SURJECTIVITY OF MOD ℓ REPRESENTATIONS ATTACHED TO ELLIPTIC CURVES AND CONGRUENCE PRIMES

IMIN CHEN

Abstract. For a modular elliptic curve E/ℚ, we show a number of links between the primes ℓ for which the mod ℓ representation of E/ℚ has projective dihedral image and congruence primes for the newform associated to E/ℚ.

1. Introduction

Let E/ℚ be an elliptic curve. Denote by \overline{ρ}_{E/ℚ,ℓ} : G_{ℚ} → GL_{2}(𝔽_{ℓ}) its mod ℓ representation, i.e. the representation obtained by the action of the absolute Galois group G_{ℚ} of ℚ on the ℓ-torsion points of E/ℚ for ℓ prime. Let S_{E/ℚ} =

\{ℓ prime | \overline{ρ}_{E/ℚ,ℓ} is not surjective \}.

Theorem 1.1 (Serre, [9]). The set S_{E/K} is finite if E/K does not have complex multiplication.

In the same paper [9], the following question was asked.

Question 1.2. Is S_{ℚ} = ⋃_{E/ℚ} S_{E/ℚ} finite as E/ℚ runs through elliptic curves without complex multiplication?

This question is usually analyzed according to the nature of the image of \overline{ρ}_{E/ℚ,ℓ}. If \overline{ρ}_{E/ℚ,ℓ} is not surjective, then by a classification of the subgroups of GL_{2}(𝔽_{ℓ}) we have that im \overline{ρ}_{E/ℚ,ℓ} is contained the normalizer N′ or N of a non-split or split Cartan subgroup, a Borel subgroup B, or a subgroup D with projective image S_{4}. The former three subgroups can be conjugated into one of the following standard forms (under the assumption ℓ is odd in case of N′), respectively,

N′ = \{(\begin{matrix}α & λβ \\ β & α \end{matrix}) , (\begin{matrix}α & λβ \\ −β & −α \end{matrix}) | α, β ∈ 𝔽_{ℓ}, (α, β) ≠ (0, 0)\}

N = \{(\begin{matrix}a & 0 \\ 0 & b \end{matrix}) , (\begin{matrix}0 & b \\ a & 0 \end{matrix}) | a, b ∈ 𝔽_{ℓ}\}

B = \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) | a, d ∈ 𝔽_{ℓ}^{×}, b ∈ 𝔽_{ℓ}\} ,

where λ is a non-square in 𝔽_{ℓ}^{×}.

Date: 1 August 2000.

1991 Mathematics Subject Classification. Primary: 11G05, Secondary: 11F80; Torsion points of elliptic curves, Galois representations, congruence primes, Serre tori, grossencharacters, nonsplit Cartan.

This research was supported by CICMA (1998-99) and MPIM (2000).

Let S_{E/ℚ}^{H} = \{ℓ prime | im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ H\}. For H being conjugate to one of

N′, N, B, D, one can ask whether S_{ℚ}^{H} = ⋃_{E/ℚ} S_{E/ℚ}^{H} is finite as E/ℚ runs through elliptic curves without complex multiplication.

Mazur’s [3] results on rational isogenies of prime degree show that

S_{ℚ}^{B} ⊂ {p prime | p ≤ 37} .

Momose shows [6] that an E/ℚ with im(\overline{ρ}_{E/ℚ,ℓ}) contained in a conjugate of N has potentially good reduction at all odd primes if ℓ > 13. These results rely on studying the associated modular curves and bounding their ℚ-rational points via the arithmetic and geometry of their jacobians. Finally, Serre shows that S_{ℚ}^{D} ⊂ {p prime | p ≤ 13} using local methods (c.f. [3] p. 36).

The case of N′ is the most difficult to study using jacobians of modular curves because the jacobians in question do not have a non-trivial quotient with finitelymany ℚ-rational points.

In this paper, we investigate more carefully the sets S_{E/ℚ}^{N} and S_{E/ℚ}^{N′} for a fixed elliptic curve E/ℚ from the point of view of modular forms. We briefly recall one such connection implicit in work of Ribet [7] and Kraus [2].

Suppose E/ℚ is such that im(\overline{ρ}_{E/ℚ,ℓ}) is contained in H where H = N′, N and ℓ is odd. Let C′ and C denote the split and non-split Cartan subgroups which are normalized by N′ and N, respectively.

Let ϵ_{E/ℚ,ℓ} be the character obtained by composing \overline{ρ}_{E/ℚ,ℓ} with the map to the quotients N/C ≅ N′/C′ ≅ {±1}. The character ϵ_{E/ℚ,ℓ} is non-trivial in the case H = N′ as complex conjugation cannot be sent to an element in C′ under \overline{ρ}_{E/ℚ,ℓ}. In the case H = N, we may assume without loss of generality that ϵ_{E/ℚ,ℓ} is nontrivial or else we are back in the H = B case. Thus, the character ϵ_{E/ℚ,ℓ} cuts out a quadratic extension K of ℚ which is imaginary in the case H = N′.

The representation \overline{ρ}_{E/ℚ,ℓ} ≅ Ind_{K}^{ℚ} χ is induced from a character χ : G_{K} → 𝔽^{×}, where 𝔽 = 𝔽_{ℓ^{2}} or 𝔽_{ℓ} in the cases H = N′ or N, respectively. It thus has the property \overline{ρ}_{E/ℚ,ℓ} ⊗ ϵ_{E/ℚ,ℓ} ≅ \overline{ρ}_{E/ℚ,ℓ}. The following lemma can then be shown.

