ON SERRE’S CONJECTURE FOR MOD ℓ GALOIS REPRESENTATIONS OVER TOTALLY REAL FIELDS

KEVIN BUZZARD, FRED DIAMOND, AND FRAZER JARVIS

Abstract. In 1987 Serre conjectured that any mod ℓ two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the “weight” part of Serre’s conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy”. Using ideas of Emerton and Vignéras, we formulate a mod ℓ local-global principle for the group D^{∗}, where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture.

1. Introduction

Serre conjectured in [63] that if ℓ is prime and

ρ : G_{Q} → GL_{2}(\overline{F}_{ℓ})

is a continuous, odd, irreducible representation, then ρ is modular in the sense that it arises as the reduction of an ℓ-adic representation associated to a Hecke eigenform in the space S_{k}(Γ_{1}(N)) of cusp forms of some weight k and level N. Let us refer to this (incredibly strong) conjecture as “the weak conjecture”. Serre goes on to formulate a refined conjecture which predicts the minimal weight and level of such an eigenform subject to the constraints k ≥ 2 and ℓ ∤ N; let us call this “Serre’s refined conjecture”. Note that Serre explicitly excludes weight 1 modular forms, although a further reformulation was made by Edixhoven in [24] to include them, and we refer to Edixhoven’s reformulation as “Edixhoven’s refined conjecture”. Through the work of Ribet [58], Gross [40], Coleman-Voloch [15] and others, the equivalence between the weak conjecture and Serre’s refinement was known for ℓ > 2 (see [20]), and also when ℓ = 2 in many cases (see [10]). The equivalence of Serre’s refined conjecture and Edixhoven’s refined conjecture is also essentially known, although the question does not appear to have been completely resolved: for ℓ = 2, there still appears to be an issue regarding constructing a weight 1 form in every case that Edixhoven predicts that such a form exists.

The aim of this paper is to formulate a generalisation of Serre’s refined conjecture to the context of two-dimensional representations of G_{K} where K is a totally real field. The details of such a formulation (assuming ℓ unramified in K) were worked out by one of us (F.D.) stemming from correspondence and conversations among the authors in 2002, and the first version of this paper appeared in 2004. At that time the weak conjecture (for G_{Q}) appeared out of reach. Since then there has been startling progress, culminating in its recent proof by Khare and Wintenberger [51, 52], building on ideas developed by Dieulefait and themselves [23, 72, 50, 49] and relying crucially on potential modularity and modularity lifting methods and results of Taylor, Wiles and Kisin [71, 68, 67, 53, 54]. Their result also resolves the remaining case for ℓ = 2 of Serre’s refined conjecture.

Date: April 2010.

Since the first version of this paper appeared, there has also been significant progress towards proving the equivalence between the “weak” and “refined” conjectures we presented over G_{K}. Partial results already followed from work of one of the authors [44, 46], Fujiwara [32] and Rajaei [57], and further results were subsequently obtained by Schein [61] and Gee [33]. For the most part the techniques were generalisations of ones already used in the case K = Q and seemed severely limited with respect to establishing the “weight part” of the refined conjecture. However in [35, 36] Gee presented a new, much more promising approach; it remains to be seen how far the ideas there can be pushed towards a complete proof of the equivalence between weak and refined conjectures. Note also that these ideas have been extended in [38] to cover cases where ℓ is ramified in K.

Another important development related to Serre’s conjecture has been the recent progress on constructing ℓ-adic and mod ℓ Langlands correspondences, especially the work of Breuil, Colmez and Emerton. In particular, a correspondence between two-dimensional ℓ-adic representations of G_{Q_{ℓ}} and certain ℓ-adic representations of GL_{2}(Q_{ℓ}) was constructed by Colmez [16], and a conjectural compatibility with a global correspondence was formulated and proved in many cases by Emerton in [26] (see also [25]). There is also a mod ℓ version of this compatibility, which we refer to as “Emerton’s refined conjecture” (see also [27]). Most cases of Serre’s refined conjecture follow from Emerton’s; in particular the specification of the weight is essentially a description of the GL_{2}(Z_{ℓ})-socle of the representation of GL_{2}(Q_{ℓ}) arising as a local factor at ℓ associated to ρ. The current version of this paper includes a partial generalisation of Emerton’s refined conjecture to certain forms of GL_{2} over K. The sense in which it is “partial” is that at primes over ℓ we only describe the Jordan-Hölder constituents of the socle of a maximal compact for the local factor, and even that only for primes unramified over ℓ. The relevance of this socle and the difficulty of generalising the mod ℓ local correspondence to extensions of Q_{ℓ} is clearly illustrated by the recent work of Breuil and Paskunas [9].

We now explain our set-up and aims in a little more detail. Suppose that K is a totally real field. Let O denote its ring of integers and let S_{K} be the set of embeddings of K in R. Suppose that k⃗ ∈ Z^{S_{K}} with k_{τ} ≥ 1 for all τ ∈ S_{K} and furthermore assume that all of the k_{τ} are of the same parity. Let 𝔫 be a non-zero ideal of O. The space of Hilbert modular cusp forms of weight k⃗ and level 𝔫, denoted Sk⃗(U_{1}(𝔫)), is a finite-dimensional complex vector space equipped with an action of commuting Hecke operators T_{𝔪}, indexed by the non-zero ideals 𝔪 of O (to fix ideas, let us normalise our spaces and Hecke operators as in [66]). Fix once and for all embeddings \overline{Q} → C and \overline{Q} → \overline{Q}_{ℓ}, and let 0 ≠ f ∈ Sk⃗(U_{1}(𝔫)) be an eigenform for all the T_{𝔪}. A construction of Rogawski-Tunnell, Ohta and Carayol [60, 56, 13], completed by Taylor and Jarvis [66, 43], associates to f an ℓ-adic representation

ρ_{f} : G_{K} → GL_{2}(\overline{Q}_{ℓ})

such that if 𝔭 is a prime of O not dividing ℓ𝔫, then ρ_{f} is unramified at 𝔭 and, if Frob_{𝔭} denotes a geometric Frobenius, then trρ_{f}(Frob_{𝔭}) is the eigenvalue of T_{𝔭} on f (note that Taylor does not need to specify whether his Frobenius elements are arithmetic or geometric, so we shall assume that they are geometric). Fixing an identification of the residue field of \overline{Q}_{ℓ} with \overline{F}_{ℓ}, we obtain a representation

\overline{ρ}_{f} : G_{K} → GL_{2}(\overline{F}_{ℓ})

defined as the semisimplification of the reduction of ρ_{f}. It is natural to expect the following “folklore” generalisation of Serre’s weak conjecture to hold:

Conjecture 1.1. Suppose that ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd. Then ρ is isomorphic to \overline{ρ}_{f} for some Hilbert modular eigenform f.

Here “totally odd” means that det(ρ(c)) = −1 for all [K : Q] complex conjugations c. Note that one could instead have defined ρ_{f} to be the representation with the property that the trace of an arithmetic Frobenius was equal to the corresponding Hecke eigenvalue, which is the same as replacing ρ_{f} by its dual, but the “geometric” conjecture above is trivially equivalent to the “arithmetic” version (ρ is geometrically modular if and only if its dual is arithmetically modular). Although the Khare-Wintenberger approach to Serre’s original conjecture may shed light on Conjecture 1.1 for a few explicit totally real fields K, it seems not (in its present form) to be able to attack the case of a general K because it relies on an induction and the fact that for certain small primes ℓ there are no 2-dimensional irreducible odd mod ℓ representations of G_{Q} unramified outside ℓ; however for a totally real field the analogous fact is not in general true—there are plenty of elliptic curves over totally real fields with everywhere good reduction, and the ℓ-torsion in these curves will generally give rise to such representations. As an explicit example, one can check easily using a computer that if ϵ = \frac{5+\sqrt{29}}{2} then the elliptic curve

y^{2} + xy + ϵ^{2}y = x^{3}

has everywhere good reduction over Q(\sqrt{29}) and that the Galois representation on the 2-torsion is absolutely irreducible. ^{1}

The main aim of this paper is to refine Conjecture 1.1 along the lines of Serre’s refinement for the case K = Q, in the special case where the prime ℓ is unramified in K. Perhaps surprisingly, this is not as simple as it sounds. The main difficulty is in specifying the weight where, even in this unramified situation, we encounter several subtleties not present in Serre’s original work. Note first of all that there is no obvious notion of a minimal weight. Moreover the possible weights and level structures at primes over ℓ are intertwined, and, contrary to the case K = Q, one does not always expect a representation as in Conjecture 1.1 to arise from a classical Hilbert modular form of level prime to ℓ. Indeed, the mod ℓ representation attached to such a form has determinant equal to the product of a finite order character unramified at ℓ and some power of the mod ℓ cyclotomic character, and it is not hard to construct a mod ℓ Galois representation whose determinant is not of this form. To deal with these issues, we introduce the notion of a Serre weight, namely an irreducible \overline{F}_{ℓ}-representation σ of GL_{2}(O/ℓ), and define what it means for ρ to be modular of weight σ. Such a notion of weight is implicit in work of Ash and Stevens [3, 4], its relation to Serre’s conjecture underlies Khare’s paper [48], and its role in generalizing the conjecture to GL_{n} over Q is evident in [1, 2]. Our aim is to describe all possible Serre weights of forms giving rise to a representation ρ.

1Prof. J.-P. Serre informs us that this example of a curve with everywhere good reduction was found by Tate, and notes that it is discussed in §5.10 of Serre’s 1972 Inventiones article “Propriétés galoisiennes des points d’ordre fini des courbes elliptiques” (Oeuvres 94).

When working with classical modular forms, certain choices for normalisations and conventions have now become standard. In the Hilbert case there are various possibilities for these choices, and experience has shown the authors that things become ultimately less confusing if one works with holomorphic automorphic representations as in [13] rather than Hilbert modular forms, the advantage of this approach being that now the only choice one has to make is the normalisation of the local Langlands correspondence. We follow Carayol in our approach and use Hecke’s normalisation rather than Langlands’. We explain our conventions more carefully later on.

Our weight conjecture (Conjecture 3.14 below) then takes the form of a recipe for the set of weights σ for which ρ is modular. Our notion of modularity is formulated in terms of ρ arising in the Jacobian (or equivalently cohomology) of Shimura curves associated to quaternion algebras over K; the weight recipe is given in terms of the local behaviour of ρ at primes over ℓ. The precise recipe is motivated by the expected compatibility with the (labelled) Hodge-Tate weights in [0, ℓ] of crystalline lifts of twists of ρ|_{G_{K𝔭}} for primes 𝔭|ℓ. One can use Fontaine-Laffaille theory [29] to describe the reductions of crystalline representations with Hodge-Tate weights in the range [0, ℓ−1]; that the resulting recipe extends to the “boundary” is supported by numerical evidence collected by one of the authors, Dembélé and Roberts [19].

