Remarks on a conjecture of Fontaine and Mazur

Richard Taylor ^{1} Department of Mathematics,

Harvard University, Cambridge, MA 02138,

U.S.A.

May 23, 2000

1Partially supported by NSF Grant DMS-9702885

Introduction

Fontaine and Mazur have made the following extremely influential conjecture (see [FM]).

Conjecture A Suppose that

ρ : Gal (ℚ^{ac}/ℚ) ⟶ GL_{n}(ℚ_{l}^{ac})

is a continuous irreducible representation such that

1. ρ is ramified at only finitely many primes

2. and the restriction of ρ to the decomposition group at l is potentially semi-stable in the sense of Fontaine.

Then ρ occurs in the l-adic cohomology (with respect to a Tate twist of the constant sheaf) of some variety defined over ℚ.

We remark that it is now known that if ρ does occur in the l-adic cohomology of some variety defined over ℚ then (1) and (2) must hold. We also remark that it would follow from this conjecture that there is an integer w (depending on ρ) such that for almost all p the eigenvalues of ρ(Frob_{p}) are algebraic and for each embedding into ℂ have absolute value p^{w/2}. Finally we remark that, combining this conjecture with conjectures of Langlands, one further expects that ρ has the same L-series as a cuspidal automorphic representation of GL_{n}(𝔸), and so in particular its L-series has holomorphic continuation to ℂ (except for a possible pole at s = 1 when n = 1) and satisfies a functional equation (which can be made precise).

The case n = 1 of the conjecture has been known to be true for some time. Besides this the only known cases are for n = 2 where the methods of Wiles [W], Taylor-Wiles [TW] and Skinner-Wiles [SW1] have been used to verify some cases of the conjecture. Except for a couple of isolated examples (see [SBT] and [Dic]) these methods have been restricted to the case where ρ has pro-soluble image. The purpose of this paper is to verify the Fontaine-Mazur conjecture in a significant number of cases where the image of ρ is not pro-soluble. More precisely we prove the following theorem and its corollaries. (Here, and in the rest of this paper, c denotes complex conjugation and ϵ the l-adic cyclotomic character.)

Theorem B Let l be an odd prime and

ρ : G_{ℚ} ⟶ GL_{2}(ℚ_{l}^{ac})

a continuous irreducible representation such that • ρ is unramified at all but finitely many primes,

• det ρ(c) = −1,

• and

ρ|_{G_{l}} ∼ ( \begin{matrix}ϵ^{n}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

for some n ∈ ℤ_{>0} and some finitely ramified characters χ_{1}, χ_{2} for which (ϵ^{n}χ_{1}χ_{2}^{−1})(I_{l}) is not pro-l.

Then there is a totally real field E, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{E}) and a place λ of the field of coefficients of π above l such that ρ_{π,λ} (the λ-adic representation associated to π) is equivalent to ρ.

Combining this with a result of Brylinski-Labesse [BL], Langlands’ cyclic base change [L] and a theorem of Brauer we obtain the following corollary.

Corollary C Keep the assumptions of theorem B and choose an isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ. For all but finitely many primes p the trace and determinant of ρ(Frob_{p}) lie in ℚ^{ac} and we have

|i(tr ρ(Frob_{p}))| ≤ 2p^{n/2}.

We define the L-function of ρ with respect to i to be

L(iρ, s) = (1 − iχ_{1,I_{l}}(Frob_{l})/l^{s−n})^{−1}(1 − iχ_{2,I_{l}}(Frob_{l})/l^{s})^{−1}
∏_{p≠l} i det(1 − ρ_{I_{p}}(Frob_{p})/p^{s})^{−1},

ecept we drop the factor (1 − iχ_{1,I_{l}}(Frob_{l})/l^{s−n})^{−1} if n = 1 and χ_{1} = χ_{2}. This converges uniformly absolutely for the real part of s sufficiently large. We also define the conductor N(ρ) to be the product

N(χ_{1})N(χ_{2}) ∏_{p≠l} N(ρ|_{G_{p}}),

except we replace N(χ_{1} by l if n = 1 and χ_{1} = χ_{2} is unramified, and we drop the factor N(χ_{1}) if n = 1 and χ_{1} = χ_{2} is ramified. (Here N(ρ|_{G_{p}}) (resp. N(χ_{i})) is the usual conductor of ρ|_{G_{p}} (resp. χ_{i}).)

The function L(iρ, s) has unique meromorphic continuation to the whole complex plane and satisfies a functional equation

\begin{aligned}N(ρ)^{s/2}(2π)^{−s}Γ(s)L(iρ, s) & = & W N(ρ)^{(n+1−s)/2}(2π)^{s−1−n}Γ(n + 1 − s) \\ & & L(i(ρ ⊗ ϵ^{n}(det ρ)^{−1}), n + 1 − s), \end{aligned}

where |W | = 1.

In particular this has the following consequence.

Corollary D Suppose that A/ℚ is an abelian variety, M is a number field with [M : ℚ] = dim A and that j : O_{M} ↪ End (A/ℚ). Then the L-function of A (relative to an embedding M ↪ ℂ) has meromorphic continuation to the whole complex plane and satisfies a functional equation

N(A)^{s/2}(2π)^{−s}Γ(s)L(A, s) = W N(A)^{(2−s)/2}(2π)^{s−2}Γ(2 − s)L(A^{∨}, 2 − s),

where N(A) denotes the conductor of A and where |W | = 1.

Alternatively combining theorem B with a result of Blasius and Rogawski [BR] and restriction of scalars, we obtain the following corollary.

Corollary E Keep the assumptions of theorem B and if n = 1 further assume that

• for some prime p ≠ l we have

ρ|_{G_{p}} ∼ ( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} ) .

Then ρ occurs in the l-adic cohomology (with coefficients in some Tate twist of the constant sheaf) of some variety over ℚ. If n = 1 then there exists a number field M, a prime λ of M above l, an abelian variety A/ℚ of dimension [M : ℚ] and an embedding O_{M} ↪ End (A/ℚ) such that ρ is equivalent to the representation on the λ-adic Tate module of A.

Combining this corollary with (a slight generalisation of) the main result of [Ram] we get the following theorem, which may lend some support to a very important conjecture of Serre (see [S2]).

