MOD pq GALOIS REPRESENTATIONS AND SERRE’S CONJECTURE

CHANDRASHEKHAR KHARE & IAN KIMING

Abstract. We consider linear representations of the Galois groups of number fields in 2 different characteristics and examine conditions under which they arise simultaneously from a motive.

1. Introduction.

It is a typical feature of motives (and automorphic forms of arithmetic type) that the Galois representations they give rise to occur in compatible families: these compatible families are of p-adic (see [8]) or mod p (see [5]) representations with p varying. While the representations that occur in such a family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial.

In this paper we investigate when a pair of mod p and mod q representations of the absolute Galois group of a number field K simultaneously arises from an automorphic motive: we do this in the 1-dimensional (Section 2) and 2-dimensional (Section 3: this time assuming K = ℚ) cases. In Section 3 we formulate a mod pq version of Serre’s conjecture refining in part the question of Barry Mazur and Ken Ribet in [11] and some of the considerations in [4].

1.1. Notation. Throughout the paper, p and q denote 2 distinct, fixed prime numbers.

We fix the following notation:

K : a finite extension of ℚ
E : group of global units in K
G_{K} : the absolute galois group of K
C_{K} : the idele class group of K
Σ : the set of embeddings K ↪ \overline{ℚ}

By ‘Hecke character’ of K we shall mean a grossencharacter of type A_{0} on C_{K}. We fix embeddings ι_{∞}: \overline{ℚ} ↪ ℂ, and for every prime ℓ an embedding ι_{ℓ}: \overline{ℚ} ↪ \overline{ℚ}_{ℓ},

and an extension v_{ℓ} of the ℓ-adic valuation on ℚ_{ℓ} to \overline{ℚ}_{ℓ}.

Thus, an element σ ∈ Σ gives rise to an embedding K ↪ \overline{ℚ}_{ℓ} via composition with ι_{ℓ}. Abusing notation, with denote again this embedding by σ.

If λ is a finite prime of K of residue characteristic ℓ we denote by:

𝔭_{λ} : maximal ideal of ring of integers in K_{λ}
k_{λ} : residue field of λ
U_{λ} : corresponding unit group

and introduce the subset Σ(λ) ⊆ Σ:

Σ(λ) := {σ ∈ Σ | v_{ℓ} ∘ σ ≅ λ}.

Thus, if σ ∈ Σ(λ) we have a canonical extension to an embedding K_{λ} ↪ \overline{ℚ}_{ℓ} inducing an embedding:

\overline{σ}_{λ}: k_{λ} ↪ \overline{𝔽}_{ℓ}.

Notice that the sets Σ(λ) form a partition of Σ when λ runs over the prime divisors of ℓ.

Since we have fixed a prime of \overline{ℚ} above λ, we can speak of the inertia group I_{λ} ≤ G_{K} at λ. We denote by

θ_{λ}: I_{λ} ⟶ k_{λ}^{×}

the character (factoring through tame inertia) that gives the action of Gal(\overline{K}_{λ}/K_{λ}) on a (#k_{λ} − 1)’st root of a uniformizer of K_{λ}. Thus, if #k_{λ} = ℓ^{n} the fundamental characters of level n over K_{λ} are the n characters:

θ_{λ}^{ℓ^{i}}, for i = 0, …, n − 1.

We denote by θ̃_{λ} the Teichmüller lift of θ_{λ}, and – as we have fixed an embedding \overline{ℚ} ↪ \overline{K}_{λ} – can regard it as taking values in either \overline{ℚ}^{×}, or in \overline{K}_{λ}^{×}.

In order to reduce the amount of notation, we allow ourselves to use class field theory implicitly. Thus, if χ : G_{K} → \overline{𝔽}_{ℓ}^{×} is a homomorphism we may find occasion to view χ as a homomorphism on C_{K}.

Generally, if χ is a Hecke character of K and if ℓ is a prime number, there is attached to the pair (χ, ℓ) an ℓ-adic galois representation χ_{ℓ} : G_{K} → \overline{ℚ}_{ℓ}^{×} satisfying appropriate reciprocity laws (cf. for instance [9], [8], or the original construction [14]). The character χ_{ℓ} lands in the group of units. Reducing ‘mod ℓ’ we obtain a representation \overline{χ}_{ℓ}:

\overline{χ}_{ℓ} : G_{K} ⟶ \overline{𝔽}_{ℓ}^{×}.

In particular, we use these notations ε_{ℓ} and ε̄_{ℓ} if ε is just a finite-order complex character of C_{K}.

Finally, if x is a complex number, we denote by x^{c} the complex conjugate of x.

2. One dimensional mod pq representations.

Given representations ρ : G_{K} → \overline{𝔽}_{p}^{×} and ρ′ : G_{K} → \overline{𝔽}_{q}^{×} we would like to determine when ρ and ρ′ arise from one Hecke character simultaneously with respect to the embeddings fixed above. There is no reasonable notion of finitely many mod p representations with p varying, such as ρ and ρ′, being compatible (see [5]) but their arising from a Hecke character (see Section 4 of [5]) is a special circumstance which we will explore below.

2.1. Hecke characters and Galois representations. In this subsection we wish to recall some properties of Hecke characters, in particular properties of their attached mod ℓ Galois representations. We shall follow mainly the exposition of [9], section 3.4.

2.1.1. Hecke characters. A Hecke character of K is a continuous homomorphism χ : C_{K} → ℂ^{×} whose restriction to the connected component of 1 at infinity has form:

x ↦ ∏_{σ real} x_{σ}^{n_{σ}} · ∏_{σ complex} x_{σ}^{n_{σ}}(x_{σ}^{c})^{n_{σ^{c}}}

with integers n_{σ}, σ ∈ Σ. We have used σ^{c} to denote the element of Σ given by σ^{c}(x) = σ(x)^{c}.

We can consider the restriction of χ to U_{λ} ↪ C_{K} for finite primes λ. As χ is a continuous homomorphism, almost all of these restrictions χ|_{U_{λ}} are trivial and in any case have finite orders and hence conductors. One defines the conductor 𝔪 of χ to be the product of these local conductors.

As in [9], we define for an arbitrary prime λ of K the group U_{λ,𝔪}: If 𝔪 = ∏_{v∈S} 𝔭_{v}^{m_{v}} with S a finite set of finite primes of K, the group U_{λ,𝔪} is defined to be the connected component of 1 ∈ K_{λ}^{×} if λ is infinite, the whole of U_{λ} if λ ∉ S, and finally the group of units in U_{λ} of level ≥ m_{λ} if λ ∈ S. We may also define U_{𝔪} as the product of all U_{λ,𝔪}.

The integers n_{σ}, σ ∈ Σ, are uniquely determined by χ. We can thus refer to the formal expression ∑_{σ∈Σ} n_{σ} · σ ∈ ℤ[Σ] as the ∞-type of χ.

