IHARA’S LEMMA FOR IMAGINARY QUADRATIC FIELDS

KRZYSZTOF KLOSIN

Abstract. An analogue over imaginary quadratic fields of a result in algebraic number theory known as Ihara’s lemma is established. More precisely, we show that for a prime ideal 𝔭 of the ring of integers of an imaginary quadratic field F, the kernel of the sum of the two standard 𝔭-degeneracy maps between the cuspidal sheaf cohomology H_{!}^{1}(X_{0}, M̃_{0})^{2} → H_{!}^{1}(X_{1}, M̃_{1}) is Eisenstein. Here X_{0} and X_{1} are analogues over F of the modular curves X_{0}(N) and X_{0}(Np), respectively. To prove our theorem we use the method of modular symbols and the congruence subgroup property for the group SL_{2} which is due to Serre.

1. Introduction

Ihara’s lemma in the version stated in [6] asserts that the kernel of the map α : J_{0}(N)^{2} → J_{0}(Np) is Eisenstein if (N, p) = 1. Here J_{0}(N′) denotes the Jacobian of the compactified modular curve Γ_{0}(N′)\H, and α is the sum of the two standard p-degeneracy maps from J_{0}(N) to J_{0}(Np). The original proof of the result is due to Ihara [4] and uses algebraic geometry. In [6] Ribet gave a different proof without appealing to algebro-geometric methods. The result was later improved upon by Khare [5] to dispose of the condition that N be coprime to p. Khare also gives a rearranged proof in the case when (N, p) = 1 using the method of modular symbols (cf. [5], Remark 4). We will use his approach to generalize the result to imaginary quadratic fields, where algebro-geometric techniques are not available.

Let F denote an imaginary quadratic extension of Q and O_{F} its ring of integers. The reason why over F the algebro-geometric machinery is not available is the fact that the symmetric space on which automorphic forms are defined is the hyperbolic-3-space, the product of C and R_{+}, and the analogues X_{n} of the modular curves are not algebraic varieties (cf. section 2). However, [5] uses only group cohomology and his method may be adapted to the situation over an imaginary quadratic field. In this setting the Jacobians are replaced with certain sheaf cohomology groups H_{!}^{1}(X_{n}, M̃_{n}) and for a prime 𝔭 ⊂ O_{F} we have analogues of the two standard 𝔭degeneracy maps whose sum H_{!}^{1}(X_{0}, M̃_{0})^{2} → H_{!}^{1}(X_{1}, M̃_{1}) we will call α. (For precise definitions see section 2.) The main result of this note (Theorem 3.1) then asserts that the kernel of α is Eisenstein (for definition of “Eisenstein” see section 3).

Originally Ihara’s lemma had been used by Ribet [6] to prove the existence of congruences between modular forms of level N and those of level Np. His result, valid for forms of weight 2, was later generalized to arbitrary weight by Diamond [2], who used the language of cohomology like we chose to. A crucial ingredient in Diamond’s proof is the self-duality of H^{1}(Γ_{0}(N), M). Over imaginary quadratic fields, as over Q, there is a connection between the space of automorphic forms and the cohomology groups H_{!}^{1}(X_{n}, M̃_{n}) called the Eichler-Shimura-Harder isomorphism (cf. [10]). However, there seems to be no obvious way to adapt the approach of Ribet and Diamond to our situation as H_{!}^{1}(X_{n}, M̃_{n}) is not self-dual.

Date: August 22, 2007.

Ihara’s lemma was also used in the proof of modularity of Galois representations attached to elliptic curves over Q ([11], [1]). Thanks to the work of Taylor [9] one can attach Galois representations to a certain class of automorphic forms on Res_{F/Q}(GL_{2/F}). One could hope that Ihara’s lemma in our formulation could be useful in proving the converse to Taylor’s theorem, i.e., that ordinary Galois representations of Gal(\overline{F} /F) (satisfying appropriate conditions) arise from automorphic forms, but at this moment this is a mere speculation as too many other important ingredients of a potential proof seem to be missing.

The author would like to thank Trevor Arnold, Tobias Berger, Brian Conrad, Chandrashekhar Khare and Chris Skinner for many helpful and inspiring discussions.

2. Preliminaries

Let F be an imaginary quadratic extension of Q and denote by O_{F} its ring of integers. Let 𝔑 be an ideal of O_{F} such that the Z-ideal 𝔑 ∩ Z has a generator greater than 3. Let 𝔭 be a prime ideal such that 𝔭 ∤ 𝔑. Denote by Cl_{F} the class group of F and choose representatives of distinct ideal classes to be prime ideals 𝔭_{i}, i = 1, …, # Cl_{F}, relatively prime to both 𝔑 and 𝔭. Let π, (resp. π_{i}) be a uniformizer of the completion F_{𝔭} (resp. F_{𝔭_{i}}) of F at the prime 𝔭 (resp. 𝔭_{i}), and put π̃ (resp. π̃_{i}) to be the idele (…, 1, π, 1, …) ∈ A_{F}^{×} (resp. (…, 1, π_{i}, 1, …) ∈ A_{F}^{×}), where π (resp. π_{i}) occurs at the 𝔭-th place (resp. 𝔭_{i}-th place). We also put O_{(𝔭)} := ⋃_{j=0}^{∞} 𝔭^{−j}O_{F}.

