Modularity of Q-curves: Galois Representations and Elliptic Curves, Papers of Quantitative Techniques

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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/ Q¯ is an elliptic curve that is also a quotient of J 1 (N )/ Q¯. Then E is a Q-curve over some number field.

A Q-curve which is a quotient of J 1 (N )/ Q¯ 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 Eσ^ is a Galois conjugate of E. Then there exists a non-zero K-isogeny μ : Eσ^ → E, and so if p is a prime dividing the square-free part of the degree of μ then the Gal( K/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( Q¯/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 3- adic 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 = EA,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 = Cp. (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( Q¯/Q) let μσ : Eσ^ → E be a non-zero isogeny. Then we define bE ∈ H^2 (Gal( Q¯/Q), ±1) by

bE (σ, τ ) = sgn(μσ μστ μ− στ^1 ).

Denote by (bE ) 3 the restriction of bE to H^2 (Gal( Q¯ 3 /Q 3 ), ±1). Furthermore, we associate to E an -adic Galois representation ρE, of Gal( Q¯/Q) and a quadratic character ψ¯E, 3 of Gal( Q¯/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 (bE ) 3 is trivial. Then E is modular.

Theorem. Suppose E/K is a Q-curve such that, for some (whence every) prime > 3 , the projec- tive representation PρE, associated to ρE,` is unramified at 3. Then E is modular.

We can weaken the condition on ρE,` in the second theorem above, at the expense of introducing some technical conditions. Denote by q 3 ,∞ the unique class in H^2 (Gal( Q¯/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 tame- ly ramified over Q 3. Suppose further that (bE ) 3 is trivial, and that the four classes ( ψ¯E, 3 , −1), bE q 3 ,∞( ψ¯E, 3 , −1), q 3 ,∞( ψ¯E, 3 , 3), and bE ( ψ¯E, 3 , 3) are all nontrivial in H^2 (Gal( Q¯/Q), ±1). Finally, suppose that deg μσ can be chosen to be prime to 3 for all σ ∈ Gal( Q¯/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

√

√

73 − 414461880

√

5257 − 4973542560

√

73 − 1089620282520)x

√

√

73 + 5349307626327168

√

5257 (1.3)

+

√

73 + 32065347994985088

which is the specialization to a = 2^2 · 32 · 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;
• PρE,` is ramified at 3 (because E/Q(

√

−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 ρE, 3 is residually irreducible, but the image of the restriction ρE, 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 ρE,`’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 ρE, 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 χK for the quadratic character of Gal( Q¯/Q) associ- ated to K. Any two quadratic characters χ and χ′^ of Gal( Q¯/Q) give classes in H^1 (Gal( Q¯/Q), ±1). We write (χ, χ′) for their cup product. This is an element in H^2 (Gal( Q¯/Q), ±1). If d is an element of Q∗/(Q∗)^2 , we write (χ, d) to mean the cup product (χ, χQ(√d)).

We write elements in H^2 (Gal( Q¯/Q), ±1) multiplicatively. Thus if c 1 , c 2 ∈ H^2 (Gal( Q¯/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( Q¯/Q).

We take an embedding ν : Q¯ ↪→ Q¯to be fixed for each, and denote the resulting decomposition subgroup (resp. inertia subgroup) of Gal( Q¯/Q) by G(resp. I). To be completely precise, we need to define two such embeddings: one in order to define decomposition subgroups of Gal( Q¯/Q), and the other in order to make sense of the scalar action of Q¯ on -adic vector spaces like TA ⊗ZQ^ ¯. Write ι : Q¯ ↪→ Q¯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 Q¯ ↪→ C to be fixed. This determines a complex conjugation c. We denote by Gt, I`t , Iellw^ the tame quotient of the decomposition group, the tame inertia group, and the wild inertia group respectively.

For a rational prime, we denote by χ : Gal( Q¯/Q) → Z∗ the cyclotomic character, and by χ¯ : Gal( Q¯/Q) → F∗ the mod 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 -adic and mod Galois representations attached to a Q-curve. We also define Galois cohomology classes cE , bE and ψ¯E,` 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 bE.

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( Q¯/Q), there exists a non-zero K-isogeny

μσ : Eσ^ → E.

We may, and do, suppose that μσ is the identity morphism for all σ ∈ Gal( Q¯/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 ` be a rational prime, and define

φE,: Gal( K/K¯ ) → GL 2 (Z)

to be the representation of Gal( K/K¯ ) on the -adic Tate module TE of E. (We have fixed an isomorphism TE ∼= Z^2 .) In the following proposition we describe an extension of φE,` to a repre- sentation of the whole group Gal( Q¯/Q).