Theorem F Suppose that l is an odd prime and that

\overline{ρ} : G_{ℚ} ⟶ GL_{2}(𝔽_{l}^{ac})

is a continuous irreducible representation such that det \overline{ρ}(c) = −1 and

\overline{ρ}|_{G_{l}} ∼ ( \begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

with χ_{1}|_{I_{l}} ≠ χ_{2}|_{I_{l}}. Then there exists a number field M, a prime λ of M above l, an abelian variety A/ℚ of dimension [M : ℚ] and an embedding O_{M} ↪ End (A/ℚ) such that \overline{ρ} is equivalent to the representation of G_{ℚ} on A[λ].

We remark that we have not tried to optimise the conditions in these results and some improvement is certainly possible.

Let us briefly describe the proof of theorem B. Let \overline{ρ} denote a reduction of ρ. The case where \overline{ρ} is reducible is the main result of [SW1]. The case where \overline{ρ} is irreducible but soluble follows from the results of [L], [Tu], [W], [TW],

[Dia1] and [SW3]. In this paper we treat the case where \overline{ρ} has insoluble image. By the methods of [W] and [TW] and their extension to totally real fields by Diamond ([Dia2]), Fujiwara ([F]) and Skinner and Wiles ([SW2] and [SW3]), the key point here is to prove that \overline{ρ}|_{Gal (E^{ac}/E)} is modular for some totally real field E.

To describe how we do this, let us for simplicity assume that \overline{ρ} has cyclotomic determinant. We find totally real fields E and M, a rational prime p and an abelian variety A/E such that

• p and l are unramified in E,

• dim A = [M : ℚ],

• there is an embedding i : O_{M} ↪ End (A/E),

• there is a prime λ|l of O_{M} such that A[λ](E^{ac}) is equivalent to \overline{ρ}|_{Gal (E^{ac}/E)} as a Gal (E^{ac}/E)-module,

• A has good ordinary reduction at all primes of O_{E} above p,

• there is a prime ℘ of O_{M} above p such that the action of Gal (E^{ac}/E) on A[℘](E^{ac}) is of the form Ind _{Gal (L^{ac}/L)}^{Gal (E^{ac}/E)}θ for some totally imaginary quadratic extension L/E not contained in E(ζ_{p}) and some character θ of Gal (L^{ac}/L).

Given such E,M, p and A the above mentioned results of Diamond, Fujiwara and Skinner and Wiles show that the ℘-adic Tate module of A is modular and hence that \overline{ρ} is modular.

Having made a suitable choice for M and p the problem of finding a suitable E and A comes down to a problem of constructing points on certain Hilbert-Blumental modular varieties over totally real fields in which p and l are unramified. To this end we employ the following general criterion of Moret-Bailly [M] which reduces the problem to local problems at ∞, l and p.

Theorem G (Moret-Bailly) Let K be a number field and S a finite set of places of K. There is a unique maximal extension K_{S}/K (inside a given algebraic closure of K) in which all places of S split completely. (For example, ℚ_{{∞}} is the maximal totally real field.) Suppose that X/Spec K is a geometrically irreducible smooth quasi-projective scheme and that, for all v ∈ S, X(K_{v}) is non-empty. Then X(K_{S}) is Zariski dense in X.

Notation

Throughout this paper l will be an odd rational prime.

If K is a perfect field we will let K^{ac} denote its algebraic closure and G_{K} denote its absolute Galois group Gal (K^{ac}/K). If moreover p is a prime number different from the characteristic of K then we will let ϵ_{p} : G_{K} → ℤ_{p}^{×} denote the p-adic cyclotomic character and ω_{p} the Teichmuller lift of ϵ_{p} mod p. In the case p = l we will drop the subscripts and write simply ϵ = ϵ_{l} and ω = ω_{l}. If K is a local field we will let W_{K} denote the Weil group of K. If K is a number field and x is a finite place of K we will write G_{x} for a decomposition group above x, I_{x} for the inertia subgroup of G_{x} and Frob_{x} for an arithmetic Frobenius element in G_{x}/I_{x}. We will also let O_{K} denote the integers of K, 𝔡_{K} the different of K and k(x) denote the residue field of O_{K} at x. We will let c denote complex conjugation on ℂ.

We will write μ_{N} for the group scheme of N^{th} roots of unity. We will write W (k) for the Witt vectors of k. If G is a group, H a normal subgroup of G and ρ a representation of G, then we will let ρ^{H} (resp. ρ_{H}) denote the representation of G/H on the H-invariants (resp. H-coinvariants) of ρ.

Suppose that A/K is an abelian variety with and action of O_{M} for some number field M over a perfect field K. Suppose also that X is a finite index O_{M}-submodule of a free O_{M}-module. If X is free with basis e_{1}, ..., e_{r} then by A ⊗_{O_{M}} X we shall simply mean A^{r}. Note that for any ideal 𝔞 of O_{M} we have a canonical isomorphism

(A ⊗_{O_{M}} X)[𝔞] ≅ A[𝔞] ⊗_{O_{M}} X.

In general if Y ⊃ X ⊃ 𝔞Y with Y free and 𝔞 a non-zero ideal of O_{M} then we will set

(A ⊗_{O_{M}} X) = (A ⊗_{O_{M}} 𝔞Y)/(A[𝔞] ⊗_{O_{M}} X/𝔞Y).

This is canonically independent of the choice of Y ⊃ X and again we get an identification

(A ⊗_{O_{M}} X)[𝔞] ≅ A[𝔞] ⊗_{O_{M}} X.

If X has an action of some O_{M} algebra then A ⊗_{O_{M}} X canonically inherits such an action. We also get a canonical identification (A ⊗_{O_{M}} X)^{∨} ≅ A^{∨} ⊗_{O_{M}} Hom (X, ℤ). Suppose that μ : A → A^{∨} is a polarisation which induces an involution c on M. Note that c equals complex conjugation for every embedding M ↪ ℂ. Suppose also that f : X → Hom (X, ℤ) is c-semilinear for the action of O_{M}. If for all x ∈ X − {0}, the totally real number f(x)(x) is totally strictly positive then λ ⊗ f : A ⊗_{O_{M}} X → (A ⊗_{O_{M}} X)^{∨} is again a polarisation.