For each n ∈ Z_{≥0}, we define compact open subgroups of GL_{2}(A_{F,f})

K_{n} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ ∏_{𝔮∤∞} GL_{2}(O_{F,𝔮}) | c ∈ 𝔑𝔭^{n}\} .

Here A_{F,f} denotes the finite adeles of F and O_{F,𝔮} the ring of integers of F_{𝔮}. For

n ≥ 0 we also set K_{n}^{𝔭} = [π̃ ; 1] K_{n} [π̃ ; 1]^{−1}.

For any compact open subgroup K of GL_{2}(A_{F,f}) we put

X_{K} = GL_{2}(F) \ GL_{2}(A_{F})/K · U_{2}(C) · Z_{∞},

where Z_{∞} = C^{×} is the center of GL_{2}(C) and U(2) := \{M ∈ GL_{2}(C) | M\overline{M}^{t} = I_{2}\} (here ‘bar’ denotes complex conjugation and I_{2} stands for the 2×2-identity matrix). If K is sufficiently large (which will be the case for all compact open subgroups we will consider) this space is a disjoint union of # Cl_{F} connected components X_{K} = ∐_{i=1}^{# Cl_{F}} (Γ_{K})_{i} \ Z, where Z = GL_{2}(C)/U_{2}(C)C^{×} and (Γ_{K})_{i} = GL_{2}(F) ∩

[π̃_{i} ; 1] K [π̃_{i} ; 1]^{−1}. To ease notation we put X_{n} := X_{K_{n}}, X_{n}^{𝔭} := X_{K_{n}^{𝔭}}, Γ_{n,i} :=

(Γ_{K_{n}})_{i} and Γ_{n,i}^{𝔭} := (Γ_{K_{n}^{𝔭}})_{i}.

We have the following diagram:

(2.1)  \xymatrix{… \ar[r]^{⊂} & K_{n+1} \ar[r]^{⊂} \ar[d]^{≀} & K_{n} \ar[r]^{⊂} \ar[d]^{≀} & K_{n−1} \ar[r]^{⊂} \ar[d]^{≀} & … \\ … \ar[r]^{⊂} & K_{n+1}^{𝔭} \ar[r]^{⊂} \ar@{-->}[ur]^{⊂} & K_{n}^{𝔭} \ar[r]^{⊂} \ar[ur]^{⊂} & K_{n−1}^{𝔭} \ar[r]^{⊂} & …}

where the horizontal and diagonal arrows are inclusions and the vertical arrows

are conjugation by the maps [π̃ ; 1]. Diagram (2.1) is not commutative, but it is

“vertically commutative”, by which we mean that given two objects in the diagram, two directed paths between those two objects define the same map if and only if the two paths contain the same number of vertical arrows.

Diagram (2.1) induces the following vertically commutative diagram of the corresponding symmetric spaces:

(2.2)  \xymatrix{… \ar[r] & X_{n+1} \ar[r] \ar[d]^{≀} & X_{n} \ar[r] \ar[d]^{≀} & X_{n−1} \ar[r] \ar[d]^{≀} & … \\ … \ar[r] & X_{n+1}^{𝔭} \ar[r] \ar@{-->}[ur] & X_{n}^{𝔭} \ar[r] \ar[ur] & X_{n−1}^{𝔭} \ar[r] & …}

The horizontal and diagonal arrows in diagram (2.2) are the natural projections and the vertical arrows are maps given by (g_{∞}, g_{f}) ↦ (g_{∞}, g_{f} [ π̃ ; 1 ]^{−1}).

Let M be a torsion abelian group of exponent relatively prime to #O_{F}^{×} endowed with a GL_{2}(F)-action. Denote by M̃_{K} the sheaf of continuous sections of the topological covering GL_{2}(F) \ (GL_{2}(A_{F})/K · U_{2}(C) · Z_{∞}) × M → X_{K}, where GL_{2}(F) acts diagonally on (GL_{2}(A_{F})/KU_{2}(C)Z_{∞})×M. Here M is equipped with the discrete topology. Since we will only be concerned with the case when M is a trivial GL_{2}(F)-module, we assume it from now on. This means that M̃_{K} is a constant sheaf. As above, we put M̃_{n} := M̃_{K_{n}} and M̃_{n}^{𝔭} := M̃_{K_{n}^{𝔭}}.

Given a surjective map ϕ : X_{K} → X_{K′}, we get an isomorphism of sheaves ϕ^{−1}M̃_{K′} ⟶^{∼} M̃_{K}, which yields a map on cohomology

H^{q}(X_{K′}, M̃_{K′}) → H^{q}(X_{K}, ϕ^{−1}M̃_{K′}) ≅ H^{q}(X_{K}, M̃_{K}).