Loosely speaking, Conjecture 1.1 can be thought of in the context of a “mod ℓ Langlands philosophy”, with Conjecture 3.14 predicting a local-global compatibility at primes over ℓ. An interesting feature of the recipe for the weights is that if ℓ is inert in K and ρ|_{G_{K_{ℓ}}} is semisimple, then the set of weights that we associate to ρ are the Jordan–Hölder factors of the reduction of an irreducible \overline{Q}_{ℓ}-representation of GL_{2}(O/ℓ). This is proved in [21], where it is further shown that this association establishes a correspondence between 2-dimensional Galois representations of a local field in its residue characteristic and representations of GL_{2} of its residue field in characteristic zero. Herzig [41] has shown that this phenomenon does not persist in the context of GL_{n} for n > 2, but rather is a property particular to n ≤ 2 of a more general relation that he conjectures between the set of Serre weights and the reduction of a characteristic zero representation associated to ρ.

In [26, 27], Emerton made precise the sense in which Serre’s refined conjecture could be viewed as part of a mod ℓ Langlands philosophy (in the case K = Q). Using automorphic forms, he associates to ρ an \overline{F}_{ℓ}-representation π(ρ) of GL_{2}(A_{Q}), which is non-zero by the theorem of Khare and Wintenberger. Emerton conjectures, and shows under some technical hypotheses, that it factors as a restricted tensor product of local factors π_{p}, where π_{p} is a smooth admissible representation of GL_{2}(Q_{p}) determined by ρ|_{G_{Qp}}. Serre’s refined conjecture can then be recovered from properties of the π_{p}; moreover, results such as those in [22] and [48] describing the possible weights and levels of forms giving rise to ρ can also be extracted. We go on to formulate a conjecture in the spirit of Emerton’s in the context of certain quaternion algebras over K. In order to do so, we need to associate a local factor to ρ|_{G_{K𝔭}} when 𝔭 is a prime not dividing ℓ. This was already done by Emerton if the quaternion algebra is split at ℓ; we augment this with a treatment of the case where it is ramified using results of Vignéras [69].

This paper is structured as follows. In §2 we introduce the notion of a Serre weight and our notation and conventions regarding automorphic representations for GL_{2} over K; we explain what it means for ρ to be modular of a given Serre weight and relate this notion to the existence of automorphic representations π such that ρ ∼ \overline{ρ}_{π}. In §3 we formulate Conjecture 3.14 giving a recipe for the set of Serre weights for which ρ is modular, and we show that it follows from known results on Serre’s Conjecture in the case K = Q (Theorem 3.17). Finally, in §4 we state our partial generalisation of Emerton’s refined conjecture and derive some consequences. Acknowledgements: Much of the research was carried out while one of the authors (F.D.) was at Brandeis University, with support from NSF grants #9996345, 0300434. K.B. was partially supported by an EPSRC Advanced Research Fellowship. We are grateful to R. Taylor for raising our interest in the questions considered here. The last section of the paper was greatly influenced by discussions at a workshop at the American Institute of Mathematics on p-adic Representations in February 2006 attended by K.B. and F.D.; we heartily thank AIM and the workshop organisers and participants, especially M. Emerton. The influence of his ideas on that part of this work is apparent; we are also grateful to him for subsequent correspondence, in particular communicating details of [26, 27]. We also benefited from conversations and correspondence with C. Breuil, B. Conrad, L. Dembélé and T. Gee in the course of writing this paper. Finally the authors thank F. Herzig, M. Schein and R. Torrey for their helpful comments and corrections on an earlier version of the paper, and of course the referees, who between them pointed out numerous typos and made several remarks which helped improve the presentation. Any remaining errors are of course the fault of the authors.

2. Serre weights

Suppose K is a totally real field (we allow K = Q) and let O denote its ring of integers. Let ℓ be a rational prime, which we assume from the outset is unramified in K (although some of this section certainly could be made to work in more generality). Recall that we have fixed embeddings \overline{Q} → C and \overline{Q} → \overline{Q}_{ℓ}, and also an identification of \overline{F}_{ℓ} with the residue field of \overline{Q}_{ℓ}. Let S_{K} denote the embeddings K → R and let us fix once and for all a preferred embedding τ_{0} : K → R.

Consider the group

G = GL_{2}(O/ℓO) ≅ ∏_{𝔭|ℓ} GL_{2}(O/𝔭).

A Serre weight is an isomorphism class of irreducible \overline{F}_{ℓ}-representations of G. These can be described explicitly as follows. For each prime 𝔭 of K dividing ℓ, set k_{𝔭} = O/𝔭, f_{𝔭} = [k_{𝔭} : F_{ℓ}] and let S_{𝔭} be the set of embeddings τ : k_{𝔭} → \overline{F}_{ℓ}. Then every irreducible \overline{F}_{ℓ}-representation of GL_{2}(k_{𝔭}) is equivalent to one of the form

V_{a⃗,b⃗} = ⨂_{τ∈S_{𝔭}}(det^{a_{τ}} ⊗_{k_{𝔭}} Symm^{b_{τ} −1}k_{𝔭}^{2}) ⊗_{τ} \overline{F}_{ℓ},

where a_{τ}, b_{τ} ∈ Z and 1 ≤ b_{τ} ≤ ℓ for each τ ∈ S_{𝔭}. Moreover we can assume that 0 ≤ a_{τ} ≤ ℓ − 1 for each τ ∈ S_{𝔭} and that a_{τ} < ℓ − 1 for some τ, in which case the resulting (ℓ^{f_{𝔭}} − 1)ℓ^{f_{𝔭}} representations V_{a⃗,b⃗} are also inequivalent. The irreducible representations of G are thus of the form V = ⊗_{{𝔭|ℓ}}V_{𝔭}, where the tensor product is over \overline{F}_{ℓ} and each V_{𝔭} is of the form V_{a⃗,b⃗} for (a⃗, b⃗) as above.

If n is an integer then we let F_{ℓ}(n) denote the 1-dimensional F_{ℓ}-vector space with left G-action defined by letting g ∈ G act via N(det(g))^{n}, where N : (O/ℓO)^{×} → (Z/ℓZ)^{×} is the norm. If F is a field of characteristic ℓ and V is an F-representation of G space then we define V (n) to be the F-representation V (n) := V ⊗_{F_{ℓ}} F_{ℓ}(n). Note that V_{a⃗,b⃗}(n) = V_{a⃗+n.1,b⃗}, where 1 = (1, 1, …, 1).

Suppose that D is a quaternion algebra over K split at τ_{0} and at no other infinite places. Fix an isomorphism D ⊗_{K,τ_{0}} R = M_{2}(R); this induces an isomorphism of (D ⊗_{τ_{0}} R)^{×} with GL_{2}(R), which acts on ℌ^{±} := C\R in the usual way. Consider K as a subfield of R (and hence of C) via the embedding τ_{0}. If A_{K}^{f} denotes the finite adeles of K and U is an open compact subgroup of (D ⊗_{K} A_{K}^{f})^{×} then there is a Shimura curve Y_{U} over K, a smooth algebraic curve whose complex points (via τ_{0} : K → C) are naturally identified with

D^{×}\ ((D ⊗_{K} A_{K}^{f})^{×} × ℌ^{±}) /U

and such that Y_{U} is a canonical model for this space, in the sense of Deligne. These canonical models have the useful property that if U′ is a normal compact open subgroup of U, then the natural right action of U/U′ on Y_{U′}(C) is induced by an action of U/U′ on Y_{U′} (that is, the action is defined over K).

Unfortunately there is more than one convention for these canonical models, and the choice that we make genuinely affects our normalisations. To fix ideas, we shall follow the conventions of Carayol in [12] and in particular our “Hodge structure” h will be that of section 0.1 of [12]. This corresponds to the choice ϵ = −1 in the notation of [17]. See Section 3.3 of [17] for a discussion of the differences between this choice and the other natural choice—the key one being (Lemma 3.12 of [17]) that the choice does affect the Galois action on the adelic component group, by a sign. That this ambiguity exists is not surprising: for example in the elliptic curve case the modular curve Y (ℓ) parametrising elliptic curves equipped with generators of their ℓ-torsion exists (for ℓ > 2) as a moduli space over Q, and the Weil pairing gives a natural morphism Y (ℓ) → Spec(Q(ζ_{ℓ})), but the two ways of normalising the Weil pairing give different morphisms.

If K = Q and D is split (we refer to this case as “the split case”), we let X_{U} denote the standard compactification of the modular curve Y_{U}; otherwise (that is, if K ≠ Q or if K = Q but D is non-split) we simply set X_{U} = Y_{U}. Then X_{U} is a smooth projective algebraic curve over K. Note that X_{U}, considered as a scheme over K, will be connected (see section 1.3 of [12]) but not in general geometrically connected. Note also that, with notation as above, the natural action of U/U′ on Y_{U′} extends to an action on X_{U′}. Opting to include the split case does sometimes increase the length of a proof (we have to verify that “all errors are Eisenstein”) but is arguably morally better than presenting proofs only in the non-split case and then merely asserting that they may be modified to deal with the split case too.

If U is a compact open subgroup of (D ⊗_{K} A_{K}^{f})^{×} as above, then let Pic^{0}(X_{U}) denote the identity component of the relative Picard scheme of X_{U} → Spec(K). This definition is chosen specifically to deal with the fact that X_{U} may not be geometrically connected. In more concrete terms, if K_{U} denotes the ring Γ(X_{U}, O_{X_{U}}) of globally-defined functions on X_{U}, then K_{U} is a number field and a finite abelian extension of K, the curve X_{U} is geometrically connected when regarded as a scheme over Spec(K_{U}), and Pic^{0}(X_{U}) is canonically isomorphic to the restriction of scalars (from K_{U} to K) of the Jacobian of X_{U}/K_{U}. In particular, Pic^{0}(X_{U}) is an abelian variety over K.

