂ D_{𝔮} for the inertia group.

Let Σ be a finite set of places of K. Then K_{Σ} will denote the maximal Galois extension of K unramified outside the primes in Σ. We also write G_{Σ} for G_{F_{Σ}}. Moreover, for a positive integer n, denote by μ_{n} the group of nth roots of unity.

2.2. Hecke characters. For a number field K, denote by A_{K} the ring of adeles of K and set A = A_{Q}. By a Hecke character of K we mean a continuous homomorphism

λ : K^{×} \ A_{K}^{×} → C^{×}.

For a place 𝔮 of K write λ^{(𝔮)} for the restriction of λ to K_{𝔮} and λ^{(∞)} for the restriction of λ to ∏_{𝔮|∞} K_{𝔮}. The latter will be called the infinity type of λ. We also usually write λ(ϖ_{𝔮}) to mean λ^{(𝔮)}(ϖ_{𝔮}). Given λ there exists a unique ideal 𝔣_{λ} of K largest with respect to the following property: λ^{(𝔮)}(x) = 1 for every finite place 𝔮 of K and x ∈ O_{K,𝔮}^{×} such that x − 1 ∈ 𝔣_{λ}O_{K,𝔮}. The ideal 𝔣_{λ} is called the conductor of λ. If K = F, there is only one archimedean place, which we will simply denote by ∞. For a Hecke character λ of F, one has λ^{(∞)}(z) = z^{m}\overline{z}^{n} with m, n ∈ R. If m, n ∈ Z, we say that λ is of type (A_{0}). We always assume that our Hecke characters are of type (A_{0}). Write L(s, λ) for the Hecke L-function of λ.

Let λ be a Hecke character of infinity type z^{a} (\frac{z}{\overline{z}} )^{b} with conductor prime to p. Assume a, b ∈ Z and a > 0 and b ≥ 0. Put

L^{alg}(0, λ) := Ω^{−a−2b} (\frac{2π}{\sqrt{d_{F}}} )^{b} Γ(a + b) · L(0, λ),

where Ω is a complex period. In most cases, this normalization is integral, i.e., lies in the integer ring of a finite extension of F_{𝔭}. See [Ber08] Theorem 3 for the exact statement. Put

L^{int}(0, λ) = \begin{cases}L^{alg}(0, λ) & if val_{p}(L^{alg}(0, λ)) ≥ 0 \\ 1 & otherwise. \end{cases}

For z ∈ C we write \overline{z} for the complex conjugate of z. The action of complex conjugation extends to an automorphism of A_{F}^{×} and we will write \overline{x} for the image of x ∈ A_{F}^{×} under that automorphism.

For a Hecke character λ of F, we denote by λ^{c} the Hecke character of F defined by λ^{c}(x) = λ(\overline{x}).

2.3. Galois representations. For a field K and a topological ring R, by a Galois representation we mean a continuous homomorphism ρ : G_{K} → GL_{n}(R). If n = 1 we usually refer to ρ as a Galois character. We write K(ρ) for the fixed field of ker ρ and call it the splitting field of ρ. If ρ is a Galois character and M is an R-module, we denote by M(ρ) the R[G_{K}]-module M with the G_{K}-action given by ρ. If K is a number field and 𝔮 is a finite prime of K with inertia group I_{𝔮} we say that ρ is unramified at 𝔮 if ρ|_{I_{𝔮}} = 1. If Σ is a finite set of places of K such that ρ is unramified at all v ∉ Σ, then ρ can be regarded as a representation of G_{K_{Σ}}.

Let E be a finite extension of Q_{p}. Every Galois representation ρ : G_{K} → GL_{n}(E) can be conjugated (by an element M ∈ GL_{n}(E)) to a representation ρ_{M} : G_{K} → GL_{n}(O_{E}). We denote by \overline{ρ}_{M} : G_{K} → GL_{n}(F_{E}) its reduction modulo ϖ_{E}O_{E}. It is sometimes called a residual representation of ρ. The isomorphism class of its semisimplification \overline{ρ}_{M}^{ss} is independent of the choice of M and we simply write \overline{ρ}^{ss}.

Let ϵ : G_{F} → Z_{p}^{×} denote the p-adic cyclotomic character. For any subgroup G ⊂ G_{F} we will also write ϵ for ϵ|_{G}. Our convention is that the Hodge-Tate weight of ϵ at 𝔭 is 1.

Let λ be a Hecke character of F of type (A_{0}) and Σ_{λ} = {𝔮 | p𝔣_{λ}}. We define (following Weil) a 𝔭-adic Galois character

λ_{𝔭} : G_{Σ_{λ}} → \overline{F} _{𝔭}^{×}

associated to λ by the following rule: For a finite place 𝔮 ∤ p𝔣_{λ} of F, put λ_{𝔭}(Frob_{𝔮}) = i_{𝔭}(i_{∞}^{−1}(λ(ϖ_{𝔮}))) where Frob_{𝔮} denotes the arithmetic Frobenius at 𝔮. It takes values in the integer ring of a finite extension of F_{𝔭}.

Definition 2.1. For a topological ring R we call a Galois representation ρ : G_{Σ} → GL_{2}(R) ordinary if

ρ|_{D_{𝔮}} ≅ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix})

for 𝔮 | p, where χ_{i}|_{I_{𝔮}} = ϵ^{k_{i}} with integers k_{1} ≥ k_{2} depending on 𝔮.

2.4. Automorphic representations of A_{F} and their Galois representations. Set G = Res_{F/Q} GL_{2}. For K_{f} an open compact subgroup of G(A_{f}), denote by S_{2}(K_{f}) the space of cuspidal automorphic forms of G(A) of weight 2, right-invariant under K_{f} (for more details see Section 3.1 of [Urb95]). For ψ a finite order Hecke character write S_{2}(K_{f}, ψ) for the forms with central character ψ. This is isomorphic as a G(A_{f})-module to ⨁ π_{f}^{K_{f}} for automorphic representations π of a certain infinity type (see Theorem 2.2 below) with central character ψ. Here π_{f} denotes the restriction of π to GL_{2}(A_{f}) and π_{f}^{K_{f}} stands for the K_{f}-invariants.

For g ∈ G(A_{f}) we have the usual Hecke action of [K_{f}gK_{f}] on S_{2}(K_{f}) and S_{2}(K_{f}, ψ). For primes 𝔮 such that the vth component of K_{f} is GL_{2}(O_{F,v}) we

define T_{𝔮} = [K_{f} [\begin{matrix}ϖ_{𝔮} \\ & 1 \end{matrix}] K_{f}].

Combining the work of Taylor, Harris, and Soudry with results of Friedberg-Hoffstein and Laumon/Weissauer, one can show the following (see [BH07] for general case of cuspforms of weight k):

Theorem 2.2 ([BH07] Theorem 1.1). Given a cuspidal automorphic representation π of GL_{2}(A_{F}) with π_{∞} isomorphic to the principal series representation corresponding to

\left[{t_{1} ∗ \atop t_{2}}\right] ↦ (\frac{t_{1}}{|t_{1}|}) (\frac{|t_{2}|}{t_{2}})

and cyclotomic central character ψ (i.e., ψ^{c} = ψ), let Σ_{π} denote the set consisting of the places of F lying above p, the primes where π or π^{c} is ramified, and the primes ramified in F/Q.

Then there exists a finite extension E of F_{𝔭} and a Galois representation

ρ_{π} : G_{Σ_{π}} → GL_{2}(E)

such that if 𝔮 ∉ Σ_{π}, then ρ_{π} is unramified at 𝔮 and the characteristic polynomial of ρ_{π}(Frob_{𝔮}) is x^{2} − a_{𝔮}(π)x + ψ(ϖ_{𝔮})(#k_{𝔮}), where a_{𝔮}(π) is the Hecke eigenvalue corresponding to T_{𝔮}. Moreover, ρ_{π} is absolutely irreducible.

Urban studied in [Urb98] the case of ordinary automorphic representations π, and together with results in [Urb05] on the Galois representations attached to ordinary Siegel modular forms showed:

Theorem 2.3 (Corollary 2 of [Urb05]). Let 𝔮 be a prime of F lying over p. If π is unramified at 𝔮 and ordinary at 𝔮, i.e., |a_{𝔮}(π)|_{𝔮} = 1, then the Galois representation ρ_{π} is ordinary at 𝔮. Moreover

ρ_{π}|_{D_{𝔮}} ≅ \left[{Ψ_{1} ∗ \atop Ψ_{2}}\right] ,

where Ψ_{2}|_{I_{𝔮}} = 1 and Ψ_{1}|_{I_{𝔮}} = det ρ_{π}|_{I_{𝔮}} = ϵ.

We conjecture that the assumption on the central character, inherent to the method of proof of Theorem 2.2, is not necessary (see also [CD06] Conjecture 3.2).

Conjecture 2.4. Let π be a cuspidal automorphic representation as in Theorem 2.2 but without the assumption on the central character. Then the representation ρ_{π} exists as above and the conclusion of Theorem 2.3 remains valid for it.

