MODULAR GALOIS REPRESENTATIONS OF “NEBEN” TYPE

HARUZO HIDA

1. Introduction

In 1995 in [W] Theorem 0.2, A. Wiles proved, under some mild conditions (and relying on a ring theoretic result on Hecke algebras in [TW]), that if a p–adic Galois representation is congruent modulo p to that of an elliptic cusp form, then the representation itself is associated to a cusp form. After that, I have been thinking of characterizing residually dihedral Galois representations, because a 2-dim induced Galois representation is always modular. In this context, this paper is a continuation of the investigation done in [H98] and [HM97]. We have shown in [H98] (downloadable at www.math.ucla.edu/ hida) that for modular Galois representations of weight k ≥ 2 with real quadratic Neben type, the (odd) prime p giving residually dihedral representation is always a factor of N(ε^{k−1} − 1) for the positive fundamental unit ε of the quadratic extension, generalizing a result of Shimura in 1972 for weight 2. A simple application of this fact to the base-change problem was discussed in [HM97]. At the same time, I made a conjecture describing the universal p–ordinary deformation ring of a dihedral representation, which has been basically solved now by Cho-Vatsal [CV] (downloadable at www.math.ubc.ca/ vatsal). However the knowledge of the deformation ring does not give us an immediate method of constructing residually dihedral Galois representations. Here we would like to present a simple and (very elementary) constructive method, which actually gives all residually dihedral representations. As an obvious application of our result, we can show automatically the modularity of many non-ℚ–rational elliptic ℚ–curves, ℚ–HBAV or rank two ℚ–motives, although we only discuss some examples of elliptic ℚ–curves in details in this paper (see Section 5). Some more background materials of this type of approach to the modularity problems of motives will be included in my forthcoming book [H00]. Also one of my students: Ami Fischman is now trying to use the idea here to make the result in [HM97] more systematic. Hereafter in this paper the prime p is assumed to be odd.

2. Taylor-Wiles theorem

Let Σ be a finite set of primes including the fixed odd prime p and ℚ^{Σ} be the maximal extension unramified outside Σ and ∞. Let 𝔊 = Gal(ℚ^{Σ}/ℚ). We pick an absolutely irreducible odd representation \overline{ρ} : 𝔊 → GL_{2}(𝔽) for a finite field 𝔽 of characteristic p. All Galois representations are supposed to be continuous, and all valuation rings will be finite flat over ℤ_{p}.

We consider the following conditions:

A summary of Sections 5.1.1 and 5.2.1-2 from my book: “Geometric Modular Forms and Elliptic Curves” from World Scientific, 2000.

(Ordinarity) \overline{ρ}|_{D_{p}} ≅ ( \begin{matrix}\overline{ε} & ∗ \\ 0 & \overline{δ} \end{matrix} ) with \overline{δ} unramified and \overline{δ} ≠ \overline{ε} on the decomposition group D_{p} at p; (Flatness) \overline{ρ} restricted to the decomposition group at p is isomorphic to a Galois module associated to a locally free group scheme over ℤ_{p} of rank |𝔽|^{2}; (Irreducibility) \overline{ρ} restricted to Gal(ℚ^{Σ}/ℚ(μ_{p})) remains absolutely irreducible; (Modularity) \overline{ρ} is associated to an elliptic cusp form.

We now quote a theorem from [W] Theorem 0.2 and [D]:

Theorem 2.1 (Wiles-Taylor-Diamond). Suppose modularity and irreducibility and either ordinarity or flatness of \overline{ρ}. Let ρ : 𝔊_{Σ} → GL_{2}(O) be a Galois representation for a DVR O such that

1. ρ ≡ \overline{ρ} mod 𝔪_{O};

2. det ρ = ν_{p}^{k−1} up to finite order characters for k ≥ 2, where ν_{p} is the p–adic cyclotomic character;

3. ρ|_{D_{p}} ≅ ( \begin{matrix}ε & ∗ \\ 0 & δ \end{matrix}) for an unramified character δ ≡ \overline{δ} mod 𝔪_{O} when \overline{ρ} is ordinary;

4. When \overline{ρ} is flat, k = 2 and det ρ|_{I_{p}} = ν_{p}|_{I_{p}} and ρ is associated to a p–divisible

group over ℤ_{p} in the sense of Tate. Then there exist a positive integer N and a Hecke eigenform f ∈ S_{k}(Γ_{1}(N)) such that ρ ≅ ρ_{f}.

The above theorem was proven in [W] as Theorem 0.2 under an additional condition on auxiliary ramification outside p, which was later removed by Diamond in [D]. Wiles proved the result assuming the complete intersection property of a certain Hecke algebra in the minimal ramification case, which was in turn proved in [TW]. The flatness condition is now eased by Conrad-Diamond-Taylor to potential flatness over an extension of ℤ_{p} with ramification index ≤ p − 1. Also there is a generalization of the above theorem to Hilbert modular forms by K. Fujiwara.

3. Residually dihedral non-dihedral representations

We always write A for a DVR. Let F be a quadratic extension of ℚ, and fix a finite set S of rational primes including p. We look into a little smaller G = Gal(F^{S}/ℚ) and H = Gal(F^{S}/F). Here F^{S}/F is the maximal extension of F unramified outside S and ∞. First we explain how to create a Galois representation ρ_{G} : G → GL_{2}(B) (for a canonical B ⊃ A) residually induced from a Galois representation ρ = ρ_{H} : H → GL_{2}(A).

Put V = A^{2}, and let H act on V via ρ_{H}. If ρ_{H} is absolutely irreducible as a representation with coefficients in the field of fractions K of A, we may assume V = A[H]v for ∃v ∈ V without changing its isomorphism class over K (replacing V by A[H]v if necessary). We fix an element δ ∈ G which induces a non-trivial automorphism on F. We assume that ρ_{H} satisfies the following two conditions: (I0) V = V (ρ_{H}) is cyclic over A[H]; (I1) ρ_{H} is absolutely irreducible over the quotient field of A; (I2) We have an A–linear endomorphism T : V → V such that T (hx) = δhδ^{−1}T (x)

and det T ≠ 0. The existence of T assures us the existence of an extension ρ_{G} of ρ_{H}. Here is how to create an universal extension ρ_{G}. Consider the induced module: W = A[G]⊗_{A[H]}V, which is free of rank 4 over A. Define an A–linear endomorphism T̃ : W → W by T̃(δ^{i} ⊗ v) = δ^{i−1} ⊗ T (v) for i ∈ ℤ/2ℤ. By

T̃(hδ^{i}⊗v) = δ^{i−1}⊗T (δ^{−i}hδ^{i}v) = δ^{i−1}⊗δ^{1−i}hδ^{i−1}T (v) = hδ^{i−1}⊗T (v) = hT̃(δ^{i}⊗v),

T̃ commutes with A[H] and obviously commutes with δ and hence commutes with A[G]. Thus B = End_{A[G]}(W) is bigger than A.

Dividing T by a suitable element in A, we may assume that \overline{T} = (T mod 𝔪_{A}) for the maximal ideal 𝔪_{A} of A does not vanish. Then by definition, ρ_{H}(δ^{−2})T^{2} commutes with ρ_{H} and hence is a scalar t by Schur’s lemma. The elements t ∈ A is uniquely determined in O = (A − {0})/(A^{×})^{2}, and we call the class Ob(ρ_{H}) ∈ O the obstruction class of ρ_{H}. Then it is easy to see:

Proposition 3.1. Suppose (I0-2). We have 1. B ≅ A[\sqrt{t}] = A[X]/(X^{2} − t);

2. W is free of rank 2 over B;

3. If Ob(ρ_{H}) ∈ 𝔪_{A}, then B is a local ring; so, write ρ_{G} : G → GL_{2}(B) for the representation realized on W;

4. If Ob(ρ_{H}) ∈ 𝔪_{A} and \overline{ρ} = (ρ_{G} mod 𝔪_{B}) is absolutely irreducible, then there exists a character ξ : H → (B/𝔪_{B})^{×} such that \overline{ρ} ≅ Ind_{H}^{G} ξ.

Proof. All assertions can be easily proven. We first prove (1-3).

Let B′ be the subalgebra of B generated over A by T̃. Since T̃^{2} = t = ρ_{H}(δ^{−2})T^{2}, we see B′ ≅ A[\sqrt{t}]. If either t is not square in A or t ∈ 𝔪_{A}, by our construction, B′ is a local ring and W/𝔪_{B′}W is two dimensional over the residue field of B′. Then Nakayama’s lemma shows that W is free of rank 2 over B′, and hence B = B′. When t is a square unit in A, obviously B ≅ A ⊕ A, and the assertion (1-3) again follows.

The assertion (4) follows from the fact that \overline{T} ≢ 0 mod 𝔪_{A} but det \overline{T} = 0. Thus we have an exact sequence of H–modules: 0 → Ker(\overline{T}) → V/𝔪_{A}V → Im(\overline{T}) → 0; so, \overline{ρ}_{H} is reducible with semi-simplification isomorphic to ξ ⊕ξ^{δ} (ξ^{δ}(h) = ξ(δhδ^{−1})) by (I2). Mackey’s criterion of irreducibility of induced representations now tells us that ξ^{δ} ≠ ξ and \overline{ρ} ≅ Ind_{H}^{G} ξ, because \overline{ρ} is supposed to be absolutely irreducible.

The ring B has an involution σ :⟳ B such that ρ_{G} ⊗ χ ≅ σ ∘ ρ_{G} and σ(\sqrt{t}) = −\sqrt{t} (χ : G/H ≅ {±1}).

There is a converse of the above proposition: Start with a Galois representation φ : G → GL_{2}(O) for a DVR O with irreducible \overline{φ} = (φ mod 𝔪_{O}) ≅ Ind_{H}^{G} ξ; so, \overline{φ}(δ) cannot be in the center. We assume that φ_{H} = φ|_{H} is absolutely irreducible. We may assume (and will assume) that O is the normalization of the subring generated by Tr(φ) over ℤ_{p} because of the absolute irreducibility of \overline{φ} (see [N] and [H99] II.2.1-2). Since, for χ : G/H ≅ {±1},

ϕ ≅ Ind_{H}^{G} η ∃η ⟺ ϕ ⊗ χ ≅ ϕ,

we can divide our consideration into two cases:

(a) There exists an involution σ of O such that σ ∘ φ ≅ φ ⊗ χ;

(b) No such involution. In Case (b), we may regard Φ ≅ φ ⊕ (φ ⊗ χ) as representations into GL_{2}(B′) for the subring B′ ⊂ O ⊕ O generated over O by Tr(Φ(g)) for all g ∈ G, because we can find such Φ in the isomorphism class of φ ⊕ (φ ⊗ χ) by means of pseudorepresentations of Wiles (see [H99] II.2.1-2 for generality of pseudo-representations). Since φ is residually induced, B′ is a local ring. Then we define σ ∈ Aut(B′) by σ(x, y) = (y, x). In Case (a), we write B′ for O and Φ for φ. Then we have

Proposition 3.2. Let A_{0} = H^{0}(⟨σ⟩, B′). Then there exist a DVR A unramified over A_{0} and ρ_{H} : H → GL_{2}(A) such that

1. Tr ρ_{H} = Tr φ_{H};

2. Ob(ρ_{H}) ∈ 𝔪_{A};

3. We have an isomorphism ι : B ↪ A ⊗_{A_{0}} B′ such that ι ∘ ρ_{G} ≅ φ or Φ

according as we are in Case (a) or (b), where ρ_{G} is the representation constructed in the previous proposition.

Proof. Since Tr(Φ ⊗ χ) = Tr(σ ∘ Φ), we have Φ ⊗ χ ≅ σ ∘ Φ over B′ again by the result of Carayol. Thus we have an automorphism T : V (Φ) → V (Φ) for the representation space V (Φ) of Φ such that T Φ(g) = χ(g)σ(Φ(g))T for all g ∈ G. The automorphism T is uniquely determined up to scalar (cf. [H99] Lemma II.1.12). If T is scalar, then Φ = χ ⊗ (σ ∘ Φ). Since we have Tr(Φ) ≡ Tr(σ ∘ Φ) mod 𝔪_{B′}, σ induces the identity map on B′/𝔪_{B′}. Thus Φ(g) ≡ χ(g)Φ(g) mod 𝔪_{B′} for all g ∈ G, which is impossible. Thus T is non-scalar.

Let \overline{T} = T mod 𝔪_{B′}. Since σ is trivial modulo 𝔪_{B′}, \overline{T}^{2} has to be a scalar, because \overline{ρ} is absolutely irreducible (by Schur’s lemma: [H99] Lemma II.1.5). Similarly σ(T )T commutes with Φ and hence is a scalar u ∈ B′^{×}. Thus T commutes with σ(T )^{−1} and hence commutes with σ(T ). In particular, u is in the subring A_{0} of B′ fixed by σ. By our construction, A_{0} is a DVR and is the normalization of the subring generated (over ℤ_{p}) by traces over H of the representation Φ. Note that \overline{φ} ⊗ χ ≅ \overline{φ}; so, σ is the identity on the residue field of B′. In Case (a), O is a ramified quadratic extension of A_{0}, and in Case (b), A_{0} = O.

If u is not a square in A_{0}^{×}, we define A = A_{0}[\sqrt{u}], otherwise, we rewrite A = A_{0}. Since p > 2, A is a discrete valuation ring unramified over A_{0}. Then we write B for B′[\sqrt{u}]. We extend σ to a unique automorphism of B fixing A. Then changing T by ζT for ζ ∈ A with ζ^{2} = u^{−1}, we may assume that σ(T )T = 1. Let K be the quotient field of A. Then K[T ] is a semi-simple quadratic extension of K. By the Hilbert theorem 90, applied to K[T ], we find S ∈ K[T ] such that T = σ(S)^{−1}S. Then ρ′ = SΦS^{−1} satisfies ρ′(g) = χ(g)σ(ρ′(g)) for all g ∈ G. If G is a compact group and φ is continuous, the image of Φ has to be in a maximal compact subgroup, which is a conjugate of GL_{2}(A). So further conjugating ρ′|_{H} by an element in GL_{2}(K), we find ρ : H → GL_{2}(A) such that

1. ρ has values in GL_{2}(A);

2. V (ρ) is generated by a single element over A[H];

3. ρ is isomorphic to φ|_{H} over the quotient field of O. The third point follows from the trace identity of ρ and φ|_{H}. The point (2) is achieved by replacing V (ρ) by A[H]v for v ≠ 0 in V (ρ). By (3) and the assumption (I1), ρ is absolutely irreducible over K, and hence, A[H]v is free of rank two over A.

Out of this choice of ρ, we go back to the process as in the proof of Proposition 3.1: Let W = A[G] ⊗_{A[H]} V (ρ). Supposing that the ring B = End_{A[G]}(W) is isomorphic to A ⊕ A, we would like to show \overline{ρ} ⊗ χ ≅ \overline{ρ̸}. This actually leads to a contradiction as we will see after proving the non-congruence. By our choice of ρ, V (ρ) = A[H]v for v ∈ V (ρ). Thus W = A[G]v. Therefore, for \overline{W} = W ⊗_{A} 𝔽 with 𝔽 = A/𝔪_{A}, we have A[G]\overline{v} = \overline{W} for \overline{v} = (v mod 𝔪_{A}W). Thus we conclude that 𝔽[\overline{G}]\overline{v} = \overline{W}, where \overline{G} = G/ Ker(\overline{ρ}) ≅ Im(\overline{ρ}). It is known (e.g. [H99] II.1.5) that

𝔽[\overline{G}] ≅ ⨁_{π∈ˆ\overline{G}} R(π),

where ˆ \overline{G} is the set of irreducible representations of the finite group \overline{G}, and R(π) modulo its nilradical is a simple algebra isomorphic to Im(π). Since \overline{W} ≅ V (\overline{ρ}) ⊕ V (\overline{ρ} ⊗ χ) is semi-simple by our assumption, the fact: \overline{W} = 𝔽[\overline{G}]\overline{v} tells us that the nilradical of 𝔽[\overline{G}] kills \overline{W}, and \overline{W} is a surjective image of a single component R(π) if \overline{ρ} ≅ \overline{ρ} ⊗ χ. In this case, End_{𝔽[\overline{G}]}(\overline{W}) = M_{2}(𝔽), which does not have two distinct idempotents commuting each other. Since we have two distinct commuting idempotents in End_{A[G]}(\overline{W}), we conclude the non-congruence: \overline{ρ} ⊗ χ ≅ \overline{ρ̸}. This is a contradiction against our assumption of residual dihedralness and residual irreducibility, because the congruence is a characterization of induced representations (see [H99] Lemma II.1.14). Thus B has to be a local ring, and hence Ob(ρ) ∈ 𝔪_{A}.

We have now rediscovered the ring B, which is either isomorphic to B′ or B′ ⊗_{A_{0}} A. The resulting representations ρ_{G} and Φ are isomorphic each other by the result of Carayol, because they have equal trace and absolutely irreducible reduction modulo the maximal ideal (cf. [H99] Proposition II.1.13).

4. Modularity Problems

We start with an irreducible Galois representation ρ_{H} : H → GL_{2}(A) satisfying the following three conditions in addition to (I1-2): (I3) We have Ob(ρ_{H}) ∈ 𝔪_{A}; (I4) \overline{ρ}_{G} = ρ_{G} mod 𝔪_{B} is absolutely irreducible; (I5) det ρ_{G}(c) = −1 for complex conjugation c. Combining the two propositions with Theorem 2.1, we get

Theorem 4.1. Suppose the five conditions (I1-5) and that the representation ρ_{G} is either p–ordinary with det ρ_{G} = ν_{p}^{k−1} up to finite order characters for k ≥ 2 or flat with det ρ|_{I_{p}} = ν_{p}|_{I_{p}}. Then for any extension φ : G → GL_{2}(O) of ρ_{H} for a DVR O over A, if \overline{φ} = (φ mod 𝔪_{O}) is absolutely irreducible on Gal(F^{S}/ℚ(μ_{p})), then there exist an integer N > 0 and a Hecke eigenform f : S_{k}(Γ_{1}(N)) such that φ ≅ ρ_{f}.

The integer N of course can be taken to be the conductor of the compatible system of Galois representations attached to f as above.

An immediate corollary of this theorem is:

Corollary 4.2. Let φ = {φ_{𝔩}}_{𝔩} be a compatible system of 2 dimensional Galois representations of H with coefficients in a number field E. Here we assume that E is a minimal possible choice. If one of the members φ_{𝔭} satisfies the five conditions (I1-5) and the assumptions of Theorem 4.1, then the system can be extended to a compatible system Φ of two dimensional Galois representations in exactly two ways, one is Φ and the other is Φ ⊗ χ, and E is either a CM field or a totally real field.

When φ is associated to an abelian variety, the conclusion of the corollary follows from the solution of the Tate conjecture by Faltings [F]. However the result is new for 2–dimensional compatible systems associated to general motives.

We now describe a sufficient condition that \overline{ρ} = Ind_{H}^{G} ξ in (I4) remains irreducible over K = Gal(F^{S}/ℚ(μ_{p})). By Frobenius reciprocity law (cf. [H99] II.1.6), reducibility of \overline{ρ} on K is equivalent to the reducibility of \overline{ρ}|_{H′} for H′ = Gal(F^{S}/ℚ(\sqrt{(−1)^{(p−1)/2}p})), where κ = ℚ(\sqrt{(−1)^{(p−1)/2}p}) is the unique quadratic extension in ℚ(μ_{p}). This happens only when \overline{ρ}(H′) modulo center is abelian.

Since the image of \overline{ρ} modulo center is a dihedral group of order 2n for the order n of ξ^{−1}ξ^{δ}, we study when a dihedral group D_{2n} of order 2n has an abelian normal subgroup of index 2. A dihedral group D_{2n} is abelian only when n ≤ 2. A dihedral group of order 2n with n ≥ 3 has a unique normal cyclic subgroup C_{n} of order n. We see easily that the maximal abelian quotient D_{2n}^{ab} of D_{2n} is either isomorphic to ℤ/2ℤ or (ℤ/2ℤ)^{2} according as n is odd or even. Therefore when n ≥ 3 is odd, the cyclic subgroup C_{n} is the unique subgroup of index 2.

Now we suppose that n is even. Since D_{2n}^{ab} ≅ (ℤ/2ℤ)^{2}, there are 3 normal subgroups X of index 2. If X ≠ C_{n}, choose d ∈ X − C_{n}. Then d^{2} = 1 and dcd^{−1} = c^{−1} for c ∈ C_{n}. Thus X ≅ D_{n}, which is non-abelian if n > 4. Thus we have an abelian normal subgroup of index two in D_{2n} if and only if n = 2 and 4.

Suppose that n = 2m is even. The conjugation action of D_{2n}/C_{n} on C_{n} fixes element by element the unique subgroup C_{n} of order 2. Thus for any character ξ of C_{n} of order n, ξ^{−1}ξ^{δ} for δ ∈ D_{2n} − C_{n} has order m. Thus if m ≥ 4, we still see that the restriction of Ind_{C_{n}}^{D_{2n}} ξ to any subgroup of index 2 is irreducible. This settles the case where the order of ξ^{−1}ξ^{δ} is equal to 4. We thus conclude

Proposition 4.3. Let the notation be as in (I4) and p be an odd prime. Suppose that F ⊄ ℚ(μ_{p}) and that \overline{ρ} = Ind_{H}^{G} ξ is absolutely irreducible. Then if either the order of ξ^{−1}ξ^{δ} is larger than 2 or there exists σ ∈ Ker(\overline{ρ}) acting non-trivially on κ, \overline{ρ} remains absolutely irreducible over ℚ(μ_{p}).

5. Elliptic ℚ–Curves

An elliptic curve E defined over a number field is called a ℚ–curve if for any σ ∈ Gal(\overline{ℚ}/ℚ), we have an isogeny μ_{σ} : E → E^{σ}. By a result of Elkies, all elliptic ℚ-curves have a model over a (2, 2, …, 2)–extension of ℚ. Here we treat the case where E itself has a model over a quadratic extension F/ℚ. Presumably an inductive argument should give a similar result to more general cases. The following construction of an example of φ as in Theorem 4.1 is a version of an argument of Shimura in [S] Sections 9-10:

Corollary 5.1. Let E be an elliptic curve defined over F. Suppose that End(E_{/\overline{ℚ}}) = ℤ and that we have an isogeny θ : E → E^{δ} defined over \overline{ℚ} with the following property for a prime 5 ≤ p ∤ D:

p| deg(θ) and the p–primary part of Ker(θ) is cyclic.

If E has a semi-stable reduction at p over F, then there exists a positive integer N such that E shows up as a factor over a finite abelian extension L of F of the jacobian of the modular curve X_{1}(N). When θ is defined over F, L can be taken to be F itself.

We are going to prove that the Tate module T_{p}(E) is reducible modulo p, giving rise to two characters ξ, ξ′ : H → 𝔽_{p}^{×} and satisfies the assumptions of Theorem 4.1. It will be clear from the following proof that if either p ≡ 1 mod 4 or ξ^{−1}ξ′ ramifies at a prime q ≠ p, we do not need to assume any condition on reduction of E modulo p to show the assertion of the corollary.

Proof. We first assume that θ is defined over F. We need to check the five conditions: (I1-5) and the assumptions of Theorem 4.1 for V (φ) = T_{p}(E). If this is done, we have by Theorem 4.1 a non-trivial ℤ_{p}[H]–linear homomorphism from the p–divisible group J_{1}(N)[p^{∞}] into E[p^{∞}]; so, again by a theorem of Faltings already quoted, we have a surjection J_{1}(N) → E defined over F.

Since End(E_{/\overline{ℚ}}) = ℤ, by the theorem of Faltings as above, the representation φ is absolutely irreducible over F (I1).

Since θ^{δ} ∘ θ is an endomorphism of E, it is equal to an integer d ≠ 0.

Now assume that p divides deg(θ) and that p ≥ 5. Here the set S is the union of {p} and the set of rational primes where E has bad reduction. We may identify E^{δ}[p^{∞}] with E[p^{∞}] by x ↦ δx. Thus we have a natural identification of V (φ^{δ}) and V (φ). Writing this identification as i : V (φ) ≅ V (φ^{δ}), we have i(h(x)) = δhδ^{−1}i(x). Then the isogeny θ induces T = i ∘ θ : V (φ) → V (φ) such that T (hx) = δhδ^{−1}T (x) (I2).

Since θ is cyclic at p, Coker(T) is isomorphic to the p–Sylow part of the group Ker(θ) which is cyclic, and hence Coker(T) is non-trivial by our choice of p. Writing d_{p} for the p–primary part of d, we find that Ob(φ_{H}) = d_{p} ∈ pℤ_{p} (I3).

Let ξ be the character of H through which H acts on Ker(θ) ⊗_{ℤ} 𝔽_{p}. Then by definition, the action of H on Ker(θ^{δ}) ⊗_{ℤ} 𝔽_{p} gives rise to ξ^{δ}(h) = ξ(δhδ^{−1}).

If x ∈ E[d_{p}], then θ^{δ} ∘ θ(x) = d_{p}x = 0. Thus θ(x) ∈ Ker(θ^{δ}). By counting the order, we find that θ(E[d_{p}]) = Ker(θ^{δ}) ⊗_{ℤ} ℤ_{p}. Thus we have an exact sequence of H–modules:

0 → V (ξ) → E[p] ⟶^{θ} V (ξ^{δ}) → 0.

Since ξξ^{δ} = ω_{p} for the mod p Teichmüller character ω_{p} of Gal(\overline{ℚ}/ℚ) and since ω_{p}|_{H} has order p − 1 if p ∤ D or (p − 1)/2 when p|D, if p ≥ 5 (under our assumption: p ∤ D), the order of ξ is larger than 2.

For the moment, suppose that F is real. Then for complex conjugation c, we have det φ(c) = −1, and hence, ξξ^{δ}(c) = −1. This shows that ξ(c) ≠ ξ^{δ}(c) if p is odd.

We now give an argument proving ξ^{δ} ≠ ξ valid both imaginary and real F. Since ξξ^{δ} = ω_{p}, if ξ = ξ^{δ}, we have ω_{p} = ξ^{2}. Restrict these characters to the inertia group at I_{p}. By local class field theory, we may regard these characters as characters of O_{𝔭}^{×} for a prime factor 𝔭|p in F. Here O is the integer ring of F. Since ξ has values in 𝔽_{p}^{×}, ξ factors through (O/𝔭)^{×}. If O/𝔭 ≅ 𝔽_{p} and p ∤ D, the identity ξ^{2} = ω_{p} is impossible, because ω_{p} has order p − 1 (if p ∤ D). If O/𝔭 ≅ 𝔽_{p^{2}}, then ξ^{2} = ω_{p} implies ξ has order 2(p − 1), which is impossible since ξ has values in 𝔽_{p}^{×}. Thus ξ ≠ ξ^{δ} on (O/𝔭)^{×} again in this case. However ξ ≠ ξ^{δ} on (O/𝔭)^{×} implies that ξ has values in 𝔽_{p^{2}} not in 𝔽_{p}; so, this case actually never happens. Thus we conclude that the prime p has to split in F.

Thus if ξ is unramified at one of the prime factors 𝔭 of p (⇐ semi-stable reduction of E at p), we are in the p–ordinary case, and ξ^{−1}ξ^{δ} has order at least p − 1 ≥ 4. Even if ξ and ξ^{δ} both ramify at 𝔭|p, we choose a Dirichlet character ϕ such that ϕ|_{I_{𝔭}} = ξ|_{I_{𝔭}}. Then φ ⊗ ϕ^{−1} is p–ordinary. Suppose that ξξ^{−δ} is of order 2, then ξ^{4} = (ξξ^{δ})^{2}(ξξ^{−δ})^{2} = ω_{p}^{2}, which is impossible if p ≡ 1 mod 4, and still ξξ^{−δ} has order > 2 in this case. Thus φ satisfies (I4) and also the irreducibility condition over Gal(F^{S}/ℚ(μ_{p})) (a condition of Theorem 4.1) if either E has semi-stable reduction at p or p ≡ 1 mod 4 or ξξ^{−δ} ramifies at a prime different from p (see Proposition 4.3).

The condition det(ρ(c)) = −1 (I5) is automatic in the elliptic curve case by the self-dual pairing ⟨ , ⟩ on E[p]. When ξ is unramified at one of prime factor 𝔭 of p, we directly find f ∈ S_{2}(Γ_{1}(N)) by Theorem 4.1 such that ρ_{f} ≅ φ over H. When ξ and ξ^{δ} both ramify at 𝔭|p, we find g ∈ S_{2}(Γ_{1}(N′)) such that ρ_{g} ≅ φ ⊗ ϕ^{−1} over H. Then the automorphic twist f = g ⊗ ϕ is associated to E.

We now assume that θ is not defined over F. Choose a nowhere vanishing differential ω ∈ H^{0}(E, Ω_{E/F}) (always possible). Then θ^{∗}ω^{δ} = cω for c ∈ \overline{ℚ}. For σ ∈ Gal(\overline{ℚ}/F), (θ^{σ})^{∗}ω^{δ} = c^{σ}ω. Then θ^{−1} ∘ θ^{σ} = \frac{n}{m} for mutually prime integers m and n, because End(E_{/\overline{ℚ}}) = ℤ. Thus n^{2} deg(θ) = m^{2} deg(θ^{σ}); so, m = ±n by the fact: deg(θ) = deg(θ^{σ}). Thus θ^{σ} = ±θ, and hence c^{σ} = ±c. Therefore, θ is defined over F (c), which is at most a quadratic extension of F.

We have (θ∘θ^{δ})^{∗}ω^{δ} = cc^{δ}ω^{δ}. By End(E_{/\overline{ℚ}}) = ℤ, θ∘θ^{δ} is an integer in End\overline{ℚ}(E^{δ}); hence, cc^{δ} ∈ ℤ for any extension of δ to \overline{ℚ}. Thus Gal(F (c)/ℚ) is either isomorphic to (ℤ/2ℤ)^{2} or (ℤ/2ℤ).

By our assumption now, we have F (c) ≠ F. Consider T : T_{p}(E) ≅ T_{p}(E^{δ}) defined by T (x) = δ(θ(x)). Then T (hx) = ε(h)δhδ^{−1}T (x) for h ∈ H, where ε(h) = (c^{h}/c).

