On the modularity of ℚ-curves

Jordan S. Ellenberg and Chris Skinner

12 Apr 2000

Abstract

A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. Ribet [16] asked whether every ℚ-curve is modular, and showed that a positive answer would follow from Serre’s conjecture on mod p Galois representations. We answer Ribet’s question in the affirmative, subject to certain local conditions at 3.

MSC classification: 11G18 (14G35,14H52)

1 Introduction

Let K be a number field, Galois over ℚ. A ℚ-curve over K is an elliptic curve E/K which is isogenous over K to each of its Galois conjugates. Our interest in ℚ-curves is motivated by the following theorem of Ribet.

Theorem ([16, §5]). Suppose E/ℚ̄ is an elliptic curve that is also a quotient of J_{1}(N)/ℚ̄. Then E is a ℚ-curve over some number field.

A ℚ-curve which is a quotient of J_{1}(N)/ℚ̄ is called modular; Ribet has conjectured that in fact every ℚ-curve is modular. The modularity of various ℚ-curves has been verified by Roberts and Washington [17], by Hasegawa, Hashimoto, and Momose [11], and by Hida [12]. In this article we establish the modularity of a large class of ℚ-curves, including infinitely many curves not treated in the aforementioned papers (but not including every curve treated there.)

Suppose E/K is a ℚ-curve, and E^{σ} is a Galois conjugate of E. Then there exists a non-zero K-isogeny μ : E^{σ} → E, and so if p is a prime dividing the square-free part of the degree of μ then the Gal(K̄/K) module E[p] is reducible. The arguments employed in [11] and [12] use this reducibility to associate to E a p-adic representation of Gal(ℚ̄/ℚ) whose reduction mod p has dihedral image, and is therefore modular (in the sense that it arises from a modular form). Consequently, the results in [11] and [12] depend on the existence of a prime p ≥ 5 dividing the square-free part of the degree of some K-rational isogeny between E and one of its Galois conjugates. Moreover, their results require E to satisfy certain local conditions at p.

In contrast, the arguments we employ in the present paper make use of the mod 3 and 3adic representations attached to a ℚ-curve. We thus obtain a theorem which does not require the existence of a rational isogeny of large degree (but which does require local conditions at 3.) This allows us, for instance, to prove the modularity of the ℚ-curve

E = E_{A,B,C} : y^{2} = x^{3} + 2(1 + i)Ax^{2} + (B + iA^{2})x  (1.1)

discussed by Darmon in [4] in connection with the generalized Fermat equation

A^{4} + B^{2} = C^{p}.  (1.2)

We will discuss in a later paper the consequences of the present result regarding solutions of (1.2).

In order to state the main theorems of this paper we introduce a few definitions. Let E/K be a ℚ-curve, and for each σ ∈ Gal(ℚ̄/ℚ) let μ_{σ} : E^{σ} → E be a non-zero isogeny. Then we define b_{E} ∈ H^{2}(Gal(ℚ̄/ℚ), ±1) by

b_{E}(σ, τ) = sgn(μ_{σ}μ_{τ}^{σ}μ_{στ}^{−1}).

Denote by (b_{E})_{3} the restriction of b_{E} to H^{2}(Gal(ℚ̄_{3}/ℚ_{3}), ±1). Furthermore, we associate to E an ℓ-adic Galois representation ρ_{E,ℓ} of Gal(ℚ̄/ℚ) and a quadratic character ψ̄_{E,3} of Gal(ℚ̄/ℚ) (cf. Proposition 2.3 and Definition 2.16).

Theorem. Suppose E/K is a ℚ-curve with potentially ordinary or multiplicative reduction at a prime of K over 3, and such that (b_{E})_{3} is trivial. Then E is modular.

Theorem. Suppose E/K is a ℚ-curve such that, for some (whence every) prime ℓ > 3, the projective representation ℙρ_{E,ℓ} associated to ρ_{E,ℓ} is unramified at 3. Then E is modular.

We can weaken the condition on ρ_{E,ℓ} in the second theorem above, at the expense of introducing some technical conditions.

Denote by q_{3,∞} the unique class in H^{2}(Gal(ℚ̄/ℚ), ±1) ramified exactly at 3 and ∞. We prove the following theorem.

Theorem. Suppose E/K is a ℚ-curve which acquires semistable reduction over a field tamely ramified over ℚ_{3}. Suppose further that (b_{E})_{3} is trivial, and that the four classes (ψ̄_{E,3}, −1), b_{E}q_{3,∞}(ψ̄_{E,3}, −1), q_{3,∞}(ψ̄_{E,3}, 3), and b_{E}(ψ̄_{E,3}, 3) are all nontrivial in H^{2}(Gal(ℚ̄/ℚ), ±1). Finally, suppose that deg μ_{σ} can be chosen to be prime to 3 for all σ ∈ Gal(ℚ̄/ℚ). Then E is modular.

We remark that the ℚ-curve (1.1) satisfies the hypotheses of both the second and third theorems above.

There are infinitely many ℚ-curves which are not proved to be modular by the theorems in this paper. An instructive example is the curve

E : y^{2} = x^{3} + (−994708512\sqrt{5257}\sqrt{73} − 414461880\sqrt{5257} − 4973542560\sqrt{73} − 1089620282520)x
+36601957546560\sqrt{5257}\sqrt{73} + 5349307626327168\sqrt{5257}  (1.3)
+55021459817878848\sqrt{73} + 32065347994985088

which is the specialization to a = 2^{2} · 3^{2} · 73 of the family of ℚ-curves described by Quer in [13, §6]. One checks that

• The reduction of E over 3 is potentially supersingular;

• ℙρ_{E,ℓ} is ramified at 3 (because E/ℚ(\sqrt{−2}) does not have good reduction at 3);

• There is an isogeny of degree 3 between E and one of its Galois conjugates.

Thus, E does not satisfy the hypotheses of any of the theorems above. The 3-adic representation ρ_{E,3} is residually irreducible, but the image of the restriction ρ_{E,3}|G_{3} does not have trivial centralizer. To prove that such a representation is modular is beyond the reach of existing technology in deformation theory, including the recent result of Breuil, Conrad, Diamond, and Taylor.

The modularity of a ℚ-curve E is equivalent to the modularity of any one of the ρ_{E,ℓ}’s. (The modularity of the latter means that it is a representation associated to a modular form.) Most of the present paper is devoted to proving the modularity of the ρ_{E,3}’s. This is essentially done by showing that under the hypotheses stated in the theorem these representations satisfy the main theorems of either [2], [21], or [22].

The authors wish to thank Brian Conrad, Fred Diamond, Matthew Emerton, Jordi Quer, Ken Ribet, Richard Taylor, and Andrew Wiles for helpful discussions.

Some Notation

If K is a quadratic extension of ℚ, we write χ_{K} for the quadratic character of Gal(ℚ̄/ℚ) associated to K. Any two quadratic characters χ and χ′ of Gal(ℚ̄/ℚ) give classes in H^{1}(Gal(ℚ̄/ℚ), ±1). We write (χ, χ′) for their cup product. This is an element in H^{2}(Gal(ℚ̄/ℚ), ±1). If d is an element of ℚ^{∗}/(ℚ^{∗})^{2}, we write (χ, d) to mean the cup product (χ, χ_{ℚ(\sqrt{d})}).

We write elements in H^{2}(Gal(ℚ̄/ℚ), ±1) multiplicatively. Thus if c_{1}, c_{2} ∈ H^{2}(Gal(ℚ̄/ℚ), ±1), then c_{1}c_{2} is the class such that (c_{1}c_{2})(σ, τ) = c_{1}(σ, τ)c_{2}(σ, τ) for all σ, τ ∈ Gal(ℚ̄/ℚ).

We take an embedding ν : ℚ̄ ↪ ℚ̄_{ℓ} to be fixed for each ℓ, and denote the resulting decomposition subgroup (resp. inertia subgroup) of Gal(ℚ̄/ℚ) by G_{ℓ} (resp. I_{ℓ}). To be completely precise, we need to define two such embeddings: one in order to define decomposition subgroups of Gal(ℚ̄/ℚ), and the other in order to make sense of the scalar action of ℚ̄ on ℓ-adic vector spaces like T_{ℓ}A ⊗_{ℤ_{ℓ}} ℚ̄_{ℓ}. Write ι : ℚ̄ ↪ ℚ̄_{ℓ} for the second embedding. We may think of ν as fixed through the course of the paper; on the other hand, we will occasionally want to vary ι.

We also take an embedding ℚ̄ ↪ ℂ to be fixed. This determines a complex conjugation c.

We denote by G_{ℓ}^{t}, I_{ℓ}^{t}, I_{e}ll^{w} the tame quotient of the decomposition group, the tame inertia group, and the wild inertia group respectively.

For ℓ a rational prime, we denote by χ_{ℓ} : Gal(ℚ̄/ℚ) → ℤ_{ℓ}^{∗} the cyclotomic character, and by χ̄_{ℓ} : Gal(ℚ̄/ℚ) → 𝔽_{ℓ}^{∗} the mod ℓ cyclotomic character. If ρ : G → GL_{2}(F) is a representation of a group G over a field F, we write ℙρ for the composition of ρ with the natural projection GL_{2}(F) → PGL_{2}(F).

2 ℚ-curves and Galois representations

In this section we describe the ℓ-adic and mod ℓ Galois representations attached to a ℚ-curve. We also define Galois cohomology classes c_{E}, b_{E} and ψ̄_{E,ℓ} which are naturally attached to a ℚ-curve E. The definitions and results of this section, with the exception of Proposition 2.13, are not original to this paper. The basic framework is laid down in Ribet’s foundational paper [16]. The interested reader should also consult Quer’s preprint [13], which alerted us to the relevance of the class b_{E}.

Let K be a number field Galois over ℚ.

Definition 2.1. A ℚ-curve E/K is an elliptic curve E/K, such that, for each σ ∈ Gal(ℚ̄/ℚ), there exists a non-zero K-isogeny

μ_{σ} : E^{σ} → E.

We may, and do, suppose that μ_{σ} is the identity morphism for all σ ∈ Gal(ℚ̄/K).

Remark 2.2. Throughout this paper, it will be understood that all ℚ-curves are elliptic curves without complex multiplication. This assumption is not restrictive from our point of view, since ℚ-curves with complex multiplication are known to be modular [20].

Let ℓ be a rational prime, and define

ϕ_{E,ℓ} : Gal(K̄/K) → GL_{2}(ℤ_{ℓ})

to be the representation of Gal(K̄/K) on the ℓ-adic Tate module T_{ℓ}E of E. (We have fixed an isomorphism T_{ℓ}E ≅ ℤ_{ℓ}^{2}.) In the following proposition we describe an extension of ϕ_{E,ℓ} to a representation of the whole group Gal(ℚ̄/ℚ).

Proposition 2.3. There exists a representation

ρ_{E,ℓ} : G_{ℚ} → ℚ̄_{ℓ}^{∗} GL_{2}(ℚ_{ℓ})

such that ℙρ_{E,ℓ}|_{Gal(K̄/K)} ≅ ℙϕ_{E,ℓ}. This representation is odd, continuous, and ramified at only finitely many primes.

Proof. For each non-zero isogeny μ : E′ → E, we write μ^{−1} to mean

(1/ deg μ)μ^{∨} ∈ Hom(E′, E) ⊗_{ℤ} ℚ,

where μ^{∨} is the dual isogeny.

Let σ and τ be elements of Gal(ℚ̄/ℚ). Following [16, §6], we define

c_{E}(σ, τ) = μ_{σ}μ_{τ}^{σ}μ_{στ}^{−1} ∈ (Hom(E, E) ⊗_{Z} ℚ)^{∗} = ℚ^{∗}.

Then c_{E} determines a class in H^{2}(Gal(ℚ̄/ℚ), ℚ^{∗}). Tate showed that H^{2}(Gal(ℚ̄/ℚ), ℚ̄^{∗}) is trivial, where ℚ̄^{∗} is acted on trivially by Gal(ℚ̄/ℚ); [18, Thm. 4]. It follows that there exists a continuous map α : Gal(ℚ̄/ℚ) → ℚ̄^{∗} such that

c_{E}(g, h) = α(g)α(h)α(gh)^{−1}  (2.4)

We can now define an action of Gal(ℚ̄/ℚ) on ℚ̄_{ℓ} ⊗_{ℤ_{ℓ}} T_{ℓ}E by

ρ_{E,ℓ}(g)(1 ⊗ x) = α^{−1}(g) ⊗ μ_{g}(x^{g}).  (2.5)

It is clear from the above definition that ℙρ_{E,ℓ}|_{Gal(K̄/K)} ≅ ℙϕ_{E,ℓ}. In particular, ρ_{E,ℓ}|_{Gal(K̄/K)} and ϕ_{E,ℓ} differ by the continuous character α|_{Gal(K̄/K)}. It follows that ρ_{E,ℓ} is continous and unramified away from finitely many primes. It remains to show that ρ_{E,ℓ} is odd; that is, that det ρ_{E,ℓ}(c) = −1, where c is our fixed complex conjugation.

Define a map ϵ_{E} : Gal(ℚ̄/ℚ) → ℚ̄^{∗} by

ϵ_{E}(σ) = α^{2}(σ)/(deg μ_{σ}),

and let ϵ_{E,ℓ} be the composition of ϵ_{E} with the chosen embedding ℚ̄ ↪ ℚ̄_{ℓ}. That this map is a character follows from the observation that

c_{E}(σ, τ)^{2} = \frac{(deg μ_{σ})(deg μ_{τ})}{deg μ_{στ}}.

It also follows immediately from (2.5) that

det ρ_{E,ℓ} = ϵ_{E,ℓ}^{−1}χ_{ℓ}.  (2.6)

Write μ for μ_{c}. We may write the complexification E/ℂ as the quotient of ℂ by a lattice Λ. Then μ is given by multiplication by a complex number z such that zΛ̄ ⊂ Λ. The composition μμ^{c} is then given by zz^{c}, a positive real number. Since the degree of μμ^{c} is (deg μ)^{2}, we conclude that

μμ^{c} = deg μ.

Therefore,

ϵ_{E}(c) = α^{2}(c)/ deg μ = c_{E}(c, c)/ deg μ = μμ^{c}/ deg μ = 1,

and

det ρ_{E,ℓ}(c) = ϵ_{E,ℓ}(c)χ_{ℓ}(c) = −1.

Since α^{2}(c) = c_{E}(c, c) = μμ^{c} = deg μ, the proposition follows from the definition of ϵ_{E,ℓ} and (2.6).

Remark 2.4. It will occasionally be useful to work directly with the homomorphism

ρ̂_{E,ℓ} : Gal(ℚ̄/ℚ) → ℚ̄^{∗} GL_{2}(ℚ_{ℓ})

defined by (2.5). More precisely: suppose M is a number field such that ρ̂_{E,ℓ} takes values in M^{∗} GL_{2}(ℚ_{ℓ}). Let λ be the prime of M defined by ι : ℚ̄ → ℚ̄_{ℓ}, and let λ = λ_{1}, …, λ_{r} be the set of all primes of M dividing ℓ. Write ρ_{E,λ_{i}} for the composition of ρ̂_{E,ℓ} with the map

M^{∗} GL_{2}(ℚ_{ℓ}) → M_{λ_{i}}^{∗} GL_{2}(ℚ_{ℓ}).

So ρ_{E,λ} is just another name for ρ_{E,ℓ}.

Remark 2.5. While ρ_{E,ℓ} and ϵ_{E} depend on our choice of α, the projective representation ℙρ_{E,ℓ} depends only on the isomorphism class of E/K. Moreover, ℙρ_{E,ℓ} is independent of the choice of ι.

Remark 2.6. We can choose α in such a way that the image of ϵ_{E} has 2-power order, by the following argument. Let n = 2^{a}m be the order of the image of ϵ_{E}, where m is odd. If m ≠ 1, replace α by αϵ_{E}^{(m−1)/2}; this has the effect of replacing ϵ_{E} by ϵ_{E}^{m}, whose image has 2-power order.

The reason for introducing the representations ρ_{E,ℓ} is found in the following proposition.

Proposition 2.7. A ℚ-curve E/K is modular if there exists a (normalized) eigenform f and a prime ℓ such that

ρ_{E,ℓ} ≅ ρ_{f,ℓ}.

Here, f is a holomorphic Hecke eigenform on the complex upper-half plane and ρ_{f,ℓ} is the Galois representation ρ_{f,ℓ} : Gal(ℚ̄/ℚ) → GL_{2}(ℚ̄_{ℓ}) such that if f(z) = ∑_{n=1}^{∞} a(n)e(nz) (a(1) = 1), then trace ρ_{f,ℓ}(Frob_{p}) = a(p) for almost all primes p.

Proof. Suppose ρ_{E,ℓ} ≅ ρ_{f,ℓ} for some eigenform f of level N. Then there exists some finite extension L/K such that

ϕ_{E,ℓ}|_{Gal(L̄/L)} ≅ ρ_{f,ℓ}|_{Gal(L̄/L)}  (2.7)

and the weight of f must be two, as can be seen by comparing determinants. Let ρ_{N,ℓ} be the representation of Gal(ℚ̄/ℚ) on T_{ℓ}J_{1}(N) ⊗_{ℤ_{ℓ}} ℚ̄_{ℓ}, where T_{ℓ}J_{1}(N) is the ℓ-Tate module of J_{1}(N). We have

ρ_{N,ℓ} ≃ ⊕ρ_{g,ℓ}  (2.8)

where the sum is over all the eigenforms g of level N and weight 2 (this can be deduced from [19, Thm. 7.11]). From (2.7) and (2.8) it follows that ϕ_{E,ℓ} is a Gal(L̄/L)-quotient of ρ_{N,ℓ}. It then follows that Hom_{L}(T_{ℓ}J_{1}(N), T_{ℓ}E) is non-zero. By a theorem of Faltings [7] we can conclude from this that Hom_{L}(J_{1}(N), E) is non-zero.

We next define some cohomological invariants associated to E. Let b_{E} ∈ H^{2}(Gal(ℚ̄/ℚ), ±1) be the composition of c_{E} with the sign map ℚ^{∗} → ±1. Then b_{E} can be computed from ϵ_{E}. Consider the exact sequence in Galois cohomology

Hom(Gal(ℚ̄/ℚ), ℤ̄^{∗}) → Hom(Gal(ℚ̄/ℚ), ℤ̄^{∗}) →^{δ} H^{2}(Gal(ℚ̄/ℚ), ±1)  (2.9)

arising from the short exact sequence of Galois modules (with trivial action)

0 → ±1 → ℤ̄^{∗} → ℤ̄^{∗} → 0.

Proposition 2.8. b_{E} = δ(ϵ_{E}).

Proof. Let χ : Gal(ℚ̄/ℚ) → ℤ̄^{∗} be a character, and for each σ ∈ Gal(ℚ̄/ℚ) let χ̃(σ) be a square root of χ(σ). Then δ(χ) is defined by

δ(χ)(σ, τ) = \frac{χ̃(σ)χ̃(τ)}{χ̃(στ)}.

To compute δ(ϵ_{E}), we may choose

ϵ̃_{E}(σ) = α(σ)/\sqrt{deg μ_{σ}}

where the √ sign signifies positive square root. We now have

δ(ϵ_{E})(σ, τ) = \frac{α(σ)α(τ)}{α(στ)} \sqrt{deg μ_{στ}} ; \sqrt{deg μ_{σ}}\sqrt{deg μ_{τ}} = c_{E}(σ, τ)/\sqrt{c_{E}^{2}(σ, τ)} = b_{E}(σ, τ).

Remark 2.9. Note that the class c_{E} is the inflation of a class in H^{2}(Gal(K/ℚ), ±1). Quer [13, Th. 2.4] has proven the converse: if E/K′ is a ℚ-curve over some extension of K, and if c_{E} is the inflation of a class in H^{2}(Gal(K/ℚ), ±1), then there exists a ℚ-curve E_{0}/K such E_{0} ×_{K} K′ is geometrically isogenous to E.

We will need the fact that the representation ρ_{E,ℓ} can also be viewed as the ℓ-adic representation attached to a certain abelian variety over ℚ.

Proposition 2.10. Let E/K be a ℚ-curve, and let α : Gal(ℚ̄/ℚ) → ℚ̄^{∗} be a 1-cochain with coboundary c_{E}, as in (2.4). Define ρ_{E,ℓ} as in (2.5). Let M be the number field generated by the α(g) for all g ∈ Gal(ℚ̄/ℚ).

There exists an abelian variety A_{α}/ℚ satisfying the following conditions.

• There exists an injection M ↪ End(A_{α}/ℚ) ⊗_{ℤ} ℚ;

• Let λ_{1}, …, λ_{r} be the primes of M lying over ℓ. Then the rational Tate module V_{ℓ}A_{α} decomposes as

V_{ℓ}A_{α} = ⨁_{i} V_{λ_{i}}A_{α}

and V_{λ_{i}}A_{α} is isomorphic, as M_{λ_{i}}[Gal(ℚ̄/ℚ)]-module, to ρ_{E,λ_{i}}. In particular, V_{λ}A_{α} ≅ ρ_{E,ℓ}.

Proof. The desired A_{α} is the one constructed by Ribet in [16, §6]. We briefly recall this construction. First, enlarge K if necessary so that α is the inflation of a function on Gal(K/ℚ). Let R be the algebra generated by elements λ_{σ} for each σ ∈ Gal(K/ℚ), with the multiplication table

λ_{στ} c_{E}(σ, τ) = λ_{σ}λ_{τ} .

Then R acts on the abelian variety

Res_{ℚ}^{K} E ×_{ℚ} K ≅ ⨁_{σ∈Gal(K/ℚ)} E^{σ}

by the rule

λ_{σ}(P) = μ_{σ}^{τ}(P)  (2.10)

for any P ∈ E^{τσ}(K̄). This action descends to an action of R on Res_{ℚ}^{K} E.

Our choice of α defines a homomorphism ω : R → M. Now define

A_{α} = Res_{ℚ}^{K} E ⊗_{R} M

in the category of abelian varieties up to isogeny. To be more precise, let π ∈ R be the projector onto M; then A_{α} is the image of mπ, where m is an integer large enough to make mπ an actual endomorphism (not only a rational endomorphism) of Res_{ℚ}^{K} E.

Then A_{α} admits the desired injection M ↪ End(A_{α})⊗_{ℤ} ℚ, and the rational λ_{i}-adic Tate module V_{λ_{i}}A_{α} is a 2-dimensional vector space over M_{λ_{i}} (see [15, Th. 2.1.1]); one then has from (2.10) that Gal(ℚ̄/ℚ) acts on V_{λ_{i}}A_{α} via ρ_{E,λ_{i}}.

Remark 2.11. We emphasize that the construction of A_{α} is independent of ℓ.

Proposition 2.12. Let ℓ > 2. Suppose ℓ does not divide deg μ_{g} for any g ∈ Gal(ℚ̄/ℚ), and suppose α is chosen so that ϵ_{E} has 2-power order (Remark 2.6.) Let λ = λ_{1}, …, λ_{i} be the set of primes of M dividing ℓ.

Then the full ring of integers of M ⊗_{ℤ} ℤ_{ℓ} ≅ ⊕_{i}M_{λ_{i}} acts on T_{ℓ}A_{α}, and the ℓ-divisible group T_{ℓ}A_{α} breaks up as a direct sum of ℓ-divisible groups

⨁_{i} T_{λ_{i}}A_{α}  (2.11)

where the λ_{i} range over the primes of M dividing ℓ. Moreover, T_{λ_{i}}A_{α} is a free O_{M_{λ_{i}}} -module of rank 2.

Proof. Let ℤ[α] be the ring generated by the α(g). Then it follows by the definition of A_{α} that ℤ_{ℓ} ⊗_{ℤ} ℤ[α] acts on T_{ℓ}A_{α}. Since deg μ_{g} is an ℓ-adic unit, and since α(g)/\sqrt{deg μ_{g}} is a 2-power root of unity, we can obtain ℤ_{ℓ} ⊗_{ℤ} ℤ[α] by successively adjoining square roots of ℓ-adic units to ℤ_{ℓ}; it follows that ℤ_{ℓ} ⊗_{ℤ} ℤ[α] is étale over ℤ_{ℓ}, and therefore

ℤ_{ℓ} ⊗_{ℤ} ℤ[α] ≅ ⨁_{i} O_{M_{λ_{i}}} .

The decomposition of T_{ℓ}A_{α} now follows immediately from the decomposition (2.11).

We now want to define a mod ℓ representation attached to E. We begin with a general result about Galois representations.

Proposition 2.13. Let L be a totally ramified extension of ℚ_{ℓ} and F an unramified extension of L. Let ρ be a continuous representation of Gal(ℚ̄/ℚ) (or any compact group) with image in F^{∗} GL_{2}(ℚ_{ℓ}). Then ρ is conjugate in GL_{2}(F) to a representation with image in O_{F}^{∗} GL_{2}(O_{L}).

