Ordinary representations and modular forms

1. Introduction

Let p be a prime, and fix an embedding of Q̄ into Q̄_p. Suppose that f is a newform of weight k ≥ 2, level N, and character ψ. For each prime ℓ ∤ N, let T_ℓ be the Hecke operator associated to ℓ and suppose that T_ℓf = c_ℓ(f)·f. Eichler and Shimura (for k = 2) and Deligne (for k > 2) have shown that there is a continuous representation ρ_f : Gal(Q̄/Q) → GL₂(Q̄_p) that is unramified at the primes not dividing pN and such that

for each prime ℓ ∤ pN.

This paper establishes criteria for a representation ρ : Gal(Q̄/Q) → GL₂(Q̄_p) to be “modular” in the sense that there is a newform f such that ρ ≅ ρ_f. Fontaine and Mazur have conjectured that ρ is modular provided it is unramified outside of a finite set of primes, irreducible, has odd determinant, and the restriction to a decomposition group D_p at p satisfies certain conditions. The following conjecture is a special case of [FM95, ref. 1, Conjecture 3c].

Conjecture 1.1. Suppose ρ : Gal(Q̄/Q) → GL₂(Q̄_p) is continuous, irreducible, and unramified outside of a finite set of primes (including ∞). If

1.1

and if det ρ = ψɛ^k−1 is odd, where ɛ is the cyclotomic character, k ≥ 2 is an integer, and ψ is a finite character, then ρ is modular.

A representation satisfying condition 1.1 is said to be ordinary (at p).

Suppose that ρ is as in the Conjecture. From the compactness of Gal(Q̄/Q), it follows that after choosing a suitable basis, ρ takes values in GL2(𝒪) for some ring of integers 𝒪 of a finite extension of Qp. Let λ be a uniformizer of 𝒪, and let ρ̄ = ρ mod λ be the reduction of ρ (in general, this is only well-defined up to semisimplification). If ρ̄ is irreducible, then ρ is residually irreducible. If ρ̄ (more precisely, its semisimplification) is isomorphic to the reduction of some ρf, then ρ is residually modular. If ρ̄|Dp ≅ ( χ1 χ2∗), χ1 ≠ χ2, then ρ is Dp-distinguished. Recently, the above conjecture (and some more general conjectures) have been shown to be true provided p is odd and ρ is residually irreducible, residually modular, and Dp-distinguished ([Wil95, ref. 2], but see also [Dia96, ref. 3] and [Fuj, ref. 4]).

In this paper, we consider representations ρ which are residually reducible and ordinary. In this case, the semisimplification ρ̄^ss of ρ̄ satisfies ρ̄^ss ≅ χ₁ ⊕ χ₂ with χ₂ unramified at p. We prove the following theorem, which establishes many new cases of the conjecture.

Theorem 1.2. Suppose p is an odd prime, and suppose ρ : Gal(Q̄/Q) → GL₂(Q̄_p) is a continuous, irreducible representation unramified away from ∞ and a finite set Σ of primes. Suppose also that ρ̄^ss ≅ χ₁ ⊕ χ₂ with χ₂ unramified at p. If

(i)  χ = χ₁χ₂⁻¹ is ramified at p and odd,

(ii)  the χ-eigenspace of the p-part of the class group of the splitting field of χ is trivial,

(iii)  for q ∈ Σ either χ is ramified at q or χ(q) ≠ q,

(iv)  det ρ = ψɛ^k−1, where k ≥ 2 is an integer and ψ has finite order, and

(v)  ρ is ordinary,

then ρ is modular.

This theorem is a consequence of a more general result, Theorem 6.1, identifying certain universal deformation rings with Hecke rings. In spirit, both the statement and proof of Theorem 6.1 resemble those of the main theorems of [Wil95, ref. 2]. Here, too, we first establish a “minimal case” of the theorem, subsequently deducing the general result from it. However, instead of resorting to the patching argument of [TW95, ref. 5] to prove this minimal case, we directly establish the numerical criteria of [Wil95, ref. 2, Appendix] and [Len95, ref. 6], proving that the order of a certain cohomology group is equal to the size of a congruence module for a certain Hecke ring (technical complications arise when including the prime 2 in Σ, but these are circumvented later). The Galois cohomology group that is computed is the Selmer group of the adjoint of a reducible represention. The relevant Galois module has a filtration whose Jordan-Hölder pieces are one-dimensional, and the computation of its cohomology boils down to class field theory and some simple consequences of the main conjecture of Iwasawa theory (for Q). The congruence module that arises measures congruences between cusp forms and an Eisenstein series and is closely related to the “Eisenstein ideals” studied in [MW84, ref. 7]. Indeed, the ideas of [MW84, ref. 7] provide the means to estimate the order of this module.

It is unfortunate that the theorem stated above places restrictions on the primes contained in Σ. Of course, this is merely a failure of our approach. In fact, the theorem is essentially the best that can be obtained from trying to identify deformation rings with Hecke rings. For if Σ contained some prime for which χ(q) = q, then, as is easily checked, the corresponding universal, ordinary deformation ring (see Section 2) sometimes contains a component having dimension greater than 3, coming from the various reducible deformations. However, the corresponding Hecke ring has dimension at most 2. Matters are made worse by the fact that in this case there may not be a natural map from the deformation ring to the Hecke ring; there may not be a representation defined over the Hecke ring.