Proposition 2.3. There exists a representation

ρE,: GQ → Q¯∗ GL 2 (Q`)

such that PρE,|Gal( K/K¯ ) ∼= PφE,. 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( Q¯/Q). Following [16, §6], we define

cE (σ, τ ) = μσ μστ μ− στ^1 ∈ (Hom(E, E) ⊗Z Q)∗^ = Q∗.

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

cE (g, h) = α(g)α(h)α(gh)−^1 (2.4)

We can now define an action of Gal( Q¯/Q) on Q¯⊗Z T`E by

ρE,`(g)(1 ⊗ x) = α−^1 (g) ⊗ μg (xg^ ). (2.5)

It is clear from the above definition that PρE,|Gal( K/K¯ ) ∼= PφE,. In particular, ρE,|Gal( K/K¯ ) and φE, differ by the continuous character α|Gal( K/K¯ ). It follows that ρE,is continous and unramified away from finitely many primes. It remains to show that ρE, is odd; that is, that det ρE,`(c) = −1, where c is our fixed complex conjugation.

Define a map E : Gal( Q¯/Q) → Q¯∗^ by

E (σ) = α^2 (σ)/(deg μσ ),

and let E,be the composition of E with the chosen embedding Q¯ ↪→ Q¯. That this map is a character follows from the observation that

cE (σ, τ )^2 =

(deg μσ )(deg μτ ) deg μστ

.

It also follows immediately from (2.5) that

det ρE,= − E,^1 χ`. (2.6)

Write μ for μc. 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 z Λ¯ ⊂ Λ. The composition μμc is then given by zzc, a positive real number. Since the degree of μμc^ is (deg μ)^2 , we conclude that

μμc^ = deg μ.

Therefore,

E (c) = α^2 (c)/ deg μ = cE (c, c)/ deg μ = μμc/ deg μ = 1,

and

det ρE,(c) = E,(c)χ`(c) = − 1.

Since α^2 (c) = cE (c, c) = μμc^ = deg μ, the proposition follows from the definition of E,` and (2.6).

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

ρˆE,: Gal( Q¯/Q) → Q¯∗^ GL 2 (Q)

defined by (2.5). More precisely: suppose M is a number field such that ˆρE,takes values in M ∗^ GL 2 (Q). Let λ be the prime of M defined by ι : Q¯ → Q¯, and let λ = λ 1 ,... , λr be the set of all primes of M dividing. Write ρE,λi for the composition of ˆρE,` with the map

M ∗^ GL 2 (Q) → M (^) λ∗i GL 2 (Q).

So ρE,λ is just another name for ρE,`.

Remark 2.5. While ρE,and E depend on our choice of α, the projective representation PρE, depends only on the isomorphism class of E/K. Moreover, PρE,` is independent of the choice of ι.

Remark 2.6. We can choose α in such a way that the image of E has 2-power order, by the following argument. Let n = 2am be the order of the image of E , where m is odd. If m 6 = 1, replace α by

α (m−1)/ 2 E ; this has the effect of replacing^ E^ by^  m E , whose image has 2-power order. The reason for introducing the representations ρE,` 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 such that ρE, ∼= ρf,`.

Here, f is a holomorphic Hecke eigenform on the complex upper-half plane and ρf,is the Galois representation ρf, : Gal( Q¯/Q) → GL 2 ( Q¯`) such that if f (z) =

∑∞

n=1 a(n)e(nz) (a(1) = 1), then trace ρf,`(Frobp) = a(p) for almost all primes p.

Proof. Suppose ρE,∼= ρf, for some eigenform f of level N. Then there exists some finite extension L/K such that

φE,|Gal( L/L¯ ) ∼= ρf,|Gal( ¯L/L) (2.7)

and the weight of f must be two, as can be seen by comparing determinants. Let ρN,be the representation of Gal( Q¯/Q) on TJ 1 (N ) ⊗ZQ^ ¯, where TJ 1 (N ) is the-Tate module of J 1 (N ). We have

ρN,' ⊕ρg, (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 φE,is a Gal( L/L¯ )-quotient of ρN,. It then follows that HomL(TJ 1 (N ), TE) is non-zero. By a theorem of Faltings [7] we can conclude from this that HomL(J 1 (N ), E) is non-zero.

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

Hom(Gal( Q¯/Q), Z¯∗) → Hom(Gal( Q¯/Q), Z¯∗) δ → H^2 (Gal( Q¯/Q), ±1) (2.9)

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

0 → ± 1 → Z¯∗^ → Z¯∗^ → 0.

Proposition 2.8. bE = δ(E ).

Proof. Let χ : Gal( Q¯/Q) → Z¯∗^ be a character, and for each σ ∈ Gal( Q¯/Q) let ˜χ(σ) be a square root of χ(σ). Then δ(χ) is defined by

δ(χ)(σ, τ ) = χ˜(σ) ˜χ(τ ) χ˜(στ )

.