Definition 2.5. Let E be a finite extension of F_{𝔭} and ρ : G_{Σ} → GL_{2}(E) a Galois representation for a finite set of places Σ. We say that ρ is modular if there exists a cuspidal automorphic representation π as in Theorem 2.2, such that ρ ≅ ρ_{π} (possibly after enlarging E).

From now on we fix a finite extension E of F_{𝔭} which we assume to be sufficiently large (see Section 4.2 and Remark 4.5, where this condition is made more precise). To simplify notation we put O := O_{E}, F = F_{E} and ϖ = ϖ_{E}.

3. Uniqueness of a residual Galois representation

Let p = 𝔭\overline{𝔭} as before be a split prime. Let Σ be a finite set of finite primes of F containing 𝔭. In this paper we will be studying deformations of a certain class of residual Galois representations which we now describe.

From now on let

ρ_{0} : G_{Σ} → GL_{2}(F)

be a Galois representation satisfying both of the following conditions:

(Red): ρ_{0} = \left[{1 ∗ \atop χ_{0}}\right] for a Galois character χ_{0} : G_{Σ} → F^{×};

(Sc): ρ_{0} has scalar centralizer.

Definition 3.1. We will say that a Galois character χ_{0} : G_{Σ} → F^{×} is Σ-admissible if

(1) χ_{0} is ramified at 𝔭 and at \overline{𝔭};

(2) if 𝔮 ∈ Σ, then either χ_{0} is ramified at 𝔮 or χ_{0}(Frob_{𝔮}) ≠ (#k_{v})^{±1} (as elements of F);

(3) dim_{F}(H^{1}(G_{Σ}, F(χ_{0}))) = dim_{F}(H^{1}(G_{Σ}, F(χ_{0}^{−1}))) = 1.

Remark 3.2. It follows from the global Euler characteristic formula ([Mil06] Theorem 5.1), that the cohomology spaces in condition (3) of Definition 3.1 are at least one-dimensional.

An important consequence of Σ-admissibility of χ_{0} is the uniqueness of ρ_{0} up to isomorphism. We record this fact as a proposition.

Proposition 3.3. Suppose ρ′ : G_{Σ} → GL_{2}(F) is a Galois representation satisfying condition (Red) and (Sc) for a Σ-admissible character χ_{0}. Then ρ′ ≅ ρ_{0}.

Proof. As ρ_{0} and ρ′ correspond to non-zero elements in Ext_{\overline{F}_{p}[G_{Σ}]}(1, χ_{0}) ≅ H^{1}(G_{Σ}, \overline{F}_{p}(χ_{0}^{−1})), the proposition follows from condition (3) in Definition 3.1. □

We will be interested in studying deformations of a representation ρ_{0} satisfying conditions (Red) and (Sc) for a Σ-admissible character χ_{0}.

Let us shortly explain the significance of these assumptions to the deformation problem we study in the coming sections. Condition (Sc) is necessary to ensure that the deformation functor studied in the following sections is representable. The assumption that χ_{0} is Σ-admissible is crucial for our method. On the one hand Proposition 3.3 guarantees that modular Galois representations which we construct (and which are defined only up to semisimplification) are deformations of ρ_{0}. On the other hand, Σ-admissibility of χ_{0} is essential for ensuring that the ideal of reducibility of the universal deformation is principal, which is crucial for our commutative algebra criterion (Proposition 6.9) that we use to prove modularity of Σ-minimal deformations of ρ_{0} (Theorem 6.10).

We will now formulate some conditions (see Theorem 3.5 below) that ensure Σadmissibility of a Galois character (for details see [BK09], Section 3). At the same time we want to emphasize that there is no reason why every Σ-admissible character should satisfy all of these conditions, however, it is in the context of Theorem 3.5 that we were able to construct examples of Σ-admissible characters (see Remark 3.7 below). Also, it is only under condition (3) of Theorem 3.5 that one can relate (by twisting, see [Ber09] Lemma 8) to cusp forms with cyclotomic character for which one knows how to attach Galois representations to automorphic forms on GL_{2}(A_{F}) by Theorem 2.2. Conjecturally, however, this condition should be unnecessary (see Conjecture 2.4).

Definition 3.4. Let R be a commutative ring, J ⊂ R an ideal, M a free R-module and N a submodule of M. We will say that N is saturated with respect to J if

N = {m ∈ M | mJ ⊂ N}.

Let χ_{0} : G_{Σ} → F^{×} be a Galois character. Let S_{p} be the set of primes of F (χ_{0}) lying over p. Write M_{χ_{0}} for ∏_{𝔮∈S_{p}}(1 + 𝔓_{v}) and T_{χ_{0}} for its torsion submodule. The quotient M_{χ_{0}}/T_{χ_{0}} is a free Z_{p}-module of finite rank. Let \overline{E}_{χ_{0}} be the closure in M_{χ_{0}}/T_{χ_{0}} of the image of E_{χ_{0}}, the group of units of the ring of integers of F (χ_{0}) which are congruent to 1 modulo every prime in S_{p}.

In [BK09] the authors proved the following theorem (see Theorem 3.5 and the proof of Corollary 3.6 in [loc. cit.]).

Theorem 3.5. Let χ_{0} : G_{Σ} → F^{×} be a Galois character. Assume that χ_{0} satisfies all of the following conditions:

(1) χ_{0} is ramified at 𝔭;

(2) if 𝔮 ∈ Σ, then either χ_{0} is ramified at 𝔮 or χ_{0}(Frob_{𝔮}) ≠ (#k_{𝔮})^{±1} (as elements of F);

(3) χ_{0} is anticyclotomic, i.e., χ_{0}(cσc) = χ_{0}(σ)^{−1} for every σ ∈ G_{Σ} and c the generator of Gal(F/Q);

(4) the Z_{p}-submodule \overline{E}_{χ_{0}} ⊂ M_{χ_{0}}/T_{χ_{0}} is saturated with respect to the ideal pZ_{p},

(5) The χ_{0}^{−1}-eigenspace of the p-part of Cl_{F (χ_{0})} is trivial.

Then χ_{0} is Σ-admissible.

Remark 3.6. Note that conditions (1) and (3) of Theorem 3.5 imply that χ_{0} is also ramified at \overline{𝔭}. Observe that χ_{0} satisfies conditions (1)-(5) of Theorem 3.5 if and only if so does χ_{0}^{−1}. Indeed, conditions (1)-(4) in Theorem 3.5 are invariant under taking the inverse. Moreover, since χ_{0} is anticyclotomic, the extension F (χ_{0})/Q is Galois, hence the χ_{0}^{−1}-eigenspace of the p-part of Cl_{F (χ_{0})} vanishes if and only if the χ_{0}-eigenspace does.

Remark 3.7. Condition (4) in Theorem 3.5 implies that every ϵ ∈ \overline{E}_{χ_{0}} which is a pth power of an element of M_{χ_{0}}/T_{χ_{0}} must be a pth power of an element of \overline{E}_{χ_{0}}. In particular the map

\overline{E}_{χ_{0}} ⊗_{Z_{p}} \overline{F}_{p} → (M_{χ_{0}}/T_{χ_{0}}) ⊗_{Z_{p}} \overline{F}_{p}

is injective. One way to check condition (4) in practice is to compute the p-part C of the class group of F (χ_{0})(μ_{p}) as a Gal(F (χ_{0})(μ_{p})/F (χ_{0}))-module. In particular, if ω denotes the character of Gal(F (χ_{0})(μ_{p})/F (χ_{0})) giving the action on μ_{p}, then Kummer theory implies that Condition (4) is satisfied if the ω-part of C is trivial. In [BK09], Section 6, the authors exhibited an example of a Galois representation whose mod ϖ-reduction satisfied conditions (Red) and (Sc) for a Galois character satisfying conditions (1)-(5) of Theorem 3.5.

4. Deformations of ρ_{0}

In this section we will fix a residual representation ρ_{0} arising as the reduction of modular Galois representations ρ : G_{Σ} → GL_{2}(O) and define a deformation problem.

4.1. Eisenstein congruences. Let ϕ_{1}, ϕ_{2} be two Hecke characters with infinity types ϕ_{1}^{(∞)}(z) = z and ϕ_{2}^{(∞)}(z) = z^{−1}. Put γ = ϕ_{1}ϕ_{2}.

Denote by 𝔖 the finite set of places where both ϕ_{i} are ramified, but ϕ is unramified. Write 𝔐_{i} for the conductor of ϕ_{i}. For an ideal 𝔑 in O_{F} and a finite place 𝔮 of F put 𝔑_{𝔮} = 𝔑O_{F,𝔮}. We define

K^{1}(𝔑_{𝔮}) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(O_{F,𝔮}) | a − 1, c ≡ 0 mod 𝔑_{𝔮}\} ,

and

U^{1}(𝔑_{𝔮}) = {k ∈ GL_{2}(O_{F,𝔮}) | det(k) ≡ 1 mod 𝔑_{𝔮}}.