The organization of this paper is as follows. Section 2 describes various deformation problems, each more restrictive than the last, culminating with the minimal cases. It also discusses the relations between these deformation problems and singles out some distinguished deformations. In Section 3 we define the Selmer group associated to a distinguished minimal deformation and estimate its order. Section 4 introduces the Hecke rings, and in Section 5 we estimate the size of a certain congruence module for the minimal Hecke rings. Finally, in Section 6, we prove our main result, Theorem 6.1, and deduce from it the theorem stated above.

2. Deformations and Deformation Rings

This section introduces the deformation problems with which this paper is concerned. It also includes some simple, but essential, observations about the corresponding universal deformation rings, relations between them, and certain distinguished deformations.

Let p be an odd prime. For Σ a finite set of primes including p, let Q_Σ be the maximal extension of Q unramified outside of Σ and ∞. We fix once and for all embeddings of Q̄ into Q̄_q for each rational prime q and into C. This fixes a choice of decomposition group D_q and inertia group I_q for each prime q and a choice of complex conjugation, hereafter denoted by c. Suppose that k is a finite field of characteristic p and that χ : Gal(Q_Σ/Q) → k^× is an odd character ramified at p. Suppose also that

2.1

is a continuous representation satisfying

2.2

and having scalar centralizer (i.e., ρ₀ is reducible, but not semisimple).

Henceforth χ and Σ satisfy the following conditions. The χ-eigenspace of the p-part of the class group of the splitting field of χ is trivial. This is always satisfied, for example, by χ = ω and by χ = ω⁻¹, where ω is the character giving the action of Gal(Q̄/Q) on the pth roots of unity [Was80, ref. 8, Proposition 6.16]. The set Σ is such that if q ∈ Σ, then either χ is ramified at q or χ(q) ≠ q. (Note that Σ must contain all the primes at which χ is ramified.) For such χ and Σ, ρ₀ is essentially unique, as we now explain. Let Q(χ) be the splitting field of χ, and let L₀(Σ) be the maximal abelian p-extension of Q(χ) unramified outside Σ, having exponent p, and such that Gal(Q(χ)/Q) acts on Gal(L₀(Σ)/Q(χ)) via the irreducible F_p-representation associated to χ. The hypotheses on χ and Σ imply that as a Gal(Q(χ)/Q)-module, Gal(L₀(Σ)/Q(χ)) is isomorphic to exactly one copy of the F_p-representation associated to χ. Therefore, if ρ′ is any representation satisfying 2.1 and 2.2, then ρ′ ≅ ρ₀. Furthermore, the extension L₀(Σ)/Q(χ) is ramified at all places above p and nowhere else. We single out ρ₀ by requiring that

and

for some fixed g₀ ∈ I_p.

Let 𝒪 be a local complete Noetherian ring with residue field k. An 𝒪-deformation of ρ₀ is a local complete Noetherian 𝒪-algebra A with residue field k and maximal ideal 𝔪_A together with an equivalence class of continuous representations ρ : Gal(Q_Σ/Q) → GL₂(A) satisfying ρ₀ = ρ mod 𝔪_A. We often write “deformation” instead of “𝒪-deformation” when this will cause no confusion. We usually denote a deformation by a single member of its equivalence class. We require all of our deformations to satisfy

with χ₂ unramified. Such a deformation is said to be ordinary. An ordinary deformation satisfying

is called Selmer, while one satisfying

is called strong. Here, χ̃ is the Teichmüller lift of χ, and ɛ is the cyclotomic character. Finally, a strong deformation satisfying

for all primes q ∈ Σ such that q is congruent to 1 modulo p is called Σ-minimal.

There exists a local complete Noetherian 𝒪-algebra R_Σ,𝒪^min and a universal Σ-minimal 𝒪-deformation

We omit the precise formulation of the universal property as well as the proof of existence as these are now standard (see [Maz89, ref. 9], [Ram93, ref. 10], and [Wil95, ref. 2]). Similarly, there exist universal ordinary, Selmer, and strong 𝒪-deformations

and

respectively.

We note that R_Σ,𝒪^Sel can be realized as a quotient of R_Σ,𝒪^ord. For if γ ∈ I_p is such that ɛ(γ) = 1 + p, and if 1 + T = det ρ_Σ,𝒪^ord(γ)(1 + p)⁻¹, then

2.3

There is also a simple relation between R_Σ,𝒪^str and R_Σ,𝒪^Sel. For q ∈ Σ let Δ_q be the Sylow p-subgroup of (Z/q)^×, and let δ_q be a generator. We write Δ_Σ for the product of the Δ_q’s. Let χ_q denote the character

and let χ_Σ = ∏ χ_q. The deformation

is Selmer, and, using the universal properties of R_Σ,𝒪^Sel and R_Σ,𝒪^str, one checks that

2.4

The relation between R_Σ,𝒪^str and R_Σ,𝒪^min is also easily described. If q ∈ Σ is a prime congruent to 1 modulo p, then arguing as in [TW95, ref. 5, Lemma, p. 569] shows that

where ϕ_q factors through χ_q (i.e., there is a unique map 𝒪[Δ_q] → R_Σ,𝒪^str taking χ_q to ϕ_q). Let 𝔞_Σ be the ideal in R_Σ,𝒪^str generated by the set {δ_q − 1}, where q runs over all primes in Σ that are congruent to 1 modulo p. Then