The classification of those elements in ℤ[Σ] that occur as the ∞-type of some Hecke character is known, cf. [14], [8], but we shall not need to go into this.

2.1.2. Galois representations attached to Hecke characters. Suppose that χ is a Hecke character of ∞-type ∑_{σ∈Σ} n_{σ} ·σ and conductor 𝔪. Let ℓ be a prime number. To χ is attached an ℓ-adic representation χ_{ℓ} of G_{K}. The character χ_{ℓ} takes its values in the group of units of \overline{ℚ}_{ℓ}, and as we have fixed a valuation of this field we can speak of the mod ℓ reduction \overline{χ}_{ℓ} of χ_{ℓ}. We shall recall, cf. [9], section 3.4, the description of χ_{ℓ} and \overline{χ}_{ℓ} on inertia groups, i.e. on local units groups if we view these characters as characters on C_{K}.

Let λ be a finite prime. Define the character ε_{λ} of U_{λ} as the restriction χ|_{U_{λ}}. Then by construction of χ_{ℓ}, cf. loc. cit., we have on U_{λ}:

χ_{ℓ}(x) := \{ \begin{matrix}ε_{λ}(x) & , for x ∈ U_{λ}, λ ∤ ℓ \\ ε_{λ}(x) · ∏_{σ∈Σ(λ)} σ_{λ}(x^{−1})^{−n_{σ}} & , for x ∈ U_{λ}, λ | ℓ. \end{matrix}

(Notice that there are 2 different ways of establishing a correspondence between the point of view of [9] and our view of a Hecke character as a character of C_{K}. We could have changed signs on the n_{σ}’s but would then have to define ε_{λ} as the inverse of the restriction of χ to U_{λ}).

Notice, that as a consequence of definitions we have for a totally positive global unit u that

∏_{λ finite} ε_{λ}(u) = ∏_{σ∈Σ} σ(u)^{−n_{σ}}.

The following lemma reverses this line of reasoning.

Lemma 1. Suppose that for each finite prime λ of K a continuous character

ε_{λ}: U_{λ} ⟶ ℂ^{×}

is given such that only finitely many ε_{λ} are non-trivial. Suppose also that integers n_{σ}, σ ∈ Σ, are given.

Then there exists a Hecke character χ on C_{K} of ∞-type ∑_{σ∈Σ} n_{σ} · σ whose restriction to U_{λ} is ε_{λ} for all λ if and only if

∏_{λ finite} ε_{λ}(u) = ∏_{σ∈Σ} σ(u)^{−n_{σ}}

for every totally positive global unit u.

Proof. As we saw above, the condition is necessary.

To prove sufficiency, consider the group

U := ∏_{λ finite} U_{λ}

as a compact subgroup of C_{K}. The ε_{λ} define a continuous character

ε := ∏_{λ finite} ε_{λ}

on U. There is an extension χ of ε to a continuous character of C_{K}. The question is whether there is an extension which has the desired shape on the connected component D_{K} of 1 in C_{K}.

Consider first the case where all n_{σ} = 0. The question is then whether ε extends to a continuous character of C_{K} vanishing on the closed subgroup D_{K}. This will be the case if and only if ε vanishes on the intersection U ∩ D_{K}. Now, the known structure of D_{K}, see [1], shows that this intersection consists precisely of the embeddings into U of totally positive global units. So, the desired vanishing is implied by the condition we have imposed.

In the general case let 𝔪 be the conductor of ε. A consequence of our hypothesis is then that

∏_{σ∈Σ} σ(u)^{n_{σ}} = 1

for every u ∈ E_{𝔪} := E ∩ U_{𝔪}. This is sufficient to ensure the existence of some Hecke character χ on C_{K} with ∞-type ∑_{σ∈Σ} n_{σ} · σ. The question is then whether the character ε · (χ|_{U})^{−1} extends to a character of finite order on C_{K}, and we are reduced to the previous case. □

Returning now to the discussion before the lemma, we find for the mod ℓ reduction \overline{χ}_{ℓ} of the galois representation χ_{ℓ} that

\overline{χ}_{ℓ}(x) = (ε_{λ} mod ℓ)(x) · \{ \begin{matrix}1 & , for x ∈ U_{λ}, λ ∤ ℓ \\ ∏_{σ∈Σ(λ)} \overline{σ}_{λ}((x mod λ)^{−1})^{−n_{σ}} & , for x ∈ U_{λ}, λ | ℓ. \end{matrix}

So, we may view this as describing the galois representation \overline{χ}_{ℓ} on inertia. Viewing the homomorphisms

(x mod λ) ↦ \overline{σ}_{λ}((x mod λ)^{−1})

as characters on tame inertia above λ, these characters are fundamental characters of level n if #k_{λ} = ℓ^{n}, with ℓ the residue characteristic of λ, cf. again [9]. So, for each σ ∈ Σ(λ) we have a well-defined number κ(λ, σ) ∈ {0, …, n − 1} such that:

\overline{σ}_{λ} = θ_{λ}^{ℓ^{κ(λ,σ)}}.

Summing up, and viewing \overline{χ}_{ℓ} as a character on G_{K}, we have a description of \overline{χ}_{ℓ} on inertia groups:

\overline{χ}_{ℓ}|_{I_{λ}} = \{ \begin{matrix}(ε_{λ} mod ℓ) & , for λ ∤ ℓ \\ (ε_{λ} mod ℓ) · θ_{λ}^{− ∑_{σ∈Σ(λ)} n_{σ}·ℓ^{κ(λ,σ)}} & , for λ | ℓ \end{matrix}

with ε_{λ} the complex character obtained by restricting χ to U_{λ}, viewed as a character on I_{λ}.

2.2. Preliminary lemmas. We begin by disposing off the easy case of Artin lifts.

Lemma 2. Let G be a profinite group and let τ : G → \overline{𝔽}_{p}^{×} and τ′ : G → \overline{𝔽}_{q}^{×} be two continuous (the target is given the discrete topology), one-dimensional characters of G. Then there is a complex character ε : G → ℂ^{×} which lifts τ and τ′ (w.r.t. the embeddings ι_{p} and ι_{q} fixed earlier) if and only if for any two Artin lifts τ̃ and τ̃′ of τ and τ′ the order of the character τ̃τ̃′^{−1} divides a power of pq.

If a lift ε exists it is unique and in fact ε = τ̃ψ′ = τ̃′ψ if τ̃ and τ̃′ are Artin lifts and τ̃τ̃′^{−1} = ψψ′^{−1} with ψ and ψ′ characters of p- and q-power order, respectively.

Proof. This follows from the fact that the only roots of unity that have trivial reduction mod p are those that have order a power of p. □

Lemma 3. Let G be a profinite group and G′ a subgroup of finite index. Then for a (finite order, continuous) ε′ : G′ → ℂ^{×} to be the restriction of a (finite order, continuous) character ε : G → ℂ^{×} it is necessary and sufficient that \overline{ε}′_{p} and \overline{ε}′_{q}, the reductions of ε′ mod p and mod q, arise by restriction from characters of G.