Now put

(4.1)  K_{f} := ∏_{𝔮∈𝔖} U^{1}(𝔐_{1,𝔮}) ∏_{𝔮∉𝔖} K^{1}((𝔐_{1}𝔐_{2})_{𝔮}) ⊂ G(A_{f}).

From now on, let Σ be a finite set of places of F containing

S_{ϕ} := {𝔮 | 𝔐_{1}𝔐_{1}^{c}𝔐_{2}𝔐_{2}^{c}} ∪ {𝔮 | pd_{F} }.

We denote by T(Σ) the O-subalgebra of End_{O}(S_{2}(K_{f}, γ)) generated by the Hecke operators T_{𝔮} for all places 𝔮 ∉ Σ. Following [Tay88] (p. 107) we define idempotents e_{𝔭} and e\overline{𝔭}, commuting with each other and with T(Σ) acting on S_{2}(K_{f}, γ). They are characterized by the property that any element h ∈ X := e_{𝔭}e\overline{𝔭} S_{2}(K_{f}, γ) which is an eigenvector for T_{𝔭} and T\overline{𝔭} satisfies |a_{𝔭}(h)|_{p} = |a\overline{𝔭}(h)|_{p} = 1, where a_{𝔭}(h) (resp. a\overline{𝔭}(h)) is the T_{𝔭}-eigenvalue (resp. T\overline{𝔭}-eigenvalue) corresponding to h. Let T^{ord}(Σ) denote the quotient algebra of T(Σ) obtained by restricting the Hecke operators to X.

Let J(Σ) ⊂ T(Σ) be the ideal generated by

{T_{𝔮} − ϕ_{1}(ϖ_{𝔮}) · #k_{𝔮} − ϕ_{2}(ϖ_{𝔮}) | 𝔮 ∉ Σ}.

Definition 4.1. Denote by 𝔪(Σ) a maximal ideal of T^{ord}(Σ) containing the image of J(Σ). We set T_{Σ} := T^{ord}(Σ)_{𝔪(Σ)}. Moreover, set J_{Σ} := J(Σ)T_{Σ}. We refer to J_{Σ} as the Eisenstein ideal of T_{Σ}.

Theorem 4.2 ([Ber05], Theorem 6.3, [Ber09] Theorem 14). Let ϕ be an unramified Hecke character of infinity type ϕ^{(∞)}(z) = z^{2}. There exist Hecke characters ϕ_{1}, ϕ_{2} with ϕ_{1}/ϕ_{2} = ϕ such that their conductors are divisible only by ramified primes or inert primes not congruent to ±1 mod p and such that

#(T_{Σ}/J_{Σ}) ≥ #(O/(L^{int}(0, ϕ))).

Proof. [Ber09] Theorem 14 states this inequality for the Hecke algebra T(Σ). However, the Eisenstein cohomology class used in the proof of [Ber09] Theorem 14 is ordinary because by [Ber08] Lemma 9 its T_{𝔭}-eigenvalue (resp. T\overline{𝔭}-eigenvalue) is the p-adic unit pϕ_{1}(𝔭) + ϕ_{2}(𝔭) (resp. pϕ_{1}(\overline{𝔭}) + ϕ_{2}(\overline{𝔭})). Therefore one can prove the statement for the ordinary cuspidal Hecke algebra. □

Remark 4.3. If ϕ is unramified then \overline{ϕ_{𝔭}ϵ} is anticyclotomic (see [Ber09] Lemma 1). The condition on the conductor of the auxiliary character ϕ_{1} together with our assumption on the discriminant of F therefore ensure that for χ_{0} = \overline{ϕ_{𝔭}ϵ} conditions (1)-(3) of Theorem 3.5 are automatically satisfied for Σ = S_{ϕ}.

The assumption on the ramification of ϕ can be relaxed. For example, Proposition 16 and Theorem 28 of [Ber08] and Proposition 9 and Lemma 11 of [Ber09] imply the following:

Theorem 4.4. Let ϕ_{1}, ϕ_{2} be as at the start of this section with p ∤ #(O_{F} /𝔐_{1}𝔐_{2})^{×}. Assume both 𝔐_{1} and 𝔐_{2} are coprime to (p) and the conductor of ϕ := ϕ_{1}/ϕ_{2} and divisible only by primes split in F/Q. Suppose \frac{L(0,\overline{ϕ})}{L(0,ϕ)} ∈ O. If the torsion part of H_{c}^{2}(S_{K_{f}} , Z_{p}) is trivial, where

S_{K_{f}} = G(Q)\G(A)/K_{f}U(2)C^{×}

then

#(T_{Σ}/J_{Σ}) ≥ #(O/(L^{int}(0, ϕ_{1}/ϕ_{2}))).

Remark 4.5. In fact, by replacing Z_{p} by the appropriate coefficient system, the result is true for characters ϕ_{1}, ϕ_{2} of infinity type z\overline{z}^{−m} and z^{−m−1}, respectively, for m ≥ 0. For Theorems 4.2 and 4.4, the field E needs to contain the values of the finite parts of ϕ_{1} and ϕ_{2} as well as L^{int}(0, ϕ_{1}/ϕ_{2}).

We will from now on assume that we are either in the situation of Theorem 4.2 or 4.4 and fix the characters ϕ_{1}, ϕ_{2} and ϕ = ϕ_{1}/ϕ_{2}, with corresponding conditions on the set Σ and definitions of K_{f}, T_{Σ}, and J_{Σ}. We also assume from now on that val_{p}(L^{int}(0, ϕ)) > 0, however our main result (Theorem 6.13) remains vacuously true when L^{int}(0, ϕ) is a p-adic unit (see Remark 5.10). Put χ_{0} = \overline{ϕ_{𝔭}ϵ} and assume that χ_{0} is Σ-admissible. If we are in the situation of Theorem 4.4 then suppose also that 𝔐_{1} and 𝔐_{2} are not divisible by any primes 𝔮 such that #k_{𝔮} ≡ 1 mod p. (This last assumption is only used in the proof of Theorem 4.9.)

4.2. Definition of ρ_{0}. Write

S_{2}(K_{f}, γ)_{𝔪(Σ)} = ⨁_{π∈Π_{Σ}} π_{f}^{K_{f}}

for a finite set Π_{Σ} of ordinary cuspidal automorphic representations with central character γ, such that π_{f}^{K_{f}} ≠ 0. The set Π_{Σ} is non-empty by Theorem 4.2 under our assumption that val_{p}(L^{int}(0, ϕ)) > 0.

Let π ∈ Π_{Σ}. Let ρ_{π} : G_{Σ} → GL_{2}(E) be the Galois representation attached to π by Conjecture 2.4 (This is another point where we assume that E is large enough). If we assume that ϕ^{c} = \overline{ϕ} then the condition on the central character in Theorem 2.2 can be satisfied (after possibly twisting with a finite character), see [Ber09] Lemma 8, so we do not need to refer to Conjecture 2.4 in this case.

The representation ρ_{π} is unramified at all 𝔮 ∉ S_{ϕ}, and satisfies

tr ρ_{π}(Frob_{𝔮}) = a_{𝔮}(π)

and

det ρ_{π}(Frob_{𝔮}) = γ(ϖ_{𝔮}) · #(k_{𝔮}).

By definition, T_{Σ} injects into ∏_{π∈Π_{Σ}} End_{O}(π^{K_{f}}). Since T_{𝔮} acts on π by multiplication by a_{𝔮}(π) ∈ O the Hecke algebra T_{Σ} embeds, in fact, into B = ∏_{π∈Π_{Σ}} O (in particular T_{Σ} has no Z-torsion).

Observe that (tr ρ_{π}(σ))_{π∈Π_{Σ}} ∈ T_{Σ} ⊂ B for all σ ∈ G_{Σ}. This follows from the Chebotarev Density Theorem and the continuity of ρ_{π} (note that T_{Σ} is a finite O-algebra).

Fix π ∈ Π_{Σ} for the rest of this subsection. Define ρ′_{π} := ρ_{π} ⊗ ϕ_{2,𝔭}^{−1}. Then ρ′_{π} satisfies

tr ρ′_{π}(Frob 𝔮) ≡ 1 + (ϕ_{𝔭}ϵ)(Frob 𝔮) (mod ϖ) for 𝔮 ∉ S_{ϕ},

and

det ρ′_{π} = γ · ϕ_{2,𝔭}^{−2} · ϵ = ϕ_{𝔭}ϵ.

By choosing a suitable lattice Λ one can ensure that ρ′_{π} has image inside GL_{2}(O). The Chebotarev Density Theorem and the Brauer-Nesbitt Theorem imply that

(\overline{ρ}′_{π})^{ss} ≅ 1 ⊕ \overline{ϕ}_{𝔭}\overline{ϵ}.

