## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# 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*_{ℓ}*f* =
*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**)
→ GL_{2}(**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**) →
GL_{2}(**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**) →
GL_{2}(**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
GL_{2}(𝒪) for some ring of integers 𝒪 of a finite
extension of **Q**_{p}. 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 ρ_{f},
then ρ is *residually modular*. If
ρ̄|_{Dp} ≅
(_{ }^{χ1} _{χ2}^{∗}),
χ_{1} ≠ χ_{2}, then ρ is
*D*_{p}-*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 *D*_{p}-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} ≅ χ_{1} ⊕
χ_{2} with χ_{2} 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**) →
GL_{2}(**Q̄**_{p}) *is**a**continuous*,
*irreducible**representation**unramified**away**from* ∞ *and**a**finite**set*
Σ *of**primes*. *Suppose**also**that* ρ̄^{ss} ≅
χ_{1} ⊕ χ_{2}*with* χ_{2}*unramified**at**p*. *If*

(*i*) χ =
χ_{1}χ_{2}^{−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., ρ_{0} 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 *p*th 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, ρ_{0} is
essentially unique, as we now explain. Let **Q**(χ) be the
splitting field of χ, and let *L*_{0}(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(*L*_{0}(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(*L*_{0}(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′ ≅ ρ_{0}.
Furthermore, the extension
*L*_{0}(Σ)/**Q**(χ) is ramified at all places
above *p* and nowhere else. We single out ρ_{0} by
requiring that
and
for some fixed *g*_{0} ∈ *I*_{p}.

Let 𝒪 be a local complete Noetherian ring with residue field
*k*. An 𝒪-deformation of ρ_{0} 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**) →
GL_{2}(*A*) satisfying ρ_{0} = 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 χ_{2} 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
ρ_{0}. Suppose Ψ = (ψ_{1}, ψ_{2}) is
a pair of **Q̄**_{p}-valued characters of
Gal(**Q**_{Σ}/**Q**) such that
ψ_{2} is unramified at *p* and
ψ_{1}ψ_{2} = χ̃ω^{−1}ψɛ
with ψ a character of finite, *p*-power order. Let
𝒪_{Ψ} be the 𝒪-algebra generated by the values of
ψ_{1} and ψ_{2}. This is a finite local
𝒪-algebra with residue field *k* and uniformizer, say, λ.
Suppose also that ψ_{1} = χ mod λ and ψ_{2} =
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
ψ_{1}ψ_{2}^{−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
ρ_{0} 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**) →
GL_{2}(𝒪_{Ψ}) of ρ_{0} 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^{−1}ɛ, 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^{−1}ɛ, 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*_{Σ}^{2}, 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 ⊆
U_{1} ⊆ U, where U_{1} is
the rank one, free 𝒪-submodule on which
Gal(**Q**_{Σ}/**Q**) acts via ϕɛ. The
quotient U_{2} = U/U_{1} 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*_{Σ}^{1}(**Q**, W) yields
an upper bound for
#(*I*_{Σ}/*I*_{Σ}^{2}).

Let Σ_{1} ⊆ Σ comprise those primes in Σ that are
congruent to 1 modulo *p* together with *p*. Let
W_{1} =
Hom_{𝒪}(*U*_{2}, *U*)
⊗_{𝒪} *K*/𝒪, and let W_{2} =
Hom_{𝒪}(*U*_{1}, *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*_{Σ}^{1}(**Q**, W)
therefore follows from upperbounds for #ker(α) and #ker(γ):
3.1
Now, W_{1} 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, W_{2} 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(*f*_{1}) = 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, *B*_{2}(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
Σ_{2} 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 Γ_{0}(*pN*_{Σ}) →
(**Z**/*pN*_{Σ})^{×}. Also, let
Γ_{Σ,1} be Γ_{1}(*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
Γ_{0}(*pN*′_{Σ}) →
(**Z**/*pN*′_{Σ})^{×}, and let
Γ_{Σ,Sel} be
Γ_{1}(*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,𝒪}^{1}, etc., would not
be different if we omitted *r*.

Proposition 4.2. rank_{𝒪}**T**_{Σ,𝒪}^{Sel} =
rank_{𝒪}**T**_{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 *H*^{1}(·, 𝒪) is
acted upon by the relevant Hecke operators. Let
*H*_{Σ}^{1}(·, 𝒪) be the
maximal direct summand of the localized cohomology group
*H*^{1}(·,
𝒪)_{𝔪Σ} such that
*U*_{q} acts nilpotently on
*H*_{Σ}^{1}(·, 𝒪) for all
*q* ≠ *p* or *r*, and *U*_{p} −
1 and *U*_{r} − 1 act nilpotently on
*H*_{Σ}^{1}(·, 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} _{1}^{0}) 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_{Σ}^{0}(·, 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*_{Σ}^{1}(*Y*_{Σ}^{Sel},
𝒪)^{−} is a free 𝒪[Δ_{Σ}]-module of
rank equal to the 𝒪-rank of
*H*_{Σ}^{1}(*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_{𝒪}**T**_{Σ,𝒪}^{str} =
rank_{𝒪}**T**_{Σ,𝒪}^{min}·#Δ_{Σ}.

*Proof:* Again, the proposition is obvious if χ = ω,
so assume otherwise. Let *Y*_{Σ} and
*Y*_{Σ}^{1} be the open curves over
**C** corresponding to Γ_{Σ} and
Γ_{Σ,1}, respectively, and let *X*_{Σ}
and *X*_{Σ}^{1} be the respective
compactifications. For · = *Y*_{Σ},
*Y*_{Σ}^{1}, *X*_{Σ}, or
*X*_{Σ}^{1}, let
*H*_{Σ}^{1}(·, 𝒪) be the
maximal direct summand of *H*^{1}(·,
𝒪)_{𝔪Σ} such that on
*H*_{Σ}^{1}(·, 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*_{Σ}^{1} 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
Γ_{1}(*p*^{m}*N*′_{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* ↦ γ_{0}
− 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*^{m}*N*′_{Σ}, and has Nebentypus
character χ̃ω^{−1}ψ, 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*_{Σ}^{1} be the curves defined in the
proof of *Proposition 4.2*. Let
X_{/Zp}_{[ζp}_{]}^{1}
be the regular model for *X*_{Σ}^{1} as
described in [MW84, ref. 7, Chapter 2, §8]. The special fiber
X_{/Fp}^{1}
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}_{]} =
*X*^{1}/Δ_{Σ}
is a regular model for *X*_{Σ} (cf. [MW84, ref. 7,
Chapter 2, §7]). The covering
X^{1} →
X is étale, and the normalization
of the special fiber
X_{/Fp}
consists of the two smooth curves
Σ_{0}^{ét} =
Σ^{ét}/Δ_{Σ} and
Σ_{0}^{μ} =
Σ^{μ}/Δ_{Σ}. That the cover is étale
away from the cusps is a consequence of Γ_{Σ} being
contained in Γ_{1}(ℓ) for some prime ℓ > 4. That it is
étale at the cusps is a simple computation. The cover π :
Σ^{ét} → Σ_{0}^{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
Pic^{0}(Σ_{0}^{ét}) and
Pic^{0}(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 [_{1}^{0}] be the zero cusp
of *X*_{Σ}^{1} and consider
ℭ_{𝔪}^{(1)} =
*R*_{𝔪}^{(1)}·[_{1}^{0}]
as on [MW84, ref. 7, p. 298] (the action of
*R*^{(1)} is via the diamond operators). Similarly,
let *R*_{0}^{(1)} =
**Z**_{p}[(**Z**/*pN*_{Σ})^{×}/±
Δ_{Σ}] and let
ℭ_{0,𝔪}^{(1)} =
*R*_{0,𝔪}^{(1)}·[_{1}^{0}]
(here [_{1}^{0}] is the zero cusp of
*X*_{Σ}). The map κ :
*R*_{0}^{(1)} → *R*^{(1)} given by
κ(*x*) = Σ_{δ∈ΔΣ} [δ]*x* is
compatible with the canonical imbedding π* :
Pic^{0}(Σ_{0}^{ét}) ↪
Pic^{0}(Σ^{ét}) in the sense
that
π*(*h*·[_{1}^{0}]) =
κ(*h*)·[_{1}^{0}].