We henceforth assume that D is split at all primes of K above ℓ, and we fix an isomorphism D ⊗_{Q} Q_{ℓ} ≅ M_{2}(K ⊗ Q_{ℓ}). We can now regard GL_{2}(O ⊗ Z_{ℓ}) as a subgroup of (D ⊗_{K} A_{K}^{f})^{×}. If U is a compact open subgroup of (D ⊗_{K} A_{K}^{f})^{×} as above, and if GL_{2}(O ⊗ Z_{ℓ}) is contained in U, then we say that U has level prime to ℓ. In this case, the natural map U → GL_{2}(O/ℓO) = G is a surjection. Let U′ denote its kernel. Then U/U′ = G acts naturally on the right on Y_{U′} and on X_{U′}, and hence naturally on the left on Pic^{0}(X_{U′}) and Pic^{0}(X_{U′})[ℓ](\overline{K}). Let us say that U is sufficiently small if it has level prime to ℓ and the map Y_{U′} → Y_{U} is étale of degree equal to the order of G. Note that any U of level prime to ℓ contains a compact open subgroup that is sufficiently small—this follows easily from 1.4.1.1–1.4.1.3 of [12] or Lemma 12.1 of [44]. The induced map X_{U′} → X_{U} will then be finite of degree equal to the order of G (but it may not be étale in the split case—there will usually be ramification at the cusps).

Definition 2.1. Suppose that ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is a continuous, irreducible representation and V is a finite-dimensional \overline{F}_{ℓ}-vector space with a left action of G. We say that ρ is modular of weight V if there is a quaternion algebra D over K split at the primes above ℓ, at τ_{0} and no other infinite places of K, and a sufficiently small open compact subgroup U of (D ⊗_{K} A_{K}^{f})^{×} of level prime to ℓ, such that ρ is an \overline{F}_{ℓ}G_{K}-subquotient of (Pic^{0}(X_{U′})[ℓ](\overline{K}) ⊗ V )^{G}, where U′ = ker(U → G), G acts diagonally on the tensor product, and G_{K} acts trivially on V.

Note that we allow V to be a reducible G-representation (out of convenience at this point in the exposition: we will see in Lemma 2.4(b) that V can ultimately be assumed irreducible with no loss of generality). Note also that we allow subquotients with respect to the Galois action although using Hecke operators and the Eichler-Shimura relation on X_{U′}, one can show that if we replace “G_{K}- subquotient” by “G_{K}-submodule” then the resulting definition is equivalent. On the other hand, we really want to demand that ρ is a G_{K}-subquotient of the Ginvariants of Pic^{0}(X_{U′})[ℓ](\overline{K})⊗V rather than an \overline{F}_{ℓ}[G_{K} ×G]-subquotient on which G acts trivially. Our conjecture would not be correct were we to use G-subquotients; a general Galois representation would then be modular of more weights and we would not recover important subtleties of the refined conjecture.

Say V is any finite-dimensional \overline{F}_{ℓ}-vector space equipped with a left G-action. The right action of G on Y_{U′} enables us to identify G with a quotient of π_{1}(Y_{U}, x) for x any geometric point of Y_{U}. Now a standard construction (see for example A I.7 of [30]) associates to V a locally constant étale sheaf F_{V} on Y_{U}, with (amongst other things) the property that the pullback of F_{V} to Y_{U′} is just the constant sheaf associated to the vector space V. We abuse notation slightly by also using F_{V} to refer to the pullback of F_{V} to Y_{U,\overline{K}}, the base change of Y_{U} to \overline{K}.

Later on we will also need an ℓ-adic variant of this construction, for which a good reference is sections 2.1.3 and 2.1.4 of [13]. Let U ⊂ (D ⊗_{K} A_{K}^{f})^{×} be compact and open, and let Y_{U} denote the associated Shimura curve over K. If k⃗ ∈ Z_{≥2}^{S_{K}} and w ∈ Z with w ≡ k_{τ} mod 2 for all τ, then we would like as in loc. cit. to define an ℓ-adic sheaf corresponding to the representation ⊗_{𝔭|ℓ} ⊗_{τ∈S_{𝔭}} det^{(w−k_{τ} +2)/2} Symm^{k_{τ} −2}(ξ_{τ}) of D^{×}, where ξ_{τ} is the tautological 2-dimensional representation of (D ⊗_{F,τ} C)^{×} induced by a fixed isomorphism D ⊗_{K,τ} C = M_{2}(C). We do this as follows. If E ⊂ \overline{Q} is a number field, Galois over Q, splitting D and containing all embeddings of K into \overline{Q}, then the representation above can be realised on an E-vector space. If λ|ℓ is a prime of E, then (possibly after replacing U by a finite index subgroup to remove any problems with elliptic points) Carayol defines an O_{E_{λ}}-sheaf F_{λ,U}^{0} on Y_{U} associated to the representation above, and an E_{λ}-sheaf F_{λ,U} = F_{λ,U}^{0} ⊗Q_{ℓ}. Strictly speaking the sheaf F_{λ,U}^{0} depends on a choice of lattice, which we always take to be the one arising from tensor products of symmetric powers of O_{E_{λ}}^{2}. Furthermore, if we choose λ so that it is the prime of E above ℓ induced by our embedding E → \overline{Q} → \overline{Q}_{ℓ} then there is an induced map O_{E_{λ}}/λ → \overline{F}_{ℓ} and the induced \overline{F}_{ℓ}- sheaf F_{λ,U}^{0}/λ ⊗ \overline{F}_{ℓ} is isomorphic to the sheaf F_{W} associated to the representation W = ⊗_{𝔭|ℓ} ⊗_{τ∈S_{𝔭}} det^{(w−k_{τ} +2)/2} Symm^{k_{τ} −2} k_{𝔭}^{2} ⊗_{τ} \overline{F}_{ℓ}.

We begin by noting that G_{K} acts via an abelian quotient on many of the cohomology groups that show up in forthcoming arguments.

Lemma 2.2. (a) Let U be any compact open subgroup of (D⊗_{K} A_{K}^{f})^{×} and let F be a locally constant torsion sheaf on Y_{U} corresponding to a continuous representation of U/U′ for some normal open compact U′ ⊆ U such that Y_{U′} → Y_{U} is étale with covering group U/U′. Then for i ∈ {0, 2} the action of G_{K} on the cohomology groups H^{i}(Y_{U,\overline{K}}, F) and H_{c}^{i}(Y_{U,\overline{K}}, F) factors through an abelian quotient.

(b) If F_{λ,U} is the sheaf associated to the data (k⃗, w) as above, and if H_{p}^{1}(Y_{U,\overline{K}}, F_{λ,U}) denotes the image of H_{c}^{1}(Y_{U,\overline{K}}, F_{λ,U}) in H^{1}(Y_{U,\overline{K}}, F_{λ,U}), then the action of G_{K} on H^{1}(Y_{U,\overline{K}}, F_{λ,U})/H_{p}^{1}(Y_{U,\overline{K}}, F_{λ,U}) is via an abelian quotient.

(c) The action of G_{K} on the cokernel of the natural inclusion H^{1}(X_{U,\overline{K}}, F_{ℓ}) → H^{1}(Y_{U,\overline{K}}, F_{ℓ}) is via an abelian quotient.

Proof. (a) The pullback of F to Y_{U′} is constant and H^{0}(Y_{U,\overline{K}}, F) can be identified with a subspace of H^{0}(Y_{U′,\overline{K}}, F). The Galois action on this latter space is abelian, as the geometric components of Y_{U′} are defined over an abelian extension of K (see for example §1.2 of [12]). This proves the result for H^{0}, and the result for H_{c}^{0} follows as H_{c}^{0} is a subgroup of H^{0}. For i = 2 the result follows from the i = 0 case and Poincaré duality, which pairs H^{0} with H_{c}^{2} and H^{2} with H_{c}^{0}.

(b) The quotient is trivial in the non-split case, so we are instantly reduced to the case K = Q, D = M_{2}(Q) and E_{λ} = Q_{ℓ}. In this case the result is surely wellknown, but we sketch the proof for lack of a reference. The sheaf associated to the data (k, w) is Q_{ℓ}-dual to the sheaf associated to (k, −w), so by Poincaré duality, this is equivalent to showing the Galois action factors through an abelian quotient on the kernel of the map (with w replaced by 2−w). Recall that H_{c}^{1}(Y_{U,\overline{Q}}, F_{λ,U}) = H^{1}(X_{U,\overline{Q}}, j_{!}F_{λ,U}) where j : Y_{U} → X_{U} is the natural inclusion, and that our map factors as

H^{1}(X_{U,\overline{Q}}, j_{!}F_{λ,U}) → H^{1}(X_{U,\overline{Q}}, j_{∗}F_{λ,U}) → H^{1}(Y_{U,\overline{Q}}, F_{λ,U}),

the first map being surjective, the second injective. Therefore it suffices to prove that the action of G_{Q} on

H^{0}(X_{U,\overline{Q}}, j_{∗}F_{λ,U}/j_{!}F_{λ,U}) = H^{0}(Z_{U,\overline{Q}}, i^{∗}j_{∗}F_{λ,U})

factors through an abelian quotient, where i : Z_{U} → X_{U} is the reduced closed subscheme defined by the cusps. Shrinking U if necessary, we can assume there is a universal generalised elliptic curve over X_{U} (in the sense of [18]), and we let s : E_{U} → X_{U} denote its restriction to the open subscheme whose fibres over closed points are its identity components. Then we find that j_{∗}F_{λ,U} is isomorphic to Symm^{k−2}(R^{1}s_{∗}Q_{ℓ})((w + k − 2)/2), and since E_{U} ×_{X_{U}} Z_{U} is isomorphic to G_{m,Z_{U}}, we conclude that i^{∗}j_{∗}F_{λ,U} is isomorphic to Q_{ℓ}((w − k + 2)/2). The assertion now follows from the fact that the cusps are defined over an abelian extension of Q.

(c) is similar to (b), but simpler. □

Let D, U, U′, G and V be as in Definition 2.1. If ψ is a continuous character of G_{K} with values in \overline{F}_{ℓ}^{×} or \overline{Q}_{ℓ}^{×}, then we let ψ_{A} denote the character of A_{K}^{×} corresponding to ψ by class field theory (with uniformizers corresponding to geometric Frobenius elements).

Lemma 2.3. Suppose that ψ : G_{K} → \overline{F}_{ℓ}^{×} is a continuous character such that ψ_{A} is trivial on det(U′), and let χ denote the restriction of ψ_{A} to (O ⊗ Z_{ℓ})^{×}. Then H^{1}(Y_{U,\overline{K}}, F_{V ⊗χ∘det}) is isomorphic to H^{1}(Y_{U,\overline{K}}, F_{V})(ψ) as G_{K}-modules.