If λ is an ideal of O_{M} prime to the characteristic of K we will write \overline{ρ}_{A,λ} for the representation of G_{K} on A[λ](K^{ac}). If λ is prime we will write T_{λ}A for the λ-adic Tate module of A, V_{λ}A for T_{λ}A ⊗_{ℤ} ℚ and ρ_{A,λ} for the representation of G_{K} on V_{λ}A. We have a canonical isomorphism T_{λ}(A⊗_{ℂ_{M}} X) →^{∼} (T_{λ}A)⊗_{O_{M}} X.

Suppose that F is a totally real number field and that π is an algebraic cuspidal automorphic representation of GL_{2}(𝔸_{F}) with field of definition (or coefficients) M ⊂ ℂ. In some cases, including the cases that π_{∞} is regular and the case π_{∞} is weight (1, ..., 1), then it is known that M is a CM number field and that for each prime λ of O_{M} there is a continuous irreducible representation

ρ_{π,λ} : G_{F} → GL_{2}(M_{λ})

canonically associated to π. (See [Ta1] for details.) We may always conjugate ρ_{π,λ} so that it is valued in GL_{2}(O_{M,λ}) and then reduce it to get a continuous representation G_{F} → GL_{2}(O_{M}/λ). If for one such choice of conjugate the resulting representation is irreducible then it is independent of the choice of conjugate and we will denote it \overline{ρ}_{π,λ}.

1 A potential version of Serre’s conjecture

Suppose that l is an odd prime and that k/𝔽_{l} is a finite extension. Suppose also that F is a totally real field and that

\overline{ρ} : G_{F} ⟶ GL_{2}(k)

is a continuous representation such that

• \overline{ρ} has insoluble image,

• for every place v of F above l we have

\overline{ρ}|_{G_{v}} ∼ ( \begin{matrix}ϵχ_{v}^{−1} & ∗ \\ 0 & χ_{v} \end{matrix} )

• and for every complex conjugation c, det \overline{ρ}(c) = −1.

For v a prime of F above l let F̃_{v} denote the smallest totally tamely ramified extension of F_{v} over which χ_{v} becomes unramified.

Let ζ denote a primitive #k^{×} root of unity and let N_{0} = ℚ(ζ, \sqrt{1 − 4l}). Note that l is unramified in N_{0} and that each prime of N_{0} above l has residue field isomorphic to k. Choose a prime λ_{0} of N_{0} above l and an isomorphism O_{N_{0}}/λ_{0} ≅ k. For v a prime of F above l set

β_{v} = ζ^{b_{v}}((1 + \sqrt{1 − 4l})/2)^{[k(v):𝔽_{l}]}

with b_{v} chosen so that β_{v} ≡ χ_{v}(ϕ_{v}) mod λ_{0} for ϕ_{v} ∈ GF̃_{v} a lift of Frob_{v}. Let χ̃_{v} denote the unique character from W_{F_{v}} to N_{0}^{×} which,

• if χ_{v}^{2} ≠ 1, takes ϕ_{v} to β_{v} and on inertia is the Teichmuller lift of χ_{v}

• and, if χ_{v}^{2} = 1, is the Teichmuller lift of χ_{v}.

Choose an odd prime p ≠ l such that

• at all primes w of F above p, \overline{ρ} is unramified and \overline{ρ}(Frob_{w}) has distinct eigenvalues,

• p splits completely in the Hilbert class field of N_{0},

• p splits completely in (F^{ac})^{ker(ϵ^{−1} det \overline{ρ})}

• and p is coprime to β_{v} − β_{v}^{c} for all places v of F above l.

Also choose a prime ℘_{0} of N_{0} above p. For each place w of F above p choose α′_{w} ∈ ℤ[(1 + \sqrt{1 − 4l})/2] with norm p (possible because p splits completely in the Hilbert class field of ℚ((1 + \sqrt{1 − 4l})/2)) and set

α_{w} = ζ^{a_{w}}α′_{w}

with a_{w} chosen so that α_{w} is congruent modulo λ to an eigenvalue of \overline{ρ}(Frob_{w}).

Lemma 1.1 Let p be a rational prime, O the integers of a finite extension of ℚ_{p} and 𝔽 the residue field of O. Let K be a totally real field and L/K a totally imaginary quadratic extension in which every place of K above p splits. Let S be a finite set of finite places of K which split in L and suppose S contains all places of K above p. Let S_{L} be a set of places of L above S which contains exactly one place above every element of S.

Let ϕ : G_{K} → O^{×} be a continuous homomorphism

• which takes every complex conjugation to −1

• and which is of the form ϵ_{p}^{n} times a finite order character for some n ∈ ℤ.

Also for each x ∈ S_{L} let \overline{ψ}_{x} : G_{L_{x}} → 𝔽^{×} be a continuous homomorphism.

Then there is a finite extension of the fraction field of O with integers O′ and residue field k′, and a continuous character ψ : G_{L} → (O′)^{×} such that

• for all x ∈ S_{L}, ψ|_{G_{Lx}} is finitely ramified and reduces to \overline{ψ}_{x}

• and det Ind _{G_{L}}^{G_{K}}ψ = ϕ.

Proof: We can choose a character ψ_{0} : G_{L} → (O′)^{×} such that

• ϵ_{p}^{−1} det Ind _{G_{L}}^{G_{K}}ψ_{0} has finite order

• and ψ_{0}|_{I_{x}} is finite order for all x ∈ S_{L}.

Looking for ψ of the form ψ_{0}^{n}ψ′ we reduce to the case that n = 0. We may also suppose that S_{L} generates the class group of L.

For x ∈ S_{L} let ψ_{x} : L_{x}^{×} → O^{×} be the character corresponding by class field theory to the Teichmuller lift of \overline{ψ}_{x}. Let ϕ′ be the character of 𝔸_{K}^{×} associated by class field theory to ϕ times the quadratic character of G_{K} with kernel G_{L}. We must find a character ψ : 𝔸_{L}^{×}/L^{×} → (O′)^{×} which restricts to ϕ′ on 𝔸_{K}^{×} and to ψ_{x} on L_{x}^{×} for all x ∈ S_{L}. Let L_{S}^{×} (resp. K_{S}^{×}) denote the subgroup of L^{×} (resp. K^{×}) which is supported on S_{L} (resp. S). Let T denote the set of finite places of K which are not in S and at which ϕ′ is ramified. For x ∈ T choose an extension ψ_{x} of ϕ′|_{O_{K,x}^{×}} to O_{L,x}^{×}. Let

ψ_{0} : (∏_{x∈S} L_{x}^{×} × ∏_{x∈T} O_{L,x}^{×})/K_{S}^{×} ⟶ (O′)^{×}

denote the unique character which

• coincides with ϕ′ on (∏_{x∈S} K_{x}^{×} × ∏_{x∈T} O_{K,x}^{×})/K_{S}^{×},

• coincides with ψ_{x} on L_{y}^{×} for y ∈ S_{L}

• and which coincides with ψ_{x} on O_{L,x}^{×} for x ∈ T.