To compute δ(E ), we may choose

˜E (σ) = α(σ)/

√

deg μσ

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

δ(E )(σ, τ ) =

α(σ)α(τ ) α(στ )

√

deg μστ √ deg μσ

√

deg μτ

= cE (σ, τ )/

√

c^2 E (σ, τ ) = bE (σ, τ ).

Remark 2.9. Note that the class cE 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 cE 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 ρE,can also be viewed as the-adic representation attached to a certain abelian variety over Q.

Proposition 2.10. Let E/K be a Q-curve, and let α : Gal( Q¯/Q) → Q¯∗^ be a 1 -cochain with coboundary cE , as in (2.4). Define ρE,` as in (2.5). Let M be the number field generated by the α(g) for all g ∈ Gal( Q¯/Q). There exists an abelian variety Aα/Q satisfying the following conditions.

• There exists an injection M ↪→ End(Aα/Q) ⊗Z Q;
• Let λ 1 ,... , λr be the primes of M lying over . Then the rational Tate module VAα decom- poses as

V`Aα =

⊕

i

Vλi Aα

and Vλi Aα is isomorphic, as Mλi [Gal( Q¯/Q)]-module, to ρE,λi. In particular, VλAα ∼= ρE,`.

Proof. The desired Aα 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 λσ for each σ ∈ Gal(K/Q), with the multiplication table

λστ cE (σ, τ ) = λσ λτ.

Then R acts on the abelian variety

ResK Q E ×Q K ∼=

⊕

σ∈Gal(K/Q)

Eσ

by the rule

λσ (P ) = μτσ (P ) (2.10)

for any P ∈ Eτ σ^ ( K¯). This action descends to an action of R on ResK Q E. Our choice of α defines a homomorphism ω : R → M. Now define

Aα = ResK Q 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 Aα is the image of mπ, where m is an integer large enough to make mπ an actual endomorphism (not only a rational endomorphism) of ResK Q E. Then Aα admits the desired injection M ↪→ End(Aα)⊗Z Q, and the rational λi-adic Tate module Vλi Aα is a 2-dimensional vector space over Mλi (see [15, Th. 2.1.1]); one then has from (2.10) that Gal( Q¯/Q) acts on Vλi Aα via ρE,λi.

Remark 2.11. We emphasize that the construction of Aα is independent of `.

Proposition 2.12. Let > 2. Suppose does not divide deg μg for any g ∈ Gal( Q¯/Q), and suppose α is chosen so that E has 2 -power order (Remark 2.6.) Let λ = λ 1 ,... , λi be the set of primes of M dividing . Then the full ring of integers of M ⊗Z Z ∼= ⊕iMλi acts on TAα, and the-divisible group TAα breaks up as a direct sum of-divisible groups ⊕

i

Tλi Aα (2.11)

where the λi range over the primes of M dividing `. Moreover, Tλi Aα is a free OMλi -module of rank 2.

Proof. Let Z[α] be the ring generated by the α(g). Then it follows by the definition of Aα that Z⊗Z Z[α] acts on TAα. Since deg μg is an `-adic unit, and since α(g)/

√

deg μg is a 2-power root of unity, we can obtain Z⊗Z Z[α] by successively adjoining square roots of-adic units to Z; it follows that Z ⊗Z Z[α] is ´etale over Z`, and therefore

Z` ⊗Z Z[α] ∼=

⊕

i

OMλi.

The decomposition of T`Aα now follows immediately from the decomposition (2.11).

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

Proposition 2.13. Let L be a totally ramified extension of Qand F an unramified extension of L. Let ρ be a continuous representation of Gal( Q¯/Q) (or any compact group) with image in F ∗^ GL 2 (Q). Then ρ is conjugate in GL 2 (F ) to a representation with image in O F∗ GL 2 (OL).

Proof. Let S, T be a basis for F ⊕^2 with respect to which the image of ρ lies in F ∗^ GL 2 (Q`), and let L 0 be the lattice OF S + OF T generated by S, T. There are only finitely many images of L 0 under the action of the compact group Gal( Q¯/Q). Each such image Li is of the form

x(OF (aS + bT ) + OF (cS + dT ))

with x ∈ F ∗^ and a, b, c, d ∈ Q`. Let L be the lattice generated by all the Li; then L is preserved by the action of ρ(Gal( Q¯/Q)). Because F/L is unramified, we may write x = yu, with y ∈ L∗^ and u ∈ O F∗. So each Li can be rewritten as

y(OF (aS + bT ) + OF (cS + dT )).

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

L′ i = y(OL(aS + bT ) + OL(cS + dT )),

and let

L′^ = OL(αS + βT ) + OL(γS + δT ),

with α, β, γ, δ ∈ L, be the lattice generated by all the L′ i. Then L = L′^ ⊗OL OF. Let S′^ = αS + βT and T ′^ = γS + δT , and write

ρ : Gal( Q¯/Q) → GL 2 (F )

with respect to the basis elements S′^ and T ′. Since ρ(Gal( Q¯/Q)) preserves the lattice OF S′^ + OF T ′, we have ρ(Gal( Q¯/Q)) ∈ GL 2 (OF ). Since S′, T ′^ lie in LS + LT , we have ρ(Gal( Q¯/Q)) ∈ F ∗GL 2 (L). Combining these two facts yields the desired result.

The representation ρE,produced in Proposition 2.3 takes values in M (^) λ∗ GL 2 (Q), where Mλ is the extension of Qλ generated by the values of ι(α(σ)) for all σ ∈ Gal( Q¯/Q). Recall from the proof of Proposition 2.3 that E,(σ) = α^2 (σ)/ deg μσ is a Dirichlet character. So Mλ is contained in an extension generated by square roots and roots of unity; it is thus an abelian extension of Q. It follows from local class field theory that there exists an abelian extension F of Qcontaining M and such that F also contains a subextension L totally ramified over Q 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 ρE,` takes images in O∗ F GL 2 (OL).

Definition 2.14. We denote by

ρ¯E,: Gal( Q¯/Q) → ¯F∗ GL 2 (F`).

the representation obtained by choosing a basis of F ⊕^2 as above and reducing the resulting repre- sentation

ρE,` : Gal( Q¯/Q) → O∗ F GL 2 (OL)

modulo the maximal ideal mF of OF. The reduced representation ¯ρE,is then well defined up to semisimplification and conjugation by GL 2 (F¯).

We observe that

det ¯ρE,= ¯E, χ¯`

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

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

Proposition 2.15. bE = δ¯(¯E,`).

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 ` be an odd prime. Then we define a quadratic Dirichlet character

ψ¯E,= det Pρ¯E, : Gal( Q¯/Q) → F∗ /(F∗ )^2 ∼= ± 1.

The character ψ¯E,`, like the cohomology class bE , depends only on the isomorphism class of E/K.

Remark 2.17. The invariants bE and ψ¯E,` 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 μμτ^ is multiplication by n. Then

bE =

{

0 if n positive χK if n negative.

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

ψ¯E,` =

{

η n ∈ (Q∗ )^2 ηχK n /∈ (Q∗ )^2.

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 Kv for one (whence every) prime v of K above 3 ;
• bE ∈ H^2 (Gal( Q¯/Q), ±1) has trivial projection to H^2 (G 3 , ±1);
• Either PρE,|G 3 is unramified for some (whence every) 6 = 3, or deg μσ is not a multiple of 3 for any σ ∈ Gal( Q¯/Q);
• The restriction of ρ¯E, 3 to Gal( Q¯/Q[

√

−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 Dia- mond [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 ρE, 3 is an odd, continuous representation unramified away from finitely many primes. Write Gv , Iv for the absolute Galois group and the inertia group of Kv. Write Iw 3 for the subgroup of wild inertia in I 3.

Lemma 3.2. ρ¯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 ¯ρE, 3 lies in ¯F∗ 3 GL 2 (F 3 ). We suppose without loss of generality that the chosen extension of the 3-adic valuation of Q to Q[

√

−2] is given by the prime (1 +

√

−2).

We can define a homomorphism

ι : F¯∗ 3 GL 2 (F 3 ) → μ∞ GL 2 (Z[

√

−2]),

where μ∞ denotes the group of roots of unity, as follows: set

ι

[

− 1 1

− 1 0

]

=

[

− 1 1

− 1 0

]

,

ι

[

1 − 1

]

=

[

1 − 1

−

√

− 2 −1 +

√

− 2

]

,

and, for each scalar a ∈ F¯∗ 3 , define ι(a) to be the preimage, under the chosen embedding Q¯ ↪→ Q¯ 3 , of the Teichm¨uller lift of a. Let F be a number field such that the image of ι ◦ ρ¯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 ◦ ρ¯E, 3 with reduction mod w is ¯ρE, 3. The composition ι◦ ρ¯E, 3 is a continuous complex representation of Gal( Q¯/Q), odd and irreducible because ¯ρE, 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=

bnqn

such that

bp = Tr(ι ◦ ρ¯E, 3 (Frobp))

for almost all p. Let F ′^ be a number field containing all the bn. 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 Tn(gE) is congruent mod w to bngE, 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=

anqn

of weight 2 with an ∈ F ′^ and an ≡ bn mod w for all n. In particular, ap = Tr(¯ρE, 3 (Frobp)) for almost all p. So ¯ρE, 3 is the mod w′^ representation associated to f.