Lemma 1.3. Let E/ℚ be a modular elliptic curve whose associated newform is f ∈ S_{2}(Γ_{0}(N_{E})). Suppose im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ H with H = N′, N and ℓ is odd. Let E′ be the twist of E by ϵ_{E/ℚ,ℓ}, and let f′ be the corresponding twist of f ∈ S_{2}(Γ_{0}(N_{E})). Then N_{E′} = N_{E} and f′ ∈ S_{2}(Γ_{0}(N_{E})) is a newform.

Proof. Kraus shows in [2] that the type of reduction (good, multiplicative, additive) of E and E′ are the same, i.e. the tame exponents ϵ_{p} of E and E′ are the same. On the other hand, the wild exponent δ_{p} of E depends only on the restriction of \overline{ρ}_{E/ℚ,ℓ} and \overline{ρ}_{E′,ℓ} = \overline{ρ}_{E/ℚ,ℓ} ⊗ ϵ_{E/ℚ,ℓ} to the wild inertia group at p. For p ≥ 3, the restrictions are the same as ϵ_{E/ℚ,ℓ} is trivial on the wild inertia at p. For p = 2, the restrictions still have the same image.

We say that two eigenforms f, g ∈ S_{2}(Γ_{0}(N)) are congruent modulo λ if a_{p}(f) ≡ a_{p}(g) (mod λ) for p ∤ ℓN where λ is a prime above ℓ of ℚ(a_{p}(f), a_{p}(g)). We say that ℓ is a congruence prime for newform f ∈ S_{2}(Γ_{0}(N)), if there exists an eigenform g in the (Petersson) orthogonal complement of f such that g is congruent to f modulo λ above ℓ.

The property that ρ_{E′/ℚ,ℓ} ≅ \overline{ρ}_{E/ℚ,ℓ} ⊗ ϵ_{E/ℚ,ℓ} ≅ \overline{ρ}_{E/ℚ,ℓ} implies the two newforms f, f′ are congruent modulo ℓ. Thus the following proposition holds.

Proposition 1.4. Let E/ℚ be a modular elliptic curve whose associated newform is f ∈ S_{2}(Γ_{0}(N_{E})). Suppose ℓ is odd and im(\overline{ρ}_{E/ℚ,ℓ}) is contained in N′, or N but not C. Then ℓ is a congruence prime for f.

In this paper, we will show that there are additional congruences between f and CM-forms in the case N′ and discuss how the character of these CM-forms can be controlled under certain hypotheses.

Theorem 1.5. Let E/ℚ be a modular elliptic curve whose associated newform is f ∈ S_{2}(Γ_{0}(N_{E})). Suppose im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ N′ for 3 < ℓ ∤ N_{E}. Then there exists a newform g ∈ S_{2}(Γ_{1}(M)) which is induced from a grossencharacter on K and is congruent to f modulo λ a prime above ℓ where M | N_{E} is the Artin conductor of \overline{ρ}_{E/ℚ,ℓ}.

Theorem 1.6. Let E/ℚ be a modular elliptic curve whose associated newform is f ∈ S_{2}(Γ_{0}(N_{E})) and 16 ∤ N_{E}. Suppose im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ N′ for 3 < ℓ ∤ N_{E}. Then there exists a newform g ∈ S_{2}(Γ_{0}(M)) which is induced from a grossencharacter on K and is congruent to f modulo λ a prime above ℓ where M | N_{E} is the Artin conductor of \overline{ρ}_{E/ℚ,ℓ}.

I would like to thank F. Momose for mentioning to me the connection between elliptic curves with im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ N′ and grossencharacters on imaginary quadratic fields (c.f. also the paper [5] from which the case of prime power N_{E} in Theorem 1.6 follows).

2. Congruences with CM-forms

2.1. Algebraic characters. Fix an algebraic closure \overline{ℚ} of ℚ and an algebraic closure \overline{ℚ_{ℓ}} of ℚ_{ℓ}. Let K ⊂ \overline{ℚ} ⊂ \overline{ℚ_{ℓ}} be a number field and denote by D_{K} the set of embeddings of K into \overline{ℚ}.

Let T/ℚ = Res_{ℚ}^{K}(𝔾_{m}/K) be the restriction of scalars of 𝔾_{m}/K to ℚ. This is a commutative algebraic group over ℚ, isomorphic over \overline{ℚ} to 𝔾_{m}^{[K:ℚ]}, with the following properties.

1. T (ℚ) = K^{×} and T (ℚ_{ℓ}) = (K ⊗ ℚ_{ℓ})^{×} = ∏_{v|ℓ} K_{v}

2. For all σ ∈ D_{K}, there is an algebraic character [σ] : T/\overline{ℚ} → GL_{1} /\overline{ℚ} such that composite

K^{×} = T (ℚ) ⊂ T (\overline{ℚ}) ⟶^{[σ]} GL_{1}(\overline{ℚ}) = \overline{ℚ}^{×}

is given by the embedding σ.

3. Every algebraic homormorphism f : T/\overline{ℚ} → GL_{1} /\overline{ℚ} is of the form f = ∏_{σ∈D_{K}} [σ]^{n(σ)} where n(σ) ∈ ℤ. The element ∑_{σ∈D_{K}} n(σ)σ ∈ ℤ[D_{K}] is called the weight of f and completely determines f. Given a weight k ∈ ℤ[D_{K}], let [k] denote the algebraic homomorphism determined by k.