It suffices to find a continuous character

ψ : (∏_{x∈S} L_{x}^{×} × ∏_{y∈T} O_{L,y}^{×} × ∏_{y∉S∪T}(O_{L,y}^{×}/O_{K,y}^{×}))/L_{S}^{×} ⟶ (O′)^{×}

which extends ψ_{0}. Equivalently we must find a continuous character

∏_{y∉S∪T}(O_{L,y}^{×}/O_{K,y}^{×}) ⟶ (O′)^{×}

which coincides with ψ_{0} on L_{S}^{×}/K_{S}^{×}.

As ψ_{0} has finite order it suffices to show that any finite index subgroup of L_{S}^{×}/K_{S}^{×} contains the preimage of some open subgroup of ∏_{y∉S∪T} O_{L,y}^{×}/O_{K,y}^{×}. Considering the commutative diagram

L_{S}^{×}/K_{S}^{×}  ↪^{c−1}  L_{S}^{×}
↓  ↓
∏_{y∉S∪T} O_{L,y}^{×}/O_{K,y}^{×} ↪^{c−1} ∏_{y∉S∪T} O_{L,y}^{×},

and recalling that L_{S}^{×} is a finitely generated abelian group, we see that we only need prove that for any positive integer n the subgroup (L_{S}^{×})^{n} ⊂ L_{S}^{×} contains the preimage of some open subgroup of ∏_{y∉S∪T} O_{L,y}^{×}. This is presumably well known, see for instance lemma 2.1 of [Ta2]. □

Thus we may choose a quadratic extension L/F and a continuous character ψ : G_{L} → (N_{0,℘_{0}}^{ac})^{×} such that

• L is a totally imaginary field not contained in F adjoin a primitive p^{th} root of 1;

• each place v of F above l splits as v_{1}v_{1}^{c} in L and ψ|_{W_{Lv_{1}}} = χ̃_{v} in (O_{N_{0}}/℘_{0})^{ac};

• each place w of F above p splits as w_{1}w_{1}^{c} in L and ψ|_{G_{w_{1}}} is unramified and takes arithmetic Frobenius to a lift of α_{w} ∈ O_{N_{0}}/℘_{0};

• and det Ind _{G_{L}}^{G_{F}}ψ = ϵ_{p}.

Let \overline{ψ} : G_{L} → ((O_{N_{0}}/℘_{0})^{ac})^{×} denote the reduction of ψ. Note that for any prime v of F above l we have \overline{ψ}|_{G_{v_{1}}} ≠ \overline{ψ}^{c}|_{G_{v_{1}}} (as β_{v} − β_{v}^{c} is coprime to p). Choose N/N_{0} be a Galois CM extension such that

• primes above l split in N/N_{0},

• primes above p are unramified in N/N_{0}

• and there is a prime ℘ above ℘_{0} such that \overline{ψ} has image in O_{N}/℘.

Let λ denote a prime of O_{N} above λ_{0} and let M denote the maximal totally real subfield of N.

By an ordered invertible O_{M}-module we shall mean an invertible O_{M}- module X together with a choice of connected component X_{x}^{+} of (X⊗M_{x})−{0} for each infinite place x of M. By the standard ordered invertible O_{M}-module O_{M}^{+} we shall mean (O_{M}, {(M_{x}^{×})^{0}}), where (M_{x}^{×})^{0} denotes the connected component of 1 in M_{x}^{×}. By an M-HBAV over a field K we shall mean a triple (A, i, j) where

• A/K is an abelian variety of dimension [M : ℚ],

• i : O_{M} ↪ End (A/K)

• and j : O_{M}^{+} →^{∼} P(A, i) is an isomorphism of ordered invertible O_{M}- modules.

Here P(A, i) is the invertible O_{M} module of symmetric (i.e. f^{∨} = f) homomorphisms f : (A, i) → (A^{∨}, i^{∨}) which is ordered by taking the unique connected component of (P(A, i) ⊗ M_{x}) which contains the class of a polarisation. (See section 1 of [Rap].)

Lemma 1.2 For each place v of F above l we can find an M-HBAV (A_{v}, i_{v}, j_{v}) over F_{v} such that

• A_{v} either has potentially good ordinary reduction or potentially multiplicative reduction,

• the action of G_{v} on A_{v}[λ|_{M}] is equivalent to \overline{ρ}|_{G_{v}}

• and the action of G_{v} on A_{v}[℘|_{M}] is equivalent to \overline{ψ}_{v_{1}} ⊕ \overline{ψ}_{v_{1}^{c}}.

Proof: First suppose that χ_{v}^{2} = 1, so that (by twisting) we may suppose that χ_{v} = 1. The extension \overline{ρ}|_{G_{v}} is described by a class in

\overline{q} ∈ H^{1}(G_{v}, k(ϵ)) ≅ F_{v}^{×}/(F_{v}^{×})^{l} ⊗_{𝔽_{l}} k ≅ F_{v}^{×} ⊗_{ℤ} 𝔡_{M}^{−1}/λ𝔡_{M}^{−1}.

We may choose

q_{0} ∈ F_{v}^{×} ⊗_{ℤ} ℘𝔡_{M}^{−1} ⊂ F_{v}^{×} ⊗_{ℤ} 𝔡_{M}^{−1}

such that q_{0} reduces to \overline{q} ∈ F_{v}^{×}⊗_{ℤ}𝔡_{M}^{−1}/λ𝔡_{M}^{−1}. Now set q = q_{0}q_{1}, where we choose q_{1} ∈ F_{v}^{×} ⊗_{ℤ} ℘λ𝔡_{M}^{−1} such that tr_{M/ℚ}(av(q_{1})) > −tr_{M/ℚ}(av(q_{0})) for all totally positive elements a ∈ O_{M}. According to section 2 of [Rap] there is a M-HBAV (A_{v}, i_{v}, j_{v})/F_{v} such that A_{v}(F_{v}^{ac}) ≅ ((F_{v}^{ac})^{×} ⊗ 𝔡_{M}^{−1})/O_{M}q as a O_{M}[G_{v}]-module. This triple suffices to prove the lemma in this case.