Proof. As \overline{ε}′_{p} arises by restriction of a mod p character \overline{ε}_{p} of G it follows by choosing any lift of \overline{ε}_{p} to a complex character that ε′^{p^{α}} arises by restriction from G for some non-negative integer α. Similarly ε′^{q^{β}} arises by restriction from G. Choosing integers a and b such that ap^{α} + bq^{β} = 1 we are done. □

Remark 1. It will be interesting to resolve the following question: Let L/K be a finite Galois extension of K such that ρ|_{G_{L}} and ρ′|_{G_{L}} arise from a Hecke character χ′. Then do they also arise from a Hecke character χ ∘ Nm_{L/K} for some Hecke character χ for K? The case when χ′ is of finite order is dealt with in by Lemma 3, and then χ′ itself arises by base change from K which certainly will not be true in general.

Lemma 4. A necessary condition for ρ and ρ′ to arise from a Hecke character is that for all primes ℓ of K not above p and q, and any two Artin lifts ρ̃|_{I_{ℓ}} and ρ̃′|_{I_{ℓ}} of ρ|_{I_{ℓ}} and ρ′|_{I_{ℓ}} the order of the character ρ̃|_{I_{ℓ}} ρ̃′|_{I_{ℓ}}^{−1} divides a power of pq.

Proof. This is clear from the construction of λ-adic representations from Hecke characters (see [8]) and Lemma 2 (see Section 4 of [5]). The point here is that if χ is a Hecke character and if χ_{p} is the 1-dimensional p-adic representation of G_{K} associated to it, then the image of a inertia group at any prime ℓ not above p is finite and does not depend on the p of the compatible system of p-adic representations that χ gives rise to as long as the residue characteristics of p and ℓ are coprime. □

Lemma 5. Assume the necessary conditions above. Then there is an Artin character ε : G_{K} → ℂ^{×} such that \overline{ε}_{p}^{−1}ρ and \overline{ε}_{q}^{−1}ρ′ are unramified outside primes above p and q and the infinite places.

Proof. Let L be the finite abelian extension of K that is the compositum of the fixed fields of the kernels of ρ and ρ′. Let K′ be the fixed field of L under the action of all the inertia groups I_{v} for v a finite place of K not above p or q. As we are assuming the necessary conditions of Lemma 4, we deduce that there are lifts ρ̃ and ρ̃′ of ρ|_{G_{K′}} and ρ′|_{G_{K′}}, respectively, such that the order of ρ̃ρ̃′^{−1} is of the form p^{α}q^{β}. Then by Lemma 2, there is an Artin character ε′ of G_{K′} such that \overline{ε}′_{p} = ρ|_{G_{K′}} and \overline{ε}′_{q} = ρ|_{G_{K′}}. Using Lemma 3, we get an Artin character ε of G_{K} such that \overline{ε}_{p} = ρ|_{G_{K′}} and \overline{ε}_{q} = ρ|_{G_{K′}}. Now as \overline{ε}_{p}^{−1}ρ|_{G_{K′}} and \overline{ε}_{q}^{−1}ρ′|_{G_{K′}} are trivial and K′/K is unramified outside primes dividing pq and the infinite places we are done. □

Lemma 6. If ρ, ρ′ as above are unramified outside primes above p and q and the infinite places, then any Hecke character χ that gives rise to both ρ and ρ′ is of conductor that divides p^{α}q^{β} for some non-negative integers α, β. More precisely, the conductor of any Hecke character that gives rise to ρ and ρ′ divides 𝔪 = ∏_{v_{i}} l.c.m{𝔭_{v_{i}}, cond(ρ′|_{I_{v_{i}}})} ∏_{v′_{j}} l.c.m{𝔭_{v′_{j}}, cond(ρ|_{I_{v′_{j}}})}. Here, the v_{i} and the

v′_{j} are the places of K above p and q respectively.

Proof. This is again clear from the construction of λ-adic representations from Hecke characters (see [8] or the above review in section 2.1.2) and Lemma 2, by considering the restrictions ρ|_{I_{v′_{j}}} and ρ′|_{I_{v_{i}}}. □

Lemmas 5 and 6 reduce us to examining ρ, ρ′ that are unramified outside primes above p and q and the infinite places, and looking for lifts by Hecke characters that are of conductor divisible only by primes above p and q.

2.3. The case K = ℚ. Before studying the general case we first study the case K = ℚ that will highlight the essential features. Assume that p, q are odd primes. As notation, for any prime p we denote by θ_{p} := θ_{𝔽_{p}} the mod p cyclotomic character and by θ̃_{p} its Teichmüller lift (we will regard the latter as a character taking values either in \overline{ℚ}^{×} or in \overline{ℚ}_{p}^{×}).

Proposition 1. Assume K = ℚ, and assume the conditions of the Lemma 4 above. Let k_{p} be an integer defined mod p − 1 such that ρ|I_{p} = θ_{p}^{k_{p}} and let a_{p} be an integer well-defined mod A_{p} := (the prime to q part of p − 1) such that ρ′|_{I_{p}} is the reduction mod q of θ̃_{p}^{a_{p}} · ψ_{p} where ψ_{p} is a character of conductor and order a power of p. Define k_{q}, a_{p}, B_{q} similarly (where this time for instance B_{q} is the prime to p part of q − 1). The representations ρ and ρ′ arise simultaneously from a Hecke character if and only if there is an integer k that is simultaneously congruent to k_{p} − a_{p} mod A_{p} and k_{q} − a_{p} mod B_{q}. Note that the above conditions depends only on ρ|_{I_{p}}, ρ|_{I_{q}}, ρ′|_{I_{q}}, ρ′|_{I_{p}}.

Proof. We will assume (w.l.o.g. because Lemma 5 ensures this situation after twisting by an Artin character) above that ρ and ρ′ are unramified outside p, q and the infinite place.

The Hecke characters for ℚ have a particularly simple description: they are of the form ε · Nm^{k} where ε is an Artin character, Nm is the norm character, and k is an integer. The proof will be accomplished by analyzing mod p and mod q representations which arise from a character χ := εε′Nm^{k} where ε is primitive of conductor a power of p, and ε′ is primitive of conductor a power of q (we need only study such characters because of Lemma 6). The mod p and mod q (i.e., more precisely w.r.t. embeddings ι_{p}, ι_{q} fixed above) characters of G_{K} that arise from χ are \overline{χ}_{p} := \overline{ε}_{p}\overline{ε}′_{p}θ_{p}^{k} and \overline{χ}_{q} := \overline{ε}_{q}\overline{ε}′_{q}θ_{q}^{k}.

We want to determine conditions on ρ and ρ′ such that there is a triple (ε, ε′, k) and the corresponding Hecke character χ = εε′Nm^{k} has the property \overline{χ}_{p} = ρ and \overline{χ}_{q} = ρ′, i.e., gives rise to ρ and ρ′. It will be useful first to determine to what extent \overline{χ}_{p} and \overline{χ}_{q} determines the triple (ε, ε′, k). For this note that \overline{χ}_{q}|_{I_{p}} = \overline{ε}_{q}|_{I_{p}}, and thus the Artin character ε which is of conductor a power of p is determined up to characters (of conductor a power of p) that have order a power of q. This determines the wild part of ε, and determines the tame part up to characters of order a power of q.

By considering ρ′|_{I_{p}} we get an integer a_{p} that is well-defined mod A_{p} (the prime to q part of p − 1) such that \overline{χ}_{q}|_{I_{p}} is the reduction mod the place above q (fixed by ι_{q}) of θ̃_{p}^{a_{p}} where now a_{p} can vary mod A_{p} (the reduction of a power of θ̃_{p} mod q is constant when the power varies in a congruence class mod A_{p}). A similar analysis applies to ε′ and gives an integer b_{q} that is well defined mod B_{q} (the prime to p part of q − 1) and such that \overline{χ}_{p}|_{I_{q}} is the reduction mod the place above p (fixed by ι_{p}) of θ̃_{p}^{b_{q}} where now b_{q} can vary mod B_{q}. Consider \overline{χ}_{p}|_{I_{p}} = θ_{p}^{k_{p}} and \overline{χ}_{q}|_{I_{q}} = θ_{q}^{k_{q}} where k_{p} and k_{q} are integers that are well-defined modulo p − 1 and q − 1 respectively by these equations.

We deduce from the above analysis, using further the fact that the abelian extension L of ℚ which is the compositum of the fixed fields of the kernels of ρ and ρ′ is generated by the inertia groups above p and q in Gal(L/ℚ) (ℚ has no non-trivial unramified extensions), that there is a Hecke character χ with \overline{χ}_{p} = ρ and \overline{χ}_{q} = ρ′, if and only if there exists an integer k that is congruent to k_{p} − a_{p} mod A_{p} and k_{q} − b_{q} mod B_{q}. Thus we have completely explicit necessary and sufficient conditions for the existence of a χ such that χ_{p} = ρ and χ_{q} = ρ′ and these conditions depend only on ρ|_{I_{p}}, ρ|_{I_{q}}, ρ′|_{I_{q}}, ρ′|_{I_{p}}. □

Remark 2. By examining the proof we see that we have a classification of the lifts of ρ and ρ′ by Hecke characters.

2.4. The case of general K. We retain of course the notation of 1.1, but also of section 2.1.2 above.

The following theorem will give a general criterion for the existence of a Hecke character χ of K, such that

\overline{χ}_{p} ≅ ρ ⊗ ϕ and \overline{χ}_{q} ≅ ρ′ ⊗ ϕ′

for some unramified characters ϕ, ϕ′ on G_{K}.

Because of Lemma 5 we may, and will, restrict ourselves to the case where ρ and ρ′ are both unramified outside p and q.

Before stating the theorem we introduce some data attached to the given representations ρ and ρ′:

Let v_{1}, …, v_{s} and v′_{1}, …, v′_{t} be the primes of K above p and q, respectively. Define the natural numbers A_{i}, B_{j} by the requirements:

#k_{v_{i}} − 1 = A_{i} · (power of q), q ∤ A_{i},

and similarly

#k_{v′_{j}} − 1 = B_{j} · (power of p), p ∤ B_{j}.

Let k_{i} and k′_{j} be integers such that:

ρ|_{I_{v_{i}}} = θ_{v_{i}}^{k_{i}}, i = 1, …, s,
ρ′|_{I_{v′_{j}}} = θ_{v′_{j}}^{k′_{j}}, j = 1, …, t.

Also, there are integers b_{j}, well-defined modulo B_{j}, and a complex character ψ_{j} : I_{v′_{j}} → ℂ^{×} of q-power order such that:

(ρ|_{I_{v′_{j}}}) = (θ̃_{v′_{j}}^{b_{j}} mod p) · (ψ_{j} mod p),

and similarly we have integers a_{i}, well-defined modulo A_{i}, such that:

(ρ′|_{I_{v_{i}}}) = (θ̃_{v_{i}}^{a_{i}} mod q) · (ψ′_{i} mod q),

with a complex character ψ′_{i} : I_{v_{i}} → ℂ^{×} of p-power order.

Finally, we shall denote by rec_{λ} the reciprocity map K_{λ}^{×} → Gal(K_{λ}^{ab}/K_{λ}).

Theorem 1. Consider the above situation and suppose that integers n_{σ}, σ ∈ Σ, are given. Then there exists a Hecke character χ of ∞-type ∑_{σ∈Σ} n_{σ} ·σ, and unramified characters

ϕ: G_{K} ⟶ \overline{𝔽}_{p}^{×}, ϕ′: G_{K} ⟶ \overline{𝔽}_{q}^{×},

such that

(∗)  \overline{χ}_{p} ≅ ρ ⊗ ϕ  and \overline{χ}_{q} ≅ ρ′ ⊗ ϕ′,

if and only if the following conditions hold:

(1)  k_{i} − a_{i} ≡ − ∑_{σ∈Σ(v_{i})} n_{σ} · p^{κ(v_{i},σ)} =: ξ_{i} mod A_{i},  for i = 1, …, s,

and

(1′) k′_{j} − b_{j} ≡ − ∑_{σ∈Σ(v_{j})} n_{σ} · q^{κ(v_{j},σ)} =: ξ′_{j} mod B_{j},  for j = 1, …, t.

(2) We have

∏_{i}(θ̃_{v_{i}}^{k_{i}−ξ_{i}} · ψ′_{i}) ∘ rec_{v_{i}}(u) · ∏_{j}(θ̃_{v′_{j}}^{k′_{j}−ξ′_{j}} · ψ_{j}) ∘ rec_{v′_{j}}(u) = ∏_{σ∈Σ} σ(u)^{−n_{σ}},

for every totally positive global unit u.

Proof. The structure of the proof is as follows. We first seek to determine the restriction of χ to local unit groups U_{λ} for primes λ lying above p or q. This is done via matching the description we have from section 2.1.2 with the information coming from the behaviour of ρ and ρ′ at local inertia groups. Then Lemma 2 is invoked to give local conditions which will turn out as the conditions (1) and (1′). Once these restrictions of χ have been determined, the existence of χ amounts to the global condition of Lemma 1 which will turn into condition (2).

Proof of necessity: Assume that χ exists with the stated properties. According to lemma 6, χ is unramified outside the primes dividing p or q, and the infinite primes.