2.2. Grossencharacters of type A_{0}. Let K ⊂ \overline{ℚ} be a number field. For a place v of K, let K_{v} be the completion of K at v, π_{v} a uniformizer of K_{v}, and O_{v} the ring of integers of K_{v} in the case v is a finite place. Let J_{K} be the ideles of K and C_{K} = J_{K}/K^{×} the idèle class group. For a modulus 𝔪 of K let U_{𝔪} = ∏_{v} U_{𝔪,v} where

U_{𝔪,v} = \begin{cases}ker(O_{v}^{×} → (O_{v}/𝔪O_{v})^{×}) & if v ∤ ∞ \\ the connected component of 1 & if v | ∞ \end{cases}

Let E_{𝔪} = ker(K^{×} → J_{K}/U_{𝔪}) denote the units congruent to 1 modulo 𝔪, and C_{𝔪} = C/U_{𝔪}K^{×} be the ray class group of modulus 𝔪.

Let χ : C_{K} → \overline{ℚ_{ℓ}}^{×} be a continuous character. This can be written in the form χ = ∏_{v} χ_{v} where χ_{v} |_{O_{v}^{×}}= 1 for all but finitely-many v. The homormorphism χ : C_{K} → \overline{ℚ_{ℓ}} is said to be locally algebraic of weight k ∈ ℤ[D_{K}] if χ_{ℓ} = ∏_{v|ℓ} χ_{v}

coincides with the algebraic character [−k] : ∏_{v|ℓ} K_{v} = T (\overline{ℚ_{ℓ}}) → GL_{1}(\overline{ℚ_{ℓ}}) = \overline{ℚ_{ℓ}}^{×} of weight −k on the subgroup ∏_{v|ℓ} U_{𝔪,v}. We say χ has modulus 𝔪 if χ_{ℓ} coincides with [−k] on ∏_{v|ℓ} U_{𝔪,v} and χ_{v} |_{U_{𝔪,v}}= 1 for v ∤ ℓ. The smallest modulus for χ is called the conductor of χ.

When ℓ = ∞, a locally algebraic character χ of modulus 𝔪 and weight k coincides with the notion of a grossencharacter of type A_{0} of modulus 𝔪 and weight k.

Theorem 2.1 (Weil, [11]). Let χ be a grossencharacter of type A_{0}. The extension ℚ(χ(π_{v}) | v ∤ ∞𝔪) is a finite extension of ℚ called the field generated by χ.

Proposition 2.2. Let k ∈ ℤ[D_{K}] be a weight. There exists a non-trivial grossencharacter of type A_{0} of weight k and modulus 𝔪 if and only if [k](E_{𝔪}) = 1. If this holds then there are h_{𝔪} such grossencharacters where h_{𝔪} is the order of the class group C_{𝔪}.

There is a natural grossencharacter of type A_{0} of conductor O_{K} and weight ∑_{σ∈D_{K}} σ. This is given by

ω_{K} : C_{K} → ℝ^{>0} ⊂ ℂ^{×}
x ↦ ∏_{v} ||x_{v}||

where ||x_{v}|| = |x_{v}|^{[K_{v}:ℚ_{p}]} and for p finite, |π_{v}| = 1/p^{1/e_{v}}, and e_{v} is the ramification index of v | p. The character ω_{K} is trivial on K^{×} by the product formula.

2.3. Fundamental characters. For v | ℓ let \overline{K_{v}} be a fixed algebraic closure of K_{v}. This fixes an algebraic closure \overline{k_{v}} of the residue field k_{v}. Let I_{K_{v}} denote the inertia subgroup of G_{K_{v}} and I_{K_{v},t} its tame quotient.

A character \overline{χ} : I_{K_{v},t} → \overline{k_{v}}^{×} is called a tame character. For all q = ℓ^{n}, there is a tame character

Θ_{q−1} : I_{K_{v},t} → 𝔽_{q}^{×} ⊂ \overline{k_{v}}^{×}

called the fundamental tame character of level n which is surjective to 𝔽_{q}^{×}.

A tame character is said to have level n if its image is contained in 𝔽_{q}^{×} ⊂ \overline{k_{v}}^{×}, q = ℓ^{n}, but no smaller finite field. The fundamental tame character of level n has the property that any character \overline{χ} of level ≤ n can be expressed as a power of Θ_{q−1}.

Suppose \overline{χ} is a tame character of level n and \overline{χ} = Θ_{q−1}^{a} with 0 ≤ a < q − 1. Because of the assumption that \overline{χ} has level n, not all possible a arise. We may write the integer a uniquely in the form a = a_{0} +a_{1}ℓ+…+a_{n−1}ℓ^{n−1} where 0 ≤ a_{i} ≤ ℓ−1 and hence \overline{χ} = Θ_{q−1}^{a_{0}}Θ_{q−1}^{ℓa_{1}} … Θ_{q−1}^{ℓ^{n−1}a_{n−1}}.