Secondly suppose that χ_{v}^{2} ≠ 1. By the theory of Honda and Tate we can find a simple ordinary abelian variety A_{0}/k(v) of dimension [ℚ(β_{v}) : ℚ]/2 and an isomorphism i_{0} : O_{ℚ(β_{v})} →^{∼} End (A/k(v)) such that A_{0}[l](k(v)^{ac}) is isomorphic to O_{ℚ(β_{v})}/(β_{v}^{c}) and β_{v} is the Frobenius endomorphism of A_{0}/k(v). Choose a polarisation μ_{0} : A_{0} → A_{0}^{∨}. The corresponding Rosati involution must correspond to complex conjugation on O_{ℚ(β_{v})}. Set A_{1} = A_{0} ⊗_{O_{ℚ(βv)}} O_{N}, an ordinary abelain variety of dimension [M : ℚ] over k(v) with an embedding i_{1} : O_{N} ↪ End (A_{1}/k(v)). Then A_{1}[l](k(v)^{ac}) ≅ O_{N}/(β_{v}^{c}) and β_{v} is the Frobenius endomorphism. The polarisation μ_{0} and the pairing

\begin{matrix}O_{N} × O_{N} & ⟶ & O_{ℚ(α)} \\ (a, b) & ⟼ & tr_{N/ℚ(α)}(ab^{c}) \end{matrix}

defines a polarisation μ_{1} : A_{1} → A_{1}^{∨} such that the μ_{1}-Rosati involution acts as c on O_{N}. The choice of λ_{1} makes Hom_{O_{M}} (A_{1}, A_{1}^{∨}) isomorphic to a fractional ideal 𝔞 of O_{N}. The symmetric elements then correspond to 𝔞∩M with the order structure coming from the subset of totally positive elements. If we replace A_{1} by A_{1}/A_{1}[𝔟] for some ideal 𝔟 of O_{N} then P(A_{1}, i_{1}) is replaced by 𝔞𝔟𝔟^{c} ∩ M with the order structure coming from the totally positive elements. The norm map from the class group of N to the class group of M is surjective, because the Hilbert class field of M is contained in that of N and being totally positive is disjoint from N over M. Thus replacing A_{1} by A_{1}/A_{1}[𝔟] for a suitable 𝔟 we may suppose that (A_{1}, i_{1}|_{O_{M}}) extends to a M-HBAV (A_{1}, i_{1}|_{O_{M}} , j_{1}).

Let χ̃′_{v} denote the unique continuous unramified extension of χ̃_{v}|_{WF̃v} to a character

G_{v} ⟶ O_{N,(β_{v}^{c})}^{×} ≅ O_{M,l}^{×}.

Serre-Tate theory tells us that liftings of the triple (A_{1}, i_{1}|_{O_{M}} , j_{1}) to OF̃_{v} are parametrised by the extensions of M_{l}/O_{M,l}(χ̃′_{v}) by μ_{l^{∞}} ⊗ O_{M}((χ̃′_{v})^{−1}) as Barsotti-Tate groups over OF̃_{v}; that is by a subgroup

H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})) ⊂ H^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})).

(If χ̃_{v}^{2} ≠ 1 then

H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})) = H^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})),

while if χ̃_{v}^{2} = 1 then H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})) corresponds by Kummer theory to

(O× ; F̃_{v})^{∧} ⊗ O_{M,l} ⊂ (F̃_{v}^{×})^{∧} ⊗ O_{M,l},

where X^{∧} denotes l-adic completion of X.) We will write (A_{x}, i_{x}, j_{x}) for the lift corresponding to an element x of the group H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})). If σ ∈ Gal (F̃_{v}/F_{v}) then

σ(A_{x}, i_{x}, j_{x}) = (A_{σx}, i_{σx}, j_{σx}).

If γ ∈ O_{N} then i_{1}(γ) lifts to a homomorphism from (A_{x}, i_{x}, j_{x}) to (A_{y}, i_{y}, j_{y}) if and only if γγ^{c} = 1 and γ^{2}x = y, where we let O_{N} acts on O_{M,l} via the map O_{N} → O_{N,β_{v}^{c}} ≅ O_{M,l}. Thus to give a triple (A, i, j) over F_{v} which restricts to some lift of (A_{1}, i_{1}, j_{1}) over F̃_{v} is the same as giving a character ψ : Gal (F̃_{v}/F_{v}) → μ_{∞}(N) and an element x ∈ H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵ(χ̃′_{v})^{−2})) such that σx = ψ(σ)^{2}x for all σ ∈ Gal (F̃_{v}/F_{v}), i.e. by continuous characters χ′ : G_{v} → O_{M,l}^{×} with χ′|_{WF̃v} = χ̃_{v}|_{GF̃v} and elements

x ∈ H^{1}(GF̃_{v}, O_{M,l}(ϵ(χ′)^{−2}))^{Gal (L/K)} ∩ H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵχ̃_{v}^{−2})).

Note that

H^{1}(GF̃_{v}, O_{M,l}(ϵ(χ′)^{−2}))^{Gal (L/K)} ∩ H_{f}^{1}(GF̃_{v}, O_{M,l}(ϵχ̃_{v}^{−2}))

is a subgroup of H^{1}(G_{v}, O_{M,l}(ϵ(χ′)^{−2})), and equals H^{1}(G_{v}, O_{M,l}(ϵ(χ′)^{−2})) if (χ′_{v})^{2} ≠ 1.

Choose x ∈ H^{1}(G_{v}, O_{M,l}(ϵχ̃_{v}^{−2})) so that its λ-component

x_{λ} ∈ H^{1}(G_{v}, O_{M,λ}(ϵχ̃_{v}^{−2}))

maps to the class of the extension \overline{ρ}|_{G_{v}} in H^{1}(G_{v}, O_{M}/λ(ϵχ_{v}^{−2})). This is possible as H^{2}(G_{v}, O_{M,λ}(ϵχ̃_{v}^{−2})) = (0) (as it is dual to H^{0}(G_{v}, M_{λ}/O_{M,λ}(χ̃_{v}^{2}))). Finally let (A_{v}, i_{v}, j_{v}))/K correspond to (χ̃_{v}, x). □

Lemma 1.3 For each place w of F above p there is an M-HBAV (A_{w}, i_{w}, j_{w}) over F_{w} such that

• A_{w} has good ordinary reduction,

• the action of G_{w} on A_{w}[λ|_{M}] is equivalent to \overline{ρ}|_{G_{w}}

• and the action of G_{w} on both A_{v}[℘|_{M}] is equivalent to \overline{ψ}_{w_{1}} ⊕ \overline{ψ}_{w_{1}^{c}}.