2.5

Suppose now that 𝒪 is the ring of integers of a finite extension of Q_p with residue field k. We consider various reducible ordinary 𝒪-deformations of ρ₀. Suppose Ψ = (ψ₁, ψ₂) is a pair of Q̄_p-valued characters of Gal(Q_Σ/Q) such that ψ₂ is unramified at p and ψ₁ψ₂ = χ̃ω⁻¹ψɛ with ψ a character of finite, p-power order. Let 𝒪_Ψ be the 𝒪-algebra generated by the values of ψ₁ and ψ₂. This is a finite local 𝒪-algebra with residue field k and uniformizer, say, λ. Suppose also that ψ₁ = χ mod λ and ψ₂ = 1 mod λ. Let Q(Ψ) be the splitting field of the pair Ψ, and let L_Ψ(Σ) be the maximal abelian pro-p-extension of Q(Ψ) unramified outside Σ and such that Gal(Q(Ψ)/Q) acts on Gal(L_Ψ(Σ)/Q(Ψ)) via ψ₁ψ₂⁻¹. The hypotheses on χ and Σ imply that Gal(L_Ψ(Σ)/Q(Ψ)) is a free Z_p-module and that

This is essentially Kummer theory (cf. [Coa77, ref. 11, Theorem 1.8]). It follows from this together with our description of ρ₀ that there is some τ ∈ I_p such that τ generates Gal(L_Ψ(Σ)/Q(Ψ)) as a Gal(Q(Ψ)/Q)-module and

Fix such a τ.

One can write down a reducible 𝒪-deformation ρ_Ψ : Gal(Q_Σ/Q) → GL₂(𝒪_Ψ) of ρ₀ as follows. First, project onto Gal(L_Ψ/(Σ)/Q), and then choose a lift H of Gal(Q(Ψ)/Q) to Gal(L_Ψ(Σ)/Q) containing c. Put

and put

Since H and τ topologically generate Gal(L_Ψ(Σ)/Q), this determines the representation. This representation is obviously ordinary. Corresponding to ρ_Ψ is an ideal I_Ψ of R_Σ,𝒪^ord. If ρ_Ψ is Selmer or strong, we also denote by I_Ψ the corresponding ideal of R_Σ,𝒪^Sel or R_Σ,𝒪^str. The pair Ψ = (χ̃ω⁻¹ɛ, 1) is the unique pair such that ρ_Ψ is Σ-minimal. We denote by I_Σ the corresponding ideal of R_Σ,𝒪^min and refer to it as the Eisenstein ideal of R_Σ,𝒪^min. We write ρ_Σ,𝒪^Eis for the corresponding representation.

We conclude this section with a brief analysis of the Eisenstein ideal of R_Σ,𝒪^min. Again, 𝒪 is the ring of integers of some finite extension of Q_p with residue field k. Let S be any finite set of primes containing Σ. The Eisenstein ideal contains the ideal I_S generated by the set

We claim that these ideals are equal. Choose a basis of ρ_Σ,𝒪^min such that

where τ ∈ I_p is chosen for the pair Ψ = (χ̃ω⁻¹ɛ, 1) as in the preceeding discussion. For each σ ∈ Gal(Q_Σ/Q) write

It is clear that

It follows from these identities that

so ρ_S = ρ_Σ,𝒪^min mod I_S satisfies

One sees that with respect to the chosen basis the matrix entries of ρ_S are in 𝒪, and this representation is in fact ρ_Σ,𝒪^Eis. The universal property of R_Σ,𝒪 now implies that I_S = I_Σ. For ease of reference we record this as a proposition.

Proposition 2.1. If S is any finite set of primes containing Σ, then the Eisenstein ideal I_Σ is generated by the set

Finally, let R_S,𝒪^min,tr be the closed 𝒪-subalgebra of R_Σ,𝒪^min generated by the elements {trace(ρ_Σ,𝒪^min(Frob_ℓ)) : ℓ ∉ S}. We define R_S,𝒪^ord,tr, R_S,𝒪^Sel,tr, and R_S,𝒪^str,tr similarly.

Corollary 2.2. For · = min, str, Sel, or ord,

It follows from Proposition 2.1 that

One easily deduces from this that R_S,𝒪^min,tr = R_Σ,𝒪^min (see [Mat86, ref. 12, Theorem 8.4]. The remaining cases are proved similarly using the relations 2.3, 2.4, and 2.5. □

3. Some Galois Cohomology

In this section we give an upper bound for the size of the 𝒪-module I_Σ/I_Σ², where I_Σ is the Eisenstein ideal of R_Σ,𝒪^min defined in the previous section. We maintain the notation of Section 2 with the restriction that 𝒪 is always the ring of integers of some finite extension K of Q_p with residue field k. We often write G_Σ for Gal(Q_Σ/Q).

Let ϕ = χ̃ω⁻¹, and let U be the representation space for ρ_Σ,𝒪^Eis. Then U is a free 𝒪-module of rank two having a filtration 0 ⊆ U₁ ⊆ U, where U₁ is the rank one, free 𝒪-submodule on which Gal(Q_Σ/Q) acts via ϕɛ. The quotient U₂ = U/U₁ is a rank one, free 𝒪-module on which Gal(Q_Σ/Q) acts trivially. Let V = Hom_𝒪(U, U) be the adjoint representation, and let

We write W and W^Sel for V ⊗_𝒪 K/𝒪 and V^Sel ⊗_𝒪 K/𝒪, respectively. Let

and for those q ∈ Σ different from p let

We define the Selmer group to be

Following [Wil95, ref. 2, Proposition 2.1] one proves that

Therefore, an upper bound for #H_Σ¹(Q, W) yields an upper bound for #(I_Σ/I_Σ²).