Let \overline{χ} : G_{K_{v}^{ab}} → \overline{k_{v}}^{×} be a character and consider its restriction to I_{K_{v}^{ab}}. This restriction factors to I_{K_{v,t}^{ab}} to yield a tame character \overline{χ} which we also denote by \overline{χ}. The local class field homomorphism r_{v} : K_{v}^{×} → G_{K_{v}^{ab}} induces an isomorphism k_{v}^{×} ≅ I_{K_{v,t}^{ab}} so that the tame character \overline{χ} has level ≤ n where q = ℓ^{n} = #k_{v}. We denote by \overline{χ} |_{k_{v}^{×}} the character on k_{v}^{×} obtained by precomposing the tame character \overline{χ} with the local class field homomorphism. Let D_{K_{v}} denote the set of embeddings σ_{v} : K_{v} → \overline{K_{v}}. For each such embedding σ_{v}, let \overline{σ_{v}} : k_{v} → \overline{k_{v}} denote the associated embedding of residue fields. We can therefore write the tame character in the form as above \overline{χ} = ∏_{σ_{v}∈D_{Kv}} Θ_{q−1}^{\overline{σ_{v}}a(σ_{v})} where 0 ≤ a(σ_{v}) ≤ ℓ − 1. The element ∑_{σ_{v}∈D_{Kv}} a(σ_{v})σ_{v} ∈ ℤ[D_{K_{v}}] is called the optimal weight of \overline{χ} at v.

A calculation in [9] shows that the composition

k_{v}^{×} ≅ I_{K_{v}^{ab}} ⟶^{Θ_{q−1}} k_{v}^{×}

corresponds with the character x ↦ x^{−1}.

3. Adjustment to optimal level and weight

Proposition 3.1. Let ℚ ⊂ K ⊂ \overline{ℚ} be an imaginary quadratic field whose set of embeddings to \overline{ℚ} is denoted by D_{K} = \{1, τ\}. Let \overline{χ} : G_{K} → 𝔽_{ℓ^{2}}^{×} be a continuous character with Artin conductor 𝔪 and let χ̃ : C_{K} → ℂ^{×} be its Teichmüller lift considered as a continuous character of C_{K}. Suppose that

1. ℓ is inert in K

2. \overline{χ} |_{k_{ℓ}^{×}}= \overline{[1]}^{−1}.

Then there exists a grossencharacter χ of type A_{0} with conductor 𝔪 and weight 1 and a prime λ above ℓ in the field generated by χ̃ and χ such that χ̃(π_{v}) ≡ χ(π_{v}) (mod λ) for all v ∤ ∞ℓ𝔪.

Proof. By the global class field homomorphism r_{K} : C_{K} → G_{K^{ab}} we may regard both \overline{χ} and χ̃ as continuous characters of C_{K} and can write \overline{χ} = ∏_{v} \overline{χ}_{v} and χ̃ = ∏_{v} χ̃_{v} where \overline{χ}_{v} and χ̃_{v} are characters of K_{v}^{×}. By comparing \overline{χ}_{v} and χ̃_{v} place by place we see that χ̃ has conductor 𝔪ℓ and weight 0.

Let u ∈ E_{𝔪} = K^{×} ∩ U_{𝔪}. Since \overline{χ} is trivial on K^{×}, \overline{χ}(u) = 1. On the other hand, we also have \overline{χ} |_{k_{ℓ}^{×}} (u) = \overline{[1]}^{−1}(u) = \overline{u}^{−1} and \overline{χ} |_{U_{𝔪,v}} (u) = 1 for v ≠ ℓ. Thus, we have that u ≡ 1 (mod ℓ). As K is imaginary quadratic and ℓ ≥ 5, this implies u = 1.

Since E_{𝔪} is trivial, there exists a grossencharacter ϕ of type A_{0} with modulus 𝔪 and weight 1. Write ϕ = ∏_{v} ϕ_{v}. As ϕ has weight 1, ϕ_{∞}(z) = \overline{z}. Let δ : J_{K} → \overline{ℚ_{ℓ}}^{×}, δ = ∏_{v} δ_{v} be defined as follows. For v ∤ ∞ℓ, let δ_{v} = ϕ_{v}, and define δ_{∞} = 1, δ_{ℓ} = ϕ_{v}[1]^{−1}. By construction, δ factors to a character of C_{K}. Let \overline{δ} : C_{K} → \overline{𝔽_{ℓ}}^{×} be the reduction of δ modulo a prime λ′ above ℓ of the field generated by δ (which is the same as the field generated by δ), and let δ̃: C_{K} → ℂ^{×} be the Teichmüller lift of \overline{δ}

The desired grossencharacter of type A_{0} is then χ = χ̃δ̃^{−1}ϕ. The weight of χ is 1 and it evidently has modulus 𝔪ℓ. In fact, χ has conductor 𝔪. Since χ_{ℓ} =

χ̃_{ℓ}δ̃_{ℓ}^{−1}ϕ_{ℓ} = ˜[1]^{−1}˜[1]ϕ̃_{ℓ}^{−1}ϕ_{ℓ} we see that χ_{ℓ} is trivial on O_{ℓ}^{×}. Thus, χ has modulus 𝔪. To see that χ has conductor precisely 𝔪, consider the character χ : C_{K} → \overline{ℚ_{ℓ}}^{×} given by χ = χ̃δ̃^{−1}δ which has the same conductor as χ. Since χ reduces modulo λ to \overline{χ} having Artin conductor 𝔪, it follows that 𝔪 divides the Artin conductor of χ as the Artin conductor can only decrease under reduction modulo λ.

For v ∤ ∞ℓ𝔪, χ(π_{v}) = χ̃_{v}(π_{v})δ̃_{v}^{−1}(π_{v})ϕ_{v}(π_{v}) = χ̃_{v}(π_{v})ϕ̃_{v}^{−1}(π_{v})ϕ_{v}(π_{v}) ≡ χ̃_{v}(π_{v}) (mod λ) where λ is a prime of the field generated by χ̃ and δ̃ above λ′.