Proof: This is proved in the same way as lemma 1.2 but is much easier so we leave the details to the reader. □

Lemma 1.4 For each infinite place x of F there is an M-HBAV (A_{x}, i_{x}, j_{x}) over F_{x}.

Proof: Choose an elliptic curve E/F_{x} and set A_{x} = E ⊗_{ℤ}O_{M}. Let i_{x} be the canonical action of O_{M} on A_{x}. Finally A_{x} has a polarisation corresponding to the unique principal polarisation on E and the pairing O_{M} × O_{M} → ℤ which sends (a, b) ↦ tr (ab). This shows that P(A_{x}, i_{x}) ≅ O_{M}^{+}. □

Let V_{λ}/F be the two dimensional (O_{M}/λ)-vector space scheme corresponding to \overline{ρ} and fix an alternating isomorphism a_{λ} of V_{λ} with its Cartier dual. Also let V_{℘} be the two dimensional (O_{M}/℘)-vector space scheme corresponding to Ind _{G_{L}}^{G_{F}} \overline{ψ} and fix an alternating isomorphism a_{℘} of V_{℘} with its Cartier dual. As in section 1 of [Rap] we see that there is a fine moduli space X/F for quintuples (A, i, j, m_{λ}, m_{℘}) where (A, i, j) is an M-HBAV, m_{λ} : V_{λ} →^{∼} A[λ] and m_{℘} : V_{℘} →^{∼} A[℘] such that a_{λ} corresponds to the j(1)-Weil pairing on A[λ] and a_{℘} corresponds to the j(1)-Weil pairing on A[℘]. (To define the moduli problem over a general base one must proceed as in section 1 of [Rap]. To see that the moduli space is fine note that ker(GL_{2}(O_{M,λ}) → → GL_{2}(O_{M}/λ)) has no element of finite order other than the identity.) As in section 1 of [Rap] one can see that X is smooth and one can describe for any infinite place x of F the complex manifold X(F ⊗_{F_{x}} ℂ) as a quotient of the product of [M : ℚ] copies of the upper half complex plane and deduce that X is geometrically connected.

It follows from lemmas 1.2, 1.3 and 1.4 that for any place x of F above l, p or infinity we have X(F_{v}) ≠ ∅. (Note that ρ|_{G_{x}} and (Ind _{G_{L}}^{G_{F}} \overline{ψ})|_{G_{x}} are reducible and so any alternating isomorphisms of V_{λ}×F_{x} or V_{℘}×F_{x} with its Cartier dual are equivalent.) Applying a theorem of Moret-Bailly [M], which we recalled in the introduction (theorem G), we obtain a totally real field E/F in which every place above l and p split completely and a M-HBAV (A, i, j)/E such that

• the representation of G_{E} on A[λ] is equivalent to \overline{ρ}|_{G_{E}}

• and the representation of G_{E} on A[℘] is equivalent to (Ind _{G_{L}}^{G_{F}} \overline{ψ})|_{G_{E}}.

Note that (Ind _{G_{L}}^{G_{F}} \overline{ψ})|_{G_{E}} is absolutely irreducible, because for any place x of E above p the restriction of \overline{ψ} to the two places of LE above x are different. Also note that A has semi-stable reduction at any prime x of E above p, because A[λ] is unramified at x and ker(GL_{2}(O_{M,λ}) → → GL_{2}(O_{M}/λ)) has no element of finite order other than the identity. Finally note that T_{℘}A is ordinary at any prime x of E above p, because A is semistable at x, E_{x} ≅ ℚ_{p} and the I_{x}-coinvariants of A[℘] are non-trivial.

In some cases we can conclude a little more.

Lemma 1.5 Suppose that v is an unramified place of F above l and that x is a place of E above v. Suppose also that χ_{v}^{2}|_{I_{v}} = ϵ^{n}|_{I_{v}} for some integer 0 ≤ n < l − 1. Suppose finally that if \overline{ρ}|_{G_{v}} is semisimple then n ≠ 1. Then the representation of G_{x} on T_{λ}A ⊗ ℚ_{l} has the form

( \begin{matrix}ϵ(χ′_{v})^{−1} & ∗ \\ 0 & χ′_{v} \end{matrix} )

where χ′_{v} is a tamely ramified lift of χ_{v}.

Proof: To prove the lemma we may first replace A by A ⊗_{O_{M}} O_{N} and then replace A by a twist so that

• the representation of G_{x} on A[λ] has the form

( \begin{matrix}ϵχ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

with χ_{2} unramified and χ_{1}|_{I_{x}} ∼ ϵ^{−n}

• and the representation of G_{x} on A[℘] has the form ψ_{1} ⊕ ψ_{2} with ψ_{2} unramified and ψ_{1}|_{I_{x}} = ω^{−n}.

We must show that the representation of G_{x} on T_{λ}A ⊗ ℚ_{l} has the form

( \begin{matrix}ϵχ′_{1} & ∗ \\ 0 & χ′_{2} \end{matrix} )

where χ′_{2} is an unramified lift of χ_{2}.

Looking at the action of G_{x} on T_{℘}A we see that either A has multiplicative reduction over E_{x} or it has good reduction over E_{x}(ζ_{l}). If it has multiplicative reduction then χ_{1} is unramified and the result is clear.

Suppose it has good reduction over E_{x}(ζ_{l}). We will also denote by A the Neron model of A over W (k(x)^{ac})[ζ_{l}]. The only possible simple subquotients of the finite flat group scheme A[λ]/W (k(x)^{ac})[ζ_{l}] are ℤ/lℤ and μ_{l}. As there are no non-trivial extensions of ℤ/lℤ by ℤ/lℤ nor of μ_{l} by μ_{l} over W (k(x)^{ac})[ζ_{l}] we see that there is a short exact sequence

(0) ⟶ μ_{l}^{[O_{N} /λ:𝔽_{l}]} ⟶ A[λ] ⟶ (ℤ/lℤ)^{[O_{N} /λ:𝔽_{l}]} ⟶ (0)

over W (k(x)^{ac})[ζ_{l}]. (We are using connected-etale exact sequence and the fact that Lie (A[λ] × 𝔽_{l}^{ac}) has dimension [O_{N}/λ : 𝔽_{l}].) In particular A has ordinary reduction. If \overline{ρ}|_{G_{v}} is not semi-simple we are done.