Proof. Let S, T be a basis for F^{⊕2} with respect to which the image of ρ lies in F^{∗} GL_{2}(ℚ_{ℓ}), and let L_{0} be the lattice O_{F} S + O_{F} T generated by S, T. There are only finitely many images of L_{0} under the action of the compact group Gal(ℚ̄/ℚ). Each such image L_{i} is of the form

x(O_{F} (aS + bT) + O_{F} (cS + dT))

with x ∈ F^{∗} and a, b, c, d ∈ ℚ_{ℓ}. Let L be the lattice generated by all the L_{i}; then L is preserved by the action of ρ(Gal(ℚ̄/ℚ)). Because F/L is unramified, we may write x = yu, with y ∈ L^{∗} and u ∈ O_{F}^{∗}. So each L_{i} can be rewritten as

y(O_{F} (aS + bT) + O_{F} (cS + dT)).

Let L′_{i} be the lattice in L^{2} defined by

L′_{i} = y(O_{L}(aS + bT) + O_{L}(cS + dT)),

and let

L′ = O_{L}(αS + βT) + O_{L}(γS + δT),

with α, β, γ, δ ∈ L, be the lattice generated by all the L′_{i}. Then L = L′ ⊗_{O_{L}} O_{F}. Let S′ = αS + βT and T′ = γS + δT, and write

ρ : Gal(ℚ̄/ℚ) → GL_{2}(F)

with respect to the basis elements S′ and T′.

Since ρ(Gal(ℚ̄/ℚ)) preserves the lattice O_{F} S′ + O_{F} T′, we have ρ(Gal(ℚ̄/ℚ)) ∈ GL_{2}(O_{F}). Since S′, T′ lie in LS + LT, we have ρ(Gal(ℚ̄/ℚ)) ∈ F^{∗}GL_{2}(L). Combining these two facts yields the desired result.

The representation ρ_{E,ℓ} produced in Proposition 2.3 takes values in M_{λ}^{∗} GL_{2}(ℚ_{ℓ}), where M_{λ} is the extension of ℚ_{λ} generated by the values of ι(α(σ)) for all σ ∈ Gal(ℚ̄/ℚ). Recall from the proof of Proposition 2.3 that ϵ_{E,ℓ}(σ) = α^{2}(σ)/ deg μ_{σ} is a Dirichlet character. So M_{λ} is contained in an extension generated by square roots and roots of unity; it is thus an abelian extension of ℚ_{ℓ}. It follows from local class field theory that there exists an abelian extension F of ℚ_{ℓ} containing M and such that F also contains a subextension L totally ramified over ℚ_{ℓ} over which F is unramified. Then F and L satisfy the conditions of Proposition 2.13, so there exists a basis of F^{⊕2} with respect to which ρ_{E,ℓ} takes images in O_{F}^{∗} GL_{2}(O_{L}).

Definition 2.14. We denote by

ρ̄_{E,ℓ} : Gal(ℚ̄/ℚ) → 𝔽̄_{ℓ}^{∗} GL_{2}(𝔽_{ℓ}).

the representation obtained by choosing a basis of F^{⊕2} as above and reducing the resulting representation

ρ_{E,ℓ} : Gal(ℚ̄/ℚ) → O_{F}^{∗} GL_{2}(O_{L})

modulo the maximal ideal 𝔪_{F} of O_{F}. The reduced representation ρ̄_{E,ℓ} is then well defined up to semisimplification and conjugation by GL_{2}(𝔽̄_{ℓ}).

We observe that

det ρ̄_{E,ℓ} = ϵ̄_{E,ℓ}χ̄_{ℓ}

where the overlines indicate the reductions of the ℓ-adic characters to mod-ℓ characters. Let δ̄ be the reduction mod ℓ of the coboundary map δ in (2.9).

From this point on, we assume that ℓ > 2.

Proposition 2.15. b_{E} = δ̄(ϵ̄_{E,ℓ}).

Proof. Immediate from Proposition 2.8.

When R is a domain we abuse notation and denote by ‘det’ the determinant character from PGL_{2}(R) to R^{∗}/(R^{∗})^{2}.

Definition 2.16. Let ℓ be an odd prime. Then we define a quadratic Dirichlet character

ψ̄_{E,ℓ} = det ℙρ̄_{E,ℓ} : Gal(ℚ̄/ℚ) → 𝔽_{ℓ}^{∗}/(𝔽_{ℓ}^{∗})^{2} ≅ ±1.

The character ψ̄_{E,ℓ}, like the cohomology class b_{E}, depends only on the isomorphism class of E/K.

Remark 2.17. The invariants b_{E} and ψ̄_{E,ℓ} are easy to compute in practice. For instance, suppose K is a quadratic extension of ℚ and E/K is a ℚ-curve. Let τ be the non-trivial element of Gal(K/ℚ), and let n be the integer such that μμ^{τ} is multiplication by n. Then

b_{E} = \{ \begin{matrix}0 & if n positive \\ χ_{K} & if n negative. \end{matrix}

Suppose, for simplicity, that ℓ does not divide n. Let η : Gal(ℚ̄/ℚ) → ±1 be the quadratic character ramified only at ℓ. Then

ψ̄_{E,ℓ} = \{ \begin{matrix}η & n ∈ (ℚ_{ℓ}^{∗})^{2} \\ ηχ_{K} & n ∉ (ℚ_{ℓ}^{∗})^{2}. \end{matrix}

3 The potentially supersingular, residually irreducible case

Theorem 3.1. Suppose K is tamely ramified over 3. Let E/K be a ℚ-curve such that

• E has good supersingular reduction over K_{v} for one (whence every) prime v of K above 3;

• b_{E} ∈ H^{2}(Gal(ℚ̄/ℚ), ±1) has trivial projection to H^{2}(G_{3}, ±1);

• Either ℙρ_{E,ℓ}|G_{3} is unramified for some (whence every) ℓ ≠ 3, or deg μ_{σ} is not a multiple of 3 for any σ ∈ Gal(ℚ̄/ℚ);

• The restriction of ρ̄_{E,3} to Gal(ℚ̄/ℚ[\sqrt{−3}]) is absolutely irreducible.

Then E is modular.

Proof. The basic tool will be the theorem of Wiles and Taylor-Wiles [24, 23], as refined by Diamond [6] and by Conrad, Diamond, and Taylor [2]. In particular, our argument follows closely the proof of Theorem 7.2.1 of [2].

We have from Proposition 2.3 that ρ_{E,3} is an odd, continuous representation unramified away from finitely many primes.

Write G_{v}, I_{v} for the absolute Galois group and the inertia group of K_{v}. Write I_{3}^{w} for the subgroup of wild inertia in I_{3}.

Lemma 3.2. ρ̄_{E,3} is modular.

Proof. We follow closely the usual argument that the 3-division points of an elliptic curve over ℚ form a modular representation–see [8, I.1] for more details.

The image of ρ̄_{E,3} lies in 𝔽̄_{3}^{∗} GL_{2}(𝔽_{3}). We suppose without loss of generality that the chosen extension of the 3-adic valuation of ℚ to ℚ[\sqrt{−2}] is given by the prime (1 + \sqrt{−2}).

We can define a homomorphism

ι : 𝔽̄_{3}^{∗} GL_{2}(𝔽_{3}) → μ_{∞} GL_{2}(ℤ[\sqrt{−2}]),

where μ_{∞} denotes the group of roots of unity, as follows: set

ι [ \begin{matrix}−1 & 1 \\ −1 & 0 \end{matrix} ] = [ \begin{matrix}−1 & 1 \\ −1 & 0 \end{matrix} ] ,

ι [ \begin{matrix}1 & −1 \\ 1 & 1 \end{matrix} ] = [ \begin{matrix}1 & −1 \\ −\sqrt{−2} & −1 + \sqrt{−2} \end{matrix} ] ,

and, for each scalar a ∈ 𝔽̄_{3}^{∗}, define ι(a) to be the preimage, under the chosen embedding ℚ̄ ↪ ℚ̄_{3}, of the Teichmüller lift of a.

Let F be a number field such that the image of ι ∘ ρ̄_{E,3} lies in GL_{2}(F), and let w be the chosen extension of the 3-adic valuation to F. Then the composition of ι ∘ ρ̄_{E,3} with reduction mod w is ρ̄_{E,3}.