We will show that ρE, 3 satisfies the conditions of Theorem 7.1.1 of [2]. Recall that, for any `,

φE,: Gal( K/K¯ ) → GL 2 (Z)

is the Galois representation attached to E/K as elliptic curve, and that PφE,and PρE, are iso- morphic projective representations of Gal( K/K¯ ), by Proposition 2.3. The representation ρE,produced by Proposition 2.3 depends on a choice of α : Gal( Q¯/Q) → Q¯∗, a cochain whose coboundary is cE. We begin by observing that α can be chosen so as to impart to ρE, some useful arithmetic properties.

Lemma 3.3. There exists a choice of α : GQ → Q¯∗^ such that

• for all 6 = 3, the representation ρE,|G 3 is tamely ramified;
• for all , det ρE,|G 3 = χ`|G 3 ;
• E has 2 -power order.

Proof. Since K is tamely ramified, cE |G 3 is the inflation of an element of

H^2 (Gt 3 , Q∗).

The cohomology group

H^2 (Gt 3 , Q¯∗)

is trival, as can be seen by placing Gt 3 in the exact sequence

0 → I 3 t → Gt 3 → 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 → Q¯∗^ such that

cE (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( Q¯/Q) → Q^ ¯∗^ be any cochain whose coboundary is cE. Then (α′|G 3 )α− 3 1 is a character θ 3 of G 3. Let θ be a character of Gal( Q¯/Q) whose restriction to G 3 is θ 3. Then define

α = α′θ−^1.

So the coboundary of α is cE , 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 φE,is tamely ramified at 3 for all 6 = 3. It follows from the definition (2.5) that ρE,|G 3 is tamely ramified for all 6 = 3. By Proposition 2.8 and (2.6), the assumption that bE |G 3 is trivial means that

E |G 3 = χ^2

for some character χ : G 3 → Q¯∗. The character E,is tamely ramified for any 6 = 3, because ρE,`|G 3 is tamely ramified; it follows that E , whence also χ, is tamely ramified. Replacing α by αχ now yields the first two desired conditions. In particular, E |G 3 is trivial. So we can modify α by any power of E without affecting the first two conditions. Now we can force  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 ρE,|G 3 satisfies the conditions in Lemma 3.3. We now take as fixed some > 3. From Proposition 2.10, we have an abelian variety Aα/Q such that

ρE,` ∼= VλAα

where λ|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 GL ⊂ G 3 the absolute Galois group of L, and by IL the inertia subgroup of GL. Let ψ be the ramified quadratic character of GL, and write Aψα /L for the twist of Aα ×Q L by ψ. From this point on, we take as fixed some > 3.

Lemma 3.4. Either

• ρE,|GL and P ρE,|G 3 are unramified, and Aα/L has good reduction; or
• ρE,`|GL ⊗ ψ is unramified, and Aψα /L has good reduction.

Proof. By Lemma 3.3, wild inertia is killed by ρE,. Let τ be a topological generator of I 3 t, and define m = ρE,(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 Q¯∗ GL 2 (Q). Since det(m) = χ(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 ρE,|GL is unramified. In the latter case, (ρE,|GL) ⊗ ψ is unramified. Suppose the eigenvalues of m are (1, 1) or (− 1 , −1); equivalently, PρE,|G 3 is unramified. Recall from Remark 2.5 that PρE,does not depend on the choice of ι. So ρE,λi (τ ) is scalar for any i, whence IL acts trivially on Vλi Aα for any prime λi of M dividing. It then follows from Proposition 2. that IL acts trivially on V`Aα, and so Aα/L has good reduction. Likewise, if the eigenvalues of m are (i, −i), then ρE,λi (τ 2 ) = −1 for all i, so ρE,λi |GL ⊗ ψ is unramified for all i, and Aψα /L has good reduction.

Suppose PρE,|G 3 is unramified. Then ρE,|GL is unramified. Since det ρE,|I 3 is trivial, the image ρE,(I 3 ) is either trivial or ±1. So, in fact, either Aα/Q 3 or its ramified quadratic twist has good reduction over Q 3. Therefore, either ρE, 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 (ρE,`|GL) ⊗ ψ is unramified, so that Aψα /L has good reduction. In this case, by the hypotheses of our theorem, 3 does not divide deg μg for any g ∈ Gal( Q¯/Q).

In this case, Aψα /L has good reduction. Therefore, if ψ′^ is a ramified quadratic character of Gal( Q¯/Q[

√

−3]), the twist Aψ

′ α /Q[

√

−3] has good reduction at the prime over 3. Let θ be the prime of M determined by the chosen embedding M ↪→ Q¯ 3. Then, by Proposi- tion 2.12, we can define a finite flat group scheme Aα[θ] as the 3-torsion (equivalently, the θ-torsion) of the 3-divisible group Tθ Aψ

′ α /Q[

√

−3]. Because ¯ρE, 3 is absolutely irreducible when restricted to Gal( Q¯/Q[

√

−3]), and because

ρE, 3 ⊗ ψ′^ ∼= Vθ Aψ

′ α ,

we have an isomorphism of (OMθ /θ)[Gal( Q¯/Q[

√

−3])]-modules

ρ¯E, 3 ⊗ ψ′^ ∼= Aψ

′ α [θ].

