AN ANALOGUE OF SERRE’S CONJECTURE FOR GALOIS REPRESENTATIONS AND HECKE EIGENCLASSES IN THE MOD-p COHOMOLOGY OF GL(n, ℤ)

AVNER ASH^{1} AND WARREN SINNOTT

THE OHIO STATE UNIVERSITY DEPARTMENT OF MATHEMATICS 231 W. 18TH AVE COLUMBUS, OH 43210

               e-mail:  ash@math.ohio-state.edu,
                  sinnott@math.ohio-state.edu

1. Introduction

Let p be a prime number and 𝔽 an algebraic closure of the finite field 𝔽_{p} with p elements. Let n and N denote positive integers, N prime to p. We are interested in representations of the Galois group G_{ℚ} = Gal(ℚ̄/ℚ) into GL(n, 𝔽), unramified at all finite primes not dividing pN. (We shall say “ρ is unramified outside pN.”) In this paper, “representation” will always mean “continuous, semisimple representation.”

We choose for each prime l not dividing pN a Frobenius element Frob_{l} in G_{ℚ}. We also fix a complex conjugation Frob_{∞} ∈ G_{ℚ}. For every prime q we fix a decomposition group G_{q} with its filtration by its ramification subgroups G_{q,i}. We denote the whole inertia group G_{q,0} by T_{q}.

Our aim is to make a conjecture about when such a representation should be attached to a cohomology class of a congruence subgroup of level N of GL(n, ℤ). Then we exhibit such evidence for the conjecture as we are able to.

Set Γ_{0}(N) to be the subgroup of SL(n, ℤ) consisting of those matrices whose first row is congruent to (∗, 0, …, 0) modulo N. Let S_{N} be the subsemigroup of the integral matrices in GL(n, ℚ) whose first row is congruent to (∗, 0, …, 0) modulo N and with determinant positive and prime to N.

We denote by H(N) the 𝔽-algebra of double cosets Γ_{0}(N)S_{N}Γ_{0}(N). It is commutative. This algebra acts on the cohomology and homology of Γ_{0}(N) with any coefficient 𝔽S_{N} - module. When a double coset is acting on cohomology, we call it a Hecke operator. H(N) contains all double cosets of the form Γ_{0}(N)D(l, k)Γ_{0}(N), where D(l, k) is the diagonal matrix with k 1’s followed by (n − k) l’s, and l is a prime not dividing N. We use the notation T (l, k) for the corresponding Hecke operator.

1RESEARCH PARTIALLY SUPPORTED BY NSF GRANT DMS-9531675

Definition 1.1: Let V be an H(pN)-module and suppose v ∈ V is an eigenclass for the action of H(pN) with T (l, k)v = a(l, k)v for some a(l, k) ∈ 𝔽 for all k = 0, …, n and all l prime to pN. Let ρ be a representation ρ : G_{ℚ} → GL(n, 𝔽) unramified outside pN such that

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

for all l not dividing pN. Then we shall say that ρ is attached to v.

For example, theorems of Eichler, Shimura, Deligne and Deligne-Serre imply that if n = 2 and V is the Hecke-module of classical modular forms mod p of some positive weight k then there exists a ρ attached to any Hecke eigenform v ∈ V. Such a ρ is always “odd”, i.e. detρ(Frob_{∞}) = −1.

Conversely, the weak form of Serre’s conjecture [Se] states that given a mod-p 2-dimensional odd Galois representation ρ of the Galois group G_{ℚ} there exists a mod-p modular Hecke eigenform f with ρ attached. In the strong form of Serre’s conjecture, formulas for a weight (≥ 2), level, and nebentype character are given in terms of ρ and it is conjectured that f may be taken to have that weight, level, and nebentype character . For a survey of the current state of Serre’s conjecture, see [Da],[Di] and [E].

The purpose of this paper is to explore an analogue of Serre’s conjecture for general n, especially n = 3. We will make a conjecture in the “niveau 1” case and present some experimental evidence for it. B. Gross has made a similar conjecture in [G4] for finite arithmetic groups. The case of niveau greater than 1 is under study.

Our conjectures use cohomology classes instead of modular forms. By the Eichler-Shimura theorem, the modular form f in Serre’s conjecture can be replaced by a Hecke eigenclass in the cohomology of a congruence subgroup of SL(2, ℤ). When n = 2 our conjecture will imply Serre’s conjecture in the case of “niveau 1”.

By way of background, we first state Conjecture 1.2, to which our Serretype conjecture is a partial converse. Conjecture 1.2 is a slightly weaker version of Conjecture B in [A], restated here in a form suitable for this paper. As proved in [A3], Conjecture B is true when n = 1, 2: by Class Field Theory when n = 1 and the theorems of Eichler, Shimura and Deligne mentioned above when n = 2.

Let V be a right 𝔽[GL(n, ℤ/N)]-module, assumed to be finite dimensional over 𝔽. We view V as an 𝔽S_{N}-module by making the elements of S_{N} act via their reductions mod N.

Conjecture 1.2: Suppose β ∈ H^{∗}(Γ_{0}(N), V) is an eigenclass for the action of the Hecke algebra H(pN). Then there exists a representation ρ : G_{ℚ} → GL(n, 𝔽) unramified outside pN attached to β.

We state our analogues to the weak and strong forms of Serre’s conjecture below as Conjectures 2.1 and 2.2. Note that they include the case where ρ is reducible. In fact, Serre’s original strong conjecture could have been made in the reducible case also, with a few modifications necessitated by the fact that there is no modular form on SL(2, ℤ) of level 1 and weight 2. For example if N = 1 and ρ is a sum of the trivial character and the cyclotomic character mod p, the predicted weight has to be raised from 2 to p + 1.

In section 3 we discuss some theoretical evidence for Conjectures 2.1 and 2.2, i.e. cases where it can be proven they hold: (1) certain monomial ρ when n = p−1; (2) the tensor product of two odd 2-dimensional Galois representations; (3) symmetric squares of odd 2-dimensional Galois representations; and (4) 3-dimensional reducible ρ’s attached to cohomology classes that restrict nontrivially to the boundary of the Borel-Serre compactification M of the locally symmetric space for the relevant congruence subgroup of SL(3, ℤ).

We also have numerical evidence produced by computer. Because of the difficulty of producing Galois extensions of ℚ with large irreducible Galois groups, all of our numerical evidence for Conjectures 1.2, 2.1, and 2.2 as found in [AM], [AAC] and in this paper involves reducible ρ’s.

Our most interesting experimental evidence concerns ρ’s which are sums of an even two-dimensional representation and a one-dimensional representation. In these cases our conjecture gives a cohomological reciprocity law for the even two-dimensional representations. These even representations are not covered by Serre’s conjecture. The corresponding cohomology classes in the cohomology of GL(3, ℤ) must restrict to 0 on the Borel-Serre boundary of M.

Implicit in the definition of “attached” (Definition 1.1) is a choice of a normalization of the Satake isomorphism, called η in [G2]. In our case η is the one parameter subgroup

η(t) = (\begin{matrix}t^{n−1} & 0 & ⋯ & 0 & 0 \\ 0 & t^{n−2} & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & 1 \end{matrix})

According to [G3] we should then expect that ρ(Frob_{∞}) = ±η(−1). This is in fact verified in practice in the examples found in [AM],[AAC] and [A4]. We include this parity requirement in Conjectures 2.1 and 2.2. It is the analogue of the oddness condition in Serre’s conjecture.

When ρ is reducible, we have found experimentally that a finer parity requirement on ρ(Frob_{∞}) is needed. This requirement is formulated below as the “strict parity condition”.

We should stress that almost all of the mod-p cohomology classes for which we test the conjectures lift to torsion classes in characteristic 0 cohomology, and therefore they have no obvious connection with automorphic cohomology or automorphic representations.

We thank B. Gross for explaining his work to us, H. Jacquet for telling us of Ramakrishnan’s work and B. Mazur for many helpful conversations. Thanks also to G. Allison and especially E. Conrad for continuing help with the computations. Thanks to J. Quer for his assistance with Â_{4} - extensions.

2. Statement of the Serre-type conjectures

Let ω denote the cyclotomic character mod p of G_{ℚ}. For any non-negative integer g let V_{g} denote the right GL(n, 𝔽)-module of homogeneous polynomials of degree g in n variables with coefficients in 𝔽.

Irreducible GL(n, 𝔽_{p})-modules are parametrized by “good” n-tuples of integers (b_{1}, b_{2}, …, b_{n}) where “good” means that 0 ≤ b_{1} − b_{2}, …, b_{n−1} − b_{n} ≤ p − 1 and 0 ≤ b_{n} ≤ p − 2. We denote the corresponding module by F (b_{1}, b_{2}, …, b_{n}). It is the restriction to GL(n, 𝔽_{p}) of the unique simple submodule of the dual Weyl module of GL(n, 𝔽) with highest weight (b_{1}, b_{2}, …, b_{n}). The representation F (b_{1}, b_{2}, …, b_{n}) can always be embedded into V_{g} for g = b_{1} + pb_{2} + ⋯ + p^{n−1}b_{n} [DW].