Proof. The restriction of ψ to G_{K_{U}} corresponds to χ via the isomorphism

G_{K_{U}} /G_{K_{U′}} ≅ det(U)/ det(U′) ≅ (O/ℓ)^{×}.

Recall from section 1.1.2 of [13] that we have a commutative diagram

Y_{U′} → Spec K_{U′}
↓  ↓
Y_{U} → Spec K_{U}

such that the action of G = U/U′ on Y_{U′} is compatible via det with that of det(U)/ det(U′) on Spec K_{U′}. So if we let F_{χ} denote the sheaf on Spec K_{U} corresponding to the character χ, we see that F_{χ∘det} is isomorphic to the pull-back of F_{χ} by the map Y_{U} → Spec K_{U}. As this map induces a bijection on geometric components, it follows that H^{0}(Y_{U,\overline{K}}, F_{χ∘det}) is isomorphic to H^{0}(Spec K_{U} ×_{K} \overline{K}, F_{χ}) as G_{K}-modules, which in turn is isomorphic to Ind_{G_{K_{U}}}^{G_{K}} \overline{F}_{ℓ}(χ). Therefore Hom_{\overline{F}_{ℓ}[G_{K}]}(\overline{F}_{ℓ}(ψ), H^{0}(Y_{U,\overline{K}}, F_{χ∘det})) is one-dimensional. Let α be the image in H^{0}(Y_{U,\overline{K}}, F_{χ∘det}) of a non-trivial element. Note that the restriction of α to each component of Y_{U,\overline{K}} is non-trivial, so cupping with α defines an isomorphism

H^{1}(Y_{U,\overline{K}}, F_{V}) ⊗_{\overline{F}_{ℓ}} \overline{F}_{ℓ}α → H^{1}(Y_{U,\overline{K}}, F_{V ⊗χ∘det}).
□

Lemma 2.4. Let ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) be continuous, irreducible and totally odd.

(a) ρ is a G_{K}-subquotient of (Pic^{0}(X_{U′})[ℓ](\overline{K}) ⊗ V )^{G} if and only if ρ is a G_{K}- subquotient of H^{1}(Y_{U,\overline{K}}, F_{V (1)}).

(b) ρ is modular of weight W (an arbitrary finite-dimensional \overline{F}_{ℓ}[G]-module) if and only if ρ is modular of weight V for some Jordan-Hölder factor of W.

Proof. (a) First recall that Pic^{0}(X_{U′})[ℓ](\overline{K}) = H^{1}(X_{U′,\overline{K}}, F_{ℓ}) ⊗ μ_{ℓ} as G_{K} × Gmodules. By Lemma 2.2(c), the action of G_{K} on the cokernel of the natural injection

H^{1}(X_{U′,\overline{K}}, F_{ℓ}) → H^{1}(Y_{U′,\overline{K}}, F_{ℓ})

factors through an abelian quotient. It follows that ρ is modular of weight V if and only if ρ is an \overline{F}_{ℓ}[G_{K}]-subquotient of (H^{1}(Y_{U′,\overline{K}}, F_{ℓ}) ⊗ V)^{G} ⊗ μ_{ℓ}.

The Hochschild-Serre spectral sequence now gives us an exact sequence of G_{K}- modules

H^{1}(G, H^{0}(Y_{U′,\overline{K}}, F_{ℓ}) ⊗ V) → H^{1}(Y_{U,\overline{K}}, F_{V}) →
→ (H^{1}(Y_{U′,\overline{K}}, F_{ℓ}) ⊗ V)^{G} → H^{2}(G, H^{0}(Y_{U′,\overline{K}}, F_{ℓ}) ⊗ V).

By Lemma 2.2(a), the action of G_{K} on the first and last terms factors through an abelian quotient. It follows that ρ is modular of weight V if and only if ρ is an \overline{F}_{ℓ}[G_{K}]-subquotient of H^{1}(Y_{U,\overline{K}}, F_{V}) ⊗ μ_{ℓ}. Finally note that by Lemma 2.3, we have

H^{1}(Y_{U,\overline{K}}, F_{V (1)}) ≅ H^{1}(Y_{U,\overline{K}}, F_{V}) ⊗ μ_{ℓ},

and part (a) of the lemma follows.

(b) If 0 → W_{1} → W_{2} → W_{3} → 0 is a short exact sequence of finite-dimensional \overline{F}_{ℓ}[G]-modules then 0 → F_{W_{1}} → F_{W_{2}} → F_{W_{3}} → 0 is a short exact sequence of étale sheaves on Y_{U}, and (b) now follows from (a) and Lemma 2.2(a).

□

Our chosen embeddings \overline{Q} → C and \overline{Q} → \overline{Q}_{ℓ} and identification of the residue field of \overline{Z}_{ℓ} with \overline{F}_{ℓ} allow us to identify S_{K} with ⋃_{𝔭|ℓ} S_{𝔭} (because we are assuming that ℓ is unramified in K). We now recall how this notion of modularity is related to the existence of automorphic representations for GL_{2} /K giving rise to ρ. We start by establishing some conventions.

When associating Galois representations to automorphic representations we shall follow Carayol’s conventions in [13]. In particular our normalisations of local and global class field theory will send geometric Frobenius elements to uniformisers and our local-global compatibility will be Hecke’s rather than Langlands’ (the one that preserves fields of definition rather than the one that behaves well under all functorialities; the difference is a dual and a twist). We summarise Carayol’s theorem (in fact we only need a weaker form which is essentially due to Langlands and Ohta), and its strengthening by Taylor and Jarvis.

For k ≥ 2 and w integers of the same parity let D_{k,w} denote the discrete series representation D_{k,w} of GL_{2}(R) with central character t ↦ t^{−w} defined in section 0.2 of [13]. For k = 1 and w an odd integer we define D_{1,w} to be the irreducible principal series Ind(μ, ν) where the induction is unitary induction and μ and ν are the (quasi-)characters of R^{×} defined by μ(t) = |t|^{−w/2} sgn(t) and ν(t) = |t|^{−w/2}. Now for k⃗ ∈ Z^{S_{K}} with each k_{τ} ≥ 1 and of the same parity, and w ∈ Z of the same parity as the k_{τ}, let us say that a cuspidal automorphic representation π of GL_{2}(A_{K}) is holomorphic of weight (k⃗, w) (or “has weight (k⃗, w)” for short), if π_{τ} ≅ D_{k_{τ} ,w} for each τ ∈ S_{K} (of course many cuspidal automorphic representations will not have any weight—we are picking out the ones that correspond to holomorphic Hilbert modular forms). The theorem of Eichler, Shimura, Deligne, Deligne–Serre, Langlands, Ohta, Carayol, Taylor, Blasius–Rogawski, Rogawski–Tunnell and Jarvis associates a Galois representation ρ_{π} : G_{K} → GL_{2}(\overline{Q}_{ℓ}) to a cuspidal automorphic representation π for GL_{2} /K of weight (k⃗, w), and the correspondence π ↦ ρ_{π} satisfies Hecke’s local-global compatibility at the finite places of K of characteristic not dividing ℓ for which π is unramified. We remark that local-global compatibility at the ramified places away from ℓ does not quite appear to be known in this generality, although Carayol and Taylor establish it if k_{τ} ≥ 2 for all τ. See Theorem 7.2 of [43] for the current state of play if k_{τ} = 1 for some τ.

From this compatibility one deduces easily (see section 3 of [13] for example) that if π is holomorphic of weight (k⃗, w) then the determinant of ρ_{π} is the product of a finite order character and ω^{w−1}, where ω denotes the cyclotomic character. Note that twisting by an appropriate power of the norm character gives bijections between the automorphic representations of weight (k⃗, w) and (k⃗, w + 2n) for any integer n; this corresponds to twisting by an appropriate power of the cyclotomic character on the Galois side.

Let \overline{ρ}_{π} : G_{K} → GL_{2}(\overline{F}_{ℓ}) denote the semisimplification of the reduction of ρ_{π}. Our goal now in this section is to relate two notions of being modular: the first is the notion of being modular of some weight (in the sense introduced above), and the second is the notion of being modular in the sense of being isomorphic to \overline{ρ}_{π} for some holomorphic π.

Proposition 2.5. Let (k⃗, w) ∈ Z_{≥2}^{S_{K}} ×Z be integers all of the same parity. Suppose ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd. Then ρ ∼ \overline{ρ}_{π} for some cuspidal automorphic representation π for GL_{2} /K of weight (k⃗, w) and level prime to ℓ if and only if ρ is modular of weight V for some Jordan-Hölder constituent V of

V_{k⃗,w} := ⨂_{𝔭|ℓ} ⨂_{τ∈S_{𝔭}} det^{(w−k_{τ})/2}Symm^{k_{τ} −2}k_{𝔭}^{2} ⊗_{τ} \overline{F}_{ℓ}.

Remark 2.6. Note that the representation V_{k⃗,w} above differs from the representation ψ in section 2.1.1 of [13] by a twist due to the fact that we are using Jacobians rather than étale cohomology.

Proof. Say V is a Jordan-Hölder constituent of V_{k⃗,w}, and that ρ is modular of weight V. Then, by definition, there is a quaternion algebra D/K satisfying the usual conditions, and a level structure U prime to ℓ such that ρ is a subquotient of (Pic^{0}(X_{U′})[ℓ] ⊗ V)^{G} (with notation as above). By Lemma 2.4(a) and (b) and the remarks before Lemma 2.2, ρ is a G_{K}-subquotient of H^{1}(Y_{U,\overline{K}}, F_{λ,U}^{0}/λ) ⊗ \overline{F}_{ℓ}, where F_{λ,U}^{0} is the sheaf that Carayol associates to (k⃗, w). Recall that by results of Jacquet-Langlands and Carayol [13], H_{p}^{1}(Y_{U,\overline{K}}, F_{λ,U}) is a direct sum of irreducible 2-dimensional ℓ-adic representations ρ_{π} for π as in the statement of the proposition. So by Lemma 2.2 and a standard cohomological argument, we see that ρ ≅ \overline{ρ}_{π} for some cuspidal automorphic representation π on GL_{2} /K.

The reverse implication is not quite so straightforward because given π one needs to see \overline{ρ}_{π} in the cohomology of a Shimura curve (the problem being that K might have even degree and π might be principal series at all finite places). However, arguments of Wiles and Taylor show that this problem is surmountable via “levelraising.” Indeed, Theorem 1 of [66] and the remarks following it show that \overline{ρ}_{π} ≅ \overline{ρ}_{π′} for π′ an automorphic representation of GL_{2} /K that is special at a finite place, and the Galois representation associated to π′ does indeed show up in the cohomology of a Shimura curve by the above-mentioned results of Jacquet-Langlands and Carayol.