Restricting this isomorphism to GL yields an isomorphism of (OMθ /θ)[GL]-modules

ρ¯E, 3 |GL ⊗ ψ ∼= Aψα [θ]. In particular, (¯ρE, 3 |GL) ⊗ ψ is flat. Recall that a representation of GL 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 OL. In the case, the finite flat group scheme in question is (Aψα )[θ].

Lemma 3.5. The centralizer of ρ¯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 OL is G = (Aψα )[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 Dieudonn´e 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 Aψα. The exactness condition is automatically satisfied because G is the 3-torsion in the 3-divisible group Tθ Aψα.

Let F be a finite extension of Q. Recall that an-adic representation ρ of the Galois group of F is said to be Barsotti-Tate if it arises from the generic fiber of an -divisible group, and to be potentially Barsotti-Tate if some restriction of ρ to a finite-index subgroup of Gal( F /F¯ ) is Barsotti- Tate. (See [2, §1.1].) The representation ρE, 3 |G 3 is potentially Barsotti-Tate, because it is realized on the θ-adic Tate module of Aα, which has potentially good reduction. From now on, we will abuse notation and refer to the local representation ρE, 3 |G 3 simply as ρE, 3. Let V be a d-dimensional vector space over a finite extension F ′^ of Q. One can associate to any potentially Barsotti-Tate representation ρ : Gal( F /F¯ ) → GL(V ) a continuous representation

W D(ρ) : WF → GL(D)

of the Weil group of F on a Q¯`-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(ρE, 3 ) is strongly acceptable for ρ¯E, 3.

Proof. We take F ′^ = Mλ. Let τ be the restriction of W D(ρE, 3 ) to I 3. It follows from Proposition 2.10 and [2, Prop. B.4.2] that ρE, 3 is Barsotti-Tate over L′^ for any finite extension L′/Q` such that τ is trivial. Our choice of α in Lemma 3.3 guarantees that det ρE, 3 = χ 3 and det ¯ρE, 3 = ¯χ 3. It follows that ρE, 3 is a deformation of ¯ρE, 3 of type τ , according to the definition in [2, §1.2]. We know that (ρE, 3 |GL) ⊗ ψ is Barsotti-Tate, because it is associated to the 3-divisible group Tθ Aψα. So

W D((ρE, 3 |GL) ⊗ ψ) = W D(ρE, 3 |GL) ⊗ W D(ψ)

is unramified, so τ |IL = W D(ψ)|IL. We know that W D(ψ) = ψ|WL ⊗MλQ^ ¯` ([2, §B.2]); that is, W D(ψ)|IL is a non-trivial quadratic character of IL. 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 ρE, 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 ∼= ˜ω^22 ⊕ ω˜^62 ,

where ˜ω 2 : It 3 → Q¯∗ 3 is the Teichm¨uller 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 ¯ρE, 3. We have by [1, Theorem 4.2.1] that either

• (¯ρE, 3 |I 3 ) ⊗F 3 ¯F 3 ∼= ωm 2 ⊕ ω 23 m, where m = 1 or 5;
• ρ¯E, 3 |I 3 ∼=

[

χ¯m 3 ∗ 0 χ¯n 3

]

, where (m, n) = (0, 1) or (1, 0) and ∗ is peu ramifi´e.

In either case, it follows from the criterion of [2, §1.2] that τ is strongly acceptable for ¯ρE, 3.

Now, combining Lemmas 3.5 and 3.6, we can apply [2, Theorem 7.1.1] and conclude that ρE, 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( Q¯/Q[

√

−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( Q¯/Q[

√

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 ¯ρE,. 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 Galois representations.

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

Pρ¯ : Gal( Q¯/Q) → PGL 2 (F`)

be a projective mod Galois representation. Let χ : Gal( Q¯/Q) → ± 1 be a quadratic Dirichlet character (possibly trivial). Let G be the subgroup of matrices in F¯∗ GL 2 (F) having determinant 1 , and let γ ∈ H^2 (PGL 2 (F), ±1) be the class of the extension

1 → ± 1 → G → PGL 2 (F`) → 1.

Let ψ¯ = det Pρ.¯ Finally, suppose that either

(a) the image of Pρ¯ lies in the normalizer N of a Cartan subgroup C of PGL 2 (F`), and the quadratic character Gal( Q¯/Q) → N/C is equal to χ, or

(b) the image of Pρ¯ lies in a Borel subgroup of PGL 2 (F`), and χ is trivial.

Then either ( ψ, χ¯ ψ¯) or Pρ¯∗γ( ψ, χ¯ ψ¯)(χ, χ) is the trivial class in H^2 (Gal( Q¯/Q), ±1).

Proof. First, suppose the image of Pρ¯ lies in the normalizer N of a Cartan subgroup C of PGL 2 (F). Write N¯ for the group N/C^2. Let π be the natural projection of N onto N¯. Then N¯ ∼= (Z/ 2 Z)⊕^2 ; a choice of isomorphism can be fixed by requiring that the first copy of Z/ 2 Z be π(C) and the second be the kernel of det : N¯ → F∗ /(F∗ ` )^2. We then have

π ◦ P¯ρ = ψ¯ ⊕ χ : Gal( Q¯/Q) → (Z/ 2 Z)⊕^2. (4.12)

We consider two cases.

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

N → Nˆ → N ,¯

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

H^1 (Gal( Q¯/Q), Nˆ ) → H^1 (Gal( Q¯/Q), N¯ ) d → H^2 (Gal( Q¯/Q), ±1).

The isomorphism N¯ ∼= (Z/ 2 Z)⊕^2 then tells us that d′( ψ¯ ⊕ χ) vanishes in

H^1 (Gal( Q¯/Q), D 4 ) → H^1 (Gal( Q¯/Q), (Z/ 2 Z)⊕^2 ) d

′ → H^2 (Gal( Q¯/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/ 2 Z). 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 ( ψ, χ¯ ψ¯) = 0, as desired.

Case 2: |C^2 | is odd. In this case, the inflation map

π∗^ : H^2 ( N ,¯ ±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/ 2 Z)⊕^2 ; in fact, any such subgroup is the image of an injection s : N¯ → N such that π ◦ s is the identity. Write ι for the inclusion of N in PGL 2 (F`). Let M be the subgroup of G lying over s( N¯ ). Then s∗ι∗γ is the class c ∈ H^2 ( N ,¯ ±1) corresponding to the extension

1 → ± 1 → M → N¯ → 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 = ( ψ¯ ⊕ χ)∗c = ( ψ, χ¯ ψ¯)(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( Q¯/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 Pρ¯ 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 ` representations attached to Q-curves.

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

(i) the image of Pρ¯E,lies in the normalizer N of a Cartan subgroup C of PGL 2 (F), and the quadratic character Gal( Q¯/Q) → N/C is equal to χ, or

(ii) the image of Pρ¯E,lies in a Borel subgroup of PGL 2 (F), and χ is trivial.

Then either ( ψ¯E,, χ ψ¯E,) or bE q,∞( ψ¯E,, χ ψ¯E,`)(χ, χ) is the trivial class in H^2 (Gal( Q¯/Q), ±1).

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

Pρ¯∗ E,γ = bE q,∞.

Let G be as in the statement of Proposition 4.1. For each σ ∈ Gal( Q¯/Q) let dσ ∈ F¯∗ be a square root of det(¯ρE,(σ)). Then gσ = d− σ 1 ρ¯E,is a set-theoretic lift of Pρ¯E, to G. To this lift one associates a 2-cocycle c given by the rule

c(σ, τ ) = gσ gτ g στ−^1 = d− σ 1 d− τ 1 dστ.

But this is just a 2-cocycle representing the class ¯δ(det ¯ρE,), where ¯δ is defined as in Proposition 2.15. From that proposition and from the fact that ¯δ( ¯χ) = q`,∞, one has

¯δ(det ¯ρE,) = bE δ¯( ¯χ) = bE q`,∞.

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 (bE ) 3 denotes the restriction of bE 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 (bE ) 3 is trivial. Then E is modular.

Proof. First, suppose that ¯ρE, 3 is absolutely reducible when restricted to Gal( Q¯/Q[

√

−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 ρE, 3 :

(i) ρE, 3 is continuous, irreducible, and odd;

(ii) det ρE, 3 (Frob) = ψ()k−^1 for some finite character ψ, some integer k ≥ 2, and almost all primes;

(iii) ρE, 3 |G 3 ∼=

[

φ 1 ∗ φ 2

]

with φ 2 |I 3 finite;

(iv) the reductions φ¯ 1 and φ¯ 2 are distinct;

(v) ¯ρE, 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 φE, 3 , and hence ρE, 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 φE, 3 to a decomposition group Gv at a prime v|3 of K satisfies

φE, 3 |Gv^ ∼=

[

θ 1 ∗ θ 2

]

with θ 2 having finite order on inertia. We claim that the same is true of ρE, 3 |G 3. Suppose otherwise. From the fact that ρE, 3 |Gal( K/K¯ ) is isomorphic to a twist of φE, 3 it follows that there is a quadratic

extension, say L, of Q 3 such that the restriction of ρE, 3 to Gal( L/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 ρE, 3 , is infinitely ramified, so must be one, and hence both, of these characters. But this contradicts the above description of φE, 3 |Gv. Write

ρE, 3 |G 3 ∼=

[

φ 1 ∗ φ 2

]

.