If ℓ is a prime number and λ a finite prime of K, we have – retaining the notation of section 2.1.2 – that

\overline{χ}_{ℓ}|_{I_{λ}} = \{ \begin{matrix}(ε_{λ} mod ℓ) & , for λ ∤ ℓ \\ (ε_{λ} mod ℓ) · θ_{λ}^{− ∑_{σ∈Σ(λ)} n_{σ}·ℓ^{κ(λ,σ)}} & , for λ | ℓ \end{matrix}

In particular we have for i = 1, …, s:

\overline{χ}_{q}|_{I_{v_{i}}} = (ε_{v_{i}} mod q) = ρ′|_{I_{v_{i}}} = (θ̃_{v_{i}}^{a_{i}} mod q) · (ψ′_{i} mod q)

and

\overline{χ}_{p}|_{I_{v_{i}}} = (ε_{v_{i}} mod p) · θ_{v_{i}}^{ξ_{i}} = ρ|_{I_{v_{i}}} = θ_{v_{i}}^{k_{i}}.

We conclude that the mod p and mod q characters

θ_{v_{i}}^{k_{i}−ξ_{i}}  and  (θ̃_{v_{i}}^{a_{i}} mod q) · (ψ′_{i} mod q)

of I_{v_{i}} simultaneously lift to a (uniquely determined) complex character of I_{v_{i}}, namely ε_{v_{i}}. On the other hand, as individual lifts of these characters are

θ̃_{v_{i}}^{k_{i}−ξ_{i}}  and  θ̃_{v_{i}}^{a_{i}} · ψ′_{i}

where ψ′_{i} has p-power order, and as θ̃_{v_{i}} has order prime to p, we deduce from lemma 2 that the character

θ̃_{v_{i}}^{a_{i}−k_{i}+ξ_{i}}

has q-power order. This implies (1) by the definition of A_{i}. Lemma 2 also gives:

(∗∗)  ε_{v_{i}} = θ̃_{v_{i}}^{k_{i}−ξ_{i}} · ψ′_{i}.

Similarly, (1′) follows from the fact χ gives a simultaneous lift of the representations ρ and ρ′ restricted to I_{v′_{j}}, and we see that in fact

(∗ ∗ ∗)  ε_{v′_{j}} = θ̃_{v′_{j}}^{k′_{j}−ξ′_{j}} · ψ_{j}.

Now, as χ is unramified outside pq and the infinite primes, condition (2) follows from lemma 1, (∗∗) and (∗ ∗ ∗).

Proof of sufficiency: Define complex characters on inertia at the primes v_{i}, i = 1, …, s, and v′_{j}, j = 1, …, t, according to (∗∗) and (∗ ∗ ∗) above. Define also ε_{λ} to be the trivial character on I_{λ} if λ is a finite prime not above p or q.

If these characters are viewed as characters on local unit groups U_{λ}, then (2) combined with Lemma 1 imply that these local characters are in fact restrictions to the U_{λ} of a Hecke character χ with ∞-type ∑_{σ∈Σ} n_{σ} · σ.

For such a χ we deduce, utilizing (1) and (1′), and reversing the pertinent reasoning in the proof of necessity above, that the characters \overline{χ}_{p}^{−1} ·ρ and \overline{χ}_{q}^{−1} ·ρ′ vanish on every inertia group, and hence are both globally unramified. □

Remark 3. As will be seen from the proof of the theorem, the contribution at the place v_{i} to the conductor of a lift χ is ‘essentially’

l.c.m{𝔭_{v_{i}}, cond(ψ′_{i})} = l.c.m{𝔭_{v_{i}}, cond(ρ′|_{I_{v_{i}}})}.

We say ‘essentially’ because this is definitely true if ψ′_{i} is nontrivial. If however ψ′_{i} is trivial the conductor is 1 or 𝔭_{v_{i}} depending on whether k_{i} − ξ_{i} is divisible by #k_{v_{i}} − 1 or not; however, this condition depends on the n_{σ}’s. Similar remarks apply to the places dividing q of course.

Remark 4. We chose to formulate a theorem considering only representations unramified outside pq and the infinite primes. The reason is first of all that this is the crucial case, and secondly that we wanted to avoid an unnecessarily complicated statement. However, the observant reader will notice that our reduction to the above case via lemma 5 is not completely explicit: The behavior of an ε from lemma 5 at primes above p or q is not given explicitly.

However, this situation can easily be rectified, and we shall limit ourselves to some indications: If one wants to prove a statement as in theorem 1 for arbitrarily ramified representations, there are 2 possible, essentially equivalent, ways of accomplishing this. First, an inspection of the proof of theorem 1 reveals immediately that the method extends without any difficulty to the general case of arbitrary ramification: First, one would have conditions like (1), (1′) for each ramification point w not dividing pq · ∞; these are purely local conditions coming directly from lemma 2. Secondly, condition (2) would change by incorporating certain factors on the left hand side; these factors would depend on the restrictions ρ|_{I_{w}} and ρ′|_{I_{w}} for each w as above. We shall leave it to the reader to work out the exact form of these factors.

The second way to extend theorem 1 to the case of arbitrary ramification is to be more precise about the behavior of an ε as in lemma 5 at the primes dividing pq. A possible way to do this would be to use lemma 1 for the case n_{σ} = 0 – under the assumptions of lemma 5 – to ‘shift’ (via twisting) any ramification outside pq · ∞ to a prime above p (say). After this, one could plug the data into theorem 1 and get explicit conditions. As the reader will however quickly ascertain, this method would result in exactly the same modified conditions (1), (1′) and (2) that result from the first method.

Remark 5. Notice that if χ_{1} and χ_{2} are 2 Hecke characters with the same ∞-type which are both ‘solutions’ to (∗), i.e. have their mod p and mod q representations isomorphic up to twist by unramified to ρ and ρ′, respectively, then χ_{1} and χ_{2} differ by a globally unramified Artin character. For the Hecke character ε := χ_{1}χ_{2}^{−1} has trivial ∞-type, i.e. is an Artin character, and its mod p and mod q reductions are by hypothesis both unramified whence the claim by Lemma 2.

Remark 6. The reason why we consider only ‘lifting up to unramified’ as in the Theorem above is to be found in the nature of the conditions we impose: As an analysis of the argument quickly reveals, in order to discuss exact lifting, and not just ‘up to unramified’, we would have to consider some sort of analogue to Lemma 1 but with given restrictions to decomposition groups instead of inertia groups. This ties up with the Grunwald-Wang theorem where of course the problem is that one cannot in general lift without introducing additional ramification. We have not been able to find a reasonable characterization of pairs (ρ, ρ′) with ‘exact’ lifts and doubt whether there is one. The following example – a simple counting argument – shows that in general one cannot lift an arbitrary pair of unramified characters.

Example: Suppose that K is an imaginary quadratic field containing no other roots of unity that ±1. Let p and q be odd primes that split in K. Fix an embedding σ : K ↪ ℂ and consider an ∞-type m · σ + n · σ^{c}, with integers m and n. We apply Theorem 1 to find the conditions on m and n necessary and sufficient for the existence of a Hecke character χ with ∞-type m · σ + n · σ^{c} and both \overline{χ}_{p} and \overline{χ}_{q} globally unramified. So, we apply Theorem 1 for the case ρ = ρ′ = 1 and hence k_{i} = k′_{j} = a_{i} = b_{j} = 0, and ψ_{i} = ψ′_{j} = 1 for all i, j.

The conditions (1) and (1′) of the theorem amount then to 4 congruence conditions on n and m which are seen to boil down to:

n, m ≡ 0 mod C

where C := l.c.m.{A, B} with A := A_{1} = A_{2} and B := B_{1} = B_{2} as in the definitions preceding the theorem. Now, C is an even number so if n and m are both divisible by C, then condition (2) of the theorem is automatically satisfied as ±1 are the only units in K.

So there exist Hecke characters χ_{1} and χ_{2} with ∞-types C · σ and C · σ^{c}, respectively, such that

((\overline{χ_{i}})_{p}, (\overline{χ_{i}})_{q}) = (ϕ_{i}, ϕ′_{i}), i = 1, 2

where ϕ_{i}, ϕ′_{i}, i = 1, 2, are unramified characters. Now, if χ is any Hecke character such that \overline{χ}_{p} and \overline{χ}_{q} are both unramified, its ∞-type has shape μC · σ + νC · σ^{c} with integers μ and ν. Consequently, χ = εχ_{1}^{μ}χ_{2}^{ν} with ε an Artin character which must be globally unramified (cf. Remark 5 above).

So any pair (ρ, ρ′) of unramified mod p and mod q characters that lifts exactly to a Hecke character necessarily has form:

(ρ, ρ′) = (\overline{ε}_{p}ϕ_{1}^{μ}ϕ_{2}^{ν}, \overline{ε}_{q}(ϕ′_{1})^{μ}(ϕ′_{2})^{ν})

with an unramified Artin character ε. If the class group of K has exponent α and order h, it follows that the number of such pairs (ρ, ρ′) is bounded by α^{2} · h. On the other hand, if h is not divisible by p or q, then the total number of pairs of unramified mod p and mod q characters is at least h^{2}. Hence, if further h > α^{2} there necessarily exist at least 1 such pair that does not lift ‘exactly’ to a Hecke character. As a concrete example, one can take K = ℚ(\sqrt{−3 · 5 · 7 · 11}), which has class group isomorphic to (ℤ/ℤ2)^{3}, and for p and q any two distinct odd primes that split in K.

3. Serre’s conjecture mod pq

This will be a speculative section on issues surrounding mod pq versions of Serre’s conjectures in [10]. Let p and q be odd primes and fix as above embeddings ι_{p} and ι_{q} of \overline{ℚ} in \overline{ℚ}_{p} and \overline{ℚ}_{q} as before. Barry Mazur and Ken Ribet had investigated the question of when a pair of odd continuous irreducible mod p and mod q Galois representations

ρ : G_{ℚ} → GL_{2}(\overline{𝔽}_{p})

and

ρ′ : G_{ℚ} → GL_{2}(\overline{𝔽}_{q})

arise from a newform f ∈ S_{2}(Γ_{1}(N)) for some level N and with respect to the embeddings ι_{p} and ι_{q} that have been fixed. William Stein made some computations ([11]) towards this question in which the level N was varied in order to discover a newform f ∈ S_{2}(Γ_{0}(N)) of the desired sort.

There are certain local constraints that one must impose, and which we will come to later (see Conjecture 1 below), to expect to have an affirmative answer. It must be pointed out here that even assuming Serre’s original conjecture [10], i.e., assuming that ρ and ρ′ are individually modular, proving that they are simultaneously modular is a different ball game. Even after having imposed these necessary local constraints what makes this question difficult to address even computationally is that there seems no a priori way of guessing what levels N one should be looking at (though one does know by the analysis in [3] what levels one should not be looking at as local constraints preclude some primes diving the levels of newforms which give rise to either of the ρ or ρ′!). Unlike the situation in Serre’s conjecture where he did specify a minimal level at which a representation like ρ should be found (before Ribet et al, cf. [7], proved this ϵ-conjecture one did not even know whether there existed a minimal level even without caring about what it exactly was), there does not seem to be a “minimal level” N_{ρ,ρ′} at which ρ and ρ′ should arise from a newform f ∈ S_{2}(Γ_{1}(N_{ρ,ρ′})), i.e., all other levels M at which ρ and ρ′ arise from a newform in S_{2}(Γ_{1}(M)) should be divisible by N_{ρ,ρ′}.

There is one natural guess for which levels N one can look at which unfortunately is very unlikely to be correct. Let N(ρ), resp., N(ρ′), be the prime to p, resp., prime to q, part of the Artin conductor of ρ, resp. ρ′. Then we can try to look for the desired newform from which ρ, ρ′ arise simultaneously in S_{2}(Γ_{1}(N(ρ)N(ρ′)p^{α}q^{β})) where α, β vary. The reason why this is unlikely to work is that there are only finitely many α, β such that there will exist a newform f ∈ S_{2}(Γ_{1}(N(ρ)N(ρ′)p^{α}q^{β}))^{new} that can give rise both ρ and ρ′. This follows from the analysis in [3] which gives that if ρ (resp., ρ′) arises from S_{2}(Γ_{1}(N(ρ)N(ρ′)p^{α}q^{β}))^{new} then β (resp., α) is bounded. (Note that on the contrary by Lemme 1 of [3] ρ, resp. ρ′, does arise from S_{2}(Γ_{1}(N(ρ)p^{α}))^{new}, resp., S_{2}(Γ_{1}(N(ρ′)q^{β}))^{new} for all α ≥ 2, resp., β ≥ 2.) Further again unlike the situation for a single representation ρ, if there does exist a newform f which gives rise to ρ and ρ′, it is not clear that there are infinitely many, i.e., there is no known way that one can systematically raise levels mod pq unlike in the case of mod p representations where a complete study is available because of the work of Ribet (cf. [6]) and others.

In the questions raised by Mazur and Ribet and the subsequent computations of William Stein only the level aspect of this question was considered. In the following paragraphs we discuss the “weight aspect”.