Let Σ₁ ⊆ Σ comprise those primes in Σ that are congruent to 1 modulo p together with p. Let W₁ = Hom_𝒪(U₂, U) ⊗_𝒪 K/𝒪, and let W₂ = Hom_𝒪(U₁, U) ⊗_𝒪 K/𝒪. There is a commutative diagram of Gal(Q_Σ/Q)-modules

having exact rows and inducing the following commutative diagram of cohomology groups:

where

The rows in this diagram are exact, so there is an exact sequence

An upper bound for #H_Σ¹(Q, W) therefore follows from upperbounds for #ker(α) and #ker(γ):

3.1

Now, W₁ fits into the short exact sequence

The associated long exact cohomology sequence yields the exact sequence

Since W^Sel ≅ K/𝒪(ϕɛ), one easily checks that the hypotheses on χ and Σ imply that the second arrow is surjective. It follows that

3.2

Similarly, W₂ fits into the short exact sequence

The associated long exact cohomology sequence yields the commutative diagram

having exact exact rows. It follows that

3.3

Class field theory alone shows that #ker(f₁) = 1 and together with the “main conjecture” of Iwasawa theory [MW84, ref. 7, Theorem, p. 214] and [MW84, ref. 7, Proposition 1, p. 193] implies that

3.4

where

3.5

Here, B₂(ϕ) is the second generalized Bernoulli number for ϕ. Substituting 3.4 into 3.3 and combining the result with 3.2 and 3.1 yields the following proposition.

Proposition 3.1. Write ϕ = χ̃ω⁻¹. Let η(ϕ, Σ) be as in 3.5. Then

4. Hecke Rings

In this section we introduce the Hecke rings that we will later relate to the deformation rings of the second section. We keep the notation of the previous sections. In particular, 𝒪 is the ring of integers of some finite extension of Q_p with residue field k and uniformizer λ.

As before, let ϕ = χ̃ω⁻¹. Let Σ₂ be the set of primes q ∈ Σ / {p} such that either χ is ramified at q, or χ|_Dq = ω⁻¹, or q is congruent to 1 modulo p. If χ ≠ ω or ω⁻¹, then let r be a prime not contained in Σ and such that r is greater than 4, r is not congruent to 1 modulo p, and χ|_Dr ≠ ω, ω⁻¹, or 1. This is always possible. If χ = ω or ω⁻¹, then put r = 1. For each prime q, let

Put

We identify Δ_Σ with the Sylow p-subgroup of (Z/pN_Σ)^×. Let Γ_Σ be the inverse image of Δ_Σ under the usual homomorphism Γ₀(pN_Σ) → (Z/pN_Σ)^×. Also, let Γ_Σ,1 be Γ₁(pN_Σ). We denote by T(Γ_Σ) and T(Γ_Σ,1) the finite 𝒪-algebras generated by the Hecke operators {T_ℓ, 〈ℓ〉 : ℓ ∉ Σ ∪ {r}} acting on the spaces of weight 2 modular forms invariant under the standard action of Γ_Σ and Γ_Σ,1, respectively. We write 𝔪_Σ for the maximal ideal of T(Γ_Σ) generated by λ (a uniformizer of 𝒪) and by T_ℓ − 1 − χ̃(ℓ) for all primes ℓ ∉ Σ ∪ {r}. Let E_2,ϕ be the Eisenstein series whose associated L-series is ζ(s)L(s − 1, ϕ). Then 𝔪_Σ is the maximal ideal of T(Γ_Σ) associated to E_2,ϕ. The inverse image of 𝔪_Σ under the surjection T(Γ_Σ,1) → T(Γ_Σ) is also denoted by 𝔪_Σ. Let

Now put

let Γ_Σ,str be the inverse image of Δ_Σ under the usual map Γ₀(pN′_Σ) → (Z/pN′_Σ)^×, and let Γ_Σ,Sel be Γ₁(pN′_Σ). We denote by T(Γ_Σ,str) and T(Γ_Σ,Sel) the 𝒪-algebras generated by the Hecke operators {T_ℓ, 〈ℓ〉 : ℓ ∉ Σ ∪ {r}} acting on the spaces of weight 2 modular forms invariant under the standard action of Γ_Σ,str and Γ_Σ,Sel, respectively. We also denote by 𝔪_Σ the maximal ideal of T(Γ_Σ,str) and of T(Γ_Σ,Sel) associated to the modular form E_2,ϕ. Let

We denote by T_Σ,𝒪^min,0, T_Σ,𝒪^1,0, T_Σ,𝒪^str,0, and T_Σ,𝒪^Sel,0 the quotient algebras obtained by restricting the Hecke operators to the corresponding spaces of cusp forms. Note that these rings may be trivial.

Remark 4.1: When χ ≠ ω or ω⁻¹, we have introduced the auxiliary prime r to ensure that Γ_Σ and Γ_Σ,str have no elliptic points. It is easy to see that T_Σ,𝒪^min, T_Σ,𝒪¹, etc., would not be different if we omitted r.

Proposition 4.2. rank_𝒪T_Σ,𝒪^Sel = rank_𝒪T_Σ,𝒪^str·#Δ_Σ.