□ Corollary 2.7. If ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, totally odd, irreducible, and ρ ≅ \overline{ρ}_{π} for some automorphic representation π of GL_{2} /K of level prime to ℓ and weight (k⃗, w) ∈ Z_{≥1}^{S_{K}} × Z, then ρ is modular of weight V for some weight V.

Proof. If k_{τ} ≥ 2 for all τ then this follows immediately from the previous proposition. If k_{τ} = 1 for one or more τ then it suffices to construct an automorphic representation π′ of level prime to ℓ with \overline{ρ}_{π} ≅ \overline{ρ}_{π′}, and with π′ of weight (k⃗′, w) with k′_{τ} ≥ 2 for all τ. This is done in [43] (via multiplication by an appropriate modular form congruent to 1 mod ℓ: in particular the result follows from the Deligne–Serre lemma and Lemma 5.2 of loc. cit.). □

We can furthermore predict the local behaviour at primes over ℓ of the automorphic representations of weight (k⃗, w) giving rise to ρ. Before we start on this, here is a simple lemma that will be of use to us later.

Lemma 2.8. If k is a finite field of characteristic ℓ and if V is an irreducible \overline{F}_{ℓ}- representation of GL_{2}(k) then there is an irreducible \overline{Q}_{ℓ}-representation of GL_{2}(k) whose reduction has V as a Jordan-Hölder factor. Furthermore there is a 1dimensional \overline{F}_{ℓ}-representation χ of GL_{2}(k) and an irreducible \overline{Q}_{ℓ}-representation of GL_{2}(k) with a fixed vector for the subgroup ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} ) whose reduction has χ ⊗ V as a Jordan-Hölder factor.

Proof. For 1-dimensional V the result is clear (use the Teichmüller lift) and for V of dimension equal to the size of k, the Steinberg representation does the job. For other V the lemma follows immediately from Proposition 1.1 of [21] (with J = S in the notation of that paper). □

We now introduce the following rather naive version of a type. If L is a finite extension of Q_{ℓ}, with integers O_{L}, if π is a smooth irreducible complex representation of GL_{2}(L) and if σ is a smooth irreducible representation of GL_{2}(O_{L}) (so σ is finite-dimensional and its kernel contains an open subgroup of GL_{2}(O_{L})), then we say that π is of type σ if the restriction of π to a representation of GL_{2}(O_{L}) contains a subspace isomorphic to σ (note that this is a much weaker and simpler version of the usual notion of a type).

We now need a mild refinement of a level-raising result of Richard Taylor; unfortunately this refinement does not appear to be in the literature, so we sketch a proof here. The reader who wants to follow the details is advised to have a copy of Taylor’s paper [66] handy.

Lemma 2.9. Suppose that [K : Q] is even and π is a cuspidal automorphic representation of GL_{2} /K which has weight (k⃗, w). Suppose moreover that k_{τ} ≥ 2 for all infinite places τ. For all primes 𝔭 | ℓ choose a smooth irreducible representation σ_{𝔭} of GL_{2}(O_{K_{𝔭}}) such that π_{𝔭} is of type σ_{𝔭}.

Then there is a prime 𝔮 of K not dividing ℓ and a cuspidal automorphic representation π′ of GL_{2} /K, also of weight (k⃗, w), and such that

• \overline{ρ}_{π} ∼ \overline{ρ}_{π′};

• π′_{𝔮} is an unramified special representation;

• π′_{𝔭} is of type σ_{𝔭} each 𝔭 | ℓ.

Proof. (sketch). The main idea of the argument is contained in the proof of Theorem 1 of [66], but it would be a little disingenuous to cite this result without further comment because, as written, the proof does not keep track of types. It is not difficult to change the argument so that it does, however. We indicate what needs to be changed in order to prove the result we want. Because of our assumptions about the weight of π and the degree of K, the lemma can be deduced from a purely combinatorial statement about automorphic forms for the group of units of the totally definite quaternion algebra D over K of discriminant 1. We need to check that the system of eigenvalues associated to π_{D} (the transfer of π to D^{×}) is congruent to the system of eigenvalues associated to an automorphic representation π′_{D} which is Steinberg at some place 𝔮 (this much is done in [66]) and furthermore such that π′_{D,𝔭} has type σ_{𝔭} at all places above ℓ. We do this by mimicking Taylor’s argument with the following changes. Instead of working at level Γ_{1}(n) as in [66], we work with a more general compact open level structure U, assumed for simplicity to be a product of local factors U_{𝔭} ⊂ GL_{2}(K_{𝔭}), for 𝔭 running over the finite places of K. For 𝔭 | ℓ we further assume that U_{𝔭} is a normal subgroup of GL_{2}(O_{𝔭}), with O_{𝔭} the integers in K_{𝔭} (all this can be achieved by shrinking U if necessary). We define G_{ℓ} to be the finite group ∏_{𝔭|ℓ}(GL_{2}(O_{𝔭})/U_{𝔭}); the group G_{ℓ} then acts on the space of automorphic forms of level U. Let σ = ⊗_{𝔭|ℓ}σ_{𝔭}, so σ is a finite-dimensional smooth complex irreducible representation of G_{ℓ}, and fix a number field N which contains F, splits D, contains the trace of σ(g) for all g ∈ G_{ℓ} and furthermore contains the values of the (algebraic) central character ψ of π_{D}. As in Taylor’s paper we define S_{(k⃗,w)}^{D}(U) = S_{(k⃗,w)}^{D}(U; C), the finite-dimensional complex

vector space of weight (k⃗, w) automorphic forms of level U for D^{×}. Note that our k⃗ is Taylor’s k and our w is his μ, and that strictly speaking Taylor only considers the case w = max{k_{τ} − 2}, but his arguments never assume this. We define S_{(k⃗,w)}^{D}(U)_{ψ} to be the subspace of S_{(k⃗,w)}^{D}(U) where the centre Z of D^{×}(A_{K}) acts via

the character ψ. Note that for a fixed U and (k⃗, w) there are only finitely many characters ψ for which S_{(k⃗,w)}^{D}(U)_{ψ} is non-zero by finite-dimensionality of S_{(k⃗,w)}^{D}(U). Furthermore the infinity type of such a ψ is determined by w; the character ψ is an analogue of the Dirichlet character associated to a classical modular form. For R a subring of C containing the integers of N we define S_{(k⃗,w)}^{D}(U; R) as in Taylor’s

paper, and set S_{(k⃗,w)}^{D}(U; R)_{ψ} = S_{(k⃗,w)}^{D}(U; R) ∩ S_{(k⃗,w)}^{D}(U)_{ψ}. These spaces all have an

action of G_{ℓ}; we define S_{(k⃗,w)}^{D}(U)_{σ,ψ} to be the σ-eigenspace of S_{(k⃗,w)}^{D}(U)_{ψ} (that is,

the C[G_{ℓ}]-direct summand of S_{(k⃗,w)}^{D}(U)_{ψ} cut out by the idempotent in C[G_{ℓ}] cor-

responding to σ), and we define S_{(k⃗,w)}^{D}(U; R)_{σ,ψ} to be S_{(k⃗,w)}^{D}(U; R) ∩ S_{(k⃗,w)}^{D}(U)_{σ,ψ}.

Note that the natural projection map S_{(k⃗,w)}^{D}(U) → S_{(k⃗,w)}^{D}(U)_{σ,ψ} typically does not

induce a map S_{(k⃗,w)}^{D}(U; R) → S_{(k⃗,w)}^{D}(U; R)_{σ,ψ} because of denominators, but some positive integer multiple of it will do. More precisely, define N(U) to be the product of the order of G_{ℓ} and the order of the finite group K^{×} det(U)\A_{K}^{×}/(K_{∞})^{o}. The composite of the projection map above and multiplication by N(U) induces a map e : S_{(k⃗,w)}^{D}(U; R) → S_{(k⃗,w)}^{D}(U; R)_{σ,ψ} such that the composite S_{(k⃗,w)}^{D}(U; R)_{σ,ψ} ⊆

S_{(k⃗,w)}^{D}(U; R) → S_{(k⃗,w)}^{D}(U; R)_{σ,ψ} is multiplication by N(U). Note crucially that

N(U) = N(U ∩ U_{0}(𝔮)), and that e^{2} = N(U)e as endomorphisms of S_{(k⃗,w)}^{D}(U; R).

Contrary to what Taylor implicitly asserts, there is in general no SL_{2}(R)-invariant perfect pairing on Symm^{a}(R^{2}) if R is not a Q-algebra. However if R is a subring of C then there is an SL_{2}(R)-invariant injection from Symm^{a}(R^{2}) to its R-dual, with cokernel killed by some positive integer C′ (which depends on a but not on R), and this induces a pairing on Symm^{a}(R^{2}) which is not perfect but which will suffice to prove the result we need (some of Taylor’s constants need to be modified by this constant). Taylor uses this pairing to produce a perfect pairing on S_{(k⃗,w)}^{D}(U; C) and the analogue of this pairing that we shall need is the induced per-

fect pairing between S_{(k⃗,w)}^{D}(U; C)_{σ,ψ} and S_{(k⃗,w)}^{D}(U; C)_{σ^{∗},ψ^{∗}}, where σ^{∗} = σ(χ ∘ det) and ψ^{∗} = ψ(χ ∘ det), where χ is the finite order Hecke character associated to π_{D} on p.272 of [66] and det is the reduced norm D^{×} → GL_{1}. The reason for this twist is that the SL_{2}-invariant pairing on the coefficient sheaves is not GL_{2}-invariant.

We now run through Taylor’s argument on pp.272–276 of loc. cit., making the following changes. If S_{k}^{D}(U_{1}(n); R) occurs on the left hand side of a pairing, we replace it by S_{(k⃗,w)}^{D}(U; R)_{σ,ψ}; if it occurs on the right hand side then we replace it

by S_{(k⃗,w)}^{D}(U; R)_{σ^{∗},ψ^{∗}}. We replace U_{1}(n; 𝔮) with U ∩ Γ_{0}(𝔮) (with Γ_{0}(𝔮) denoting the usual level structure, namely the matrices which are upper triangular modulo 𝔮), and replacing the Hecke algebra T_{k}^{D}(n) in [66] by the sub-Z-algebra T_{(k⃗,w)}(U)_{σ,ψ} of End_{C}(S_{(k⃗,w)}^{D}(U)_{σ,ψ}) generated only by Hecke operators T_{𝔮} at the unramified primes 𝔮. These Hecke algebras are as big as we shall need for our application—we do not need to consider the operators S_{𝔮} as we have fixed a central character, and we also do not need to consider Hecke operators at the ramified places.