The composition ι∘ρ̄_{E,3} is a continuous complex representation of Gal(ℚ̄/ℚ), odd and irreducible because ρ̄_{E,3} is odd and absolutely irreducible. It follows from the theorem of Langlands and Tunnell that there exists a weight 1 eigenform, of some level and Dirichlet character,

g = ∑_{n=1}^{∞} b_{n}q^{n}

such that

b_{p} = Tr(ι ∘ ρ̄_{E,3}(Frob_{p}))

for almost all p. Let F′ be a number field containing all the b_{n}. If E is a weight 1 Eisenstein series whose Fourier expansion is congruent to 1 mod 3, then gE is a weight 2 cusp form, of some level and Dirichlet character, such that T_{n}(gE) is congruent mod w to b_{n}gE, for some prime w′ of F′ above w. It then follows from an argument of Deligne and Serre [5, §6.10] that there exists an eigenform

f = ∑_{n=1}^{∞} a_{n}q^{n}

of weight 2 with a_{n} ∈ F′ and a_{n} ≡ b_{n} mod w for all n. In particular, a_{p} = Tr(ρ̄_{E,3}(Frob_{p})) for almost all p. So ρ̄_{E,3} is the mod w′ representation associated to f.

We will show that ρ_{E,3} satisfies the conditions of Theorem 7.1.1 of [2].

Recall that, for any ℓ,

ϕ_{E,ℓ} : Gal(K̄/K) → GL_{2}(ℤ_{ℓ})

is the Galois representation attached to E/K as elliptic curve, and that ℙϕ_{E,ℓ} and ℙρ_{E,ℓ} are isomorphic projective representations of Gal(K̄/K), by Proposition 2.3.

The representation ρ_{E,ℓ} produced by Proposition 2.3 depends on a choice of α : Gal(ℚ̄/ℚ) → ℚ̄^{∗}, a cochain whose coboundary is c_{E}. We begin by observing that α can be chosen so as to impart to ρ_{E,ℓ} some useful arithmetic properties.

Lemma 3.3. There exists a choice of α : G_{ℚ} → ℚ̄^{∗} such that

• for all ℓ ≠ 3, the representation ρ_{E,ℓ}|G_{3} is tamely ramified;

• for all ℓ, det ρ_{E,ℓ}|G_{3} = χ_{ℓ}|G_{3};

• ϵ_{E} has 2-power order.

Proof. Since K is tamely ramified, c_{E}|G_{3} is the inflation of an element of

H^{2}(G_{3}^{t}, ℚ^{∗}).

The cohomology group

H^{2}(G_{3}^{t}, ℚ̄^{∗})

is trival, as can be seen by placing G_{3}^{t} in the exact sequence

0 → I_{3}^{t} → G_{3}^{t} → G_{3}/I_{3}

and computing the initial terms of the Hochschild-Serre spectral sequence [18, §6.1]. Therefore, there is a cochain a_{3} : G_{3} → ℚ̄^{∗} such that

c_{E}(g, h) = α_{3}(g)α_{3}(h)α_{3}(gh)^{−1}

for all g, h ∈ G_{3}, and such that α_{3} vanishes on the wild inertia group I_{3}^{w}. Now let α′ : Gal(ℚ̄/ℚ) → ℚ̄^{∗} be any cochain whose coboundary is c_{E}. Then (α′|G_{3})α_{3}^{−1} is a character θ_{3} of G_{3}. Let θ be a character of Gal(ℚ̄/ℚ) whose restriction to G_{3} is θ_{3}. Then define

α = α′θ^{−1}.

So the coboundary of α is c_{E}, and α|G_{3} = α_{3}; in particular, α vanishes on wild inertia. Since E obtains good reduction after a tame extension of K, we know ϕ_{E,ℓ} is tamely ramified at 3 for all ℓ ≠ 3. It follows from the definition (2.5) that ρ_{E,ℓ}|G_{3} is tamely ramified for all ℓ ≠ 3.

By Proposition 2.8 and (2.6), the assumption that b_{E}|G_{3} is trivial means that

ϵ_{E}|G_{3} = χ^{2}

for some character χ : G_{3} → ℚ̄^{∗}. The character ϵ_{E,ℓ} is tamely ramified for any ℓ ≠ 3, because ρ_{E,ℓ}|G_{3} is tamely ramified; it follows that ϵ_{E}, whence also χ, is tamely ramified. Replacing α by αχ now yields the first two desired conditions. In particular, ϵ_{E}|G_{3} is trivial. So we can modify α by any power of ϵ_{E} without affecting the first two conditions. Now we can force ϵ to have 2-power order by the argument of Remark 2.4.

For the rest of the proof, it is understood that α is chosen so that ρ_{E,ℓ}|G_{3} satisfies the conditions in Lemma 3.3. We now take as fixed some ℓ > 3. From Proposition 2.10, we have an abelian variety A_{α}/ℚ such that

ρ_{E,ℓ} ≅ V_{λ}A_{α}

where λ|ℓ is the prime of M (the number field generated by the α(g)) determined by ι.

Denote by L the ramified quadratic extension of ℚ_{3}, by G_{L} ⊂ G_{3} the absolute Galois group of L, and by I_{L} the inertia subgroup of G_{L}. Let ψ be the ramified quadratic character of G_{L}, and write A_{α}^{ψ}/L for the twist of A_{α} ×_{ℚ} L by ψ.

From this point on, we take as fixed some ℓ > 3.

Lemma 3.4. Either

• ρ_{E,ℓ}|G_{L} and P ρ_{E,ℓ}|G_{3} are unramified, and A_{α}/L has good reduction; or

• ρ_{E,ℓ}|G_{L} ⊗ ψ is unramified, and A_{α}^{ψ}/L has good reduction.

Proof. By Lemma 3.3, wild inertia is killed by ρ_{E,ℓ}. Let τ be a topological generator of I_{3}^{t}, and define m = ρ_{E,ℓ}(τ). Since m has finite order, it is diagonalizable.

Note that τ and τ^{3} are conjugate in G_{3}. So m and m^{3} are conjugate in ℚ̄_{ℓ}^{∗} GL_{2}(ℚ_{ℓ}). Since det(m) = χ_{ℓ}(τ) = 1, we conclude that the eigenvalues of m must be either (1, 1), (−1, −1) or (i, −i). In the former two cases, we see that ρ_{E,ℓ}|G_{L} is unramified. In the latter case, (ρ_{E,ℓ}|G_{L}) ⊗ ψ is unramified.

Suppose the eigenvalues of m are (1, 1) or (−1, −1); equivalently, ℙρ_{E,ℓ}|G_{3} is unramified. Recall from Remark 2.5 that ℙρ_{E,ℓ} does not depend on the choice of ι. So ρ_{E,λ_{i}}(τ) is scalar for any i, whence I_{L} acts trivially on V_{λ_{i}}A_{α} for any prime λ_{i} of M dividing ℓ. It then follows from Proposition 2.10 that I_{L} acts trivially on V_{ℓ}A_{α}, and so A_{α}/L has good reduction.

Likewise, if the eigenvalues of m are (i, −i), then ρ_{E,λ_{i}}(τ^{2}) = −1 for all i, so ρ_{E,λ_{i}}|G_{L} ⊗ ψ is unramified for all i, and A_{α}^{ψ}/L has good reduction.

Suppose ℙρ_{E,ℓ}|G_{3} is unramified. Then ρ_{E,ℓ}|G_{L} is unramified. Since det ρ_{E,ℓ}|I_{3} is trivial, the image ρ_{E,ℓ}(I_{3}) is either trivial or ±1. So, in fact, either A_{α}/ℚ_{3} or its ramified quadratic twist has good reduction over ℚ_{3}. Therefore, either ρ_{E,3}|G_{3} or its ramified quadratic twist is associated to a 3-divisible group, and E is modular by [6, Theorem 5.3].

We therefore assume from now on that (ρ_{E,ℓ}|G_{L}) ⊗ ψ is unramified, so that A_{α}^{ψ}/L has good reduction. In this case, by the hypotheses of our theorem, 3 does not divide deg μ_{g} for any g ∈ Gal(ℚ̄/ℚ).

In this case, A_{α}^{ψ}/L has good reduction. Therefore, if ψ′ is a ramified quadratic character of Gal(ℚ̄/ℚ[\sqrt{−3}]), the twist A_{α}^{ψ′}/ℚ[\sqrt{−3}] has good reduction at the prime over 3.