Let *k*_{0} be the minimal field of definition for
χ (i.e., the subfield of *k* generated by the values of
χ), and let *W*(*k*_{0}) be the Witt vectors of
*k*_{0}. By the choice of 𝔪,
*R*_{𝔪}^{(1)} and
*R*_{0,𝔪}^{(1)} are natural
*W*(*k*_{0})-algebras. Put
5.1
and put
5.2
Clearly,
**T**_{Σ,𝒪}^{min}
acts on ℭ_{0}. Let
*Proof of Lemma 5.2:* The action of
**T**_{Σ,𝒪}^{min}
on ℭ_{0} factors through
**T**_{Σ,𝒪}^{min,0},
so *I*_{0} contains *I*^{cusp}.
Furthermore, by the choice of 𝔪, 〈ℓ〉 acts as ϕ(ℓ)
on ℭ_{0}, 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 ℭ_{0} as 1 +
ϕɛ(ℓ). This proves that *I*_{0} 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. #(*R*_{0}/*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*(*k*_{0})-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*·[_{1}^{0}] ∼
0 (hence
κ(*g*)·[_{1}^{0}] ∼
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*)·[_{1}^{0}]
= 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(δ*s*) +
ℓ(−δ*s*) 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} :
*R*_{1} → 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 *B*_{2}(ϕ) is the second generalized
Bernoulli number for ϕ. Here, we note that
*B*_{2}(ϕ) is 𝒪-integral if ϕ ≠ 1 or
ω^{−2} and that
*B*_{2}(ω^{−2})·(1 −
ω^{−2}(*q*)*q*^{2}) 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(*R*_{0})) =
#Δ_{Σ}·𝒪, and since
by **5.5**, it follows that # (*R*_{0}/*I*) ≥
[ρ_{ϕ}(κ(*R*_{0})) :
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 ρ_{0} 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**) →
GL_{2}(*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*_{f}^{2} be a
Gal(**Q̄**/**Q**)-stable
*A*_{f}-lattice. Choosing a suitable basis for
*L* yields a representation ρ_{L} :
Gal(**Q̄**/**Q**) →
GL_{2}(*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_{L}mod
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
ρ_{0}, so, after possibly choosing a new basis for
*L*_{f}, we may assume that
ρ̄_{f} = ρ_{0}. 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^{min}mod 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* ρ_{0}*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,𝒪min}
ker π =
Ann_{TΣ,𝒪min}*I*^{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,𝒪min}ker
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*_{Σ}^{2})
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
Σ_{0} = Σ / {2}. As just shown,
*R*_{Σ0},_{𝒪min ≅
TΣ0},_{𝒪}^{min}.
Fix a lift σ of Frob_{2} 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 1}^{1}). Clearly, (*c*) =
ker{*R*_{Σ,𝒪}^{min} →
*R*_{Σ0},_{𝒪min}, so
TΣ,𝒪min/c
=
TΣ0},_{𝒪min.
Put x = (β/α − 2) + c. The map
πmin} 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_{𝒪}**T**_{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 ρ_{1} =
ρ_{⊗}χ̃_{2}^{−1}, and let
*F* be a finite extension of **Q**_{p}
such that ρ_{1} takes values in
GL_{2}(*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 ρ_{1} and
ρ_{L} is an ordinary *A*-deformation of
a representation ρ_{0} :
Gal(**Q**_{Σ}/**Q**) →
GL_{2}(*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} ≅ ρ_{1},
and therefore ρ_{f} ⊗ χ_{2}
≅ ρ.

## Acknowledgments

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.

- Copyright © 1997, The National Academy of Sciences of the USA

## References

- ↵
- Coates J,
- Yau S T

- Fontaine J-M,
- Mazur B

- ↵
- Wiles A

- ↵
- Diamond D

- ↵
- Fujiwara K

- ↵
- Taylor R,
- Wiles A

- ↵
- Coates J,
- Yau S T

- Lenstra H W

- ↵
- Mazur B,
- Wiles A

- ↵
Washington, L. (1980)
*Introduction to Cyclotomic Fields*, Graduate Texts in Math. 83 (Springer, New York). - ↵
- Mazur B

**Q**(Springer, New York). - ↵
- Ramakrishna R

- ↵
- Fröhlich A

- Coates J

- ↵
Matsumura, H. (1986)
*Commutative Ring Theory*, Camb. Studies in Adv. Math., 8 (Cambridge Univ. Press, Cambridge, U.K.). - ↵
- Shimura G

- ↵
- Hida H

- ↵
- Hida H

- ↵
Carayol, H. (1986)
*Ann. Sci. Ec. Nor. Sup*., IV, Ser. 19, 409–468. - ↵
- Wiles A

## Citation Manager Formats

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.