The analogues of the assertions about direct sums in Lemma 1 of [66] are still true on the (σ, ψ)-component of S_{(k⃗,w)}^{D}(U). Taylor’s map i commutes with the action of G_{ℓ} and with the action of Z, and the analogue of Taylor’s Lemma 2 holds (indeed the proof given remains valid when S_{k}^{D}(n) is replaced by S_{(k⃗,w)}^{D}(U)_{σ,ψ} etc., as the

map i^{†} also commutes with the G_{ℓ} and Z-action). The analogue of Lemma 3 that we need is that for a fixed compact open subgroup X ⊂ GL_{1}(A_{K}^{f}) there are positive integer constants C_{1} and C_{2} such that for any compact open U with det(U) = X we have

C_{1}⟨S_{(k⃗,w)}^{D}(U; R)_{σ,ψ}, S_{(k⃗,w)}^{D}(U; R)_{σ^{∗},ψ^{∗}}⟩ ⊆ R

and for f ∈ S_{(k⃗,w)}^{D}(U; C)_{σ,ψ} with

⟨f, S_{(k⃗,w)}^{D}(U; R)_{σ^{∗},ψ^{∗}}⟩ ⊆ R

we have C_{2}f ∈ S_{(k⃗,w)}^{D}(u; R)_{σ,ψ}. The statement about C_{1} follows from Taylor’s Lemma 3, and that about C_{2} can also be deduced from Taylor’s result, the comments about C′ above, the fact that the pairing ⟨, ⟩ restricts to the zero pairing between the (σ, ψ)-eigenspace and the (σ′, ψ′)-eigenspace if (σ′, ψ′) ≠ (σ^{∗}, ψ^{∗}) (using the argument on the bottom of p.272 of [66]) and the existence of the “projector” e above.

We need the (σ^{∗}, ψ^{∗})-analogue of Lemma 4 of [66] and this is true—indeed it can be deduced from Lemma 4 of [66] by restricting to the (σ^{∗}, ψ^{∗})-eigenspace.

We can now prove the analogue of Theorem 1 of [66] (where we replace Taylor’s Hecke algebras with ours as indicated above); we simply mimic Taylor’s beautiful proof on p.276 of loc. cit.; the assiduous reader can check that we have explained the analogues of all the ingredients that we need. Now using a standard Cebotarev argument we deduce that given π_{D} of weight (k⃗, w), we can find a prime 𝔮 ∤ ℓ of K at which π_{D} is unramified principal series and such that the associated system of eigenvalues of T_{(k⃗,w)}(U)_{σ,ψ} is congruent (modulo some prime above ℓ) to a system of eigenvalues occurring in T_{(k⃗,w)}(U_{0}(𝔮))_{σ,ψ}^{new}. □

Using this beefed-up version of Taylor’s level-raising theorem, we can deduce a beefed-up version of Proposition 2.5. Suppose that k⃗ ∈ Z^{S_{K}} and w ∈ Z with k_{τ} ≥ 2 and of the same parity as w for all τ. For each 𝔭|ℓ, suppose that σ_{𝔭} is a smooth irreducible representation of GL_{2}(O_{K,𝔭}). Via our fixed embeddings \overline{Q} → C and \overline{Q} → \overline{Q}_{ℓ}, and our identification of the residue field of \overline{Q}_{ℓ} with \overline{F}_{ℓ}, we can unambiguously define the semisimplification \overline{σ}_{𝔭} of the mod ℓ reduction of σ_{𝔭}; so \overline{σ}_{𝔭} is a representation of GL_{2}(O_{K,𝔭}) on a finite-dimensional \overline{F}_{ℓ}-vector space. Define σ := ⊗_{𝔭|ℓ}σ_{𝔭} and \overline{σ} := ⊗_{𝔭|ℓ}\overline{σ}_{𝔭}. Finally let G_{ℓ} denote a finite quotient of GL_{2}(O_{K} ⊗ Z_{ℓ}) through which σ factors.

Proposition 2.10. For an irreducible representation ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}), the following are equivalent:

• ρ ∼ \overline{ρ}_{π} for some cuspidal holomorphic weight (k⃗, w) automorphic representation π of GL_{2}(A_{K}) such that π_{𝔭} has type σ_{𝔭} for each 𝔭|ℓ.

• ρ is modular of weight V for some Jordan-Hölder constituent V of

W := \overline{σ}^{∨} ⊗ ⨂_{𝔭|ℓ} ⨂_{τ∈S_{𝔭}} (det^{(w−k_{τ})/2}Symm^{k_{τ} −2}k_{𝔭}^{2} ⊗_{τ} \overline{F}_{ℓ}) .

Proof. Lemma 2.9 shows that ρ ∼ \overline{ρ}_{π} for some π as above if and only if there are D and U as usual, and a compact open subgroup U′′ of U with U/U′′ = G_{ℓ}, such that ρ is the mod ℓ reduction of an irreducible 2-dimensional \overline{Q}_{ℓ}-representation ρ̃ which is a direct summand (equivalently, a subquotient) of Hom_{G_{ℓ}}(σ, H_{p}^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′})), with F_{λ,U′′} the E_{λ}-sheaf constructed by Carayol on X_{U′′} corresponding to (k⃗, w) (where here we assume U is small enough for the projection ϕ : Y_{U′′} → Y_{U} to be étale with covering group G_{ℓ}, and furthermore that our coefficient field E_{λ} is assumed large enough to satisfy the conditions that Carayol requires of it, and also large enough to afford a model for σ and to ensure that all the irreducible subquotients of H^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′}) have dimension either 1 or 2). By Lemma 2.2(b) the latter condition on ρ is equivalent to ρ being isomorphic to the mod ℓ reduction of an irreducible 2-dimensional subquotient of Hom_{G_{ℓ}}(σ, H^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′})).

Now let let F_{σ^{∨}} denote the E_{λ}-sheaf on Y_{U} associated to the dual of σ, and let F_{λ,U} denote the E_{λ}-sheaf constructed by Carayol on Y_{U} and corresponding to weight (k⃗, w). Then

Hom_{G_{ℓ}}(σ, H^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′})) = (H^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′}) ⊗ σ^{∨})^{G_{ℓ}}
= H^{1}(Y_{U′′,\overline{K}}, F_{λ,U′′} ⊗ ϕ^{∗}F_{σ^{∨}})^{G_{ℓ}}
= H^{1}(Y_{U′′,\overline{K}}, ϕ^{∗}(F_{λ,U} ⊗ F_{σ^{∨}}))^{G_{ℓ}}
= H^{1}(Y_{U,\overline{K}}, F_{λ,U} ⊗ F_{σ^{∨}}),

the last equality coming, for example, from the Hochschild-Serre spectral sequence and the fact that the order of G_{ℓ} is invertible in the (characteristic zero) field E_{λ}.

Let O_{λ} denote the integers in E_{λ}, let F_{λ,U}^{0} and F_{σ^{∨}}^{0} be O_{λ}-lattices in F_{λ,U} and F_{σ^{∨}} and set F^{0} := F_{λ,U}^{0} ⊗ F_{σ^{∨}}^{0}. We then deduce that an irreducible ρ is the reduction of an irreducible \overline{ρ}_{π} as above if and only if it is a subquotient of H^{1}(Y_{U,\overline{K}}, F^{0})^{tf}/λ, where λ denotes the maximal ideal of O_{λ} and tf denotes the maximal torsion-free quotient. Now a standard argument shows that this is so if and only if ρ is a subquotient of H^{1}(Y_{U,\overline{K}}, F^{0}/λ) (the torsion in H^{1}(Y_{U,\overline{K}}, F^{0}) is a subquotient of H^{0}(Y_{U,\overline{K}}, F^{0}/ℓ^{N}) for some N and hence the Galois representations arising as subquotients of it are all 1-dimensional by Lemma 2.2(a), and the cokernel of the injection H^{1}(Y_{U,\overline{K}}, F^{0})/λ → H^{1}(Y_{U,\overline{K}}, F^{0}/λ) is contained in H^{2}(Y_{U,\overline{K}}, F^{0})[λ] and hence in H^{2}(Y_{U,\overline{K}}, F^{0}/λ^{N}) for some N, and the irreducible subquotients of this group are also all 1-dimensional by Lemma 2.2(a)). To finish the proof it suffices by Lemma 2.4 to check that F^{0}/λ ⊗ \overline{F}_{ℓ} and W (1) have the same Jordan-Hölder factors, which follows immediately from the definitions. □

For τ ∈ S_{𝔭}, we denote by ω_{τ} the fundamental character of I_{K_{𝔭}} defined by composing τ with the homomorphism I_{K_{𝔭}} → k_{𝔭}^{×} obtained from local class field theory (with the convention that uniformizers correspond to geometric Frobenius elements). We then have the following compatibility among determinants, central characters and twists.

Corollary 2.11. (1) If ρ is modular of weight V and V has central character ⊗_{𝔭|ℓ} ∏_{τ∈S_{𝔭}} τ^{c_{τ}}, then

det ρ|_{I_{K𝔭}} = ∏_{τ∈S_{𝔭}} ω_{τ}^{c_{τ} +1}

for each 𝔭|ℓ.

(2) Let χ : G_{K} → \overline{F}_{ℓ}^{×} be such that χ|_{I_{K𝔭}} = ∏_{τ∈S_{𝔭}} ω_{τ}^{c_{τ}} for each 𝔭|ℓ. Then ρ is modular of weight V if and only χρ is modular of weight V ⊗ V_{χ}, where

V_{χ} = ⨂_{𝔭|ℓ} ⨂_{τ∈S_{𝔭}} det^{c_{τ}} k_{𝔭}^{2} ⊗_{τ} \overline{F}_{ℓ}.