Suppose for simplicity for this paragraph that ρ and ρ′ arise individually (with respect to the embeddings ι_{p}, ι_{q}) from newforms of level N in S_{k}(Γ_{0}(N)) with N squarefree and (N, pq) = 1. (This ensures that the necessary local conditions of Conjecture 1 below are satisfied.) Then we can ask if they arise simultaneously from a newform f ∈ S_{k′}(Γ_{0}(N)) for some k′ >> 0. Unlike the previous paragraph this is more difficult to rule out as for all k′ congruent to k mod l.c.m.(p − 1, q − 1) the ρ and ρ′ individually do arise from S_{k′}(Γ_{0}(N)). This of course can be seen by multiplication by powers of the normalized Eisenstein series E_{p−1} and E_{q−1} (Hasse invariants mod p and mod q) that are congruent to 1 mod p and 1 mod q respectively and then applying the mod p or mod q Deligne-Serre lifting lemma, or better still in the present context by multiplication by a suitable power of the souped up “mod pq Hasse invariant”

E_{l.c.m.(p−1,q−1)} = 1 − \frac{2(l.c.m.(p − 1, q − 1))}{B_{l.c.m.(p−1,q−1)}}. ∑_{n≥1} σ_{l.c.m.(p−1,q−1)−1}(n)q^{n}

(where by B_{k} we mean the kth Bernoulli number and σ_{k}(n) = ∑_{d|n} d^{k}) which by the von-Staudt-Clausen theorem is congruent to 1 mod pq. Unfortunately there is no mod pq Deligne-Serre lifting lemma, and so we do not know and perhaps do not even expect that there is any periodicity with respect to weight of newforms in S_{k}(Γ_{0}(N)) which give rise to ρ and ρ′ simultaneously.

It will be interesting to investigate the issues raised in the paragraph computationally. What makes the possibility of ρ and ρ′ arising simultaneously from S_{k′}(Γ_{0}(N)) attractive is that arithmetically the weight is just one parameter (or automorphically, the component at the archimedean place) while if we were to look at S_{k}(Γ_{0}(N)) for fixed k and varying N we would have to allow N to be divisible by varying primes (as follows from the analysis in [3]), so arithmetically vary infinitely many independent parameters (or automorphically, vary infinitely many non-archimedean local components).

In [4] one set of local constraints for ρ and ρ′ to arise simultaneously from a newform were highlighted: this may be called the semistable case. We will formulate here a finer mod pq Serre conjecture.

Given a (algebraic) Weil-Deligne parameter (τ_{ℓ}, N_{ℓ}) (for this see [12]) with τ_{ℓ} : W_{ℚ_{ℓ}} → GL_{2}(\overline{ℚ}) a continuous representation of W_{ℚ_{ℓ}}, the Weil group of ℚ_{ℓ}, with the target given the discrete topology, and N_{ℓ} a nilpotent matrix ∈ M_{2}(\overline{ℚ}), and p is a prime ≠ ℓ then the corresponding 2-dimensional p-adic representation (τ_{ℓ}, N_{ℓ})_{p} : Gal(\overline{ℚ}_{ℓ}/ℚ_{ℓ}) → GL_{2}(\overline{ℚ}_{p}) is:

• N_{ℓ} = 0, τ_{ℓ} reducible: τ_{ℓ} = ε_{1} ⊕ ε_{2}, the sum of a pair of quasicharacters of the Weil group W_{ℚ_{ℓ}} taking values in \overline{ℚ}^{×}, then (τ_{ℓ}, N_{ℓ})_{p} := ε_{1,p} ⊕ ε_{2,p}, and ε_{1,p}, ε_{2,p} the corresponding p-adic characters arising via the fixed embedding ι_{p} : \overline{ℚ} → \overline{ℚ}_{p}.

• N_{ℓ} = 0, τ_{ℓ} irreducible: then up to twisting by a character ε : W_{ℚ_{ℓ}} → \overline{ℚ}^{×}, τ_{ℓ} has finite image. Then (τ_{ℓ}, N_{ℓ})_{p} := (τ_{ℓ} ⊗ ε)_{p} ⊗ ε_{p}^{−1} with (τ_{ℓ} ⊗ ε)_{p} the embedding of the finite image of τ_{ℓ} ⊗ ε in GL_{2}(\overline{ℚ}) into GL_{2}(\overline{ℚ_{p}}) and ε_{p} the p-adic character corresponding to ε arising via the fixed embedding ι_{p} : \overline{ℚ} → \overline{ℚ}_{p}.

• N_{ℓ} ≠ 0: then τ_{ℓ} is forced to be of the form ε ⊕ ε|| || with ε a quasicharacter of the Weil group W_{ℚ_{ℓ}} taking values in \overline{ℚ}^{×} and || || the “norm character” (i.e., unramified character with ||Frob_{ℓ}|| = ℓ) then (τ_{ℓ}, N_{ℓ})_{p} is

(\begin{matrix}ε_{p}|| ||_{p} & 0 \\ 0 & ε_{p} \end{matrix}) exp(N_{ℓ}t_{ℓ}) : here ε_{p} is the p-adic character corresponding to

ε w.r.t. the embedding ι_{p} : \overline{ℚ} → \overline{ℚ}_{p} as before, t_{ℓ} is the homomorphism of the Galois group of the maximal tamely ramified extension of Q_{ℓ} which arises by projecting first to I_{ℓ}^{t}, the tame inertia at ℓ (i.e., after choosing a Frobenius σ, σ^{n}τ → τ where τ ∈ I_{ℓ}^{t}), followed by a ℤ_{p}-valued homomorphism of I_{ℓ}^{t} which is the projection to the unique ℤ_{p} quotient of I_{ℓ}^{t} and || ||_{p} is the p-adic cyclotomic character (the isomorphism class of (τ_{ℓ}, N_{ℓ})_{p} is independent of the choices made).

Conjecture 1. Let ρ : G_{ℚ} → GL_{2}(\overline{𝔽}_{p}) and ρ′ : G_{ℚ} → GL_{2}(\overline{𝔽}_{q}) be continuous, odd, absolutely irreducible representations and assume that ρ is unramified at q and ρ′ is unramified at p. Let k(ρ) and k(ρ′) be the Serre invariants of ρ, ρ′ as in [10]. Then ρ and ρ′ are the reductions mod p and mod q of the p-adic and q-adic representations (w.r.t. ι_{p}, ι_{q}) attached to a newform in S_{k}(Γ_{1}(N)) for some integer k ≥ 2 and some positive integer N with (N, pq) = 1 if and only if

• there is an integer congruent to k(ρ) mod p − 1 and k(ρ′) mod q − 1, • and further for every prime ℓ ≠ p, q, there is a (algebraic) Weil-Deligne parameter (τ_{ℓ}, N_{ℓ}) and a choice of integral models for (τ_{ℓ}, N_{ℓ})_{p} and (τ_{ℓ}, N_{ℓ})_{q} such that their mod p and mod q reductions are isomorphic to ρ|_{D_{ℓ}} and ρ′|_{D_{ℓ}} respectively.

Discussion of the conjecture: The main feature of the conjecture is that the conditions are local (recall that in [10] the definition of k(ρ), k(ρ′) is purely in terms of ρ|_{I_{p}}, ρ′|_{I_{q}}). The condition in the conjecture at primes ℓ is empty for ℓ that are ≠ p, q and unramified in both ρ and ρ′, and thus there are only finitely many conditions that need to be checked as per the conjecture. The restriction that ρ is unramified at q and ρ′ is unramified at p is a technical restriction which arises from the difficulties of studying mod p representation restricted to D_{p} arising from newforms with levels divisible by powers of p (there are some results towards this in [2] and [13]): it will be nice to remove this technical restriction and refine the above conjecture to allow ρ and ρ′ to be ramified at q and p respectively.