4. Proof of Theorem 1

Let E/ℚ be a modular elliptic curve whose associated newform is f ∈ S_{2}(Γ_{0}(N_{E})). Suppose im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ N′ for 3 < ℓ ∤ N_{E}. Then \overline{ρ}_{E/ℚ,ℓ} ≅ Ind_{K}^{ℚ} \overline{χ} is induced from a character \overline{χ} : G_{K} → 𝔽_{ℓ^{2}}^{×} on the imaginary quadratic field K associated to such a \overline{ρ}_{E/ℚ,ℓ}. Since ℓ ∤ N_{E}, the argument in [9] p. 317 shows that ℓ is inert in K.

Let us briefly recall the definition of the Serre’s optimal weight attached to this particular \overline{ρ}_{E/ℚ,ℓ} [10]. Identifying G_{ℚ_{ℓ}} with a decomposition subgroup at ℓ of G_{ℚ}, the restriction of \overline{ρ}_{E/ℚ,ℓ} to I_{ℚ_{ℓ}} factors through its tame quotient I_{ℚ_{ℓ},t} and is semisimple. Since ℓ is unramified in K, I_{ℚ_{ℓ}} ⊂ G_{K} so that

\overline{ρ}_{E/ℚ,ℓ} |_{I_{ℚ_{ℓ}}} ≅ (\begin{matrix}\overline{χ} & 0 \\ 0 & \overline{χ}′ \end{matrix})

where \overline{χ}′(g) = \overline{χ}(τ^{−1}gτ).

Both \overline{χ} and \overline{χ}′ are tame characters of level 2 so that \overline{χ}^{ℓ} = \overline{χ}′. Write \overline{χ} = Θ_{q−1}^{a_{0}+ℓa_{1}} where q = ℓ^{2} = #k_{v}^{×} and 0 ≤ a_{0}, a_{1} ≤ ℓ − 1. Since \overline{ρ}_{E/ℚ,ℓ} is induced from either \overline{χ} or \overline{χ}′, up to switching \overline{χ}′ for \overline{χ} we may assume a > b. The optimal weight is defined as k = 1 + a_{0} + ℓa_{1}. Since ℓ ∤ N_{E}, Proposition 4 of [10] implies that k = 2 and so a_{0} = 1, a_{1} = 0 in our situation. Thus, we see that \overline{χ} |_{k_{v}^{×}}= \overline{[1]}^{−1}.

Let χ̃ be the Teichmüller lift of \overline{χ}. By Proposition 3.1, there exists a grossencharacter χ of type A_{0} with conductor 𝔪 and weight 1 such that χ(π_{v}) ≡ χ̃(π_{v}) (mod λ) where λ is a prime above ℓ of the field generated by χ̃ and χ.

Let I(𝔪) denote the group of fractional ideals of K prime to 𝔪. For an ideal 𝔞 ∈ I(𝔪) denote by [𝔞] the idèle ∏_{v∤∞𝔪} π_{v}^{e_{v}} associated to the ideal 𝔞 = ∏_{v∤∞𝔪} 𝔭_{v}^{e_{v}}, 𝔭_{v} is the prime of K associated to the finite place v, and π_{v} is any choice of uniformizer for K_{v}.

Theorem 4.1 (Hecke). Let K ⊂ \overline{ℚ} be an imaginary quadratic field with discriminant d_{K} and let D_{K} = \{1, τ\} denote its embeddings into \overline{ℚ}. Let χ be a grossencharacter of type A_{0} on K with conductor 𝔪 and weight k = u · 1 ∈ ℤ[D_{K}], u > 0. Consider g(z) = ∑_{𝔞∈I(𝔪)} χ([𝔞])q(z)^{N_{K}(𝔞)} where q(z) = e^{2πiz}. Then g is a newform on S_{u+1}(Γ_{0}(M), ξ) where M = N_{K}(𝔪) |d_{K}| and ξ : (ℤ/Mℤ)^{×} → ℂ^{×} is defined by ξ = ϵ_{K} \frac{χ∘Ver}{ω_{ℚ}^{u}}. Here ϵ_{K} : C_{ℚ} → \{±1\} is the character defining K and Ver : C_{ℚ} → C_{K} is the Verlagerung map.

Proof. c.f. Theorem 4.8.2 [4] (but note Miyake normalizes his grossencharacters so they are unitary)

Let g(z) = ∑_{𝔞∈I(𝔪)} χ([𝔞])q(z)^{N_{K}(𝔞)} ∈ S_{2}(Γ_{1}(M)) be the newform constructed from χ as in the theorem above. Let p ∤ ℓM be a prime and Fr_{p} ∈ G_{ℚ} a Frobenius element at p. If p is inert in K, then Fr_{p} ∉ G_{K} so that a_{p}(f) ≡ tr \overline{ρ}_{E/ℚ,ℓ}(Fr_{p}) ≡ tr(Ind_{K}^{ℚ} \overline{χ})(Fr_{p}) ≡ 0 = a_{p}(g) (mod λ). If p is split in K with v_{i} | p being the two places above p, then Fr_{p} ∈ G_{K} so that a_{p}(f) ≡ tr \overline{ρ}_{E/ℚ,ℓ}(Fr_{p}) ≡ tr(Ind_{K}^{ℚ} \overline{χ})(Fr_{p}) ≡ \overline{χ}(Fr_{p})+\overline{χ}(τ Fr_{p} τ^{−1}) ≡ \overline{χ}(π_{v_{1}})+\overline{χ}(π_{v_{2}}) ≡ χ(π_{v_{1}})+χ(π_{v_{2}}) = a_{p}(g) (mod λ). Thus, a_{p}(f) ≡ a_{p}(g) (mod λ) for p ∤ ℓM.