If we can find a finite order character η : Gal(\overline{ℚ}/F) → \overline{ℚ}^{×} such that ε = η^{δ−1}, then for ϕ = φ ⊗ η, we find T ϕ = ϕ^{δ}T. Thus once we find such η, the twist ϕ satisfies (I2).

We now show that η with ε = η^{δ−1} always exists. Let X be a number field. By class field theory, any continuous character of Gal(\overline{ℚ}/X) can be regarded as a continuous idele character: C_{X} = X_{𝔸}^{×}/X^{×} → 𝕋, where

𝕋 = {z ∈ ℂ||z| = 1} .

A given continuous character of C_{X} is of finite order if and only if it is trivial on the identity component of the infinite part F_{∞}^{×} of F_{𝔸}^{×} (cf. [H99] Proposition II.1.2). By Artin reciprocity, any finite order character of C_{X} can be viewed as a Galois character of Gal(\overline{ℚ}/X) canonically. Looking at the exact sequence:

1 → F^{×} → F_{𝔸}^{×} → C_{F} → 1,

by Hilbert’s theorem 90 applied to F^{×} and Gal(F/ℚ), we find

H^{0}(Gal(F/ℚ), C_{F}) = C_{ℚ}.

Thus the kernel of δ − 1 : x ↦ x^{δ−1} is given by C_{ℚ}. A character ψ : C_{F} → 𝕋 is of the form ψ = η^{δ−1} if and only if ψ is trivial on C_{ℚ}. Since Gal(F (c)/ℚ) ≅ (ℤ/2ℤ)^{2}, we can write F (c) = F [\sqrt{m}] for m ∈ ℤ. Thus ε = μ ∘ N_{F/ℚ} for the quadratic character μ associated to ℚ[\sqrt{m}]/ℚ. This shows that ε(x) = μ(xx^{δ}) = μ(x^{2}) = 1 for x ∈ C_{ℚ}. Thus we can write ε = η^{δ−1} for a character η : C_{F} → 𝕋. To show that η can be chosen to be of finite order, we need to show that η restricted to the identity component of F_{∞}^{×} can be chosen to be trivial.

Suppose that F is imaginary, and write η_{∞}(z) = |z|^{s}z^{m} (z ∈ ℂ^{×}) for m ∈ ℤ and s ∈ ℂ. Since η_{∞}^{δ−1} = 1, we have m = 0. Thus multiplying η by | |_{𝔸}^{−s/2}, we may assume that η is of finite order.

Suppose that F is real. Then we have η_{∞}(x, x′) = |x|^{s}(x/|x|)^{u}|x′|^{t}(x′/|x′|)^{v} for (x, x′) ∈ (ℝ^{×})^{2} (s, t ∈ \sqrt{−1}ℝ and u, v ∈ ℤ/2ℤ). By η^{δ−1}(x, x′) = ε_{∞}(x, x′) = (x/|x|)^{w}(x′/|x′|)^{w} for w ∈ ℤ/2ℤ. This shows that s = t and u + v = w. Thus multiplying η again by | |_{𝔸}^{−s}, we may assume that η is of finite order. Thus we have basically shown the following facts: (EI1) There exists a finite order continuous character η of Gal(\overline{ℚ}/F) such that

T_{p}(E) ⊗ η satisfies (I2). (EI2) We can choose η unramified outside a given finite set of primes which are

unramified over F in the minimal field M = F (c) of definition of θ : E → E^{δ}. We will leave to the reader to give the detail of the proof of (EI2), since we do not use it in the sequal.

By our construction, T_{p}(E) ⊗ η satisfies (I1-5) except possibly for (I4). We now prove (I4) for T_{p}(E) ⊗ η. Note that we have an exact sequence of H–modules:

0 → V (ξ) → E[p] → V (εξ^{δ}) → 0.

Thus εξξ^{δ} = ω_{p}. If ξη = (ξη)^{δ}, then ξ^{1−δ} = η^{δ−1} = ε. This shows ξ^{2} = ω_{p}, which is impossible if p ∤ D as we have already seen. We also conclude that p splits in F under this circumstance. Thus ξη ≠ (ξη)^{δ}. Since p splits in F, we can always find a Dirichlet character ϕ such that ϕ|_{I_{𝔭}} = η|_{I_{𝔭}} for the inertia group I_{𝔭} at 𝔭. Then replacing η by η(ϕ ∘ N_{F/ℚ})^{−1}, we may assume that η is unramified at 𝔭. Then we know that the order of ηξ|_{I_{𝔭}} and (ηξ)(ηξ)^{−δ} = εξξ^{−δ} are greater than 2 as before. Thus the extension of φ ⊗ η to G is residually irreducible over Gal(\overline{ℚ}/ℚ(μ_{p})) by Proposition 4.3.

Thus we have a cusp form f ∈ S_{2}(Γ_{1}(N)) such that T_{p}(E) ⊗ η ≅ V (ρ_{f}) as H– modules. The abelian variety A_{f} associated to f in Shimura’s sense has a factor isogenous to E over the field L such that η : Gal(L/F) ≅ Im(η). By definition L ⊃ F (c).