By Theorem 2.2 ρ′_{π} is irreducible, and by Theorem 2.3 ρ′_{π} is ordinary so a standard argument (see e.g. Proposition 2.1 in [Rib76]) shows the lattice Λ may be chosen in such a way that \overline{ρ}′_{π} is not semi-simple and

(4.2)  \overline{ρ}′_{π} = \left[{1 ∗ \atop \overline{ϕ}_{𝔭}\overline{ϵ}}\right]

and is ordinary. We put

(4.3)  ρ_{0} := \overline{ρ}′_{π}.

Note that the isomorphism class of ρ_{0} is in fact independent of the choice of π ∈ Π_{Σ} by Proposition 3.3.

Remark 4.6. The representation ρ_{0} satisfies conditions (Red) and (Sc) of Section 3. The fact that ρ′_{π} is ordinary combined with (4.2) implies that

(4.4)  \overline{ρ}′_{π}|_{D\overline{𝔭}} ≅ [\begin{matrix}1 \\ & (\overline{ϕ}_{𝔭}\overline{ϵ})|_{D\overline{𝔭}} \end{matrix}] .

Note that in fact (4.4) together with condition (2) in Definition 3.1 implies that the extension F (ρ_{0})/F (χ_{0}) is unramified away from 𝔭.

Remark 4.7. In Remark 4.6 of [BK09] we pointed out that the extension F (ρ_{0})/F (χ_{0}) under the conditions of Theorem 3.5 is totally ramified at 𝔭, so, in particular, there exists τ ∈ I_{𝔭} such that

ρ_{0}(τ) = \left[{1 1 \atop 1}\right] .

This implies that no twist of ρ_{0} by a character is invariant under c ∈ Gal(F/Q) and so no character twists of ρ_{0} arise from base change. However, we would like to withdraw our additional statement in Remark 4.6 of [BK09] (and corresponding remark in Section 4.6 of [Ber09]) that this implies that (twists of) deformations of ρ_{0} cannot arise from base change. The G_{Σ}-invariant lattice in E^{2} that we consider need not be invariant under complex conjugation. In fact, Hida proved in [Hid82] the existence of deformations of a twist of ρ_{0} for anticyclotomic characters χ_{0} starting from congruences of CM forms over Q.

4.3. Deformation problem. Recall that (p) = 𝔭\overline{𝔭} is split. Let Σ, ϕ, χ_{0} and ρ_{0} be as above. Recall that we have assumed that χ_{0} is Σ-admissible and have shown in Section 4.2 that ρ_{0} satisfies conditions (Red) and (Sc) of Section 3. Hence by Proposition 3.3, ρ_{0} is unique up to isomorphism. By (4.4) the extension F (ρ_{0})/F (χ_{0}) splits at \overline{𝔭}. In this section we study deformations of ρ_{0}.

Denote the category of local complete Noetherian O-algebras with residue field F by LCN(E). An O-deformation of ρ_{0} is a pair consisting of A ∈ LCN(E) and a strict equivalence class of continuous representations ρ : G_{Σ} → GL_{2}(A) such that ρ_{0} = ρ (mod 𝔪_{A}), where 𝔪_{A} is the maximal ideal of A. As is customary we will denote a deformation by a single member of its strict equivalence class. Note that the Hodge-Tate weights of ϕ_{𝔭}ϵ are −1 at 𝔭 and +1 at \overline{𝔭}.

Following [SW97] we make the following definition:

Definition 4.8. We say that an O-deformation ρ : G_{Σ} → GL_{2}(A) of ρ_{0} is Σminimal if ρ is ordinary,

det ρ = ϕ_{𝔭}ϵ,

and for all primes 𝔮 ∈ Σ such that #k_{𝔮} ≡ 1 (mod p) one has

ρ|_{I_{𝔮}} ≅ [\begin{matrix}1 \\ & ϕ_{𝔭}|_{I_{𝔮}} \end{matrix}] .

Note that by our assumption on the conductor of ϕ, we in fact have ϕ_{𝔭}|_{I_{𝔮}} = 1 for 𝔮 as above. Also the ordinarity condition means in this case that

ρ|_{I_{𝔭}} ≅ \left[{1 ∗ \atop ϵ^{−1}}\right]

and

ρ|_{I\overline{𝔭}} ≅ \left[{ϵ ∗ \atop 1}\right] .

Note that ρ_{0} satisfies all the conditions in Definition 4.8. Since ρ_{0} has a scalar centralizer and Σ-minimality is a deformation condition in the sense of [Maz97], there exists a universal deformation ring which we will denote by R_{Σ} ∈ LCN(E), and a universal Σ-minimal O-deformation ρ_{Σ,O} : G_{Σ} → GL_{2}(R_{Σ}) such that for every A ∈ LCN(E) there is a one-to-one correspondence between the set of O-algebra maps R_{Σ} → A (inducing identity on F) and the set of Σ-minimal deformations ρ : G_{Σ} → GL_{2}(A) of ρ_{0}.

The arguments from Section 4.2 together with the uniqueness of ρ_{0} (Proposition 3.3 - recall that we are all the time assuming that χ_{0} is Σ-admissible) can now be reinterpreted as:

Theorem 4.9. For any π ∈ Π_{Σ} there is an O-algebra homomorphism r_{π} : R_{Σ} ↠ O inducing ρ′_{π}.

Proof. The only property left to be checked is Σ-minimality. This is clear since ρ_{π} is unramified away from S_{ϕ}, and no 𝔮 ∈ S_{ϕ} satisfies #k_{𝔮} ≡ 1 (mod p) by construction (if we are in the case of Theorem 4.2) or assumption (in the case of Theorem 4.4). □

The next three propositions were proved in [BK09] (see Propositions 5.4, 5.5 and Lemma 5.6 in [loc. cit.]).

Proposition 4.10. There does not exist any non-trivial upper-triangular Σ-minimal deformation of ρ_{0} to GL_{2}(F[x]/x^{2}).

Proposition 4.11. The universal deformation ring R_{Σ} is generated as an Oalgebra by traces.

Proposition 4.12. The image of the map R_{Σ} → ∏_{π∈Π_{Σ}} O given by x ↦ (r_{π}(x))_{π} is T_{Σ}.

Corollary 4.13. There exists a surjective O-algebra homomorphism r : R_{Σ} ↠ T_{Σ}.

Proof. This is a direct consequence of Proposition 4.12. □

5. Galois cohomology calculations

In this section we bound the Krull dimension of the universal deformation ring R_{Σ} via its tangent space. We conclude that R_{Σ} is a quotient of O[[X]], which will later allow us to prove that R_{Σ} is a complete intersection (see Corollary 6.14). We also study reducible deformations of ρ_{0} to O/ϖ^{n}.

5.1. The Krull dimension of R_{Σ} is less than 3. Let ρ_{0} be as in Section 4 and let U be the representation space for ρ_{0}. Then U is a two-dimensional F-vector space with a G_{Σ}-stable filtration

0 ⊆ U_{1} ⊆ U,

where U_{1} is the 1-dimensional F-vector space on which G_{Σ} acts trivially. The quotient U_{2} = U/U_{1} is a 1-dimensional F-vector space on which G_{Σ} acts via χ_{0}. Let W = adρ_{0} = Hom_{F}(U, U) be the adjoint representation, and let

W^{Sel} = {f ∈ W : f(U) ⊆ U_{1}, f(U_{1}) = 0}.

Recall that ρ_{0} splits when restricted to I\overline{𝔭} (cf. (4.4)). Define the I\overline{𝔭}-submodule

W^{Sel,t} = {f ∈ W : f(U) ⊆ U_{2}, f(U_{2}) = 0}.

For 𝔮 ∈ Σ let

H_{𝔮} = \{ \begin{matrix}H^{1}(I_{𝔭}, W/W^{Sel}) & if 𝔮 = 𝔭, \\ H^{1}(I\overline{𝔭}, W/W^{Sel,t}) & if 𝔮 = \overline{𝔭}, \\ H^{1}(I_{𝔮}, W) & if #k_{𝔮} ≡ 1 mod p, \\ 0 & otherwise. \end{matrix}

Define the following Selmer group:

H_{Σ}^{1}(F, W) = ker(H^{1}(G_{Σ}, W) → ⨁_{𝔮∈Σ} H_{𝔮}).

Proposition 5.1.

dim_{F}H_{Σ}^{1}(F, W) ≤ 1.