Lemma 4.2. Let ℓ be a prime which is inert in an imaginary quadratic field K ⊂ \overline{ℚ} with its set of embeddings denoted by D_{K} = \{1, τ\}. Let \overline{χ} : G_{K} → 𝔽_{ℓ^{2}}^{×} be a character with Artin conductor 𝔪(\overline{χ}) prime to ℓ and suppose \overline{ρ} = Ind_{K}^{ℚ} \overline{χ} is irreducible. If we denote by N(\overline{ρ}) the Artin conductor of \overline{ρ}, then

N(\overline{ρ}) = (d_{K})N_{K}(𝔪(\overline{χ})).

Proof. Let χ̃ : G_{K} → L^{×} ⊂ ℂ^{×}, L = ℚ(ζ_{n}), n = ℓ^{2} − 1 be the Teichmüller lift of \overline{χ}, and let ρ̃ = Ind_{K}^{ℚ} χ̃. By [8] VI.3 Proposition 6, N(ρ̃) = (d_{K})N_{K}(𝔪(χ̃)) = (d_{K})ℓ^{2}N_{K}𝔪(\overline{χ}).

Let us compare N(\overline{ρ}) = ∏_{p≠ℓ} p^{\overline{e}^{p}} and N(ρ̃) = ∏_{p≠ℓ} pẽ_{p} (as ℓ ∤ N(ρ̃)). The quantities \overline{e}_{p} and ẽ_{p} are defined as

\overline{e}_{p} = ∑_{i=0}^{∞} \frac{#\overline{ρ}(G_{p,i})}{#\overline{ρ}(G_{p,0})}(2 − dim \overline{ρ}^{G_{p,i}})

ẽ_{p} = ∑_{i=0}^{∞} \frac{#ρ̃(G_{p,i})}{#ρ̃(G_{p,0})}(2 − dim ρ^{G_{p,i}})

where G_{p,i} denotes the i-th ramification group of a decomposition group at p, indexed so that G_{p,0} is the inertia subgroup at p.

Our aim is to show that N(\overline{ρ}) is the prime to ℓ-part of N(ρ̃) and hence equal to N(ρ̃) = (d_{K})N_{K}(𝔪(\overline{χ})). It suffices from the definitions of \overline{e}_{p} and ẽ_{p} to show that dim \overline{ρ}^{H} = dim ρ^{H} for any given subgroup H of G_{ℚ}. Let Ṽ = L ⊕ Lτ be the representation space of ρ̃ and let Λ = O_{L} ⊕ O_{L}τ be the natural G_{ℚ}-invariant lattice lying inside Ṽ. For any prime λ above ℓ of L, the 𝔽_{ℓ^{2}}-vector space Λ/λΛ is isomorphic to \overline{ρ}.

If H is a given subgroup of G_{ℚ}, then we see from the description of \overline{ρ} as a reduction of ρ̃ that dim ρ^{H} ≤ dim \overline{ρ}^{H}. To show equality, we first show that given a non-zero \overline{v} ∈ \overline{V}^{H} it is possible to find a lift ṽ ∈ Λ^{H}, i.e. ṽ ∈ Λ^{H}, ṽ ≡ \overline{v} (mod λΛ). To do this write \overline{v} = \overline{x} + \overline{y}τ and let H_{1} = H ∩ G_{K} and H_{2} = H ∩ τG_{K}.

Suppose both \overline{x}, \overline{y} ≠ 0. For every h ∈ H_{1}, \overline{ρ}(h)(\overline{v}) = \overline{χ}(h)\overline{x} + \overline{χ}′(h)\overline{y}τ = \overline{x} + \overline{y}τ = \overline{v}. It follows that \overline{χ}(h) = \overline{χ}′(h) = 1 for all h ∈ H_{1}, and hence ρ̃(h) = 1 for all h ∈ H_{1} so that any lift of \overline{v} is invariant under h ∈ H_{1}. For every h = τσ ∈ H_{2}, \overline{ρ}(h)(\overline{v}) = \overline{χ}(σ)\overline{y} + \overline{χ}′(σ)\overline{x}τ = \overline{x} + \overline{y}τ = \overline{v}. Thus, \overline{χ}(σ)\overline{y} = \overline{x} and \overline{χ}′(σ)\overline{x} = \overline{y} for all h = τσ ∈ H_{2}. Note this implies that \overline{χ}(σ), \overline{χ}′(σ), χ̃(σ), χ′(σ) are constant as h = τσ varies in H_{2}. Let ỹ ∈ O_{L} be any lift of \overline{y}. Define x̃ = χ̃(σ)ỹ and ṽ = x̃ + ỹτ. Then also χ′(σ)x̃ = ỹ, and hence ρ̃(h)(ṽ) = ṽ for all h ∈ H_{2}.