Here is some remark on the above result:

1. If one could ease the irreducibility condition of \overline{ρ} over Gal(\overline{ℚ}/ℚ(μ_{p})) in Theorem 2.1 to the absolutely irreducibility over Gal(\overline{ℚ}/ℚ), one can remove the semi-stability assumption at p for E from the above corollary and ease the assumption on p to 3 ≤ p ∤ D.

2. The base-change lift f̂ of f as above to GL(2)_{/F} (established by Jacquet [J]; see also [L]) is a cohomological cusp form. We write g for f̂if F = F (c). When F ≠ F (c), we twist back f̂⊗ η^{−1} and write it as g, which is the automorphic twist of f̂ by the finite order Hecke character η. The above proof shows that the cusp form g is associated to T_{p}(E) restricted to H. This shows the existence of a cusp form g on GL_{2}(F_{𝔸}) associated to E_{/F}.

Here is how to find a lot of examples of Elliptic ℚ–curves as in the above corollary. Pairs (E, C) of an elliptic curve and a cyclic subgroup of order p is classified by the modular curve X_{0}(p). For each point y ∈ X_{0}(p)(\overline{ℚ}) represented by (E, C), ℚ(y) is characterized as the field of moduli of the pair (E, C) defined over \overline{ℚ}: ℚ(y) is the fixed field of

G(E, C) = \{σ ∈ Gal(\overline{ℚ}/ℚ)|(E^{σ}, C^{σ}) ≅ (E, C) over \overline{ℚ}\} ,

Take a point y with ℚ(y) = F. As is well known, we can choose a model of E defined over F. For this model, we have the following two possibilities:

1. (E, C) is defined over F;

2. C is defined over a (2, 2)–extension of ℚ containing F. These two cases are covered by Corollary 5.1 if 5 ≤ p ∤ D and End(E_{/\overline{ℚ}}) = ℤ. Thus the main point is to find y ∈ X_{0}(p) quadratic over ℚ. The functorial correspondence (E, C) ↦ (E/C, E[p]/C) induces an involution τ of X_{0}(p). We make a quotient curve X^{∗}(p) = X_{0}(p)/⟨τ⟩. If x ∈ X^{∗}(p)(ℚ) and y ∈ X_{0}(p) is over x, ℚ(y) is either ℚ or a quadratic extension F.

Here is a list of primes p ≥ 5 for which X^{∗}(p) ≅ P_{/ℚ}^{1}:

p = 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71.

By a work of Mazur ([M1] and [M2]) if p ≥ 11 in the above list, there are no non-CM non-cuspidal rational points of X_{0}(p) (actually there is at most one non-cuspidal ℚ–rational point in these cases). Thus all points on X_{0}(p) over infinitely many (non-CM) ℚ–rational points of X^{∗}(p) yield examples of type (1) or (2).

Even if the genus of X_{0}(p) is 0, by Hilbert’s irreducibility theorem, for infinitely many x ∈ X^{∗}(ℚ), we find ℚ(y) quadratic over ℚ (for y ∈ X_{0}(p) over x) as explicitly seen by Shimura in [S] for p = 5.

Here are some concluding remarks:

1. We could have stated the result for ℚ–HBAV (or possibly rank 2 ℚ–motives) in place of elliptic ℚ–curves. In the case of ℚ–HBAV, we need to start with ℚ–HBAV A defined over a quadratic field F/ℚ with an isogeny θ : A → A^{δ} and assume the HBAV-version of the assumption of the corollary and that A has semi-stable reduction at p (this last assumption could presumably be removed). Then A is a factor of J_{1}(N) for a sutable N over an abelian extension L/F.

2. Almost all modular compatible systems of quadratic Neben types (up to character twists) satisfies the conditions (I1-5) and the assumptions of Theorem 4.1 for an odd prime p. In particular, when weight is two, every geometric factor of modular jacobian is a ℚ–HBAV or its quaternionic analog. In the case of higher weight, modular compatible system is obtained from motivic analogs of ℚ–HBAV, which we call “rank 2 ℚ–motive”.

3. If f is a Hecke eigenform with quadratic Neben type, its field of Hecke eigenvalues is a CM field ℚ(f) over a totally real field ℚ_{+}(f), as long as f does not have complex multiplication ( ⟺ ρ_{f} is non-dihedral). In this case, the relative different of ℚ(f) over ℚ_{+}(f) is spanned by Ob(ρ_{f}|_{H}). Strictly speaking, we should have said that the relative different of the order of ℚ_{p}(f) generated by Hecke eigenvalues over the integer ring of the closure of ℚ_{+}(f) in ℚ_{p}(f) is spanned by Ob(ρ_{f}|_{H}), where ℚ_{p}(f) is the subfield of \overline{ℚ}_{p} generated over ℚ_{p} by Tr(ρ_{f}). For a given f, there always exists a prime p for which the relative different is non-trivial. Therefore we could say that almost all f with quadratic Neben type can be included into the frame of our theory. We said “almost all”, because we still have some problem if the prime p is small, like, 2 and 3, or dividing D.

4. The above arguments can be basically carried out for Hilbert modular forms in place of elliptic modular forms using Fujiwara’s generalization of Theorem 2.1 to Hilbert modular case (cf. [HM97]).

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