Proof. (1) Let T denote the Teichmüller lift of the norm map N : (O/ℓ)^{×} → F_{ℓ}^{×}. By Lemma 2.7 there is an irreducible \overline{Q}_{ℓ}-representation σ of G such that the reduction of σ^{∨}⊗(T^{−1}∘det) contains V as a Jordan-Hölder factor. By Proposition 2.10 we see that if ρ is modular of weight V then ρ ∼ \overline{ρ}_{π} for some π of weight (2, 0) and type σ. Now by section 5.6.1 of [13] we see that det(ρ_{π}) = χ_{π}^{−1}ω^{−1} where χ_{π} is the central character of π and ω is the cyclotomic character. Now χ_{π} can be computed on O_{K_{𝔭}}^{×} because it is the central character of σ_{𝔭}, which is the inverse of the central character α_{𝔭} of σ_{𝔭}^{∨}. We deduce that det(ρ)|I_{𝔭} = α_{𝔭}\overline{ω}^{−1} = α_{𝔭}N_{𝔭}^{−1}, where N_{𝔭} = ∏_{τ∈S_{𝔭}} is the map I_{K_{𝔭}} → k_{𝔭}^{×} → F_{ℓ}. Finally the fact that V is a Jordan-Hölder factor of the reduction of σ^{∨} ⊗ (T^{−1} ∘ det) implies that the central character of V (considered as a representation of GL_{2}(k_{𝔭})) is α_{𝔭}.N_{𝔭}^{−2} and the result follows.

(2) is simpler and could have been deduced earlier; in fact it is immediate from Lemma 2.3 and Lemma 2.4(a). □

Recall that for K = Q, every modular ρ arises from a form of level prime to ℓ and some weight k ≥ 2. Moreover, after twisting ρ, one may take the weight k to be in the range 2 ≤ k ≤ ℓ + 1 (see [3]). This is in general false for larger K. Indeed if ρ arises from a form of weight (k⃗, w) and level prime to ℓ, then we see from the preceding results that det ρ|_{I_{𝔭}} = ω^{w−1} for all 𝔭|ℓ, and it is easy to construct representations none of whose twists have this property: choose for example an odd prime ℓ inert in a real quadratic K, and a totally odd ρ such that det ρ|_{I_{ℓ}} = ω_{τ}^{a} for some odd integer a (where τ : O_{K}/ℓ → \overline{F}_{ℓ}); see [19] for some explicit examples. On the other hand, it is still the case that every modular ρ arises from a form of weight (2, …, 2) and some level not necessarily prime to ℓ. Moreover, after twisting ρ, we can assume the form has level dividing 𝔫ℓ for some 𝔫 prime to ℓ.

Corollary 2.12. For an irreducible ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) the following are equivalent:

(1) ρ ∼ \overline{ρ}_{π} for some holomorphic cuspidal automorphic representation π of GL_{2}(A_{K});

(2) ρ ∼ \overline{ρ}_{π} for some cuspidal automorphic representation π of GL_{2}(A_{K}) which is holomorphic of weight (2, 0);

(3) χρ ∼ \overline{ρ}_{π} for some character χ and some cuspidal automorphic representation π of GL_{2}(A_{K}) which is holomorphic of weight (2, 0) and level U = U^{ℓ}.U_{1}(ℓ) (the adelic analogue of “level Γ_{1}(ℓ) at ℓ”);

(4) ρ is modular of weight V for some Serre weight V.

Proof. It is clear that (3) ⇒ (2) ⇒ (1). Proposition 2.10 shows that (1) ⇒ (4) if k_{τ} ≥ 2 for all τ; if some of the k_{τ} are equal to 1 then one has to first multiply by an Eisenstein series to increase the weight k⃗: more formally one uses the Deligne–Serre lemma and Lemma 5.2 of [43]. Finally we need to show that (4) ⇒ (3). If (4) holds then by Lemma 2.8 there is an irreducible \overline{Q}_{ℓ}-representation σ of G with a ( \begin{matrix}∗ & ∗ \\ 0 & 1 \end{matrix} )- fixed vector such that the reduction of σ^{∨} ⊗ ⨂_{𝔭|ℓ} ⊗_{τ∈S_{𝔭}} det^{−1} contains some twist of V. We deduce (3) from Lemma 2.11(2) and the case (k⃗, w) = (2, 0) of 2.10. □

3. The weight conjecture

Suppose that ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd. The aim of this section is to provide a conjectural recipe for the set of V such that ρ is modular of weight V.

For each prime 𝔭 of K dividing ℓ, we will define a set of representations W_{𝔭}(ρ) of GL_{2}(k_{𝔭}) depending only on ρ|_{I_{K𝔭}}, and then define the conjectural weight set W (ρ) as the set of Serre weights of the form ⊗_{\overline{F}_{ℓ}}V_{𝔭} with V_{𝔭} ∈ W_{𝔭}(ρ).

We need some more notation before defining W_{𝔭}(ρ). With our prime 𝔭 dividing ℓ fixed for now, we write simply k, f and S for k_{𝔭}, f_{𝔭} and S_{𝔭}. Fix an embedding \overline{K} → \overline{K}_{𝔭} and identify D = G_{K_{𝔭}} and I = I_{K_{𝔭}} with subgroups of G_{K}. Let K′_{𝔭} be the unramified quadratic extension of K_{𝔭} in \overline{K}_{𝔭} and let k′ denote its residue field. We let S′ denote the set of embeddings k′ → \overline{F}_{ℓ}, let D′ = G_{K′_{𝔭}} and define a map π : S′ → S by τ′ ↦ τ′|_{k}.

Suppose that L ⊂ \overline{K}_{𝔭} is a finite unramified extension of K_{𝔭} and σ is an embedding of its residue field O_{L}/ℓO_{L} in \overline{F}_{ℓ}. We denote by ω_{σ} the fundamental character of I = I_{L} defined by composing σ with the homomorphism I_{L} → (O_{L}/ℓO_{L})^{×} gotten from local class field theory.

In defining W_{𝔭}(ρ), we treat separately the cases where ρ|_{D} is irreducible and where it is reducible.

3.1. The irreducible case. If ρ|_{D} is irreducible, we define W_{𝔭}(ρ) by the following rule:

(1)

ρ|_{I} ∼ ∏_{τ∈S} ω_{τ}^{a_{τ}} (\begin{matrix}∏_{τ′∈J} ω_{τ′}^{b_{π(τ′)}} & 0 \\ 0 & ∏_{τ′∉J} ω_{τ′}^{b_{π(τ′)}} \end{matrix})

V_{a⃗,b⃗} ∈ W_{𝔭}(ρ) ⟺

for some J ⊂ S′ such that π : J⟶^{∼}S.

Since ρ|_{D} is irreducible, there is a character ξ : D′ → \overline{F}_{ℓ}^{×} such that ρ|_{D} ∼ Ind_{D′}^{D}ξ. We define

W′(ξ) = { (V_{a⃗,b⃗}, J) | J ⊂ S′, π : J⟶^{∼}S, ξ|_{I} = ∏_{τ∈S} ω_{τ}^{a_{τ}} ∏_{τ′∈J} ω_{τ′}^{b_{π(τ′)}} }.

Thus W_{𝔭}(ρ) = { V | (V, J) ∈ W′(ξ) for some J }. (Note that replacing ξ by its conjugate under D/D′ replaces J by its complement.) We shall see that the projection maps W′(ξ) → W_{𝔭}(ρ) and W′(ξ) → { J ⊂ S′ |π : J⟶^{∼}S } are typically bijections, so that typically (but not always) |W_{𝔭}(ρ)| = 2^{f}.

We now choose an element of S′ which we denote τ′_{0}, and then let τ′_{i} = τ′_{0} ∘ Frob_{ℓ}^{i} and τ_{i} = π(τ′_{i}). Note that S = { τ_{i} | i ∈ Z/fZ } and S′ = { τ′_{i} | i ∈ Z/2fZ }. Letting ω = ω_{τ_{0}} and ω′ = ω_{τ′_{0}}, we have ω_{τ_{i}} = ω^{ℓ^{i}}, ω_{τ′_{i}} = (ω′)^{ℓ^{i}} and ω = (ω′)^{ℓ^{f} +1}. Note that ξ|_{I} = (ω′)^{n} for some n mod ℓ^{2f} − 1, and since ρ|_{D} is irreducible, n is not divisible by ℓ^{f} + 1.

For B ⊂ {0, …, f − 1} (where the symbol ⊂ includes the case of equality), let J_{B} = {τ′_{i}|i ∈ B} ∪ {τ′_{f+i}|i ∉ B}. If a ∈ Z/(ℓ^{f} − 1)Z, b⃗ = (b_{0}, …, b_{f−1}) with each b_{i} ∈ {1, …, ℓ} and B ⊂ {0, …, f − 1}, let

n′_{a,b⃗,B} = a(ℓ^{f} + 1) + ∑_{i∈B} b_{i}ℓ^{i} + ∑_{i∉B} b_{i}ℓ^{f+i} mod ℓ^{2f} − 1.

Then W′(ξ) is in bijection with the set of triples (a, b⃗, B) as above with n ≡ n′_{a,b⃗,B} mod ℓ^{2f} − 1. Now note for each B ⊂ {0, …, f − 1}, there is a unique such

triple (a, b⃗, B) with this property for each solution of

n ≡ ∑_{i∈B} b_{i}ℓ^{i} − ∑_{i∉B} b_{i}ℓ^{i} mod ℓ^{f} + 1

with b_{0}, …, b_{f−1} ∈ {1, …, ℓ}. But the values of ∑_{i∈B} b_{i}ℓ^{i} − ∑_{i∉B} b_{i}ℓ^{i} are the ℓ^{f} consecutive integers from n′_{B} − ℓ^{f} to n′_{B} − 1 where

n′_{B} = ∑_{i∈B} ℓ^{i+1} − ∑_{i∉B} ℓ^{i} + 1,

so there is a solution as long as n ≢ n′_{B} mod ℓ^{f} + 1 and this solution is unique. We have thus shown that W′(ξ) is in bijection with the set of B such n ≢ n′_{B} mod ℓ^{f} +1; moreover the projection W′(ξ) → { J ⊂ S′ |π : J⟶^{∼}S } is injective.

Note that if f is odd and B is either {0, 2, 4, …, f −1} or B = {1, 3, 5, …, f −2}, then n′_{B} ≡ 0 mod ℓ^{f} + 1. To see that the converse holds as well, observe that

−(ℓ^{f} + 1) < −ℓ\frac{ℓ^{f−1} − 1}{ℓ − 1} ≤ n′_{B} ≤ \frac{ℓ^{f+1} − 1}{ℓ − 1} < 2(ℓ^{f} + 1).

Thus if n′_{B} ≡ 0 mod ℓ^{f} + 1, then n′_{B} = 0 or ℓ^{f} + 1. If n′_{B} = 0, then solving

∑_{i∈B} ℓ^{i+1} − ∑_{i∉B} ℓ^{i} + 1 ≡ 0 mod ℓ^{r}

by induction on r, we find that f is odd and B = {1, 3, …, f − 2}. Similarly if n′_{B} = ℓ^{f} + 1, then f is odd and B = {0, 2, …, f − 1}.