Suppose one of \overline{x}, \overline{y} = 0. If there exists an element h = τσ ∈ H_{2}, then arguing as above, we have that \overline{χ}(σ)\overline{y} = \overline{x} and \overline{χ}′(σ)\overline{x} = \overline{y}. But then implies both \overline{x}, \overline{y} = 0 contradicting \overline{v} ≠ 0. Hence, we must have H ⊂ G_{K}. Again, arguing as above, we see that ρ̃(h) = 1 for all h ∈ H and so any lift ṽ of \overline{v} lies in Λ^{H}.

The equality dim \overline{ρ}^{H} = dim ρ^{H} now follows by picking a lift as above for each element of a basis of \overline{V}^{H} to form a basis for Ṽ^{H} of the same size.

From the above lemma, it follows that M = N_{K}(𝔪) |d_{K}| is the Artin conductor of \overline{ρ}_{E/ℚ,ℓ} which divides N_{E}.

5. Proof of Theorem 2

In this section, we show that the grossencharacter χ used to prove Theorem 1 can adjusted (in certain situations) so that it has the additional property that the character ξ : (ℤ/Mℤ)^{×} → ℂ^{×} of the the associated newform g(z) = ∑_{𝔞∈I(𝔪)} χ([𝔞])q(z)^{N_{K}(𝔞)} is trivial.

The Verlagerung map Ver : C_{ℚ} → C_{K} is defined by Ver = ∏_{p} Ver_{p}, where Ver_{p} is the natural map ℚ_{p}^{×} → (K ⊗ ℚ_{p})^{×} (here p = ∞ is included). By Theorem 4.1, the character ξ is given by the formula ξ = ϵ_{K} \frac{χ∘Ver}{ω_{ℚ}^{u}}. If χ is a grossencharacter on K of weight k = a_{0}+a_{1}τ and modulus 𝔪, then χ∘Ver is a grossencharacter on ℚ of weight a_{0}+a_{1} and modulus N_{K}(𝔪). Thus, the expression for ξ is indeed a grossencharacter of ℚ of weight 0 and modulus M and hence factors to C_{ℚ,M} ≅ (ℤ/Mℤ)^{×}.

Proposition 5.1. Let H ⊂ G be finite abelian groups. Let v be the unique valuation of \overline{ℚ_{ℓ}} extending that of ℚ_{ℓ}. Suppose f : H → \overline{ℚ_{ℓ}}^{×} is a character such that such that v(f(h) − 1) > 0 for all h ∈ H. Then there exists a character f′ : G → \overline{ℚ_{ℓ}}^{×} extending f such that v(f′(g) − 1) > 0 for all g ∈ G.

Proof. The main idea of the proof is to mimic the proof of Baer’s criterion (c.f. [1]). We shall write the abelian groups H ⊂ G additively. The first step is to show the following intermediate result.

Let f : mℤ → \overline{ℚ_{ℓ}}^{×} be a homomorphism such that f(m) is a root of unity and v(f(m) − 1) > 0. Then there exists a homomorphsim \overline{f} : ℤ → \overline{ℚ_{ℓ}}^{×} extending f such that \overline{f}(1) is a root of unity and v(\overline{f}(1) − 1) > 0. To show that \overline{f} exists, choose a root of unity x ∈ \overline{ℚ_{ℓ}} such that x^{m} = f(m). Let L = ℚ_{ℓ}(x), K = ℚ_{ℓ}(f(m)), with λ | ℓ the unique primes of L, ℚ_{ℓ} corresponding to the restrictions of v to these fields. Let \overline{x} be the reduction of x modulo λ and let x̃ denote the Teichmüller lift of this reduction to \overline{ℚ_{ℓ}}^{×}. Since x^{m} = f(m) ≡ 1 (mod λ), we see that x^{m} = 1, so x̃ is an m-th root of unity in \overline{ℚ_{ℓ}}. Now, (x/x̃)^{m} is also equal to f(m) but x/x̃ is congruent to 1 modulo λ. We define \overline{f} by \overline{f}(1) = x/x̃.

Let f : H → \overline{ℚ_{ℓ}}^{×} be given such that v(f(h)−1) > 0 for all h ∈ H. There exists a maximal extension \overline{f} : \overline{H} → \overline{ℚ_{ℓ}}^{×} extending f : H → \overline{ℚ_{ℓ}}^{×} such that v(\overline{f}(\overline{h}) − 1) > 0 for all \overline{h} ∈ \overline{H}. If \overline{H} = G, then we are done. If \overline{H} is strictly contained in G, then let a ∈ G such that a ∉ \overline{H}. Consider the ideal 𝔞 = \{r ∈ ℤ : ra ∈ \overline{H}\} of ℤ. We define a homomorphism f_{0} : 𝔞 → \overline{ℚ_{ℓ}}^{×} by f_{0}(r) = \overline{f}(ra). By the intermediate result above, there is an extension \overline{f}_{0} : ℤ → \overline{ℚ_{ℓ}}^{×} such that v(\overline{f}_{0}(1) − 1) > 0. Let u = \overline{f}_{0}(1). We now define f′(x + ra) = \overline{f}(x) · u^{r}, where x ∈ \overline{H}, and r ∈ ℤ. This is well-defined since if x + ra = 0, then r ∈ 𝔞, and hence \overline{f}(x) · ru = \overline{f}(x) · \overline{f}(ra) = \overline{f}(x + ra) = 0. Now, f′ extends \overline{f} to H′ =< \overline{H}, a > still keeping the property v(f′(h′) − 1) > 0 for all h′ ∈ H′, contradicting the maximality of \overline{f}.

Let \overline{Ver} : C_{ℚ,M} → C_{K,𝔪} be the homomorphism induced by Ver on ray class groups.