Proof. We proceed in a similar manner to the proof of Proposition 3.1 in [SW97] Section 3. Let Σ_{1} ⊂ Σ comprise those primes in Σ such that #k_{𝔮} ≡ 1 mod p, together with 𝔭 and \overline{𝔭}. Let W_{1} = Hom_{F}(U_{2}, U), and let W_{2} = Hom_{F}(U_{1}, U). There is a commutative diagram of G_{Σ}-modules

\xymatrix{0 \ar[r] & W_{1} \ar[r] \ar[d] & W \ar[r] \ar[d] & W_{2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & F \ar[r] & W/W^{Sel} \ar[r] & W_{2} \ar[r] & 0}

having exact rows. Similarly, we have a commutative diagram of I\overline{𝔭}-modules

\xymatrix{0 \ar[r] & W_{1} \ar[r] \ar[d] & W \ar[r] \ar[d] & W_{2} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W_{1} \ar[r] & W/W^{Sel,t} \ar[r] & F \ar[r] & 0}

having exact rows. (The third vertical arrow is induced by a splitting of the exact sequence for W_{2} as I\overline{𝔭}-modules below.) Together these induce the following commutative diagram of cohomology groups:

\xymatrix{0 \ar[r] & H^{1}(G_{Σ}, W_{1}) \ar[r] \ar[d]^{α} & H^{1}(G_{Σ}, W) \ar[r] \ar[d]^{β} & H^{1}(G_{Σ}, W_{2}) \ar[d]^{γ} \\ 0 \ar[r] & X \ar[r] & ⨁_{𝔮∈Σ_{1}} H_{𝔮} \ar[r] & Y} ,

where

X = H^{1}(I_{𝔭}, F) ⊕ ⨁_{𝔮∈Σ_{1}\{𝔭}} H^{1}(I_{𝔮}, W_{1})

and

Y = H^{1}(I\overline{𝔭}, F) ⊕ ⨁_{𝔮∈Σ_{1}\\{\overline{𝔭}\}} H^{1}(I_{𝔮}, W_{2}).

Note that the rows in the diagram are exact. Indeed, the only thing to check is the exactness on the left, which can be verified easily by observing that the H^{0}-terms form a short exact sequence in each of the cases. Hence by the snake lemma there is an exact sequence

0 → ker(α) → ker(β) = H_{Σ}^{1}(F, W) → ker(γ).

We claim now that ker(α) = 0 so that ker(β) ↪ ker(γ). First observe that W_{1} fits into the short exact sequence

0 → W^{Sel} → W_{1} ⟶^{f} F → 0.

Since W_{1} ≅ ρ_{0} ⊗ χ_{0}^{−1}, it is clear that W_{1}^{G_{Σ}} = 0 and hence the associated long exact cohomology sequence yields the exact sequence

0 → F → H^{1}(G_{Σ}, W^{Sel}) → H^{1}(G_{Σ}, W_{1}) ⟶^{H^{1}(f)} H^{1}(G_{Σ}, F).

Since W^{Sel} ≅ F(χ_{0}^{−1}), it follows from the Σ-admissibility of χ_{0} that the second arrow is surjective. Hence H^{1}(f) is injective.

Lemma 5.2.

ker(α) ↪ ker(H^{1}(G_{Σ}, F) → ⨁_{𝔮∈Σ_{1}} H^{1}(I_{𝔮}, F))) = 0.

Proof. Consider the following diagram

\xymatrix{H^{1}(G_{Σ}, W_{1}) \ar[r]^{H^{1}(f)} \ar[d]^{α} & H^{1}(G_{Σ}, F) \ar[d]^{ϕ:=⊕_{q∈Σ_{1}} res_{q}} \\ X \ar[r]^{f′} & ⨁_{q∈Σ_{1}} H^{1}(I_{q}, F)}

where f′ is the identity on H^{1}(I_{p}, F) and is induced by f on the rest of the components. By definitions of the maps involved it is clear that the diagram is commutative. Since H^{1}(f) is injective as observed above, if c ∈ ker α, its image must lie in ker ϕ, hence the inclusion of the kernels in the statement of the lemma follows. Now consider c ∈ ker ϕ, i.e., c is unramified at all q ∈ Σ_{1}. But since F is a p-power order abelian group, we see that c is also unramified at all q ∉ Σ_{1}. Hence it is unramified everywhere. Since H^{1}(G_{Σ}, F) = Hom(G_{Σ}, F) and by our assumption p ∤ #Cl_{F} it is clear that c = 0. □

Similarly, W_{2} fits into the short exact sequence

0 → F → W_{2} → Hom(U_{1}, U_{2}) ≅ F(χ_{0}) → 0.

The associated long exact cohomology sequence yields the commutative diagram

\xymatrix{H^{1}(G_{Σ}, F) \ar@{^{(}->}[r] \ar[d]^{f_{1}} & H^{1}(G_{Σ}, W_{2}) \ar[r] \ar[d]^{γ} & H^{1}(G_{Σ}, F(χ_{0})) \ar[d]^{f_{2}} \\ ⨁_{𝔮∈Σ_{1}} H^{1}(I_{𝔮}, F) \ar@{^{(}->}[r] & Y \ar[r] & ⨁_{𝔮∈Σ_{1}\\{\overline{𝔭}\}} H^{1}(I_{𝔮}, F(χ_{0}))}

having exact rows. As above we see that ker(f_{1}) = 0 and therefore

ker(γ) ↪ ker(f_{2}) ⊂ H^{1}(G_{Σ}, F(χ_{0})).

Since the character χ_{0} is Σ-admissible we have that dim_{F} H^{1}(G_{Σ}, F(χ_{0})) = 1. This concludes the proof of Proposition 5.1. □

Proposition 5.3. We have

H_{Σ}^{1}(F, ad^{0} ρ_{0}) = Hom(𝔪_{R_{Σ}}/(𝔪_{R_{Σ}}^{2} + ϖR_{Σ}), F) = Hom_{O−alg}(R_{Σ}, F[x]/x^{2}).

Proof. See for example [dS97], Theorem 15 which together with Lemma 5.4 yields the Proposition. □

Lemma 5.4. H\overline{𝔭} is the cohomological condition corresponding to ordinarity at \overline{𝔭} of an infinitesimal deformation, i.e., a deformation to GL_{2}(F[X]/X^{2}).

Proof. Let ρ be an infinitesimal deformation of ρ_{0} with det(ρ) = Ψ = χ_{0}. Write

ρ(g)ρ_{0}(g)^{−1} = 1 + ϵc_{ρ}(g).

Then c defines a 1-cocycle with values in ad^{0}ρ_{0}. For g ∈ I\overline{𝔭}, we have for a suitable basis (with respect to which ρ_{0} splits and ρ is lowertriangular)

c_{ρ}(g) ∈ \{(\begin{matrix}0 & 0 \\ ∗ & 0 \end{matrix})\} = W^{Sel,t},

so the image of [c_{ρ}] in H\overline{𝔭} is zero, as desired. Conversely, if this condition is satisfied then ρ|_{I\overline{𝔭}} is lower-triangular , hence so is ρ|_{D\overline{𝔭}} and ρ is ordinary at \overline{𝔭}. □

Corollary 5.5. We have R_{Σ} = O[[X]]/I for an ideal I.

Proof. This is an application of Nakayama, see e.g. [dS97] Theorem 15 or [Til96], Lemma 5.1. Note that we have H_{Σ}^{1}(F, ad ρ_{0}) = H_{Σ}^{1}(F, ad^{0} ρ_{0}) since p ∤ # Cl_{F} by our assumption. □

5.2. No reducible deformation beyond p-valuation of L-value. Set Ψ := ϕ_{𝔭}ϵ and write Ψ_{r} for Ψ mod ϖ^{r} with r > 0. Note that Gal(F (Ψ)/F) = Γ × Δ with Γ ≅ Z_{p} and Δ a finite abelian group. Assume p ∤ #Δ. Set χ̃_{0} := Ψ|_{Δ}.

For a finite abelian extension K of F write M(K) for the maximal abelian prop-extension of K unramified away from the primes lying over 𝔭. For a character φ : Gal(K/F) → O^{×} write M(K)_{φ} for the maximal abelian pro-p-extension of K unramified away from the primes lying over 𝔭 such that Gal(K/F) acts on Gal(M(K)_{φ}/K) via φ^{−1}.

Lemma 5.6. Any Σ-minimal uppertriangular deformation ρ_{r} of ρ_{0} to O/ϖ^{r} must have the form

ρ_{r} = \left[{1 ∗ \atop Ψ_{r}}\right] .

Proof. We will prove this by induction on r. It is true for r = 1, so assume this holds for r = k. So, we can write

ρ_{k+1} = \left[{1 + αϖ^{k} b \atop d}\right] ,

where α : G_{Σ} → F is a group homomorphism (the group operation on F being addition). Arguing as in the proof of Proposition 5.4 of [BK09] we see that α can only be ramified at 𝔭 and \overline{𝔭}. Ordinarity at 𝔭 (respectively at \overline{𝔭}) forces ρ_{k+1}|_{I_{𝔭}} (respectively ρ_{k+1}|_{I\overline{𝔭}}) to have a free O/ϖ^{k+1}-submodule (respectively quotient) on which I_{𝔭} (respectively I\overline{𝔭}) acts trivially. This together with the fact that χ_{0} is ramified at 𝔭 and at \overline{𝔭} easily imply that α|_{I_{𝔭}} = α|_{I\overline{𝔭}} = 0. Finally, using the assumption that p ∤ # Cl_{F} we conclude that α = 0. Since det ρ_{k+1} = Ψ_{k+1}, we get d = Ψ_{k+1}. □

Write e for the ramification index of O over Z_{p}. Note that the exponent of O/ϖ^{r} is p^{⌈r/e⌉}, where for a real number α, ⌈α⌉ denotes the smallest integer n ≥ α. By a slight abuse of terminology we will say that a deformation ρ of ρ_{0} is upper-triangular if there exists a member of the strict equivalence of ρ that is upper-triangular.

Lemma 5.7. Suppose there exists a Σ-minimal uppertriangular deformation ρ_{r} of ρ_{0} to GL_{2}(O/ϖ^{r}). Then there exists a surjective map of groups

Gal(M(F (Ψ_{r}))_{Ψ_{r}}/F (Ψ_{r})) ↠ Z/p^{⌈r/e⌉}.

Proof. Consider the b-entry of ρ_{r} as a function b : G_{Σ} → O/ϖ^{r}. Suppose that the restriction b|_{Gal(F (ρ_{r})/F (Ψ_{r}))} ∈ ϖO/ϖ^{r}O. This means that the mod ϖ reduction of b is zero after we split Ψ_{r}, so that F (ρ_{0}) ⊂ F (Ψ_{r}), but this is impossible as F (ρ_{0})/F is non-abelian. So, we get that b|_{Gal(F (ρ_{r})/F (Ψ_{r}))} ∉ ϖO/ϖ^{r}O. Hence Gal(F (ρ_{r})/F (Ψ_{r})) has a cyclic quotient of order p^{⌈r/e⌉}. Since F (ρ_{r}) ⊂ M(F (Ψ_{r})_{Ψ_{r}}), we are done. □

Write L^{int}(0, ϕ) = uϖ^{n}, where u is a unit. One has val_{p}(ϖ) = 1/e. We also have for any x ∈ O, that val_{p}(x) = \frac{1}{e} val_{ϖ}(x).

Remark 5.8. Note that we have the following sequence of equalities:

(5.1) #O/L^{int}(0, ϕ) = #O/ϖ^{n} = p^{\frac{n[O:Zp]}{e}} = p^{n[F:F_{p}]} =

= p^{val_{ϖ}(L^{int}(0,ϕ))[F:F_{p}]} = p^{val_{p}(L^{int}(0,ϕ))[O:Z_{p}]}.

Proposition 5.9. Let n be as above. There is no upper-triangular Σ-minimal deformation ρ_{n+1} : G_{Σ} → GL_{2}(O/ϖ^{n+1}).

Proof. Suppose ρ_{n+1} exists. By Lemma 5.7 there exists an element

x ∈ Gal(M(F (Ψ_{n+1}))_{Ψ_{n+1}}/F (Ψ_{n+1}))

of order p^{⌈(n+1)/e⌉}. Note that since p ∤ #Δ, the character χ̃_{0} is the Teichmüller lift of a character with values in F^{×}. Let F_{0} be the smallest subfield F_{p} ⊂ F_{0} ⊂ F such that the image of (the reduction of) χ̃_{0} is contained in F_{0}^{×}. Let O_{0} be the corresponding extension of Z_{p}, i.e., Z_{p} ⊂ O_{0} ⊂ O. Note that we can assume without loss of generality that O_{0} is unramified over Z_{p}. By Lemma 5.6, ρ_{n+1} : G_{Σ} → GL_{2}(O/ϖ^{n+1}) has the form

ρ_{n+1}(σ) = \left[{1 b(σ) \atop Ψ_{n+1}(σ)}\right] .

Let y ∈ Δ be a generator (note that Δ is cyclic as a group isomorphic to a subgroup of F_{0}^{×}). Then it is easy to see that χ̃_{0}^{−1}(y^{k})b(x) are all elements of order p^{⌈(n+1)/e⌉}, and they generate a subgroup H of b(Gal(F (ρ_{n+1})/F (Ψ_{n+1})) of order (p^{⌈(n+1)/e⌉})^{[F_{0}:F_{p}]} = (p^{⌈(n+1)/e⌉})^{[O_{0}:Z_{p}]} - the last equality is true because O_{0}/Z_{p} is unramified. (This subgroup still has exponent only p^{⌈(n+1)/e⌉}.)

Now we have

(H ⊗_{Z_{p}} O)^{χ̃_{0}^{−1}} = (H ⊗_{Z_{p}} O_{0} ⊗_{O_{0}} O)^{χ̃_{0}^{−1}} = (H ⊗_{Z_{p}} O_{0})^{χ̃_{0}^{−1}} ⊗_{O_{0}} O,

and hence

(5.2) #(H ⊗_{Z_{p}} O)^{χ̃_{0}^{−1}} = (#(H ⊗_{Z_{p}} O_{0})^{χ̃_{0}^{−1}})^{[O:O_{0}]} ≥ (#H)^{[O:O_{0}]} = p^{⌈(n+1)/e⌉[O:Z_{p}]} =

p^{⌈(n+1)/e⌉e[F:F_{p}]} ≥ p^{(n+1)[F:F_{p}]} = #O/ϖ^{n+1}.

But the group H can be regarded as a subgroup of Gal(F (ρ_{n+1})/F (Ψ_{n+1})). We have F (ρ_{n+1}) ⊂ M(F (Ψ_{n+1}))_{Ψ}, hence we arrive at a contradiction by Theorem 5.11 below. □

Remark 5.10. Note that the proof of Proposition 5.9 is valid also for n = 0. Hence it shows that in the case when L^{int}(0, ϕ) is a p-adic unit, ρ_{0} as in Section 3 is not Σ-minimal.

This implies that when n = 0 Theorem 6.13 remains vacuously true, since condition (5) in that theorem can never be satisfied.

Theorem 5.11. One has

#(Gal(M(F (Ψ_{n+1}))_{Ψ}/F (Ψ_{n+1})) ⊗ O)^{χ̃_{0}^{−1}} ≤ #O/L^{int}(0, ϕ) = #(O/ϖ^{n}).

Proof. One estimates the order of the left hand side by relating it to the Selmer group of a Hecke character. Using standard methods one can bound the order of the latter from above by the L-value applying the main conjecture proven by Rubin. For details see Lemma 5.14 and 5.15 of [BK09], where this is done for n = 1. The proofs of both of these lemmas generalize easily to arbitrary n. □

6. R_{Σ} = T_{Σ}

In this section we will prove the main result which asserts that the surjection in Corollary 4.13 is an isomorphism. As a consequence we obtain a result on modularity of Galois representations and deduce some properties of the Hecke algebra and cyclicity of a certain Galois group. As in Section 5.2 in what follows we assume that p ∤ #Δ. As before, throughout this section we assume that χ_{0} is Σ-admissible.

6.1. The ideal of reducibility. We briefly recall some general facts about Eisenstein representations from Section 3 of [Cal06] and Section 2 of [BC06]: Let (R, 𝔪_{R}, F) be a local p-adically complete ring. Let G be a topological group and consider a continuous representation ρ : G → GL_{2}(R) such that tr (ρ) mod 𝔪_{R} is the sum of two distinct characters τ_{i} : G → F^{×}, i = 1, 2. Moreover, assume that

dim_{F} Ext_{cts,F[G]}^{1}(τ_{1}, τ_{2}) = dim_{F} Ext_{cts,F[G]}^{1}(τ_{2}, τ_{1}) = 1.

Definition 6.1. The ideal of reducibility of R is the smallest ideal I of R such that tr (ρ) mod I is the sum of two characters. We will denote the ideal of reducibility of R_{Σ} by I_{re}.

Proposition 6.2. The ideal I_{re} is principal.

Proof. See [Cal06], Proof of Lemma 3.4. This uses in a crucial way condition (3) of Σ-admissibility of χ_{0} (see Definition 3.1). □

Theorem 6.3 (Urban). Let (R, 𝔪_{R}, F) be a local Artinian ring. Let ρ_{1}, ρ_{2}, and ρ be three representations of a topological group G with coefficients in R (with ρ having image in GL_{m}(R)). Assume the following are true:

• ρ and ρ_{1} ⊕ ρ_{2} have the same characteristic polynomials;

• The mod 𝔪_{R}-reductions \overline{ρ}_{1} and \overline{ρ}_{2} of ρ_{1} and ρ_{2} respectively are absolutely irreducible and non-isomorphic;

• The mod 𝔪_{R}-reduction \overline{ρ} of ρ is indecomposable and the subrepresentation

of \overline{ρ} is isomorphic to \overline{ρ}_{1}. Then there exists g ∈ GL_{m}(R) such that

ρ(h) = g \left[{ρ_{1}(h) ∗ \atop ρ_{2}(h)}\right] g^{−1}

for all h ∈ G.

Proof. This is Theorem 1 in [Urb99]. □

Corollary 6.4. Let R ∈ LCN(E) and suppose ρ : G_{Σ} → GL_{2}(R) is a Σ-minimal deformation of ρ_{0}. Let I ⊂ R be an ideal such that R/I ∈ LCN(E) and is an Artin ring. Then I contains the ideal of reducibility of R if and only if ρ mod I is an upper-triangular deformation of ρ_{0} to GL_{2}(R/I).

Proof. If ρ mod I is isomorphic to an upper-triangular deformation of ρ_{0} to GL_{2}(R/I), then clearly tr ρ mod I is a sum of two characters (which are distinct, since they must reduce to χ_{0} and 1 modulo 𝔪_{R}), so I contains the ideal of reducibility. The converse is an easy consequence of Theorem 6.3. □

Proposition 6.5. Assume n = val_{ϖ}(L^{int}(0, ϕ)) > 0. Then one has R_{Σ}/I_{re} = O/ϖ^{r} where 0 < r ≤ n.

Proof. Write S for R_{Σ}/I_{re}. Then S is a local complete ring. Moreover, by Corollary 5.5 we have that S is a quotient of O[[X]], and hence R_{Σ}/ϖR_{Σ} (and thus S/ϖS) is a quotient of F[[X]]. But F[[X]] is a dvr, so S/ϖS = F[[X]]/X^{m} for some m ∈ Z_{+} ∪ {∞}. (By F[[X]]/X^{∞} we mean F[[X]].) We will first show that m = 1. Suppose m ≠ 1. Then Hom_{O−alg}(R_{Σ}, S/ϖS) contains at least two elements - the map R_{Σ} ↠ F ↪ S/ϖS and the surjection R_{Σ} ↠ S/ϖS. These two elements give rise to two distinct elements in Hom_{O−alg}(R_{Σ}, F[X]/X^{2}), the trivial one and the surjection R_{Σ} ↠ S/ϖS ↠ F[X]/X^{2}. By the definition of R_{Σ} there is a one-to-one correspondence between the deformations to F[X]/X^{2} and elements of Hom_{O−alg}(R_{Σ}, F[X]/X^{2}). The trivial element corresponds to the trivial deformation to F[X]/X^{2}, i.e., with image contained in GL_{2}(F), which is clearly upper-triangular. However, the deformation corresponding to the surjection must also be upper-triangular by Corollary 6.4 since ker(R_{Σ} ↠ S/ϖS ↠ F[X]/X^{2}) contains I_{re} and F[X]/X^{2} is Artinian. But we know by Proposition 4.10 that ρ_{0} does not admit any non-trivial Σ-minimal upper-triangular deformations to F[X]/X^{2}. Hence we must have m = 1.

Thus by the complete version of Nakayama’s Lemma ([Eis95], Exercise 7.2) we know that S is generated (as a O-module) by one element. So S = O/ϖ^{r} with r ∈ Z_{+} ∪ {∞}. Finally we must have 0 < r ≤ n, r ≠ ∞ since by Corollary 6.4 if r > n or r = ∞, then there would be an upper-triangular Σ-minimal deformation of ρ_{0} to O/ϖ^{n+1}, which is impossible by Proposition 5.9. □

Remark 6.6. Note that the fact that S is a quotient of O[[X]] is actually not necessary for the proof of Proposition 6.5. Indeed, it suffices to know that S (and hence S/ϖS) is topologically finitely generated as an O-algebra. It is an easy fact that then S/ϖS = F if and only if S/ϖS has no quotient isomorphic to F[X]/X^{2}.

Corollary 6.7. Let n = val_{ϖ}(L^{int}(0, ϕ)) > 0. Suppose there exists π ∈ Π_{Σ} such that tr ρ′_{π}(Frob_{𝔮}) = 1 + (ϕ_{𝔭}ϵ)(Frob_{𝔮}) mod ϖ^{n} for all 𝔮 ∉ Σ. Write r_{π,n} for the composite R_{Σ} ↠ O ↠ O/ϖ^{n} (the first arrow being r_{π}). Then I_{re} = ker r_{π,n}.

Proof. By the definition of the ideal of reducibility we have I_{re} ⊂ ker r_{π,n}. But, by Proposition 6.5, R_{Σ}/I_{re} = O/ϖ^{r} for 0 < r ≤ n, so we have O/ϖ^{r} = R_{Σ}/I_{re} ↠ R_{Σ}/ ker r_{π,n} = O/ϖ^{n} and the corollary follows. □

6.2. A commutative algebra criterion and R=T theorem. Let R and S denote complete local Noetherian O-algebras with residue field F. Suppose that S is finitely generated as a module over Z_{p} and has no Z-torsion. Write 𝔪_{R} and 𝔪_{S} for the maximal ideals of R and S respectively.

Lemma 6.8. Let x ∈ S be such that #(S/xS) < ∞. Then multiplication by x is injective.

Proof. Consider the exact sequence

0 → ker x → S ⟶^{·x} S → S/xS → 0.

Since S is finitely generated as a module over Z_{p} we can tensor the exact sequence with Q_{p} and conclude that since S/xS ⊗Q_{p} = 0, we must also have ker x⊗Q_{p} = 0. Since ker x is a finitely generated Z_{p}-module it must be a finite group. Hence if a ∈ ker x, p^{m}a = 0 for some m ∈ Z_{≥0}. Thus a is a torsion element, which implies a = 0 by our assumption on S. □

Proposition 6.9. Suppose there exists a surjective O-algebra map ϕ : R ↠ S inducing identity on the residue fields and an element π ∈ R such that the bottom map in the following commutative diagram

\xymatrix{R \ar[r]^{ϕ} \ar[d] & S \ar[d] \\ R/πR \ar[r] & S/ϕ(π)S}

is an isomorphism. If R/πR = O/ϖ^{r} for some positive integer r, then ϕ is an isomorphism.

Proof. Write x = ϕ(π). We have ϕ(π^{k}) = x^{k}. First, we are going to show that R/π^{k}R = S/x^{k}S for every positive integer k. Indeed, the map R/π^{k}R → S/ϕ(π^{k})S is surjective because R → S is. So, it remains to prove injectivity. The map π^{k−1}R/π^{k}R → π^{k}R/π^{k+1}R given by multiplication by π is clearly surjective, so we get

#(R/πR) ≥ #(πR/π^{2}R) ≥ ….

Moreover, we have the short exact sequence

0 → π^{k−1}R/π^{k}R → R/π^{k}R → R/π^{k−1}R → 0,

so #(R/π^{k}R) = #(R/π^{k−1}R)#(π^{k−1}R/π^{k}R). Combining this equation with the previous sequence of inequalities, we get by induction on k that #(R/π^{k}R) ≤ (#(R/πR))^{k}. On the other hand, the maps

S/xS ↠^{·x} xS/x^{2}S ↠^{·x} x^{2}S/x^{3}S ↠^{·x} … ↠^{·x} x^{k−1}S/x^{k}S

are all injective because multiplication by x on S is injective by Lemma 6.8. Thus we get

#(S/x^{k}S) = #(S/xS)^{k} = #(R/πR)^{k} ≥ #(R/π^{k}R).

This proves injectivity of the map R/π^{k}R ↠ S/x^{k}S.

Hence

\underleftarrow{lim}_{k} R/π^{k}R = \underleftarrow{lim}_{k} S/x^{k}S.

So, it suffices to show that R = \underleftarrow{lim}_{k} R/π^{k}R and S = \underleftarrow{lim}_{k} S/x^{k}S. The first follows

from Lemma 7.14 on page 197 in [Eis95] since for every power of 𝔪_{R}, say 𝔪_{R}^{k} there is a power π^{s} of π such that (π)^{s} ⊂ 𝔪_{R}^{k} (indeed, take s = k) and for every power of (π), say (π)^{k} there is a power of 𝔪_{R}^{s} of 𝔪_{R} such that 𝔪_{R}^{s} ⊂ (π)^{k} (indeed, since 𝔪_{R} = (π, ϖ), we get 𝔪_{R}^{r} ⊂ (π), so s = kr works). The situation for S is analogous. □

We will now state some consequences of Proposition 6.9.

Theorem 6.10. The map r : R_{Σ} ↠ T_{Σ} in Corollary 4.13 is an isomorphism.

Proof. Write ρ_{J} for the G_{Σ}-representation induced by R_{Σ} → T_{Σ}/J_{Σ}. By definition of J_{Σ} and the Chebotarev density theorem we see that tr ρ_{J} = 1 + ϕ_{𝔭}ϵ, hence r^{−1}(J_{Σ}) ⊃ I_{re} and hence J_{Σ} ⊃ r(I_{re}). By Proposition 6.2 the ideal I_{re} is principal and we fix a generator π of I_{re}. Thus the surjection

R_{Σ}/πR_{Σ} ↠ T_{Σ}/r(π)T_{Σ} ↠ T_{Σ}/J_{Σ},

must be an isomorphism since R_{Σ}/πR_{Σ} = O/ϖ^{r} with r ≤ n by Proposition 6.5 and #(T_{Σ}/J_{Σ}) ≥ #O/ϖ^{n} by Theorems 4.2 and 4.4. Here n = val_{ϖ}(L^{int}(0, ϕ)). Hence r induces the commutative diagram in Proposition 6.9 with R = R_{Σ}, S = T_{Σ} (note that T_{Σ} is Z-torsion-free and finitely generated as a module over Z_{p} since by definition it acts faithfully on a finite dimensional vector space of characteristic zero). □

Corollary 6.11. Let n = val_{ϖ}(L^{int}(0, ϕ)). Suppose there exists π ∈ Π_{Σ} such that tr ρ′_{π}(Frob_{𝔮}) = 1 + (ϕ_{𝔭}ϵ)(Frob_{𝔮}) mod ϖ^{n} for all 𝔮 ∉ Σ. Then R_{Σ} = T_{Σ} = O, i.e., R_{Σ} = T_{Σ} is a discrete valuation ring.