We can make a more optimistic conjecture and fix a weight k ≥ 2 congruent to k(ρ) mod p−1 and k(ρ′) mod q −1, and ask that ρ and ρ′ arise from S_{k}(Γ_{1}(N)) for some (variable) N prime to pq (as in [11]), or we can fix N prime to pq and divisible by N(ρ) and N(ρ′), the prime-to-p and prime-to-q parts of the Artin conductors of ρ and ρ′ respectively, and ask that ρ and ρ′ arise from S_{k}(Γ_{1}(N)) for some (variable) k as in the discussion above. We can also weaken the conjecture and instead of asking that ρ and ρ′ arise from a common newform with respect to fixed embeddings ι_{p}, ι_{q}, allow these embeddings to vary, or alternatively ask that ρ and ρ′ arise simultaneously (w.r.t. the fixed embeddings ι_{p} and ι_{q}) from the Galois orbit of a newform f.

One can also formulate a conjecture entirely analogous to Conjecture 1 for compatible lifts of ρ and ρ′ to p-adic and q-adic representations of G_{ℚ}, which we leave as an exercise for the interested reader. As was pointed out in [4] such a conjecture has relevance to Serre’s original conjecture in [10] because of the modularity lifting theorems of Wiles et al.

In connection with Conjecture 1 it is important to notice that the weight k may very well depend on the choice of primes of \overline{ℚ} over p and q. The following simple example illustrates this.

Example: Suppose that f and f′ are two newforms on SL_{2}(ℤ) of some weight k and let ρ and ρ′ be the mod p and mod q representations attached to f and f′, respectively. The conditions of Conjecture 1 are then satisfied so that the Conjecture would have ρ and ρ′ arising from a newform F ∈ S_{m}(Γ_{1}(N)) for some m ≥ 2 and some N prime to pq.

To illustrate with a simple example that the possible weights m may depend on our choice of primes over p and q, suppose that p = 5, q = 7, and that f and f′ are the 2 newforms of weight 24 and level 1:

f = 24αΔ^{2} + Q^{3}Δ  and  f′ = 24α′Δ^{2} + Q^{3}Δ

with \{α, α′\} := \{−13/2 ± \sqrt{144169}/2\}. The primes 5 and 7 both split in the quadratic field ℚ(\sqrt{144169}), say 5 = 𝔭_{5} · 𝔭′_{5} and 7 = 𝔭_{7} · 𝔭′_{7}. Now, Δ is congruent mod 𝔭_{7} to either f or f′. Let us assume notation so that Δ ≡ f (𝔭_{7}). As Q ≡ 1 (5) we have then

Δ ≡ f (𝔭_{5}), Δ ≡ f′ (𝔭′_{7}).

On the other hand, as αα′ is divisible by 5 we have

f′ ≡ f (𝔭_{5}), f′ ≡ f′ (𝔭_{7})

and f′ is the solution to these congruences with smallest possible weight.

Remark 7. Unlike the case of 1-dimensional mod pq representations (see Lemma 6) one would expect that there are modular forms of levels divisible by arbitrarily many primes that give rise to ρ and ρ′ if the local conditions are met (this is substantiated by the tables in [11]). Here the point is that one can attempt to raise levels at primes that are “Steinberg” for ρ and ρ′ and Steinberg lifts have the property that the image of inertia is unipotent and hence infinite. Thus Lemma 2 on which Lemma 6 relies does not apply.

Remark 8. A mod pq descent result of the type suggested in Remark 1 is not true for 2-dimensional representations even in the simplest case when K/ℚ is a quadratic extension. The principle for constructing a counterexample is the following: Consider representations ρ and ρ′ as above such that ρ is ramified at a prime ℓ ≠ p, q and > 2, and ℓ is not congruent to ±1 either mod p or mod q, with the image of inertia ρ(I_{ℓ}) non-trivial and unipotent, ρ′ unramified at ℓ and such that ρ′(Frob_{ℓ}) has eigenvalues with ratio −ℓ. Let ρ_{ℓ} := ρ|_{D_{ℓ}} and ρ′_{ℓ} := ρ|_{D_{ℓ}}. Then it is easy from local computations at ℓ that use structure of tame inertia to see that while ρ_{ℓ} and ρ′_{ℓ} cannot arise from a single Weil-Deligne representation (τ, N) as above, the restriction of ρ_{ℓ} and ρ′_{ℓ} to the absolute Galois group of the unramified quadratic extension of ℚ_{ℓ} can arise in this way. From this it is easy to construct counterexamples remarking that if ρ and ρ′ are irreducible on restriction to a quadratic extension K of ℚ (with ℓ inert in K), the extensions of ρ|_{G_{K}} and ρ′|_{G_{K}} to G_{ℚ} are unique up to twisting by characters, and the condition that ρ′(Frob_{ℓ}) has eigenvalues with ratio −ℓ is invariant under twisting. (The case of solvable mod p automorphic descent for 2-dimensional representations is the main theorem of [4].)

References

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[3] H. Carayol, Sur les représentations galoisiennes modulo l attachées aux formes modulaires, Duke Math. J. 59 (1989), 785–801.

[4] C. Khare, Mod p descent for Hilbert modular forms, Mathematical Research Letters 7 (2000), 455-462.

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[6] K. Ribet, Congruence relations between modular forms, Proceedings of the ICM, Warsaw (1983), 503–514.

[7] K. Ribet, Report on mod ℓ representations of Gal(\overline{ℚ}/ℚ), in Motives, Proc. Sympos. Pure Math. 55, Part 2 (1994), 639–676.

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[9] J.-P. Serre, Propriétés galoisiennes des points d’ordre finides courbes elliptiques, Invent. Math. 15 (1972), 259–331.

[10] J.-P. Serre, Sur les représentations modulaires de degré 2 de Gal(\overline{ℚ}/ℚ), Duke Math. J. 54 (1987), 179–230.

[11] W. Stein, Serre’s conjecture mod pq?, available at http://modular.fas.harvard.edu/Tables/serremodpq/

[12] J. Tate, Number theoretic background, in Automorphic forms, Representations and Lfunctions, Proc. Sympos. Pure Math. 33, Part 2, (1979), 3–26.

[13] D. Ulmer, Slopes of modular forms and congruences, Ann. Inst. Fourier 46 (1996), 1–32.

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C.K.: School of Mathematics, TIFR, Homi Bhabha Road, Mumbai 400 005, India.

E-mail address: shekhar@math.tifr.res.in

I.K.: Department of Mathematics, Copenhagen, Denmark.

E-mail address: kiming@math.ku.dk