Corollary 5.2. Let ξ : C_{ℚ,M} → ℂ^{×} be a character such that ξ is trivial on the kernel of \overline{Ver} and ξ ≡ 1 (mod λ) for λ a prime above ℓ of the field generated by ξ. Then there exists a character ψ : C_{ℚ,𝔪} → ℂ^{×} such that ψ ∘ \overline{Ver} = ξ and ψ ≡ 1 (mod λ).

Let χ be as in the proof of Theorem 1 and let ξ = ϵ_{K} \frac{χ∘Ver}{ω_{ℚ}}. Assume ξ^{−1} satisfies the requirements of the corollary above and let ψ be the character extending the character ξ^{−1} from the corollary. The character χ′ = χψ also satisfies the requirements for Theorem 1, but now the character of the associated newform g′ becomes

ξ′ = ϵ_{K} \frac{χψ ∘ Ver}{ω_{ℚ}} = ϵ_{K} \frac{χ ∘ Ver}{ω_{ℚ}}ψ ∘ Ver = ξξ^{−1} = 1.

Since \overline{ρ}_{f,ℓ} ≅ \overline{ρ}_{g,λ} (mod λ), it follows that ξ ≡ 1 (mod λ) as the character of f is trivial. Thus, to prove Theorem 2, we need only verify that ξ is trivial on the kernel of \overline{Ver}.

Let π_{N} : C_{ℚ} → C_{ℚ,M} denote the quotient map. Let x ∈ C_{ℚ} such that \overline{Ver}(π_{N}(x)) = 1. This means that Ver(x) = u · k ∈ U_{K,𝔪} · K^{×}. Now, χ(u · k) = χ(u) = χ_{∞}(u_{∞}) = u_{∞}. On the other hand, ω_{ℚ} = ω_{K}^{1/2} and ω_{K}(u · k) = ω_{ℚ}(u) = ω_{K,∞}(u_{∞}) = u_{∞}^{2}. Hence, \frac{χ∘Ver}{ω_{ℚ}} considered as a character of C_{ℚ,M} is trivial on the kernel of \overline{Ver}.

Thus, it remains to show that ϵ_{K} is trivial on the kernel of \overline{Ver}. The class group C_{ℚ,M} ≅ (ℤ/Mℤ)^{×}. Given an element g ∈ C_{ℚ,M}, there exist infinitely many primes q such that \overline{q} = g (mod M) by the Chebaterov density theorem. The character ϵ_{K} considered as a character of C_{ℚ,M} can then be described by q ↦ (\frac{d_{K}}{q}). Let g ∈ C_{ℚ,M} be such that \overline{Ver}(g) = 1 and let us represent g = \overline{q} for an odd prime q. The property \overline{Ver}(\overline{q}) = 1 implies that q ≡ 1 (mod p) for every prime p ∤ N_{K}(𝔪).

Assume now that 2 ∤ d_{K} so that d_{K} ≡ 1 (mod 4) is square-free. From [9] §5.8, we deduce that

1. The character ϵ_{K} = ϵ_{E/ℚ,ℓ} is unramified outside p | N_{E} because of the condition 3 < ℓ ∤ N_{E}.

2. Furthermore, if p ∤ d_{K}, then p^{2} | N_{E}.

Thus, since d_{K} is square-free, it follows that if p | d_{K}, then p | N_{E}/d_{K}. But then p | N_{K}(𝔪) as only semi-stable primes can be stripped from N_{E} (c.f. [10] §4.6). Thus, we have

ϵ_{K}(q) = (\frac{d_{K}}{q}) = (−1)^{\frac{q−1}{2} \frac{d_{K} −1}{2}} (\frac{q}{d_{K}}) = 1

as d_{K} ≡ 1 (mod 4) and q ≡ 1 (mod p) for each p | N_{K}(𝔪). Given the following lemma, Theorem 2 is now proved.

Lemma 5.3. Let E/ℚ be an elliptic curve with conductor N_{E} and suppose im(\overline{ρ}_{E/ℚ,ℓ}) ⊂ N′ for ℓ odd. Let K be the imaginary quadratic field associated to such \overline{ρ}_{E/ℚ,ℓ}. If 16 ∤ N_{E} then 2 ∤ d_{K}.

Proof. Let Φ_{2} = \overline{ρ}_{E/ℚ,ℓ}(I_{2}) be the image of inertia at 2. Then Φ_{2} can be considered as a subgroup of SL_{2}(𝔽_{3}) with order restricted to 1, 2, 3, 4, 6, 8, 24 [9]. If #Φ_{2} = 1, 2, 3, 6, then under the assumption ℓ odd, we have that 2 ∤ d_{K} by [9] §5.8. In fact, if #Φ_{2} = 24, then 2 ∤ d_{K} as SL_{2}(𝔽_{3}) cannot be embedded into the normalizer of a non-split Cartan subgroup N′.

If #Φ_{2} = 4 and 2 | d_{K} then #\overline{ρ}_{E/ℚ,ℓ}(G_{2,0}) = #\overline{ρ}_{E/ℚ,ℓ}(G_{2,1}) = 4 which implies the Artin exponent e_{p} of \overline{ρ}_{E/ℚ,ℓ} is ≥ 4. Similarly, if #Φ_{2} = 8, then also e_{p} ≥ 4.

6. Acknowledgements

I would like to thank CICMA and the MPIM for their support and hospitality during my stay there 1998-99 and 2000 respectively.

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