Hence diagram (2.2) gives rise to a vertically commutative diagram of cohomology groups: (2.3)

\xymatrix{… & H^{q}(X_{n+1}, M̃_{n+1}) \ar[l] & H^{q}(X_{n}, M̃_{n}) \ar[l]_{α_{1}^{n,n+1}} \ar[dl]_{α_{1}^{n,(n+1)𝔭}} & H^{q}(X_{n−1}, M̃_{n−1}) \ar[l]_{α_{1}^{n−1,n}} \ar[dl]_{α_{1}^{n−1,n𝔭}} & … \ar[l] \\ … & H^{q}(X_{n+1}^{𝔭}, M̃_{n+1}^{𝔭}) \ar[l] \ar[u]_{≀}^{α_{𝔭}^{n+1}} & H^{q}(X_{n}^{𝔭}, M̃_{n}^{𝔭}) \ar[l]_{α_{1}^{n𝔭,(n+1)𝔭}} \ar[u]_{≀}^{α_{𝔭}^{n}} & H^{q}(X_{n−1}^{𝔭}, M̃_{n−1}^{𝔭}) \ar[l]_{α_{1}^{(n−1)𝔭,n𝔭}} \ar[u]_{≀}^{α_{𝔭}^{n−1}} & … \ar[l]}

These sheaf cohomology groups can be related to the group cohomology of Γ_{n,i} and Γ_{n,i}^{𝔭} with coefficients in M. In fact, for each compact open subgroup K with corresponding decomposition X_{K} = ∐_{i=1}^{# Cl_{F}} (Γ_{K})_{i} \ Z, we have the following commutative diagram in which the horizontal maps are inclusions:

(2.4)  \xymatrix{H_{!}^{q}(X_{K}, M̃_{K}) \ar[r] & H^{q}(X_{K}, M̃_{K}) \\ ⨁_{i=1}^{# Cl_{F}} H_{P}^{q} ((Γ_{K})_{i}, M) \ar[r] \ar[u] & ⨁_{i=1}^{# Cl_{F}} H^{q}((Γ_{K})_{i}, M) \ar[u]}

Here H_{!}^{q}(X_{K}, M̃_{K}) denotes the image of the cohomology with compact support H_{c}^{q}(X_{K}, M̃_{K}) inside H^{q}(X_{K}, M̃_{K}) and H_{P}^{q} denotes the parabolic cohomology, i.e., H_{P}^{q} ((Γ_{K})_{i}, M) := ker(H^{q}((Γ_{K})_{i}, M) → ⨁_{B∈B} H^{q}((Γ_{K})_{i,B}, M)), where B is the set of Borel subgroups of GL_{2}(F) and (Γ_{K})_{i,B} := (Γ_{K})_{i} ∩ B. The vertical arrows in diagram (2.4) are isomorphisms provided that there exists a torsion-free normal subgroup of (Γ_{K})_{i} of finite index relatively prime to the exponent of M. If K = K_{n} or K = K_{n}^{𝔭}, n ≥ 0, this condition is satisfied because of our assumption that 𝔑 ∩ Z has a generator greater than 3 and the exponent of M is relatively prime to #O_{F}^{×} (cf. [10], section 2.3). In what follows we may therefore identify the sheaf cohomology with the group cohomology. Note that all maps in diagram (2.3) preserve parabolic cohomology. The maps α_{1}^{∗,∗} are the natural restriction maps on group cohomology, so in particular they preserve the decomposition ⨁_{i=1}^{# Cl_{F}} H^{q}((Γ_{K})_{i}, M).

Using the identifications of diagram (2.4) we can prove the following result which will be useful later:

Lemma 2.1. The map α_{1}^{0,1} : H_{!}^{1}(X_{0}, M̃_{0}) → H_{!}^{1}(X_{1}, M̃_{1}) is injective.

Proof. Using the isomorphism between group and sheaf cohomology all we need to prove is that the restriction maps res_{i} : H^{1}(Γ_{0,i}, M) → H^{1}(Γ_{1,i}, M) are injective. Since M is a trivial Γ_{0,i}-module, the cohomology groups are just Homs, so it is enough to show the following statement: if G denotes the smallest normal subgroup of Γ_{0,i} containing Γ_{1,i}, then G = Γ_{0,i}. For this we use the decomposition

Γ_{0,i} = ∐_{\begin{matrix}k∈R(O_{F} /𝔭) \\ k∈𝔑 \end{matrix}} Γ_{1,i} \left[{1 \atop k 1}\right] ⊔ Γ_{1,i} [\begin{matrix}A & B \\ C & D \end{matrix}] ,

where the matrix [ \begin{matrix}A & B \\ C & D \end{matrix}] is chosen so that C and D are relatively prime elements of O_{F} with C ∈ 𝔑, D ∈ 𝔭, and A ∈ O_{F}, B ∈ 𝔭_{i} satisfy AD − BC = 1. Here R(O_{F} /𝔭) denotes a set of representatives in O_{F} of the distinct residue classes of O_{F} /𝔭. Let