If (a_{1}, a_{2}, …, a_{n}) is any n-tuple of integers, we denote by (a_{1}, a_{2}, …, a_{n})′ a good n-tuple (b_{1}, b_{2}, …, b_{n}) which is congruent to (a_{1}, a_{2}, …, a_{n}) mod p − 1. If none of the consecutive differences a_{i} − a_{i+1} is divisible by p − 1, (a_{1}, a_{2}, …, a_{n})′ is uniquely determined. In the ambiguous cases, one must interpret statements involving (a_{1}, a_{2}, …, a_{n})′ as saying that they are true for some choice of (b_{1}, b_{2}, …, b_{n}) as above. When n = 2 and ρ is irreducible, the ambiguous cases are those in which a discrimination must be made between a Galois representation’s being “peu ramifiée” and “très ramifiée” in Serre’s sense [Se]. Since we have no numerical evidence for such cases when n > 2 we do not pursue this discrimination here. We shall see that this ambiguity also comes into play at level 1 in another way when ρ is reducible, in Proposition 2.7.

Conjecture 2.1: Let there be given a representation (as always continuous and semisimple) ρ : G_{ℚ} → GL(n, 𝔽) unramified outside pN. We assume that ρ(Frob_{∞}) has eigenvalues ±(1, −1, 1, −1, …). Then there exists some integer N′ divisible only by the primes dividing pN and an 𝔽S_{pN′} - module V on which the S_{pN′} - action factors through reduction modulo pN′ and an H(pN′) - eigenclass β ∈ H^{∗}(Γ_{0}(N′), V) such that ρ attached to β.

This is the weak form of our conjecture. Note that we allow N′ to be divisible by p in this form. The necessity of the “parity” condition on ρ(Frob_{∞}) is strongly indicated by the experimental evidence below. In addition, from Theorem 4.3 of [AAC] and its proof one can list all possible Galois representations that can be attached to a Hecke eigenclass in the cohomology of GL(3, ℤ) with coefficients in an 𝔽S_{p} module in which the action factors through reduction mod p, where p = 5, 7. They all satisfy the “parity” condition on ρ(Frob_{∞}). There are strong indications that the same is true for p = 3, 11.

We shall next give a stronger conjecture in which we predict precise analogues of the level, nebentype character, and weight in Serre’s conjecture. We only venture to predict the “weight” when p is odd and ρ restricted to the inertia subgroup T_{p} at p is “niveau 1” in the sense of Serre [Se].

The level: For each prime q dividing N (in particular not equal to p) let G denote the image of a chosen decomposition group at q under ρ and let G_{i} be its ramification subgroups. Denote the cardinality of G_{i} by g_{i}. Imitating the definition of the Artin conductor (as Serre does in [Se]) we set f_{q} = ∑(g_{i}/g_{0})dimM/M^{G_{i}} where M is the vector space 𝔽^{n} viewed as a G-module under ρ. We then set N(ρ) = ∏ q^{f_{q}}. Note that if ρ ≈ σ_{1} ⊕⋯⊕σ_{r} with σ_{i} irreducible for all i then N(ρ) = ∏ N(σ_{i}).

The nebentype character: Again following [Se] we factor detρ into ϵω^{d} for some integer d, where ϵ is unramified at p. By Class Field Theory we can view ϵ also as an 𝔽^{×}-valued Dirichlet character ϵ : (ℤ/N(ρ))^{×} → 𝔽^{×} of conductor dividing N(ρ), and as such we call it ϵ(ρ).

Let N = N(ρ). We have a homomorphism from S_{pN} to (ℤ/N)^{×} sending a matrix to its upper left-hand entry mod N. We use this to pull back ϵ to a character from S_{pN} to 𝔽^{×}, and we denote by 𝔽_{ϵ} the one-dimensional space on which S_{pN} acts via ϵ.

If V is any GL(n, ℤ/p) module, we denote by V (ϵ) the S_{pN} module V ⊗𝔽_{ϵ} where the action on the first factor is via reduction mod p of the matrices in S_{pN}.

The weight: The natural generalization of the weight is an irreducible GL(n, ℤ/p)-module F (b_{1}, …, b_{n}). Every such module can be embedded into V_{g} for some g. When n = 2, the Eichler-Shimura theorem relates the cohomology of V_{g} to the modular forms of weight g + 2.

When ρ is reducible there is an additional requirement needed in our stronger conjecture which we shall call the “strict parity” condition. We consider ρ as a given representation, not merely up to equivalence.

Suppose ρ is isomorphic to the direct sum of irreducible representations σ_{1}, …, σ_{k} of degrees d_{1}, …, d_{k} respectively. We may assume there exists a standard Levi subgroup L of GL(n) of type (d_{1}, …, d_{k}) such that the image of ρ lies in L(𝔽). “Standard” means that if E = {e_{1}, …, e_{n}} is the standard basis of n-space then there exists a partition of E into k parts of sizes d_{1}, …, d_{k} respectively such that L is the simultaneous stabilizer of the k subspaces each of which is spanned by the basis vectors in one part of the partition.

Definition of the strict parity condition: With L and ρ as above, we say that ρ satisfies the strict parity condition if and only if ρ(Frob_{∞}) is conjugate inside L(𝔽) to the diagonal matrix ±diag(1, −1, 1, −1, …).

Conjecture 2.2: Continue the hypotheses of Conjecture 2.1. Assume p is odd and ρ satisfies the strict parity condition. Set N = N(ρ) and ϵ = ϵ(ρ). Let T denote the chosen inertia subgroup at p. Suppose that ρ|T is conjugate inside L(𝔽) to a matrix of the form

ρ|T ∼ (\begin{matrix}ω^{a_{1}} & ∗ & ⋯ & ∗ & ∗ \\ 0 & ω^{a_{2}} & ⋯ & ∗ & ∗ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & ω^{a_{n}} \end{matrix} )

for integers a_{1}, … a_{n}. Then we may take N′ = N and

V = F (a_{1} − (n − 1), a_{2} − (n − 2), …, a_{n})′(ϵ).

Remarks: (1) If the degree ∗ of cohomology is the virtual cohomological dimension of GL(n, ℤ) and if p is prime to the orders of all torsion elements of Γ_{0}(N) then we may replace V by V_{g} for any g such that V embeds into V_{g}. This is because in this case, H^{∗}(Γ_{0}(N′), V) embeds into H^{∗}(Γ_{0}(N′), V_{g}) as Hecke-module. For example, if (a_{1} − (n − 1), a_{2} − (n − 2), …, a_{n})′ = (b_{1}, …, b_{n}), we can take g = b_{1} + b_{2}p + ⋯ + b_{n}p^{n−1}.

We use this in our experimental testing when n = 3 and ∗ = 3 = vcd(GL(3, ℤ)). The torsion primes of GL(3, ℤ) are 2 and 3, and we will always have p > 3 in our examples. We compute the cohomology of a congruence subgroup Γ of GL(3, ℤ) using a cellulation of a deformation retract of minimal dimension of the Borel-Serre compactification M of the locally symmetric space X/Γ, where X is the symmetric space of SL(3, ℝ). For testing the conjectures we concentrate on the degree 3 cohomology classes. This is because cohomology Hecke eigenclasses which restrict nontrivially to the Borel-Serre boundary ∂M are known [AAC] to satisfy Conjecture 2.1 with reducible attached Galois representations. (It would still be of interest to verify Conjecture 2.2 for them.) On the other hand, those cohomology classes that restrict to 0 on ∂M (the “interior part” of the cohomology) only occur in degrees 2 or 3. And the degree 2 and degree 3 interior parts are dual to each other by Lefschetz duality. So for testing the conjectures on the interior cohomology, it is enough to look at the degree 3 space. See [AAC] and [AT] for details, where the quotient of homology dual to the interior cohomology is called the “quasicuspidal homology”. See also Proposition 2.8 below.

Since 3 is the top dimension of the retract of M used to compute the cohomology, it is much easier to work in that dimension, and our programs were written only to compute H^{3}.

Also, we worked with the easiest coefficient systems to handle algorithmically. Hence our computer programs only compute cohomology with coefficients in V_{g} for variable g.

In fact, models for the irreducible modules F (b_{1}, …, b_{n}) are not known in general. However, it is a result of Doty and Walker [DW] that the irreducible GL(n, 𝔽_{p})-module F (b_{1}, b_{2}, …, b_{n}) can be embedded in V_{g} where g = b_{1} + b_{2}p + ⋯ + b_{n}p^{n−1}. This is not always the minimal embedding degree, and certainly not the minimal degree in which F (b_{1}, b_{2}, …, b_{n}) occurs as a subquotient of V_{g} in general. We don’t know what the minimal g should be in general.

Embedding in V_{g} does cause problems when g is too big, as happens with irreducible representations ρ. Conjecture 2.2 will predict some weight F (a, b, c). By twisting we can always insure that c = 0. Reducible representations give us an extra degree of freedom that also lets us take b to be small. That is why all our numerical examples below involve reducible ρ.