Let θ be the prime of M determined by the chosen embedding M ↪ ℚ̄_{3}. Then, by Proposition 2.12, we can define a finite flat group scheme A_{α}[θ] as the 3-torsion (equivalently, the θ-torsion) of the 3-divisible group T_{θ}A_{α}^{ψ′}/ℚ[\sqrt{−3}]. Because ρ̄_{E,3} is absolutely irreducible when restricted to Gal(ℚ̄/ℚ[\sqrt{−3}]), and because

ρ_{E,3} ⊗ ψ′ ≅ V_{θ}A_{α}^{ψ′},

we have an isomorphism of (O_{M_{θ}}/θ)[Gal(ℚ̄/ℚ[\sqrt{−3}])]-modules

ρ̄_{E,3} ⊗ ψ′ ≅ A_{α}^{ψ′}[θ].

Restricting this isomorphism to G_{L} yields an isomorphism of (O_{M_{θ}}/θ)[G_{L}]-modules

ρ̄_{E,3}|G_{L} ⊗ ψ ≅ A_{α}^{ψ}[θ].

In particular, (ρ̄_{E,3}|G_{L}) ⊗ ψ is flat. Recall that a representation of G_{L} with finite image is said to be flat if the attached finite flat group scheme over L is the generic fiber of a finite flat group scheme over O_{L}. In the case, the finite flat group scheme in question is (A_{α}^{ψ})[θ].

Lemma 3.5. The centralizer of ρ̄_{E,3}(G_{3}) consists entirely of scalars.

Proof. The result follows from a theorem of Conrad [1, Theorem 4.2.1]. As above, the relevant finite flat group scheme over O_{L} is G = (A_{α}^{ψ})[θ]. To apply Conrad’s theorem, we need only to verify that G is connected and has connected Cartier dual, and that G satisfies a certain exactness condition on Dieudonné modules. The connectedness of G and its dual follow from the fact that G is a closed subgroup scheme of the 3-torsion subscheme of the supersingular abelian variety A_{α}^{ψ}. The exactness condition is automatically satisfied because G is the 3-torsion in the 3-divisible group T_{θ}A_{α}^{ψ}.

Let F be a finite extension of ℚ_{ℓ}. Recall that an ℓ-adic representation ρ of the Galois group of F is said to be Barsotti-Tate if it arises from the generic fiber of an ℓ-divisible group, and to be potentially Barsotti-Tate if some restriction of ρ to a finite-index subgroup of Gal(F̄/F) is Barsotti-Tate. (See [2, §1.1].) The representation ρ_{E,3}|G_{3} is potentially Barsotti-Tate, because it is realized on the θ-adic Tate module of A_{α}, which has potentially good reduction. From now on, we will abuse notation and refer to the local representation ρ_{E,3}|G_{3} simply as ρ_{E,3}.

Let V be a d-dimensional vector space over a finite extension F′ of ℚ_{ℓ}. One can associate to any potentially Barsotti-Tate representation ρ : Gal(F̄/F) → GL(V) a continuous representation

W D(ρ) : W_{F} → GL(D)

of the Weil group of F on a ℚ̄_{ℓ}-vector space D of dimension d, as in Conrad, Diamond, and Taylor [2, Appendix B]. In the lemma that follows, we will freely use definitions and facts from that paper, especially §1.2, §2.3, and Appendix B.

Lemma 3.6. The type of W D(ρ_{E,3}) is strongly acceptable for ρ̄_{E,3}.

Proof. We take F′ = M_{λ}.

Let τ be the restriction of W D(ρ_{E,3}) to I_{3}. It follows from Proposition 2.10 and [2, Prop. B.4.2] that ρ_{E,3} is Barsotti-Tate over L′ for any finite extension L′/ℚ_{ℓ} such that τ is trivial. Our choice of α in Lemma 3.3 guarantees that det ρ_{E,3} = χ_{3} and det ρ̄_{E,3} = χ̄_{3}. It follows that ρ_{E,3} is a deformation of ρ̄_{E,3} of type τ, according to the definition in [2, §1.2].

We know that (ρ_{E,3}|G_{L}) ⊗ ψ is Barsotti-Tate, because it is associated to the 3-divisible group T_{θ}A_{α}^{ψ}. So

W D((ρ_{E,3}|G_{L}) ⊗ ψ) = W D(ρ_{E,3}|G_{L}) ⊗ W D(ψ)

is unramified, so τ|I_{L} = W D(ψ)|I_{L}. We know that W D(ψ) = ψ|W_{L} ⊗_{M_{λ}} ℚ̄_{ℓ} ([2, §B.2]); that is, W D(ψ)|I_{L} is a non-trivial quadratic character of I_{L}. We also know that the determinant of τ is trivial on I_{3}, because the W D functor commutes with exterior products, and the determinant of ρ_{E,3} is the cyclotomic character χ_{3}; the character W D(χ_{3}) is shown to be unramified in [2, §B.2]. We conclude that

τ ≅ ω̃_{2}^{2} ⊕ ω̃_{2}^{6},

where ω̃_{2} : I_{3}^{t} → ℚ̄_{3}^{∗} is the Teichmüller lift of ω_{2}, the fundamental tame character of level 2. It now follows from Corollary 2.3.2 of [2] that τ is acceptable for ρ̄_{E,3}. We have by [1, Theorem 4.2.1] that either

• (ρ̄_{E,3}|I_{3}) ⊗_{𝔽_{3}} 𝔽̄_{3} ≅ ω_{2}^{m} ⊕ ω_{2}^{3m}, where m = 1 or 5;

• ρ̄_{E,3}|I_{3} ≅ [ \begin{matrix}χ̄_{3}^{m} & ∗ \\ 0 & χ̄_{3}^{n} \end{matrix} ], where (m, n) = (0, 1) or (1, 0) and ∗ is peu ramifié.

In either case, it follows from the criterion of [2, §1.2] that τ is strongly acceptable for ρ̄_{E,3}.

Now, combining Lemmas 3.5 and 3.6, we can apply [2, Theorem 7.1.1] and conclude that ρ_{E,3}, whence E, is modular.

4 More on residual representations

In [24], Wiles deals with the case where the 3-adic representation associated to an elliptic curve C is residually reducible by executing a “3-5 switch”. That is, he replaces C with another elliptic curve C′, such that the mod 3 representation attached to C′ is absolutely irreducible when restricted to Gal(ℚ̄/ℚ[\sqrt{−3}]), and such that C and C′ have isomorphic mod 5 Galois representations. Aside from a finite set of exceptions, the common mod 5 Galois representation is absolutely irreducible when restricted to Gal(ℚ̄/ℚ[\sqrt{5}]). This coincidence of mod 5 Galois representations is enough to show that modularity of C′ is equivalent to modularity of C, and the modularity of C′ follows from the condition on the mod 3 Galois representation of C′. This argument relies on the fact that, given an elliptic curve C, there are plenty of elliptic curves C′ whose mod 5 representations are isomorphic to that of C. This fact, in turn, depends on the fact that the modular curve X(5)/ℚ is isomorphic to ℙ^{1}/ℚ. In general, the modular curve parametrizing ℚ-curves with full level 5 structure will not have genus 0, rendering a 3-5 switch impossible.

We are left with two methods of treating the residually reducible cases. One method is to generalize the lifting theorems of Wiles, Taylor Wiles, et. al. to the residually reducible situation.

Several theorems in this direction have been proven by Andrew Wiles and the second author [21],[22] in the case where the reduction of E is ordinary or multiplicative. We will apply those theorems to the present situation in Theorem 5.1 below.

Another method is to exploit the fact that, in contrast with the case of elliptic curves over ℚ, there are often cohomological obstructions to the reducibility of ρ̄_{E,ℓ}. These obstructions can be computed explicitly in terms of the invariants described in section 2. We begin with a general fact about reducible projective mod ℓ Galois representations.

Proposition 4.1. Let ℓ be an odd prime, and let

ℙρ̄: Gal(ℚ̄/ℚ) → PGL_{2}(𝔽_{ℓ})

be a projective mod ℓ Galois representation. Let χ : Gal(ℚ̄/ℚ) → ±1 be a quadratic Dirichlet character (possibly trivial). Let G be the subgroup of matrices in 𝔽̄_{ℓ}^{∗} GL_{2}(𝔽_{ℓ}) having determinant 1, and let γ ∈ H^{2}(PGL_{2}(𝔽_{ℓ}), ±1) be the class of the extension

1 → ±1 → G → PGL_{2}(𝔽_{ℓ}) → 1.

Let ψ̄ = det ℙρ̄. Finally, suppose that either

(a) the image of ℙρ̄ lies in the normalizer N of a Cartan subgroup C of PGL_{2}(𝔽_{ℓ}), and the quadratic character Gal(ℚ̄/ℚ) → N/C is equal to χ, or

(b) the image of ℙρ̄ lies in a Borel subgroup of PGL_{2}(𝔽_{ℓ}), and χ is trivial.

Then either (ψ̄, χψ̄) or ℙρ̄^{∗}γ(ψ̄, χψ̄)(χ, χ) is the trivial class in H^{2}(Gal(ℚ̄/ℚ), ±1).

Proof. First, suppose the image of ℙρ̄ lies in the normalizer N of a Cartan subgroup C of PGL_{2}(𝔽_{ℓ}). Write N̄ for the group N/C^{2}. Let π be the natural projection of N onto N̄. Then N̄ ≅ (ℤ/2ℤ)^{⊕2}; a choice of isomorphism can be fixed by requiring that the first copy of ℤ/2ℤ be π(C) and the second be the kernel of det : N̄ → 𝔽_{ℓ}^{∗}/(𝔽_{ℓ}^{∗})^{2}. We then have

π ∘ ℙρ̄ = ψ̄ ⊕ χ : Gal(ℚ̄/ℚ) → (ℤ/2ℤ)^{⊕2}.  (4.12)

We consider two cases.

Case 1: |C^{2}| is even. Then π factors as

N → N̂ → N̄,

where N̂ is a dihedral group of order 8 whose cyclic subgroup of order 4 is the preimage of π(C). So π ∘ ℙρ̄ lifts to a homomorphism from Gal(ℚ̄/ℚ) to N̂, which means that d(π ∘ ℙρ̄) vanishes in the cohomology sequence

H^{1}(Gal(ℚ̄/ℚ), N̂) → H^{1}(Gal(ℚ̄/ℚ), N̄) →^{d} H^{2}(Gal(ℚ̄/ℚ), ±1).

The isomorphism N̄ ≅ (ℤ/2ℤ)^{⊕2} then tells us that d′(ψ̄ ⊕ χ) vanishes in

H^{1}(Gal(ℚ̄/ℚ), D_{4}) → H^{1}(Gal(ℚ̄/ℚ), (ℤ/2ℤ)^{⊕2}) →^{d′} H^{2}(Gal(ℚ̄/ℚ), ±1),

where D_{4} is a dihedral group of order 8 whose cyclic subgroup of order 4 is the preimage of the first copy of (ℤ/2ℤ). It is well known that, for any two characters χ_{1}, χ_{2}, we have d′(χ_{1} ⊕ χ_{2}) = (χ_{1}, χ_{1}χ_{2}) [10, Prop. 3.10]. So (ψ̄, χψ̄) = 0, as desired.

Case 2: |C^{2}| is odd.

In this case, the inflation map

π^{∗} : H^{2}(N̄, ±1) → H^{2}(N, ±1)  (4.13)

is an isomorphism. The subgroup of N generated by an involution in C and any element of N\C is isomorphic to (ℤ/2ℤ)^{⊕2}; in fact, any such subgroup is the image of an injection s : N̄ → N such that π ∘ s is the identity. Write ι for the inclusion of N in PGL_{2}(𝔽_{ℓ}). Let M be the subgroup of G lying over s(N̄). Then s^{∗}ι^{∗}γ is the class c ∈ H^{2}(N̄, ±1) corresponding to the extension

1 → ±1 → M → N̄ → 1.

It follows from the fact that a non-scalar element of G whose square is a scalar has exact order 4 that M is the quaternion group of order 8. Now, from (4.12) and [10, Th. 3.11], one gets

ℙρ^{∗}π^{∗}c = (ψ̄ ⊕ χ)^{∗}c = (ψ̄, χψ̄)(χ, χ),

The isomorphism (4.13) implies that π^{∗}s^{∗} acts as the identity on H^{2}(N, ±1). In particular, we have π^{∗}c = ι^{∗}γ. Pulling back both of these by ℙρ (or, more precisely, by the homomorphism f : Gal(ℚ̄/ℚ) → N such that ι ∘ f = ℙρ) one obtains the equality

ℙρ^{∗}γ = ℙρ^{∗}π^{∗}c.

which yields the desired result.

The only case remaining is that where the image of ℙρ lies in a Borel subgroup but not necessarily in the normalizer of a Cartan. In this case, the semisimplification of ℙρ̄ has image lying in a split Cartan subgroup, and we are in the case already discussed.

We now apply Proposition 4.1 to the case of mod ℓ representations attached to ℚ-curves.

Proposition 4.2. Let E/K be a ℚ-curve and ℓ an odd prime. Let χ : Gal(ℚ̄/ℚ) → ±1 be a quadratic Dirichlet character (possibly trivial). Let q_{ℓ,∞} ∈ H^{2}(Gal(ℚ̄/ℚ), ±1) be the Brauer class of the quaternion algebra ramified only at ℓ and ∞. Suppose that either

(i) the image of ℙρ̄_{E,ℓ} lies in the normalizer N of a Cartan subgroup C of PGL_{2}(𝔽_{ℓ}), and the quadratic character Gal(ℚ̄/ℚ) → N/C is equal to χ, or

(ii) the image of ℙρ̄_{E,ℓ} lies in a Borel subgroup of PGL_{2}(𝔽_{ℓ}), and χ is trivial.

Then either (ψ̄_{E,ℓ}, χψ̄_{E,ℓ}) or b_{E}q_{ℓ,∞}(ψ̄_{E,ℓ}, χψ̄_{E,ℓ})(χ, χ) is the trivial class in H^{2}(Gal(ℚ̄/ℚ), ±1).

Proof. The proposition is an immediate corollary of Proposition 4.1. The only thing to check is that

ℙρ̄_{E,ℓ}^{∗}γ = b_{E}q_{ℓ,∞}.

Let G be as in the statement of Proposition 4.1. For each σ ∈ Gal(ℚ̄/ℚ) let d_{σ} ∈ 𝔽̄_{ℓ}^{∗} be a square root of det(ρ̄_{E,ℓ}(σ)). Then g_{σ} = d_{σ}^{−1}ρ̄_{E,ℓ} is a set-theoretic lift of ℙρ̄_{E,ℓ} to G. To this lift one associates a 2-cocycle c given by the rule

c(σ, τ) = g_{σ}g_{τ} g_{στ}^{−1} = d_{σ}^{−1}d_{τ}^{−1}d_{στ} .

But this is just a 2-cocycle representing the class δ̄(det ρ̄_{E,ℓ}), where δ̄ is defined as in Proposition 2.15. From that proposition and from the fact that δ̄(χ̄_{ℓ}) = q_{ℓ,∞}, one has

δ̄(det ρ̄_{E,ℓ}) = b_{E}δ̄(χ̄_{ℓ}) = b_{E}q_{ℓ,∞}.

The desired result follows.

Proposition 4.2 guarantees in many cases that the 3-adic representation attached to a ℚ-curve is residually absolutely irreducible, even when restricted to a quadratic field.

5 The main theorems

We are now ready to state and prove the main results of the paper. Recall that (b_{E})_{3} denotes the restriction of b_{E} to H^{2}(G_{3}, ±1).

Theorem 5.1. Suppose E/K is a ℚ-curve with potentially ordinary or multiplicative reduction at some (whence every) prime of K over 3, and such that (b_{E})_{3} is trivial. Then E is modular.

Proof. First, suppose that ρ̄_{E,3} is absolutely reducible when restricted to Gal(ℚ̄/ℚ[\sqrt{−3}]). For this case we appeal to the main theorems of [21] and [22]. In order for these theorems to apply we need only verify the following properties of the representation ρ_{E,3}:

(i) ρ_{E,3} is continuous, irreducible, and odd;

(ii) det ρ_{E,3}(Frob_{ℓ}) = ψ(ℓ)ℓ^{k−1} for some finite character ψ, some integer k ≥ 2, and almost all primes ℓ;

(iii) ρ_{E,3}|_{G_{3}} ≅ \left[{ϕ_{1} ∗ \atop ϕ_{2}}\right] with ϕ_{2}|_{I_{3}} finite;

(iv) the reductions ϕ̄_{1} and ϕ̄_{2} are distinct;

(v) ρ̄_{E,3} is modular (in the sense of Lemma 3.2) if it is absolutely irreducible.

Properties (i) and (ii) follow from Proposition 2.3 and (2.6). (Here we have again used that E does not have complex multiplication, this time to ensure that ϕ_{E,3}, and hence ρ_{E,3}, is irreducible.) We next prove that property (iii) holds.

