Reducible Galois representations and arithmetic homology for <span class="mathjax-formula">$\protect \mathrm{GL}(4)$</span>

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Avner Ash & Darrin Doud

Reducible Galois representations and arithmetic homology forGL(4) Volume 25, n^o2 (2018), p. 207-246.

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Reducible Galois representations and arithmetic homology for GL ( 4 )

Darrin Doud

Abstract

We prove that a sum of two odd irreducible two-dimensional Galois representations with squarefree relatively prime Serre conductors is attached to a Hecke eigenclass in the homology of a subgroup of GL(4,Z), with the level, nebentype, and coefficient module of the homology predicted by a generalization of Serre’s conjecture to higher dimensions. To do this we prove along the way that any Hecke eigenclass in the homology of a congruence subgroup of a maximal parabolic subgroup of GL(n,Q)has a reducible Galois representation attached, where the dimensions of the components correspond to the type of the parabolic subgroup. Our main new tool is a resolution ofZby GL(n,Q)-modules consisting of sums of Steinberg modules for all subspaces ofQⁿ.

1. Introduction

Serre’s conjecture [23] (now a theorem of Khare, Wintenberger, and Kisin [18, 19, 20]) gives a connection between odd irreducible Galois representationsρ:G_Q→GL(2,F¯p) and modular forms that are simultaneous eigenvectors of all the Hecke operators. Via the Eichler–Shimura isomorphism [24], it can be interpreted as giving a connection between such Galois representations and elements of a cohomology groupH¹(Γ₀(N),V)for an appropriate coefficient moduleV. This interpretation of Serre’s conjecture was generalized by [10] to a conjecture relating odd Galois representations ρ : G_Q → GL(n,F¯p) to eigenclasses of the Hecke operators in cohomology groups H^∗(Γ₀(n,N),V), where Γ₀(n,N)is the congruence subgroup of SL(n,Z) that generalizes Γ₀(N) ⊂ SL(2,Z). Refinements of the conjecture [8, 16] make more precise predictions concerning the proper coefficient modules.

Some proven cases of the conjectured connection between Galois representations and cohomology eigenclasses are known. In particular, the conjecture is known for two- dimensional Galois representations. In [12], the conjecture is proven for certain irreducible symmetric square representations of odd irreducible two-dimensional representations.

Our long-term goal is to prove the conjecture for any odd reducible Galois representation under the assumption that the conjecture holds for each constituent. In [3], odd Galois representations that are sums of characters are shown to correspond to cohomology eigenclasses. This work was extended in [5] to show that any sum of characters with a Galois representation satisfying the conjecture such that the resulting representation

Keywords:Galois representations, arithmetic homology.

2010Mathematics Subject Classification:11F75, 11R80.

is odd will satisfy the conjecture. In [6], it is shown that if ρ₁ and ρ₂are two Galois representations, each attached to a cohomology eigenclass with trivial coefficient module, then a twisted sum of the two representations will also be attached to a cohomology Hecke eigenclass with trivial coefficients (although not in the cohomology of a group of the formΓ₀(n,N)).

In this paper, we prove that Galois representations of the formρ₁⊕ρ₂of squarefree level N with ρ₁ and ρ₂ two-dimensional irreducible and odd are attached to Hecke eigenclasses in the cohomology ofΓ₀(4,N). Together with our previous results cited above, we have now proven the conjecture of [8] for all odd reducible four-dimensional Galois representations, as long as the constituents have squarefree pairwise relatively prime conductors, and satisfy the conjecture themselves.

The two main new tools that we use in the proof are an exact sequence of GL(n,K)- modules for any fieldKthat involves the Steinberg modules for GL(W)for all subspaces W ofKⁿ (Section 4), and Theorem 11.5 on the reducibility of Galois representations attached to the cohomology of arithmetic subgroups of parabolic subgroups of GL(n,Q). The exact sequence of Section 4 generalizes the exact sequence used in [5, 7], which only works forn=3, to arbitraryn. It is a resolution ofZby non-free modules which are induced from various parabolic subgroups. Using a certain Hecke equivariant spectral sequence arising from this exact sequence, we are able to construct a system of Hecke eigenvalues that have ρ₁⊕ρ₂attached.

Our other main tool, Theorem 11.5 proves that any Hecke eigenclass attached to the cohomology of a maximal parabolic subgroup of type(n₁,n₂)has an attached Galois representation that is a sum of (possibly reducible) Galois representationsρ₁⊕ρ₂, with ρ_i :G_Q→GL(n_i,F¯p)fori=1,2.

On the more technical side, this paper includes extensions fromn=3 to generaln of the study we made in [5, 7] of orbits of subspaces ofQⁿ under certain congruence subgroups of GL(n,Z). We also study the action of a Levi component on the groups H∗(Γ∩U,M)appearing in the Hochschild–Serre spectral sequence for

1→Γ∩U→Γ∩P→Γ∩L→1,

where U is the unipotent radical of a maximal parabolic subgroupP=LUof GL(n,Q). We show in Theorem 7.11 that under certain hypotheses, the Hecke algebra forΓ_P acts in an equivariant fashion on the spectral sequence. This is surprisingly hard to do and we don’t even know if this continues to be the case if Pis not maximal. A third generalization of our earlier work concerns the detailed action of the matrices defining the Hecke operators on the homology. The last new technical point regards the interplay of the Hecke operators and the Künneth formula in Section 10.

We expect to be able to apply these methods to higher dimensional Galois represen- tations; at present, we are not able to because we have not yet been able to prove that a certain Hochschild–Serre spectral sequence, used to construct the systems of Hecke eigenvalues that we need, degenerates whenn>4.

We know that a system of Hecke eigenvalues appears inH^k(Γ₀(n,N),V)if and only if it appears inH_k(Γ₀(n,N),V). We have discussed mostly cohomology in the introduction to conform with common usage. Our theorems and proofs below will all be stated for homology.

2. Galois representations, Hecke Operators, cohomology and homology Letp>2 be a prime number, and letF=Fp. Throughout the paper, we use Borel–Serre duality for subgroups of GL(n,Z)[13, Section 11.4], and the fact that the Borel–Serre duality isomorphism is Hecke equivariant [9, Corollary 3]; when we do this we will requirep >n+1, so that no torsion element of GL(n,Z)has order divisible byp. By a Galois representation we mean a continuous homomorphism ρ:G_Q→GL(n,F)for some positive integern. For each prime`, we fix a choice of Frobenius element Fr`∈G_Q; we use the arithmetic Frobenius, so that ifω : G_Q → F is the cyclotomic character, ω(Fr`)=`. Ifρis unramified at`, thenρ(Fr`)is defined up to similarity. Hence, forρ unramified at`, the characteristic polynomial det(I−ρ(Fr`)X)ofρ(Fr`)is well defined.

Definition 2.1. Letn>1 andN ∈N, and letpbe a prime inZ.

(1) S₀^±(n,N)consists of the set of alln×nmatrices with integer entries and nonzero determinant prime topNwhose first row is congruent to(∗,0, . . . ,0)moduloN. (2) S₀(n,N)consists of elements ofS₀^±(n,N)with positive determinant.

(3) Γ₀^±(n,N)=S₀^±(n,N) ∩GL(n,Z). (4) Γ₀(n,N)=S₀(n,N) ∩SL(n,Z).

For a given primepand positive integerNprime top,(Γ₀(n,N),S₀(n,N))is a Hecke pair (see [1]), and we denote theF-algebra of its double cosets byH_n,N. We note that H_n,N is commutative, and is generated by the double cosets

Γ₀(n,N)s(`,n,k)Γ₀(n,N),

wheres(`,n,k)=diag(1, . . . ,1, `, . . . , `)is a diagonal matrix withkcopies of`on the diagonal,`runs over all primes not dividingpN, and 0≤k≤n. The algebraH_n,N acts on the homology or cohomology ofΓ₀(n,N)with coefficients in anyF[S₀(n,N)]-module

M. When the double coset ofs(`,n,k)acts on homology or cohomology, we will denote it byTn(`,k).

Definition 2.2. LetVbe anyH_n,N-module, and suppose thatv∈V is a simultaneous eigenvector of all theTn(`,k)for` - pN, with eigenvalues a(`,k) ∈ F. Suppose that ρ:G<sub>Q</sub>→GL(n,F)is a Galois representation unramified outsidepN. We say thatρis attachedtovif, for all`-pN,

det(I−ρ(Fr`)X)= Õn

k=0

(−1)^k`<sup>k(k−1)/2</sup>a(`,k)X^k. Ifρis attached tov∈V, we will also say thatρfitsV.

3. Conjectures relating Galois representations and arithmetic homology/co- homology

Definition 3.1([7]). LetSbe a subsemigroup of the matrices in GL(n,Q)with integer entries whose determinants are prime to pN. A(p,N)-admissible S-moduleM is an F[S]-module of the formM⁰⊗F, whereM⁰is anF[S]-module on whichS∩GL(n,Q)⁺ acts via its reduction modulop, andis a character:S →F^×which factors through the reduction ofSmoduloN. HereFis the vector spaceF, withSacting as multiplication via . An admissible module is one which is(p,N)-admissible for some choice ofpandN. We can construct(p,N)-admissible modules by starting with GL(n,Fp)-modules, and lettingSact via reduction modulop. We have the following parametrization of irreducible GL(n,Fp)-modules.

Theorem 3.2([15]). Call ann-tuple of(a₁, . . . ,a_n)of integersp-restricted if0≤a_n <

p−1and, for eachi <n,0 ≤ a_i −a_i+1 ≤ p−1. Then there is a bijection between p-restrictedn-tuples of integers and irreducibleF[GL(n,Fp)]-modules, with then-tuple (a₁, . . . ,a_n)corresponding to the unique simple submodule of the dual Weyl module with highest weight(a₁, . . . ,an).

Definition 3.3. Denote byF(a₁, . . . ,a_n)the irreducibleF[GL(n,Fp)]-module correspond- ing to thep-restrictedn-tuple(a₁, . . . ,a_n).

As described above,F(a₁, . . . ,a_n)becomes anS-module on whichSacts via reduction modulop. We will relax the condition on the value ofa_n, allowing it to be an arbitrary integer; this has the effect that a given module corresponds to infinitely manyn-tuples, all congruent to somep-restrictedn-tuple modulop−1, and it allows flexibility in specifying modules. Given a character:S→F^×that factors through the reduction ofSmoduloN, we see thatF(a₁, . . . ,a_n) =F(a₁, . . . ,a_n) ⊗F is a(p,N)-admissible module.

If S = S₀(n,N) and : (Z/NZ)^× → F^× is a character, we will also denote by :S→F^×the character sendings∈Sto the image underof the modNreduction of the(1,1)entry ofs. In this case, we callanebentype character.

In order to state the main conjecture of [8], we recall the following definition.

Definition 3.4. Forn > 1, a Galois representationρ : G_Q → GL(n,F)is odd if the image of complex conjugation is similar to a matrix with alternating 1’s and−1’s on the diagonal. A Galois representation ρ:G_Q →GL(1,F)=F^×is odd if the image of complex conjugation is−1, and is called even otherwise.

Conjecture 3.5([8, Conjecture 3.1]). For any odd Galois representation ρ: G_Q → GL(n,F), we may find an integerN(called the level), an irreducibleGL(n,Fp)-module M (called the weight), and a Dirichlet character(called the nebentype), such thatρfits H^k(Γ₀(N),M).

In fact, [8, Conjecture 3.1] predicts the level, weight and nebentype from the structure of ρ. We do not give these definitions in detail in this paper; see [8] for detailed descriptions.

In this paper, we will generalize techniques developed in [3, 5, 7] to prove the following theorem.

Theorem 3.6. Letp>5. Fori=1,2, letρ_i :G_Q→GL(2,F)be odd, irreducible Galois representations with squarefree relatively prime levels. Thenρ₁⊕ρ₂fits at least one of H₆(Γ₀(n,N),M)orH₂(Γ₀(n,N),M), withN,M, andas predicted by[8].

4. A Steinberg module exact sequence

Throughout this section, fix a fieldK. In this section, we derive a resolution ofZby GL(n,K)-modules that generalizes the resolution used in [5], which only works for GL(3,K).

For a vector spaceWoverK, denote byP(W)the projective space(W− {0})/K^×of nonzero vectors inWmodulo scalar multiplication. For any collectionw₁, . . . ,w_k ∈P(W), we will denote by span(w₁, . . . ,w_k)the subspace ofW spanned by lifts of thew_i toW (we see easily that the span is independent of the choice of lifts).

We recall the definition of the sharbly complex.

Definition 4.1([2]). LetW be ak-dimensional vector space over a fieldK. Fori ≥0, define thei-sharblies Shi(W)to be the freeZ-module generated by the(k+i)-tuples (w₁, . . . ,w_k+i)forw_i ∈P(W), modulo theZ-span of the following elements:

(1) (w_σ(1), . . . ,w_σ(k+i)) − (−1)^σ(w₁, . . . ,w_k+i)for all permutationsσ∈S_k+i,

(2) (w₁, . . . ,w_k+i)if{w₁, . . . ,w_k+i}does not spanW.

We denote a basis element of thei-sharblies by the symbol[w₁, . . .w_k+i]where each w_j ∈P(W), with (1) implying that this symbol is antisymmetric in the entries, and (2) implying that it is 0 if the entries do not spanW. By considering GL(W)to act onWby right multiplication, there is a natural right action of GL(W)on Shi(W). The boundary mapd_i: Shi(W) →Shi−1(W)is given by

d_i([w₁, . . . ,w_k+i])= Õk+i

j=1

(−1)^j[w₁, . . . ,wˆ_j, . . . ,w_k+i].

Thesharbly complexis the complex of right GL(W)-modules

· · · →Shi(W)−→^di Shi−1(W) → · · · →Sh1(W)−→^d1 Sh0(W).

By [2] the Steinberg module St(W)is isomorphic to the cokernel of the mapd₁ : Sh1(W) →Sh0(W), and therefore we get a resolution of the Steinberg module:

· · · →Shj(W) →Shj−1(W) → · · · →Sh1(W) →Sh0(W) →St(W) →0. We will denote the image of a 0-sharbly[w₁, . . . ,w_k]in St(W)bynw₁, . . . ,w_ko. We note that ifWis 0-dimensional, the Steinberg module and the 0-sharblies are isomorphic toZ(with trivial GL(W)-action), generated by the empty symbolsn oand[ ]. IfW is 1-dimensional, we also have that the Steinberg module and the 1-sharblies are isomorphic toZ(with trivial GL(W)-action), generated by the symbolsnwoand[w], wherewis the unique element ofP(W).

Theorem 4.2. LetVbe ann-dimensional vector space overKwithn>0. Then there is an exact sequence ofGL(V)-modules

0→Ê

Wⁿ

St(Wⁿ)−→^δn Ê

Wⁿ⁻¹

St(Wⁿ⁻¹)−→ · · ·^δn−1 −→^δ2 Ê

W¹

St(W¹)−→^δ1 Ê

W⁰

St(W⁰) →0, where eachWⁱ runs through all subspaces ofV of dimensioni, and the map

δ_k :Ê

W^k

St(W^k) →Ê

Wk−1

St(W^k−1) is defined by

δ_k(nw₁, . . . ,w_ko)= Õk

j=1

(−1)^jnw₁, . . . ,wˆ_j, . . . ,w_ko, fornw₁, . . . ,w_ko∈St(W^k), withW^k =span(w₁, . . . ,w_k)and

nw₁, . . . ,wˆ_j, . . . ,w_ko∈St(W^k−_j ¹), whereW_j^k−1=span(w₁, . . . ,wˆ_j, . . . ,w_k).

Proof. Forγ∈GL(V), and a generatornw₁, . . . ,w_ko∈St(W^k), we define nw₁, . . . ,w_koγ=nw₁γ, . . . ,w_kγo∈St(W^kγ).

With this action, we see easily that Ê

W^k

St(W^k)

is a GL(V)-module and thatδ_kis an equivariant map of GL(V)-modules.

We now check that eachδ_k is a well defined map.

To begin, letW be a k-dimensional subspace ofV. We may write the Steinberg module ofW asA(W)/(B(W)+C(W)), whereA(W)is the freeZ-module generated by antisymmetric symbols (w₁, . . . ,w_k)withw_i ∈ P(W),B(W)is generated by all such symbols where w₁, . . . ,w_k are contained in a proper subspace ofW^k, andC(W) is generated by elements of the form

k

Õ

j=1

(−1)^j(w₁, . . . ,wˆ_j, . . . ,w_k).

Defineφ:A(W) → ⊕St(W^k−1)by (w₁, . . . ,w_k) 7→

k

Õ

j=1

(−1)^jnw₁, . . . ,wˆ_j, . . . ,w_ko.

Note that in any case where the symbolnw₁, . . . ,wˆ_j, . . . ,w_kois not in a unique Steinberg module because the dimension of the span ofw₁, . . . ,wˆ_j, . . . ,w_k is less thank−1= dimW^k−1, the symbol vanishes, regardless of which module it is considered to lie in.

Hence, sinceA(W)is free overZ,φis well-defined, and ifφmaps bothB(W)andC(W) to 0, then the mapδ_kthat it induces onA(W)/(B(W)+C(W))will be well defined.

NowφmapsC(W)to 0 by the standard argument that the boundary of the boundary is 0. Further, if(w₁, . . . ,w_k) ∈B(W), then lettingW^k−1be a(k−1)-dimensional subspace ofW containingw₁, . . . ,w_k, we find that

Õk

j=1

(−1)^j(w₁, . . . ,wˆ_j. . . ,w_k) ∈C(W^k−1).

NowC(W^k−1)maps to 0 in St(W^k−1), soφ(w₁, . . . ,w_k)=0 andφ(B(W))=0. Thus, we see thatδ_kis well defined.

It is clear that the compositionδ_k−1◦δ_kis equal to 0 fork>0. In addition,δ₁is clearly surjective, since there is only one zero-dimensional subspace ofVandδ₁(w)=−n ofor anyw∈V.

Now we wish to prove that any element of kerδ_kis contained in the image ofδ_k+1.

Suppose that for some index setA, and some integersc_afora∈ A, we have s= Õ

a∈ A

c_anw₁^a, . . . ,w_k^ao

is in the kernel ofδ_k for 0<k<n. We will show that it is in the image ofδ_k+1. Choose an arbitraryx∈P(V). Then

Õ

a

c_anx,w^a₁, . . . ,w_k^ao∈Ê

W^k+1

St(W^k+1)

(note that some of the terms may be 0, ifx∈span(w^a₁, . . . ,w_k^a)). In any case, we have

δ_k+1 Õ

a

c_anx,w₁^a, . . . ,w_k^ao

!

=−Õ

a

c_anw₁^a, . . . ,w_k^ao

−Õ

a

Õk

j=1

(−1)^jcanx,w₁^a, . . . ,wˆ^a_j, . . . ,w_k^ao. Adding this tos, we obtain a new elements⁰∈kerδ_k that differs fromsby an element of the image ofδ_k+1. It thus suffices to prove thats⁰is in the image ofδ_k+1. We note that each symbol comprisings⁰has as its first component the chosenx. Hence, (changing the w_i^a, thec_a, and indeed the index setA), we may write

s⁰= Õ

a∈ A

c_anx,w₂^a, . . . ,w_k^ao. By eliminating terms where{x,w₂^a, . . . ,w^a

k}does not span ak-dimensional space, we may also assume that for eacha, we havex<span(w₂^a, . . . ,w_k^a).

Now,

0=δ_k(s⁰)=−Õ

a

c_anw₂^a, . . . ,w_k^ao− Õk

j=1

Õ

a

(−1)^jc_anx,w₂^a, . . . ,wˆ^a_j, . . . ,w_k^ao. Since, for eacha, we havex<span(w₂^a, . . . ,w_k^a), we see that no terms of the double sum are in the same component of

Ê

W^k−1

St(W^k−1) as any term in the first sum. Hence, we must have

Õ

a

c_anw₂^a, . . . ,w_k^ao=0.

For each(k−1)-dimensional subspaceWofV, setA_W ={a: span(w₂^a, . . . ,w_k^a)=W}. Then eachais in precisely oneA_W. We see that for eachW

Õ

a∈ AW

c_anw^a₂, . . . ,w_k^ao=0.

For anyW withA_W nonempty, since the Steinberg module ofW is the cokernel of Sh1(W)−→^d1 Sh0(W), we see that there is an index setB, integerscb and elements [y₁^b, . . . ,y^b_k] ∈Sh1(W)for eachb∈ Bsuch that

Õ

a∈ A_W

c_a[w₂^a, . . . ,w_k^a]=d₁ Õ

b∈ B

c_b[y₁^b, . . . ,y^b_k]

!

=Õ

b∈ B k

Õ

j=1

(−1)^jc_b[y₁^b, . . . ,yˆ^b_j, . . . ,y_k^b].

LetWxbe the span ofWand a lift ofx. Then in Sh0(W_x)we have Õ

a∈ AW

c_a[x,w₂^a, . . . ,w_k^a]= Õ

b∈ B k

Õ

j=1

(−1)^jc_b[x,y₁^b, . . . ,yˆ^b_j, . . . ,y_k^b].

Hence, in St(W_x), Õ

a∈ A_W

c_anx,w^a₂, . . . ,w_k^ao= Õ

b∈ B k

Õ

j=1

(−1)^jc_bnx,y^b₁, . . . ,yˆ^b_j, . . . ,y_k^bo. Becausey₁^b, . . . ,y_k^bspan a(k−1)-dimensional subspace, this equals

−Õ

b

©

­

«

cbny₁^b, . . . ,y^b_ko− Õk

j=1

(−1)^jcbnx,y₁^b, . . . ,yˆ^b_j, . . . ,y_k^boª

®

¬ ,

which is equal to

−δ_k+1 Õ

b

cbnx,y₁, . . . ,y_ko

! , so that

Õ

a∈ AW

c_anx,w^a₂, . . . ,w_k^ao=δ_k+1 −Õ

b

c_bnx,y₁, . . . ,y_ko

! . Since this is true for allW, we have thats⁰=Í

acanx,w₂^a, . . . ,w^a

kois in the image of δ_k+1.

The proof thatδ_nis injective is similar, but usesd₁ : Sh1(V) →Sh0(V)in place of the nonexistentδ_n+1, and is left to the reader (see [4, Theorem 2.1] for more details).

5. Γ₀(n,N)-orbits of subspaces ofQⁿ

By considering the elements ofQⁿas row vectors, right multiplication by elements of GL(n,Q)yields a right action of GL(n,Q)onQⁿ. This action restricts to an action of Γ₀(n,N)onQⁿ. In a natural way, we may also consider GL(n,Q)(and hence alsoΓ₀(n,N)) as acting on the set ofk-dimensional subspaces ofQⁿ, for 0≤k ≤n. We wish to find explicit representatives of theΓ₀(n,N)-orbits ofk-dimensional subspaces. Note that for k =0 andk =n, there is only onek-dimensional subspace, and hence only one orbit, with a unique orbit representative.

Theorem 5.1. Let0<k<nand assume thatNis squarefree. Then theΓ₀(n,N)-orbits of k-dimensional subspaces ofQⁿare in one-to-one correspondence with the set of positive divisors ofN, where the orbit corresponding to the divisordcontains thek-dimensional subspace spanned by

e₁+de_k+1,e₂,e₃,e₄, . . . ,e_k,

whereei denotes the standard basis element ofQⁿwith a1in theith column, and0’s elsewhere.

Multiplication by elements ofS₀^±(n,N)preserves theΓ₀(n,N)-orbits.

Proof. LetW be ak-dimensional subspace, and letM be ak×nmatrix with integer entries whose rows spanW. Forγ∈Γ₀(n,N),Mγhas row spaceWγ. Left multiplication by an element of GL(k,Q)does not change the row space of a matrix, so we wish to find a canonical element of the double coset

GL(k,Q)MΓ₀(n,N).

Integer column operations on the rightmostn−1 columns ofMcan be represented by right multiplication by an element of Γ₀(n,N); similarly, arbitrary row operations correspond to left multiplication by an element of GL(k,Q). Using these operations we find that the row space ofMisΓ₀(n,N)-equivalent to the row space of

M⁰=

a 0^t_k−1 b 0 · · · 0

0k−1 I_k−1 0k−1 0k−1 · · · 0k−1

,

wherea,b∈Zand gcd(a,b)=1. Here, 0k−1denotes the column vector of all zeros and lengthk−1, and 0^t_k−1denotes its transpose.

Since gcd(a,b)= 1, we may findr andsso thatar +bs =1. Then for all` ∈ Z, a(r+b`)+b(s−a`)=1 and for alld∈Z,

(a+d(s−a`))(r+b`)+(b−d(r+b`))(s−a`)=1.

Hence, the matrix

S=

r+`b −(b−d(r+`b)) s−a` a+d(s−a`)

has determinant 1. In order for it to be inΓ₀(2,N), we needb−d(r+`b)=mNfor some m∈Z. To guarantee this, we choosed =gcd(b,N). SinceNis squarefree, we see that gcd(b,N/d)=1, so that there are integers`,mwithb`+mN/d =b/d−r. With this choice of`, we see thatS ∈Γ₀(2,N).

Form the matrixTby replacing the(1,1),(1,k+1),(k+1,1)and(k+1,k+1)entries ofInby the(1,1),(1,2),(2,1), and(2,2)entries ofS. ThenTis clearly inΓ₀(n,N). Thus, the row space ofMisΓ₀(n,N)-equivalent to the row space of

M⁰T = 1 0^t_k−1 d 0 · · · 0 0k−1 I_k−1 0k−1 0k−1 · · · 0k−1

.

Hence, everyΓ₀(n,N)-orbit of ak-dimensional subspace ofQⁿcontains a subspace of the proper form.

SupposeS∈S₀^±(n,N)takes a subspace in theΓ₀(n,N)-orbit corresponding tod|N, to a subspace in the orbit corresponding tod⁰|N. Then, after multiplyingSby an appropriate element ofΓ₀(n,N)so that it takes the representative subspace of the orbit corresponding todto the representative corresponding tod⁰, for somex∈Z, we must have

(1,0^t_k−1,d,0^t_n−k−1)S=(x,∗, . . . ,∗,xd⁰,0^t_n−k−1).

If we now denote the(1,1),(1,k+1),(k+1,1)and(k+1,k+1)entries ofSbya,bN,r, ands, respectively, and note that gcd(a,N)=1, we see that we must havea+r d=xand bN+sd=d⁰x. Hence,bN+sd=d⁰a+r dd⁰, and we see thatd|d⁰a, so thatd|d⁰. Now using that det(S) ·S⁻¹=S⁰ ∈S₀^±(n,N), replacingS byS⁰, and reversing the roles ofd andd⁰, we must similarly haved⁰|d. Hence,d=d⁰.

Now, letW₀^k be the row space of thek×nmatrix M₀ =

I_k 0

.

Letg_dbe equal to then×nidentity matrixI_n, with the(1,k+1)entry replaced byd. Let P₀^kbe the stabilizer (in GL(n,Q), acting on the right) ofW₀^k. DefineP^k_d=g⁻¹

d P₀g_d; it is the stabilizer of the row spaceW_d^kofM₀g_d, or in other words the stabilizer of the canonical representative of theΓ₀(n,N)-orbit ofk-dimensional subspaces ofQⁿcorresponding to the divisordofN. Typically, whenkis understood, we will omit it, writingP₀,Pd, orWd

rather thanP^k₀,P^k_d, orW_d^k. We will call a subgroupP_darepresentative maximal parabolic subgroup, and denote its unipotent radical byUdand its Levi quotient byLd =Pd/U_d. Note thatU_d=g⁻_d¹U₀g_d, whereU₀is the unipotent radical ofP₀.

The subgroupP₀consists of block matrices A 0 B C

in which Ais an invertiblek×kmatrix,Cis an invertible(n−k) × (n−k)matrix,B is an arbitrary(n−k) ×k matrix, and the block of zeroes isk× (n−k). For a block matrix g ∈ P₀ as above, we define ψ₀¹(g) = Aand ψ₀²(g) = C. One sees easily that ψ¹₀ :P₀ →GL(k,Q)andψ₀²:P₀ →GL(n−k,Q)are group homomorphisms.

Fors∈P_d, defineψ_dⁱ(s)=ψ₀ⁱ(g_dsg⁻_d¹).

We have the following straightforward generalization of [3, Theorem 7].

Theorem 5.2. Letdbe a positive divisor ofNand assume that(d,N/d)=1. (1) Ifs∈Pd∩S₀(n,N)^±, thenψ¹

d(s)₁₁≡s₁₁ (modd)andψ²

d(s)₁₁≡s₁₁ (modN/d).

(2) ψ_d¹(P_d∩S₀(n,N)^±) ⊂S₀(k,d)^±. (3) ψ_d²(P_d∩S₀(n,N)^±) ⊂S₀(n−k,N/d)^±. (4) There is an exact sequence of groups

1→U_d∩Γ₀^±(n,N) →P_d∩Γ₀^±(n,N)^ψ

d1×ψ²_d

−→ Γ₀^±(k,d) ×Γ₀^±(n−k,N/d) →1. Proof. Statements (1), (2), and (3) are proven as in [3, Theorem 7].

The only question in the exactness of the sequence in (4) is whether ψ_d¹ ×ψ_d² is surjective. To show this surjectivity, suppose that A=(a_{i j}) ∈Γ₀^±(k,d)andB=(b_{i j}) ∈ Γ^±₀(n−k,N/d), chooseq,r∈Zsuch thatq(N/d)+r(d)=a₁₁−b₁₁(which can be done since gcd(d,N/d)=1), and letZ be the block matrix

Z = A 0

C B

, whereC=(c_{i j})has

c_{i j} =













0 ifi>1,

a_1j/d ifi=1 and j>1, r ifi=j=1.

We note that sinceA∈Γ^±₀(k,d), each of these entries is an integer. One checks easily that g_d⁻¹Zg_d ∈Γ₀^±(n,N) ∩P_d, and that(ψ_d¹×ψ_d²)(g⁻¹

d Zg_d)=(A,B).

6. A spectral sequence

Definition 6.1. LetV be ann-dimensional vector space over a fieldKwithn>0. For 0 ≤i ≤n, letCibe the set of all subspaces ofVof dimensioni.

Definition 6.2. LetV be ann-dimensional vector space over a fieldKwithn>0. For

−1≤i≤n−1 define

X_i = Ê

W∈Ci+1

St(W).

We note that Theorem 4.2 yields an exact sequence of GL(n,K)-modules,

0→ X_n−1→ X_n−2→ · · · →X₀→Z→0, (6.1) sinceX−1Z.

LetK =Qand letFbe a field of characteristicp>n+1. Let(Γ,S)=(Γ₀(n,N),S₀(n,N)), and letMbe an admissible rightS-module overF. We apply the spectral sequence of [5, Section 8], to the exact sequence (6.1) to obtain

E_q,r¹ =Hr(Γ,Xq⊗M)=⇒Hq+r(Γ,M).

As in [5] this spectral sequence is equivariant for the action of the Hecke operators. For eachqwith 0≤q≤n−1, we choose a set ˆC_q+1ofΓ-orbit representatives ofC_q+1. For a W ∈Cˆ_q+1, letΓ_W be the stabilizer ofW inΓ. Then we may decompose

X_q ⊗M= Ê

W∈Cˆq+1

Ind^ΓΓWSt(W) ⊗M into a finite direct sum of induced modules.

Now, by the Shapiro isomorphism, we obtain E¹_q,r

Ê

W∈Cˆq+1

H_r(Γ_W,St(W) ⊗M).

Forq < n−1, we note that ˆC_q+1 consists exactly of the subspacesW_d^q, asd runs through the divisors ofN. In addition,Γ_W =Γ∩P_d, which we will denote byΓ_Pd. For q=n−1, we have that ˆC_q+1^k =V. Hence, the first page of the spectral sequence has the following terms.

E_q,r¹ =













H_r(Γ,St(V) ⊗M) ifq=n−1, Ê

d|N

H_r(Γ_Pq+1 d

,St(W_d^q+1) ⊗M) ifq<n−1.

The remainder of the paper is devoted to studying the terms of this spectral sequence, in order to show that for reducible Galois representationsρ=ρ₁⊕ρ₂satisfying certain conditions, ρfitsE¹_q,r, and that the eigenvector withρattached either survives to the

infinity page of the spectral sequence, showing thatρfitsH_q+r(Γ,M), or is killed off in such a way as to show thatρnevertheless fitsHs(Γ,M)for some value ofs.

In order to proceed with this argument, it is necessary to show that the terms of the spectral sequence are finite-dimensional vector spaces overF. For the terms withq=n−1, this follows immediately from Borel–Serre duality. Forq<n−1, we prove the following theorem.

Theorem 6.3. Let0<k<n. LetΓ=Γ₀(n,N), and assume thatp>n+1. LetP=P_d^k for somed|N, letW =W_d^k, and letM be a finite-dimensionalΓ_P-module. Then forr ≥0, the homology

H_r(Γ_P,St(W) ⊗M) is a finite dimensional vector space overF.

Proof. For convenience, we will write kerψ_d¹for kerψ_d¹∩Γ_Pin this proof. We use the Hochschild–Serre spectral sequence for the exact sequence

1→kerψ¹_d→Γ_P→Γ_P/kerψ_d¹ →1. This gives us a spectral sequence

E_{i j}² =H_i(Γ_P/kerψ¹_d,H_j(kerψ_d¹,St(W) ⊗M)) =⇒ H_i+j(Γ_P,St(W) ⊗M), and if each term of the E² page is finite dimensional, the abutment must be finite dimensional. Now kerψ¹

dacts trivially on St(W), so a termE_{i j}² may be written as Hi(Γ_P/kerψ_d¹,St(W) ⊗Hj(kerψ_d¹,M)).

We now note that Theorem 5.2(4) implies thatΓ_P/kerψ_d¹ ψ¹_d(Γ_P) = Γ₀^±(k,d), so, noting that conjugation byg_dtakes the action ofΓ_PonW to an action ofψ_d¹(Γ_P)onW₀, we have that (as a vector space),E_{i j}² is isomorphic to

H_i(Γ₀^±(k,d),St(W₀) ⊗H_j(kerψ_d¹,M)),

where the action ofγ∈Γ₀^±(k,d)onH_j(kerψ¹_d,M)is obtained by choosing anyγ⁰∈Γ_P withψ¹_d(γ⁰)=γand allowingγ⁰to act onH_j(kerψ¹_d,M). By Borel–Serre duality,E_{i j}² is then isomorphic to

Hⁱ(Γ^±₀(k,d),H_j(kerψ_d¹,M)),

which is finite dimensional since kerψ¹_d is an arithmetic group and M is finite

dimensional.

We also have the following related theorem.

Theorem 6.4. Let0<k<n. LetΓ=Γ₀(n,N), and assume thatp>n+1. LetP=P_d^k for somed|N, letW =W_d^k, and letMbe a finite-dimensionalΓ_L =Γ_P/Γ_U-module. Then forr ≥0, the homology

H_r(Γ_L,St(W) ⊗M) is a finite dimensional vector space overF.

Proof. Sinceψ¹_d is trivial onΓ_U, it yields a well defined function onΓ_L. The proof is

then basically the same as the proof of Theorem 6.3.

Remark 6.5. If(Γ,S)=(Γ₀(n,N),S₀(n,N)), and ifPis a representative maximal parabolic subgroup stabilizing a subspaceW, then the results of [5, Section 3] show that as long as(∗)given anys∈Swe can choose left coset representativess_α forΓsΓto be inS_P, there is an isomorphism H (Γ,S) H (Γ_P,S_P)and we can viewH_r(Γ_P,St(W) ⊗M) as anH (Γ,S)-module via this isomorphism. Hence, under condition(∗), in studying systems ofH (Γ,S)-eigenvalues inH_r(Γ_P,St(W) ⊗M)we may study the homology as an H (Γ_P,SP)-module. In Theorem 8.6, we show that condition(∗)holds.

7. Two cases of Hecke equivariance of the Hochschild–Serre spectral se- quence

From this section on, we require definitions of additional subsets of GL(n,Q).

Definition 7.1. LetP=P^k_dbe a representative maximal parabolic subgroup of GL(n,Q) for some d|N and let p be a prime in Z. We define the following subgroups and subsemigroups of GL(n,Q).

(1) Γ(N)={A∈GL(n,Z): A≡I (modN)}.

(2) S_k(N)is the set of matrices A ∈ GL(n,Q) with integer entries and positive determinant prime topN such that A≡ diag(1, . . . ,1,∗,1, . . . ,1,∗) (modN), with the∗’s in thekandnpositions.

(3) Γ_P(N)=Γ(N) ∩P.

(4) S_P(N)={s∈S_k(N) ∩P:ψ¹_d(s), ψ²_d(s)both have positive determinant}. We now prove two results involving the Hecke equivariance of the Hochschild–Serre spectral sequence. The first of these results (Theorem 7.11) is closely related to Section 4 of [7]; however, the results here are more general. The second (Theorem 7.15) is similar to [1, Theorem 3.1].

Definition 7.2. Let Pbe a maximal parabolic subgroup of GL(n,Q)and letUbe the unipotent radical of P. We note thatU(Q)is an abelian group, and we set r equal to the Q-dimension ofU. Let(Γ,S)be a congruence Hecke pair of levelpN (see [1, Definition 1.2]. ThenΓ_Uis a free abelian group of rankr. LetMbe a(p,N)-admissible S-module.

LetTbe the set of matricest∈GL(n,Q)with determinant prime topNsuch that all denominators of bothtandt⁻¹are prime topN andtnormalizesU(Q). LetC•be the standard resolution (also known as the bar resolution) ofZover GL(n,Q). Fort∈T, we define an action oftonC•⊗_ΓU M by setting

(c⊗m) ·t= 1 d^r

Õ

b

cu_bt⊗mu_bt,

whered is any integer prime topsuch that the right conjugation action oftonΓ_U^d is contained inΓ_U, and{u_b : 1≤b≤d^r}is a set of coset representatives forΓ^d

U insideΓ_U. We can taked=det(t), so there is adthat fits the conditions of the definition (note that the condition thatdbe prime topwas inadvertently omitted in [7]). In [7, Lemma 4.3], this action oftis shown to be well defined; in particular, it does not depend on the choice ofdor on the choice of coset representatives. It also commutes with the boundary operator.

This action of individual elements ofTdoes not normally extend to a group action ofTon C•⊗_ΓUM. However, we now show that the action does, in fact, define a semigroup action ofS_P ⊆T onH∗(Γ_U,M)under certain hypotheses, makingH∗(Γ_U,M)anS_P-module.

We note also that an element ofT actually has entries inZ⁽p)(the localization ofZat the prime ideal(p)), so there is a well defined notion of reduction modulopfor such an element.

The proof of the following lemma is clear.

Lemma 7.3. Fixtanddas in Definition 7.2. Then the largest subgroupHofΓ_Usuch thatt⁻¹Ht⊂Γ_Uis

H=Γ_U∩tΓ_Ut⁻¹,

and we haveΓ_U^d ⊆H ⊆Γ_U. Hence,[Γ_U :H]|d^r, and is thus prime top(since we may taked=det(t).

By [24, p. 51], we note that if we writeΓ_UtΓ_U =Ýet

j=1α_jΓ_U, then we havee_t=[Γ_U: H]. There is a natural action of double cosets on homology and cohomology; when we consider the double cosetΓ_UtΓUas a Hecke operator, we will denote it by[Γ_UtΓU]. Since C•⊗_ΓU M = H₀(Γ_U,C• ⊗_ZM), the Hecke operator[Γ_UtΓ_U]acts onC•⊗_ΓU M. This

action is given by

c⊗m[Γ_UtΓ_U]=

et

Õ

j=1

cα_j⊗mα_j.

Lemma 7.4. Lett∈T. Then with the notation defined above, the action oftonC•⊗M given in Definition 7.2 is equal to the action of _e¹

t[Γ_UtΓ_U].

Proof. The action of the Hecke operator[Γ_UtΓU]is independent of the choice of coset representatives{α_j}. We choose theα_jas follows: fix a collection of coset representatives {w_j: j=1, . . . ,e_t}ofHinsideΓ_U, so thatΓ_U =Ýet

j=1w_jH. Setα_j =w_jt. One checks easily that theα_jthus defined give distinct cosets ofΓ_U insideΓ_UtΓ_U; since there aree_t of them, they are a complete set of coset representatives.

Now choose a collection{v_k : k =1, . . . ,[H :Γ_U^d]}of coset representatives ofΓ_U^d insideH, so thatH=Ý

kv_kΓ_U^d. Then the set{w_jv_k}is a complete collection of coset representatives ofΓ^d

U insideΓ_U. Since the action oft is independent of the choice of coset representatives, we may choose {u_b} = {w_jv_k}. SinceΓ_U is abelian, we have w_jv_k =v_kw_j. We then obtain

(c⊗m) ·t= 1 dⁿ

dⁿ

Õ

b=1

cu_bt⊗mu_bt

= 1 dⁿ

Õ

j,k

cv_kw_jt⊗mv_kw_jt

= 1 dⁿ

Õ

j,k

cv_kα_j ⊗mv_kα_j

= 1 dⁿ

Õ

k

cv_k ⊗mv_k

!

[Γ_UtΓU]

= 1 dⁿ

Õ

k

c⊗m

!

[Γ_UtΓ_U]

= [H:Γ^d

U] [Γ_U :Γ^d

U](c⊗m)[Γ_UtΓ_U]

= 1

e_t(c⊗m)[Γ_UtΓ_U],

where thev_k vanish in the sixth line becausev_k ∈ Γ_U and the tensor product is over

Γ_U.

LetT⁰be any subgroup ofTsuch that every element ofT⁰∩U(Q)is congruent modulo pto an element ofΓ_U. See Theorem 7.10 for examples whereT⁰may be taken to be the group generated by a semigroupS_P. We will show that for any such subgroupT⁰, the individual actions of elements ofT⁰onHk(Γ_U,M)compile together into a group action.

To do this, we need to show that the composition of the actions ofs,t ∈T⁰is equal to the action ofst. We will see that this can be done on the level of the homology groups H_k(Γ_U,M), but not on the level of chainsC•⊗_ΓU M. Our first step is the following lemma, from [24, p. 51] (since we are working with right modules, we have changed right cosets to left cosets).

Lemma 7.5. Lets,t ∈T, and suppose thatΓ_UsΓ_U =Ý

is_iΓ_U andΓ_UtΓ_U =Ý

jt_jΓ_U. We may choose a finite setΞ⊂T such that

Γ_UsΓ_UtΓ_U=Þ

ξ∈Ξ

Γ_UξΓ_U.

Then in the Hecke algebra we have the equality [Γ_UsΓU][Γ_UtΓ_U]=Õ

ξ∈Ξ

m(ξ)[Γ_UξΓ_U],

wherem(ξ)=|{(i,j)|s_it_jΓ_U =ξΓ_U}|.

Lemma 7.6. With notation as in Lemma 7.5, for eachξ ∈Ξ, we haveξ =stu(ξ)for someu(ξ) ∈U(Q), and

Γ_UξΓ_U =Γ_UstΓ_Uu(ξ).

In addition,

[Γ_UstΓ_U][Γ_Uu(ξ)Γ_U]=[Γ_UξΓ_U], ande_st=e_ξ.

Proof. SinceTnormalizesU(Q), andU(Q)is abelian, this is immediate.

Lemma 7.7. Letu∈T∩U(Q), and assume that there is someu⁰∈Γ_U that is congruent toumodulopN. LetM be a(p,N)-admissibleT-module. Then[Γ_UuΓ_U]acts trivially on the homology groups

H∗(Γ_U,M).

Proof. There is only one single coset in the double cosetΓ_UuΓ_U, so the Hecke operator [Γ_UuΓ_U]acts on the homology asudoes.

Since u centralizes Γ_U, it acts on the homology via its action on M. Since M is (p,N)-admissible the action ofu⁰and the action ofuon M are the same. Hence, the action ofuand the action ofu⁰on the homology are the same. However,u⁰acts trivially

on the homology, by [14, Proposition III.8.1].

Remark 7.8. We note that this lemma fails if we apply it on the chain level. This is because the chains,C⊗_ΓU M =H₀(Γ_U,C⊗_ZM), although acted on by the Hecke operators, are the homology with coefficients inC⊗M, which is not an admissible coefficient module.

Corollary 7.9. LetT⁰be any subgroup ofT such that every element ofT⁰∩U(Q)is congruent modulopto an element ofΓ_U. Lets,t ∈T⁰. Then, with notation as in Lemma 7.5, we have, forz∈H∗(Γ_U,M),

z[Γ_UsΓ_U][Γ_UtΓ_U]=z Õ

ξ∈Ξ

m(ξ)

!

[Γ_UstΓ_U].

Also,

z 1

e_se_t[Γ_UsΓ_U][Γ_UtΓ_U]=z 1

e_st[Γ_UstΓ_U],

and we see that the action of individual elements ofT⁰onH∗(Γ_U,M)in Definition 7.2 yields a group action ofT⁰given byz·s=z_e¹

s[Γ_UsΓU].

Proof. The first displayed equation follows immediately from Lemmas 7.6 and 7.7. For the second, by [24, Proposition 3.3], Lemma 7.5, and Lemma 7.6, we have

est

Õ

ξ∈Ξ

m(ξ)=Õ

ξ∈Ξ

eξm(ξ)=deg([Γ_UsΓU][Γ_UtΓ_U])

=deg([Γ_UsΓ_U])deg([Γ_UtΓ_U])=e_se_t. Hence, by Lemma 7.4 and the above equation,

(z·s) ·t=z 1

e_se_t[Γ_UsΓU][Γ_UtΓ_U]=z Ím(ξ)

e_se_t [Γ_UstΓ_U]=z 1

e_st[Γ_UstΓU]=z· (st).

Theorem 7.10. LetP=P_d^kfor somed|Nand somek. Let(Γ,S)be either(Γ(pN),S_k(pN)) or(Γ₀(n,N),S₀(n,N)). Letz∈H∗(Γ_U,M), and lets,t∈S_P. Then the action of individual elements of S_PonH∗(Γ_U,M)in Definition 7.2 yields a semigroup action ofS_Punder whichS_U acts trivially.

Proof. By the previous lemmas, we are finished if we can show thatS_Plies in a subgroup T⁰such that every element ofT⁰∩U(Q)is congruent modulopto an element ofΓ_U. It suffices to show that every element ofS⁻_P¹S_P∩U(Q)is congruent modulopto an element ofΓ_U.

For (Γ,S) =(Γ₀(n,N),S₀(n,N)) one checks that the intersectionS⁻_P¹SP∩U(Q)is contained in the set of matrices

M=

g_d⁻¹

I_k 0 A I_n−k

g_d

,