[ \begin{matrix}a & b \\ c & d \end{matrix} ] ∈ Γ_{1,i} with d ∈ 𝔭. Then for any k ∈ 𝔑𝔭_{i}^{−1}O_{F} we have

[\begin{matrix}a & b \\ c & d \end{matrix}]^{−1} [\begin{matrix}1 + bdk & −b^{2}k \\ d^{2}k & 1 − bdk \end{matrix}] [\begin{matrix}a & b \\ c & d \end{matrix}] = \left[{1 \atop k 1}\right]

and the matrix [ \begin{matrix}1+bdk & −b^{2}k \\ d^{2}k & 1−bdk \end{matrix} ] ∈ Γ_{i,1}, hence G contains ∐\begin{matrix}k∈R(O_{F} /𝔭) \\ k∈𝔑 \end{matrix} Γ_{1,i} \left[{1 \atop k 1}\right], and

thus G = Γ_{i,0}. □

We can augment diagram (2.1) on the right by introducing one more group:

K_{−1} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ GL_{2}(F_{𝔭}) × ∏_{𝔮∤𝔭∞} GL_{2}(O_{F,𝔮}) | c ∈ 𝔑, ad − bc ∈ ∏_{𝔮∤∞} O_{F,𝔮}^{×}\} .

The group K_{−1} is not compact, but we can still define

Γ_{−1,i} := GL_{2}(F) ∩ [π̃_{i} ; 1] K_{−1} [π̃_{i} ; 1]^{−1}

for i = 1, … # Cl_{F}. After identifying the sheaf cohomology groups H_{!}^{1}(X_{0}, M̃_{0}) and H_{!}^{1}(X_{0}^{𝔭}, M̃_{0}^{𝔭}) with the groups ⨁_{i} H_{P}^{1} (Γ_{0,i}, M) and ⨁_{i} H_{P}^{1} (Γ_{0,i}^{𝔭}, M), respectively, using diagram (2.4), we can augment diagram (2.3) on the right in the following way (2.5)

\xymatrix{… & H_{!}^{q}(X_{1}, M̃_{1}) \ar[l] & H_{!}^{q}(X_{0}, M̃_{0}) \ar[l]_{α_{1}^{0,1}} \ar[dl]_{α_{1}^{0,1𝔭}} & H_{!}^{q}(X_{−1}, M̃_{−1}) \ar[l]_{α_{1}^{−1,0}} \ar[dl]_{α_{1}^{−1,0𝔭}} \\ … & H_{!}^{q}(X_{1}^{𝔭}, M̃_{1}^{𝔭}) \ar[l] \ar[u]_{≀}^{α_{𝔭}^{1}} & H_{!}^{q}(X_{0}^{𝔭}, M̃_{0}^{𝔭}) \ar[l]_{α_{1}^{0𝔭,1𝔭}} \ar[u]_{≀}^{α_{𝔭}^{0}}}

Here we put H_{!}^{q}(X_{−1}, M̃_{−1}) := ⨁_{i} H_{P}^{1} (Γ_{−1,i}, M) and the maps α_{1}^{−1,0} and α−1,0𝔭_{1}

are direct sums of the restriction maps.

The sheaf and group cohomologies are in a natural way modules over the corresponding Hecke algebras. (For the definition of the Hecke action on cohomology, see [10] or [3]). Here we will only consider the subalgebra T of the full Hecke

algebra which is generated over Z by the double cosets T_{𝔭′} := K [\begin{matrix}π′ \\ & 1 \end{matrix}] K and

T_{𝔭′,𝔭′} := K [\begin{matrix}π′ \\ & π′ \end{matrix}] K for π′ a uniformizer of F_{𝔭′} with 𝔭′ running over prime ideals

of O_{F} such that 𝔭′ ∤ 𝔑𝔭. The algebra T acts on all the cohomology groups in diagram (2.5). Moreover if 𝔭′ is principal, the induced action of T_{𝔭′} and T_{𝔭′,𝔭′} on the group cohomology respects the decomposition ⨁_{i} H_{∗}^{q}((Γ_{K})_{i}, M), where ∗ = ∅ or P.

3. Main result

We will say that a maximal ideal 𝔫 of the Hecke algebra T is Eisenstein if T_{𝔩} ≡ N𝔩 + 1 (mod 𝔫) for all ideals 𝔩 of O_{F} which are trivial as elements of the ray class group of conductor 𝔫. Such ideals 𝔩 are principal and have a generator l with l ≡ 1 (mod 𝔫). Here N𝔩 denotes the ideal norm.

From now on we fix a non-Eisenstein maximal ideal 𝔪 of the Hecke algebra T. Our main result is the following theorem.