Proof: If χ = ω, then Σ = {p}, #Δ_Σ = 1, and Γ_Σ,str = Γ_Σ,Sel, so the proposition is obvious. Assume χ ≠ ω. A simple analysis of the possible Eisenstein series associated to the maximal ideal 𝔪_Σ yields

4.1

and

4.2

For each prime q dividing pN_Σ, we denote by U_q the usual Atkin-Lehner operator. Let Y_Σ^str and Y_Σ^Sel be the open curves over C corresponding to the quotients of the complex upper half-plane by the congruence subgroups Γ_Σ,str and Γ_Σ,Sel, respectively. Let X_Σ^str and X_Σ^Sel be the respective compactifications, obtained by adjoining the cusps. For · = X_Σ^str, Y_Σ^str, etc., the singular cohomology group H¹(·, 𝒪) is acted upon by the relevant Hecke operators. Let H_Σ¹(·, 𝒪) be the maximal direct summand of the localized cohomology group H¹(·, 𝒪)_𝔪Σ such that U_q acts nilpotently on H_Σ¹(·, 𝒪) for all q ≠ p or r, and U_p − 1 and U_r − 1 act nilpotently on H_Σ¹(·, 𝒪)/λ. Using the correspondence between spaces of cusp forms and cohomology groups (cf. [Shi71, ref. 13, Chapter 8]), it is straightforward to check that

4.3

and

4.4

Here, and in what follows, the superscript minus sign denotes the −1 eigenspace for the action of (₀⁻¹ ₁⁰) on the indicated cohomology group.

The excision sequence for singular cohomology gives rise to the exact sequences

4.5

and

4.6

where the subscript Σ on Div_Σ⁰(·, cusps, 𝒪)_𝔪Σ has the same meaning as it does for the other terms in the sequences. A simple analysis of the cuspidal divisor groups, using that the cover X_Σ^Sel → X_Σ^str is unramified at the cusps, shows that

4.7

Arguing as in the proof of [TW95, ref. 5, Proposition 1] shows that H_Σ¹(Y_Σ^Sel, 𝒪)⁻ is a free 𝒪[Δ_Σ]-module of rank equal to the 𝒪-rank of H_Σ¹(Y_Σ^str, 𝒪)⁻. This, together with 4.7 and 4.5, 4.6, implies that

which combined with 4.3, 4.4 and 4.5, 4.6 yields the proposition. □

Proposition 4.3. rank_𝒪T_Σ,𝒪^str = rank_𝒪T_Σ,𝒪^min·#Δ_Σ.

Proof: Again, the proposition is obvious if χ = ω, so assume otherwise. Let Y_Σ and Y_Σ¹ be the open curves over C corresponding to Γ_Σ and Γ_Σ,1, respectively, and let X_Σ and X_Σ¹ be the respective compactifications. For · = Y_Σ, Y_Σ¹, X_Σ, or X_Σ¹, let H_Σ¹(·, 𝒪) be the maximal direct summand of H¹(·, 𝒪)_𝔪Σ such that on H_Σ¹(·, 𝒪)/λ, if χ|_Dq = ω⁻¹, then U_q − ϕ(q)q acts nilpotently, and otherwise U_q − 1 acts nilpotently. One has

and

The comparisons with the ranks of cohomology groups are proved just as are 4.3, 4.4. That the ranks of T_Σ,𝒪^str,0 and T_Σ,𝒪^1,0 are equal follows from the fact that the modular forms on which the one acts are just twists of the forms on which the other acts. Considering the excision sequences for Y_Σ and Y_Σ¹ and arguing as in the proof of Proposition 4.2 shows that

This, combined with 4.1 and the simple observation that

yields the proposition. □

Finally, for each positive integer m, let T^(m) denote the 𝒪-algebra generated by the Hecke operators {T_ℓ, 〈ℓ〉 : ℓ ∉ Σ ∪ {r}} acting on the space of weight 2 modular forms that are invariant under the usual action of Γ₁(p^mN′_Σ) and that are ordinary at p in the sense of [Hid85, ref. 14]. Note that E_2,ϕ is such a form and therefore defines a maximal ideal of T^(m), also denoted 𝔪_Σ. Put

Note that T_Σ,𝒪⁽¹⁾ = T_Σ,𝒪^Sel.

Let Λ = 𝒪[[T]]. By the work of Hida [Hid85, ref. 14, Hid88, ref. 15], T_Σ,𝒪^ord is a finite, torsion-free Λ-algebra via T ↦ γ₀ − 1, where

Hida has also shown that if k ≥ 2 is an integer, and if 𝔭 is a height one prime ideal of T_Σ,𝒪^ord containing (1 + T)^pm − (1 + p)^pm(k−2), then 𝔭 corresponds to a weight k newform f that is ordinary at p, has level dividing p^mN′_Σ, and has Nebentypus character χ̃ω⁻¹ψ, where ψ has order dividing p^m. This correspondence is given by (ℓth Fourier coefficient of f) = T_ℓ mod 𝔭.

Proposition 4.4. The canonical surjection T_Σ,𝒪^ord → T_Σ,𝒪^Sel factors through T_Σ,𝒪^ord/T.

Later we shall see that T_Σ,𝒪^Sel = T_Σ,𝒪^ord/T (see 6.4).