As pointed out to us by Tiep, usually there are workable modules of dimension smaller than that of V_{g} for g = a + pb in which we can embed a given F (a, b, 0). For example, F (a, b, 0) is always a submodule of V_{a} ⊗ V_{b}. If we are willing to work with subquotients (although we are not guaranteed that the package of Hecke eigenvalues attached to an interior cohomology class with coefficients in a subquotient of V will also appear in H^{3}(V)) then we can use the fact that F (a, b, 0) is always a subquotient of V_{a−b}⊗V_{b}^{∗} where the star denotes 𝔽-dual. We hope to use these smaller models in some future calculations.

(2) Our assumption on the restriction of ρ to inertia at p corresponds to the “niveau 1” case of Serre’s conjecture. We have no experimental evidence for higher niveaux, nor are we certain what the conjecture should look like in those cases.

(3) If the image of ρ|T is not the whole upper triangular subgroup, for instance if it is contained in the diagonal subgroup, then there may be more than one ordering possible for the characters along the diagonal. One may then choose the ordering that leads to the smallest possible value of g. Other orderings would give “companion forms”, as they are called when n = 2 [G3]. Unfortunately, in the cases we have at hand, the g for the companions is too large for us to test their existence.

(4) Let n = 3. In the case where ρ is the sum of a 1-dimensional and a 2-dimensional representation, the “strict parity condition” of the conjecture is based on computer examples discussed in Section 4 below. When ρ is the sum of three 1-dimensional representations, or for larger n, our conjecture is based on extrapolating from the 1+2 case.

(5) If ρ(Frob_{∞}) has the eigenvalues ±(1, −1, 1, …) the strict parity condition can always be obtained by conjugating ρ and choosing L appropriately. However, this will affect the order of the exponents of ω along the diagonal in ρ|T, and hence the “weight”.

(6) Following Lemma 2.3, we will prove that Conjectures 2.1 and 2.2 are stable under twisting by powers of ω and under replacing ρ with its contragredient.

To state Lemma 2.3, let χ : G_{ℚ} → GL(1, 𝔽) be a character of conductor dividing N and let χ also denote the corresponding Dirichlet character χ : (ℤ/N)^{×} → 𝔽^{×}. For any 𝔽S_{N} - module V as in Conjecture 1.2, let V (χ) denote the tensor product of V and the one-dimensional 𝔽S_{N} - module on which S_{N} acts via the determinant reduced modulo N composed with χ. We also let V (i) denote the tensor product of V and the one-dimensional 𝔽S_{N} module on which S_{N} acts via the i-th power of the determinant mod p.

Lemma 2.3: A class β ∈ H^{∗}(Γ_{0}(N), V) is an H(N) - eigenclass with ρ attached, if and only if β ∈ H^{∗}(Γ_{0}(N), V (χ)(i)) is an H(N) - eigenclass with ρ ⊗ χω^{i} attached.

Proof: Since Γ_{0}(N) is in the kernel of the determinant, it sees V and V (χ)(i) as the same module, so we can view β also as a Hecke eigenclass in H^{∗}(Γ_{0}(N), V (χ)(i)). It is easy to check that this avatar of β has ρ ⊗ χω^{i} attached. (The elements of S_{N} do see the difference between V and V (χ)(i).)

Lemma 2.4: (F (c_{1}, …, c_{n})′)(1)(ϵ) = F (c_{1} + 1, …, c_{n} + 1)′(ϵ).

Proof: First assume ϵ = 1. Let F (c_{1}, …, c_{n})′ = F (b_{1}, …, b_{n}) where (b_{1}, …, b_{n}) is a good n-tuple. Then (F (c_{1}, …, c_{n})′)(1) = F (b_{1}+1, …, b_{n}+1) if b_{n} < p−2 and (F (c_{1}, …, c_{n})′)(1) = F (b_{1}−p+2, …, b_{n}−p+2) if b_{n} = p−2. In either case the parameters on the right hand side form a good n-tuple which is congruent mod p − 1 to (c_{1} + 1, …, c_{n} + 1). Hence the right hand side, in either case, equals F (c_{1} + 1, …, c_{n} + 1)′. To get the general result, just tensor both sides with 𝔽(ϵ).

Remark: On the left hand side of Lemma 2.4 one can make any choice of F (…)′ if there is an ambiguity, and then the proof tells what choice to make on the right hand side.

Lemma 2.5: Suppose ρ : G_{ℚ} → GL(n, 𝔽) is attached to β ∈ H^{∗}(Γ_{0}(N), F (c_{1}, …, c_{n})′(ϵ)). Then ρ ⊗ ω is attached to β now viewed as a class in H^{∗}(Γ_{0}(N), F (c_{1} + 1, …, c_{n} + 1)′(ϵ)).

Proof: By Lemma 2.3, ρ ⊗ ω is attached to β viewed as a class in H^{∗}(Γ_{0}(N), F (c_{1}, …, c_{n})′(1)(ϵ)). Then we are finished by Lemma 2.4.

Proposition 2.6: Conjectures 2.1 and 2.2 are stable under twisting by ω.

Proof: Stability of Conjecture 2.1 under twisting follows immediately from Lemma 2.3. As for Conjecture 2.2, its stability follows from Lemma 2.5.

A small bit of evidence for the conjectures when n = 2 comes from the following Proposition.

Proposition 2.7: Conjectures 2.1 and 2.2 are true for reducible representations when n = 2.

Proof: By assumption ρ is semisimple, so if it is also reducible, there are two characters ψ and ϕ from G_{ℚ} to 𝔽^{×} such that ρ = ψ ⊕ ϕ. We can write ψ = xω^{a} and ϕ = yω^{b} where x, y are unramified at p and have prime-to-p conductors N and M respectively. By Lemma 2.5, we may assume that b = 0, and we may also assume that 2 ≤ a ≤ p. Conjecture 2.2 predicts that ρ is attached to some eigenclass β in H^{1}(Γ_{0}(NM), F (a − 1, 0)′(xy)).

We view x and y as the reductions mod the prime π above p of characters X and Y into a suitable finite extension A of ℤ_{p}. Letting χ denote the cyclotomic character for p, we have that ψ and ϕ are the reductions of Xχ^{a} and Y. Since X, Y are unramified at p they may be identified with A-valued Dirichlet characters having primitive conductors N and M respectively. From the parity condition on ρ(Frob_{∞}) we have that XY (−1) = (−1)^{a+1}.

Then by Proposition 1, page 3 of [W], there exists an Eisenstein series F_{X,Y} of weight a + 1, level NM and nebentype XY, which is a Hecke eigenform with Xχ^{a} ⊕Y as attached Galois representation. This Eisenstein series corresponds à la Eichler-Shimura (cf. Theorem 2.3 in [AS]) to a cohomology eigenclass in H^{1}(Γ_{0}(NM), Sym^{a−1}(A^{2})(XY)). The reduction modulo π of this class is the desired β.

Conjectures 2.1 and 2.2 are compatible with duality in the following sense. For any group G and 𝔽[G]-module V, let V^{∗} denote the dual module, i.e. Hom(V, 𝔽) where g in G acts on a functional f by fg(v) = f(vg^{−1}).

Proposition 2.8: Let ρ be attached to a Hecke eigenclass α in H^{i}(Γ_{0}(N), V (ϵ)), where V is an irreducible GL(n, ℤ/p)-module over 𝔽. Let σ be defined by σ(g) = ^{t}ρ(g^{−1}). Then σ is attached to a Hecke eigenclass β in H^{i}(Γ_{0}(N), V^{∗}(ϵ^{−1})).

Proof: If the invariants of ρ are N, ϵ, V, then the invariants of σ are N, ϵ^{−1}, V^{∗}. This is a simple exercise, given the facts that V^{∗} is isomorphic to the contragredient of V, because V is irreducible (see Lemma 4.6 of [AT]) and that the contragredient of F (a_{1}, …, a_{n})′ is F (−a_{n}, …, −a_{1})′.

Let κ be the character of H supported by α. Define λ to be the character of H given by λ(ΓsΓ) = κ(Γs^{−1}Γ). Let m be the matrix diag(N, 1, 1). Consider the outer automorphism of GL(3) that sends x to m^{−1} ·^{t}x^{−1} ·m. It preserves Γ_{0} and induces an isomorphism of 𝔽-vector spaces from H^{i}(Γ_{0}(N), V (ϵ)) to H^{i}(Γ_{0}(N), V^{∗}(ϵ^{−1})). It sends α to a Hecke eigenclass β supporting λ. It is a straightforward but amusing exercise to show that σ is attached to β.

Remark: Using Lefschetz duality (as in the proof of Theorem 5.1 in [AT]) provided α is an interior class, one can also show the existence of a Hecke eigenclass γ in H^{j}(Γ_{0}(N), V^{∗}(ϵ^{−1})) supporting λ where i+j = n(n+1)/2−1 is the dimension of the locally symmetric space X/Γ_{0}(N).

For absolutely irreducible representations when n = 2 we have the following:

Proposition 2.9: Conjectures 2.1 and 2.2 are compatible with Serre’s conjecture for irreducible representations when n = 2.

Proof: This is easily checked using the fact that F (a, b) embeds into V_{a+pb}, and the Eichler-Shimura isomorphism between cohomology with coefficients in V_{g} and modular forms of weight g + 2.