So suppose \overline{ρ}|_{G_{v}} is semi-simple. Then I_{x} either acts on Lie (A[λ] × 𝔽_{l}^{ac}) by ϵ^{−n} or ϵ^{−1}, according as A[λ]^{0} ∼ ϵχ_{1} or χ_{2} as G_{x}-modules. (See section 5 of [Ed].) If it acted by ϵ^{−1} then it would also act by ω^{−1} on some subquotient of A[℘] (see appendix B of [CDT]). Hence χ_{1}|_{I_{x}} = ω, which we are assuming does not occur when A[λ] is semi-simple as a G_{x}-module. □

Because (Ind _{G_{L}}^{G_{F}}ψ)|_{G_{E}} is modular we may apply theorem 5.1 of [SW3] to deduce that T_{℘}A is modular and hence that T_{λ}A is modular. Thus we have proved the following theorem in the case that \overline{ρ} has insoluble image. The case that \overline{ρ} has soluble image follows from known cases of the strong Artin conjecture (see [Tu], and [RT] for how to use congruences to ensure the regularity of π).

Theorem 1.6 Suppose that l is an odd prime and that k/𝔽_{l} is a finite extension. Suppose also that F is a totally real field and that

\overline{ρ} : G_{F} ⟶ GL_{2}(k)

is a continuous irreducible representation such that

1. for every place v of F above l we have

\overline{ρ}|_{G_{v}} ∼ ( \begin{matrix}ϵχ_{v}^{−1} & ∗ \\ 0 & χ_{v} \end{matrix} )

2. and for every complex conjugation c we have det \overline{ρ}(c) = −1.

Then there is a finite Galois totally real extension E/F in which every prime of F above l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{E}) and a place λ′ of the field of coefficients of π above l such that \overline{ρ}_{π,λ′} ∼ \overline{ρ}|_{G_{E}}.

Moreover E, π and λ′ may be chosen so that the following holds. If x is an unramified prime of E above l such that

• χ_{x}^{2}|_{I_{x}} = ϵ^{n}|_{I_{x}}

• and \overline{ρ}(I_{x}) does not consist of scalar matrices,

then

ρ_{π,λ′}|_{G_{x}} ∼ ( \begin{matrix}ϵ(χ′_{x})^{−1} & ∗ \\ 0 & χ′_{x} \end{matrix} )

where χ′_{x} is a tamely ramified lift of χ_{x}.

Corollary 1.7 Suppose that l is an odd prime and that k/𝔽_{l} is a finite extension. Suppose also that F is a totally real field and that

\overline{ρ} : G_{F} ⟶ GL_{2}(k)

is a continuous irreducible representation such that for every complex conjugation c we have det \overline{ρ}(c) = −1. Then there is a finite Galois totally real extension E/F in which every prime of F above l is unramified with inertial degree at most 2, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{E}) and a place λ′ of the field of coefficients of π above l such that ρ_{π,λ′} ∼ \overline{ρ}|_{G_{E}}.

Moreover let T denote the set of unramified primes v of F above l such that \overline{ρ}(I_{v}) does not consist of scalar matrices and

\overline{ρ}|_{G_{v}} ∼ ( \begin{matrix}χ_{v,1} & ∗ \\ 0 & χ_{v,2} \end{matrix} )

with (χ_{2}χ_{1}^{−1})|_{I_{v}} = ϵ^{n}|_{I_{v}} for some n ∈ ℤ/(l − 1)ℤ. Then we may choose E, π and λ′ so that for any place x of E above a place v ∈ T we have

ρ_{π,λ′}|_{G_{x}} ∼ ( \begin{matrix}χ′_{x,1} & ∗ \\ 0 & χ′_{x,2} \end{matrix} )

where χ′_{x,2} is a tamely ramified lift of χ_{v,2}.

2 Applications

Theorem 2.1 Let l be an odd prime and

ρ : G_{ℚ} ⟶ GL_{2}(ℚ_{l}^{ac})

a continuous irreducible representation such that

• ρ is unramified at all but finitely many primes,

• det ρ(c) = −1,

• and

ρ|_{G_{l}} ∼ ( \begin{matrix}ϵ^{n}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

for some n ∈ ℤ_{>0} and some finitely ramified characters χ_{1}, χ_{2} for which (ϵ^{n}χ_{1}χ_{2}^{−1})(I_{l}) is not pro-l.

Then there is a finite Galois totally real extension E/ℚ, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{E}) and a place λ′ of the field of coefficients of π above l such that ρ_{π,λ′} ∼ ρ.

Proof: Let \overline{ρ} denote a reduction of some conjugate of ρ which is valued in GL_{2}(O_{ℚ_{l}^{ac}}). If \overline{ρ} is reducible then the theorem follows from theorem A of [SW1]. If \overline{ρ} is induced from a character of a real quadratic field then \overline{ρ} is modular (of weight 1) and so the theorem follows from theorem 5.1 of [SW3]. Otherwise \overline{ρ} remains irreducible on restriction to any totally real field. In this case the theorem follows from combining corollary 1.7 with theorem 5.1 of [SW3]. □

Corollary 2.2 Keep the assumptions of theorem 2.1 and choose an isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ. For all but finitely many primes p the trace and determinant of ρ(Frob_{p}) lie in ℚ^{ac} and we have

|i(tr ρ(Frob_{p}))| ≤ 2p^{n/2}.

We define the L-function of ρ with respect to i to be

L(iρ, s) = (1 − iχ_{1,I_{l}}(Frob_{l})/l^{s−n})^{−1}(1 − iχ_{2,I_{l}}(Frob_{l})/l^{s})^{−1}
∏_{p≠l} i det(1 − ρ_{I_{p}}(Frob_{p})/p^{s})^{−1},

ecept we drop the factor (1 − iχ_{1,I_{l}}(Frob_{l})/l^{s−n})^{−1} if n = 1 and χ_{1} = χ_{2}. This converges uniformly absolutely for the real part of s sufficiently large. We also define the conductor N(ρ) to be the product

N(χ_{1})N(χ_{2}) ∏_{p≠l} N(ρ|_{G_{p}}),

except we replace N(χ_{1} by l if n = 1 and χ_{1} = χ_{2} is unramified, and we drop the factor N(χ_{1}) if n = 1 and χ_{1} = χ_{2} is ramified. (Here N(ρ|_{G_{p}}) (resp. N(χ_{i})) is the usual conductor of ρ|_{G_{p}} (resp. χ_{i}).)