5. The Eisenstein Ideal

One can define an “Eisenstein ideal” of T_Σ,𝒪^min analogous to the Eisenstein ideal I_Σ of R_Σ,𝒪^min. Let I^Eis be the ideal of T_Σ,𝒪^min generated by the set {T_ℓ − 1 − ϕɛ(ℓ), 〈ℓ〉 − ϕ(ℓ) : ℓ ∉ Σ ∪ {r}}. This is just the minimal prime ideal of T_Σ,𝒪^min corresponding to the Eisenstein series E_2,ϕ. The following proposition is the main result of this section.

Proposition 5.1. Write ϕ = χ̃ω⁻¹. Let η(ϕ, Σ) be as in 3.5. Suppose that if 2 ∈ Σ, then χ(2) ≠ 2⁻¹. Then

Our proof of this proposition relies heavily on ideas and results of [MW84, ref. 7], and for many definitions and details in the following arguments we refer to this paper. In particular, we adopt the conventions of [MW84, ref. 7] regarding models of modular curves and their cusps.

The proposition is obvious if ϕ is the trivial character or if ϕ = ω⁻² and Σ = {p}, so throughout the rest of this section we assume otherwise. Let X_Σ and X_Σ¹ be the curves defined in the proof of Proposition 4.2. Let X_/Zp[ζp]¹ be the regular model for X_Σ¹ as described in [MW84, ref. 7, Chapter 2, §8]. The special fiber X_/Fp¹ consists of two smooth curves Σ^ét and Σ^μ (in the notation of [MW84, ref. 7]) intersecting transversally at the supersingular points. The quotient X_/Zp[ζp] = X¹/Δ_Σ is a regular model for X_Σ (cf. [MW84, ref. 7, Chapter 2, §7]). The covering X¹ → X is étale, and the normalization of the special fiber X_/Fp consists of the two smooth curves Σ₀^ét = Σ^ét/Δ_Σ and Σ₀^μ = Σ^μ/Δ_Σ. That the cover is étale away from the cusps is a consequence of Γ_Σ being contained in Γ₁(ℓ) for some prime ℓ > 4. That it is étale at the cusps is a simple computation. The cover π : Σ^ét → Σ₀^ét is étale with Galois group Δ_Σ. The usual actions of the Hecke operators T_n and 〈n〉, (n, pN_Σ) = 1, extend to actions on Pic⁰(Σ₀^ét) and Pic⁰(Σ^ét).

Let R⁽¹⁾ = Z_p[(Z/pN_Σ)^×/± 1], and let 𝔪 be the component of R⁽¹⁾ corresponding to ϕ (see [MW84, ref. 7, Chapter 1, §3]). Let [₁⁰] be the zero cusp of X_Σ¹ and consider ℭ_𝔪⁽¹⁾ = R_𝔪⁽¹⁾·[₁⁰] as on [MW84, ref. 7, p. 298] (the action of R⁽¹⁾ is via the diamond operators). Similarly, let R₀⁽¹⁾ = Z_p[(Z/pN_Σ)^×/± Δ_Σ] and let ℭ_0,𝔪⁽¹⁾ = R_0,𝔪⁽¹⁾·[₁⁰] (here [₁⁰] is the zero cusp of X_Σ). The map κ : R₀⁽¹⁾ → R⁽¹⁾ given by κ(x) = Σ_δ∈ΔΣ [δ]x is compatible with the canonical imbedding π* : Pic⁰(Σ₀^ét) ↪ Pic⁰(Σ^ét) in the sense that π*(h·[₁⁰]) = κ(h)·[₁⁰].

Let k₀ be the minimal field of definition for χ (i.e., the subfield of k generated by the values of χ), and let W(k₀) be the Witt vectors of k₀. By the choice of 𝔪, R_𝔪⁽¹⁾ and R_0,𝔪⁽¹⁾ are natural W(k₀)-algebras. Put

5.1

and put

5.2

Clearly, T_Σ,𝒪^min acts on ℭ₀. Let

Proof of Lemma 5.2: The action of T_Σ,𝒪^min on ℭ₀ factors through T_Σ,𝒪^min,0, so I₀ contains I^cusp. Furthermore, by the choice of 𝔪, 〈ℓ〉 acts as ϕ(ℓ) on ℭ₀, and since the action of T_ℓ on the cuspidal group ℭ_0,𝔪⁽¹⁾ is via 1 + 〈ℓ〉ℓ (see [MW84, ref. 7, p. 238]), T_ℓ acts on ℭ₀ as 1 + ϕɛ(ℓ). This proves that I₀ contains I^Eis. □

Now, let

It follows from Lemma 5.2 that

5.3

Combining 5.3 with the following lemma yields the proposition.

Lemma 5.3. #(R₀/I) ≥ #(𝒪/η(ϕ, Σ)).

Proof of Lemma 5.3: Let I′ = Ann_{R0,𝔪⁽¹⁾}(ℭ_0,𝔪⁽¹⁾). Since 𝒪 is a flat W(k₀)-module, it follows from 5.1 and 5.2 that

5.4

Suppose that the conductor of ϕ is pN_Σ/m and suppose that h ∈ I′. We can find g ∈ R_0,𝔪⁽¹⁾ such that g ≡ h (mod p^M) for M arbitrarily large and such that g·[₁⁰] ∼ 0 (hence κ(g)·[₁⁰] ∼ 0). By the arguments in the last paragraph on [MW84, ref. 7, p. 299], there is an integer e, coprime to p, such that e·κ(g)·[₁⁰] = div(f^ℓ) for some f^ℓ ∈ 𝔉^(m) (notation as in [MW84, ref. 7, Chapter 4]). Moreover, f^ℓ is invariant under the action of Δ_Σ (as div(f^ℓ) is). The proofs of [MW84, ref. 7, Propositions 3 and 3^(m), p. 298] show that if f^ℓ = c·∏ f_0,s^ℓ(s) is invariant under Δ_Σ, then ℓ(s) + ℓ(−s) = ℓ(δs) + ℓ(−δs) for all δ ∈ Δ_Σ. It follows from [MW84, ref. 7, (8), p. 298] that

where the second sum is over a complete set of representatives for Z/pN_Σ modulo the equivalence

and where ϑ̂^(m)(s; pN_Σ) is the “hatted” Stickelberger element defined in [MW84, ref. 7, Chapter 1, §1]. As ℓ(s) = 0 if (p, s) ≠ 1, this shows that κ(h) is contained in the ideal

By 5.4, κ(I) is contained in the same ideal.

Now, consider the homomorphism ρ_ϕ : R₁ → 𝒪 induced by ϕ. Arguing as in [MW84, ref. 7, Propositions 2 and 4, pp. 201–205] shows that if (p, s) = 1, then

5.5

where B₂(ϕ) is the second generalized Bernoulli number for ϕ. Here, we note that B₂(ϕ) is 𝒪-integral if ϕ ≠ 1 or ω⁻² and that B₂(ω⁻²)·(1 − ω⁻²(q)q²) is 𝒪-integral for any prime q distinct from p (recall that if ϕ = ω⁻², then we have assumed that Σ ≠ {p}). Since ρ_ϕ(κ(R₀)) = #Δ_Σ·𝒪, and since

by 5.5, it follows that # (R₀/I) ≥ [ρ_ϕ(κ(R₀)) : ρ_ϕ(κ(I))] ≥ # (𝒪/η(ϕ, Σ)). □

6. The Main Theorem

In this section we prove our main results, relating the deformation rings of Section 2 to the Hecke rings of Section 4. In particular, we show that any deformation of ρ₀ that “looks modular” is modular in the sense that its semisimplification is equivalent to a representation associated to a modular form.

Each minimal prime ideal of T_Σ,𝒪^min corresponds to a newform of weight 2 and level dividing pN_Σ. Let G be the set of all newforms corresponding to prime ideals of T_Σ,𝒪^min, let F be the subset consisting of cusp forms, and let E be the Eisenstein series (E = {E_2,ϕ}). For f ∈ G, we denote by A_f the normalization of T_Σ,𝒪^min/𝔭_f (𝔭_f being the prime ideal corresponding to f). This is a local complete finite 𝒪-algebra with residue field k, maximal ideal 𝔪_f, and fraction field F_f. It is well known that there is a semisimple representation ρ_f : Gal(Q̄/Q) → GL₂(F_f) such that ρ_f is unramified at all primes not dividing pN_Σ, det ρ_f = ϕɛ, and

6.1

Let L ⊆ F_f² be a Gal(Q̄/Q)-stable A_f-lattice. Choosing a suitable basis for L yields a representation ρ_L : Gal(Q̄/Q) → GL₂(A_f) whose trace satisfies 6.1 and such that ρ_L(c) = (⁻¹ ₁). Reducing the trace modulo 𝔪_f shows that ρ̄_L = ρ_Lmod 𝔪_f satisfies

where the superscript ss denotes the corresponding semisimplification. If f ∈ F, then ρ_L is irreducible, and there are three possibilities for ρ̄_L; either

where ∗ is not identically zero. It is not hard to see that the latter two possibilities always occur for some choice of L. Fix a lattice L_f for which the last possibility occurs. Abusing notation, we call the corresponding representation ρ_f. As remarked in Section 2, since ρ̄_f satisfies 2.1 and 2.2, ρ̄_f is equivalent to ρ₀, so, after possibly choosing a new basis for L_f, we may assume that ρ̄_f = ρ₀. The work of Carayol, Deligne, Langlands, and Wiles (see [Car86, ref. 16] and [Wil88, ref. 17]) relating the level and Hecke eigenvalues of f to the representation ρ_f shows that ρ_f is a Σ-minimal 𝒪-deformation. (In particular, if χ ≠ ω or ω⁻¹, then ρ_f is unramified at the auxiliary prime r and the level of f is coprime to r.) There is therefore a homomorphism π_f : R_Σ,𝒪^min → A_f inducing ρ_f. If f ∈ E, then

and we take for π_f the map R_Σ,𝒪^min → 𝒪 = A_f given by reduction modulo I_Σ. It follows from Corollary 2.2 that the image of the map R_Σ,𝒪^min → ∏_f∈G A_f given by x ↦ (π_f(x)) is T_Σ,𝒪^min. Thus, there is a (local) surjection π^min : R_Σ,𝒪^min → T_Σ,𝒪^min of 𝒪-algebras such that π_f = π^minmod 𝔭_f.

In the same manner, one shows that there are similar surjections π^str : R_Σ,𝒪^str → T_Σ,𝒪^str, π^Sel : R_Σ,𝒪^Sel → T_Σ,𝒪^Sel, and π^(m) : R_Σ,𝒪^ord → T_Σ,𝒪^(m), only now to define π_f for f an Eisenstein series one uses the reducible deformations described in Section 2. Let π^ord : R_Σ,𝒪^ord → T_Σ,𝒪^ord be

This is a homomorphism of Λ-algebras (Λ = 𝒪[[T]]).

