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

This paper establishes criteria for a representation ρ : Gal(/Q) → GL₂(_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(/Q) → GL₂(_p) is continuous, irreducible, and unramified outside of a finite set of primes (including ∞). If

1.1

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(/Q), it follows that after choosing a suitable basis, ρ takes values in GL2() for some ring of integers of a finite extension of Qp. Let λ be a uniformizer of , and let = ρ mod λ be the reduction of ρ (in general, this is only well-defined up to semisimplification). If is irreducible, then ρ is residually irreducible. If (more precisely, its semisimplification) is isomorphic to the reduction of some ρf, then ρ is residually modular. If |Dp (χ1χ2), 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 ^ss of satisfies ^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(/Q) → GL₂(_p) is a continuous, irreducible representation unramified away from ∞ and a finite set Σ of primes. Suppose also that ^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 = ψ^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.

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 into _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

2.2

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(/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

Let be a local complete Noetherian ring with residue field k. An -deformation of ρ₀ is a local complete Noetherian -algebra A with residue field k and maximal ideal _A together with an equivalence class of continuous representations ρ : Gal(Q_Σ/Q) → GL₂(A) satisfying ρ₀ = ρ mod _A. We often write “deformation” instead of “-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

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

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

2.3

Σ,

Σ,

Σ,

Σ,

2.4

Σ,

Σ,

Σ,

Σ,

2.5

One can write down a reducible -deformation ρ_Ψ : Gal(Q_Σ/Q) → GL₂(_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

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

We conclude this section with a brief analysis of the Eisenstein ideal of R_Σ,^min. Again, 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

Σ,

Σ,

Σ,

Σ,

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

S,

Σ,

Σ,

S,

S,

S,

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

S,

Σ,

In this section we give an upper bound for the size of the -module I_Σ/I_Σ², where I_Σ is the Eisenstein ideal of R_Σ,^min defined in the previous section. We maintain the notation of Section 2 with the restriction that 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 = ω⁻¹, and let U be the representation space for ρ_Σ,^Eis. Then U is a free -module of rank two having a filtration 0 U₁ U, where U₁ is the rank one, free -submodule on which Gal(Q_Σ/Q) acts via ϕ. The quotient U₂ = U/U₁ is a rank one, free -module on which Gal(Q_Σ/Q) acts trivially. Let V = Hom(U, U) be the adjoint representation, and let

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

3.1

3.2

having exact exact rows. It follows that

3.3

3.4

3.5

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

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, is the ring of integers of some finite extension of Q_p with residue field k and uniformizer λ.

As before, let ϕ = ω⁻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

Σ,

Σ,

Σ,

Σ,

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,^min, T_Σ,¹, etc., would not be different if we omitted r.

Proposition 4.2. rankT_Σ,^Sel = rankT_Σ,^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 _Σ yields

4.1

4.2

_Σ

4.3

4.4

0

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

4.5

4.6

_Σ

4.7

Proposition 4.3. rankT_Σ,^str = rankT_Σ,^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(·, ) be the maximal direct summand of H¹(·, )_Σ such that on H_Σ¹(·, )/λ, if χ|_Dq = ω⁻¹, then U_q − ϕ(q)q acts nilpotently, and otherwise U_q − 1 acts nilpotently. One has

Σ,

Σ,

Finally, for each positive integer m, let T^(m) denote the -algebra generated by the Hecke operators {T, : 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 _Σ. Put

Σ,

Σ,

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

Σ,

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

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

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

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

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() for some prime > 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 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 ⁽¹⁾ = R⁽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 _0,⁽¹⁾ = R_0,⁽¹⁾·[₁⁰] (here [₁⁰] is the zero cusp of X_Σ). The map κ : R₀⁽¹⁾ → R⁽¹⁾ given by κ(x) = Σ_δΔΣ [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 , R⁽¹⁾ and R_0,⁽¹⁾ are natural W(k₀)-algebras. Put

5.1

5.2

Σ,

Σ,

Σ,

0,

Now, let

5.3

Lemma 5.3. #(R₀/I) ≥ #(/η(ϕ, Σ)).

Proof of Lemma 5.3:Let I′ = Ann_R0,⁽¹⁾(_0,⁽¹⁾). Since is a flat W(k₀)-module, it follows from 5.1 and 5.2 that

5.4

0,

(s)

Now, consider the homomorphism ρ_ϕ : R₁ → 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

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,^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,^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,^min/_f (_f being the prime ideal corresponding to f). This is a local complete finite -algebra with residue field k, maximal ideal _f, and fraction field F_f. It is well known that there is a semisimple representation ρ_f : Gal(/Q) → GL₂(F_f) such that ρ_f is unramified at all primes not dividing pN_Σ, det ρ_f = ϕ, and

6.1

Σ,

Σ,

Σ,

fG

Σ,

Σ,

Σ,

In the same manner, one shows that there are similar surjections π^str : R_Σ,^str → T_Σ,^str, π^Sel : R_Σ,^Sel → T_Σ,^Sel, and π^(m) : R_Σ,^ord → T_Σ,^(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,^ord → T_Σ,^ord be

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

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

6.2

6.3

Σ,

Σ,

^min T_Σ0

Σ,

β

c1

Σ,

^min}, so T_Σ,^min/c = T_Σ0

^min. Put x = (β/α − 2) + c. The map π^min

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

6.4

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Σ,

Theorem 1.2 is an immediate consequence of Theorem 6.1. Suppose ρ is a representation satisfying the conditions of the theorem. Consider ρ₁ = ρ₂⁻¹, 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

This work was supported by a National Science Foundation Graduate Fellowship and a Sloan Foundation Doctoral Dissertation Fellowship (C.M.S.) and a National Science Foundation grant.