Proof. By Corollary 6.7 we have r_{π}^{−1}(ϖ^{n}O) = I_{re}. So, we see that r_{π} induces the commutative diagram in Proposition 6.9 with R = R_{Σ}, S = O, π a generator of I_{re}. □

Remark 6.12. Note that Corollary 6.11 implies that if n = val_{ϖ}(L^{int}(0, ϕ)) > 0 and there exists an automorphic representation π ∈ Π_{Σ} such that π has Hecke eigenvalues congruent to the Hecke eigenvalues of an Eisenstein series mod ϖ^{n}, then Π_{Σ} = {π}.

6.3. Modularity theorem. In this section we state a modularity theorem which is a consequence of the results of the previous sections. To make its statement self-contained, we explicitly include all the assumptions we have made so far.

Theorem 6.13. Let F be an imaginary quadratic field and p > 3 a rational prime which splits in F. Fix a prime 𝔭 of F over p. Assume that p ∤ # Cl_{F} and that any prime q | disc_{F} satisfies q ≢ ±1 (mod p). Let ϕ_{1}, ϕ_{2} be Hecke characters of F with split conductors and of infinity type z and z^{−1} respectively such that ϕ := ϕ_{1}/ϕ_{2} is unramified. Assume that the conductor 𝔐_{1} of ϕ_{1} is coprime to (p) and that p ∤ #(O_{F} /𝔐_{1})^{×}. Write Gal(F (ϕ_{𝔭}ϵ)/F) = Γ × Δ with Γ ≅ Z_{p} and Δ a finite group. Assume p ∤ #Δ.

Let ρ : G_{Σ} → GL_{2}(E) be a continuous irreducible representation that is ordinary at all places 𝔮 | p. Suppose \overline{ρ}^{ss} ≅ χ_{1} ⊕ χ_{2} with χ_{1} = \overline{ϕ_{1,𝔭}ϵ}, χ_{2} = \overline{ϕ_{2,𝔭}}. Set χ_{0} := χ_{1}χ_{2}^{−1}. If all of the following conditions are satisfied:

(1) Σ ⊃ {𝔮 | pd_{F} 𝔐_{1}𝔐_{1}^{c}},

(2) p ∤ #Δ

(3) χ_{0} is Σ-admissible (cf. Definition 3.1),

(4) det(ρ) = ϕ_{1}ϕ_{2}ϵ,

(5) ρ ⊗ ϕ_{2,𝔭}^{−1} is Σ-minimal, then ρ is modular in the sense of Definition 2.5.

6.4. More consequences of our result. We will now state some corollaries of the R = T theorem proved in the previous section. As before, we assume that χ_{0} is Σ-admissible. We will also discuss how our approach can be used to partially recover the results of Skinner and Wiles [SW99].

Corollary 6.14. The Eisenstein ideal J_{Σ} is a principal ideal and the Hecke algebra T_{Σ} is a complete intersection (and hence Gorenstein).

Proof. The first statement follows immediately from Theorem 6.10 and the fact that the ideal of reducibility I_{re} (which is principal - see Proposition 6.2) is mapped exactly onto the Eisenstein ideal J_{Σ} (see proof of Theorem 6.10). The fact that T_{Σ} is Gorenstein follows from Proposition 6.4 in [Bas63]. Note that the proposition is applicable since the maximal ideal of T_{Σ} is generated by ϖ and a generator of the ideal of reducibility (see Proof of Proposition 6.9) and p is clearly a non-zero divisor. Finally since R_{Σ} = T_{Σ} = O[[X]]/I with the ideal I of codimension 1, the Gorenstein condition is equivalent with principality of I ([Eis95], Corollary 21.20). Hence T_{Σ} is a complete intersection. □

Remark 6.15. The properties of the Hecke algebra and the Eisenstein ideal stated in Corollary 6.14 were proved for their counterparts over Q by Mazur ([Maz77], Theorem 11, and Chapter 2, Section 14). See also a discussion in [CE05] (page 99 and Corollary 3.17) where these properties, like here, are derived as a consequence of an R = T theorem.

Corollary 6.16. Let the notation be as in Section 5. There exists a Σ-minimal upper-triangular deformation of ρ_{0} to GL_{2}(O/ϖ^{n}). Moreover, the O-module

Gal(M(F (Ψ_{n}))_{Ψ}/F (Ψ_{n})) ⊗ O)^{χ̃_{0}^{−1}}

is isomorphic to O/ϖ^{n}.

Proof. The first statement follows from the isomorphisms R_{Σ}/I_{re} ≅ T_{Σ}/J_{Σ} ≅ O/ϖ^{n} combined with Theorem 6.3. The existence of the upper-triangular deformation together with Lemma 5.7 provides an element of Gal(M(F (Ψ_{n}))_{Ψ_{n}}/F (Ψ_{n})) of order p^{⌈n/e⌉}. As in the proof of Theorem 5.11 we see that this element generates an O-submodule of

Gal(M(F (Ψ_{n}))_{Ψ}/F (Ψ_{n})) ⊗ O)^{χ̃_{0}^{−1}}

of order #O/ϖ^{n}. Hence the second statement follows from Theorem 5.11. □

Remark 6.17. While the upper bounds on the order of the Galois group in Corollary 6.16 are predicted by the Main Conjecture of Iwasawa Theory, their exact Omodule structure is in general a mystery. Our results show that the upper bounds are in fact optimal and provide a rather definitive answer to structure question, however only in the case of a Σ-admissible character, where one assumes at the outset the cyclicity of the group of extensions (see Definition 3.1).

Finally, we will discuss to what extent our method provides an alternative to the approach of Skinner and Wiles in [SW99] for proving an R = T theorem for 2-dimensional, p-adic, residually reducible, ordinary representations of G_{Q} with unique non-semisimple reduction which are unramified outside finitely many primes. First note that the assumptions of [SW99] are weaker than ours, because they only imply that the group of extensions of 1 by the character as in Definition 3.1 is onedimensional, but not so if one reverses the order of 1 and the character. However, even with that weaker assumption in place our Proposition 4.10 carries over (with the same proof) showing that there are no upper-triangular Σ-minimal deformations of the residual representation ρ_{0} of [SW99] to GL_{2}(F[X]/X^{2}). Using this we proceed as in the proof of our Proposition 6.5 to show that the universal deformation ring (denoted in [SW99] by R_{Σ,O}^{min}) has the property that R_{Σ,O}^{min}/I_{re} ≅ O/ϖ^{k} for some k ∈ Z_{+} ∪ {∞}, where I_{re} denotes the ideal of reducibility. Note that by Remark 6.6 it is not necessary to know that R_{Σ,O}^{min} is a quotient of O[[X]]. Since [SW99] in fact do exhibit a Σ-minimal upper-triangular deformation ρ_{Σ,O}^{Eis} to GL_{2}(O), we must have k = ∞, i.e., R_{Σ,O}^{min}/I_{re} ≅ O. In particular, I_{re} is the kernel of the map R_{Σ,O}^{min} ↠ O correspoding to ρ_{Σ,O}^{Eis}, which in [SW99] is denoted by I_{Σ}. It follows therefore that the surjection of R_{Σ,O}^{min} onto the (full, not just cuspidal) Hecke algebra (that in [SW99] is denoted by T_{Σ,O}^{min}) descends to an isomorphism R_{Σ,O}^{min}/I_{re} ≅ T_{Σ,O}^{min}/I^{Eis}, where I^{Eis} denotes the Eisenstein ideal (see also Proposition 3.12 in [CE05] where an analogous statement is proved in the case when the residual representation is reducible and semi-simple). In particular our method gives an alternative proof to the statement that every reducible Σ-minimal deformation of ρ_{0} is modular. However, we cannot apply our commutative algebra criterion (Proposition 6.9) to conclude that R_{Σ,O}^{min} = T_{Σ,O}^{min}. First, as mentioned above, the assumptions of [SW99] are weaker than our assumption of Σ-admissibility, so do not imply that the ideal of reducibility is principal, and secondly, R_{Σ,O}^{min}/I_{re} is not finite. In [SW99] and [CE05] the R = T statement is proved using the numerical criterion of Wiles and Lenstra which requires one to compare the size of I_{re}/I_{re}^{2} with the size of the quotient of the cuspidal Hecke algebra by I^{Eis}.

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1University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Cambridge, CB3 0WB, United Kingdom.

2Université Paris 13, Département de Mathématiques, Institut Galilée, 99, avenue Jean-Baptiste Clément, 93430 - Villetaneuse, France