We next prove that the reductions φ¯ 1 and φ¯ 2 are distinct on G 3 ; in other words, that ρE, 3 has property (iv). To see this we note that if φ¯ 1 and φ¯ 2 were not distinct on G 3 then det ¯ρE, 3 |G 3 would be a square. Suppose this were so. Then from det ¯ρE, 3 = ¯ 3 χ¯ 3 (see (2.6)) we conclude that ¯ 3 |G 3 = φ^2 χ¯ 3 |G 3 for some character φ of G 3. It then follows from Proposition 2.15 that the restriction of bE to G 3 equals the restriction of ¯δ( ¯χ 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 ¯ρE, 3 is absolutely irreducible, then it must be dihedral and in fact induced from a character of Gal( Q¯/Q(

√

−3)) since we are assuming that ¯ρE, 3 is absolutely reducible on Gal( Q¯/Q(

√

−3)). It is a classical result that such representations are modular.

We have shown that ρE, 3 has properties (i)-(v) listed above. As mentioned before, the theorem follows. Now, suppose that ¯ρE, 3 is absolutely irreducible when restricted to Gal( Q¯/Q[

√

−3]). By the argument above,

ρE, 3 |G 3 ∼=

[

φ 1 ∗ φ 2

]

,

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 ρE, 3 by a finite-order character of GQ, we may assume φ 2 is unramified. We have already shown above that φ¯ 1 6 = φ¯ 2. Finally, ¯ρE, 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 ρE, 3 , whence E, is modular.

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

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

We have that ρE,`|I 3 is a character θ. So

θ^2 = det ρE,`|I 3 = E |I 3.

Choose α such that E has 2-power order; then E , whence also ρE,, 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 = ρE,(τ ); then m is a scalar which is conjugate to its cube, so m = ±1. In either case,

det m = E (τ ) = 1,

so E is unramified at 3, and (bE ) 3 = δ(E |G 3 ) is trivial.

From Proposition 2.10, the action of τ on T`Aα is either 1 or −1. Thus, after modifying α by a quadratic character, we may assume that Aα has good supersingular reduction at 3. Therefore, Aα[3] extends to a finite flat group scheme over R = W (¯F 3 ), to which we can apply Raynaud’s classification [14]. Let F be the fraction field of R. Let H be a Jordan-H¨older quotient of Aα[3]F. Then we have from [14, Cor. 3.4.4] that the action of τ on H( F¯ ) has eigenvalues

ψm(τ )n^3

i (i = 0,... , m − 1)

where ψm 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( F¯ ) if and only if τ 2 acts trivially. Suppose τ 2 acts trivially on H( F¯ ). Then H( F¯ ) is a 1-dimensional F 3 -vector space, and H is isomorphic to either (Z/ 3 Z)K or (μ 3 )K. It then follows from [14, Cor. 3.3.6] that Aα[3]/R has either Z/ 3 Z or μ 3 as a subquotient, which contradicts the supersingularity of Aα.

We may therefore suppose that τ 4 acts non-trivially on the F¯ -points of every subquotient of Aα[3]. In particular, ¯ρE, 3 (τ 4 ) does not have 1 as an eigenvalue. Suppose the restriction of ¯ρE, 3 to Gal( Q¯/Q[

√

−3]) is absolutely reducible. As in § 3, let L be the ramified quadratic extension of Q 3. Then

(¯ρE, 3 |IL)ss^ ∼= φ 1 ⊕ φ 2

for some characters φ 1 , φ 2 : ILt → F¯∗ 3. Since (¯ρE, 3 |IL) extends to a representation of G 3 , we have that {φ 1 , φ 2 } = {φ^31 , φ^32 }. The fact that ¯ρE, 3 |Q[

√

−3] is absolutely reducible means that in fact φ^3 i = φi for i = 1, 2; in other words, φ 1 and φ 2 are quadratic characters. In particular, ¯ρE, 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 ;
• (bE ) 3 = 1; and
• ρ¯E, 3 | Gal( Q¯/Q[

√

−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 tame- ly ramified over Q 3. Suppose further that (bE ) 3 is trivial, and that the four classes ( ψ¯E, 3 , −1), bE q 3 ,∞( ψ¯E, 3 , −1), q 3 ,∞( ψ¯E, 3 , 3), and bE ( ψ¯E, 3 , 3) are all nontrivial in H^2 (Gal( Q¯/Q), ±1). Finally, suppose that deg μσ can be chosen to be prime to 3 for all σ ∈ Gal( Q¯/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 ¯ρE, 3 to Gal( Q¯/Q[

√

−3]) is absolutely irreducible. It then follows from Theorem 3.1 that E is modular.

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