Theorem 6.1. Let χ, Σ, and ρ₀ be as in Section 2. If · = min, str, Sel, or ord, then

is an isomorphism.

Proof: Assume at first that 2 ∉ Σ if χ(2) = 2⁻¹. Let π : T_Σ,𝒪^min → 𝒪 be the homomorphism given by reduction modulo I^Eis. Since T_Σ,𝒪^min is reduced, Ann_TΣ,𝒪^min ker π = Ann_TΣ,𝒪^minI^Eis is just the intersection of all minimal prime ideals of T_Σ,𝒪^min distinct from I^Eis. In other words, the annihilator of the kernel of π is just the kernel of the projection T_Σ,𝒪^min → T_Σ,𝒪^min,0. Letting (η) denote the ideal π(Ann_TΣ,𝒪^minker π), it follows from Proposition 5.1 that

6.2

By Proposition 2.1, I^Eis = π^min(I_Σ), so, upon considering Fitting ideals (see [Wil95, ref. 2, Appendix] or [Len95, ref. 6]),

Combining this with 6.2 and the estimate for #(I_Σ/I_Σ²) provided by Proposition 3.1 shows that

6.3

The main result of [Len95, ref. 6] now applies, yielding R_Σ,𝒪^min = T_Σ,𝒪^min and that these rings are complete intersections over 𝒪. Suppose now that 2 ∈ Σ and that χ(2) = 2⁻¹. Let Σ₀ = Σ / {2}. As just shown, R_Σ0,_{𝒪^min ≅ TΣ0},_𝒪^min. Fix a lift σ of Frob₂ and a generator τ of the pro-p-part of tame inertia at 2. By the hypotheses on χ and Σ, there is a basis for ρ_Σ,𝒪^min such that ρ(σ) = (_β^α) and ρ(τ) = (_c 1¹). Clearly, (c) = ker{R_Σ,𝒪^min → R_Σ0,_{𝒪^min}, so TΣ,𝒪^min/c = TΣ0},_{𝒪^min. Put x = (β/α − 2) + c. The map π^min} induces an isomorphism R_Σ,𝒪^min/x ≅ T_Σ,𝒪^min/x. As it is easily checked that x is not a zero-divisor in T_Σ,𝒪^min, it follows that π^min is an isomorphism. By 2.5, R_Σ,𝒪^min = R_Σ,𝒪^str/𝔞_Σ. Since R_Σ,𝒪^min is a finite 𝒪-algebra, as we have just seen, R_Σ,𝒪^str is a finite 𝒪-algebra generated by at most

elements, the second equality coming from Proposition 4.3. Since T_Σ,𝒪^str is a finite free 𝒪-module and a quotient of R_Σ,𝒪^str, it follows that R_Σ,𝒪^str = T_Σ,𝒪^str. The proof for the Selmer case is similar. In 2.4 we observed that R_Σ,𝒪^Sel = R_Σ,𝒪^str ⊗_𝒪 𝒪[Δ_Σ], so it follows from the strong case of the theorem that R_Σ,𝒪^Sel is a finite free 𝒪-module and that

the last equality coming from Proposition 4.3. Since T_Σ,𝒪^Sel is a quotient of R_Σ,𝒪^Sel, equality of ranks implies that the rings are the same. Finally, the Selmer case of the theorem together with 2.3, Proposition 4.5, and the surjection π^ord : R_Σ,𝒪^ord → T_Σ,𝒪^ord yields

6.4

Therefore, R_Σ,𝒪^ord and T_Σ,𝒪^ord are finite Λ-modules, generated by rank_𝒪T_Σ,𝒪^Sel elements. Since T_Σ,𝒪^ord is a torsion-free Λ-module, and since T_Σ,𝒪^Sel = T_Σ,𝒪^ord/T is a free 𝒪-module, it is not hard to see that T_Σ,𝒪^ord is a free Λ-module, necessarily having Λ-rank equal to the 𝒪-rank of T_Σ,𝒪^Sel. It now follows, just as in the previous cases, that R_Σ,𝒪^ord = T_Σ,𝒪^ord. This completes the proof of the theorem. □

Theorem 1.2 is an immediate consequence of Theorem 6.1. Suppose ρ is a representation satisfying the conditions of the theorem. Consider ρ₁ = ρ_⊗χ̃₂⁻¹, and let F be a finite extension of Q_p such that ρ₁ takes values in GL₂(F). Let A be the ring of integers of F, and let k be the residue field of A. Using the hypotheses on χ and Σ, arguing as in the second paragraph of this section shows that there is a Gal(Q_Σ/Q)-stable A-lattice L such that ρ_L has the same trace and determinant as ρ₁ and ρ_L is an ordinary A-deformation of a representation ρ₀ : Gal(Q_Σ/Q) → GL₂(k) satisfying 2.2. Theorem 6.1 implies that there is a homomorphism

such that

Let 𝔭 be the kernel of α. Then 𝔭 is a height one prime ideal of T_Σ,A^ord containing (1 + T)^pm − (1 + p)^pm(k−2) for some m. As mentioned in Section 4, Hida [Hid85, ref. 14, Hid88, ref. 15] has shown that 𝔭 corresponds to a newform of weight k. That is, there is a weight k newform f such that the ℓth Fourier coefficient of f is α(T_ℓ). As the representation ρ_f satisfies

it must be that ρ_f ≅ ρ₁, and therefore ρ_f ⊗ χ₂ ≅ ρ.