The function L(iρ, s) has unique meromorphic continuation to the whole complex plane and satisfies a functional equation

\begin{aligned}N(ρ)^{s/2}(2π)^{−s}Γ(s)L(iρ, s) & = & W N(ρ)^{(n+1−s)/2}(2π)^{s−1−n}Γ(n + 1 − s) \\ & & L(i(ρ ⊗ ϵ^{n}(det ρ)^{−1}), n + 1 − s), \end{aligned}

where |W | = 1.

Proof: We will simply sketch the proof. the first assertion follows on combining theorem 2.1 with theorem 3.4.6 of [BL]. This implies the uniform absolute convergence of the L-function in some right half-plane.

By Brauer’s theorem (see for instance [S1], theorems 16 and 19 in sections 8.5 and 10.5 respectively), we may find field F_{j} ⊂ E such that Gal (E/F_{j}) is soluble, characters χ_{j} : Gal (E/F_{j}) → (ℚ^{ac})^{×} and integers n_{j} such that the trivial representation of Gal (E/ℚ) has the form

∑_{j} n_{j}Ind _{G_{F_{j}}}^{G_{ℚ}} χ_{j}.

Let χ_{j} also denote the corresponding character of 𝔸_{F_{j}}^{×}/F_{j}^{×}. By the argument of the last paragraph of the proof of theorem 2.4 of [Ta2], we see that there is a regular algebraic cuspidal automorphic representation π_{j} of GL_{2}(𝔸_{F_{j}}) such that ρ|_{G_{F_{j}}} ∼ ρ_{π_{j},l}. Then

L(iρ, s) = ∏_{j} L(π_{j} ⊗ (χ_{j} ∘ det), s)^{n_{j}}.

□

Corollary 2.3 Suppose that A/ℚ is an abelian variety, M is a number field with [M : ℚ] = dim A and that j : O_{M} ↪ End (A/ℚ). Then the L-function of A (relative to an embedding M ↪ ℂ) has meromorphic continuation to the whole complex plane and satisfies a functional equation

N(A)^{s/2}(2π)^{−s}Γ(s)L(A, s) = W N(A)^{(2−s)/2}(2π)^{s−2}Γ(2 − s)L(A^{∨}, 2 − s),

where N(A) denotes the conductor of A and where |W | = 1.

Proof: By the last corollary it suffices to find a prime λ of M such tha T_{λ}A is ordinary at l. Fix a prime μ of M. Using the Weil bound, we see that it suffices to find a prime l > 3 which is unramified in M, at which A has good reduction, which does not divide the residue characteristic of μ and such that

tr ρ_{A,μ}(Frob_{l}) ≠ 0.

The construction of such a prime l is a standard application of the Cebotarev density theorem. □

Corollary 2.4 Keep the assumptions of theorem 2.1 and if n = 1 further assume that • for some prime p ≠ l we have

ρ|_{G_{p}} ∼ ( \begin{matrix}ϵχ & ∗ \\ 0 & χ \end{matrix} ) .

Then ρ occurs in the l-adic cohomology (with coefficients in some Tate twist of the constant sheaf) of some variety over ℚ. If n = 1 then there exists a number field M, a prime λ of M above l, an abelian variety A/ℚ of dimension [M : ℚ] and an embedding O_{M} ↪ End (A/ℚ) such that

ρ_{A,λ} ∼ ρ.

Proof: The first part follows by combining theorem 2.1 with theorem 2.5.1 of [BR] and using restriction of scalars.

By theorem 2.1 and (for instance) theorem 4.12 (and proposition 2.5) of [H] there is a a totally real field E, a number field N, a prime λ′ of N above l, an abelian variety B/E of dimension [N : ℚ] and an embedding O_{N} ↪ End (B/E) such that

ρ_{B,λ′} ∼ ρ|_{G_{E}}.

Let C denote the restriction of scalars from E to ℚ of B. Then

End_{O_{N}} (C/ℚ) ⊗_{ℤ} ℚ ≅ P ⊕ ⨁_{i=1}^{r} M_{i},

where all simple constituents of P are non-abelian and where M_{i} are finite extensions of N. We have a corresponding decomposition up to isogeny

C ∼ A_{P} ⊕ ⨁_{i=1}^{r} A_{i},

where O_{M_{i}} →^{∼} End_{O_{N}} (A_{i}/ℚ). Note that

V_{λ′}C ≅ Ind _{G_{E}}^{G_{ℚ}}V_{λ′}B ≅ X ⊕ Y

where X ∼ ρ but X is not equivalent to any subquotient of Y. By Faltings theorem (End_{O_{N}} (C/ℚ) ⊗_{ℤ} ℚ)_{λ′} has a corresponding decomposition P_{X} ⊕ P_{Y} where P_{X} ↪ End (X) and P_{Y} ↪ End (Y). Thus for some choice of i = 1, ..., r and some prime λ_{i} of M_{i} above λ′ we have V_{λ_{i}}A_{1} = X. Take M = M_{i}, λ = λ_{i} and A = A_{i}. □

Our final theorem results by combining the last with a beautiful result of Ramakrishna [Ram] (but see theorem 1.3 of [Ta2] for the precise formulation we are using here).

Theorem 2.5 Suppose that is an odd prime and that

\overline{ρ} : G_{ℚ} ⟶ GL_{2}(𝔽_{l}^{ac})

is a continuous irreducible representation such that det \overline{ρ}(c) = −1 and

\overline{ρ}|_{G_{l}} ∼ ( \begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix} )

with χ_{1}|_{I_{l}} ≠ χ_{2}|_{I_{l}}. Then there exists a number field M, a prime λ of M above l, an abelian variety A/ℚ of dimension [M : ℚ] and an embedding O_{M} ↪ End (A/ℚ) such that \overline{ρ} is equivalent to the representation of G_{ℚ} on A[λ].

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