From the possibilities for the reduction type of E it follows that the restriction of ϕ_{E,3} to a decomposition group G_{v} at a prime v|3 of K satisfies

ϕ_{E,3}|_{G_{v}} ≅ \left[{θ_{1} ∗ \atop θ_{2}}\right]

with θ_{2} having finite order on inertia. We claim that the same is true of ρ_{E,3}|_{G_{3}}. Suppose otherwise. From the fact that ρ_{E,3}|_{Gal(K̄/K)} is isomorphic to a twist of ϕ_{E,3} it follows that there is a quadratic extension, say L, of ℚ_{3} such that the restriction of ρ_{E,3} to Gal(L̄/L) is the direct sum of two characters that are interchanged by the action of Gal(L/ℚ_{3}). Since the product of these two characters, being the restriction of det ρ_{E,3}, is infinitely ramified, so must be one, and hence both, of these characters. But this contradicts the above description of ϕ_{E,3}|_{G_{v}}. Write

ρ_{E,3}|_{G_{3}} ≅ \left[{ϕ_{1} ∗ \atop ϕ_{2}}\right] .

We next prove that the reductions ϕ̄_{1} and ϕ̄_{2} are distinct on G_{3}; in other words, that ρ_{E,3} has property (iv). To see this we note that if ϕ̄_{1} and ϕ̄_{2} were not distinct on G_{3} then det ρ̄_{E,3}|_{G_{3}} would be a square. Suppose this were so. Then from det ρ̄_{E,3} = ϵ̄_{3}χ̄_{3} (see (2.6)) we conclude that ϵ̄_{3}|_{G_{3}} = ϕ^{2}χ̄_{3}|_{G_{3}} for some character ϕ of G_{3}. It then follows from Proposition 2.15 that the restriction of b_{E} to G_{3} equals the restriction of δ̄(χ̄_{3}) to G_{3}. But the latter is non-trivial, hence so is the former, contradicting hypothesis (ii) of the theorem.

It remains to prove that property (v) holds. If ρ̄_{E,3} is absolutely irreducible, then it must be dihedral and in fact induced from a character of Gal(ℚ̄/ℚ(\sqrt{−3})) since we are assuming that ρ̄_{E,3} is absolutely reducible on Gal(ℚ̄/ℚ(\sqrt{−3})). It is a classical result that such representations are modular.

We have shown that ρ_{E,3} has properties (i)-(v) listed above. As mentioned before, the theorem follows.

Now, suppose that ρ̄_{E,3} is absolutely irreducible when restricted to Gal(ℚ̄/ℚ[\sqrt{−3}]). By the argument above,

ρ_{E,3}|_{G_{3}} ≅ \left[{ϕ_{1} ∗ \atop ϕ_{2}}\right] ,

where ϕ_{2} has finite image on inertia and ϕ_{1}|_{I_{3}} = ηχ_{3}|_{I_{3}}, with η a finite-order character. After twisting ρ_{E,3} by a finite-order character of G_{ℚ}, we may assume ϕ_{2} is unramified. We have already shown above that ϕ̄_{1} ≠ ϕ̄_{2}. Finally, ρ̄_{E,3} is modular by Lemma 3.2 (which does not use the assumption of supersingular reduction in Theorem 3.1.) It now follows from Theorem 5.3 of [6] that ρ_{E,3}, whence E, is modular.

Theorem 5.2. Suppose E/K is a ℚ-curve such that, for some (whence every) prime ℓ > 3, the projective representation ℙρ_{E,ℓ} associated to ρ_{E,ℓ} is unramified at 3. Then E is modular.

Proof. If E has potentially ordinary or multiplicative reduction, the modularity follows from Theorem 5.1. We therefore assume that the reduction of E is potentially supersingular.

We have that ρ_{E,ℓ}|I_{3} is a character θ. So

θ^{2} = det ρ_{E,ℓ}|I_{3} = ϵ_{E}|I_{3}.

Choose α such that ϵ_{E} has 2-power order; then ϵ_{E}, whence also ρ_{E,ℓ}, is tamely ramified. We may choose K to be a compositum of quadratic fields [13, Cor. 2.5], in which case it follows that E obtains good reduction over a tamely ramified extension of ℚ_{3}.

Let τ be a topological generator of tame inertia, and let m = ρ_{E,ℓ}(τ); then m is a scalar which is conjugate to its cube, so m = ±1. In either case,

det m = ϵ_{E}(τ) = 1,

so ϵ_{E} is unramified at 3, and (b_{E})_{3} = δ(ϵ_{E}|G_{3}) is trivial.

From Proposition 2.10, the action of τ on T_{ℓ}A_{α} is either 1 or −1. Thus, after modifying α by a quadratic character, we may assume that A_{α} has good supersingular reduction at 3.

Therefore, A_{α}[3] extends to a finite flat group scheme over R = W (𝔽̄_{3}), to which we can apply Raynaud’s classification [14]. Let F be the fraction field of R. Let H be a Jordan-Hölder quotient of A_{α}[3]_{F}. Then we have from [14, Cor. 3.4.4] that the action of τ on H(F̄) has eigenvalues

ψ_{m}(τ)^{n3^{i}}(i = 0, …, m − 1)

where ψ_{m} is a fundamental tame character of I_{3}, and n is an integer whose base-3 expansion contains only 0’s and 1’s. In particular, τ^{4} acts trivially on H(F̄) if and only if τ^{2} acts trivially.

Suppose τ^{2} acts trivially on H(F̄). Then H(F̄) is a 1-dimensional 𝔽_{3}-vector space, and H is isomorphic to either (ℤ/3ℤ)_{K} or (μ_{3})_{K}. It then follows from [14, Cor. 3.3.6] that A_{α}[3]/R has either ℤ/3ℤ or μ_{3} as a subquotient, which contradicts the supersingularity of A_{α}.

We may therefore suppose that τ^{4} acts non-trivially on the F̄-points of every subquotient of A_{α}[3]. In particular, ρ̄_{E,3}(τ^{4}) does not have 1 as an eigenvalue.

Suppose the restriction of ρ̄_{E,3} to Gal(ℚ̄/ℚ[\sqrt{−3}]) is absolutely reducible. As in § 3, let L be the ramified quadratic extension of ℚ_{3}. Then

(ρ̄_{E,3}|I_{L})^{ss} ≅ ϕ_{1} ⊕ ϕ_{2}

for some characters ϕ_{1}, ϕ_{2} : I_{L}^{t} → 𝔽̄_{3}^{∗}. Since (ρ̄_{E,3}|I_{L}) extends to a representation of G_{3}, we have that \{ϕ_{1}, ϕ_{2}\} = \{ϕ_{1}^{3}, ϕ_{2}^{3}\}. The fact that ρ̄_{E,3}|ℚ[\sqrt{−3}] is absolutely reducible means that in fact ϕ_{i}^{3} = ϕ_{i} for i = 1, 2; in other words, ϕ_{1} and ϕ_{2} are quadratic characters. In particular, ρ̄_{E,3}(τ^{4}) is unipotent, which is a contradiction.

To sum up: we have shown that under the hypotheses of the theorem, we know that

• E obtains good supersingular reduction over a tame extension of ℚ_{3};

• (b_{E})_{3} = 1; and

• ρ̄_{E,3}| Gal(ℚ̄/ℚ[\sqrt{−3}]) is absolutely irreducible.

It now follows from Theorem 3.1 that E is modular.

Theorem 5.3. Suppose E/K is a ℚ-curve which acquires semistable reduction over a field tamely ramified over ℚ_{3}. Suppose further that (b_{E})_{3} is trivial, and that the four classes (ψ̄_{E,3}, −1), b_{E}q_{3,∞}(ψ̄_{E,3}, −1), q_{3,∞}(ψ̄_{E,3}, 3), and b_{E}(ψ̄_{E,3}, 3) are all nontrivial in H^{2}(Gal(ℚ̄/ℚ), ±1). Finally, suppose that deg μ_{σ} can be chosen to be prime to 3 for all σ ∈ Gal(ℚ̄/ℚ). Then E is modular.

Proof. We may assume that the reduction of E over 3 is potentially supersingular; otherwise, E is modular by Theorem 5.1.

It follows from Proposition 4.2 that the restriction of ρ̄_{E,3} to Gal(ℚ̄/ℚ[\sqrt{−3}]) is absolutely irreducible. It then follows from Theorem 3.1 that E is modular.

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