We now show that if f is even, then the 2^{f} values of n′_{B} mod ℓ^{f} + 1 are distinct. First note that

n′_{B} ≡ −1 + (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i} mod ℓ^{f} + 1,

where B^{∗} = ({0, 2, …, f−2}∩B)∪({1, 3, …, f−1}\B). Thus n′_{B_{1}} ≡ n′_{B_{2}} mod ℓ^{f} +1 if and only if

∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} ≡ ∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} mod (ℓ^{f} + 1)/d,

where d = gcd(ℓ + 1, ℓ^{f} + 1) ≤ ℓ − 1 (so d = 2 if ℓ is odd, and d = 1 if ℓ = 2). But these two sums differ by at most (ℓ^{f} − 1)/(ℓ − 1) < (ℓ^{f} + 1)/d, so the above congruence holds if and only if equality holds, in which case B_{1}^{∗} = B_{2}^{∗}, so B_{1} = B_{2}.

Next we show that if f and ℓ are odd, then the 2^{f} −2 non-zero values of n′_{B} mod ℓ^{f} + 1 are distinct. In this case we have

n′_{B} ≡ (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i} mod ℓ^{f} + 1,

where B^{∗} = ({0, 2, …, f−1}∩B)∪({1, 3, …, f−2}\B). Thus n′_{B_{1}} ≡ n′_{B_{2}} mod ℓ^{f} +1 if and only if

∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} ≡ ∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} mod (ℓ^{f} + 1)/(ℓ + 1).

But these two sums differ by at most (ℓ^{f} − 1)/(ℓ − 1) < 2(ℓ^{f} + 1)/(ℓ + 1), so if the above congruence holds then either equality holds, in which case B_{1} = B_{2}, or

∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} = ∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} + ∑_{i=0}^{f−1}(−1)^{i}ℓ^{i},

exchanging B_{1} and B_{2} if necessary. Solving mod ℓ^{r} inductively on r, we see that the only possibility is that B_{1}^{∗} = ∅ and B_{2}^{∗} = {0, 1, …, f −1}, but these are precisely the cases where n′_{B} ≡ 0 mod ℓ^{f} + 1.

Finally suppose that f is odd and ℓ = 2. In this case we have

n′_{B} ≡ 3 ∑_{i∈B^{∗}}(−1)^{i}2^{i} mod 2^{f} + 1,

where B^{∗} = ({0, 2, …, f − 1} ∩ B) ∪ ({1, 3, …, f − 2} \ B). In particular n′_{B} ≡ 0 mod 3. Moreover we have the inequality |n′_{B_{1}} − n′_{B_{2}}| < 3(2^{f} + 1), showing that each congruence class mod 2^{f} + 1 arises as n′_{B} for at most 3 values of B. It follows that each of the (2^{f} − 2)/3 non-zero multiples of 3 mod 2^{f} + 1 arises as n′_{B} for exactly 3 values of B.

We have thus proved the following propositions:

Proposition 3.1. Suppose that ℓ is odd. If f is even, then the congruence classes mod ℓ^{f} + 1 of the form

−1 + (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i}

are distinct and non-zero as B^{∗} runs through all subsets of {0, 1, …, f − 1}. If f is odd, then the congruence classes mod ℓ^{f} + 1 of the form

(ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i}

are distinct and non-zero as B^{∗} runs through all non-empty proper subsets of {0, 1, …, f − 1}. Letting A denote the set of such classes in each case, we have

|W′(ξ)| = \{ \begin{matrix}2^{f}, & if n ∉ A, \\ 2^{f} − 1, & if n ∈ A, \end{matrix}

where ξ|_{I} = (ω′)^{n}.

Proposition 3.2. Suppose that ℓ = 2 and ξ|_{I} = (ω′)^{n}. Then

|W′(ξ)| = \{ \begin{matrix}2^{f} − 1, & if f is even, \\ 2^{f}, & if f is odd and 3 ∤ n, \\ 2^{f} − 3, & if f is odd and 3 | n. \end{matrix}

Multiple B can occur with the same (a, b⃗); for example if f = 3 and n = 1, then (−ℓ^{2}, (1, ℓ, 1), {0, 1}) and (−ℓ^{2}, (1, ℓ, 1), {0, 2}) are both in W′(ξ), so the map W′(ξ) → W_{𝔭}(ρ) is not injective. (In this case in fact, |W′(ξ)| = 8, but |W_{𝔭}(ρ)| = 6.)

Proposition 3.3. The map W′(ξ) → W_{𝔭}(ρ) fails to be injective if and only if ℓ^{r}n ≡ m mod ℓ^{f} + 1 for some integers r, m with |m| ≤ ℓ(ℓ^{f−2} − 1)/(ℓ − 1).

Proof. Suppose first that W′(ξ) → W_{𝔭}(ρ) is not injective. This means that for some a, b⃗ and B_{1} ≠ B_{2}, we have

(2)  n ≡ n′_{a,b⃗,B_{1}} ≡ n′_{a,b⃗,B_{2}} mod ℓ^{2f} − 1.

First we note that B_{2} cannot be the complement \overline{B}_{1} of B_{1} in \{0, …, f −1\} since

n′_{a,b⃗,B_{1}} ≡ n′_{a,b⃗,\overline{B}_{1}} ≡ ℓ^{f}n′_{a,b⃗,B_{1}} mod ℓ^{2f} − 1

would imply n ≡ 0 mod ℓ^{f} +1, contradicting the irreducibility of ρ|_{D}. We thus have J_{2} ≠ S′ \ J_{1}, where J_{1} = J_{B_{1}} and J_{2} = J_{B_{2}}. One now checks that (possibly after switching J_{1} and J_{2}) we may find t ∈ Z/2fZ so that τ′_{t−1} ∈ J_{1} \J_{2} and τ′_{t} ∈ J_{1} ∩J_{2}. We then have

ℓ^{f−t}n ≡ n′_{a′,b⃗′,B′_{1}} ≡ n′_{a′,b⃗′,B′_{2}} mod ℓ^{2f} − 1,

where a′ ≡ ℓ^{−t}a mod ℓ^{f} − 1, b′_{i} = b_{i+t mod f} for i ∈ {0, …, f − 1} and B′_{ν} is such that J_{B′_{ν}} = J_{ν} ∘ Frob_{ℓ}^{f−t} for ν = 1, 2. Replacing n with ℓ^{f−t}n, we may thus assume that τ′_{f−1} ∈ J_{1}\J_{2} and τ′_{f} ∈ J_{1}∩J_{2}, or equivalently, f −1 ∈ B_{1}\B_{2} and 0 ∉ B_{1}∪B_{2}.

Returning to the congruence (2), we have

∑_{i∈B_{1}} b_{i}ℓ^{i} + ℓ^{f} ∑_{i∉B_{1}} b_{i}ℓ^{i} ≡ ∑_{i∈B_{2}} b_{i}ℓ^{i} + ℓ^{f} ∑_{i∉B_{2}} b_{i}ℓ^{i} mod ℓ^{2f} − 1,

or equivalently,

∑_{i∈B_{1}\B_{2}} b_{i}ℓ^{i} ≡ ∑_{i∈B_{2}\B_{1}} b_{i}ℓ^{i} mod ℓ^{f} + 1.

Since each sum is less than 2(ℓ^{f} + 1), they must either be equal or differ by ℓ^{f} + 1, and since 0 ∉ B_{1} ∪ B_{2}, each sum is divisible by ℓ, so in fact equality holds. Since f − 1 ∈ B_{1} \ B_{2}, we have

b_{f−1}ℓ^{f−1} ≤ ∑_{i∈B_{1}\B_{2}} b_{i}ℓ^{i} = ∑_{i∈B_{2}\B_{1}} b_{i}ℓ^{i} ≤ ∑_{i=2}^{f−1} ℓ^{i} < 2ℓ^{f−1},

so b_{f−1} = 1. Moreover if b_{f−2}, b_{f−3}, …, b_{s+1} are all less than ℓ for some s < f − 2, then we must have b_{f−2} = b_{f−3} = ⋯ = b_{s+1} = ℓ−1 and f −2, f −3 …, s ∈ B_{2}\B_{1}, for if either fails, we find that

∑_{i∈B_{2}\B_{1}} b_{i}ℓ^{i} ≤ (ℓ − 1) ∑_{i=s+1}^{f−2} ℓ^{i} + ∑_{i=2}^{s} ℓ^{i} < ℓ^{f−1}.

Since 0 ∉ B_{2} \ B_{1}, we conclude that for some s with 0 < s < f − 1, we have (b_{s}, b_{s+1}, …, b_{f−1}) = (ℓ, ℓ − 1, …, ℓ − 1, 1) and s, s + 1, …, f − 2 ∈ B_{2} \ B_{1}. It follows that

n ≡ ∑_{i∈B_{1}} b_{i}ℓ^{i} − ∑_{i∉B_{1}} b_{i}ℓ^{i} ≡ ∑_{i∈B_{1},i<s} b_{i}ℓ^{i} − ∑_{i∉B_{1},i<s} b_{i}ℓ^{i} mod ℓ^{f} + 1,

and this last difference has absolute value at most ℓ(ℓ^{f−2} − 1)/(ℓ − 1).

Conversely suppose that ℓ^{r}n ≡ m mod ℓ^{f} + 1 for some r, m with |m| ≤ ℓ(ℓ^{f−2} − 1)/(ℓ − 1). Replacing r by r + f if necessary, we may assume m > 0 and then

\frac{ℓ^{s} − 1}{ℓ − 1} ≤ m ≤ ℓ · \frac{ℓ^{s} − 1}{ℓ − 1}

for some s with 0 < s < f − 1. We can then write m = ∑_{i=0}^{s−1} b_{i}ℓ^{i} for some b_{0}, b_{1}, …, b_{s−1} ∈ {1, …, ℓ}. We can then choose a ∈ Z/(ℓ^{f} − 1)Z so that

ℓ^{r}n ≡ n_{a,b⃗,B_{1}} ≡ n_{a,b⃗,B_{2}} mod ℓ^{2f} − 1,

where b⃗ = (b_{0}, b_{1}, …, b_{s−1}, ℓ, ℓ − 1, …, ℓ − 1, 1), B_{1} = {0, 1, …, f − 2} and B_{2} = {0, 1, …, s − 1, f − 1}. We conclude that

n ≡ n′_{a′,b⃗′,B′_{1}} ