On the Meromorphic Continuation

of Degree Two L-Functions

To John Coates on the occasion of his 60^{th} birthday, with much gratitude.

Richard Taylor ^{1}

Received: January 9, 2005 Revised: June 28, 2006

Abstract. We prove that the L-function of any regular (distinct Hodge numbers), irreducible, rank two motive over the rational numbers has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

2000 Mathematics Subject Classification: 11R39, 11F80, 11G40. Keywords and Phrases: Galois representation, modularity, L-function, meromorphic continuation.

Introduction

In this paper we extend the results of [Tay4] from the ordinary to the crystalline, low weight (i.e. in the Fontaine-Laffaille range) case. The underlying ideas are the same. However this extension allows us to prove the meromorphic continuation and functional equation for the L-function of any regular (i.e. distinct Hodge numbers) rank two “motive” over ℚ. We avoid having to know what is meant by “motive” by working instead with systems of l-adic representations satisfying certain conditions which will be satisfied by the l-adic realisations of any “motive”. More precisely by a rank 2 weakly compatible system of l-adic representations R over ℚ we shall mean a 5-tuple (M, S, {Q_{p}(X)}, {ρ_{λ}}, {n_{1}, n_{2}}) where • M is a number field;

1This material is based upon work partially supported by the National Science Foundation under Grant Nos. 9702885 and 0100090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Documenta Mathematica · Extra Volume Coates (2006) 729–779

• S is a finite set of rational primes;

• for each prime p ∉ S of ℚ, Q_{p}(X) is a monic degree 2 polynomial in M[X];

• for each prime λ of M (with residue characteristic l say)

ρ_{λ} : G_{ℚ} ⟶ GL_{2}(M_{λ})

is a continuous representation such that, if l ∉ S then ρ_{λ}|_{G_{l}} is crystalline, and if p ∉ S ∪ {l} then ρ_{λ} is unramified at p and ρ_{λ}(Frob_{p}) has characteristic polynomial Q_{p}(X); and

• n_{1}, n_{2} are integers such that for all primes λ of M (lying above a rational prime l) the representation ρ_{λ}|_{G_{l}} is Hodge-Tate with numbers n_{1} and n_{2}, i.e. ρ_{λ}⊗_{ℚ_{l}} ℚ̂_{l}^{ac} ≅ (M_{λ}⊗_{ℚ_{l}} ℚ̂_{l}^{ac})(−n_{1})⊕(M_{λ}⊗_{ℚ_{l}} ℚ̂_{l}^{ac})(−n_{2}) as M_{λ}⊗_{ℚ_{l}} ℚ̂_{l}^{ac}- modules with M_{λ}-linear, ℚ̂_{l}^{ac}-semilinear G_{ℚ_{l}}-actions.

We call R regular if n_{1} ≠ n_{2} and det ρ_{λ}(c) = −1 for one (and hence all) primes λ of M. We remark that if R arises from a regular (distinct Hodge numbers) motive then one can use the Hodge realisation to check that det ρ_{λ}(c) = −1 for all λ. Thus we consider this oddness condition part of regularity. It is not difficult to see that if one of the ρ_{λ} is absolutely reducible so are all the others. In this case we call R reducible, otherwise we call it irreducible. (If ρ_{λ_{0}}^{ss} is the sum of two characters these characters are Hodge-Tate and hence by results of [S1] themselves fit into compatible systems. The elements of these compatible systems provide the Jordan-Hölder factors of the other ρ_{λ}.) We will call R strongly compatible if for each rational prime p there is a Weil-Deligne representation WD_{p}(R) of W_{ℚ_{p}} such that for primes λ of M not dividing p, WD_{p}(R) is equivalent to the Frobenius semi-simplification of the Weil-Deligne representation associated to ρ_{λ}|_{G_{p}}. (WD_{p}(R) is defined over \overline{M}, but it is equivalent to all its Gal (\overline{M}/M)-conjugates.) If R is strongly compatible and if i : M ↪ ℂ then we define an L-function L(iR, s) as the infinite product

L(iR, s) = ∏_{p} L_{p}(iWD_{p}(R)^{∨} ⊗ |Art^{−1}|_{p}^{−s})^{−1}

which may or may not converge. Fix an additive character Ψ = ∏ Ψ_{p} of 𝔸/ℚ with Ψ_{∞}(x) = e^{2π\sqrt{−1}x}, and a Haar measure dx = ∏ dx_{p} on 𝔸 with dx_{∞} the usual measure on ℝ and with dx(𝔸/ℚ) = 1. If, say, n_{1} > n_{2} then we can also also define an ϵ-factor ϵ(iR, s) by the formula

ϵ(iR, s) = \sqrt{−1}^{1+n_{1}−n_{2}} ∏_{p} ϵ(iWD_{p}(RS)^{∨} ⊗ |Art^{−1}|_{p}^{−s}, Ψ_{p}, dx_{p}).

(See [Tat] for the relation between l-adic representations of G_{ℚ_{p}} and Weil-Deligne representations of W_{ℚ_{p}}, and also for the definition of the local L and ϵ-factors.)

Theorem A Suppose that R = (M, S, {Q_{x}(X)}, {ρ_{λ}}, {n_{1}, n_{2}})/ℚ is a regular, irreducible, rank 2 weakly compatible system of l-adic representations with n_{1} > n_{2}. Then the following assertions hold.

1. If i : M ↪ ℂ then there is a totally real Galois extension F/ℚ and a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) such that L(iR|_{G_{F}} , s) = L(π, s).

2. For all rational primes p ∉ S and for all i : M ↪ ℂ the roots of i(Q_{p}(X)) have absolute value p^{−(n_{1}+n_{2})/2}.

3. R is strongly compatible.

4. For all i : M ↪ ℂ, the L-function L(iR, s) converges in Re s > 1 − (n_{1} + n_{2})/2, has meromorphic continuation to the entire complex plane and satisfies a functional equation

(2π)^{−(s+n_{1})}Γ(s + n_{1})L(iR, s) = ϵ(iR, s)(2π)^{s+n_{2}−1}Γ(1 − n_{2} − s)L(iR^{∨}, 1 − s).

More precisely we express L(iR, s) as a ratio of products of the L-functions associated to Hilbert modular forms over different subfields of F. (See section 6 for more details.) For example suppose that X/ℚ is a rigid Calabi-Yau 3-fold, where by rigid we mean that H^{2,1}(X(ℂ), ℂ) = (0). Then the zeta function ζ_{X}(s) of X has meromorphic continuation to the entire complex plane and satisfies a functional equation relating ζ_{X}(s) and ζ_{X}(4 − s). A more precise statement can be found in section six. Along the way we prove the following result which may also be of interest. It partially confirms the Fontaine-Mazur conjecture, see [FM].

Theorem B Let l > 3 be a prime and let 2 ≤ k ≤ (l + 1)/2 be an integer. Let ρ : G_{ℚ} → GL_{2}(ℚ_{l}^{ac}) be a continuous irreducible representation such that

• ρ ramifies at only finitely many primes,

• det ρ(c) = −1,

• ρ|_{G_{l}} is crystalline with Hodge-Tate numbers 0 and 1 − k.

Then the following assertions hold.

1. There is a Galois totally real field F in which l is unramified, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ of the field of rationality of π into ℚ_{l}^{ac} such that

• ρ_{π,λ} ∼ ρ|_{G_{F}}, • π_{x} is unramified for all places x of E above l, and

• π_{∞} has parallel weight k.

2. If ρ is unramified at a prime p and if α is an eigenvalue of ρ(Frob_{p}) then α ∈ ℚ^{ac} and for any isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ we have

|iα|^{2} = p^{(k−1)/2}.

3. Fix an isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ. There is a rational function L_{l,i}(X) ∈ ℂ(X) such that the product

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

converges in Re s > (k + 1)/2 and extends to a meromorphic function on the entire complex plane which satisfies a functional equation

(2π)^{−s}Γ(s)L(iρ, s) = W N(ρ)^{k/2−s}(2π)^{s−k}Γ(k − s)L(i(ρ^{∨} ⊗ ϵ^{k−1}), k − s),

where ϵ denotes the cyclotomic character, where N(ρ) denotes the conductor of ρ (which is prime to l), and where W is a complex number. (W is given in terms of local ϵ-factors in the natural way. See section 6 for details.)

4. If k = 2 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.

Again we actually show that L(iρ, s) is a ratio of products of the L-functions associated to Hilbert modular forms over different subfields of F. (See section 6 for more details.) For further discussion of the background to these results and for a sketch of the arguments we use we refer the reader to the introduction of [Tay4]. The first three sections of this paper are taken up generalising results of Wiles [W2] and of Wiles and the author [TW] to totally real fields. Previous work along these lines has been undertaken by Fujiwara [Fu] (unpublished) and Skinner and Wiles [SW2]. However the generalisation we need is not available in the literature, so we give the necessary arguments here. We claim no great originality, this is mostly a technical exercise. We hope, however, that other authors may find theorems 2.6, 3.2 and 3.3 of some use. In the fourth and fifth sections we generalise some of our results from [Tay4] about a potential version of Serre’s conjecture. This is the most original part of this paper. The main result is theorem 5.7. Finally in section six we combine theorems 3.3 and 5.7 to deduce the main results of this paper which we have summarised above. We would like to apologise for the long delay in submitting this paper (initially made available on the web in 2001) for publication. We would also like to thank the referee for reading the paper very carefully and making several useful suggestions.

Notation

Throughout this paper l will denote a rational prime, usually assumed to be odd and often assumed to be > 3. 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 Teichmüller 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}. We will let c denote complex conjugation on ℂ. If K is an l-adic field we will let | |_{K} denote the absolute value on K normalised to take uniformisers to the inverse of the cardinality of the residue field of K. We will let I_{K} denote the inertia subgroup of G_{K}, W_{K} denote the Weil group of K and Frob_{K} ∈ W_{K}/I_{K} denote an arithmetic Frobenius element. We will also let Art : K^{×} →^{∼} W_{K}^{ab} denote the Artin map normalised to take uniformisers to arithmetic Frobenius elements. Please note these unfotunate conventions. We apologise for making them. (They are inherited from [CDT].) By an ndimensional Weil-Deligne representation of W_{K} over a field M we shall mean a pair (r, N) where r : W_{K} → GL_{n}(M) is a homomorphism with open kernel and where N ∈ M_{n}(M) satisfies

r(σ)Nr(σ)^{−1} = |Art^{−1}σ|_{K}^{−1}N

for all σ ∈ W_{K}. We call (r, N) Frobenius semi-simple if r is semi-simple. For n ∈ ℤ_{>0} we define a character ω_{K,n} : I_{K} → (K^{ac})^{×} by

ω_{K,n}(σ) = σ( \sqrt[l^{n}−1]{l})/ \sqrt[l^{n}−1]{l}.

We will often write ω_{n} for ω_{ℚ_{l},n}. Note that ω_{K,1} = ω. Now suppose that K/ℚ_{l} is a finite unramified extension, that O is the ring of integers of a finite extension of K with maximal ideal λ and that 2 ≤ k ≤ l − 1. Let MF_{K,O,k} denote the abelian category whose objects are finite length O_{K} ⊗_{ℤ_{l}} O-modules D together with a distinguished submodule D^{0} and Frob_{K} ⊗ 1-semilinear maps φ_{1−k} : D → D and φ_{0} : D^{0} → D such that

• φ_{1−k}|_{D^{0}} = l^{k−1}φ_{0}, and

• Im φ_{1−k} + Im φ_{0} = D.

Also let MF_{K,O/λ^{n},k} denote the full subcategory of objects D with λ^{n}D = (0). If D is an object of MF_{K,O,k} we define D^{∗}[1 − k] by • D^{∗}[1 − k] = Hom (D, ℚ_{l}/ℤ_{l});

• D^{∗}[1 − k]^{0} = Hom (D/D^{0}, ℚ_{l}/ℤ_{l});

• φ_{1−k}(f)(z) = f(l^{k−1}x + y), where z = φ_{1−k}(x) + φ_{0}(y);

• φ_{0}(f)(z) = f(x mod D^{0}), where z ≡ φ_{1−k}(x) mod (φ_{0}D^{0}).

There is a fully faithful, O-length preserving, exact, O-additive, covariant functor 𝕄 from MF_{K,O,k} to the category of continuous O[G_{K}]-modules with essential image closed under the formation of sub-objects. (See [FL], especially section 9. In the notation of that paper 𝕄(D) = \underline{U}_{S}(D^{∗}), where D^{∗} is D^{∗}[1−k] with its filtration shifted by k − 1. The reader could also consult section 2.5 of [DDT], where the case k = 2 and K = ℚ_{l} is discussed.) If K is a number field and x is a finite place of K we will write K_{x} for the completion of K at x, k(x) for the residue field of x, ϖ_{x} for a uniformiser in K_{x}, 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 and 𝔡_{K} the different of K. If S is a finite set of places of K we will write K_{S}^{×} for the subgroup of K^{×} consisting of elements which are units outside S. We will write 𝔸_{K} for the adeles of K and || || for ∏_{x} | |_{F_{x}} : 𝔸_{K}^{×} → ℝ^{×}. We also use Art to denote the global Artin map, normalised compatibly with our local normalisations. 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. We will also let ρ^{ss} denote the semisimplification of ρ, ad ρ denote the adjoint representation and ad^{0}ρ denote the kernel of the trace map from ad ρ to the trivial representation. Suppose that A/K is an abelian variety over a perfect field K with an action of O_{M} defined over K, for some number field M. Suppose also that X is a finite torsion free O_{M}-submodule. The functor on K-schemes S ↦ A(S) ⊗_{O_{M}} X is represented by an abelian variety A ⊗_{O_{M}} X. (If X is free with basis e_{1}, ..., e_{r} then we can take A ⊗_{O_{M}} X = A^{r}. Note that for any ideal 𝔞 of O_{M} we then 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 principal ideal of O_{M} prime to the characteristic of K then we can take

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

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, O_{M}). 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_{O_{M}} (X, O_{M}) is c-semilinear. If for all x ∈ X − {0}, the totally real number f(x)(x) is totally strictly positive then μ ⊗ f : A ⊗_{O_{M}} X → (A ⊗_{O_{M}} X)^{∨} is again a polarisation which induces c on M. 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⊗_{O_{M}} X) →^{∼} (T_{λ}A)⊗_{O_{M}} X. Suppose that M is a totally real field. 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. If 𝔞 is a fractional ideal in M then we will denote by 𝔞^{+} the invertible ordered O_{M}- module (𝔞, {(M_{x}^{×})^{0}}), where (M_{x}^{×})^{0} denotes the connected component of 1 in M_{x}^{×}. By an M-HBAV (‘Hilbert-Blumenthal abelian variety’) 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 : (𝔡_{M}^{−1})^{+} →^{∼} 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].) If λ is a prime of M and if x ∈ 𝔡_{M}^{−1} then j(x) : A → A^{∨} gives rise to an alternating pairing

e_{j,x,0} : T_{λ}A × T_{λ}A ⟶ ℤ_{l}(1).

This corresponds to a unique O_{M,λ}-bilinear alternating pairing

e_{j,x} : T_{λ}A × T_{λ}A ⟶ 𝔡_{M,λ}^{−1}(1),

which are related by e_{j,x,0} = tr ∘ e_{j,x}. The pairing x^{−1}e_{j,x} is independent of x and gives a perfect O_{M,λ}-bilinear alternating pairing

e_{j} : T_{λ}A × T_{λ}A ⟶ O_{M,λ}(1),

which we will call the j-Weil pairing. (See section 1 of [Rap].) Again using the trace, we can think of e_{j} as an O_{M,λ}-linear isomorphism

ẽ_{j} : T_{λ}A ⊗ 𝔡_{M}^{−1} ⟶ Hom_{ℤ_{l}}(T_{λ}A, ℤ_{l}(1)).

More precisely

ẽ_{j}(a ⊗ y)(b) = tr (ye_{j}(a, b)) = e_{j,x,0}(x^{−1}ya, b).

The same formula (for x ∈ 𝔡_{M}^{−1}−𝔞𝔡_{M}^{−1}) gives rise to an O_{M,λ}-linear isomorphism

ẽ_{j} : A[𝔞] ⊗_{O_{M}} 𝔡_{M}^{−1} ⟶ A[𝔞]^{∨},

which is independent of x and which we will refer to as the j-Weil pairing on A[𝔞]. Suppose that F is a totally real number field and that π is an algebraic (see for instance [Cl]) cuspidal automorphic representation of GL_{2}(𝔸_{F}) with field of definition (or coefficients) M ⊂ ℂ. (That is M is the fixed field of the group of automorphisms σ of ℂ with σπ^{∞} = π^{∞}. By the strong multiplicity one theorem this is the same as the fixed field of the group of automorphisms σ of ℂ with σπ_{x} ≅ π_{x} for all but finitely many places x of F.) We will say that π_{∞} has weight (k⃗, w⃗) ∈ ℤ_{>0}^{Hom (F,ℝ)} × ℤ^{Hom (F,ℝ)} if for each infinite place τ : F ↪ ℝ the representation π_{τ} is the (k_{τ} − 1)^{st} lowest discrete series representation of GL_{2}(F_{x}) ≅ GL_{2}(ℝ) (or in the case k_{τ} = 1 the limit of discrete series representation) with central character a ↦ a^{2−k_{τ} −2w_{τ}}. Note that w = k_{τ} + 2w_{τ} must be independent of τ. If π_{∞} has weight ((k, ..., k), (0, ..., 0)) we will simply say that it has weight k. In some cases, including the cases that π_{∞} is regular (i.e. k_{τ} > 1 for all τ) and the case π_{∞} has weight 1, it is known that M is a CM number field and that for each rational prime l and each embedding λ : M ↪ ℚ_{l}^{ac} there is a continuous irreducible representation

ρ_{π,λ} : G_{F} → GL_{2}(M_{λ})

canonically associated to π. For any prime x of F not dividing l the restriction ρ_{π,λ}|_{G_{x}} depends up to Frobenius semi-simplification only on π_{x} (and λ). (See [Tay1] for details. To see that M is a CM field one uses the Peterssen inner product

(f_{1}, f_{2}) = ∫_{GL_{2}(F)(ℝ_{>0}^{×})^{Hom (F,ℝ)}} f_{1}(g)^{c}(f_{2}(g))|| det g||^{w−2}dg.

