MODULAR REPRESENTATIONS OF GL_{3}(𝔽_{p}) AND MOD-p GALOIS REPRESENTATIONS

AVNER ASH AND PHAM HUU TIEP

Abstract. We prove certain cases of nonabelian reciprocity between mod p Galois representations and Hecke eigenclasses in the mod p cohomology of GL(3, ℤ), using the symmetric square liftings from GL(2) and congruences on Hecke eigenvalues. We need to develop some modular representation theory for GL(3, ℤ/p). In some cases we also need to invoke certain computer calculations to finish the proof.

J. Algebra 222(1999), 376 - 399

1. Introduction

The origin of this paper is a conjecture about Galois representations attached to mod p cohomology classes of a congruence subgroup Γ of GL(n, ℤ), p being a rational prime.

In brief, the conjecture states that to any Hecke eigenclass in the mod p cohomology (possibly with coefficients) of Γ there is attached a mod p representation of the absolute Galois group G_{ℚ} of ℚ such that for almost all primes ℓ, the Hecke polynomial at ℓ equals the characteristic polynomial of a Frobenius element at ℓ. The exact conjecture is given below. It should be thought of as giving mod p nonabelian reciprocity laws.

The conjecture has been experimentally tested for Γ ⊂ GL(3, ℤ) and trivial coefficients in [AM1] and for Γ = GL(3, ℤ) and certain twisted coefficients in [AAC]. In a number of examples, Galois representations were found which appeared to satisfy the conjecture, in the sense that the equality in Conjecture 1.1 below was verified for small ℓ’s. The first example of this appeared already in [APT]. However in no case could we prove that the representation was really attached to the eigenclass, i.e. that the equality persisted for all ℓ.

Date: April 20, 1998.

Key words and phrases. Galois representations, mod-p cohomology.

The first author was supported in part by NSA grant MDA-904-94-2030 and NSF grant DMS-9531675.

Part of this work was done while the second author was participating in the special semester on Representations of Algebraic Groups and Related Finite Groups at the Isaac Newton Institute (Cambridge). It is a pleasure to thank the Institute for its hospitality.

The authors are grateful to S. Doty, R. Taylor, and G. Walker and especially B. Mazur for helpful discussions.

In this paper, we will prove the conjecture for certain cohomology eigenclasses that are related modulo p to automorphic forms on GL(3) that arise from GL(2) via the “symmetric square lifting”. Among these are classes that were computed in [AAC].

The symmetric square lifting is a theorem of Gelbart and Jacquet [GJ]. In the context in which we use it, it associates to a classical holomorphic newform ϕ of weight k ≥ 2 and level 1 on the upper half plane an automorphic representation of GL(3)/ℚ. From there we get a Hecke eigenclass in the cohomology of SL(3, ℤ) with coefficients in some finite dimensional module W over ℂ. By choosing a lattice in W and reducing mod p to get W̄, we get a Hecke eigenclass in the homology of GL(3, ℤ) with coefficients in W̄, whose eigenvalues in a finite extension of 𝔽_{p} := ℤ/pℤ are given by a simple function of the Hecke eigenvalues of ϕ. This is all explained in [AS1] and recalled in section 2 below. The application of the symmetric square lifting to cohomology in characteristic 0 of arithmetic groups over general number fields may be found in [LS].

At this point the problem arises that the twisted coefficients used in [AAC] are V_{g} := homogeneous polynomials in three variables of degree g with coefficients in 𝔽_{p}, whereas W mod p is not isomorphic to any of the V_{g}’s. Thus we are led to study the question of comparing constituents of various 𝔽_{p}[GL(3, 𝔽_{p})]-modules. That is the main topic of this paper.

Our results are far from complete, but in cases where k and g are sufficiently small we can control the situation. It sometimes helps to introduce a character χ : 𝔽^{×} → 𝔽^{×}. When we can show that W̄ ⊗ (χ ∘ det) is either a sub or a quotient of V_{g} (or it is located in a “good way” in V_{g}, see Proposition 4.7 and Theorem 5.1) for some g then we can prove that the “symmetric squares” eigenvalues (after twisting by χ) occur on some eigenclass in H_{3}(GL(3, ℤ), V_{g}).

More precisely, for any g, 0 ≤ g ≤ p−1, we determine what is the minimal possible value d_{min} of d such that a twist F of the simple 𝔽_{p}GL(3, 𝔽_{p})-module F (2g, g, 0) (with highest weight (2g, g, 0)) is a constituent of V_{d}. If 0 ≤ g ≤ (p − 3)/2 or g = p − 1, then F is a submodule of V_{d_{min}}. If g = (p − 1)/2, then F is a quotient of V_{d_{min}}. If (p − 1)/2 < g < p − 1, then we don’t know whether F is a submodule or a quotient of V_{d_{min}}, but yet we can show that F is located in a good way in V_{d_{min}}. In all cases, this information is enough to show that any “irreducible” system of Hecke eigenvalues occurring in H_{3}(GL(3, ℤ), F) also shows up in H_{3}(GL(3, ℤ), V_{d_{min}}).

We have the unconditional result of Corollary 5.3 below. This says that there exists Galois representations attached to certain Hecke eigenclasses in H_{3}(GL(3, ℤ), V_{d_{min}}) where g = h(p + 2) and 0 ≤ h < (p − 1)/2 or h = p − 1. In these cases the Galois representation is the symmetric square of the mod p Galois representation attached to a classical cuspform of weight h + 2. However, none of these examples turned up in the computations of [AAC].

COHOMOLOGY OF GL(3, ℤ) AND GALOIS REPRESENTATIONS 3

We understand next best the situation when p ≡ 1(mod4) and g = 3(p − 1)/2. In this case, to finish the proof that the Galois representations are actually attached to the Hecke eigenclasses in question, we must invoke computer aided calculations that rule out a contribution to the cohomology by an unwanted constituent of W̄. In this way, we have proved that the assumption (⋆) in Corollary 5.4 holds in the cases where p = 29,37, and 41. These are the three classes found in [AAC] which appeared to be symmetric squares. Now we know they are. In retrospect we see why just these examples were discovered first experimentally: these are the cases with smallest possible d_{min} among those that occur in the homology of GL(3, ℤ). We thank Eric Conrad for the computer programs he wrote to test assumption (⋆).

When p ≡ 3(mod4) and g = 3(p − 1)/2, we have to turn to SL(3, ℤ) in place of GL(3, ℤ) and we have weaker results. This is because in this case we are forced to go to a larger k and there are more constituents in W̄ to keep track of. In the case of other g’s matters only get worse.

One more technical point remains to be mentioned in proving Conjecture 1.1 for the classes in H_{3}(GL(3, ℤ), V_{g}) that arise from symmetric squares in the way outlined above. If the constituent held in common by W̄ and the appropriate V_{g} actually embeds into V_{g}, there is no problem. Otherwise we have to invoke a duality theorem, that says that the Hecke operators behave in a certain way under Poincare duality. This is covered in section 3.

We wish to emphasize the point that modulo p various “algebraically defined” modules for GL(n) have constituents in common in a way that has no reflection among their characteristic 0 versions. Therefore “the symmetric squares lift mod p” of any particular cuspform mod p appears in the cohomology of SL(3, ℤ) for many different coefficient modules.

To finish this introduction, let us state the conjecture referred to above in the version relevant to us here. For the statement of the full conjecture, see [A1].

We fix p to be a prime number and 𝔽 a field of characteristic p. Let n and N be positive integers, and ℚ_{(N)} the ring of rational numbers with denominators prime to N.

Let (Γ, S) be the Hecke pair consisting of Γ = SL(n, ℤ) and S the subsemigroup S of GL(n, ℚ_{(p)}) of elements with positive determinant. An admissible 𝔽S-module V is a right 𝔽S-module, finite dimensional over 𝔽, on which the elements of S act via their reductions mod p. We will also need to use the subsemigroup S′ of S consisting of the integral matrices in S. For generalities about Hecke algebras and their action on cohomology see [Shi], [AS1] and [A1].

Let ℓ be a rational prime and D(ℓ, k) the diagonal matrix diag(\underbrace{1, …, 1}_{n−k}, \underbrace{ℓ, …, ℓ}_{k}),

and let T (ℓ, k) be the Hecke operator corresponding to the double coset ΓD(ℓ, k)Γ. Given a Hecke eigenelement in cohomology, we will let a(ℓ, k) denote its eigenvalue under T (ℓ, k).

Conjecture 1.1. Let V be an admissible 𝔽S-module. Suppose β ∈ H^{∗}(Γ, V) is an eigenclass for the action of the Hecke algebra with eigenvalues a(ℓ, k) ∈ 𝔽. Then there exists a continuous semisimple representation ρ : G_{ℚ} → GL(n, 𝔽) unramified outside p such that

∑_{k=0}^{n}(−1)^{k}ℓ^{k(k−1)/2}a(ℓ, k)X^{k} = det(I − ρ(Frob_{ℓ})^{−1}X)

for all ℓ other than p.

For commentary and basic results on the conjecture see [A1], [A2], and [A3]. In particular, the conjecture holds for n = 1 or 2. For a converse conjecture when n = 2 we have Serre’s conjecture [Ser]. See [AM1] and [AAC] for some numerical evidence when n = 3, and [AMa] for some theoretical results when n = a multiple of p − 1.

2. Symmetric squares mod p

In this paper we will be concerned with n = 3 and V = V_{g}(𝔽) where V_{g} denotes the right GL(3)-module of homogeneous polynomials in column vectors of three variables of degree g. It has highest weight (g, 0, 0). We fix the upper triangular matrices and diagonal matrices respectively as (B, T) when naming highest weights.

We define some GL(n)-representations: V_{h} was defined just above. Denote by V_{h}^{∗} the dual representation. Let W_{h} := W (h + p − 1, p − 1, p − 1 − h) be the dual Weyl module of GL(3) with highest weight (2h, h, 0) tensored by the determinant raised to the −h power. It is the largest component of V_{h} ⊗ V_{h}^{∗}. It has dimension (h + 1)^{3} and trivial central character. Let U_{h} denote the right GL(2)-module of homogeneous polynomials of degree h in column vectors of two variables.

Take a classical holomorphic cuspform ϕ of weight h + 2 for the modular group SL(2, ℤ). Assume that ϕ is an eigenform for all the Hecke operators. In particular, ϕ ≠ 0 so that h is even. Let the eigenvalue of T_{ℓ} be c_{ℓ}. From the Eichler-Shimura theorem, we know that there exists a Hecke eigenclass θ in the parabolic cohomology H_{!}^{1}(SL(2, ℤ), U_{h}(ℂ)). (For GL(2), parabolic cohomology is the same as the interior cohomology as defined below in section 3.) Then T_{ℓ}θ = c_{ℓ}θ.

In [AS1] the symmetric square lifting is applied to the automorphic representation corresponding to θ. Lemma 3.4.3 of [AS1] states that there exists a Hecke eigenclass Θ ∈ H_{!}^{3}(SL(3, ℤ), W_{h}(ℂ)) such that T (ℓ, 1)Θ = T (ℓ, 2)Θ = ℓ^{−h}(c_{ℓ}^{2} − ℓ^{h+1}). In the language of Conjecture 1.1 above, we have for Θ the eigenvalues a(ℓ, 1) = a(ℓ, 2) = ℓ^{−h}(c_{ℓ}^{2} − ℓ^{h+1}). Note that a(ℓ, 3) = 1.

It is easy to check that Θ satisfies Conjecture 1.1 when one takes ρ to be the symmetric square of the Galois representation attached to θ by Deligne. See the proof of Proposition 3.5.1 in [AS1] for details.

Over the complex numbers this is basically the whole story with similar statements for higher level. However, if we introduce coefficients over the field 𝔽 of characteristic p there is more room to maneuver. There are many families of classical newforms whose systems of Hecke eigenvalues are congruent to each other mod p within the family. Their symmetric squares thus give families of GL(3)-cohomology classes with the same property.

Thus, from Theorems 3.4 and 3.5 of [AS2], one can begin with a newform ψ of level p and weight 2, for example, and find a θ of appropriate weight whose system of Hecke eigenvalues are congruent up to a twist to those of ψ mod p. We then obtain Θ whose Hecke eigenvalues are the “symmetric squares” of those of ψ. In fact, this was the original example leading to this paper. Namely, it was experimentally observed in [AAC] that the Hecke eigenvalues of the quasicuspidal homology class in H_{3}(SL(3, ℤ), V_{42}(𝔽_{29})) were the “symmetric squares” of those for a classical newform of level 29, weight 2, and quadratic nebencharacter. The latter system of eigenvalues is congruent mod 29 to those of the newform of level 1 and weight 16 (no twist needed in this case).

The advantage of working with θ rather than ψ arises from the greater simplicity of the representation theory. The local component at p of the complex automorphic representation corresponding to θ is irreducible and unramified, so it’s relatively easy to take its symmetric square lift locally, and track the result in Lie algebra cohomology. The local component at p for ψ and the reduction mod p of its symmetric square lift are much more complicated.

Another way of creating a cohomology class of larger weight whose Hecke eigenvalues are congruent mod p to those of the original class is to use the GL(2, 𝔽_{p})-invariant polynomial σ = X^{p}Y − XY^{p}.

By Lemma 3.3 and Theorem 3.4(c) of [AS1], as long as p − 1 does not divide h, multiplication by σ takes an eigenclass θ in H^{1}(SL(2, ℤ), U_{h}(𝔽)) with Hecke eigenvalues c_{ℓ} to an eigenclass θ(1) in H^{1}(SL(2, ℤ), U_{h+p+1}(𝔽)) with eigenvalues ℓc_{ℓ}. (If p − 1 divides h then multiplication by σ has a one-dimensional kernel.) We refer to multiplication by σ as “twisting”.

Lemma 2.1. An eigenclass in H^{1}(SL(2, ℤ), U_{h}(𝔽)) and its twists have symmetric squares whose systems of Hecke eigenvalues are identical.

Proof. Suppose the original class has eigenvalues c_{ℓ}, so its symmetric square has the eigenvalues ℓ^{−h}(c_{ℓ}^{2} − ℓ^{h+1}). If we twist it a times, we get a class in

H^{1}(SL(2, ℤ), U_{h+a(p+1)}(𝔽))

with eigenvalues ℓ^{a}c_{ℓ}. Its symmetric square has eigenvalues

ℓ^{−h−a(p+1)}(ℓ^{2a}c_{ℓ}^{2} − ℓ^{h+a(p+1)+1}).

Since p + 1 ≡ 2 mod (p − 1), we see that these eigenvalues are the same as the old ones.

We will use the preceding lemma later in the case when p − 3 is divisible by 4. After this digression, go back to the Hecke eigenclass Θ ∈ H_{!}^{3}(SL(3, ℤ), W_{h}(ℂ)).

Recall that S′ denotes the semigroup of integral matrices with positive determinant. Choose an S′-invariant ℤ-lattice Λ in W_{h}(ℂ) ⊗ det^{h}. (For example, there are obvious S′-invariant ℤ-lattices M_{h} and M′_{h} in B_{h} := V_{h}(ℂ) and B′_{h} := V_{h}(ℂ)^{∗} ⊗ det^{h} respectively. One can realize W_{h}(ℂ) ⊗ det^{h} as the kernel of a trace morphism t from B_{h} ⊗ B′_{h} onto B_{h−1} ⊗ B′_{h−1} and take Λ to be the kernel of t restricted to M_{h} ⊗ M′_{h}.) By the universal coefficient theorem, there exists a non-torsion Hecke eigenclass Θ′ ∈ H_{!}^{3}(SL(3, ℤ), Λ) with the same Hecke eigenvalues as Θ, except twisted h times, i.e. a(ℓ, k) is multiplied by ℓ^{kh}. Let Λ̄ denote the reduction of Λ mod p. As a 𝔽_{p}[GL(3, 𝔽_{p})]-module it depends on the choice of Λ, but its composition series is independent of that choice.

Theorem 2.2. Let θ be a Hecke eigenclass in H^{1}(SL(2, ℤ), U_{h}(𝔽)) with eigenvalues c_{ℓ}. Then there exists an irreducible constituent V of Λ̄ and a Hecke eigenclass in H^{3}(SL(3, ℤ), V) with eigenvalues a(ℓ, 1) = (c_{ℓ}^{2} − ℓ^{h+1}) and a(ℓ, 2) = ℓ^{h}(c_{ℓ}^{2} − ℓ^{h+1}).

Proof. First suppose that θ is an interior class, i.e corresponds to a cuspform. Consider the map given by reduction mod p

H^{3}(SL(3, ℤ), Λ) → H^{3}(SL(3, ℤ), Λ̄).

It follows from Proposition 1.2.2 of [AS1] that there exists a Hecke eigenclass in H^{3}(SL(3, ℤ), Λ̄) whose Hecke eigenvalues are congruent mod p to those of Θ′. Lemma 2.1 of [AS1] states that for any system of Hecke eigenvalues occurring in the cohomology with coefficients in a module E, it also occurs in the cohomology of the same group with coefficients in some irreducible constituent of E.

If θ corresponds to an Eisenstein series, a similar argument can be made, starting with a symmetric squares lifting that is constructed explicitly.

Later in the paper we will be concerned with ascertaining which irreducible constituent is V. Then we will ask for which g is V a constituent of V_{g}(𝔽), when is it a submodule and when a quotient.

3. Duality and Hecke operators

In this section we define interior cohomology and quasi-cuspidal cohomology for a compact manifold M with boundary, and recall Lefschetz duality as induced by the cup product. We apply this when M is the Borel-Serre compactification of a symmetric space by an arithmetic group. We then state and prove a formula showing how the Hecke operators behave under Lefschetz duality.

In this section we assume that Γ is a torsion-free arithmetic group that acts on the associated symmetric space without reversing orientation. If Γ′ contains Γ as a normal subgroup of finite index, our results can be immediately applied to the Γ′-invariants in the homology or cohomology of Γ.

In particular, we will apply these results to the GL(3, ℤ) or SL(3, ℤ)-invariants of the (co)homology of a torsion-free congruence subgroup Δ of SL(3, ℤ) with coefficients in a vector space V over a field of characteristic p. If the index of Δ in SL(3, ℤ) is prime to p then this is the same as the (co)homology of GL(3, ℤ) or SL(3, ℤ) itself. Even if the index is divisible by p, one should note that what we computed in [AAC], was the GL(3, ℤ)-invariants in H_{3}(Δ, V), not H_{3}(GL(3, ℤ), V). (In [AAC] we showed that the Hecke module of invariants is independent of the choice of Δ.)

If S is a group, R a ring and V a right R[S] module, we denote by V^{∗} the right R[S] module Hom(V, R) with the action (ϕs)(v) = ϕ(vs^{−1}). Sometimes we may write the natural pairing ϕ(v) as i(v, ϕ).

Let k be a field. If M is an oriented compact manifold of dimension n with boundary and V a coefficient system on M defined over k, the pairing

(·, ·) : H^{q}(M, V) × H^{(n−q)}(M, ∂M, V^{∗}) → k

given by cup product composed with the map induced by i on the coefficients, followed by evaluation on the fundamental class ζ , is perfect.

We define the “interior” cohomology of M to be the image of H^{q}(M, ∂M) in H^{q}(M), and we denote it by H_{!}^{q}(M). Then (·, ·) induces a perfect pairing

(·, ·) : H_{!}^{q}(M, V) × H_{!}^{(n−q)}(M, V^{∗}) → k.

Dually, we define the “quasi-cuspidal homology” of M to be the quotient of H_{q}(M) by the image of H_{q}(∂M) and denote it by H_{q}^{qc}(M). Via Lefschetz duality, (·, ·) induces a perfect pairing

(·, ·) : H_{q}^{qc}(M, V) × H_{(n−q)}^{qc}(M, V^{∗}) → k.

Let G be a reductive ℚ-group, Γ a torsion-free arithmetic subgroup of G(ℚ), S a subgroup of G(ℚ) containing Γ, and V a kS-module. Denote the Borel-Serre compactification of the symmetric space X for G(ℝ) by X̄. Then G(ℝ) acts on X̄ on the left. We fix an orientation on X̄.

Set M = Γ\X̄ and denote again by V the coefficient system on M induced by the kΓ-module V. For any torsion-free subgroup Δ of G(ℚ), we denote by ζ_{Δ} ∈ H_{n}(Δ\X̄, ∂Δ\X̄, k) the fundamental class.

The goal of this section is to show that the pairings induced by cup product are compatible with the Hecke operators, i.e. with the action of the double cosets ΓsΓ. Since G(ℝ) acts on X̄ on the left, group elements will act on homology on the left (in particular on the fundamental class) while they and the Hecke operators will act on cohomology on the right. Note that because Γ is torsion-free, the group (co)homology of Γ with coefficients in V is the same as the (co)homology of M with coefficients in V.

We recall some standard facts on group cohomology from Sections 8 and 9 of Chapter III of [Br]. If G and G′ are any two groups with right modules V and V′ respectively, then a morphism from (G, V) to (G′, V′) is a pair of morphisms α : G → G′ and f : V′ → V such that f(v′α(g)) = f(v′)g for g ∈ G, v′ ∈ V′. Given such a morphism, there exists a map on cohomology (α, f)^{∗} : H^{∗}(G′, V′) → H^{∗}(G, V). In case V = V′ and f = id_{V}, we will write α^{∗} for (α, f)^{∗}.

We also use the notation α_{∗} for the transfer map H^{∗}(G, V′) → H^{∗}(G′, V′), where α : G ↪ G′ is injective with cofinite image and V′ is given a G-module structure via the map α.

Definition of the Hecke Operator T_{s} on H^{∗}(Γ, V):

For s ∈ S, consider the morphism (α, f) from the pair (s^{−1}Γs, V) into (Γ, V), given by α(γ) = sγs^{−1}, f(v) = vs. We will denote the map (α, f)^{∗} by s^{∗}.

Also set Γ(s) = s^{−1}Γs ∩ Γ. We have a morphism (j, f) of the pair (Γ(s), V) into (s^{−1}Γs, V) given by j(g) = g, f(v) = v. Let i : Γ(s) → Γ denote the inclusion map. Then for any β ∈ H^{∗}(Γ, V), we define the Hecke operator T_{s} as follows:

T_{s}(β) = i_{∗} j^{∗} s^{∗}(β).

This definition can be reinterpreted in the cohomology of M. In fact, let Y be X̄ or ∂X̄. To define T_{s} acting on H^{∗}(Γ\Y, V), consider the following commutative diagram :

Γ(s)\Y ⟶ Γ(s^{−1})\Y
↘  ↙
Γ\Y

where Γ(s^{−1}) = sΓs^{−1} ∩ Γ, the horizontal arrow is induced by the left action of s on Y, and the diagonal arrows are the natural projections, π and π̃ respectively. Then we define T_{s} as

T_{s} = π_{∗} ∘ s ∘ π̃^{∗}.

Similarly, T_{s} acts on H^{∗}(M, ∂M, V) and the cohomology exact sequence of the pair (M, ∂M) is T_{s}-equivariant, as is the isomorphism between H^{∗}(M, V) and H^{∗}(Γ, V).

We can compute T_{s} on the level of cochains as follows. Triangulate M and lift to a Γ-invariant triangulation of X̄. With respect to this triangulation, let C^{q}(·) denote the degree q cochains on whatever space appears in the parentheses. For any torsion-free subgroup Δ of G(ℚ), we denote by C_{∗}(Δ\X̄, ∂Δ\X̄, k) the k-valued qchains on Δ\X̄ modulo the boundary and we let Z_{Δ} ∈ C_{n}(Δ\X̄, ∂Δ\X̄, k) denote a representative of the fundamental class ζ_{Δ}.

Let π : X̄ → M denote the projection. Then for any cochain A ∈ C^{q}(M) or C^{q}(M, ∂M), π^{∗}(A) is a Γ-invariant cochain on X̄.

Write the double coset ΓsΓ as a finite disjoint union of single cosets Γs_{i} where s_{i} = sγ_{i}.

For A as above, define

T_{s}(A) = ∑_{i} π̃^{∗}(A)s_{i}.

Of course this depends on the choice of coset representatives. However, if A is a cocycle representing the cohomology class a, then T_{s}(a) is represented by the Γinvariant cocycle T_{s}(A) pushed down to M.

We can now give a formula for the pairing

(·, ·) : H^{q}(M, V) × H^{(n−q)}(M, ∂M, V^{∗}) → k

on the level of cochains. Suppose we are given cohomology classes a ∈ H^{q}(M, V) and b ∈ H^{n−q}(M, ∂M, V^{∗}). Let A ∈ C^{q}(M, V) and B ∈ C^{n−q}(M, ∂M, V^{∗}) represent a, b respectively. Then

(a, b) = i(A ∪ B)(Z_{Γ}).

Recall that i is the map on cohomology induced from the natural contraction i : V ⊗ V^{∗} → k.

We can now state and prove the result of this section:

Theorem 3.1. With notation as above, we have

(T_{s}(a), b) = (a, T_{s^{−1}}(b)) .

Proof. This proof is modeled on the proof of Proposition 3.39, p.75 of [Shi]. We have (a, b) = (T_{s}(A) ∪ B)(Z_{Γ}). Then

(T_{s}(a), b) = i(T_{s}(a) ∪ b)(ζ_{Γ})
= i(∑ π^{∗}Asγ_{i} ∪ π^{∗}B)(Z_{Γ})

which, because π^{∗}B is Γ-invariant

= i(∑_{i} π^{∗}As ∪ π^{∗}B)(γ_{i}Z_{Γ})
= i(∑ π^{∗}As ∪ π^{∗}B)(Z_{Γ(s)}).

(1)

This is because if F is a fundamental domain for Γ on X̄ then the union of γ_{i}F is a fundamental domain F(s) for Γ(s). Hence ∑_{i} γ_{i}Z_{Γ} = Z_{Γ(s)}. Furthermore, sF(s) = F(s^{−1}) where F(s^{−1}) is a fundamental domain for Γ(s^{−1}). So the right hand side of (1) equals

i(∑ π^{∗}A ∪ π^{∗}Bs^{−1})(sZ_{Γ}(s)) = i(∑ π^{∗}A ∪ π^{∗}Bs^{−1})(Z_{Γ(s^{−1})}) = (a, T_{s^{−1}}(b)).

The last equality follows by switching A with B and s with s^{−1} in equality (1).

4. Modular representations of GL(3, 𝔽_{p}) in natural characteristic

Let p be a prime and let G := GL(3, 𝔽_{p}) denote the general linear group of degree 3 over the prime field 𝔽_{p}. First we recall some basic facts about representations of G in characteristic p (cf. [DW1]).

Let 𝔽 = \overline{𝔽_{p}} and let G := GL(3, 𝔽). Then we have the G-modules W (α) (dual Weyl module) and F (α) (simple module) associated with each non-increasing sequence of non-negative integers α = (a, b, c). For i = 1, 2, we write α = (a_{1}, …, a_{i}) instead of (a_{1}, …, a_{i}, \underbrace{0, …, 0}_{3−i}). For example, if α = (d) = (d, 0, 0) then W (α) = V_{d}, the

G-module of all homogeneous polynomials of degree d in three variables x, y, z over 𝔽. It is known that W (α) has a unique simple submodule isomorphic to F (α), and all simple G-modules are obtained in this way. We can extend the W (α)’s and the F (α)’s to modules over the full matrix semigroup M_{3}(𝔽), and W (a, b, c) ≃ W (a − i, b − i, c − i) ⊗ det^{i} (if 0 ≤ i ≤ c ≤ b ≤ a) as M_{3}(𝔽)-modules. Similarly for the F (α)’s.

By restricting the scalars to the prime subfield 𝔽_{p} of 𝔽, the modules W (α) and F (α) give G-modules, which we denote by the same symbols. A sequence α = (a, b, c) is called p-restricted if 0 ≤ a − b, b − c, c ≤ p − 1. We call a p-restricted sequence (a, b, c) good if c < p − 1. The set of p^{2}(p − 1) modules F (α) with good α is a complete set of simple G-modules. The set of p^{2} modules F (a, b, 0), where (a, b, 0) are p-restricted, is a complete set of simple SL(3, 𝔽_{p})-modules (observe that F (a, b, 0) is the simple module with the highest weight (a−b)ω_{1} +bω_{2}, where ω_{1} and ω_{2} are the fundamental dominant weights.)

If A, B are G-modules, then we write A ↪ B if A is isomorphic to a submodule of B, A ≺ B if A is a subquotient of B (i.e. there are submodules B′ ⊂ B′′ of B such

that A ≃ B′′/B′),

A ≈ B if A and B have the same irreducible composition factors and with the same multiplicity. It is known that every simple G-module occurs as a composition factor of the symmetric algebra V = ∑_{d=0}^{∞} V_{d}. On the other hand, let T be the factor algebra of V by the ideal generated by x^{p}, y^{p}, and z^{p}, and let T_{i} = T ∩ V_{i} for 0 ≤ i ≤ 3(p − 1). Then each T_{i} is a simple module for M := M_{3}(𝔽_{p}), G and S := SL(3, 𝔽_{p}); namely,

T_{i} ≃ F (β_{i}), where β_{i} := (\underbrace{p − 1, …, p − 1}_{k}, l) and i = k(p − 1) + l with 0 ≤ k ≤ 3 and

0 ≤ l < p − 1. Moreover, a theorem of Carlisle and Kuhn [CK] says that

V_{d} ≈ ⨁_{λ=(i_{1},... ,i_{s})∈P (d)} T_{i_{1}} ⊗ T_{i_{2}} ⊗ … ⊗ T_{i_{s}}, (2)

where P (d) is the set of all sequences λ = (i_{1}, …, i_{s}) such that 0 ≤ i_{j} ≤ 3(p − 1) and d = i_{1} + pi_{2} + … + p^{s−1}i_{s}. Finally, if i ≥ j − 1 and 1 ≤ j < p then there is an epimorphism of M-modules

ϕ_{i,j} : T_{i} ⊗ T_{j} → T_{i+1} ⊗ T_{j−1}.

The kernel of ϕ_{i,j} is decribed by Theorem 2.12 [DW1].

We are interested in the simple G-modules F := F (γ) = F (2g + h, g + h, h), where g, h are given integers with 0 ≤ g ≤ p − 1 and 0 ≤ h < p − 1. Using the results of [DW1], we determine what is the smallest value d_{min} of d such that F ≺ V_{d}.

Proposition 4.1. Let p be an odd prime, and let 0 ≤ g ≤ p − 1, 0 ≤ h < p − 1. Suppose that F := F (γ) = F (2g+h, g+h, h) is a composition factor of the G-module V_{d} for some d in the range 0 ≤ d ≤ p^{2} + p − 1. Suppose d_{min} is the smallest value of d with this property. Then one of the following holds.

(i) h = 0, 0 ≤ g < (p − 1)/2, d_{min} = g(p + 2). (ii) h = 0, (p − 1)/2 ≤ g ≤ p − 1, d_{min} = g(2p + 1) − (p − 1)^{2}. (iii) h = 1, 0 ≤ g ≤ (p − 5)/2, d_{min} = g(2p + 1) + 3p. (iv) h = p − 2 − 2g, 0 ≤ g ≤ (p − 3)/2, d_{min} = (p + 2)(p − 2 − g).

(v) h = p − 1 − g, 1 ≤ g ≤ p − 2, d_{min} = (g + 3)(p − 1).

Proof. 1) First we show that, if F ≺ W (α) for a good sequence α = (a, b, c), then either

• α = γ, or

• α = (h + p − 2, g + h, 2g + h − p + 2) and p − 2 ≤ 2g + h ≤ 2p − 4, g ≤ (p − 3)/2, or

• α = (h + 2p − 3, g + h + p − 1, 2g + h + 1) and 2g + h ≤ p − 3. Indeed, suppose that α ≠ γ. Then by Proposition 2.11 [DW1], a − c ≥ p − 1, and W (α) has two composition factors: F (α) and F (p − 2 + c, b, a − p + 2). The latter is isomorphic to F as G-modules by our assumption. Hence we arrive at one of the above possibilities. (Incidentally, if a − c < p − 1 then F (α) = W (α).)

2) Here we show that: if F ≺ Ker ϕ_{i,j} for some i, j with 0 < j ≤ p − 1 and j ≤ i ≤ 3p − 3, then one of the following holds:

(a) h = 0 and (i, j) = (2g, g);

(b) h = 0, (i, j) = (p − 1 + g, 2g) and g ≤ (p − 1)/2;

(c) h = 0, (i, j) = (p − 2 + g, 2g + 1) and g ≤ (p − 3)/2;

(d) h = 1, (i, j) = (p + g, 2g + 2) and g ≤ (p − 3)/2;

(e) h = p − 2 − 2g, (i, j) = (2(p − 2 − g), p − 2 − g) and g ≤ (p − 3)/2;

(f) h = 0, (i, j) = (p − 1 + g, 2g − p + 1) and (p − 1)/2 < g < p − 1;

(g) h = p − 1 − g, (i, j) = (3p − 3 − g, g) and g > 0;

(h) h = p − 1 − g, (i, j) = (2p − 1 + g, p − 2 − g) and 0 < g ≤ (p − 3)/2;

(j) h = 0, (i, j) = (2p − 1, p − 2) and g = 0. For, suppose that 0 ≤ j ≤ i ≤ p − 2. By Theorem 2.12 [DW1] Ker ϕ_{i,j} = F (α) with α = (i, j) being good. Applying 1) we get (a). Next suppose that 0 ≤ i−j < p−1 ≤ i. By Theorem 2.12 [DW2] and Young’s rule (see e.g. p. 88 of [JK]), one has

Ker ϕ_{i,j} ≈ W (i, j) + W (j − 1, i − p + 1) ⊗ V_{1} ≈

≈ W (i, j) + W (j, i − p + 1) + W (j − 1, i − p + 2) + W (j − 1, i − p + 1, 1).

This formula involves four summands of type W (β), where either β is good, or β is not non-increasing and then W (β) = 0. Applying 1) we arrive at one of the possibilities (a) – (e). Now suppose that 0 ≤ j ≤ i − p + 1 < p − 1. By Theorem 2.12 [DW1] Ker ϕ_{i,j} = F (α) with α = (j + p − 1, i − p + 1) being good. Hence we come to (f). Finally, suppose that 2p − 2 ≤ i ≤ 3p − 3. By Theorem 2.12 [DW1] Ker ϕ_{i,j} = W (α) with α = (j + p − 1, p − 1, i − 2p + 2). If i < 3p − 3 then α is good and we obtain (g), (h) and (j) because of 1). If i = 3p − 3 then W (α) ≃ W (j) = F (j) and therefore j = 0, a contradiction.

3) Suppose that F ≺ T_{i} for some i. Then either i = g = h = 0, or (i, g, h) = ((3p − 3)/2, (p − 1)/2, 0). To see it, it is enough to identify T_{i} with F (β_{i}).

4) Suppose that F ≺ V_{d} for some d ≤ p^{2} + p − 1. Then we can use formula (2). Since d < p^{2} + p + 1, each λ in (2) contains at most two positive entries. Hence F ≺ T_{i} ⊗ T_{j} with 0 ≤ j ≤ i ≤ 3p − 3 and d = i + pj or d = j + pi. In particular, (p + 1)j ≤ d ≤ p^{2} + p − 1, whence j ≤ p − 1. We want to show that d ≥ d_{min}, where the parameter d_{min} is given in the formulation of the proposition. Recall that we have a sequence of epimorphisms of G-modules

T_{i} ⊗ T_{j} → T_{i+1} ⊗ T_{j−1} → … → T_{i+j−1} ⊗ T_{1} → T_{i+j}

(and T_{k} = 0 if k > 3p − 3). Observe that

j + pi ≥ i + pj > (i + 1) + p(j − 1) > (i + 2) + p(j − 2) > …

Hence it suffices to show that i + pj ≥ d_{min}, provided that F ≺ T_{i} or F ≺ Ker ϕ_{i,j}. In the former case we get (i, g, h) = (0, 0, 0), ((3p − 3)/2, (p − 1)/2, 0), hence we arrive at (i) or (ii). In the latter case we can use the results of 2). In the cases (a) – (c) and (j) (and g < (p − 1)/2) we have h = 0 and i + pj ≥ p(g + 2), so (i) holds. In the cases (a), (f) (and g > (p − 1)/2) we have h = 0 and i + pj ≥ g(2p + 1) − (p − 1)^{2}, so (ii) holds. In case (d) we get (iii), or (iv) with g = (p − 3)/2. In case (e) we get (iv). In the cases (g) and (h) we arrive at (v) if g < p − 1 and (ii) if g = p − 1.

5) Finally, we show that F ≺ V_{d} if d = d_{min}. This follows from Lemma 4.5 and Proposition 4.7 (below) if h = 0. Suppose we are in case (iii). Choosing (i, j) = (p + g, 2g + 2), we have F = W (2g + 1, g + 1, 1) ≺ Ker ϕ_{i,j} with i + pj = d, whence F ≺ V_{d} because of (2). In case (iv), we take (i, j) = (2(p − 2 − g), p − 2 − g). Then i + pj = d and F ≺ W (i, j) ≺ Ker ϕ_{i,j}. In case (v) choose (i, j) = (3p − 3 − g, g). Then i + pj = d and F ≺ W (g + p − 1, p − 1, p − 1 − g) = Ker ϕ_{i,j}, and so we are done.

Remark 4.2. Keep the notation of Proposition 4.1. One can show that F = F (2g + h, g + h, h) occurs in V_{d_{min}} with multiplicity 1.

Corollary 4.3. Let 0 ≤ g ≤ p − 1 and suppose that d_{min}^{∗} is the smallest value of d such that a certain twist (by some power of the determinant) of F (2g, g, 0) is a composition factor of V_{d}. Then the twist needed is trivial and

d_{min}^{∗} = \{ \begin{matrix}g(p + 2), & if & 0 ≤ g < (p − 1)/2 \\ g(2p + 1) − (p − 1)^{2}, & if & (p − 1)/2 ≤ g ≤ p − 1 \end{matrix} .

□

Corollary 4.4. Keep the notation of Proposition 4.1. Then F ≺ V_{d} if and only if d = d_{min} + k(p − 1) with k ≥ 0.

Proof. Consider the Poincaré series P_{X}^{α}(t) = ∑_{d=0}^{∞} m_{X}^{α}(d)t^{d}, where X = G or X = M, and m_{X}^{α}(d) is the multiplicity of F (α) as a composition factor of V_{d} as an X-module. It is known that if α = (a, b, c) is p-restricted then

P_{M}^{α}(t) = M_{3}(α, t)/(1 − t^{p−1})(1 − t^{p^{2}−1})(1 − t^{p^{3}−1}),

where M_{3}(α, t) is a polynomial in t (cf. [CW]). More precisely, M_{3}(α, t) is

t^{a+b+c} · (a polynomial in T := t^{p−1}).

Furthermore, if 0 < c < p − 1 then P_{G}^{α}(t) = P_{M}^{α}(t). If c = 0 then

P_{G}^{α}(t) = P_{M}^{α}(t) + P_{M}^{α^{(p−1)}}(t),

where α^{(i)} = (a+i, b+i, c+i). The polynomials M_{3}(α, t) either are listed in Table IV of [DW1], or can be derived from those using the formula M_{3}(α, t) = t^{p^{2}+p+1}M_{3}(α^{(−1)}, t). Note that negative exponents occur in Table IV only if a = b or if b = c and that cannot happen if g ≥ 0. Similarly, negative exponents occur in Table I only if a = b or if b = 0 and that cannot happen if g ≥ 0.

In particular, one can check that, if γ = (2g + h, g + h, h) and g > 0, or if g = 0 and 1 ≤ h ≤ p − 3, then

P_{G}^{γ}(t) = G_{3}(γ, t)/(1 − t^{p−1})(1 − t^{p^{2}−1})(1 − t^{p^{3}−1}),

where G_{3}(γ, t) is a polynomial in t with non-negative integral cofficients. If (g, h) = (0, 0) or (g, h) = (0, p − 2), then (1 − t^{p−1})P_{G}^{γ}(t) is a Laurent series in t with nonnegative integral coefficients, since

P_{G}^{(0)}(t) = 1 + \frac{T^{3} + T^{p+1} + T^{2p+1} + T^{2p+4} + T^{3p+3} + T^{p^{2}+2} + T^{p^{2}+2p+3}}{(1 − T)(1 − T^{p+1})(1 − T^{p^{2}+p+1})} +

+\frac{T^{p^{2}+p+1}}{(1 − T)(1 − T^{p^{2}+p+1})},

P_{G}^{(p−2,p−2,p−2)}(t) = t^{3p−6} \frac{T^{p−2} + T^{2p−2} + T^{p^{2}−p−3} + T^{p^{2}−4} + T^{p^{2}−1} + T^{p^{2}+2p−3} + T^{p^{2}+2p}}{(1 − T)(1 − T^{p+1})(1 − T^{p^{2}+p+1})}

+t^{3p−6} \frac{T^{p^{2}+p−2}}{(1 − T)(1 − T^{p^{2}+p+1})}.

Hence we can always write P_{G}^{γ}(t) = ∑_{d=0}^{∞} a_{d}t^{d}/(1 − t^{p−1}) with a_{d} ≥ 0. From this it follows that m_{G}^{γ}(d + p − 1) − m_{G}^{γ}(d) = a_{d+p−1} ≥ 0. Thus, F ≺ V_{d} implies F ≺ V_{d+p−1}. Also, we observe that G_{3}(γ, t) is t^{3g+3h} · (a polynomial in t^{p−1}). From this it follows that m_{G}^{γ}(d) can be nonzero only if d ≡ 3(g + h) mod (p − 1). Hence F ≺ V_{d} implies d ≡ d_{min} mod (p − 1).