Theorem 3.1. Consider the map H_{!}^{1}(X_{0}, M̃_{0})^{2} ⟶^{α} H_{!}^{1}(X_{1}, M̃_{1}) defined as α : (f, g) ↦ α_{1}^{0,1}f + α_{𝔭}^{1}α_{1}^{0,1𝔭}g. The localization α_{𝔪} of α is injective.

We prove Theorem 3.1 in two steps. Define a map β : H_{!}^{1}(X_{−1}, M̃_{−1}) → H_{!}^{1}(X_{0}, M̃_{0})^{2} by g′ ↦ (−α_{𝔭}^{0}α_{1}^{−1,0𝔭}g′, α_{1}^{−1,0}g′) and note that αβ = 0 by the vertical commutativity of diagram (2.5), i.e., ker α ⊃ β(H_{!}^{1}(X_{−1}, M̃_{−1})). We first prove Proposition 3.2. ker α = β(H_{!}^{1}(X_{−1}, M̃_{−1})).

Then we show

Proposition 3.3. H_{!}^{1}(X_{−1}, M̃_{−1})_{𝔪} = 0.

Propositions 3.2 and 3.3 imply Theorem 3.1.

The idea of the proof is due to Khare [5] and uses modular symbols, which we now define. Let D denote the free abelian group on the set B of all Borel subgroups of GL_{2}(F). The action of GL_{2}(F) on B by conjugation gives rise to a Z-linear action of GL_{2}(F) on D. We sometimes identify B with P^{1}(F) = \{\frac{a}{c} | a ∈ O_{F} , c ∈ O_{F} \ {0}} ∪ {∞}, on which GL_{2}(F) acts by the linear fractional transformations. Let D_{0} := {∑ n_{i}B_{i} ∈ D | n_{i} ∈ Z, B_{i} ∈ B, ∑ n_{i} = 0} be the subset of elements of degree zero. If X_{K} = ∐_{i} Γ_{i} \ Z, then for each Γ_{i} the exact sequence

0 → Hom_{Z}(Z, M) → Hom_{Z}(D, M) → Hom_{Z}(D_{0}, M) → 0

gives rise to an exact sequence

(3.1) 0 → M → Hom_{Z[Γ_{i}]}(D, M) → Hom_{Z[Γ_{i}]}(D_{0}, M) →

→ H^{1}(Γ_{i}, M) → H^{1}(Γ_{i}, Hom_{Z}(D, M)).

The group Hom_{Z[Γ_{i}]}(D_{0}, M) is called the group of modular symbols.

Lemma 3.4. Let Γ be a group acting on the set B of Borel subgroups of GL_{2}(F) and let C denote a set representatives for the Γ-orbits of B. Then for any trivial Γ-module W,

H^{1}(Γ, Hom_{Z}(D, W)) = ⨁_{c∈C} H^{1}(Γ_{c}, W),

where Γ_{c} is the stabilizer of c in Γ.

Proof. The Γ-module structure on Hom_{Z}(D, W) is defined via ϕ^{γ}(x) = ϕ(γ^{−1}x) and on ⨁_{c∈C} Ind_{Γ_{c}}^{Γ}W via

(f_{c_{1}}, …, f_{c_{n}})^{γ}(γ_{c_{1}}, …, γ_{c_{n}}) = (f_{c_{1}}, …, f_{c_{n}})(γ_{c_{1}}γ, …, γ_{c_{n}}γ).

Note that we have a Γ-module isomorphism Φ : Hom_{Z}(D, W) ⟶^{∼} ⨁_{c∈C} Ind_{Γ_{c}}^{Γ}W, given by Φ(ϕ)_{c}(γ) = ϕ(γ^{−1}c). Thus

H^{1}(Γ, Hom_{Z}(D, W)) ≃ H^{1}(Γ, ⨁_{c∈C} Ind_{Γ_{c}}^{Γ}W ) ≃ ⨁_{c∈C} H^{1}(Γ, Ind_{Γ_{c}}^{Γ}W)

since the action of Γ stabilizes Ind_{Γ_{c}}^{Γ}W for every c ∈ C. The last group is in turn isomorphic to ⨁_{c∈C} H^{1}(Γ_{c}, W) by Shapiro’s Lemma. □

By taking the direct sum of the exact sequences (3.1) and using Lemma 3.4, we obtain the exact sequence

(3.2) 0 → ⨁_{i} M → ⨁_{i} Hom_{Z[Γ_{i}]}(D, M) →

→ ⨁_{i} Hom_{Z[Γ_{i}]}(D_{0}, M) → ⨁_{i} H_{P}^{1} (Γ_{i}, M) → 0,

where the last group is isomorphic to H_{!}^{1}(X_{K}, M̃_{K}).

Remark 3.5. The space of modular symbols ⨁_{i} Hom_{Z[Γ_{i}]}(D_{0}, M) is also a Hecke module in a natural way. In fact it can be shown (at least if 𝔑 is square-free) that the localized map (⨁_{i} Hom_{Z[Γ_{i}]}(D, M))_{𝔪} → (⨁_{i} Hom_{Z[Γ_{i}]}(D_{0}, M))_{𝔪} is an isomorphism, but we will not need this fact.

4. Proof of Proposition 3.2

Suppose (f, g) ∈ ker α, i.e., α_{1}^{0,1}f = −α_{𝔭}^{1}α_{1}^{0,1𝔭}g. Let h_{1} = −g ∈ H^{1}(X_{0}, M̃_{0}) and h_{2} ∈ H^{1}(X_{0}^{𝔭}, M̃_{0}^{𝔭}) be the pre-image of f under the isomorphism α_{0}^{𝔭}. Then α_{1}^{0,1}α_{𝔭}^{0}h_{2} = α_{𝔭}^{1}α_{1}^{0,1𝔭}h_{1}. By the vertical commutativity of diagram (2.3), we have α_{1}^{0,1}α_{𝔭}^{0} = α_{𝔭}^{1}α_{1}^{0𝔭,1𝔭}, whence α_{𝔭}^{1}α_{1}^{0𝔭,1𝔭}h_{2} = α_{𝔭}^{1}α_{1}^{0,1𝔭}h_{1}. Since α_{𝔭}^{1} is an isomorphism, we get α_{1}^{0𝔭,1𝔭}h_{2} = α_{1}^{0,1𝔭}h_{1} ∈ H^{1}(X_{1}^{𝔭}, M̃_{1}^{𝔭}).

For K ⊂ K′ two compact open subgroups of GL_{2}(A_{F,f}) for which X_{K} = ∐_{i} Γ_{i} \ Z and X_{K′} = ∐_{i} Γ′_{i} \ Z with Γ_{i}, Γ′_{i} ∈ G_{i}, Γ_{i} ⊂ Γ′_{i}, we have a commutative diagram

(4.1)  \xymatrix{Hom_{Z[Γ_{i}]}(D_{0}, M) \ar[r]^{ϕ_{Γ_{i}}} & H_{P}^{1} (Γ_{i}, M) \\ Hom_{Z[Γ′_{i}]}(D_{0}, M) \ar[r]^{ϕ_{Γ′_{i}}} \ar[u]^{inclusion} & H_{P}^{1} (Γ′_{i}, M) \ar[u]^{res}} ,

where the maps ϕ_{Γ_{i}} and ϕ_{Γ′_{i}} denote the appropriate connecting homomorphisms from exact sequence (3.2). So far we have shown that

(4.2)  −α_{1}^{0,1𝔭}g = α_{1}^{0,1𝔭}h_{1} = α_{1}^{0𝔭,1𝔭}h_{2}.

We identify g with a tuple (g_{i})_{i} ∈ ⨁_{i} H_{P}^{1} (Γ_{0,i}, M) and define (h_{1})_{i} ∈ H_{P}^{1} (Γ_{0,i}, M) and (h_{2})_{i} ∈ H_{P}^{1} (Γ_{0,i}^{𝔭}, M) similarly. Equality (4.2) translates to

(4.3)  −g_{i}|_{Γ_{1,i}^{𝔭}} = (h_{1})_{i}|_{Γ_{1,i}^{𝔭}} = (h_{2})_{i}|_{Γ_{1,i}^{𝔭}}

Fix g_{mod,i} ∈ ϕ_{Γ_{1,i}^{𝔭}}^{−1}(g) and regard it as an element of Hom_{Z}(D_{0}, M) invariant under Γ_{0,i}. Using diagram (4.1) with Γ_{i} = Γ_{1,i}^{𝔭} and Γ′_{i} = Γ_{0,i}^{𝔭} and equality (4.3) we conclude that there exists (h_{2})_{mod,i} ∈ ϕ_{Γ_{0,i}^{𝔭}}^{−1}((h_{2})_{i}) such that (h_{2})_{mod,i} = −g_{mod,i} regarded as elements of Hom_{Z}(D_{0}, M). Hence g_{mod,i} is invariant under both Γ_{0,i} and Γ_{0,i}^{𝔭}.

Lemma 4.1. For i = 1, …, # Cl_{F} the groups Γ_{0,i} and Γ_{0,i}^{𝔭} generate Γ_{−1,i}.

Proof. This is an immediate consequence of Theorem 3 on page 110 in [8]. □

Using Lemma 4.1 we conclude that g_{mod,i} ∈ Hom_{Z[Γ_{−1,i}]}(D_{0}, M). Put g′_{i} = ϕ_{Γ_{−1,i}}(g_{mod,i}). Again, by the commutativity of diagram (4.1) with Γ′_{i} = Γ_{−1,i} and Γ_{i} = Γ_{0,i} we have g′_{i}|_{Γ_{0,i}} = g_{i}. Hence g′ := (g′_{i})_{i} ∈ ⨁_{i} H_{P}^{1} (Γ_{−1,i}, M) ≅ H_{!}^{1}(X_{−1}, M̃_{−1}) satisfies g = α_{1}^{−1,0}g′.

Thus 0 = α_{1}^{0,1}f + α_{𝔭}^{1}α_{1}^{0,1𝔭}g = α_{1}^{0,1}f + α_{𝔭}^{1}α_{1}^{0,1𝔭}α_{1}^{−1,0}g′. By the vertical commutativity of diagram (2.5) we have α_{𝔭}^{1}α_{1}^{0,1𝔭}α_{1}^{−1,0} = α_{1}^{0,1}α_{𝔭}^{0}α_{1}^{−1,0𝔭}, so 0 = α_{1}^{0,1}f + α_{1}^{0,1}α_{𝔭}^{0}α_{1}^{−1,0𝔭}g′. Since α_{1}^{0,1} is injective by Lemma 2.1, this implies that f = −α_{𝔭}^{0}α_{1}^{−1,0𝔭}g′. Hence (f, g) = (−α_{𝔭}^{0}α_{1}^{−1,0𝔭}g′, α_{1}^{−1,0}g′) ∈ β(H_{!}^{1}(X_{−1}, M̃_{−1})), completing the proof of Proposition 3.2.

5. Proof of Proposition 3.3

In this section we prove that for a principal ideal 𝔩 = (l) ⊂ O_{F} such that l ≡ 1 (mod 𝔑) we have T_{𝔩}f = (N𝔩 + 1)f on elements f ∈ H_{!}^{1}(X_{−1}, M̃_{−1}) ≅ ⨁_{i} H_{P}^{1} (Γ_{−1,i}, M). For such an ideal 𝔩, the operators T_{𝔩} preserve each direct summand H_{P}^{1} (Γ_{−1,i}, M). The restriction of T_{𝔩} to H_{P}^{1} (Γ_{−1,i}, M) is given by the usual action of the double coset Γ_{−1,i} [ \begin{matrix}1 \\ & l \end{matrix}] Γ_{−1,i} on group cohomology (see, e.g., [3]). For k, l ∈ O_{F} we put σ_{k,l} := \left[{1 k \atop l}\right], σ_{l} := [ \begin{matrix}l \\ & 1 \end{matrix} ]. To describe the action of T_{𝔩} explicitly we use the following lemma.

Lemma 5.1. Let 𝔩 = (l) be a principal ideal of O_{F} and n ≥ −1. Then

Γ_{n,i} [\begin{matrix}1 \\ & l \end{matrix}] Γ_{n,i} = ∐_{\begin{matrix}k∈R(O_{F} /𝔩) \\ k∈𝔭_{i} \end{matrix}} Γ_{n,i}σ_{k,l} ⊔ Γ_{n,i}σ_{l},

where R(O_{F} /𝔩) denotes a set of representatives of O_{F} /𝔩 in O_{F}.

Proof. This is easy. □

Lemma 5.2. Let n ≥ 3 be an odd integer. Every ideal class c of F contains infinitely many prime ideals 𝔮 such that (N𝔮 − 1, n) = 1.

Proof. We assume F ∩Q(ζ_{n}) = Q, the other case being easier. Let G = Gal(F/Q), N = Gal(Q(ζ_{n})/Q) and C = Gal(H/F) ≅ Cl_{F}, where H denotes the Hilbert class field of F. We have the following diagram of fields

(5.1)  [figure: F_{n}H; C; N; F_{n} = F Q(ζ_{n}); G; N; H; C; Q(ζ_{n}); N; F; G; Q]

Choose (σ, τ) ∈ Gal(F_{n}H/H) × Gal(F_{n}H/F_{n}) ≅ N × C, such that

σ ∈ Gal(F_{n}H/H) ≅ (Z/nZ)^{×}

corresponds to an element σ̃ ∈ (Z/nZ)^{×} with σ̃ ≢ 1 modulo any of the divisors of n, and τ ∈ Gal(F_{n}H/F_{n}) ≅ C ≅ Cl_{F} corresponds to the ideal class c. By the Chebotarev density theorem there exist infinitely many primes 𝔔 of the ring of integers of F_{n}H such that Frob_{𝔔} = (σ, τ). Then the infinite set of primes 𝔮 of O_{F} lying under such 𝔔 satisfy the condition of the lemma, i.e., 𝔮 ∈ c and (N𝔮 − 1, n) = 1. □

By Lemma 5.2 we may assume that the ideals 𝔭_{i} were chosen so that N𝔭_{i} − 1 is relatively prime to the exponent of M for all i = 1, …, # Cl_{F}.

Proof of Proposition 3.3. Let f ∈ H_{P}^{1} (Γ_{−1,i}, M) and let 𝔩 = (l) be a principal ideal of O_{(𝔭)} with l ≡ 1 (mod 𝔑). We will prove that f_{𝔩} := T_{𝔩}f − (N𝔩 + 1) = 0. By the definition of parabolic cohomology, we have f_{𝔩}(Γ_{−1,i} ∩ B) = 0 for all B ∈ B.

Moreover, as the exponent of M is relatively prime to #O_{F}^{×}, it is enough to prove that f_{𝔩}(Γ̃_{i}) = 0, where Γ̃_{i} := Γ_{−1,i} ∩ SL_{2}(F). Put

SL_{2}(O_{(𝔭)})_{i} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ SL_{2}(F) | a, d ∈ O_{(𝔭)}, b ∈ 𝔭_{i}O_{(𝔭)}, c ∈ 𝔭_{i}^{−1}O_{(𝔭)}\} .

We first show that f_{𝔩} = 0 on the i-th principal congruence subgroup

Γ_{𝔑,i} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ SL_{2}(O_{(𝔭)})_{i} | b, c ∈ 𝔑O_{(𝔭)}, a ≡ d ≡ 1 mod 𝔑O_{(𝔭)}\} .

If x ∈ 𝔑𝔭_{i}O_{(𝔭)}, and g ∈ SL_{2}(O_{(𝔭)}), then g \left[{1 x \atop 1}\right] g^{−1} ∈ SL_{2}(O_{(𝔭)}) ∩ Γ_{−1,i} and

f_{𝔩} (g \left[{1 x \atop 1}\right] g^{−1}) = 0 by the definition of parabolic cohomology. So f_{𝔩} = 0 on the

smallest normal subgroup H of SL_{2}(O_{(𝔭)}) containing matrices of the form \left[{1 x \atop 1}\right]

with x ∈ 𝔑𝔭_{i}O_{(𝔭)}. By a theorem of Serre [7],

H = Γ_{𝔑𝔭_{i}} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ SL_{2}(O_{(𝔭)}) | b, c ∈ 𝔑𝔭_{i}O_{(𝔭)}, a ≡ d ≡ 1 mod 𝔑𝔭_{i}O_{(𝔭)}\} .

Thus f_{𝔩} = 0 on Γ_{𝔑𝔭_{i}}. Put

Γ′_{𝔑𝔭_{i}} := \{[\begin{matrix}a & b \\ c & d \end{matrix}] ∈ SL_{2}(O_{(𝔭)})_{i} | b, c ∈ 𝔑𝔭_{i}O_{(𝔭)}\} .

Since Γ_{𝔑𝔭_{i}} is a normal subgroup of Γ′_{𝔑𝔭_{i}} of index N𝔭_{i} − 1 we have f_{𝔩} (Γ′_{𝔑𝔭_{i}}) = 0 by Lemma 5.2 and our choice of 𝔭_{i}. On one hand f_{𝔩} is zero on the elements of

the form \left[{1 \atop c 1}\right], c ∈ 𝔑𝔭_{i}^{−1}O_{(𝔭)}, (again by the definition of parabolic cohomology)

and on the other hand elements of this form together with Γ′_{𝔑𝔭_{i}} generate Γ_{𝔑,i}, so f_{𝔩}(Γ_{𝔑,i}) = 0, as asserted.

Thus f_{𝔩} descends to the quotient Γ̃_{i}/Γ_{𝔑,i}. However, on this quotient all σ_{k,l} and σ_{l} act as the identity, since l ≡ 1 (mod 𝔑) and we can always choose k ∈ 𝔑𝔭_{i}. Thus f_{𝔩} = 0. □

References

1. Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Amer. Math. Soc., 14(4):843–939 (electronic), 2001.

2. Fred Diamond. Congruence primes for cusp forms of weight k ≥ 2. Astérisque, (196-197):6, 205–213 (1992), 1991. Curbes modulaires et courbes de Shimura (Orsay, 1987/1988).

3. Haruzo Hida. Elementary theory of L-functions and Eisenstein series, volume 26 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1993.

4. Yasutaka Ihara. On modular curves over finite fields. In Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), pages 161–202. Oxford Univ. Press, Bombay, 1975.

5. Chandrashekhar Khare. Congruences between cusp forms: the (p, p) case. Duke Math. J., 80(3):631–667, 1995.

6. Kenneth A. Ribet. Congruence relations between modular forms. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), pages 503–514, Warsaw, 1984. PWN.

7. Jean-Pierre Serre. Le problème des groupes de congruence pour SL2. Ann. of Math. (2), 92:489–527, 1970.

8. Jean-Pierre Serre. Arbres, amalgames, SL_{2}. Société Mathématique de France, Paris, 1977. Avec un sommaire anglais, Rédigé avec la collaboration de Hyman Bass, Astérisque, No. 46.

9. Richard Taylor. l-adic representations associated to modular forms over imaginary quadratic fields. II. Invent. Math., 116(1-3):619–643, 1994.

10. Éric Urban. Formes automorphes cuspidales pour GL_{2} sur un corps quadratique imaginaire. Valeurs spéciales de fonctions L et congruences. Compositio Math., 99(3):283–324, 1995.

11. Andrew Wiles. Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2), 141(3):443–551, 1995.