3. Theoretical evidence

First let n = p − 1. Conjecture 2.1 is true for the following type of ρ, as proved in [A4]. Let K be the cyclotomic extension of ℚ obtained by adjoining a primitive p-th root of unity. Let θ denote an 𝔽-valued ray class character on K viewed as a homomorphism from G_{K} to 𝔽^{×}. Then let ρ : G_{ℚ} → GL(n, 𝔽) denote the induced representation from θ, where n = p − 1.

The “weight”, level and nebentype character that come in also support Conjecture 2.2. If we assume that ρ|T has the stated form in Conjecture 2.2, it follows from Lemma 5.3 of [A4] that ρ|T ≡ diag(ω^{n−1}, …, ω, 1) so the predicted weight is F (0, …, 0)′. Indeed, at the end of section 5 of [A4] it is shown that ρ is attached to a cohomology class with trivial coefficients in this case.

As for the level and nebentype character appearing in [A4], they are consistent with Conjecture 2.2, but cannot be compared exactly because a different kind of congruence subgroup is used in that paper, rather than Γ_{0}(N). (This is because in [A4] we work above the virtual cohomological dimension, and the congruence subgroup we use must have p-torsion.)

Next, let n = 4. Let f_{1}, f_{2} be classical holomorphic modular newforms of even weights k_{1} > k_{2} for SL(2, ℤ). Let π_{i} be the corresponding automorphic representations on GL(2, 𝔸), where 𝔸 denotes the adèle group of ℚ. By the work of D. Ramakrishnan [R] one knows there exists an automorphic representation Π on GL(4, 𝔸) which lifts the tensor product π_{1} ⊗ π_{2} from GL(2) × GL(2) to GL(4).

Each f_{i} corresponds to a cohomology Hecke eigenclass in H^{1}(SL(2, ℤ), F (k_{i} − 2, 0)) with a mod p Galois representation σ_{i} attached to it. From pp. 112-3 of [Cl] one sees that a certain Tate twist of Π has an infinity type with (𝔤, K)-cohomology with a certain finite dimensional coefficient system E over ℂ. Taking a lattice Λ ⊂ E and reducing mod p, we get the existence of a cohomology Hecke eigenclass β in H^{∗}(SL(4, ℤ), Λ/pΛ) which has attached to it the Galois representation ρ = σ_{1} ⊗ σ_{2} : G_{ℚ} → GL(4, 𝔽). This verifies Conjecture 2.1 for ρ, and the same reasoning applies when the f_{i} have nontrivial level and nebentype.

However, in the case we are discussing we have gone further and verified Conjecture 2.2, i.e. we have determined that the weight is correctly predicted in case the σ_{i} are niveau 1 (so that ρ also has niveau 1). To be more precise, we have checked that the weight is correct up to a twist – to determine the twist would require following through all the normalizations involved in the parametrizations of automorphic representations by maps into the L-groups of GL(2) and GL(4). We also have to assume that k_{1} and k_{2} are small compared with p so that Λ/pΛ is irreducible. Interestingly, the requirement that k_{1} and k_{2} be distinct is necessitated by the fact otherwise Π_{∞} is not “regular” in the sense of [Cl] and therefore is not known to be connected with cohomology.

For the rest of this section and for the remainder of this paper we set n = 3. We conclude this section by discussing two series of examples that support Conjecture 2.1. The first applies to symmetric squares lifts from GL(2) to GL(3). The second example applies to reducible representations of type 1+2 where the 2-dimensional representation is odd. This type of representation we expect to be attached to cohomology classes that come from the boundary of the symmetric space. A fuller discussion of these examples, as of the preceding GL(4) example, will be more appropriate for a separate paper.

(i) Symmetric squares: Suppose π is a cuspidal irreducible automorphic representation for GL(2, 𝔸) generated by some holomorphic newform f of integer weight k greater than 1, level N and nebentype ϵ. Thus by a theorem of Eichler and Shimura, f corresponds to a cohomology Hecke eigenclass in H^{1}(Γ_{0}(N), V_{k−2}(ϵ)). Let σ denote the attached Galois representation G_{ℚ} → GL(2, 𝔽) for a suitable 𝔽.

Let A : GL(2) → GL(3) be the “symmetric squares” homomorphism, i.e. the adjoint representation of GL(2) on the 2-by-2 matrices of trace 0 multiplied by the determinant. Let Π be the symmetric square lifting of π (due to Gelbart and Jacquet [GJ]). As discussed in [AS] and [AT], the automorphic representation Π for GL(3, 𝔸) has cohomological infinity type. That is, Π_{∞} has (𝔤, K)-cohomology with a well-defined finite dimensional coefficient system E over ℂ. Taking a lattice Λ ⊂ E and reducing mod p, this allows one to deduce the existence of a cohomology Hecke eigenclass β_{f} in H^{3}(Γ_{0}(N′), (Λ/pΛ)(ϵ′)) for suitable N′ and ϵ′ which has attached to it the Galois representation G_{ℚ} → GL(3, 𝔽) given by A ∘ σ.

Now do this in reverse: suppose we are given ρ : G_{ℚ} → GL(3, 𝔽) which is the symmetric square of an irreducible 2-dimensional representation σ and satisfies the hypotheses of Conjecture 2.1. So ρ = A ∘ σ. In particular, σ must be odd. Then by Serre’s conjecture, σ is attached to some f mod p, and so by the preceding paragraph, ρ will be attached to the corresponding β_{f}. Thus Conjecture 2.1 holds for ρ if Serre’s conjecture holds for σ.

In [AT] we show that if σ is “niveau 1” and if 0 < k − 2 < (p − 1)/2, or if k − 2 = (p − 1)/2 and p = 29, 37, or 41, then Λ/pΛ can be replaced by the weight predicted for ρ by Conjecture 2.2, which is a twist of F (2k−2, k−2, 0).

(ii) Borel-Serre boundary: Let X be the symmetric space for SL(3, ℝ) and M the Borel-Serre compactification of X/Γ for an arithmetic subgroup Γ of SL(3, ℤ). Assume that p > 3. Then the cohomology of Γ with trivial coefficients is canonically isomorphic to the cohomology of M, and one can consider the restriction map of cohomology induced by the inclusion of the boundary of M into M.

From [LS] one sees that the cohomology of the boundary of M can be given in terms of classical modular forms of weights 2 and 3 and Dirichlet characters. Therefore those boundary classes which are Hecke eigenclasses, and hence also those Hecke eigenclasses in the cohomology of Γ that restrict nontrivially to the boundary, have attached p-adic Galois representations that are reducible. (Compare the statement and proof of Theorem 3.1 in [AAC] which states that any Hecke eigenclass in the cohomology of the boundary with any admissible mod p coefficient module has an attached reducible Galois representation.) We can reduce the p-adic representation mod p to get an attached ρ.

Remark: In the case where ρ is a sum of irreducible representations of dimensions 1 and 2 respectively, the 2-dimensional representation must be odd, since it is coming from a classical modular form. Therefore, to conform with Conjecture 2.2 in predicting the weight, we must take the strict parity condition into account, although there is no parity condition on the 1-dimensional character. It remains to be seen if one can prove that every sum of a 1-dimensional and an odd 2-dimensional representation, and every sum of three 1-dimensional representations, that obeys the strict parity condition is attached to a boundary cohomology class of the predicted weight, nebentype, and level.

4. Computer generated examples

Examples with N > 1

In [AM] we computed a number of cohomology Hecke eigenclasses with their Hecke eigenvalues for l = 2, 3, …, 97 for the congruence subgroups Γ_{0}(N) of SL(3, ℤ) for prime N up to about 250, trivial nebentype character and trivial coefficients ℤ/p. In some of these cases we were able to find a Galois representation ρ that appeared to be attached to the eigenclass in the sense that the Hecke polynomial for every l ≤ 97 equaled the characteristic polynomial of Frob_{l}.

In these cases, ρ was reducible. (Cases in which the data predicted an irreducible ρ had the image of ρ so large that we were unable to find it.) In each of these cases we have gone back now and checked that ρ(Frob_{∞}) has eigenvalues (1, −1, 1) and that ρ restricted to inertia at p has eigencharacters along the diagonal (ω^{2}, ω, 1) in conformity with Conjecture 2.2, in particular the strict parity condition.

These cases include the examples where p = 3 and the image of the 2dimensional component is Â_{4}, and the example where p = 5 and the ρ is a sum of three characters. In these cases, the predicted level and nebentype characters can also be checked to be correct.

One additional case that did not appear in [AM] deserves mention. There is a totally real Â_{4} extension of ℚ unramified outside 277. This leads to a test of Conjecture 2.2 for p = 277 and N = 1 as discussed below. However, Â_{4} is isomorphic to SL(3, ℤ/3), so the same extension gives rise to σ : G_{ℚ} → SL(3, 𝔽) where now p = 3. Set ρ = σ ⊕ ω with L equal to the intersection of the stabilizers of the spaces of row vectors (0, ∗, 0) and column vectors t(0, ∗, 0). Then the strict parity condition is satisfied if we put ω in the “middle”. Then ρ|T is conjugate in L(𝔽) to the diag(1, ω, 1) = diag(ω^{2}, ω, 1). The predicted weight is thus the trivial module F (0, 0, 0). It is easy to see that the predicted level is N = 277 and the predicted nebentype character is trivial. We asked Mark McConnell to run his programs to find the Hecke module H^{3}(Γ_{0}(277), ℤ/3). Indeed, there was exactly one interior class (up to scalar multiples) and its Hecke eigenvalues (for primes l ≠ 3, l ≤ 97) confirmed Conjecture 2.2 in this case.

We have included an example of a calculation showing the equality of a Hecke polynomial at l and the characteristic polynomial of Frob_{l} for p = 277, g = 90, l = 5 at the end of the discussion of the Â_{4} cases in the next section.

Examples with N = 1

Suppose that we have an irreducible representation σ : G_{ℚ} → GL(2, 𝔽) with the following properties:

(1) σ is unramified outside p.

(2) The image of σ has order relatively prime to p.

(3) σ(Frob_{∞}) is central, where Frob_{∞} is a complex conjugation in G_{ℚ}.

(4) σ(T_{p}) has order dividing p − 1.

Because of condition (3), Serre’s Conjecture does not apply to σ. However, if we let ρ = ω^{j}σ ⊕ ω^{k}, for suitably chosen integers j and k, we obtain a representation to which Conjectures 2.1 and 2.2 apply, with n = 3 and N = 1: precisely, if σ(Frob_{∞}) = 1, we should choose j and k to have opposite parity, while if σ(Frob_{∞}) = −1, we should choose j and k to have the same parity. In this section we consider numerical evidence for Conjectures 2.1 and 2.2 for representations ρ obtained in this way.

The prescription in 2.2 tells us that we should replace ρ by a conjugate representation corresponding to the embedding of GL(2)×GL(1) into GL(3) whose image L is the Levi subgroup of the form

( \begin{matrix}∗ & & ∗ \\ & ∗ \\ ∗ & & ∗ \end{matrix} );

then we will have

ρ(Frob_{∞}) ∼_{L} ± ( \begin{matrix}1 \\ & −1 \\ & & 1 \end{matrix} )

and

ρ|T_{p} ∼_{L} ( \begin{matrix}ω^{a} \\ & ω^{b} \\ & & ω^{c} \end{matrix} ) ,

for certain integers a, b, c. In these expressions “∼_{L}” represents conjugacy in L, so that b ≡ k mod p − 1, while a mod p − 1 and c mod p − 1 can be found by examining ω^{j}σ|T_{p}. Conjecture 2.2 then predicts that we should find a Hecke eigenclass corresponding to ρ in H^{∗}(SL(3, ℤ), F (a − 2, b − 1, c)′); by Remark (1) after Conjecture 2.2, there should be a Hecke eigenclass corresponding to ρ in H^{3}(SL(3, ℤ), V_{g}(𝔽_{p})), where g = a − 2 + (b − 1)p + cp^{2} (assuming that a, b, c have been chosen so that (a − 2, b − 1, c) is a “good” triple). The existence of an eigenclass for the predicted value of g is already a partial confirmation of the conjecture; we can usually go further and compare the characteristic polynomials of ρ(Frob_{l}) with the characteristic polynomials predicted by Conjecture 2.1, for a number of small primes l, and find additional confirmation in this way that Conjectures 2.1 and 2.2 are correct. The computations of cohomology and Hecke action use the programs described in [AAC].

Let σ̃ : G_{ℚ} → P GL(2, 𝔽) be the projective representation corresponding to σ, let G be the image of σ̃, and let K be the fixed field of the kernel of σ̃. Then K will be a Galois extension of ℚ, unramified outside of p, with Galois group isomorphic to G. Furthermore, K is totally real, since σ̃(Frob_{∞}) = 1.

Since G has order prime to p, it can be lifted to characteristic 0; hence G can be realized as an irreducible subgroup of P GL(2, ℂ) and so is isomorphic to A_{5}, S_{4}, A_{4}, or a dihedral group.

Thus we can organize our search for representations σ satisfying conditions (1)–(4) above by reversing these steps, as follows:

Choose a group G from the list above, and search for totally real Galois extensions K/ℚ unramified outside a single prime p |̸#G with Gal(K/ℚ) ≃ G. Given such a K, choose a projective representation σ̃ : G_{ℚ} → P GL(2, 𝔽) whose kernel has K as fixed field, and consider the problem of lifting σ̃ to a representation σ : G_{ℚ} → GL(2, 𝔽) unramified outside p. The following lemma (and its proof) is due to Serre; we heard about it from Richard Taylor:

Lifting Lemma : Let σ̃ : G_{ℚ} → P GL(2, 𝔽) be a continuous projective representation which is totally real (i.e. σ̃(Frob_{∞}) = 1) and unramified outside the prime p. Then there is a unimodular lifting σ : G_{ℚ} → SL(2, 𝔽) of σ̃ which is unramified outside p.

Proof: We show first that a unimodular lifting exists; then we investigate the ramification. Since 𝔽 is algebraically closed, the map SL(2, 𝔽) → P GL(2, 𝔽) is surjective, and its kernel is ±1; hence the obstruction to lifting σ̃ : G_{ℚ} → P GL(2, 𝔽) is the class of σ̃ in H^{2}(G_{ℚ}, ±1). From class field theory we have a short exact sequence of local and global Brauer groups

0 → H^{2}(G_{ℚ}, ℚ̄^{×}) → ⨁_{v} H^{2}(G_{ℚ_{v}}, ℚ̄_{v}^{×}) → ℚ/ℤ → 0,

which yields, taking the kernel of multiplication by 2, the following exact sequence:

0 → H^{2}(G_{ℚ}, ±1) → ⨁_{v} H^{2}(G_{ℚ_{v}}, ±1) → \frac{1}{2}ℤ/ℤ.

Here v runs over all the places of ℚ and G_{ℚ_{v}} = Gal(ℚ̄_{v}/ℚ_{v}) is identified with a decomposition group of v in G_{ℚ}. Hence it suffices to show that the restriction σ̃_{v} of σ̃ to G_{ℚ_{v}} can be lifted for each v.

If v = ∞, then σ̃_{v} is the trivial representation, since σ̃(Frob_{∞}) = 1, and so σ̃_{v} can be lifted to the trivial representation.

If v is a finite prime not equal to p, then σ̃_{v} factors through the maximal unramified extension of ℚ_{v}. Therefore σ̃_{v} can be lifted to an unramified representation of G_{ℚ_{v}} by choosing an element of SL(2, 𝔽) which maps onto σ̃_{v}(Frob_{v}), where Frob_{v} is a Frobenius automorphism in G_{ℚ_{v}}.

It follows that σ̃_{p} must also lift, by the short exact sequence above. Hence there is a representation σ : G_{ℚ} → SL(2, 𝔽) which lifts σ̃. It is clear that σ is uniquely determined up to multiplication by a quadratic character of G_{ℚ}.

The image of σ is not isomorphic to G, since A_{5}, S_{4}, and A_{4} have no faithful 2 dimensional representations, and the dihedral groups have no faithful unimodular 2 dimensional representations. Thus the fixed field of the kernel of σ is a quadratic extension of K and σ may be ramified at other primes besides p. However, multiplying σ by a suitable quadratic character of G_{ℚ} will produce a unimodular lifting unramified outside p.

This can be seen as follows. Suppose that q ≠ p is a prime which is ramified for σ. Then σ(T_{q}) is the subgroup ±I of SL(2, 𝔽), where I is the 2 × 2 identity matrix. Let G_{q} be the decomposition group of q in G_{ℚ} containing T_{q}. Since σ(G_{q})/σ(T_{q}) is cyclic and σ(T_{q}) is central, σ(G_{q}) is abelian. Let N be the commutator subgroup of G_{q}; then N ⊆ T_{q}, and T_{q}/N may be identified with Gal(ℚ_{q}(μ_{q^{∞}})/ℚ_{q}), which in turn can be identified with Gal(ℚ(μ_{q^{∞}})/ℚ). (Here μ_{q^{∞}} is the group of roots of unity of order a power of q.) Hence there is a quadratic character χ_{q} of G_{ℚ}, unramified outside q, such that

σ(τ) = χ_{q}(τ)I, for τ ∈ T_{q}.

(A simpler argument is available for q odd: then σ is tamely ramified at q, and T_{q} has a unique character of order 2, so that we may take χ_{q} to be the quadratic character of conductor q.) Let

χ = ∏_{q} χ_{q},

the product taken over all rational primes q ≠ p which are ramified for σ. Then χσ is unramified outside p.

Finally, once σ has been found, we must check that condition (4) is satisfied (it always is in the dihedral and A_{4} cases).

Suppose then that σ is a unimodular representation satisfying conditions (1)–(4) above. Then σ|T_{p} = ω^{d} ⊕ ω^{−d} for some integer d. Let ρ = ω^{j}σ ⊕ ω^{k}, where the integers j and k satisfy the parity condition needed to guarantee that the eigenvalues of ρ(Frob_{∞}) are either (1, −1, 1) or (−1, 1, −1). Then

ρ|T_{p} ≃ ω^{j+d} ⊕ ω^{j−d} ⊕ ω^{k}.

The “strict parity condition” then requires that we take b ≡ k mod p − 1 (a and c may be taken to be congruent to j + d and j − d in either order). In the examples that follow, we choose j and k so as to make g small: this means that we take j = ±d and k = 1 or 2 (the parity of k is fixed by the choice of j); this allows us to take c = 0 and b = 1 or 2.

Examples of dihedral type We take G to be a dihedral group, so that we are looking for a Galois extension K/ℚ which is totally real and unramified outside a prime p, and for which Gal(K/ℚ) is dihedral and of order prime to p. p must be odd since G has even order. Let k be the maximal abelian extension of ℚ in K. Then since k is unramifed outside p, k/ℚ is cyclic; it follows that [k : ℚ] = 2 and that the relative degree [K : k] is odd, since G is dihedral. Since k is also totally real, we have p ≡ 1 mod 4 and k = ℚ(\sqrt{p}).

Let T be an inertia group of p in G. Since p is tamely ramified in K, T is cyclic; since T surjects onto Gal(k/ℚ), T ∩ Gal(K/k) must be trivial. So T has order 2, and K/k is unramified at p.

Hence the dihedral extensions we are looking for are in 1-to-1 correspondence with the set of pairs (p, C), where p is a prime ≡ 1 mod 4 and C is a nontrivial cyclic quotient of the class group of ℚ(\sqrt{p}). (It is known that the class number of ℚ(\sqrt{p}) is odd and prime to p, so no further restrictions on C are needed; see [Sl].)

We fix such a dihedral extension K, and let σ̃ : G_{ℚ} → P GL(2, 𝔽) be a projective representation corresponding to K. In this case σ̃ lifts to a representation σ : G_{ℚ} → GL(2, 𝔽) whose image is also isomorphic to G, so that σ is again totally real and unramified outside p. We have σ|T_{p} ≃ ω^{(p−1)/2} ⊕ 1, since the image of T_{p} under σ has order 2 and det σ = ω^{(p−1)/2} (the determinant of any of the 2 dimensional irreducible representations of G is the nontrivial 1 dimensional character of G). So we may take ρ = σ⊕ω, for which we will have a = (p−1)/2, b = 1, c = 0, so that g = a−2 = (p−5)/2.

There are 6 primes p < 1000 which have class number h greater than 1; in each of these cases h itself is prime, so we obtain a unique dihedral extension unramified outside p:

\begin{matrix}p & 229 & 257 & 401 & 577 & 733 & 761 \\ h & 3 & 3 & 5 & 7 & 3 & 3 \\ g & 112 & 126 & 198 & 286 & 364 & 378 \end{matrix}

The dihedral group of order 2h has ϕ(h)/2 two dimensional faithful representations. Hence for p = 229, 257, 733, and 761, there is a unique choice of σ, while for p = 401 there are two choices for σ, and for p = 577 there are three. Using the programs described in [AAC], we find a unique interior Hecke eigenclass for p = 229 in weight g = 112, a unique interior Hecke eigenclass for p = 257 in weight g = 126, and a unique interior Hecke eigenclass for p = 733 in weight g = 364; for p = 401, we find a 2-dimensional interior Hecke eigenspace in weight g = 198, and for p = 577, there is a 3-dimensional interior Hecke eigenspace in weight 286. Moreover, as stated in [AAC], the Hecke action on the eigenclasses for p = 229 and 257 was computed for a number of small primes l (l ≤ 13 for p = 229; l ≤ 19 for p = 257) and in each case was found to be consistent with the representation ρ = σ ⊕ ω, (G ≃ S_{3} in these cases, so σ is the unique irreducible 2-dimensional representation of G). Finally, we have also computed the Hecke action for l ≤ 7 for the classes for p =401, 577, and 733, again with results consistent with Conjecture 2.2. The final pair in the table above (p = 761, g = 378) is just out of reach of our computer programs at present.

Examples of A_{4} type We take G to be A_{4}, and look for totally real Galois extensions K/ℚ, unramified outside a single prime p, with Galois group isomorphic to A_{4}. An analysis similar to that given in the dihedral case shows that such extensions are in 1-to-1 correspondence with the set of pairs (p, C), where p is prime, p ≡ 1 mod 3, and C is a Galois stable quotient of the ideal class group of the cubic subfield of ℚ(ζ_{p}) isomorphic to the Klein four-group. These fields can be found be looking for monic quartic polynomials f(x) ∈ ℤ[x] with Galois group A_{4}, four real roots, and field discriminant the square of a prime p. A search of the Bordeaux tables [B] finds 9 such extensions in the range p < 1000; the values of p for these extensions are p =163, 277, 349, 397, 547, 607, 709, 853, and 937. For each of these primes there is a unique field K. Note that the ramification index of p in K is 3.

Let p be one of these primes, and let K be the associated A_{4}-extension. Let σ̃ : G_{ℚ} → P GL(2, 𝔽) be a projective representation corresponding to K; σ̃ is unique (up to conjugacy). By the lifting lemma above, there are exactly two liftings of σ̃ to SL(2, 𝔽) which are unramified outside p; if σ : G_{ℚ} → SL(2, 𝔽) is one these liftings, ω^{(p−1)/2}σ is the other.

Let K̂ be the fixed field of the kernel of σ, and let e be the ramification index of K̂ at p. Then e is either 6 or 3 (depending on whether K̂/K is ramified at p or not); since σ(T_{p}) is abelian, we have

σ|T_{p} ≃ ω^{(p−1)/e} ⊕ ω^{−(p−1)/e}.

It follows that replacing σ by ω^{(p−1)/2}σ switches the cases e = 6 and e = 3; hence both cases do in fact occur, and we may suppose that σ has been chosen so that e is equal to 3 and K̂/K is unramified at p.

The image of σ is isomorphic to Â_{4}, the double cover of A_{4}. K̂ either totally real or totally complex, and to test Conjectures 2.1 and 2.2 we need to know which.

We do this as follows. Given the field K, described as the splitting field of a monic quartic polynomial f(x) ∈ ℤ[x], we obtained from Jordi Quer an element g ∈ K, written as a polynomial with integer coefficients in two of the roots a, b of f (since K = ℚ(a, b), for any two roots of f), with the property that K(\sqrt{g}) is an Â_{4} extension of ℚ. Quer’s programs are based on the method of Crespo [C] for constructing Â_{n}-extensions.

Lemma The set of Â_{4} extensions L of ℚ containing K is in one-to-one correspondence with the set of square-free integers: each such L can be written as L = K(\sqrt{gm}) for some unique square-free integer m.

Proof: From the short exact sequence

1 → {±1} → K^{×} → (K^{×})^{2} → 1,

induced by squaring, we obtain the exact sequence

1 → H^{1}(G, (K^{×})^{2}) → H^{2}(G, ±1),

where G = Gal(K/ℚ) ≃ A_{4}; and from

1 → (K^{×})^{2} → K^{×} → K^{×}/(K^{×})^{2} → 1,

we obtain the exact sequence

ℚ^{×} → (K^{×}/(K^{×})^{2})^{G} → H^{1}(G, (K^{×})^{2}) → 1.

Splicing these together gives us an exact sequence

ℚ^{×} → (K^{×}/(K^{×})^{2})^{G} → H^{2}(G, ±1);

by direct calculation, one sees that an element x ∈ K^{×} which representing a nontrivial class in (K^{×}/(K^{×})^{2})^{G} maps to the class in H^{2}(G, ±1) of the group extension

1 → \{±1\} → Gal(K(\sqrt{x})/ℚ → G → 1.

(Here we have identified Gal(K(\sqrt{x})/K) with \{±1\}). H^{2}(G, ±1) has order 2, and the class of Â_{4}, as an extension of A_{4} by ±1, corresponds to the nontrivial class in H^{2}(G, ±1). Hence the element g provided us by Quer lies in (K^{×}/(K^{×})^{2})^{G} and maps to the nontrivial class in H^{2}(G, ±1); and if x ∈ (K^{×}/(K^{×})^{2})^{G} is any element which maps to the nontrivial class in H^{2}(G, ±1), then x can be written in the form gmy^{2}, where m is square free integer and y ∈ K. The uniqueness of m results from the fact that ℚ^{×} ∩ (K^{×})^{2} = (ℚ^{×})^{2}, since K has no quadratic subfields.

Thus the extension K̂ is of the form K(\sqrt{gm}) for some unique square free integer m; of course, we are free to modify g by a square as well, if convenient.

To modify g so as to find a generator for K̂/K, we proceed as follows. First of all, since K(\sqrt{g}) is Galois over ℚ, it follows that g^{σ−1} is a square in K, for every σ ∈ G. Let F = ℚ(a); then [K : F] = 3, which is odd, so that r = ℕ_{K/F} g will generate the same quadratic extension of K as g. Next we find the largest rational integer N which divides r in the ring of integers O_{F} of F, and set s = r/N. Then K(\sqrt{s}) will be a possibly different Â_{4}-extension.

We now observe that K(\sqrt{s})/K is unramified at all primes except possibly those lying above p or 2. Indeed, if q ≠ p is an odd rational prime, then q does not divide s in O_{F}; hence there is some prime 𝔮 of F dividing q which does not divide s. Hence F (\sqrt{s})/F is unramified at 𝔮, and so K(\sqrt{s})/K is unramified at all primes of K above 𝔮, and therefore at all primes above q.

To decide whether K(\sqrt{s})/K is unramified at the primes above p, we need to examine the divisibility of s by the primes above p in F; in fact, in all the examples we treat below, s turns out to be prime to p. This is easily detected by calculating ℕ_{F/ℚ}(s) and seeing that this integer is not divisible by p.

At this point we can say that K̂ is either K(\sqrt{s}) or K(\sqrt{−s}), since the only freedom we have left is to change s to −s. To decide which, we have to consider ramification at 2. As with odd primes, there is a prime 𝔭 dividing 2 in F which does not divide s. We have the following lemma:

Lemma The following are equivalent:

1. K(\sqrt{s})/K is unramified at all primes of K above 2.

2. F (\sqrt{s})/F is unramified at 𝔭.

3. s ≡ u^{2} (mod 𝔭^{2}) for some u ∈ O_{F}.

Proof: (1) ⟺ (2) is immediate, since K(\sqrt{s})/ℚ is Galois and K is unramified at 2. To show that (3) ⇒ (2), we may change s by a square and suppose that s ≡ 1 (mod 𝔭^{2}); then F (\sqrt{s})/F is also generated by the roots of the polynomial x^{2} + x + (1 − s)/4, hence is unramified at 𝔭. (Note that (1 − s)/4 is integral at 𝔭 because 2 is unramified in F.) To show that (2) ⇒ (3), we first observe that (O_{F} /𝔭^{2})^{×} ≃ 𝔽(𝔭)^{×} × 𝔽(𝔭)^{+} (with 𝔽(𝔭) = O_{F} /𝔭), from which it follows that

s is a square in (O_{F} /𝔭^{2})^{×} ⟺ s^{q−1} ≡ 1 (mod 𝔭^{2}),

where q = #𝔽(𝔭) = ℕ𝔭. Now suppose that F (\sqrt{s})/F is unramified at 𝔭. If 𝔭 splits in F (\sqrt{s}), then s is a square in 𝔽(𝔭) and (3) holds. On the other hand, if 𝔭 remains prime, then the Artin symbol of 𝔭 for the extension F (\sqrt{s})/F is the non-trivial automorphism ϕ of F (\sqrt{s})/F; we have the congruence

ϕ(\sqrt{s}) = −\sqrt{s} ≡ (\sqrt{s})^{q} (mod 𝔭),

which implies on squaring that s ≡ s^{q} (mod 𝔭^{2}), so that (3) holds.

The method for checking whether s or −s satisfies condition (3) varies from case to case. Often, the simplest thing to do is to modify s by a square in K so that it becomes prime to 2, then considering s mod 4 (this amounts to considering condition (3) for all the primes dividing 2 at once). In any case, once we have found t = ±s such that K̂ = K(\sqrt{t}), we determine whether K̂ is totally real or totally complex by determining whether t is totally positive or totally negative.

The standard part of these calculations goes as follows, in GP-PARI (1.39). We have to supply the quartic polynomial f(x), the expression for g as a polynomial in two of the roots (a and b) of f, and the prime p.

f=????;
galois(f)
sturm(f)
factor(discf(f))
sqrt(disc(f)/discf(f))

a=mod(x,f); ff=f/(x-a);
b=[0,1,0; 0,0,1;-coeff(ff,0),-coeff(ff,1),-coeff(ff,2)];
g=????;
r=det(g); s=r/content(r);
p=????;
mod(norm(s),p)

The first three commands compute the Galois group of f, the number of real roots of f, and the factorization of the discriminant of the field F = ℚ(a); these are simply to confirm that we do indeed have a totally real A_{4}- extension of ℚ ramified only at a single prime. The next command calculates the index [O_{F} : ℤ[a]]. We view the second root b as the companion matrix (with entries from F) of the polynomial f(x)/(x−a), which is an irreducible cubic over F, so that the norm r = ℕ_{K/F} g can be computed by taking the determinant of the matrix of g. We obtain s by removing from r the gcd of its coefficients as a polynomial (of degree at most 3) in a. The last command checks that s is prime to p (provided that mod(norm(s),p) is nonzero, which is true for the cases examined below).

When [O_{F} : ℤ[a]] > 1, s may still be divisible by an integer dividing this index — in the cases treated below, s is sometimes still divisible by 2. This can be detected by checking to see if s/2 is an algebraic integer:

content(char(s/2,x))

—if s/2 is an integer in F, then the content of its characteristic polynomial will be 1, in which case we replace s by s/2. Of course we must then test the new s to see whether it is still divisible by 2.

We illustrate with p = 277. The field F is generated by a root a of the polynomial f(x) = x^{4} − x^{3} − 16x^{2} + 3x + 1, and the value of g obtained from Quer is 9417+787a−51a^{2}−133a^{3}+1359b−627ab−468a^{2}b+114a^{3}b−345b^{2}− 216ab^{2} +57a^{2}b^{2}. The PARI commands above confirm that the splitting field of f is a totally real A_{4}-extension of ℚ ramified only at 277, determine that the index [O_{F} : ℤ[a]] equals 4, and find that s = −150700a^{3} + 193305a^{2} + 2034265a + 347661. s is an element of the field F such that K(\sqrt{s}) is an Â_{4} extension of ℚ, and such that K(\sqrt{s})/K is unramified at all primes except possibly 2 and 277. In fact K(\sqrt{s})/K is unramified at 277, since ℕ_{F/ℚ}(s) = 10483965209607696 is not divisible by 277.

In this case, we need still to investigate whether s might be divisible by 2 (since ℤ[a] has index 4 in O_{F}): to see whether s/2 is integral, we calculate the content of its characteristic polynomial:

content(char(s/2,x))

yields the value 1/4—so s is not divisible by 2, and thus is not divisible by at least one prime of F above 2.

Using PARI, the following facts are easy to discover: since ℕ(a + 1) = 16 there is a prime 𝔭 above 2 which divides a+1, so that 𝔭 also divides a−1; the number (a−1)^{2}/(a+1) is an algebraic integer with odd norm; hence a+1 is in fact divisible by 𝔭^{2}. Note that s ≡ a^{2}+a+1 mod 4; so s ≡ 1 mod (4, a+1), and therefore s ≡ 1 mod 𝔭^{2}. So, by the lemma, K(\sqrt{s}) is unramified at all primes dividing 2, and therefore K̂ = K(\sqrt{s}). Finally, since the trace of s is 3775974, it follows that s is totally positive, and so K̂ is totally real.

Similar calculations in PARI with the remaining primes yield the following table:

\begin{matrix}p & 163 & 277 & 349 & 397 & 547 & 607 & 709 & 853 & 937 \\ K̂ & − & + & − & − & − & + & − & − & − \end{matrix}

where we have used + to stand for a totally real field, and − to stand for a totally complex field.

Let σ : G_{ℚ} → SL(2, 𝔽) be the unimodular representation associated to K̂. To get a 3-dimensional representation to which Conjectures 2.1 and 2.2 apply, we form ρ = ω^{j}σ ⊕ ω^{k} where j and k have different parity if σ(Frob_{∞}) = 1, and j and k have the same parity if σ(Frob_{∞}) = −1. Since K̂/K is unramified at p, the ramification index of p in K̂/ℚ is 3. Hence σ|T_{p} ≃ ω^{−(p−1)/3} ⊕ ω^{(p−1)/3}.

Suppose that σ(Frob_{∞}) = 1: thus K̂ is totally real, and so p = 277 or 607. Then we can take j = −(p − 1)/3 and k = 1. For ρ = ω^{−(p−1)/3}σ ⊕ ω, we can then take a = (p − 1)/3, b = 1, and c = 0, giving (according to Conjecture 2.2) g = (p − 1)/3 − 2; so g = 90 if p = 277 and g = 200 if p = 607. Using the programs discussed in [AAC] we do in fact find a unique one dimensional interior Hecke eigenspace in H^{3}(SL(3, ℤ), V_{90}(𝔽_{277})) and in H^{3}(SL(3, ℤ), V_{200}(𝔽_{607})). Furthermore, we have computed the eigenvalues of the Hecke operators T (l, k) for l = 2, 3, 5, 7, 11, 13 and k = 1, 2 and found that the characteristic polynomials of Frobenius predicted by our conjecture are those arising from the representation ρ = ω^{−(p−1)/3}σ ⊕ ω.

We can also take j = (p − 1)/3 and k = 1, yielding a = 2(p − 1)/3, b = 1, c = 0, and so (according to Conjecture 2.2) g = 2(p − 1)/3 − 2. So g = 182 if p = 277 and g = 402 if p = 607. The first case is small enough for calculation; we find a unique one dimensional interior Hecke eigenspace in H^{3}(SL(3, ℤ), V_{182}(𝔽_{277})), and can verify that Conjecture 2.1 correctly predicts the characteristic polynomials of Frobenius for l = 2, 3, 5, 7.