Lemma 4.5. Keep the notation of Proposition 4.1. If h = 0, and 0 ≤ g ≤ (p − 3)/2 or g = p − 1, then F = F (2g + h, g + h, h) is a submodule of V_{d_{min}}.

Proof. This is a consequence of Proposition 1.3 [DW2], since d_{min} = 2g + pg.

If V is any G-module, then V^{∗} := Hom_{𝔽}(V, 𝔽) denotes the dual of V, and V^{#} denotes the contragredient dual of V. As an 𝔽-space, V^{#} is the same as V, but with a new action of G: X ∈ G acts via X ∘ v = ^{t}X^{−1}(v).

Lemma 4.6. Let V be an irreducible G-module. Then the G-modules V^{∗} and V^{#} are isomorphic.

Proof. We may replace 𝔽 by a big enough finite field 𝕂 and replace G by the finite group G′ := GL_{3}(𝕂). Then V is an absolutely irreducible G′-module, since 𝕂 is a splitting field for G′. Let φ, φ^{∗}, φ^{#} denote the Brauer characters of the G′-modules V, V^{∗} and V^{#}, respectively. Then for any X ∈ G′ we have φ^{#}(X) = φ(^{t}X^{−1}) = φ^{∗}(^{t}X). But X and ^{t}X are conjugate in G′, hence φ^{∗}(^{t}X) = φ^{∗}(X). Thus the irreducible G′-modules V^{#} and V^{∗} have the same Brauer characters, hence they are isomorphic. In particular, the G-modules V^{#} and V^{∗} are isomorphic.

Proposition 4.7. Keep the notation of Proposition 4.1. Suppose h = 0 and (p − 1)/2 ≤ g < p − 1. Then the G-module V_{d_{min}} has submodules T ⊆ S such that F ≃ S/T and T^{#} ≃ T^{∗}. In particular, if g = (p − 1)/2, then S = V_{d_{min}} and so F is a quotient of V_{d_{min}}.

Proof. Given d, consider the following partial order on P (d): if λ = (λ_{1}, …, λ_{s}) and μ = (μ_{1}, …, μ_{s}) are in P (d) then we say λ ≥ μ if ∑_{i≤k} p^{i}λ_{i} ≥ ∑_{i≤k} p^{i}μ_{i} for k = 1, …, s. According to [Krop], for each λ = (λ_{1}, …, λ_{s}) ∈ P (d), V_{d} has a G-submodule J^{λ} with a unique maximal submodule r(J^{λ}) and

J^{λ}/r(J^{λ}) ≃ T_{λ_{1}} ⊗ T_{λ_{2}}^{(p)} ⊗ … ⊗ T_{λ_{s}}^{(p^{s−1})}

(where ^{(p)} denotes the Frobenius twist induced by raising the coefficients of a matrix to the pth power). Each J^{λ}/r(J^{λ}) is a simple composition factor of V_{d} as G-module, and all simple composition factors of V_{d} as G-module have multiplicity one, and arise this way. Also, if λ ≥ μ then J^{λ} ⊇ J^{μ}.

First we consider the case g = (p − 1)/2. Then d_{min} = 3(p − 1)/2. In this case P (d) = {α = (g − 1, 1) < β = (3g, 0)}. Furthermore,

V_{d_{min}} = J^{β} ⊃ r(J^{β}) = J^{α} ⊃ r(J^{α}) = 0,

and J^{β}/J^{α} = T_{3g} ≃ F (p − 1, (p − 1)/2, 0) = F as G-modules. (Cf. 3) in the proof of Proposition 4.1.) Thus we can put T = J^{α}, S = V_{d_{min}}, and so F is a quotient of V_{d_{min}}. Also, T = J^{α} is an irreducible G-module, hence T^{#} = T^{∗} by Lemma 4.6. Actually, one can show that

T ≃ T_{(p−3)/2} ⊗ T_{1} ≃ F ((p − 3)/2, 1) ⊕ F ((p − 1)/2)

as G-modules.

Next we consider the case (p − 1)/2 < g < p − 1. Then d_{min} = g(2p + 1) − (p − 1)^{2}. In this case

P (d) = {α = (g−1, 2g−p+2) < β = (g+p−1, 2g−p+1) < δ = (g+2p−1, 2g−p)}.

Furthermore,

V_{d_{min}} ⊃ J^{β} ⊃ r(J^{β}) = J^{α} ⊃ r(J^{α}) = 0,

and J^{β}/J^{α} = T_{g+p−1} ⊗T_{2g−p+1} as G-modules. Next, ϕ_{g+p−1,2g−p+1} is an epimorphism of G-modules T_{g+p−1} ⊗ T_{2g−p+1} → T_{g+p} ⊗ T_{2g−p} with kernel equal to F (2g, g) = F, cf. (iii) of Theorem 2.12 [DW1]. Thus if we take T to be J^{α} and S to be the complete inverse image of Ker ϕ_{g+p−1,2g−p+1} in J^{β}/J^{α}, then F ≃ S/T. Also, T = J^{α} is an irreducible G-module, hence T^{#} = T^{∗} by Lemma 4.6.

5. Completion of the proof of Conjecture 1.1 in certain cases

In the following we use notation from Section 1 for n = 3. In particular Γ = SL(3, ℤ). We write H_{∗}(M) for H_{∗}(Γ, M). If κ is a character of the Hecke algebra with values in 𝕂, a finite extension of 𝔽_{p}, we say that κ occurs in H_{∗}(M) if there exists an eigenelement α ∈ H_{∗}(M) such that T α = κ(T)α for all T in the Hecke algebra.

We call κ “irreducible” if the Hecke polynomials associated to it could not be equal to the characteristic polynomials of Frobenius elements (as in Conjecture 1.1) of a reducible Galois representation ρ. By Theorem 3 of [AAC], an “irreducible” κ must occur in the quasi-cuspidal homology H_{∗}^{qc}(M).

Theorem 5.1. Assume p > 3. Let 0 ≤ h ≤ p − 1, F = F (2h, h), and let d = d_{min}^{∗} as in Corollary 4.3. Suppose an “irreducible” character κ of the Hecke algebra occurs in H_{3}^{qc}(F). Then κ occurs in H_{3}(V_{d}).

Proof. Our hypothesis on p implies that H_{4}(M) = 0 for any 𝔽_{p}S-module M. Now the statement is clear in the cases 0 ≤ h < (p − 1)/2 and h = p − 1, since in these cases F is a submodule of V_{d} by Lemma 4.5. So we suppose that (p − 1)/2 ≤ h < p − 1 and apply Proposition 4.7. We have the following exact sequences 0 → T → S → F → 0 and

… → H_{3}(T) → H_{3}(S) → H_{3}(F) → H_{2}(T) → …

Assume that κ does not occur in H_{3}(V_{d}).

Since H_{3}(S) ↪ H_{3}(V_{d}), κ must show up in H_{2}(T). Here we use Proposition 1.2.2 of [AS1] on lifting systems of Hecke eigenvalues. By Theorem 3.1 there is a nondegenerate Hecke-equivariant pairing (·, ·) between H_{2}(T) and H_{3}(T^{∗}). Suppose κ is supported by v ∈ H_{2}(T). This means T_{s}(v) = κ(s)v. (We write κ(s) instead of κ(T_{s}) for short.)

Let λ be the character of the Hecke algebra defined by λ(s) = κ(s^{−1}). We claim that H_{3}(T^{∗}) contains a Hecke eigenvector supporting λ. Indeed, since the Hecke algebra is commutative H_{3}(T^{∗}) is a direct sum of generalized eigenspaces E(ν) (i.e. given any s, each u ∈ E(ν) is annihilated by a sufficiently big power of T_{s} − ν(s)). Moreover, in each E(ν) we can choose a basis in which each T_{s} is represented by an upper triangular matrix. If E(λ) ≠ 0, then the first basis vector is the desired eigenvector. Suppose that ν ≠ λ whenever E(ν) ≠ 0. Then there exists an s such that ν(s^{−1}) ≠ κ(s). Let us denote the aforementioned basis of E(ν) as (e_{1}, …, e_{m}) and let e_{0} = 0. We prove by induction on i = 0, 1, … that (v, e_{i}) = 0. This is certainly true when i = 0. For the induction step we have T_{s^{−1}}(e_{i}) = ν(s^{−1})e_{i} + ∑_{j<i} a_{j}e_{j} for some a_{j} ∈ 𝔽. According to Theorem 3.1 and by the induction hypothesis,

κ(s)(v, e_{i}) = (T_{s}(v), e_{i}) = (v, T_{s^{−1}}(e_{i})) = ν(s^{−1})(v, e_{i}) + ∑_{j<i} a_{j}(v, e_{j}) = ν(s^{−1})(v, e_{i}),

whence (v, e_{i}) = 0 as stated. We have just shown that (v, E(ν)) = 0 for all ν, hence (v, H_{3}(T^{∗})) = 0, contrary to the non-degeneracy of the pairing.

Next, T^{∗} ≃ T^{#} by Proposition 4.7. Thus λ lives in H_{3}(T^{#}). But T^{#} ↪ (V_{d})^{#}, whence λ shows up in H_{3}((V_{d})^{#}). Applying the outer automorphism X ↦ ^{t}X^{−1} of G, we obtain that μ lives in H_{3}(V_{d}), where μ(s) = λ(^{t}s^{−1}). Thus μ(s) = κ(^{t}s). In our case Γ = SL_{3}(ℤ), therefore ΓsΓ = Γ(^{t}s)Γ, which implies that κ(^{t}s) = κ(s), i.e. μ = κ occurs in H_{3}(V_{d}), a contradiction.

Lemma 5.2. Let G be a group with a central involution z and let U be an 𝔽Gmodule on which z acts as −1. Suppose that char(𝔽) ≠ 2. Then H^{1}(G, V) = 0.

Proof. Let α be an 1-cocycle of G with coefficients in V. Then for all g, h ∈ G, α(gh) = α(g)h + α(h). Taking h = z we get α(gz) = −α(g) + α(z). On the other hand, substituting g = z gives α(zh) = α(z)h + α(h). Thus −α(g) + α(z) = α(gz) = α(z)g + α(g). Hence, for v = −\frac{1}{2}α(z) we have α(g) = vg − v, i.e. α is an 1-coboundary.

Now we look carefully at the irreducible constituent V arising in Theorem 2.2. We use the notation of that theorem. First of all, by Lemma 5.2 we have to assume that h is even to guarantee that θ is nontrivial.

If 0 < h < (p − 1)/2 or h = p − 1, then Λ mod p ≃ W (2h, h, 0) is an irreducible GL(3, 𝔽_{p})-module, according to Proposition 2.11 of [DW1]. Hence V = W (2h, h, 0) = F (2h, h, 0). Also, V is a submodule of V_{h(p+2)}, due to Lemma 4.5. Thus Theorem 2.2 immediately yields Corollary 5.3. Let 0 < h < (p − 1)/2 or h = p − 1. Suppose in addition that h is even. Let θ be a Hecke eigenclass in H_{1}(SL(2, ℤ), U_{h}(𝔽)) with eigenvalues c_{ℓ}. Assume that the Galois representation attached to θ has an irreducible symmetric square. Then there exists a Hecke eigenclass in H_{3}(SL(3, ℤ), V_{h(p+2)}) with eigenvalues a(ℓ, 1) = (c_{ℓ}^{2} − ℓ^{h+1}) and a(ℓ, 2) = ℓ^{h}(c_{ℓ}^{2} − ℓ^{h+1}). □

Next we consider the situation when (p − 1)/2 ≤ h < p − 1. Then the GL(3, 𝔽_{p})- module Λ mod p ≃ W (2h, h, 0) has a simple submodule F_{1} := F (2h, h, 0), and its quotient by F_{1} is the simple module F_{2} := F (p − 2, h, 2h − p + 2). By Corollary 4.3, the minimal possible g such that F_{1} ≺ V_{g} is g := h(2p + 1) − (p − 1)^{2}. On the other hand, the minimal possible g′ such that F_{2} ≺ V_{g′} is g′ := (p + 2)h, according to Proposition 4.1 (iv). Clearly, g′ > g, hence F_{2} is not an irreducible constituent of V_{g}.

Assumption (⋆) : the V arising in Theorem 2.2 is F_{1}.

Then Theorem 2.2 and Theorem 5.1 imply

Corollary 5.4. Let (p − 1)/2 < h < p − 1 and suppose that h is even. Keep the assumption (⋆). Let θ be a Hecke eigenclass in H_{1}(SL(2, ℤ), U_{h}(𝔽)) with eigenvalues c_{ℓ}. Assume that the Galois representation attached to θ has an irreducible symmetric square. Then there exists a Hecke eigenclass in H_{3}(SL(3, ℤ), V_{g}), where g = h(2p + 1) − (p − 1)^{2}, with eigenvalues a(ℓ, 1) = (c_{ℓ}^{2} − ℓ^{h+1}) and a(ℓ, 2) = ℓ^{h}(c_{ℓ}^{2} − ℓ^{h+1}). □

Remark 5.5. Assumption (⋆) has been verified in the case h = (p − 1)/2 for p = 29, 37 and 41. Thanks to the help of Eric Conrad, we have determined by computer that H_{3}(F_{2}) = 0. The corresponding homology classes provided by Corollary 5.4 had been experimentally discovered first in [AAC].

6. The case where g = 3(p − 1)/2 and p ≡ 3(mod4)

In the last section, under Assumption (⋆), we explained the experimental findings in [AAC] of Hecke eigenclasses in H_{3}(SL(3, ℤ), V_{g}(𝔽_{p})) that appear to be “symmetric squares lifts mod p” of the Hecke eigenvalues of some holomorphic modular Hecke eigenform of level 1 and weight h + 2 where (p, g, h) = (p, 3(p − 1)/2, (p − 1)/2) and p = 29, 37, 41. Since the preparation of [AAC], with the assistance of Eric Conrad again, we have found similar classes in H_{3}(SL(3, ℤ), V_{g}(𝔽_{p})) for (p, g, h) = (43, 63, 20) and (59, 87, 28). In these cases g = 3(p − 1)/2 and p ≡ 3(mod4). In this section we explain this case.

We need to know, at least at the level of SL(3, 𝔽_{p}), the structure of the module Ω_{a} := W (ω_{a}), where ω_{a} = ((p − 3) + 2a(p + 1), (p − 3)/2 + a(p + 1)) and a ≥ 0. Of course, the SL(3, 𝔽_{p})-module Ω_{0} is irreducible.

Lemma 6.1. Suppose that p ≥ 5 and let h = (p − 1)/2. Then the SL(3, 𝔽_{p})-module Ω_{1} has at least four irreducible constituents:

F (2h + 2, h + 1), F (2h + 1, h + 2), F (2h + 2, h, 1), F (2h + 1, h + 1, 1).

Proof. First we consider the SL(3, 𝔽)-module Ω_{1} and its (unique) irreducible submodule F (ω_{1}). Observe that ω_{1} = (2h, h) + p(2, 1). Hence by Steinberg’s tensor product theorem, F (ω_{1}) ≃ F (2h, h) ⊗ F (2, 1)^{(p)} as SL(3, 𝔽)-module. In particular, F (ω_{1}) ≃ F (2h, h) ⊗ F (2, 1) as SL(3, 𝔽_{p})-module. It suffices therefore to show that

F (2h, h)⊗F (2, 1) ≈ F (2h+2, h+1)⊕F (2h+1, h+2)⊕F (2h+2, h, 1)⊕F (2h+1, h+1, 1)

as SL(3, 𝔽_{p})-module. By Proposition 2.11 [DW1], F (2h, h) ≈ W (2h, h) − W (2h − 1, h, 1). Furthermore, the epimorphism ϕ_{2,1} yields F (2, 1) ≈ V_{2} ⊗ V_{1} − V_{3}. Working over the infinite field 𝔽 and using Young’s rule, we get

W (2h, h) ⊗ V_{1} ≈ W (2h + 1, h) ⊕ W (2h, h + 1) ⊕ W (2h, h, 1).

Similarly,

\begin{aligned}W (2h + 1, h) ⊗ V_{2} & ≈ & W (2h + 3, h) ⊕ W (2h + 2, h + 1) ⊕ W (2h + 2, h, 1) \\ & & ⊕W (2h + 1, h + 2) ⊕ W (2h + 1, h + 1, 1) ⊕ W (2h + 1, h, 2), \end{aligned}

\begin{aligned}W (2h, h + 1) ⊗ V_{2} & ≈ & W (2h + 2, h + 1) ⊕ W (2h + 1, h + 2) ⊕ W (2h + 1, h + 1, 1) \\ & & ⊕W (2h, h + 3) ⊕ W (2h, h + 2, 1) ⊕ W (2h, h + 1, 2), \end{aligned}

\begin{aligned}W (2h, h, 1) ⊗ V_{2} & ≈ & W (2h + 2, h, 1) ⊕ W (2h + 1, h + 1, 1) ⊕ W (2h + 1, h, 2) \\ & & ⊕W (2h, h + 2, 1) ⊕ W (2h, h + 1, 2) ⊕ W (2h, h, 3), \end{aligned}

W (2h, h) ⊗ V_{3} ≈ W (2h + 3, h) ⊕ W (2h + 2, h + 1)

⊕W (2h + 2, h, 1) ⊕ W (2h + 1, h + 2) ⊕ W (2h + 1, h + 1, 1) ⊕ W (2h + 1, h, 2) ⊕W (2h, h + 3) ⊕ W (2h, h + 2, 1) ⊕ W (2h, h + 1, 2) ⊕ W (2h, h, 3).

As a consequence we obtain

\begin{matrix}W (2h, h) ⊗ F (2, 1) ≈ W (2h + 2, h + 1) ⊕ W (2h + 1, h + 2) ⊕ 2W (2h + 1, h + 1, 1) \\ ⊕W (2h + 2, h, 1) ⊕ W (2h + 1, h, 2) ⊕ W (2h, h + 2, 1) ⊕ W (2h, h + 1, 2). \end{matrix}

Similarly, W (2h − 1, h, 1) ⊗ F (2, 1) is

\begin{matrix}≈ W (2h + 1, h + 1, 1) ⊕ W (2h, h + 2, 1) ⊕ 2W (2h, h + 1, 2) \\ ⊕W (2h + 1, h, 2) ⊕ W (2h, h, 3) ⊕ W (2h − 1, h + 2, 2) ⊕ W (2h − 1, h + 1, 3). \end{matrix}

Hence, the SL(3, 𝔽_{p})-module F (2h, h) ⊗ F (2, 1) is

≈ W (2h + 2, h + 1) ⊕ W (2h + 1, h + 2) ⊕ W (2h + 1, h + 1, 1) ⊕ W (2h + 2, h, 1)

−W (2h, h + 1, 2) − W (2h, h, 3) − W (2h − 1, h + 2, 2) − W (2h − 1, h + 1, 3)

≈ F (2h + 2, h + 1) ⊕ F (2h + 1, h + 2) ⊕ F (2h + 2, h, 1) ⊕ F (2h + 1, h + 1, 1).

Here we used Proposition 2.11 of [DW1] again.

Note that as SL(3, 𝔽_{p})-module, all twists F (2h+a, h+a, a) of F (2h, h) are isomorphic. Now Lemma 6.1 shows that the symmetric square F (2h, h), where h = (p−1)/2, is a constituent of the SL(3, 𝔽_{p})-module Ω_{1}. This is true for all a in a certain range:

Lemma 6.2. Suppose that 1 ≤ a ≤ (p + 1)/8 and let h = (p − 3)/2 + a. Then the symmetric square F (2h, h) is a constituent of the SL(3, 𝔽_{p})-module Ω_{a}.

Proof. Again, the SL(3, 𝔽)-module Ω_{a} has a unique irreducible submodule F (ω_{a}), and ω_{a} = (2h, h) + p(2a, a). Hence, F (ω_{a}) ≃ F (2h, h) ⊗ F (2a, a) as SL(3, 𝔽_{p})- module. By Proposition 2.11 [DW1], F (2h, h) ≈ W (2h, h) − W (p − 2, h, 2a − 1). Furthermore, the epimorphism ϕ_{2a,a} yields F (2a, a) ≈ V_{2a} ⊗ V_{a} − V_{2a+1} ⊗ V_{a−1}. The condition on a guarantees that all the weights arising in our computation are good. Working over the infinite field 𝔽 and using Young’s rule, we see that the multiplicity of F (2h + a, h + a, a) in W (2h, h) ⊗ V_{2a} ⊗ V_{a} (resp. W (2h, h) ⊗ V_{2a+1} ⊗ V_{a−1}) is (a + 1)(a + 2)/2 (resp. a(a + 1)/2). Similarly, the multiplicity of F (2h + a, h + a, a) in W (p − 2, h, 2a − 1) ⊗ V_{2a} ⊗ V_{a} (resp. W (p − 2, h, 2a − 1) ⊗ V_{2a+1} ⊗ V_{a−1}) is 0 (resp. 0) if a ≥ 2, and 2 (resp. 1) if a = 1. Consequently, the multiplicity of F (2h + a, h + a, a) in F (2h, h) ⊗ F (2a, a) is a + 1 if a ≥ 2 and 1 if a = 1.

Now Lemma 2.1 applies. Starting from a Hecke eigenclass θ in H^{1}(SL(2, ℤ), U_{h}(𝔽)), where h = (p−3)/2 and p ≡ 3( mod 4), we get a twist θ(1) in H^{1}(SL(2, ℤ), U_{h+p+1}(𝔽)). Applying the symmetric square lifting to θ(1), we come to a class in H^{3}(SL(3, ℤ), Ω_{1}), where Ω_{1} = W (3p − 1, (3p − 1)/2). Lemma 6.1 tells us that Ω_{1} and V_{3(p−1)/2} has an irreducible constituent F (p − 1, (p − 1)/2, 0) in common, at least at the level of SL(3, 𝔽_{p}). This should be the reason why we get a class in H^{3}(SL(3, ℤ), V_{3(p−1)/2}(𝔽)) which resembles the symmetric square mod p of θ ∈ H^{1}(SL(2, ℤ), U_{(p−3)/2}(𝔽)). Here we assume as before that the symmetric square of the Galois representation attached to θ is irreducible.

7. Comparison of theory and experiment

We list here all the pairs (p, g, h) for which we have checked computationally that the “symmetric squares” of the Hecke eigenvalues of some holomorphic modular Hecke eigenform of level 1 and weight h + 2 appear mod p as the Hecke eigenvalues of some Hecke eigenclass in H_{3}(SL(3, ℤ), V_{g}(𝔽_{p})).

Using the programs developed in [AAC] and modified (thanks to Eric Conrad) to apply to SL(3, ℤ) as well as GL(3, ℤ), we checked that there exists (i) a Hecke eigenclass in H_{3}(SL(3, ℤ), V_{g}(𝔽_{p})) with Hecke eigenvalues a(ℓ, k); (ii) a Hecke eigenclass in H_{1}(SL(2, ℤ), U_{h}(𝔽_{p})) with Hecke eigenvalues c_{ℓ}; and (iii) a character χ : 𝔽^{×} → 𝔽^{×} such that for all ℓ for which we computed the data, and for k = 1,2,

χ(ℓ^{k})a(ℓ, k) = ℓ^{(k−1)g}(c_{ℓ}^{2} − ℓ^{g+1}).

In each case we computed the Hecke eigenvalues for at least the first three prime numbers ℓ, and usually for several more. In these examples, χ is either trivial or the quadratic character.

List of (p, g, h): (11, 88, 16), (11, 130, 10), (13, 126, 10), (13, 156, 10), (13, 168, 10), (13, 180, 10), (17, 94, 10), (19, 66, 10), (29, 42, 14), (37, 54, 18), (41, 60, 20), (43, 63, 20).

For the triple (59, 87, 28), we computed that there is quasi-cuspidal homology as expected, but we didn’t check the Hecke eigenvalues because of the extra work involved in finding the q-expansions of the newforms of weight 30, since they no longer make up a one-dimensional space.

The case (p, g, h) = (11, 130, 10) should be compared with Corollary 5.3. The cases (p, g, h) = (13, 126, 10), (17, 94, 10), (19, 66, 10), (29, 42, 14), (37, 54, 18) and (41, 60, 20) should be compared with Corollary 5.4. The cases (p, g, h) = (43, 63, 20) and (59, 87, 28) have a heuristic explanation in section 6.

The cases (p, g, h) = (13, 156, 10), (13, 168, 10) and (13, 180, 10) occur because V_{156}, V_{168} and V_{180} all have a composition factor isomorphic to a twist of F (2h, h), namely F (22, 12, 2). Moreover, by Proposition 4.1 (v) above, d_{min} = 156 for this twist. Note that 156,168 and 180 are in an arithmetic progression with difference 12 = p − 1.

In the case (p, g, h) = (11, 88, 16) the two-dimensional Galois representation attached to θ is not irreducible and the GL(3) homology class in question is not quasicuspidal. To deal with this case by the methods of this paper, one would begin with Lemma 6.1.

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Department of Mathematics, Ohio State University, Columbus, OH 43210-1174

E-mail address: ash@math.ohio-state.edu

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105

E-mail address: tiep@math.ufl.edu