For all σ ∈ Aut (ℂ) the representation ^{σ}π^{∞} extends to an algebraic automorphic representation π(σ) of GL_{2}(𝔸_{F}) with the same value for w. The pairing ( , ) gives an isomorphism ^{c}π(σ) ≅ π(σ)^{∨}|| det ||^{2−w}. Thus ^{σ^{−1}cσ}π^{∞} is independent of σ and M is a CM field.) We will write ρ_{π,λ}|_{W_{Fx}}^{ss} = WD_{λ}(π_{x}), where WD_{λ}(π_{x}) is a semi-simple two-dimensional representation of W_{F_{x}}. If π_{x} is unramified then WD_{λ}(π_{x}) is also unramified and WD_{λ}(π_{x})(Frob_{x}) has characteristic polynomial

X^{2} − t_{x}X + (Nx)s_{x}

where t_{x} (resp. s_{x}) is the eigenvalue of

[GL_{2}(O_{F_{x}}) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix} ) GL_{2}(O_{F_{x}})]

(resp. of

[GL_{2}(O_{F_{x}}) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & ϖ_{x} \end{matrix} ) GL_{2}(O_{F_{x}})])

on π_{x}^{GL_{2}(O_{Fx})}. An explicit description of some other instances of WD_{λ}(π_{x}) may be found in section 4 of [CDT]. 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}(𝔽_{l}^{ac}). 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 l-adic modular forms on definite quaternion algebras

In this section we will establish some notation and recall some facts about l-adic modular forms on some definite quaternion algebras. To this end, fix a prime l > 3 and a totally real field F of even degree in which l is unramified. Let D denote the division algebra with centre F which ramifies exactly at the set of infinite places of F. Fix a maximal order O_{D} in D and isomorphisms O_{D,x} ≅ M_{2}(O_{F,x}) for all finite places x of F. These choices allow us to identify GL_{2}(𝔸_{F}^{∞}) with (D ⊗_{ℚ} 𝔸^{∞})^{×}. For each finite place x of F also fix a uniformiser ϖ_{x} of O_{F,x}. Also let A be a topological ℤ_{l}-algebra which is either an algebraic extension of ℚ_{l}, the ring of integers in such an extension or a quotient of such a ring of integers. Let U = ∏_{x} U_{x} be an open compact subgroup of GL_{2}(𝔸_{F}^{∞}) and let ψ : (𝔸_{F}^{∞})^{×}/F^{×} → A^{×} be a continuous character. Also let τ : U_{l} → Aut (W_{τ}) be a continuous representation of U_{l} on a finite A-module W_{τ} such that

τ|_{U_{l}∩O_{F,l}^{×}} = ψ|_{U_{l}∩O_{F,l}^{×}}^{−1}.

We will write W_{τ,ψ} for W_{τ} when we want to think of it as a U(𝔸_{F}^{∞})^{×}-module with U acting via τ and (𝔸_{F}^{∞})^{×} by ψ^{−1}. We define S_{τ,ψ}(U) to be the space of continuous functions

f : D^{×}\GL_{2}(𝔸_{F}^{∞}) ⟶ W_{τ}

such that

• f(gu) = τ(u_{l})^{−1}f(g) for all g ∈ GL_{2}(𝔸_{F}^{∞}) and all u ∈ U, and

• f(gz) = ψ(z)f(g) for all g ∈ GL_{2}(𝔸_{F}^{∞}) and all z ∈ (𝔸_{F}^{∞})^{×}.

If

GL_{2}(𝔸_{F}^{∞}) = ∐_{i} D^{×}t_{i}U(𝔸_{F}^{∞})^{×}

then

\begin{matrix}S_{τ,ψ}(U) & ⟶^{∼} & ⨁_{i} W_{τ,ψ}^{(U(𝔸_{F}^{∞})^{×}∩t_{i}^{−1}D^{×}t_{i})/F^{×}} \\ f & ⟼ & (f(t_{i}))_{i}. \end{matrix}

The index set over which i runs is finite.

Lemma 1.1 Each group (U(𝔸_{F}^{∞})^{×} ∩ t_{i}^{−1}D^{×}t_{i})/F^{×} is finite and, as we are assuming l > 3 and l is unramified in F, the order of (U(𝔸_{F}^{∞})^{×} ∩ t_{i}^{−1}D^{×}t_{i})/F^{×} is not divisible by l.

Proof: Set V = ∏_{x |̸∞} O_{F,x}^{×}. Then we have exact sequences

\begin{matrix}(0) ⟶ (UV ∩ t_{i}^{−1}D^{det=1}t_{i})/\{±1\} ⟶ (U(𝔸_{F}^{∞})^{×} ∩ t_{i}^{−1}D^{×}t_{i})/F^{×} ⟶ \\ (((𝔸_{F}^{∞})^{×})^{2}V ∩ F^{×})/(F^{×})^{2} \end{matrix}

and

(0) ⟶ O_{F}^{×}/(O_{F}^{×})^{2} ⟶ (((𝔸_{F}^{∞})^{×})^{2}V ∩ F^{×})/(F^{×})^{2} ⟶ H[2] ⟶ (0),

where H denotes the class group of O_{F}. We see that (((𝔸_{F}^{∞})^{×})^{2}V ∩F^{×})/(F^{×})^{2} is finite of 2-power order. Moreover UV ∩ t_{i}^{−1}D^{det=1}t_{i} is finite. For l > 3 and l unramified in F, D^{×} and hence UV ∩ t_{i}^{−1}D^{det=1}t_{i} contain no elements of order exactly l. The lemma follows. □

Corollary 1.2 If B is an A-algebra then

S_{τ,ψ}(U) ⊗_{A} B ⟶^{∼} S_{τ⊗_{A}B,ψ}(U).

If x |̸l, or if x|l but τ|_{U_{x}} = 1, then the Hecke algebra A[U_{x}\GL_{2}(F_{x})/U_{x}] acts on S_{τ,ψ}(U). Explicitly, if

U_{x}hU_{x} = ∐_{i} h_{i}U_{x}

then

([U_{x}hU_{x}]f)(g) = ∑_{i} f(gh_{i}).

Let U_{0} denote ∏_{x} GL_{2}(O_{F,x}). Now suppose that 𝔫 is an ideal of O_{F} and that, for each finite place x of F diving 𝔫, H_{x} is a quotient of (O_{F,x}/𝔫_{x})^{×}. Then we will write H for ∏_{x|𝔫} H_{x} and we will let U_{H}(𝔫) = ∏_{x} U_{H}(𝔫)_{x} denote the open subgroup of GL_{2}(𝔸_{F}^{∞}) defined by setting U_{H}(𝔫)_{x} to be the subgroup of GL_{2}(O_{F,x}) consisting of elements

( \begin{matrix}a & b \\ c & d \end{matrix} )

with c ∈ 𝔫_{x} and, in the case x|𝔫, with ad^{−1} mapping to 1 in H_{x}. If x |̸l𝔫 then we will let T_{x} denote the Hecke operator

[U_{H}(𝔫) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix} ) U_{H}(𝔫)]

and S_{x} the Hecke operator

[U_{H}(𝔫) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)] .

If x|𝔫 and, either x |̸l or x|l but τ|_{U_{H}(𝔫)} = 1, then we will set

⟨h⟩ = [U_{H}(𝔫) ( \begin{matrix}h̃ & 0 \\ 0 & 1 \end{matrix} ) U_{H}(𝔫)]

for h ∈ H_{x} and h̃ a lift of h to O_{F,x}^{×}; and

U_{ϖ_{x}} = [U_{H}(𝔫) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix} ) U_{H}(𝔫)];

and

V_{ϖ_{x}} = [U_{H}(𝔫) ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)];

and

S_{ϖ_{x}} = [U_{H}(𝔫) ( \begin{matrix}ϖ_{x} & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)] .

For x|𝔫 we note the decompositions

U_{H}(𝔫)_{x} ( \begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix} ) U_{H}(𝔫)_{x} = ∐_{a∈k(x)} ( \begin{matrix}ϖ_{x} & ã \\ 0 & 1 \end{matrix} ) U_{H}(𝔫)_{x},

and

U_{H}(𝔫)_{x} ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)_{x} = ∐_{a∈k(x)} ( \begin{matrix}ϖ_{x} & 0 \\ ϖ_{x}ã & 1 \end{matrix} ) U_{H}(𝔫)_{x}

and

U_{H}(𝔫)_{x} ( \begin{matrix}ϖ_{x} & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)_{x} = ( \begin{matrix}ϖ_{x} & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{H}(𝔫)_{x},

where ã is some lift of a to O_{F,x}. We will let h_{τ,A,ψ}(U_{H}(𝔫)) denote the A-subalgebra of End_{A}(S_{τ,ψ}(U_{H}(𝔫))) generated by T_{x} for x |̸l𝔫 and by U_{ϖ_{x}} for x|𝔫 but x |̸l. It is commutative. We will call a maximal ideal 𝔪 of h_{τ,A,ψ}(U_{H}(𝔫)) Eisenstein if it contains T_{x} − 2 and S_{x} − 1 for all but finitely many primes x of F which split completely in some finite abelian extension of F. (The following remark may help explain the form of this definition. If \overline{ρ} : G_{F} → GL_{2}(\overline{𝔽}_{l}) is a continuous reducible representation, then there is a finite abelian extension L/F such that tr \overline{ρ}(G_{L}) = \{2\} and (ϵ_{l}^{−1} det \overline{ρ})(G_{L}) = \{1\}.) For k ∈ ℤ_{≥2} and we will let Symm^{k−2}(A^{2}) denote the space of homogeneous polynomials of degree k − 2 in two variables X and Y over A with a GL_{2}(A)- action via

(( \begin{matrix}a & b \\ c & d \end{matrix} ) f) (X, Y) = f ((X, Y) ( \begin{matrix}a & b \\ c & d \end{matrix} )) = f(aX + cY, bX + dY).

Let A be an O_{L} algebra for some extension L/ℚ_{l} containing the images of all embeddings F ↪ ℚ_{l}^{ac}. Suppose that (k⃗, w⃗) ∈ ℤ_{>1}^{Hom (F,ℚ_{l}^{ac})}×ℤ^{Hom (F,ℚ_{l}^{ac})} is such that k_{σ} +2w_{σ} is independent of σ. We will write τ_{(k⃗,w⃗),A} for the representation of GL_{2}(O_{F,l}) on W_{(k⃗,w⃗),A} = ⨂_{σ:F →ℚ_{l}^{ac}} Symm^{k_{σ}−2}(A^{2}) via

g ⟼ ⊗_{σ:F →ℚ_{l}^{ac}}(Symm^{k_{σ}−2}(σg) ⊗ det^{w_{σ}}(σg)).

We will also write S_{(k⃗,w⃗),A,ψ}(U) for S_{τ_{(k⃗,w⃗),A},ψ}(U). Let S_{(k⃗,w⃗),A,ψ}^{triv}(U) denote (0)

unless (k⃗, w⃗) = ((2, ..., 2), (w, ..., w)), in which case let it denote the subspace of S_{(k⃗,w⃗),A,ψ}(U) consisting of functions f which factor through the reduced norm. Set

S_{(k⃗,w⃗),A,ψ}(U_{l}) = lim_{→U^{l}} S_{(k⃗,w⃗),A,ψ}(U^{l} × U_{l}).

It has a smooth action of GL_{2}(𝔸_{F}^{∞,l}) (by right translation). If (k⃗, w⃗) = ((k, ..., k), (0, ..., 0)) then we will often write k in place of (k⃗, w⃗). Set

S_{2,A,ψ} = lim_{→U} S_{2,A,ψ}(U)

and

S_{2,A,ψ}^{triv} = lim_{→U} S_{2,A,ψ}^{triv}(U).

They have smooth actions of GL_{2}(𝔸_{F}^{∞}).

Lemma 1.3 Suppose that (k⃗, w⃗) ∈ ℤ_{>1}^{Hom (F,ℚ_{l}^{ac})} × ℤ^{Hom (F,ℚ_{l}^{ac})} and w = k_{σ} − 1 + 2w_{σ} is independent of σ. Also suppose that ψ : 𝔸_{F}^{×}/F^{×} → (ℚ_{l}^{ac})^{×} is a continuous character satisfying ψ(a) = (Na)^{1−w} for all a in a non-empty open subgroup of F_{l}^{×}. Choose an isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ. Define i(k⃗, w⃗) = (ik⃗, iw⃗) ∈ ℤ_{>1}^{Hom (F,ℂ)} × ℤ^{Hom (F,ℂ)} by (ik⃗)_{τ} = k⃗_{i^{−1}τ} and (iw⃗)_{τ} = w_{i^{−1}τ}. Also define ψ_{i} : 𝔸_{F}^{×}/F^{×} → ℂ^{×} by ψ_{i}(z) = i((Nz_{l})^{w−1}ψ(z^{∞}))(Nz_{∞})^{1−w}. Then we have the following assertions.

1. S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}(U_{l}) is a semi-simple admissible representation of GL_{2}(𝔸_{F}^{∞,l})

and S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}(U_{l})^{U^{l}} = S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}(U_{l} × U^{l}).

2. There is an isomorphism

(S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}(U_{l})/S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}^{triv}(U_{l})) ⊗_{ℚ_{l}^{ac},i} ℂ ≅ ⨁_{π} π^{∞,l} ⊗ π_{l}^{U_{l}}

where π runs over regular algebraic cuspidal automorphic representations of GL_{2}(𝔸_{F}) such that π_{∞} has weight (k⃗, w⃗) and such that π has central character ψ_{i}.

3. S_{2,ℚ_{l}^{ac},ψ} is a semi-simple admissible representation of GL_{2}(𝔸_{F}^{∞}) and

S_{2,ℚ_{l}^{ac},ψ}^{U} = S_{2,ℚ_{l}^{ac},ψ}(U).

4. There is an isomorphism

S_{2,ℚ_{l}^{ac},ψ} ⊗_{ℚ_{l}^{ac},i} ℂ ≅ ⨁_{χ} ℚ_{l}^{ac}(χ) ⊕ ⨁_{π} π^{∞}

where π runs over regular algebraic cuspidal automorphic representations of GL_{2}(𝔸_{F}) such that π_{∞} has weight 2 and such that π has central character ψ_{i}, and where χ runs over characters (𝔸_{F}^{∞})^{×}/F_{>>0}^{×} → (ℚ_{l}^{ac})^{×} with χ^{2} = ψ.

Proof: We will explain the first two parts. The other two are similar. Let C^{∞}(D^{×}\(D ⊗_{ℚ} 𝔸)^{×}/U_{l}, ψ_{∞}) denote the space of smooth functions

D^{×}\(D ⊗_{ℚ} 𝔸)^{×}/U_{l} ⟶ ℂ

which transform under 𝔸_{F}^{×} by ψ_{∞}. Let τ_{∞} denote the representation of D_{∞}^{×} on W_{τ_{∞}} = W_{(k⃗,w⃗),ℚ_{l}^{ac}} ⊗_{i} ℂ via

g ⟼ ⊗_{σ:F →ℚ_{l}^{ac}}(Symm^{k_{σ}−2}(iσg) ⊗ det^{w_{σ}}(iσg)).

Then there is an isomorphism

S_{(k⃗,w⃗),ℚ_{l}^{ac},ψ}(U_{l}) ⟶^{∼} Hom_{D_{∞}^{×}}(W_{τ_{∞}}^{∨}, C^{∞}(D^{×}\(D ⊗_{ℚ} 𝔸)^{×}/U_{l}, ψ_{∞}))

which sends f to the map

y ⟼ (g ⟼ y(τ_{∞}(g_{∞})^{−1}τ_{(k⃗,w⃗),ℚ_{l}^{ac}}(g_{l})f(g^{∞}))).

Everything now follows from the Jacquet-Langlands theorem. □

There is a pairing

Symm^{k−2}(A^{2}) × Symm^{k−2}(A^{2}) ⟶ A

defined by

⟨f_{1}, f_{2}⟩ = (f_{1}(∂/∂Y, −∂/∂X)f_{2}(X, Y))|_{X=Y =0}.

By looking at the pairing of monomials we see that

⟨f_{1}, f_{2}⟩ = (−1)^{k}⟨f_{2}, f_{1}⟩

and that if 2 ≤ k ≤ l + 1 then this pairing is perfect. Moreover if

u = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(A)

then

⟨uf_{1}, uf_{2}⟩

= (f_{1}(a∂/∂Y − c∂/∂X, b∂/∂Y − d∂/∂X)f_{2}(aX + cY, bX + dY))|_{X=Y =0}
= (f_{1}((det u)∂/∂W, −(det u)∂/∂Z)f_{2}(Z, W))|_{Z=W =0}
= (det u)^{k−2}⟨f_{1}, f_{2}⟩,

where Z = aX + cY and W = bX + dY. This extends to a perfect pairing W_{(k⃗,w⃗),A} × W_{(k⃗,w⃗),A} → A such that

⟨ux, uy⟩ = (N det u)^{w−1}⟨x, y⟩

for all x, y ∈ W_{(k⃗,w⃗),A} and all u ∈ GL_{2}(O_{F,l}). Here w = k_{σ} + 2w_{σ} − 1, which is independent of σ. We can define a perfect pairing S_{k,A,ψ}(U_{H}(𝔫))×S_{k,A,ψ}(U_{H}(𝔫)) → A by setting (f_{1}, f_{2}) equal to

∑_{[x]}⟨f_{1}(x), f_{2}(x)⟩ψ(det x)^{−1}(#(U_{H}(𝔫)(𝔸_{F}^{∞})^{×} ∩ x^{−1}D^{×}x)/F^{×})^{−1},

where [x] ranges over D^{×}\(D ⊗_{ℚ} 𝔸^{∞})^{×}/U_{H}(𝔫)(𝔸_{F}^{∞})^{×}. (We are using the fact that #(U_{H}(𝔫)(𝔸_{F}^{∞})^{×} ∩ x^{−1}D^{×}x)/F^{×} is prime to l.) The usual calculation shows that

([U_{H′}(𝔫′)gU_{H}(𝔫)]f_{1}, f_{2})_{U_{H′} (𝔫′)} = ψ(det g)(f_{1}, [U_{H}(𝔫)g^{−1}U_{H′}(𝔫′)]f_{2})_{U_{H}(𝔫)}.

Now specialise to the case that A = O is the ring on integers of a finite extension of ℚ_{l}. We will write simply h_{(k⃗,w⃗),ψ}(U_{H}(𝔫)) for h_{(k⃗,w⃗),O,ψ}(U_{H}(𝔫)). It follows from lemma 1.3 and the main theorem in [Tay1] that there is a continuous representation

ρ : G_{F} ⟶ GL_{2}(h_{(k⃗,w⃗),ψ}(U_{H}(𝔫)) ⊗_{O} ℚ_{l}^{ac})

such that

• if x |̸𝔫l then ρ is unramified at x and tr ρ(Frob_{x}) = T_{x}; and

• det ρ = ϵ(ψ ∘ Art^{−1}).

From the theory of pseudo-representations (or otherwise, see [Ca2]) we deduce that if 𝔪 is a non-Eisenstein maximal ideal of h_{(k⃗,w⃗),ψ}(U_{H}(𝔫)) then ρ gives rise to a continuous representation

ρ_{𝔪} : G_{F} ⟶ GL_{2}(h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪})

such that

• if x |̸𝔫l then ρ_{𝔪} is unramified at x and tr ρ_{𝔪}(Frob_{x}) = T_{x}; and

• det ρ_{𝔪} = ϵ(ψ ∘ Art^{−1}).

From the Cebotarev density theorem we see that h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪} is generated by U_{ϖ_{x}} for x|𝔫 but x |̸l and by T_{x} for all but finitely many x |̸l𝔫. (For let h denote the O-subalgebra of h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪} generated by U_{ϖ_{x}} for x|𝔫 but x |̸l and by T_{x} for all but finitely many x |̸l𝔫. The Cebotarev densitry theorem implies that tr ρ_{𝔪} is valued in h and hence

T_{x} = tr ρ_{𝔪}(Frob_{x}) ∈ h

for all x |̸𝔫l. Thus h = h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪}.) We will write \overline{ρ}_{𝔪} for (ρ_{𝔪} mod 𝔪). If ϕ : h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪} → R is a map of local O-algebras then we will write ρ_{ϕ} for ϕρ_{𝔪}. If R is a field of characteristic l we will sometimes write \overline{ρ}_{ϕ} instead of ρ_{ϕ}.

Lemma 1.4 Let (k⃗, w⃗) be as above. Suppose that x |̸𝔫 is a split place of F above l such that 2 ≤ k_{x} ≤ l − 1. If 𝔪 is a non-Eisenstein maximal ideal of h_{(k⃗,w⃗),ψ}(U_{H}(𝔫)) and if I is an open ideal of h_{(k⃗,w⃗),ψ}(U_{H}(𝔫))_{𝔪} (for the l-adic topology) then ((ρ_{𝔪} ⊗ ϵ^{−w_{x}}) mod I)|_{G_{x}} is of the form 𝕄(D) for some object D of MF_{F_{x},O,k_{x}} with D ≠ D^{0} ≠ (0).

Proof: Combining the construction of ρ_{𝔪} with the basic properties of 𝕄 listed in the section of notation, we see that it suffices to prove the following. Suppose that π is a cuspidal automorphic representation of GL_{2}(𝔸_{F}) such that π_{∞} is regular algebraic of weight (k⃗, w⃗). Let M denote the field of definition of π. Suppose that x is a split place of F above l with π_{x} unramified. Let M^{ac} denote the algebraic closure of M in ℂ and fix an embedding λ : M^{ac} ↪ ℚ_{l}^{ac}. Let τ : F ↪ M^{ac} be the embedding so that λ ∘ τ gives rise to x. Suppose that 2 ≤ k_{τ} ≤ l − 1. If I is a power of the prime of O_{M} induced by λ, then (ρ_{π,λ}⊗ϵ^{−w_{τ}})|_{G_{x}} mod I is of the form 𝕄(D) for some object D of MF_{F_{x},O_{M,λ},k_{τ}} with D ≠ D^{0} ≠ (0). By the construction of ρ_{π,λ} in [Tay1], our assumption that (ρ_{π,λ} mod λ) is irreducible, and the basic properties of 𝕄, we see that it suffices to treat the case that π_{y} is discrete series for some finite place y (cf [Tay2]). Because 2 ≤ k_{τ} ≤ l − 1, it follows from [FL] that we need only show that ρ_{π,λ} is crystalline with Hodge-Tate numbers −w_{τ} and 1 − k_{τ} − w_{τ}. In the case π_{y} is discrete series for some finite place y this presumably follows from Carayol’s construction of ρ_{π,λ} [Ca1] and Faltings theory [Fa], but for a definite reference we refer the reader to theorem VII.1.9 of [HT] (but note the different, more sensible, conventions in force in that paper). □

Corollary 1.5 Suppose that x |̸𝔫 is a split place of F. Suppose that (k⃗, w⃗) is as above and that 2 ≤ k_{x} ≤ l − 1. If 𝔪 is a non-Eisenstein maximal ideal of h_{(k⃗,w⃗),ψ}(U_{H}(𝔫)) then \overline{ρ}_{𝔪}|_{I_{x}} ∼ ω_{2}^{k_{x}−1+(l+1)w_{x}} ⊕ ω_{2}^{l(k_{x}−1)+(l+1)w_{x}} or

( \begin{matrix}ω^{k_{x}+w_{x}−1} & ∗ \\ 0 & ω^{w_{x}} \end{matrix} ) .

Proof: This follows easilly from the above lemma together with theorem 5.3, proposition 7.8 and theorem 8.4 of [FL]. □

The following lemma is well known.

Lemma 1.6 Suppose that x is a finite place of F and that π is an irreducible admissible representation of GL_{2}(F). If χ_{1} and χ_{2} are two characters of F^{×}, let π(χ_{1}, χ_{2}) denote the induced representation consisting of locally constant functions GL_{2}(F) → ℂ such that

f (( \begin{matrix}a & b \\ 0 & d \end{matrix} ) g) = χ_{1}(a)χ_{2}(b)|a/b|_{x}^{1/2}f(g)

(with GL_{2}(F)-action by right translation). Let U_{1} (resp. U_{2}) denote the subgroup of elements in GL_{2}(O_{F,x}) which are congruent to a matrix of the form

( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} ) mod (ϖ_{x})

(resp.

( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) mod (ϖ_{x}^{2})).

1. If π^{U_{1}} ≠ (0) then π is a subquotient of some π(χ_{1}, χ_{2}) where the conductors of χ_{1} and χ_{2} are ≤ 1.

2. If the conductors of χ_{1} and χ_{2} are ≤ 1 then

π(χ_{1}, χ_{2})^{U_{1}}

is two dimensional with a basis e_{1}, e_{2} such that

U_{ϖ_{x}}e_{i} = (Nx)^{1/2}χ_{i}(ϖ_{x})e_{i}

and

⟨h⟩e_{i} = χ_{i}(h)e_{i}

for h ∈ (O_{F,x}/x)^{×}.

3. If π^{U_{2}} ≠ (0) then π is either cuspidal or a subquotient of some π(χ_{1}, χ_{2}) where the conductors of χ_{1} and χ_{2} are equal and ≤ 1.

4. If π is cuspidal then dim π^{U_{2}} ≤ 1 and U_{ϖ_{x}} acts as zero on π^{U_{2}}.

5. If χ_{1} and χ_{2} have conductor 1 then π(χ_{1}, χ_{2})^{U_{2}} is one dimensional and U_{ϖ_{x}} acts on it as 0.

6. If χ_{1} and χ_{2} have conductor 0 then π(χ_{1}, χ_{2})^{U_{2}} is three dimensional and U_{ϖ_{x}} acts on it with characteristic polynomial

X(X − (Nx)^{1/2}χ_{1}(ϖ_{x}))(X − (Nx)^{1/2}χ_{2}(ϖ_{x})).

As a consequence we have the following lemma.

Lemma 1.7 Suppose that ξ : h_{k,ψ}(U_{H}(𝔫))_{𝔪} → ℚ_{l}^{ac} and that x |̸l.

1. If x(𝔫) = 1 and if ξ′ is any extension of ξ to the subalgebra of End (S_{k,O,ψ}(U_{H}(𝔫))_{𝔪}) generated by h_{k,ψ}(U_{H}(𝔫))_{𝔪} and ⟨h⟩ for h ∈ H, then

ξ(ρ_{𝔪})|_{G_{x}} ∼ ( \begin{matrix}∗ & ∗ \\ 0 & χ_{x} \end{matrix} )

where χ_{x}(Art ϖ_{x}) = ξ(U_{ϖ_{x}}) and, for u ∈ O_{F,x}^{×}, we have χ_{x}(Art u) = ξ′(⟨u⟩).

2. If x(𝔫) = 2 and H_{x} = {1} then either ξ(U_{ϖ_{x}}) = 0 or ξ(U_{ϖ_{x}}) is an eigenvalue of ξ(ρ_{𝔪})|_{G_{x}}(σ) for any σ ∈ G_{x} lifting Frob_{x}.

We also get the following corollary.

Corollary 1.8 1. If x |̸l, x(𝔫) = 1 and U_{ϖ_{x}}^{2} − (Nx)ψ(ϖ_{x}) ∉ 𝔪 then

ρ_{𝔪}|_{G_{x}} ∼ ( \begin{matrix}∗ & ∗ \\ 0 & χ_{x} \end{matrix} )

where χ_{x}(Art ϖ_{x}) = U_{ϖ_{x}} and χ_{x}(Art u) = ⟨u⟩ for u ∈ O_{F,x}^{×}. In particular ⟨h⟩ ∈ h_{k,ψ}(U_{H}(𝔫))_{𝔪} for all h ∈ H_{x}.

2. If x |̸l, x(𝔫) = 2, H_{x} = {1} and U_{ϖ_{x}} ∈ 𝔪 then U_{ϖ_{x}} = 0 in h_{k,ψ}(U_{H}(𝔫))_{𝔪}.

3. If l is coprime to 𝔫 and for all x|𝔫 we have x(𝔫) = 2, H_{x} = {1} and U_{ϖ_{x}} ∈ 𝔪, then the algebra h_{k,ψ}(U_{H}(𝔫))_{𝔪} is reduced.

Proof: The first part follows from the previous lemma via a Hensel’s lemma argument. For the second part one observes that by the last lemma ξ(U_{ϖ_{x}}) = 0 for all ξ : h_{k,ψ}(U_{H}(𝔫))_{𝔪} → ℚ_{l}^{ac}. Hence by lemma 1.6 we have that U_{ϖ_{x}} = 0 on S_{k,ℚ_{l}^{ac},ψ}(U_{H}(𝔫))_{𝔪}. The third part follows from the second (because the algebra h_{k,ψ}(U_{H}(𝔫))_{𝔪} is generated by commuting semi-simple elements). □

2 Deformation rings and Hecke algebras I

In this section we extend the method of [TW] to totally real fields. This relies crucially on the improvement to the argument of [TW] found independently by Diamond [Dia] and Fujiwara (see [Fu], unpublished). Following this advance it has been clear to experts that some extension to totally real fields would be possible, the only question was the exact extent of the generalisation. Fujiwara has circulated some unpublished notes [Fu]. Then Skinner and Wiles made a rather complete analysis of the ordinary case (see [SW2]). We will treat the low weight, crystalline case. As will be clear to the reader, we have not tried to work in maximal generality, rather we treat the case of importance for this paper. We apologise for this. It would be very helpful to have these results documented in the greatest possible generality. In this section and the next let F denote a totally real field of even degree in which a prime l > 3 splits completely. (As the reader will be able to check without undue difficulty it would suffice to assume that l is unramified in F.) Let D denote the quaternion algebra with centre F which is ramified at exactly the infinite places, let O_{D} denote a maximal order in D and fix isomorphisms O_{D,x} ≅ M_{2}(O_{F,x}) for all finite places x of F. Let 2 ≤ k ≤ l − 1. Let ψ : 𝔸_{F}^{×}/F^{×} → (ℚ_{l}^{ac})^{×} be a continuous character such that

• if x |̸l is a prime of F then ψ|_{O_{F,x}^{×}} = 1,

• ψ|_{O_{F,l}^{×}}(u) = (Nu)^{2−k}.

For each finite place x of F choose a uniformiser ϖ_{x} of O_{F,x}. Suppose that ϕ : h_{k,𝔽_{l}^{ac},ψ}(U_{0}) → 𝔽_{l}^{ac} is a homomorphism with non-Eisenstein kernel, which we will denote 𝔪. Let O denote the ring of integers of a finite extension K/ℚ_{l} with maximal ideal λ such that

• K contains the image of every embedding F ↪ ℚ_{l}^{ac},

• ψ is valued in O^{×},

• there is a homomorphism ϕ̃: h_{k,O,ψ}(U_{0})_{𝔪} → O lifting ϕ, and

• all the eigenvalues of all elements of the image of \overline{ρ}_{ϕ} are rational over O/λ.

For any finite set Σ of finite places of F not dividing l we will consider the functor D_{Σ} from complete noetherian local O-algebras with residue firld O/λ to sets which sends R to the set of 1_{2} + M_{2}(𝔪_{R})-conjugacy classes of liftings ρ : G_{F} → GL_{2}(R) of \overline{ρ}_{ϕ} such that

• ρ is unramified outside l and Σ,

• det ρ = ϵ(ψ ∘ Art^{−1}), and

• for each place x of F above l and for each finite length (as an O-module) quotient R/I of R the O[G_{x}]-module (R/I)^{2} is isomorphic to 𝕄(D) for some object D of MF_{F_{x},O,k}.

This functor is represented by a universal deformation

ρ_{Σ} : G_{F} ⟶ GL_{2}(R_{Σ}).

(This is now very standard, see for instance appendix A of [CDT].) Now let Σ be a finite set of finite places of F not dividing l such that if x ∈ Σ then

• Nx ≡ 1 mod l,

• \overline{ρ}_{ϕ} is unramified at x and \overline{ρ}_{ϕ}(Frob_{x}) has distinct eigenvalues α_{x} ≠ β_{x}.

By Hensel’s lemma the polynomial X^{2} − T_{x}X + (Nx)ψ(ϖ_{x}) splits as (X − A_{x})(X − B_{x}) in h_{k,O,ψ}(U_{0})_{𝔪}, where A_{x} mod 𝔪 = α_{x} and B_{x} mod 𝔪 = β_{x}. For x ∈ Σ we will let Δ_{x} denote the maximal l-power quotient of (O_{F} /x)^{×}. We will let 𝔫_{Σ} = ∏_{x∈Σ} x; Δ_{Σ} = ∏_{x∈Σ} Δ_{x}; U_{0,Σ} = U_{{1}}(𝔫_{Σ}); and U_{1,Σ} = U_{Δ_{Σ}}(𝔫_{Σ}). We will let 𝔪_{Σ} denote the ideal of either h_{k,ψ}(U_{0,Σ}) or h_{k,ψ}(U_{1,Σ}) generated by

• l;

• T_{x} − tr \overline{ρ}_{ϕ}(Frob_{x}) for x |̸l𝔫_{Σ}; and

• U_{ϖ_{x}} − α_{x} for x ∈ Σ.

Lemma 2.1 Let Σ satisfy the assumptions of the last paragraph.

1. If x ∈ Σ then ρ_{Σ}|_{G_{x}} ∼ χ_{α,x} ⊕ χ_{β,x} where χ_{α,x} mod 𝔪_{R_{Σ}} is unramified and takes Frob_{x} to α_{x}.

2. χ_{α,x} ∘ Art |_{O_{F,x}^{×}} factors through Δ_{x}, and these maps make R_{Σ} into a O[Δ_{Σ}]-module.

3. The universal property of R_{Σ} gives rise to a surjection of O[Δ_{Σ}]-algebras

R_{Σ} → → h_{k,ψ}(U_{1,Σ})_{𝔪_{Σ}}

under which ρ_{Σ} pushes forward to ρ_{𝔪_{Σ}}.

Proof: The first part is proved in exactly the same manner as lemma 2.44 of [DDT]. The second part is then clear. The third part is clear because for x |̸𝔫_{Σ}l we have tr ρ_{Σ}(Frob_{x}) ↦ T_{x} while for x ∈ Σ we have χ_{α,x}(ϖ_{x}) ↦ U_{ϖ_{x}}. □

Lemma 2.2 The map

\begin{matrix}η : S_{k,O,ψ}(U_{0,Σ−\{x\}})_{𝔪_{Σ−\{x\}}} & ⟶ & S_{k,O,ψ}(U_{0,Σ})_{𝔪_{Σ}} \\ f & ⟼ & A_{x}f − ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f \end{matrix}

is an isomorphism which induces an isomorphism

η^{∗} : h_{k,ψ}(U_{0,Σ})_{𝔪_{Σ}} ⟶^{∼} h_{k,ψ}(U_{0,Σ−{x}})_{𝔪_{Σ−{x}}}.

Proof: The map η is well defined because U_{ϖ_{x}} ∘ η = η ∘ A_{x}. It is injective with torsion free cokernel because the composition of η with the adjoint of the natural inclusion S_{k,O,ψ}(U_{0,Σ−{x}}) ↪ S_{k,O,ψ}(U_{0,Σ}) is (Nx)A_{x} − B_{x} ∉ 𝔪_{Σ}. As α_{x}/β_{x} ≠ (Nx)^{±1}, no lift of \overline{ρ}_{ϕ} with determinant ϵ(ψ ∘ Art^{−1}) has conductor at x exactly x. Thus

S_{k,O,ψ}(U_{0,Σ})_{𝔪_{Σ}} = (S_{k,O,ψ}(U_{0,Σ−\{x\}}) + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) S_{k,O,ψ}(U_{0,Σ−\{x\}}))_{𝔪_{Σ}}.

As

U_{ϖ_{x}}(f_{1} + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f_{2}) = (T_{x}f_{1} + (Nx)ψ(ϖ_{x})f_{2}) − ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f_{1}

and the matrix

( \begin{matrix}T_{x} & (Nx)ψ(ϖ_{x}) \\ −1 & 0 \end{matrix} )

has eigenvalues A_{x} and B_{x} which are distinct mod 𝔪, the lemma follows. □

We remark that S_{k,O,ψ}(U_{1,Σ}) is a Δ_{Σ}-module via h ↦ ⟨h⟩.

Lemma 2.3 1. ∑_{h∈Δ_{Σ}}⟨h⟩ : S_{k,O,ψ}(U_{1,Σ})_{Δ_{Σ}} →^{∼} S_{k,O,ψ}(U_{0,Σ}).

2. S_{k,O,ψ}(U_{1,Σ}) is a free O[Δ_{Σ}]-module.

Proof: The second assertion follows from the first as we can compute that

dim S_{k,O,ψ}(U_{1,Σ}) ⊗_{O} K = [U_{0,Σ} : U_{1,Σ}] dim S_{k,O,ψ}(U_{0,Σ}) ⊗_{O} K.

(We use the fact that [U_{0,Σ} : U_{1,Σ}] is coprime to #(U_{0,Σ}(𝔸_{F}^{∞})^{×} ∩ x^{−1}D^{×}x)/F^{×} for all x ∈ (D ⊗_{ℚ} 𝔸^{∞})^{×}.) Using the duality introduced above it suffices to check that the natural map

S_{k,O,ψ}(U_{0,Σ}) ⊗_{O} K/O ⟶ (S_{k,O,ψ}(U_{1,Σ}) ⊗_{O} K/O)^{Δ_{Σ}}

is an isomorphism. This is immediate from the definitions and the fact that l |̸#(U_{0,Σ}(𝔸_{F}^{∞})^{×} ∩ x^{−1}D^{×}x)/F^{×} for all x ∈ (D ⊗_{ℚ} 𝔸^{∞})^{×}. □

As S_{k,O,ψ}(U_{1,Σ})_{𝔪_{Σ}} is a direct summand of S_{k,O,ψ}(U_{1,Σ}), we deduce the following corollary.

Corollary 2.4 1. S_{k,O,ψ}(U_{1,Σ})_{𝔪_{Σ},Δ_{Σ}} →^{∼} S_{k,O,ψ}(U_{0})_{𝔪} compatibly with a map h_{k,ψ}(U_{1,Σ})_{𝔪_{Σ}} → h_{k,ψ}(U_{0})_{𝔪} sending T_{x} to T_{x} for x |̸l𝔫_{Σ}, ⟨h⟩ to 1 for h ∈ Δ_{Σ} and U_{ϖ_{x}} to A_{x} for x ∈ Σ.

2. S_{k,O,ψ}(U_{1,Σ})_{𝔪_{Σ}} is a free O[Δ_{Σ}]-module.

Suppose that ρ : G_{F} → GL_{2}(O/λ^{n}) is a lifting of \overline{ρ}_{ϕ} corresponding to some map R_{∅} → O/λ^{n}. If x is a place of F above l and if (O/λ^{n})^{2} ≅ 𝕄(D) as a G_{x}-module, then we set

H_{f}^{1}(G_{x}, ad^{0}ρ) =H^{1}(G_{x}, ad^{0}ρ)∩Im (Ext _{MF_{Fx,O/λ^{n},k}}^{1}(D, D) ⟶ H^{1}(G_{x}, ad ρ)).

Exactly as in section 2.5 of [DDT] we see that

Im (Ext _{MF_{Fx,O/λ^{n},k}}^{1}(D, D) ⟶ H^{1}(G_{x}, ad ρ)) ≅ (O/λ^{n})^{2} ⊕ H^{0}(G_{x}, ad^{0}ρ).

If two continuous O[G_{x}]-modules have the same restriction to I_{x}, then one is in the image of 𝕄 if and only if the other is. We conclude that the image of the composite

Ext _{MF_{O/λ^{n},k}}^{1}(D, D) ⟶ H^{1}(G_{x}, ad ρ) ⟶^{tr} H^{1}(G_{x}, O/λ^{n})

is at least one dimensional (coming from unramified twists) and hence that

#H_{f}^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ})|#(O/λ^{n})#H^{0}(G_{x}, ad^{0}\overline{ρ}_{ϕ}).

We will let H_{Σ}^{1}(G_{F} , ad^{0}ρ) denote the kernel of the map

H^{1}(G_{F} , ad^{0}ρ) ⟶ ⨁_{x |̸𝔫_{Σ}l} H^{1}(I_{x}, ad^{0}ρ) ⊕ ⨁_{x|l} H^{1}(G_{x}, ad^{0}ρ)/H_{f}^{1}(G_{x}, ad^{0}ρ).

The trace pairing (a, b) ↦ tr ab gives a perfect duality on ad^{0}\overline{ρ}_{ϕ}. For x|l we will let H_{f}^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ}(1)) denote the annihilator in H^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ}(1)) of H_{f}^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ}) under Tate local duality. We will also let H_{Σ}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) denote the kernel of the restriction map from H^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) to

⨁_{x |̸𝔫_{Σ}l} H^{1}(I_{x}, ad^{0}\overline{ρ}_{ϕ}(1)) ⊕ ⨁_{x∈Σ} H^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ}(1))⊕
⨁_{x|l} H^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ})/H_{f}^{1}(G_{x}, ad^{0}\overline{ρ}_{ϕ}(1)))

so that

H_{Σ}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) = ker(H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) ⟶ ⨁_{x∈Σ} H^{1}(G_{x}/I_{x}, ad^{0}\overline{ρ}_{ϕ}(1))).

A standard calculation (see for instance section 2.7 of [DDT]) shows that

H_{Σ}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}) ≅ Hom_{O}(𝔪_{R_{Σ}}/𝔪_{R_{Σ}}^{2}, O/λ),

so that R_{Σ} can be topologically generated by dim H_{Σ}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}) elements as an O-algebra. A formula of Wiles (see theorem 2.19 of [DDT]) then tells us that R_{Σ} can be topologically generated as an O-algebra by

#Σ + dim H_{Σ}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1))

elements.

Lemma 2.5 Suppose that the restriction of \overline{ρ}_{ϕ} to F (\sqrt{(−1)^{(l−1)/2}l}) is irreducible. Then for any m ∈ ℤ_{>0} we can find a set Σ_{m} of primes such that

1. #Σ_{m} = dim H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)),

2. R_{Σ_{m}} can be topologically generated by dim H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) elements as an O-algebra,

3. if x ∈ Σ_{m} then Nx ≡ 1 mod l^{m} and \overline{ρ}_{ϕ}(Frob_{x}) has distinct eigenvalues α_{x} and β_{x}.

Proof: By the above calculation we may replace the second requirement by the requirement that H_{Σ_{m}}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) = (0) (for then R_{Σ_{m}} is generated by #Σ_{m} = dim H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) elements). Then we may suppress the first requirement, because any set satisfying the modified second requirement and the third requirement can be shrunk to one which also satisfies the first requirement. (Note that for x satisfying the third requirement H^{1}(G_{x}/I_{x}, ad^{0}\overline{ρ}_{ϕ}(1)) is one dimensional.) Next, by the Cebotarev density theorem, it suffices to show that for [γ] ∈ H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) we can find σ ∈ G_{F} such that

• σ|_{F (ζ_{l^{m}})} = 1,

• \overline{ρ}_{ϕ}(σ) has distinct eigenvalues, and

• γ(σ) ∉ (σ − 1)ad^{0}\overline{ρ}_{ϕ}.

Let F_{m} denote the extension of F (ζ_{l^{m}}) cut out by ad^{0}\overline{ρ}. Finally it will suffice to show that

1. H^{1}(Gal (F_{m}/F), ad^{0}\overline{ρ}(1)) = (0); and

2. for any non-trivial irreducible Gal (F_{m}/F)-submodule V of ad^{0}\overline{ρ}_{ϕ} we can find σ ∈ Gal (F_{m}/F (ζ_{l^{m}})) such that ad^{0}\overline{ρ}_{ϕ}(σ) has an eigenvalue other than 1 but σ does have an eigenvalue 1 on V.

(Given [γ] ∈ H_{∅}^{1}(G_{F} , ad^{0}\overline{ρ}_{ϕ}(1)) the first assertion tells us that the O/λ-span of γG_{F_{m}} contains some non-trivial irreducible Gal (F_{m}/F)-submodule V of ad^{0}\overline{ρ}_{ϕ}. Let σ be as in the second assertion for this V. Then for some σ′ ∈ G_{F_{m}} we will have

γ(σ′σ) = γ(σ′) + γ(σ) ∉ (σ − 1)ad^{0}\overline{ρ}_{ϕ}.)

Because l > 3 is unramified in F, we see that [F (ζ_{l}) : F] > 2 and so, by the argument of the penultimate paragraph of the proof of theorem 2.49 of [DDT], H^{1}(Gal (F_{m}/F), ad^{0}\overline{ρ}(1)) = (0). Suppose that V is an irreducible Gal (F_{m}/F)-submodule of ad^{0}\overline{ρ}_{ϕ} and write ad^{0}\overline{ρ}_{ϕ} = V ⊕ W. If W = (0) any σ ∈ Gal (F_{m}/F (ζ_{l^{m}})) with an eigenvalue other than 1 on ad^{0}\overline{ρ}_{ϕ} will suffice to prove the second assertion. Thus suppose that W ≠ (0). If dim W = 1 then \overline{ρ}_{ϕ} is induced from a character of some quadratic extension E/F and any σ ∉ G_{E} will suffice to prove the second assertion (as E is not a subfield of F (ζ_{l^{m}})). If dim W = 2 then G_{F} acts on V via a quadratic character corresponding to some quadratic extension E/F and \overline{ρ}_{ϕ} is induced from some character χ of G_{E}. Let χ′ denote the Gal (E/F)-conjugate of χ. Then any σ ∈ G_{E(ζ_{l^{m}})} with χ/χ′(σ) ≠ 1 will suffice to prove the second assertion. (Such a σ will exist unless the restriction of ad^{0}\overline{ρ}_{ϕ} to E(\sqrt{(−1)^{(l−1)/2}l}) is trivial in which case \overline{ρ}_{ϕ} becomes reducible over F (\sqrt{(−1)^{(l−1)/2}l}), which we are assuming is not the case.) □

Combining lemma 2.5, corollary 2.4 and theorem 2.1 of [Dia] we obtain the following theorem.

Theorem 2.6 Keep the notation and assumptions of the second and fourth paragraphs of this section and suppose that the restriction of \overline{ρ}_{ϕ} to the absolute Galois group of F (\sqrt{(−1)^{(l−1)/2}l}) is irreducible. Then the natural map

R_{∅} ⟶ h_{k,ψ}(U_{0})_{𝔪}

is an isomorphism of complete intersections and S_{k,O,ψ}(U_{0})_{𝔪} is finite free as a h_{k,ψ}(U_{0})_{𝔪}-module.

3 Deformation rings and Hecke algebras II

In this section we use analogues of Wiles’ arguments from [W2] to extend the isomorphism of theorem 2.6 from ∅ to any Σ. We will keep the notation and assumptions of the last section. (Σ will again be any finite set of finite places of F not dividing l.) Let ρϕ̃ : G_{F} → GL_{2}(O)

denote the Galois representation corresponding to ϕ̃ (a chosen lift of ϕ). The universal property of R_{Σ} gives maps

R_{Σ} → → R_{∅} \underrightarrow{ϕ̃} O.

We will denote the kernel by ℘_{Σ}. A standard calculation (see section 2.7 of [DDT]) shows that

Hom_{O}(℘_{Σ}/℘_{Σ}^{2}, K/O) ≅ H_{Σ}^{1}(G_{F} , (ad^{0}ρ) ⊗ K/O),

where

H_{Σ}^{1}(G_{F} , (ad^{0}ρ) ⊗ K/O) = lim_{n^{⟶}} H_{Σ}^{1}(G_{F} , (ad^{0}ρ) ⊗ λ^{−n}/O).

In particular we see that

# ker(℘_{Σ}/℘_{Σ}^{2} → → ℘_{∅}/℘_{∅}^{2}) = #(H_{Σ}^{1}(G_{F} , (ad^{0}ρ)⊗K/O)/H_{∅}^{1}(G_{F} , (ad^{0}ρ)⊗K/O))

divides

∏_{x∈Σ} #H^{1}(I_{x}, (ad^{0}ρ) ⊗ K/O)^{G_{x}}
= ∏_{x∈Σ} #H^{0}(G_{x}, (ad^{0}ρ) ⊗ K/O(−1))
= ∏_{x∈Σ} #O/(1 − Nx)((1 + Nx)^{2} det ρ(Frob_{x}) − (Nx)(tr ρFrob_{x}))O.

Let 𝔫′_{Σ} denote the product of the squares of the primes in Σ and set U_{Σ} = U_{{1}}(𝔫′_{Σ}). Let h_{Σ} = h_{k,ψ}(U_{Σ})_{𝔪′_{Σ}} and S_{Σ} = S_{k,O,ψ}(U_{Σ})_{𝔪′_{Σ}}, where 𝔪′_{Σ} is the maximal ideal of h_{k,ψ}(U_{Σ}) generated by

• λ,

• T_{x} − tr \overline{ρ}_{ϕ}(Frob_{x}) for x |̸l𝔫′_{Σ}, and

• U_{ϖ_{x}} for x ∈ Σ.

The Galois representation ρ_{𝔪′_{Σ}} induces a homomorphism R_{Σ} → h_{Σ} which takes tr ρ_{Σ}(Frob_{x}) to T_{x} for all x |̸𝔫′_{Σ}l. Corollary 1.8 tells us that for x ∈ Σ we have U_{ϖ_{x}} = 0 in h_{Σ} and that h_{Σ} is reduced. In particular the map R_{Σ} → h_{Σ} is surjective. From lemma 1.3, lemma 1.6 and the strong multiplicity one theorem for GL_{2}(𝔸_{F}) we see that dim(S_{Σ} ⊗_{O} K)[℘_{Σ}] = 1. We can write

S_{Σ} ⊗_{O} K = (S_{Σ} ⊗_{O} K)[℘_{Σ}] ⊕ (S_{Σ} ⊗_{O} K)[Ann_{h_{Σ}}(℘_{Σ}h_{Σ})].

We set

Ω_{Σ} = S_{Σ}/(S_{Σ}[℘_{Σ}] ⊕ S_{Σ}[Ann_{h_{Σ}}(℘_{Σ}h_{Σ})]).

By theorem 2.4 of [Dia] and theorem 2.6 above, we see that

#Ω_{∅} = #℘_{∅}/℘_{∅}^{2}.

Let w_{Σ} ∈ GL_{2}(𝔸_{F}^{∞}) be defined by w_{σ,x} = 1_{2} if x ∉ Σ and

w_{Σ,x} = ( \begin{matrix}0 & 1 \\ ϖ_{x}^{2} & 0 \end{matrix} )

if x ∈ Σ. Then w_{Σ} normalises U_{Σ}. We define a new pairing on S_{k,O,ψ}(U_{Σ}) by

(f_{1}, f_{2})′ = ( ∏_{x∈Σ} ψ(ϖ_{x}))^{−1}(f_{1}, w_{Σ}f_{2}).

Because ( , ) is a perfect pairing so is ( , )′. Moreover the action of any element of h_{k,ψ}(U_{Σ}) is self adjoint with respect to ( , )′, so that ( , )′ restricts to a perfect pairing on S_{Σ}. Choose a perfect O-bilinear pairing on S_{Σ}[℘_{Σ}], let j_{Σ} denote the natural inclusion

j_{Σ} : S_{Σ}[℘_{Σ}] ↪ S_{Σ},

and let j_{Σ}^{†} denote the adjoint of j_{Σ} with respect to ( , )′ on S_{Σ} and the chosen pairing on S_{Σ}[℘_{Σ}]. Then one sees that

j_{Σ}^{†} : Ω_{Σ} ⟶^{∼} S_{Σ}[℘_{Σ}]/j_{Σ}^{†}S_{Σ}[℘_{Σ}].

If x |̸l𝔫′_{Σ} then define

i_{x} : S_{k,O,ψ}(U_{Σ}) ⟶ S_{k,O,ψ}(U_{Σ∪{x}})

by

i_{x}(f) = (Nx)ψ(ϖ_{x})f − ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) T_{x}f + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x}^{2} \end{matrix} ) f.

It is easy to check that i_{x} commutes with T_{y} for y |̸l𝔫′_{Σ∪{x}} and with U_{ϖ_{y}} for y ∈ Σ. Moreover U_{ϖ_{x}}i_{x} = 0 and so

i_{x} : S_{Σ} ⟶ S_{Σ∪{x}}.

Moreover i_{x}S_{Σ}[℘_{Σ}] ⊂ S_{Σ∪{x}}[℘_{Σ∪{x}}]. We will let i_{x}^{†} denote the adjoint of i_{x} with respect to the pairings ( , )′ on S_{Σ} and S_{Σ∪{x}}. (We warn the reader that the former is not simply the restriction of the latter.) An easy calculation shows that i_{x}^{†} equals

ψ(ϖ_{x})(Nx)[U_{Σ}U_{Σ∪\{x\}}]−T_{x}[U_{Σ} ( \begin{matrix}ϖ_{x} & 0 \\ 0 & 1 \end{matrix} ) U_{Σ∪\{x\}}]+[U_{Σ} ( \begin{matrix}ϖ_{x}^{2} & 0 \\ 0 & 1 \end{matrix} ) U_{Σ∪\{x\}}]

and hence that

i_{x}^{†} ∘ i_{x} = ψ(ϖ_{x})(Nx)(1 − Nx)(T_{x}^{2} − (1 + Nx)^{2}ψ(ϖ_{x})).

The following key lemma is often referred to as Ihara’s lemma.

Lemma 3.1 S_{Σ∪{x}}/i_{x}S_{Σ} is l-torsion free.

Proof: It suffices to check that

i_{x} : S_{k,O/λ,ψ}(U_{Σ})_{𝔪′_{Σ}} ⟶ S_{k,O/λ,ψ}(U_{Σ∪{x}})_{𝔪′_{Σ∪{x}}}

is injective, or even that the localisation at 𝔪′_{Σ} of the kernel of

\begin{matrix}S_{k,O/λ,ψ}(U_{Σ})^{3} & ⟶ & S_{k,O/λ,ψ}(U_{Σ∪\{x\}}) \\ (f_{1}, f_{2}, f_{3}) & ⟼ & f_{1} + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f_{2} + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x}^{2} \end{matrix} ) f_{3} \end{matrix}

vanishes. Let V denote the subgroup of elements u ∈ U_{Σ} with

u_{x} ≡ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) mod ϖ_{x}.

We see that

V ∩ ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) V ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} )^{−1} = U_{Σ∪\{x\}}

and that U_{Σ} is the subgroup of GL_{2}(𝔸_{F}^{∞}) generated by V and

( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} )^{−1} V ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) .

Thus the sequence

(0) → S_{k,O/λ,ψ}(U_{Σ}) → S_{k,O/λ,ψ}(V) ⊕ S_{k,O/λ,ψ}(V) → S_{k,O/λ,ψ}(U_{Σ∪{x}})
f  ↦  (( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f, −f)
(f_{1}, f_{2})  ↦f_{1} + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f_{2}

is exact. Hence it suffices to show that the localisation at 𝔪′_{Σ} of the kernel of

\begin{matrix}S_{k,O/λ,ψ}(U_{Σ})^{2} & ⟶ & S_{k,O/λ,ψ}(V) \\ (f_{1}, f_{2}) & ⟼ & f_{1} + ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) f_{2} \end{matrix}

vanishes. However if (f_{1}, f_{2}) is in the kernel then f_{1} is invariant by the subgroup of GL_{2}(𝔸_{F}^{∞}) generated by U_{Σ} and

( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) U_{Σ} ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} )^{−1} ,

i.e. by U_{Σ}SL_{2}(F_{x}). First suppose that k = 2. Then, by the strong approximation theorem, we see that f_{1} is invariant by right translation by any element of SL_{2}(𝔸_{F}^{∞}), so that f_{1} ∈ S_{k,O/λ,ψ}^{triv}(U_{σ}). Any maximal ideal of h_{2,ψ}(U_{Σ}) in the support of S_{k,O/λ,ψ}^{triv}(U_{σ}) is Eisenstein. Now suppose that 3 ≤ k ≤ l − 1. By the strong approximation theorem, given any g ∈ GL_{2}(𝔸_{F}^{∞}) and any u ∈ GL_{2}(O_{F,l}), we can find a δ ∈ D^{×} ∩ gU_{Σ}SL_{2}(F_{x})g^{−1} such that

g_{l}^{−1}δg_{l} ≡ u mod l.

Then

f_{1}(g) = f_{1}(δg) = f_{1}(g(g^{−1}δg)) = f_{1}(gu) = u^{−1}f_{1}(g),

so that

f_{1}(g) ∈ ( ⨂_{O_{F,l}→O/λ} Symm^{k−2}((O/λ)^{2}))^{GL_{2}(O_{F,l})} = (0).

Thus f_{1} = 0. □

In particular we see that i_{x}S_{Σ}[℘_{Σ}] = S_{Σ∪{x}}[℘_{Σ∪{x}}]. Thus

Ω_{Σ∪{x}} ≅ S_{Σ}[℘_{Σ}]/j_{Σ}^{†}i_{x}^{†}S_{Σ}[℘_{Σ}]
≅ S_{Σ}[℘_{Σ}]/j_{Σ}^{†}(1 − Nx)(Nx)(T_{x}^{2} − (1 + Nx)^{2}ψ(ϖ_{x}))S_{Σ}[℘_{Σ}],

and so

#Ω_{Σ∪{x}} = #Ω_{Σ}# (O/(1 − Nx)((Nx)tr ρ(Frob_{x})^{2} − (1 + Nx)^{2} det ρ(Frob_{x}))) .

We conclude that

#(℘_{Σ}/℘_{Σ}^{2})|#Ω_{Σ}

for all Σ (which contains no prime above l). Combining this with theorem 2.4 of [Dia] we see obtain the following theorem.

Theorem 3.2 Keep the notation and assumptions of the second and fourth paragraphs of section 2 and suppose that the restriction of \overline{ρ}_{ϕ} to the absolute Galois group of F (\sqrt{(−1)^{(l−1)/2}l}) is irreducible. If Σ is a finite set of finite places of F not dividing l then the natural map

R_{Σ} ⟶ h_{Σ}

is an isomorphism of complete intersections and S_{Σ} is a free h_{Σ}-module.

As an immediate consequence we have the following theorem.

Theorem 3.3 Let l > 3 be a prime and let 2 ≤ k ≤ l − 1 be an integer. Let F be a totally real field of even degree in which l splits completely. Let ρ : G_{F} → GL_{2}(O_{ℚ_{l}^{ac}}) be a continuous irreducible representation unramified outside finitely many primes and such that for each place x of F above l the restriction ρ|_{G_{x}} is crystalline with Hodge-Tate numbers 0 and 1 − k. Let \overline{ρ} denote the reduction of ρ modulo the maximal ideal of O_{ℚ_{l}^{ac}}. Assume that the restriction of \overline{ρ} to F (\sqrt{(−1)^{(l−1)/2}l}) is irreducible and that there is a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

• \overline{ρ}_{π,λ} ∼ \overline{ρ},

• π_{x} is unramified for every finite place x of F, and

• π_{∞} has weight k.

Then there is a regular algebraic cuspidal automorphic representation π′ of GL_{2}(𝔸_{F}) and an embedding λ′ : M_{π′} → ℚ_{l}^{ac} such that ρ ∼ ρ_{π′,λ′} and π′_{∞} has weight k.

Proof: We need only remark that det ρ/ det ρ_{π,λ} has finite l-power order and so by twisting π we may suppose that det ρ = det ρ_{π,λ} (as l > 2). □

4 A potential version of Serre’s conjecture

In this section we will prove the following result, which we will improve somewhat in section 5.

Proposition 4.1 Let l > 2 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous odd representation with \overline{ρ}|_{I_{l}} ∼ ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} for some integer 2 ≤ k ≤ l. (In particular \overline{ρ}|_{G_{l}} is absolutely irreducible.) Then there is a Galois totally real field F of even degree in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ}; 2. π_{∞} has weight 2; and

3. for each place x of F above l, WD_{λ}(π_{x}) is tamely ramified and

WD_{λ}(π_{x})|_{I_{x}} = ω_{2}^{k−(l+1)} ⊕ ω_{2}^{lk−(l+1)}.

We remark that the key improvement of this over results in [Tay4] is the condition that l split completely in F. This may seem minor but it will be crucial for the arguments in section 5 and the proof of theorem 5.7. We now turn to the proof of the proposition. Suppose that \overline{ρ} is valued in GL_{2}(k′) for some finite field k′ ⊂ 𝔽_{l}^{ac} and let k denote the unique quadratic extension of k′ in 𝔽_{l}^{ac}. We must have that \overline{ρ}|_{G_{l}} = Ind _{ℚ_{l^{2}}}^{ℚ_{l}} θ, where θ|_{I_{l}} = ω_{2}^{k−1} with 2 ≤ k ≤ l and so θ is not equal to its Gal (ℚ_{l^{2}}/ℚ_{l})-conjugate. Set \overline{μ} = ϵ^{−1} det \overline{ρ}, let N denote the minimum splitting field for \overline{μ} and 𝔣_{μ} its conductor. Thus N is a cyclic totally real extension of ℚ. Choose an imaginary quadratic field M in which l remains prime and which contains only two roots of unity. Let δ_{M} denote the unique non-trivial character of 𝔸^{×}/ℚ^{×}N𝔸_{M}^{×} and let 𝔣_{M} denote the conductor of δ_{M}. Choose a Galois totally real field E′′ such that E′′M contains a primitive root of unity ζ of order 2#k^{×}. Note that the degree over 𝔽_{l} of every residue field of a prime of E′′ above l is even. Choose a continuous character χ_{0} : M^{×}(M_{∞}^{×} ×∏_{q} O_{M,q}^{×}) → M^{×} which extends the canonical inclusion on M^{×} (use the fact that M has a prime x |̸2 with −1 ∉ (k(x)^{×})^{2}) and let 𝔣_{0} denote the conductor of χ_{0}. Also choose two distinct odd primes p_{1} and p_{2} such that for both i = 1, 2

• χ_{0} is unramified above p_{i};

• p_{i} ≠ l;

• \overline{ρ} is unramified at p_{i};

• \overline{ρ}(Frob_{p_{i}}) has distinct eigenvalues; and

• p_{i} splits in the Hilbert class field of M.

(We explain why this is possible. Let M′ denote the extension of M cut out by \overline{M}^{ker \overline{ρ}} and by the Hilbert class field H of M. By the Cebotarev density theorem it suffices to find σ ∈ Gal (M′/H) so that \overline{ρ}(σ) has distinct eigenvalues. A fortiori it suffices to find σ ∈ I_{l} so that \overline{ρ}(σ) has distinct eigenvalues. This is possible because (l + 1) |̸k − 1.) Set w = 2w_{E′′M}#(O_{M}/l𝔣_{M}𝔣_{μ}𝔣_{0}𝔣_{0}^{c}O_{M})^{×}, where w_{E′′M} denotes the number of roots of unity in E′′M. Let S_{1} denote set of rational primes dividing 𝔣_{M}𝔣_{μ}𝔣_{0}𝔣_{0}^{c}, let S_{2} be a finite set of rational primes disjoint from S_{1} which split in M and such that the primes of M above S_{2} generate the class group of M, and set S_{0} = S_{1} ∪S_{2} ∪{l, p_{1}, p_{2}}. As in the proof of lemma 1.1 of [Tay4] we can find an open subgroup W_{0} of ∏_{q∉S_{0}} O_{M,q}^{×}/ℤ_{q}^{×} such that W_{0} ∩ M_{S_{0}}^{×}/ℚ_{S_{0}}^{×} ⊂ (M_{S_{0}}^{×}/ℚ_{S_{0}}^{×})^{w}. Let w′ denote the index of W_{0} in ∏_{q∉S_{0}} O_{M,q}^{×}/ℤ_{q}^{×}. Then we can choose a Galois (over ℚ) totally real field E′ such that

• E′ ⊃ E′′;

• E′ contains a primitive root of 1 of order ww′; and

• χ_{0} extends to a continuous character χ_{0} : 𝔸_{M}^{×} → (E′M)^{×}.

(If χ̃_{0} : 𝔸_{M}^{×} → ℂ^{×} is any extension of χ_{0} then χ̃_{0}c(χ̃_{0}) ∏_{x |̸∞} | |_{x} has finite order

and is valued in ℝ_{>0}^{×} and so is identically 1. Hence c(χ̃_{0}) = χ_{0}^{−1} ∏_{x |̸∞} | |_{x} and χ̃_{0} is valued in a CM field.) Let E denote the maximal totally real extension of E′ which is unramified outside lp_{1}p_{2} and tamely ramified at these primes. Choose primes ℘_{1} and ℘_{2} of EM above p_{1} and p_{2} respectively. Also choose a prime λ of EM above l and an embedding k ↪ O_{EM}/λ such that the composite of the Artin map I_{l} → O_{M,l}^{×} with the natural map O_{M,l}^{×} → (O_{EM}/λ)^{×} coincides with ω_{2}^{−1} : I_{l} → k^{×} ⊂ (O_{EM}/λ)^{×}. Let μ : Gal (N/ℚ) → (EM)^{×} be the unique character reducing modulo λ to \overline{μ}. For i = 1, 2 we can find α_{i} ∈ (℘_{i} ∩ M)O_{E′′M} which reduces modulo λ to an eigenvalue of \overline{ρ}(Frob_{p_{i}}) and which satisfies α_{i}α_{i}^{c} = p_{i}. (First choose α′_{i} ∈ M ∩ ℘_{i} satisfying α′_{i}(α′_{i})^{c} = p_{i} and then multiply α′_{i} by a suitable root of unity in E′′M.)

Lemma 4.2 Let 𝔞′ denote the product of all primes of E above lp_{1}p_{2} and factor 𝔞′O_{ME} = 𝔞𝔞^{c}, where ℘_{1}℘_{2}λ|𝔞 (which is possible as p_{1} and p_{2} split in M and as the degree over 𝔽_{l} of the residue field of every prime of E above l is even). There is a unit η ∈ O_{E}^{×} with η ≡ ζ mod 𝔞.

Proof: Let \overline{ζ} denote the image of ζ in O_{E′}/(𝔞′ ∩ O_{E′}) = O_{E′M}/(𝔞 ∩ O_{E′M}). Let H denote the maximal totally real abelian extension of E which is unramified outside lp_{1}p_{2} and which is tamely ramified above each of these three primes. Thus H/E′ is Galois and Gal (H/E) is the commutator subgroup of Gal (H/E′). In particular the transfer map Gal (E/E′) → Gal (H/E) vanishes. By class field theory we can identify (O_{E′}/𝔞′)^{×}/O_{E′}^{×} as a subgroup of Gal (E/E′) and (O_{E}/𝔞)^{×}/O_{E}^{×} as a subgroup of Gal (H/E) in such a way that the natural map

(O_{E′}/𝔞′)^{×}/O_{E′}^{×} ⟶ (O_{E}/𝔞)^{×}/O_{E}^{×}

corresponds to the transfer map on Galois groups and so is trivial. The lemma follows. □

Lemma 4.3 There is a continuous character χ : 𝔸_{M}^{×} → (EM)^{×} such that

• χ|_{M^{×}} is the canonical inclusion; • χ|_{O_{M,l}^{×}} is the unique character of order prime to l with

χ|_{O_{M,l}^{×}}(x) ≡ x^{l+1−k} mod λ

for all x ∈ O_{M,l}^{×} and all primes λ of EM above l (and where O_{M} ↪ O_{EM} via the natural map);

• for i = 1, 2, χ is non-ramified above p_{i} and χ|_{M_{℘_{i}}} (p_{i}) = α_{i}; and

• χ|_{𝔸^{×}} = μδ_{M/ℚ}|| ||^{−1}i_{∞} where δ_{M/ℚ} is the unique non-trivial character of 𝔸^{×}/ℚ^{×}N𝔸_{M}^{×}, || || is the product of the usual absolute values and i_{∞} is the projection onto ℝ^{×}.

Proof: Note that χ_{0}|_{𝔸^{×}} = νδ_{M/ℚ}|| ||^{−1}i_{∞}, where ν is a finite order character of 𝔸^{×}/ℚ^{×}ℝ^{×} with conductor dividing 𝔣_{0}𝔣_{0}^{c}𝔣_{M}. We look for χ = χ_{0}χ_{1}. Thus we are required to find a finite order continuous character χ_{1} : 𝔸_{M}^{×}/M^{×} ⟶ (EM)^{×} such that

• χ_{1}|_{𝔸^{×}} = μν^{−1}, and

• χ_{1} has prescribed, finite order restriction to M_{℘_{1}}^{×}, M_{℘_{2}}^{×} and O_{M,l}^{×}, the latter compatible with μν^{−1}|_{ℤ_{l}^{×}} (because \overline{μ}|_{ℤ_{l}^{×}} takes x to (x mod l)^{2−k}).

Note that μν^{−1} has conductor dividing 𝔣_{M}𝔣_{μ}𝔣_{0}𝔣_{0}^{c}. Also note that for i = 1, 2 the unit a_{i} = α_{i}χ_{0}(ϖ_{℘_{i}})^{−1} satisfies a_{i}a_{i}^{c} = 1 for all complex conjugations c and so is a root of unity. Thus the specified restrictions have orders dividing w_{E′′M} in the first two cases and #(O_{M}/l𝔣_{M}𝔣_{μ}𝔣_{0}𝔣_{0}^{c}O_{M})^{×} in the third case. We can find a character

χ_{1,S_{0}} : ∏_{q∈S_{0}} M_{q}^{×} ⟶ (EM)^{×}

with the desired restrictions to ∏_{q∈S_{0}} ℚ_{q}^{×}, M_{℘_{1}}^{×}, M_{℘_{2}}^{×} and O_{M,l}^{×}, and with order dividing w. As

( ∏_{q∈S_{0}} M_{q}^{×} × ∏_{q∉S_{0}} O_{M,q}^{×})/M_{S_{0}}^{×} ⟶^{∼} 𝔸_{M}^{×}/M^{×}M_{∞}^{×},

it suffices to find a character

χ_{1}^{S_{0}} : ∏_{q∉S_{0}} O_{M,q}^{×}/ℤ_{q}^{×} ⟶ (EM)^{×}

which coincides with χ_{1,S_{0}}^{−1} on M_{S_{0}}^{×}/ℚ_{S_{0}}^{×}. One can choose such a character which is trivial on W_{0} and so has order dividing w′. □

We remark that as χ(c ∘ χ)|| ||(i_{∞} ∘ N_{M/ℚ})^{−1} has finite image contained in the totally positive elements of E^{×} we must have χ(c ∘ χ) = || ||^{−1}(i_{∞} ∘ N_{M/ℚ}).

If x is a place of EM above a place x′ of M, let χ_{x} denote the character

\begin{matrix}𝔸_{M}^{×}/M^{×} & ⟶ & (EM)_{x}^{×} \\ a & ⟼ & χ(a)a_{x′}^{−1} \end{matrix}

where a_{x′} denotes the x′ component of a embedded in (EM)_{x}^{×} via the natural map M_{x′} → (EM)_{x}. Set 𝔟 = λ℘_{1}℘_{2} and 𝔟_{0} = 𝔟∩E, so that O_{E}/𝔟_{0} ≅ O_{EM}/λ×O_{EM}/℘_{1}×O_{EM}/℘_{2}. Let W_{𝔟_{0},0}/ℚ denote the finite free group scheme with O_{E}-action which has

W_{𝔟_{0},0}(ℚ^{ac}) ≅ O_{E}/𝔟_{0}(1) ⊕ O_{E}/𝔟_{0}.

By the standard pairing on W_{𝔟_{0},0} we shall mean the map W_{𝔟_{0},0} ⊗_{O_{E}} 𝔡_{E}^{−1} → W_{𝔟_{0},0}^{∨} which corresponds to the pairing

\begin{matrix}(O_{E}/𝔟_{0}(1) ⊕ O_{E}/𝔟_{0}) & × & (O_{E}/𝔟_{0}(1) ⊕ O_{E}/𝔟_{0}) & ⟶ & O_{E}/𝔟_{0}(1) \\ (x_{1}, y_{1}) & × & (x_{2}, y_{2}) & ⟼ & y_{2}x_{1} − y_{1}x_{2}. \end{matrix}

We will let X/ℚ denote the moduli space for quadruples (A, i, j, α), where (A, i, j) is an E-HBAV and α : W_{𝔟_{0},0} →^{∼} A[𝔟_{0}] takes the standard pairing on W_{𝔟_{0},0} to the j-Weil pairing on A[𝔟_{0}]. As 𝔟_{0} is divisible by two primes with coprime residue characteristic we see that X is a fine moduli space. As in section 1 of [Rap] we see that X is smooth and geometrically connected (because of the analytic uniformization of its complex points by a product of copies of the upper half complex plane). Let Γ denote the set of pairs

(γ, ε) ∈ GL_{2}(O_{E}/𝔟_{0}) × O_{E,≫0}^{×}/(O_{E,≡1 (𝔟_{0})}^{×})^{2}

such that

ε det γ ≡ 1 mod 𝔟_{0}.

Here O_{E,≫0}^{×} denotes the set of totally positive elements of O_{E}^{×}, and O_{E,≡1 (𝔟_{0})}^{×} denotes the set of elements of O_{E}^{×} which are congruent to 1 modulo 𝔟_{0}. The group Γ acts faithfully on X via

(γ, ε)(A, i, j, α) = (A, i, j ∘ ε^{−1}, α ∘ γ^{−1}).

The action of G_{ℚ} on the group of automorphisms of X preserves Γ and we have

σ(γ, ε) = (( \begin{matrix}ϵ(σ) & 0 \\ 0 & 1 \end{matrix} ) γ ( \begin{matrix}ϵ(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} ) , ε) .

The set H^{1}(G_{ℚ}, Γ) is in bijection with the set of pairs (R, ψ) where R : G_{ℚ} → GL_{2}(O_{E}/𝔟_{0}) is a continuous representation and ψ : G_{ℚ} → O_{E,≫0}^{×}/(O_{E,≡1 (𝔟_{0})}^{×})^{2} is a continuous homomorphism with

ϵ^{−1} det R ≡ ψ^{−1} mod 𝔟_{0}.

This pair corresponds to the cocycle

(R, ψ)(σ) = (R(σ) ( \begin{matrix}ϵ(σ)^{−1} & 0 \\ 0 & 1 \end{matrix} ) , ψ(σ)) .

Thus to any such pair we can associated a twist X_{R,ψ}/ℚ of X/ℚ. Next we will give a description of the F -rational points of X_{R,ψ} for any number field F. Let N′ denote the splitting field of ψ. Let W_{R}/ℚ denote the finite free group scheme with an action of O_{E} such that

W_{R}(ℚ^{ac}) ≅ O_{E}/𝔟_{0} ⊕ O_{E}/𝔟_{0}

with Galois action via R. By the standard pairing on W_{R}/N′ we shall mean the map W_{R} ⊗_{O_{E}} 𝔡_{E}^{−1} → W_{R}^{∨} (defined over N′) which corresponds to the pairing

\begin{matrix}(O_{E}/𝔟_{0} ⊕ O_{E}/𝔟_{0}) & × & (O_{E}/𝔟_{0} ⊕ O_{E}/𝔟_{0}) & ⟶ & O_{E}/𝔟_{0} \\ (x_{1}, y_{1}) & × & (x_{2}, y_{2}) & ⟼ & y_{2}x_{1} − y_{1}x_{2}. \end{matrix}

Then F -rational points of X_{R,ψ} correspond to quadruples (A, i, j, β), where (A, i, j)/N′F is an E-HBAV and where β : W_{R} →^{∼} A[𝔟_{0}] such that

• under β the standard pairing on W_{R} and the j-Weil pairing on A[𝔟_{0}] correspond, and

• for all σ ∈ Gal (N′F/F) there is an isomorphism

κ_{σ} : σ(A, i) ⟶^{∼} (A, i)

such that σ(j) = κ_{σ}^{∗} ∘ j ∘ ψ(σ)^{∼} for some lifting ψ(σ)^{∼} ∈ O_{E}^{×} of ψ(σ) and such that for some lifting σ^{∼} ∈ G_{F} of σ

σA[𝔟_{0}] ⟶^{κ_{σ}} A[𝔟_{0}]
↑  ↑
W_{R}  ⟶^{R(σ^{∼})} W_{R}

commutes, where the left vertical arrow is σ^{∼} ∘ β and the right one is β.

We will be particularly interested in two pairs (R, ψ) defined as follows. For σ ∈ Gal (N/ℚ) we can write μ(σ) = ζ^{−2m_{σ}} for some integer m_{σ}. Define η_{σ} = (ηζ^{−1})^{m_{σ}} ∈ O_{EM,≡1 (𝔟)}^{×} and ψ(σ) = N_{EM/E}η_{σ} = η^{2m_{σ}}. As

η^{2#k^{×}} = (−η^{#k^{×}})^{2} ∈ (O_{E,≡1 (𝔟_{0})}^{×})^{2},

we see that

ψ : Gal (N/ℚ) ⟶ O_{E,≫0}^{×}/(O_{E,≡1 (𝔟_{0})}^{×})^{2}

is a homomorphism. Let

R\overline{ρ} = \overline{ρ} ⊕ Ind _{G_{M}}^{G_{ℚ}} χ_{℘_{1}} ⊕ Ind _{G_{M}}^{G_{ℚ}} χ_{℘_{2}}

and

R_{Dih} = Ind _{G_{M}}^{G_{ℚ}} χ_{λ} ⊕ Ind _{G_{M}}^{G_{ℚ}} χ_{℘_{1}} ⊕ Ind _{G_{M}}^{G_{ℚ}} χ_{℘_{2}},

so that ϵ^{−1} det R_{ρ} = ϵ^{−1} det R_{Dih} = μ. Then (R\overline{ρ}, ψ) and (R_{Dih}, ψ) define elements of H^{1}(G_{ℚ}, Γ) and we will denote the corresponding twists of X by X\overline{ρ} and X_{Dih} respectively. Note that X\overline{ρ} and X_{Dih} become isomorphic over ℚ_{l}, ℚ_{p_{1}}, ℚ_{p_{2}} and ℝ.

Lemma 4.4 Suppose that F is a number field. If X\overline{ρ} has an F -rational point then there exists an abelian variety B/F of dimension [EM : ℚ], an embedding i′ : O_{EM} ↪ End (B/F), and an isomorphism β′ between B[𝔟](F^{ac}) and R\overline{ρ}.

Proof: Suppose that (A, i, j, β)/F N is a quadruple corresponding to an F - rational point of X\overline{ρ} as above. Also, for σ ∈ Gal (NF/F) let κ_{σ} : σA →^{∼} A be the maps of the last but one paragraph. Set B = A ⊗_{O_{E}} O_{EM} and let i′ denote the natural map O_{EM} → End (B). Let β′ denote the composite

W_{R\overline{ρ}} ⟶^{β} A[𝔟_{0}] ⟶ A[𝔟_{0}] ⊗_{O_{E}} O_{EM}/𝔟 = B[𝔟].

Define f_{0} : O_{EM} → Hom_{O_{E}}(O_{EM}, O_{E}) by f_{0}(a)(b) = tr_{EM/E}ab^{c} and set f = j(1) ⊗ f_{0} a polarisation of B. Also set

κ′_{σ} = κ_{σ} ⊗ η_{σ} : σB ⟶ B.

We see that κ′_{σ} commutes with the action of O_{EM}, that σf = (κ′_{σ})^{∨}fκ′_{σ} and that for any lifting σ^{∼} ∈ G_{F} of σ

σB[𝔟] ⟶^{κ′_{σ}} B[𝔟]
↑  ↑
W_{R\overline{ρ}}  ⟶^{R\overline{ρ}(σ^{∼})} W_{R\overline{ρ}}

commutes, where the left vertical arrow is σ^{∼} ∘ β′ and the right one is β′. As the quadruple (B, i′, f, β′) has no non-trivial automorphisms (because any automorphism of (B, i′, f) has finite order and because 𝔟 is divisible by two primes with distinct residual characteristic), we see that κ′_{σ}σ(κ′_{τ}) = κ′_{στ}. Thus we can descend (B, i′) to F in such a way that β′ also descends to an isomorphism

β′ : W_{R\overline{ρ}} →^{∼} B[𝔟] over F. □

Lemma 4.5 X_{Dih} has a ℚ-rational point and hence X\overline{ρ} has rational points over ℚ_{l}, ℚ_{p_{1}}, ℚ_{p_{2}} and over ℝ.

Proof: Fix an embedding τ : M ↪ ℂ and let Φ denote the CM-type for EM consisting of all embeddings EM ↪ ℂ which restrict to τ on M. Let (𝔡_{EM}^{−1})^{−} denote the ordered O_{E}-module {d ∈ 𝔡_{EM}^{−1} : tr_{EM/E}d = 0} with (𝔡_{EM}^{−} ⊗_{E,σ}ℝ)^{+} the subset with positive imaginary part under σ⊗τ. From the theory of complex multiplication (see [Lang], particularly theorem 5.1 of chapter 5) we see that there is • an abelian variety A/M of dimension [E : ℚ];

• an embedding i : O_{EM} ↪ End (A/M);

• an isomorphism j : (𝔡_{EM}^{−1})^{−} →^{∼} P(A, i|_{O_{E}}); and

• for each prime 𝔮 of E a Galois invariant isomorphism α_{𝔮} : O_{EM,𝔮}(χ_{𝔮}) →^{∼} T_{𝔮}A

such that

• the action of EM on Lie τA is ⨁_{σ∈Φ} σ; and

• for any d ∈ (𝔡_{EM}^{−1})^{−} which is totally positive the j(d)-Weil pairing on T_{𝔮}A

is given by

x × y ⟼ tr_{EM/E}dxy^{c}.

(For the existence of j note that if f is a polarisation of τA/ℂ such that the f-Rosati involution stabilises and acts trivially on E, then the f-Rosati involution also stabilises EM and acts on it via complex conjugation. This follows from the fact that EM is the centraliser of E in End (τA/ℂ).) As χ(c ∘ χ) = (|| ||^{−1}i_{∞}) ∘ N_{M/ℚ}, we see that for σ ∈ G_{M} we have

tr_{EM/M}dα_{𝔮}(σx)α_{𝔮}(σy)^{c} = ϵ_{q}(σ)tr_{EM/M}dα_{𝔮}(x)α_{𝔮}(y)^{c}.

Thus the quadruple (A, i|_{O_{E}}, j, (∏_{𝔮} α_{𝔮}) mod 𝔟_{0}) defines a point in X_{Dih}(M). As χ(χ ∘ c) = (|| ||^{−1}i_{∞}μ) ∘ N_{M/ℚ}, we see that c ∘ χ ∘ N_{NM/M} = χ ∘ c ∘ N_{NM/M} and so over NM there is an isomorphism between (A, i, j, {α_{𝔮}}) and (cA, c ∘ i ∘ c, c ∘ j, {c ∘ α_{𝔮} ∘ c}). Thus the point in X_{Dih}(M) ⊂ X_{Dih}(NM) defined by (A, i|_{O_{E}}, j, (∏_{𝔮} α_{𝔮}) mod 𝔟_{0}) is invariant under c and so lies in X_{Dih}(ℚ). □

Combining the last two lemmas with a theorem of Moret-Bailly (see theorem G of [Tay4]) we see that we can find a Galois totally real field F of even degree in which l, p_{1} and p_{2} split completely, an abelian variety B/F of dimension [EM : M] and an embedding i : O_{EM} ↪ End (B/F) such that B[λ] realises \overline{ρ} and, for i = 1, 2, B[℘_{i}] realises Ind _{G_{M}}^{G_{ℚ}} (χ_{℘_{i}} mod ℘_{i}). As B[λ] is unramified at any prime above p_{1} we see that the action of inertia at such a prime on T_{λ}B has l-power order. As B[℘_{2}] is unramified at any prime above p_{1} we see that the action of inertia at such a prime on T_{℘_{2}}B has p_{2}-power order. Hence the action of inertia at a prime above p_{1} on T_{λ}B has both l-power order and p_{2}-power order. We conclude that T_{λ}B is unramified at primes above p_{1} and hence B has semi-stable reduction at such primes. As p_{1} splits completely in F and as B[℘_{1}] is reducible as a representation of the decomposition group of any prime of F above p_{1}, we see that T_{℘_{1}}B is an ordinary representation of the decomposition group at any prime of F above p_{1}. If x is a prime of F above l then I_{x} acts on both B[℘_{1}] and B[℘_{2}] via ω_{2}^{k−(l+1)} ⊕ ω_{2}^{lk−(l+1)} where ω̃_{2} : I_{x} → O_{EM}^{×} is tamely ramified and reduces mod λ to ω_{2}. Thus I_{x} acts on T_{℘_{1}}B by ω_{2}^{k−(l+1)} ⊕ ω_{2}^{lk−(l+1)}. Because Ind _{G_{F M}}^{G_{F}} (χ_{℘_{1}}) is modular, theorem 5.1 of [SW2] tells us that there is a algebraic, cuspidal automorphic representation π of GL_{2}(𝔸_{F}) of weight 2 and an embedding M_{π} ↪ EM such that ρ_{π,℘_{1}} is equivalent to T_{℘_{1}}B. (Alternatively one may appeal to the main theorem of [SW1], theorem 3.3 of this paper and a standard descent argument.) It follows that in addition ρ_{π,λ} is equivalent to T_{λ}B. This completes the proof of proposition 4.1. Using Langlands base change [Langl] we immediately obtain the following corollary.

Corollary 4.6 Let l > 2 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous odd representation with \overline{ρ}|_{I_{l}} ∼ ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} for some integer 2 ≤ k ≤ l. Then there is a Galois totally real field F of even degree in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ};

2. π_{∞} has weight 2;

3. the central character of π^{∞,l} is unramified; and

4. for each place x of F above l, WD_{λ}(π_{x}) is tamely ramified and

WD_{λ}(π_{x})|_{I_{x}} = ω_{2}^{k−(l+1)} ⊕ ω_{2}^{lk−(l+1)}.

5 Change of weight

In this section we will prove various refinements of proposition 4.1, but first we shall discuss some results about congruences between modular forms. Let F be a totally real field of even degree in which a prime l > 3 splits completely. Let 𝔫 denote an ideal of O_{F} coprime to l. Let ψ : (𝔸_{F}^{∞})^{×}/F^{×} → (ℚ_{l}^{ac})^{×} be a continuous character trivial on O_{F,x}^{×} if x |̸l and on (1 + lO_{F,x}) if x|l. Suppose further that there exists i ∈ (ℤ/(l − 1)ℤ) such that for a ∈ O_{F,l}^{×}, ψ(a) is congruent to (Na)^{−i} modulo the maximal ideal of O_{ℚ_{l}^{ac}}. Let D denote the division algebra with centre F ramified at exactly the infinite places of F. Let O_{D} be a maximal order in D and fix an isomorphism O_{D,x} ≅ M_{2}(O_{F,x}) for each finite place x of F. We will write

• U_{0}(𝔫, l) for U_{{1}}(𝔫l), and

• U_{1}(𝔫, l) for U_{(O_{F} /lO_{F})^{×}}(𝔫l).

(See section 1, in particular the paragraph after corollary 1.2, for this notation.) We will let \overline{η}^{i} denote the character U_{0}(𝔫, l)/U_{1}(𝔫, l) → (𝔽_{l}^{ac})^{×} which sends u, with

u_{l} = ( \begin{matrix}∗ & ∗ \\ ∗ & d \end{matrix} ) ,

to (Nd mod l)^{i}. We will also let η^{i} denote the Teichmüller lift of \overline{η}^{i}. For any O_{ℚ_{l}^{ac}}-algebra R, there is a natural embedding

S_{η^{i}⊗R,ψ}(U_{0}(𝔫, l)) ↪ S_{η^{i}⊗R,ψ}(U_{1}(𝔫, l)) = S_{2,R,ψ}(U_{1}(𝔫, l)),

which is equivariant for the action of T_{x} and S_{x} for all x |̸l𝔫, and for U_{ϖ_{x}} for x|𝔫. The image is the subset of S_{2,R,ψ}(U_{1}(𝔫, l)) where ⟨h⟩ = 1 for all h ∈ (O_{F} /lO_{F})^{×}. If ϕ : h_{η^{i},𝔽_{l},ψ}(U_{0}(𝔫, l)) → 𝔽_{l}^{ac} has non-Eisenstein kernel then for x|l we have

det \overline{ρ}_{ϕ}|_{I_{x}} = ω^{1+i}.

The operators U_{ϖ_{x}} and V_{ϖ_{x}} on S_{2,𝔽_{l}^{ac},ψ}(U_{1}(𝔫, l)) commute with the action of ⟨h⟩ for h ∈ (O_{F} /lO_{F})^{×} and hence preserve S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)). We will let h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l))′ (resp. h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l))′′) denote the commutative subalgebra of the endomorphisms of S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)) generated by h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l)) and U_{ϖ_{x}} (resp. V_{ϖ_{x}}) for all x|l. If ϕ : h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l))′ → 𝔽_{l}^{ac} and ϕ(U_{ϖ_{x}}) ≠ 0 then

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

where χ_{2} is unramified and χ_{2}(Frob_{x}) = ϕ(U_{ϖ_{x}}) (see [W1]). If f_{1} ∈ S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)) and f_{2} ∈ S_{\overline{η}^{−i},ψ^{−1}}(U_{0}(𝔫, l)) then define (f_{1}, f_{2}) to be

∑_{[x]∈D^{×}\(D⊗_{ℚ}𝔸^{∞})/U_{0}(𝔫,l)(𝔸_{F}^{∞})^{×}} f_{1}(x)f_{2}(xw)(#(U_{0}(𝔫, l)(𝔸_{F}^{∞})^{×} ∩x^{−1}D^{×}x)/F^{×})^{−1},

where

w_{x} = ( \begin{matrix}0 & 1 \\ ϖ_{x}^{x(𝔫)} & 0 \end{matrix} )

if x |̸l and w_{x} = 1_{2} if x|l. This is easily seen to be a perfect pairing. Moreover a standard calculation shows that the adjoint of S_{x} is S_{x}^{−1}, the adjoint of T_{x} is S_{x}^{−1}T_{x}, the adjoint of U_{ϖ_{x}} for x|𝔫 is S_{ϖ_{x}}^{−1}U_{ϖ_{x}} and the adjoint of U_{ϖ_{x}} for x|l is S_{ϖ_{x}}^{−1}V_{ϖ_{x}}. Thus if ϕ : h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l)) → 𝔽_{l}^{ac} then there is also a homomorphism ϕ^{∗} : h_{\overline{η}^{−i},𝔽_{l},ψ^{−1}}(U_{0}(𝔫, l)) → 𝔽_{l}^{ac} satisfying ϕ^{∗}(T_{x}) = ϕ(S_{x})^{−1}ϕ(T_{x}) and ϕ^{∗}(S_{x}) = ϕ(S_{x})^{−1}. Moreover if ϕ extends to h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l))′′ so that ϕ(V_{ϖ_{x}}) ≠ 0 then ϕ^{∗} extends to h_{\overline{η}^{−i},𝔽_{l},ψ^{−1}}(U_{0}(𝔫, l))′ with ϕ^{∗}(U_{ϖ_{x}}) ≠ 0. We deduce that \overline{ρ}_{ϕ^{∗}} = \overline{ρ}_{ϕ}^{∨}(1). Hence if ϕ : h_{\overline{η}^{i},𝔽_{l},ψ}(U_{0}(𝔫, l))′′ → 𝔽_{l}^{ac} and ϕ(V_{ϖ_{x}}) ≠ 0 then

\overline{ρ}_{ϕ}|_{G_{x}} ∼ ( \begin{matrix}ϵχ_{1} & ∗ \\ 0 & ω^{i}χ_{2} \end{matrix} )

where χ_{1} and χ_{2} are unramified. We will denote by I^{i} the induced representation from U_{0}(𝔫, l) to U_{H}(𝔫) of \overline{η}^{i}. It is a tesnor product ⨂_{x|l} I_{x}^{i} where I_{x}^{i} is the induction from U_{0}(𝔫, l)_{x} to GL_{2}(O_{F,x}) of \overline{η}^{i}. We can realise I_{x}^{i} concretely as the space of functions

θ : k(x)^{2} − {(0, 0)} ⟶ 𝔽_{l}^{ac}

such that θ(a(x, y)) = a^{i}θ(x, y) for all a ∈ k(x)^{×}. The action of GL_{2}(O_{F,x}) is via (uθ)(x, y) = θ((x, y)u). We have an isomorphism

S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)) ≅ S_{I^{i},ψ}(U_{H}(𝔫))

under which f ∈ S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)) corresponds to F ∈ S_{I^{i},ψ}(U_{H}(𝔫)) if

f(g) = F (g)((0, 1)_{x})

and

F (g)(a_{x}, b_{x}) = f(gu^{−1})

where u ∈ GL_{2}(O_{F,l}) and

u mod x = ( \begin{matrix}∗ & ∗ \\ a_{x} & b_{x} \end{matrix} )

for all x|l. Now suppose that 0 ≤ i ≤ l − 2. If x is a prime of F above l then we have an exact sequence

(0) ⟶ Symm^{i}((𝔽_{l}^{ac})^{2}) ⟶ I_{x}^{i} ⟶ Symm^{l−1−i}((𝔽_{l}^{ac})^{2}) ⊗ det^{i} ⟶ (0).

The first map is just the natural inclusion of homogeneous polynomials of degree i into the space of homogeneous functions of degree i. The second map sends a homogeneous function θ onto the polynomial

∑_{(s,t)∈ℙ^{1}(k(x))} θ(s, t)(tX − sY)^{l−1−i}.

Thus for any subset T of the set of places of F above l we have a submodule I_{T}^{i} ⊂ I^{i} with

I_{T}^{i} ≅ ⨂_{x∉T} Symm^{i}((𝔽_{l}^{ac})^{2}) ⊗ ⨂_{x∈T} I_{x}^{i}.

These give rise to subspaces

S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l)) ⊂ S_{\overline{η}^{i},ψ}(U_{0}(𝔫, l))

with

S_{\overline{η}^{i},ψ,∅}(U_{0}(𝔫, l)) ≅ S_{i+2,𝔽_{l}^{ac},ψ}(U_{H}(𝔫))

as a module for the Hecke operators T_{x} and S_{x} for all x |̸l𝔫 and for U_{ϖ_{x}} for all x|𝔫. The following lemma is a variant of an unpublished result of Buzzard (see [Bu]).

Lemma 5.1 For any set T of places of F above l and for any place x ∉ T of F above l there is an injection

κ_{x} : S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l))/S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l)) ↪ S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l)) which is equivariant for the actions of T_{y} and S_{y} for all y |̸l and for U_{ϖ_{x}} for x|𝔫, and such that the composite

S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l)) ⟶^{κ_{x}} S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l)) ↪ S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l))

coincides with V_{ϖ_{x}}.

Proof: Define U_{0}(T) ⊂ U_{H}(𝔫) by U_{0}(T)_{y} = U_{0}(𝔫, l)_{y} if y ∈ T and U_{0}(T)_{y} = U_{H}(𝔫)_{y} otherwise. Let τ_{T} denote the representation

(⨂_{y∈T} \overline{η}_{y}^{i}) ⊗ (⨂_{y∉T} Symm^{2+i}((𝔽_{l}^{ac})^{2})

of U_{0}(T)_{l}. If x ∉ T is a place of F above l, let τ_{T,x} denote the representation

(⨂_{y∈T} \overline{η}_{y}^{i}) ⊗ ( ⨂_{y∉T ∪\{x\}} Symm^{i+2}((𝔽_{l}^{ac})^{2})) ⊗ (Symm^{l−1−i}((𝔽_{l}^{ac})^{2}) ⊗ det^{i})

of U_{0}(T)_{l}. Then the exact sequence

\begin{matrix}(0) ⟶ S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l)) ⟶ S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l)) ⟶ \\ ⟶ S_{\overline{η}^{i},ψ,T ∪\{x\}}(U_{0}(𝔫, l))/S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l)) ⟶ (0) \end{matrix}

is identified to the exact sequence

(0) ⟶ S_{τ_{T} ,ψ}(U_{0}(T)) ⟶^{α} S_{τ_{T ∪{x}},ψ}(U_{0}(T ∪ {x})) ⟶^{β} S_{τ_{T,x},ψ}(U_{0}(T)) ⟶ (0),

where

α(f)(g) = f(g)(0, 1)_{x}

and

β(f)(g)(X, Y)_{x} = ∑_{(s:t)∈ℙ^{1}(k(x))} f(gu(s, t)^{−1})(tX − sY)^{l−1−i}

with u(s, t) ∈ GL_{2}(O_{F,x}) congruent to

( \begin{matrix}∗ & ∗ \\ s & t \end{matrix} )

modulo x. Now define

κ : S_{τ_{T,x},ψ}(U_{0}(T)) ⟶ S_{τ_{T ∪{x}},ψ}(U_{0}(T ∪ {x}))

by

κ(f)(g) = f(gγ)(1, 0)_{x}

where

γ = ( \begin{matrix}1 & 0 \\ 0 & ϖ_{x} \end{matrix} ) ∈ GL_{2}(F_{x}).

To see this is well defined the only slightly subtle point is that if u ∈ U_{0}(𝔫, l)_{x} then

κ(f)(gu) = f(gγ(γ^{−1}uγ))(1, 0)_{x}
= (det u)^{i}f(gγ)((1, 0)_{x}(γ^{−1}u^{−1}γ))
= (det u)^{i}f(gγ)(η_{x}(u)/ det u, 0)_{x}
= η_{x}(u)^{−i}f(gγ)(1, 0)_{x}
= η_{x}(u)^{−i}κ(f)(g).

Moreover κ is clearly injective and equivariant for the action of T_{y} and S_{y} if y |̸l𝔫 and for U_{ϖ_{x}} for x|𝔫. Finally we have

(κ ∘ β)(f)(g) = ∑_{(s:t)∈ℙ^{1}(k(x))} f(gγu(s, t)^{−1})t^{l−1−i}
= ∑_{s∈k(x)} f(gγu(s, 1)^{−1})
= (V_{ϖ_{x}}f)(g).

as we can take

u(s, 1) = ( \begin{matrix}1 & 0 \\ s & 1 \end{matrix} ) .

□

Corollary 5.2 There is a natural surjection

h_{\overline{η}^{i},𝔽_{l}^{ac},ψ}(U_{0}(𝔫, l)) → → h_{i+2,𝔽_{l}^{ac},ψ}(U_{H}(𝔫))

which takes T_{y} to T_{y} and S_{y} to S_{y} for all y |̸l𝔫 and which takes U_{ϖ_{x}} to U_{ϖ_{x}} for all x|𝔫. If 𝔪 is a maximal ideal of h_{\overline{η}^{i},𝔽_{l}^{ac},ψ}(U_{0}(𝔫, l)) such that for any x|l and any maximal ideal 𝔪′′_{x} of h_{\overline{η}^{i},𝔽_{l}^{ac},ψ}(U_{0}(𝔫, l))′′ extending 𝔪 one has V_{ϖ_{x}} ∈ 𝔪′′_{x}, then h_{i+2,𝔽_{l}^{ac},ψ}(U_{H}(𝔫))_{𝔪} ≠ (0). This assumption will be verified if 𝔪 is non-Eisenstein and the kernel of a homomorphism ϕ : h_{\overline{η}^{i},ψ}(U_{0}(𝔫, l)) → 𝔽_{l}^{ac} such that for all x|l

\overline{ρ}_{ϕ}|_{G_{x}} ≁ ( \begin{matrix}ϵχ_{1} & ∗ \\ 0 & ω^{i}χ_{2} \end{matrix} ) ,

with χ_{1} and χ_{2} unramified.

Proof: Choose a minimal T such that S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l))_{𝔪} ≠ (0). If T = ∅ then S_{k,𝔽_{l}^{ac},ψ}(U_{H}(𝔫))_{𝔪} ≠ (0) and the corollary follows. Thus suppose that x ∈ T and set T′ = T − {x}. By our minimality assumption we see that

S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l))_{𝔪}→^{∼}(S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l))/S_{\overline{η}^{i},ψ,T′}(U_{0}(𝔫, l)))_{𝔪} ↪^{κ_{x}} S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l))_{𝔪}

and the composite coincides with V_{ϖ_{x}}. Thus V_{ϖ_{x}} is an isomorphism on the space S_{\overline{η}^{i},ψ,T} (U_{0}(𝔫, l))_{𝔪} and V_{ϖ_{x}} does not lie in some maximal ideal of h_{\overline{η}^{i},𝔽_{l}^{ac},ψ}(U_{0}(𝔫, l))′′ above 𝔪, a contradiction. □

We also have the following lemma, which generalises results of Ash and Stevens [AS]. We write U_{0} for ∏_{y} GL_{2}(O_{F,y}).

Lemma 5.3 If k ∈ ℤ_{≥2} and if ϕ : h_{k,𝔽_{l}^{ac},ψ}(U_{0}) → 𝔽_{l}^{ac} is a homomorphism, then there is a homomorphism (Dϕ) : h_{k+l+1,𝔽_{l}^{ac},ψ(ϵ∘Art^{−1})}(U_{0}) → 𝔽_{l}^{ac} such that for all places y |̸l we have (Dϕ)(T_{y}) = ϕ(T_{y})(Ny) and (Dϕ)(S_{y}) = ϕ(S_{y})(Ny)^{2}.

Proof: If f ∈ S_{k,𝔽_{l}^{ac},ψ}(U_{0}) then the function

(Df)(g) = f(g)(||N det g||(N det g_{l}))^{−1},

where || || : (𝔸^{∞})^{×} → ℚ_{>0}^{×} denotes the product of the usual p-adic absolute values, lies in S_{τ_{k,𝔽{ac \atop l}} ⊗(N det),ψ(ϵ∘Art^{−1})}(U_{0}). Moreover if T_{y}f = af (resp. S_{y}f =

bf) then T_{y}(Df) = a(Ny)(Df) (resp. S_{y}(Df) = b(Ny)(Df)). Thus it suffices to exhibit an embedding

S_{τ_{k,𝔽{ac \atop l}} ⊗(N det),ψ(ϵ∘Art^{−1})}(U_{0}) ↪ S_{k+l+1,𝔽_{l}^{ac},ψ(ϵ∘Art^{−1})}(U_{0})

compatible with the action of T_{y} and S_{y} for all y |̸l. By lemma 1.1 it suffices to exhibit a GL_{2}(O_{F,l})-equivariant embedding

⨂_{x}(Symm^{k−2}(k(x)^{2}) ⊗ det) ↪ ⨂_{x} Symm^{k+l−1}(k(x)^{2}),

or simply GL_{2}(O_{F,x})-equivariant embeddings

Symm^{k−2}(k(x)^{2}) ⊗ det ↪ Symm^{k+l−1}(k(x)^{2})

for all x|l. Because l splits completely in F such an embedding simply results from multiplication by X^{l}Y − XY^{l}, as we see from the following calculation. For a, b, c, d ∈ 𝔽_{l} we have

(aX + cY)^{l}(bX + dY) − (aX + cY)(bX + dY)^{l}
= (aX^{l} + cY^{l})(bX + dY) − (aX + cY)(bX^{l} + dY^{l})
= (ad − bc)(X^{l}Y − XY^{l}).

□

We now turn to our improvements to proposition 4.1. First we have the following lemma.

Lemma 5.4 Let l > 3 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous odd representation with \overline{ρ}|_{I_{l}} ∼ ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} for some integer 2 ≤ k ≤ l. Then there is a Galois totally real field F in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ};

2. π_{∞} has weight 2; and

3. for each place x of F above l, π_{x} has conductor dividing x.

Proof: Let F, π, λ be as provided by corollary 4.6. Let ψ_{0} : (𝔸_{F}^{∞})^{×}/F^{×} → (ℚ_{l}^{ac})^{×} be the character such that ϵ(ψ_{0} ∘ Art^{−1}) equals the determinant of ρ_{π,λ}. Thus ψ_{0} is unramified away from l. Let 𝔫_{0} denote the prime to l part of the conductor of π. Let D be the division algebra with centre F which is ramified at exactly the infinite places of F. Let O_{D} be a maximal order in D and fix an isomorphism O_{D,x} ≅ M_{2}(O_{F,x}) for each finite place x of F. Let O denote the ring of integers of ℚ_{l}^{ac}. Let χ_{k} denote the character 𝔽_{l^{2}}^{×} → O^{×} which sends a to the Teichmüller lift of a^{k−l−1}. Let Θ(χ_{k}) denote a model over O of the representation of GL_{2}(ℤ_{l}) → → GL_{2}(𝔽_{l}) denoted the same way in section 3.1 of [CDT]. Let Θ_{k} denote the representation ⨂_{x|l} Θ(χ_{k}) of GL_{2}(O_{F,l}). From proposition 4.1, lemma 1.3 and lemma 4.2.4 of [CDT] we see that there is a homomorphism

ϕ_{1} : h_{Θ_{k},O,ψ_{0}}(U_{H_{0}}(𝔫_{0})) ⟶ 𝔽_{l}^{ac}

such that ker ϕ_{1} is non-Eisenstein and \overline{ρ}_{ϕ_{1}} ∼ \overline{ρ}|_{G_{F}}. By lemma 3.1.1 of [CDT] we see that Θ_{k} ⊗ 𝔽_{l}^{ac} has a Jordan-Hölder sequence with subquotients

R_{T} = ⨂_{x∉T} Symm^{k−2}((𝔽_{l}^{ac})^{2}) ⊗ ⨂_{x∈T}(Symm^{l−1−k}((𝔽_{l}^{ac})^{2}) ⊗ det^{k−1})

where T runs over sets of places of F above l, and where, if k = l, we only have one subquotient namely T = ∅. Thus for some T, ϕ_{1} factors through h_{R_{T} ,𝔽_{l}^{ac},ψ_{0}}(U_{H_{0}}(𝔫_{0})). It then follows from corollary 1.5 that for x ∈ T we must have \overline{ρ}|_{I_{x}} ∼ ω_{2}^{k−l} ⊕ ω_{2}^{kl−1} or

( \begin{matrix}1 & ∗ \\ 0 & ω^{k−1} \end{matrix} ) .

Thus in fact ϕ_{1} must factor through h_{R_{∅},𝔽_{l}^{ac},ψ_{0}}(U_{H_{0}}(𝔫_{0})) = h_{k,𝔽_{l}^{ac},ψ_{0}}(U_{H_{0}}(𝔫_{0})). It follows from the first part of corollary 5.2 that ϕ_{1} gives rise to a map

ϕ_{0} : h_{η^{k−2},O,ψ_{0}}(U_{H_{0}}(𝔫_{0})) ⟶ 𝔽_{l}^{ac}

such that ker ϕ_{0} is non-Eisenstein and \overline{ρ}_{ϕ_{0}} ∼ \overline{ρ}|_{G_{F}}. The proposition follows. □

Combining the lemma 5.4 with the main theorem of [SW1], we immediately obtain the following corollary.

Corollary 5.5 Let l > 3 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous odd representation with \overline{ρ}|_{I_{l}} ∼ ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} for some integer 2 ≤ k ≤ l. Then there is a Galois totally real field F in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that 1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ};

2. π_{∞} has weight 2;

3. for each finite place x of F not dividing l, π_{x} is unramified; and

4. for each place x of F above l, the conductor of π_{x} divides x.

Now we can use corollary 5.2 to obtain a further refinement of proposition 4.1.

Lemma 5.6 Let l > 3 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous odd representation with \overline{ρ}|_{I_{l}} ∼ ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} for some integer 2 ≤ k ≤ l. Then there is a Galois totally real field F of even degree in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ};

2. π_{∞} has weight k; and

3. π_{x} is unramified at every finite place x of F.

Proof: Now let F, π, λ be as provided by corollary 5.5. Also denote by ψ_{0} : (𝔸_{F}^{∞})^{×}/F^{×} → (ℚ_{l}^{ac})^{×} be the character such that ϵ(ψ_{0} ∘ Art^{−1}) equals the determinant of ρ_{π,λ}. Thus ψ_{0} is unramified away from l. Note also that if a ∈ O_{F,l}^{×} then ψ_{0}(a) is the Teichmüller lift of (Na)^{2−k} mod l. Let D be the division algebra with centre F which is ramified at exactly the infinite places of F. Let O_{D} be a maximal order in D and fix an isomorphism O_{D,x} ≅ M_{2}(O_{F,x}) for each finite place x of F. Let U_{0} = ∏_{y} GL_{2}(O_{F,y}). There is a homomorphism

ϕ_{0} : h_{\overline{η}^{k−2},𝔽_{l}^{ac},ψ_{0}}(U_{0}(O_{F} , l)) → 𝔽_{l}^{ac}

with ker ϕ_{0} non-Eisenstein and \overline{ρ}_{ϕ_{0}} ∼ \overline{ρ}|_{G_{F}}. By corollary 5.2 this factors through h_{k,𝔽_{l}^{ac},ψ_{0}}(U_{0}) and the proposition follows. □

Finally we have the following version of our potential version of Serre’s conjecture.

Theorem 5.7 Let l > 3 be a prime. Suppose that \overline{ρ} : G_{ℚ} → GL_{2}(𝔽_{l}^{ac}) is a continuous irreducible odd representation with \overline{ρ}|_{G_{l}} irreducible. Then there is a Galois totally real field F of even degree in which l splits completely, a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) and an embedding λ : M_{π} ↪ ℚ_{l}^{ac} such that

1. \overline{ρ}|_{G_{F}} ∼ \overline{ρ}_{π,λ};

2. π_{∞} has weight k\overline{ρ}, where k\overline{ρ} is the weight associated to \overline{ρ}|_{G_{l}} by Serre in [S2]; and

3. π_{x} is unramified for every finite place x of F.

Proof: From the definition of k\overline{ρ} we see that there is an integer 0 ≤ c < l − 1 such that 2 ≤ k\overline{ρ} −c(l +1) ≤ l and (\overline{ρ}⊗ϵ^{−c})|_{I_{l}} ∼ ω_{2}^{k\overline{ρ}−1−c(l+1)} ⊕ω_{2}^{l(k\overline{ρ}−1)−c(l+1)}. By lemma 5.6 we can find a Galois totally real field F of even degree in which l splits completely and a regular algebraic cuspidal automorphic representation π of GL_{2}(𝔸_{F}) such that

1. (\overline{ρ}⊗ϵ^{−c})|_{G_{F}} is equivalent to \overline{ρ}_{π,λ} for some prime λ|l and some embedding k(λ) ↪ 𝔽_{l}^{ac};

2. π_{∞} has weight k\overline{ρ} − c(l + 1); and

3. π_{x} is unramified at every finite place x of F.

By lemma 1.3 we can find, for some character ψ, a homomorphism

ϕ : h_{k\overline{ρ}−c(l+1),𝔽_{l}^{ac},ψ}(U_{0}) → 𝔽_{l}^{ac}

with non-Eisenstein kernel such that \overline{ρ}_{ϕ} ≅ (\overline{ρ} ⊗ ϵ^{−c})|_{G_{F}}. The theorem now follows from lemma 5.3. □

6 Applications

Combining theorem 2.1 of [Tay4], theorem 5.7, theorem 3.3 and a standard descent argument (see for example the proof of theorem 2.4 of [Tay3]) we obtain our main theorem.

Theorem 6.1 Let l > 3 be a prime and let 2 ≤ k ≤ l − 1 be an integer. Let ρ : G_{ℚ} → GL_{2}(O_{ℚ_{l}^{ac}}) be a continuous irreducible representation such that

• ρ is ramifies at only finitely many primes,

• det ρ(c) = −1,

• ρ|_{G_{l}} is crystalline with Hodge-Tate numbers 0 and 1 − k.

Let \overline{ρ} denote the reduction of ρ modulo the maximal ideal of O_{ℚ_{l}^{ac}}. If \overline{ρ}|_{G_{l}} is irreducible assume that \overline{ρ} restricted to ℚ_{l}(\sqrt{(−1)^{(l−1)/2}l}) is irreducible. (This will be the case if, for instance, 2k ≠ l + 3.) Then there is a Galois totally real field F in which l is unramified with the following property. For each subfield E ⊂ F with Gal (F/E) soluble there is a regular algebraic cuspidal automorphic representation π_{E} of GL_{2}(𝔸_{E}) and an embedding λ of the feild of coefficients of π_{E} into ℚ_{l}^{ac} such that

• ρ_{π_{E},λ} ∼ ρ|_{G_{E}},

• π_{E,x} is unramified for all places x of E above l, and

• π_{E,∞} has weight k.

(We explain the parenthetical comment. In the case discussed in that comment \overline{ρ}|_{I_{l}} = ω_{2}^{k−1} ⊕ ω_{2}^{l(k−1)} and so \overline{ρ}|_{I_{ℚ_{l}(\sqrt{±l})}} = (ω′_{2})^{2(k−1)} ⊕ (ω′_{2})^{2l(k−1)} where ω′_{2} is the fundamental character of level 2 of I_{ℚ_{l}(\sqrt{±l})}. The assumption k ≠ (l + 3)/2 tells us that (l + 1) |̸2(k − 1) so that \overline{ρ}|_{G_{ℚ_{l}(\sqrt{±l})}} is absolutely irreducible.) Combining this with the main theorem of [Bl] we deduce the following corollary.

Corollary 6.2 Keep the assumptions of theorem 6.1. If ρ is unramified at a prime p and if α is an eigenvalue of ρ(Frob_{p}) then α ∈ ℚ^{ac} and for any isomorphism i : ℚ_{l}^{ac} →^{∼} ℂ we have

|iα|^{2} ≤ p^{(k−1)/2}.

(We remark that we can deduce this corollary for all but finitely many primes p by appealing to theorem 3.4.6 of [BL] instead of the main theorem of [Bl].) Continue to assume that ρ satisfies the hypotheses of theorem 6.1. If p ≠ l and if i : ℚ_{l}^{ac} →^{∼} ℂ then we define

L_{p}(iρ, X) = i det(1 − ρ_{I_{p}}(Frob_{p})X) ∈ ℂ[X].

Corollary 6.2 tells us that

L^{l}(iρ, s) = ∏_{p≠l} L_{p}(iρ, p^{−s})^{−1}

defines a meromorphic function in Re s > (k + 1)/2. Choose a non-trivial additive character Ψ = ∏ Ψ_{p} : 𝔸/ℚ → ℂ^{×} with ker Ψ_{l} = ℤ_{l} and Ψ_{∞}(x) = e^{2π\sqrt{−1}x}. Also choose a Haar measure dx = ∏ dx_{p} on 𝔸_{F} with dx_{∞} the usual measure on ℝ, with dx_{l}(ℤ_{l}) = 1 and with dx(𝔸_{F} /F) = 1. If p ≠ l we will let WD(ρ|_{G_{p}}) denote the Weil-Deligne representation associated to ρ|_{G_{p}}. Then we define

ϵ(iρ, s) = \sqrt{−1}^{k} ∏_{p≠l} ϵ(iWD(ρ^{∨}|_{G_{p}}) ⊗ |Art^{−1}|_{p}^{−s}, Ψ_{p}, dx_{p}).

(See [Tat].) Note that ϵ(iρ, s) = W N^{k/2−s} where W is independent of s, and where N is the (prime to l) conductor of ρ. The proof of corollary 2.2 of [Tay4] then gives the following corollary.

Corollary 6.3 Keep the assumptions of theorem 6.1 and let i : ℚ_{l}^{ac} →^{∼} ℂ. There is a rational function L_{l}(ρ, X) such that if we set

L(iρ, s) = L^{l}(iρ, s)L_{l}(iρ, l^{−s})^{−1}

then L(iρ, s) has meromorphic extension to the entire complex plane and satisfies the functional equation

(2π)^{−s}Γ(s)L(iρ, s) = ϵ(iρ, s)(2π)^{s−k}Γ(k − s)L(i(ρ^{∨} ⊗ ϵ^{k−1}), k − s).

The proof of corollary 2.4 of [Tay4] also gives us the following result.

Corollary 6.4 Keep the assumptions of theorem 6.1 and if k = 2 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.

By a rank d weakly compatible system of l-adic representations R over ℚ we shall mean a 5-tuple (M, S, {Q_{p}(X)}, {ρ_{λ}}, {n_{1}, ..., n_{d}}) where

• M is a number field;

• S is a finite set of rational primes;

• for each prime p ∉ S of ℚ, Q_{p}(X) is a monic degree d polynomial in M[X];

• for each prime λ of M (with residue characteristic l say)

ρ_{λ} : G_{ℚ} ⟶ GL_{d}(M_{λ})

is a continuous representation such that, if l ∉ S then ρ_{λ}|_{G_{l}} is crystalline, if p ∉ S ∪ {l} then ρ_{λ} is unramified at p and ρ_{λ}(Frob_{p}) has characteristic polynomial Q_{p}(X); and

• {n_{1}, ..., n_{d}} is a multiset (i.e. set with multiplicities) of integers such that for all primes λ of M (lying above a rational prime l) the representation ρ_{λ}|_{G_{l}} is Hodge-Tate with numbers {n_{1}, ..., n_{d}}.

We will call {n_{1}, ..., n_{d}} the Hodge numbers of R. We will call R strongly compatible if for each rational prime p there is a Weil-Deligne representation WD_{p}(R) of W_{ℚ_{p}} such that for primes λ of M not dividing p, WD_{p}(R) is equivalent to the Frobenius semi-simplification of the Weil-Deligne representation associated to ρ_{λ}|_{G_{p}}. We will call a rank 2 weakly compatible system R regular if the Hodge numbers are distinct and for one, and hence all, primes λ of M we have det ρ_{λ}(c) = −1. We remark that whatever is meant by a “motive”, the l-adic realisations of a “motive” would give rise to weakly compatible systems of l-adic representations which are generally expected to be strongly compatible. Moreover one can use the Hodge realisation to see that if the Hodge numbers of a rank 2 “motive” are distinct then the associated system of l-adic representations is regular in the above sense. This explains the perhaps somewhat unnatural definition of regularity given above. The following lemma is an easy consequence of the characterisation of one dimensional Hodge-Tate representations of G_{ℚ}.

Lemma 6.5 If R/ℚ is a rank 2 weakly compatible system of l-adic representations and if ρ_{λ} is absolutely reducible for one λ, then ρ_{λ} is absolutely reducible for all λ.

We will call a rank 2 weakly compatible system of l-adic representations reducible if the hypothesis (and hence the conclusion) of the previous lemma holds. Otherwise we call it irreducible.

Theorem 6.6 Suppose that R = (M, S, {Q_{x}(X)}, {ρ_{λ}}, {n_{1}, n_{2}})/ℚ is a regular, irreducible, rank 2 weakly compatible system of l-adic representations with n_{1} > n_{2}.

1. There is a Galois totally real field such that for any i : M ↪ ℂ there is a regular algebraic cuspidal automorphic representation of GL_{2}(𝔸_{F}) with L(iR|_{G_{F}} , s) = L(π, s).

2. For all rational primes p ∉ S and for all i : M ↪ ℂ the roots of i(Q_{p}(X)) have absolute value p^{−(n_{1}+n_{2})/2}.

3. R is strongly compatible.

4. Fix i : M ↪ ℂ. If we define

L(iR, s) = ∏_{p} L_{p}(iWD_{p}(R)^{∨}, s)^{−1}

and

ϵ(iR, s) = i^{1+n_{1}−n_{2}} ∏_{p} ϵ(iWD_{p}(R)^{∨} ⊗ |Art^{−1}|_{p}^{−s}, Ψ_{p}, dx_{p})

then the product defining L(iR, s) converges to a meromorphic function in Re s > 1 − (n_{1} + n_{2})/2 and L(iR, s) has meromorphic continuation to the entire complex plane and satisfies a functional equation

(2π)^{−(s+n_{1})}Γ(s + n_{1})L(iR, s) = ϵ(iR, s)(2π)^{s+n_{2}−1}Γ(1 − n_{2} − s)L(iR^{∨}, 1 − s).

Proof: We may assume that n_{1} = 0. For all but finitely many primes λ of M the representation ρ_{λ} satisfies the hypotheses of theorem 6.1. The first part follows immediately from that theorem and the second part from corollary 6.2. Choose one such prime λ and fix an embedding M_{λ} ⊂ ℚ_{l}^{ac}. Let F be as in theorem 6.1 and write

1 = ∑_{j} m_{j}Ind _{Gal (F/E_{j})}^{Gal (F/ℚ)}χ_{j}

where m_{j} ∈ ℤ, Gal (F/E_{j}) is soluble and χ_{j} is a character of Gal (F/E_{j}). For each j we have a regular algebraic cuspidal automorphic representation π_{j} of GL_{2}(𝔸_{E_{j}}) with field of coefficients M_{j} and an embedding λ_{j} : M_{j} ↪ ℚ_{l}^{ac} such that

ρ_{π_{j},λ_{j}} ∼ ρ_{λ}|_{G_{E_{j}}} .

We see in particular that λ_{j} : M_{j} ↪ M. Thus any embedding λ′ : M ↪ ℚ_{l′}^{ac} gives rise to an embedding λ′_{j} : M_{j} ↪ ℚ_{l′}^{ac}. From the Cebotarev density theorem we see that

ρ_{π_{j},λ′_{j}} ∼ ρ_{λ′}|_{G_{E_{j}}}

and hence that

ρ_{λ′} = ∑_{j} m_{j}Ind _{Gal (ℚ^{ac}/E_{i})}^{Gal (ℚ^{ac}/ℚ)}ρ_{π_{j},λ′_{j}} ⊗ χ_{j}.

As the ρ_{π_{j},λ′_{j}} are strongly compatible (see [Tay1]), the same is true for the ρ_{λ′}. (To check compatibility of the nilpotent operators in the Weil-Deligne representations one notices that it suffices to check that they are equal after any finite base change.) Moreover we see that

L(iR, s) = ∏_{j} L(π_{j} ⊗ (χ_{j} ∘ Art ∘ det), s)^{m_{j}}

and that

ϵ(iR, s) = ∏_{j} ϵ(π_{j} ⊗ (χ_{j} ∘ Art ∘ det), s)^{m_{j}},

and the fourth part of the theorem follows. □

As an example suppose that X/ℚ is a rigid Calabi-Yau 3-fold. Let X /ℤ denote a model for X. Also let ζ_{X}(s) denote the zeta function of X, so that

ζ_{X}(s) = ∏_{p} ζ_{X,p}(p^{−s})^{−1},

where ζ_{X,p}(T) is a rational function of T and for all but finitely many p we have

ζ_{X,p}(T) = ∏_{x}(1 − T^{[k(x):𝔽_{p}]})

where x runs over closed points of X × 𝔽_{p}. If we set

Z_{X}(s) = ((s − 1)(s − 3))^{−1}(2π)^{−s dim H^{2}(X(ℂ),ℝ)}Γ(s − 1)^{dim H^{2}(X(ℂ),ℝ)^{c=1}} Γ(s − 2)^{dim H^{2}(X(ℂ),ℝ)^{c=−1}}ζ_{X}(s),

then we have that

Z_{X}(s) = AB^{s−2}Z_{X}(4 − s)

where B is a non-zero rational number and where A = ±1. (To see this note that

• H^{0}(X × ℚ^{ac}, ℚ_{l}) = ℚ_{l} and H^{6}(X × ℚ^{ac}, ℚ_{l}) = ℚ_{l}(−3); • H^{1}(X × ℚ^{ac}, ℚ_{l}) = H^{5}(X × ℚ^{ac}, ℚ_{l}) = (0);

• H^{2,0}(X(ℂ), ℂ) = H^{0,2}(X(ℂ), ℂ) = (0) and so by Lefschetz’s theorem there is finite dimensional ℚ-vector space W with a continuous action of G_{ℚ} such that

H^{2}(X × ℚ^{ac}, ℚ_{l}) ≅ W ⊗_{ℚ} ℚ_{l}(−1)

and

H^{4}(X × ℚ^{ac}, ℚ_{l}) ≅ W^{∨} ⊗_{ℚ} ℚ_{l}(−2)

for all rational primes l; and

• {H^{3}(X × ℚ^{ac}, ℚ_{l})} forms a regular, rank two weakly compatible system in the above sense.

Thus it suffices to combine the above theorem with the functional equation for Artin L-series.)

Corrections to [Tay4].

We are extremely grateful to Laurent Clozel for raising the following points. All the references below are to [Tay4].

• The third bulleted point on page 130 should read det \overline{ρ} = ϵ. (Without this change the choice of a_{λ} at the top of page 136 becomes impossible.)

• It would be clearer if the parenthetical comment “(as β_{v} − β_{v}^{c} is coprime to p)” read “(as β_{v}β_{v}^{c} = ψ(ϕ_{v})ψ^{c}(ϕ_{v}) = p and β_{v} − β_{v}^{c} is coprime to p)”.

• Before the “i.e.” in the middle of page 135 it would be clearer to add a parenthetical explanation “(note that End_{O_{M}} (A_{1}) is the centraliser of O_{M} in M_{[N:ℚ(β_{v})]}(O_{ℚ(β_{v})}), which is just O_{N})”.

• The superscript Gal (L/K) in the fourth displayed formula on page 135 should read Gal (F̃_{v}/F_{v}).

• After the fourth displayed formula on page 135 it would be clearer to add the parenthetical comment: “(N.B. Because [F̃_{v} : F_{v}]|#χ_{v}(I_{v})|#k^{×} and because N_{0} contains a primitive #k^{×} root of one, N contains a primitive [F̃_{v} : F_{v}] root of one.)”.

• The proof of lemma 1.4 is wrong. A correct proof can be given as follows. “Choose z ∈ (iℝ_{>0})^{Hom (M,ℝ)}. Let M act on ℂ^{Hom (M,ℝ)} by acting via τ on the τ-component. Set A = ℂ^{Hom (M,ℝ)}/(𝔡_{M}^{−1}1 + O_{M}z) (where 1 denotes the vector (1, ..., 1)). This complex torus is an abelian variety with an action of O_{M}, which is actually defined over ℝ (in such a way that complex conjugation on A(ℂ) corresponds to complex conjugation on ℂ^{Hom (M,ℝ)}). Moreover P(A, i) ≅ O_{M}^{+}, where α ∈ O_{M} corresponds to the alternating Riemannian form

E(x + yz, u + vz) = tr_{M/ℚ}α(yu − xv)

for x, y, u, v ∈ M ⊗_{ℚ} ℝ.”

• At the end of the second sentence of the paragraph before theorem 1.6 add “and det \overline{ρ} = ϵ” after “the case that \overline{ρ} has insoluble image”.

• With the above changes, specifically adding det \overline{ρ} = ϵ in two places, theorem 1.6 requires some further proof. The following will suffice: “We may assume that \overline{ρ} has insoluble image. Choose a totally real quadratic extension F′/F in which all primes above l split and a finite extension k′/k and a character ξ : G_{F′} → (k′)^{×} such that det \overline{ρ}|_{G_{F′}} = ϵξ^{2}. (This is possible as the obstruction to taking the square root of a character lies in the two part of the Brauer group.) Now work with \overline{ρ}′ = \overline{ρ} ⊗ ξ^{−1} : G_{F′} → GL_{2}(k′), and find p, N, M, λ, ℘, L, ψ, E′/F′ and A′ as above. Let E be the normal closure of E′/F. Then l and p split completely in E/F. Take A = A′ ×_{E′} E and argue as above.”

References

[AS] A.Ash and G.Stevens, Modular forms in characteristic l and special values of their L-functions, Duke Math. J. 53 (1986), 849–868.

[BL] J.-L.Brylinski and J.-P.Labesse, Cohomologie d’intersection et fonctions L de certaines variétés de Shimura, Ann. Sci. ENS 17 (1984), 361-412.

[Bl] D.Blasius, Hilbert modular forms and the Ramanujam conjecture,

  preprint available at www.math.ucla.edu/~blasius/papers.html .

[Bu] K.Buzzard, The levels of modular representations, PhD thesis, Cambridge University, 1995.

[Ca1] H.Carayol, Sur les représentations p-adiques associées aux formes modulaires de Hilbert, Ann. Sci. ENS (4) 19 (1986), 409-468.

[Ca2] H.Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, in “p-adic monodromy and the Birch and Swinnerton-Dyer conjecture”, Contemp. Math. 165, Amer. Math. Soc., 1994.

[Cl] L.Clozel, Motifs et formes automorphes: applications du principe de fonctorialité, in “Automorphic forms, Shimura varieties, and Lfunctions, Vol. I,(Ann Arbor, MI, 1988)” eds. L. Clozel and J. S. Milne, Perspect. Math. 10, Academic Press, 1990.

[CDT] B.Conrad, F.Diamond and R.Taylor, Modularity of certain potentially Barsotti-Tate Galois representations, JAMS 12 (1999), 521-567.

[DDT] H.Darmon, F.Diamond and R.Taylor, Fermat’s last theorem, in “Elliptic curves, modular forms and Fermat’s last theorem” Internat. Press, 1997.

[Dia] F.Diamond, The Taylor-Wiles construction and multiplicity one, Invent. Math. 128 (1997), 379-391.

[Ed] S.Edixhoven, The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), 563-594.

[Fa] G.Faltings, Crystalline cohomology and p-adic Galois representations, Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugural Conference, Johns-Hopkins Univ. Press (1989), 25-79.

[FL] J.-M.Fontaine and G.Laffaille, Construction de représentations padiques, Ann. Sci. Ecole Norm. Sup. (4) 15 (1982), 547–608.

[FM] J.-M.Fontaine and B.Mazur, Geometric Galois representations, in “Elliptic curves, modular forms and Fermat’s last theorem”, International Press 1995.

[Fu] K.Fujiwara, Deformation rings and Hecke algebras in the totally real case, preprint.

[HT] R.Taylor and M.Harris, The geometry and cohomology of some simple Shimura varieties, Annals of Math. Studies 151, PUP 2001.

[Lang] S.Lang, Complex multiplication, Springer 1983.

[Langl] R.Langlands, Base change for GL(2), PUP 1980.

[Rap] M.Rapoport, Compactifications de l’espace de modules de Hilbert-Blumental, Comp. Math. 36 (1978), 255-335.

[S1] J.-P.Serre, Abelian l-adic representations and elliptic curves, W.A.Benjamin 1968.

[S2] J.-P.Serre, Sur les représentations modulaires de degré 2 de Gal (\overline{ℚ}/ℚ), Duke Math. J. 54 (1987), 179-230.

[SW1] C.Skinner and A.Wiles, Base change and a problem of Serre, Duke Math. J. 107 (2001), 15–25.

[SW2] C.Skinner and A.Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. 10 (2001), 185– 215.

[Tat] J.Tate, Number theoretic background, in “Automorphic forms, representations and L-functions, part 2”, Proc. Sympos. Pure Math. XXXIII, AMS 1979.

[Tay1] R.Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265-280.

[Tay2] R.Taylor, On Galois representations associated to Hilbert modular forms II, in “Elliptic curves, modular forms and Fermat’s last theorem” eds. J.Coates and S.T.Yau, International Press 1995.

[Tay3] R.Taylor, On icosahedral Artin representations II, Amer. J. Math. 125 (2003), 549–566.

[Tay4] R.Taylor, On a conjecture of Fonatine and Mazur, Journal of the Institute of Mathematics of Jussieu 1 (2002), 125-143.

[Tu] J.Tunnell, Artin’s conjecture for representations of octahedral type, Bull. AMS 5 (1981), 173-175.

[TW] R.Taylor and A.Wiles, Ring theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995), 553-572.

[W1] A.Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math. 94 (1988), 529–573.

[W2] A.Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), 443-551.

Richard Taylor

Department of Mathematics Harvard University, Cambridge MA 02138 USA rtaylor@math.harvard.edu 780

Documenta Mathematica · Extra Volume Coates (2006)