We illustrate these calculations with p = 277, l = 5, and ρ = ω^{−92}σ ⊕ ω. We consider the Hecke operators T (5, k) (k=0, 1, 2, 3) acting on the unique one dimensional interior Hecke eigenspace in H^{3}(SL(3, ℤ), V_{90}(𝔽_{277})). T (5, 0) is the identity, and has eigenvalue 1; T (5, 3) arises from the scalar matrix diag(5, 5, 5), and has eigenvalue 5^{90} (mod 277). According to [AAC], the eigenvalues of the Hecke operators T (5, 1) and T (5, 2) are 122 and 251 (mod 277), respectively, so that Conjecture 2.1 predicts that (in 𝔽_{277}[X])

det(1 − ρ(Frob_{5})X) = 1 − 122X + 5 · 251X^{2} − 5^{3} · 5^{90}X^{3}
= 1 + 155X + 147X^{2} + 251X^{3}

ρ(Frob_{5}) can be determined by examining the factorization of 5 in K̂. Recall that K is the splitting field of f(X) = 1 + 3X − 16X^{2} − X^{3} + X^{4} and that F is the field generated by a root of f. Since f(X) factors mod 5 into the irreducible factors X + 2 and X^{3} + 2X^{2} + 3, 5 factors in F into a prime of degree 1 and a prime of degree 3, and from this it is easy to see that Frob_{5} is a 3-cycle in Gal(K/ℚ) ≃ A_{4}. Since K̂ = K(\sqrt{s}), with s = −150700a^{3} + 193305a^{2} + 2034265a + 347661 ≡ 1 mod 5, the primes above 5 in K split in K̂.

Now Â_{4} has a unique unimodular 2-dimensional representation, so we may identify σ with this representation; we find tr(σ(τ)) = −1 for any element of order 3 in Â_{4}, so that for ρ = ω^{−(p−1)/3}σ ⊕ ω = ω^{−92}σ ⊕ ω, we have det(1 − ρ(Frob_{5})X) = (1 − 5^{−92}(−1)X + 5^{−184}X^{2})(1 − 5X) = 1 + 155x + 147x^{2} + 251x^{3}.

If σ(Frob_{∞}) = −1, we can take j = −(p − 1)/3 and k = 2, giving a = (p − 1)/3, b = 2, c = 0, and g = (p − 1)/3 − 2 + p; so g = 215 if p = 163; g = 463 if p = 349, etc. The first case is small enough for calculation; again we find unique one dimensional interior Hecke eigenspace in H^{3}(SL(3, ℤ), V_{215}(𝔽_{163})), and can verify that Conjecture 2.1 correctly predicts the characteristic polynomials of Frobenius for l = 2, 3, 5, 7. The second case was too large for calculation of the action of the Hecke operators, but we were able to verify that there is a unique one-dimensional interior Hecke eigenspace in H^{3}(SL(3, ℤ), V_{463}(𝔽_{349})).

Finally, we mention several examples that led us to the “strict parity condition” of Conjecture 2.2:

• Let p = 163, and let σ be the Â_{4} representation associated to p above. We have σ(Frob_{∞}) = −1 and σ|T_{p} ≃ ω^{54} ⊕ ω^{−54}. If we take ρ = ω^{−52}σ ⊕ 1, then ρ|T_{p} ≃ ω^{2} ⊕ ω^{56} ⊕ 1; if we drop the “strict parity condition,” we would be able to take a = 56, b = 2, c = 0, which would lead to g = 217. But H^{3}(SL(3, ℤ), V_{217}(𝔽_{163})) = 0.

• Let p = 277, and let σ be the Â_{4} representation associated to p above. We have σ(Frob_{∞}) = 1 and σ|T_{p} = ω^{92} ⊕ ω^{−92}. If we take ρ = ω^{−91}σ ⊕ 1, then ρ|T_{p} ≃ ω ⊕ ω^{93} ⊕ 1; if we drop the “strict parity condition,” we would be able to take a = 93, b = 1, c = 0, which would lead to g = 91. But H^{3}(SL(3, ℤ), V_{91}(𝔽_{277})) = 0.

The remaining types The smallest examples of types S_{4} and A_{5} give values of g which are too large to verify at present. For example, the smallest totally real S_{4}-extension ramified at only one prime has prime discriminant p = 2777; in this case σ cuts out an Ŝ_{4}-extension of K̂/ℚ ramified only at 2777. The ramification index of p in K̂ is 4, so that σ|T_{p} ≃ ω^{(p−1)/4} ⊕ ω^{−(p−1)/4} = ω^{694} ⊕ ω^{−694}. Hence if K̂ is totally real, we may take ρ = ω^{694}σ ⊕ ω, leading to a predicted value of g = 1386. If K̂ is totally complex, the best we can do is to take ρ = ω^{694}σ ⊕ ω^{2}, leading to a predicted value of g = 4163.

References

[A] A. Ash, Galois representations attached to mod p cohomology of GL(n, ℤ), Duke Math. J. vol. 65, no.2, (1992), 235-255.

[A1] A. Ash, Farrell cohomology of GL(n, ℤ), Israel Math. J. 67 (1989), 327-336.

[A2] A. Ash, Galois representations and cohomology of GL(n, ℤ), Séminaire de Théorie des Nombres, Paris, 1989-90, (S. David, ed.), Birkhäuser, Boston (1992), 9-22.

[A3] A. Ash, Galois representations and Hecke operators associated with the mod p cohomology of GL(1, ℤ) and GL(2, ℤ), Proc. A.M.S. 125 (1997) 3209-3212.

[A4] A. Ash, Monomial Galois representations and Hecke eigenclasses in the mod p cohomology of GL(p − 1, ℤ), to appear in Math. Ann.

[AAC] G. Allison, A. Ash, and E. Conrad, Galois representations, Hecke operators and the mod p cohomology of GL(3, ℤ) with twisted coefficients, Experimental Math. 7 (1998) 361-390.

[AM] A. Ash and M. McConnell, Experimental indications of three-dimensional Galois representations from the cohomology of SL(3, ℤ), Experimental Math 1 (1992) 235-255.

[AS] A. Ash and G. Stevens, Cohomology of arithmetic groups and congruences between systems of Hecke eigenvalues, J. Reine Angew. Math. 365 (1986) 192-220.

[AS2] A. Ash and G. Stevens, Modular forms in characteristic l and special values of their L-functions, Duke Math. J. 53 (1986) 849-868.

[AT] A. Ash and P.H. Tiep, Symmetric squares, mod-p cohomology of GL(3, ℤ) and mod-p Galois representations, to appear in J. Alg.

[B] Laboratoire A2X, UFR de Mathématiques et Informatique, Université Bordeaux I, Tables of Number Fields (May 15, 1998) [ONLINE] Available:

      ftp://megrez.math.u-bordeaux.fr/pub/numberfields/, [January 22, 1999].

[Br] K. Brown, Cohomology of Groups, Springer-Verlag, New York, 1982.

[Cl] L. Clozel, Motifs et formes automorphes , in Shimura Varieties and L-functions, ed. L. Clozel and J.S. Milne, vol. 1, Academic Press, San Diego, 1990.

[C] T. Crespo, Explicit constructions of A_{n} type fields, J. Algebra 127 (1989) 452-461.

[CR] C. Curtis and I. Reiner, Methods of Representation Theory, Vol. 1, Wiley Interscience, New York 1981.

[Da] H. Darmon, Serre’s Conjectures , in Seminar on Fermat’s Last Theorem, ed V. Murty, CMS Conference Proceedings, Vol. 17, AMS, Providence 1995.

[Di] F. Diamond, The refined conjecture of Serre , in Elliptic Curves, Modular Forms and Fermat’s Last Theorem, eds. J. Coates and S.T. Yau, International Press, Cambridge, MA 1995.

[E] B. Edixhoven, Serre’s Conjectures , in Modular Forms and Fermat’s Last Theorem, eds. G. Cornell, J. Silverman, G. Stevens, Springer, New York 1997.

[GJ] S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3) , Ann. Sc. E.N.S. 11 (1978) 471-542.

[G1] B. Gross, A tameness criterion for Galois representations associated to modular forms (mod p), Duke J. 61 (1990) 445-517.

[G2] B. Gross, On the Satake isomorphism, Proc. Durham Symp. of L.M.S. (1996).

[G3] B. Gross, Modular forms (mod p) and Galois representations, IMRN, Duke J. (1998).

[G4] B. Gross, Algebraic modular forms, to appear Israel J.

[LS] R. Lee and J. Schwermer, Cohomology of arithmetic subgroups of SL_{3} at infinity, J. Reine Angew. Math. 330 (1982) 100-131.

[R] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series and multiplicity one for SL(2), preprint.

[Se] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(Q̄/Q), Duke J. 54 , (1987), 179-230.

[Sl] I. Sh. Slavutskij, Upper bounds and numerical calculation of the number of ideal classes of real quadratic fields, J.A.M.S. Transl. II, Ser. 82 (1969) 67-72.

[W] J. Weisinger, Some results on classical Eisenstein series and modular forms over function fields, Ph. D. Thesis, Harvard University, 1977.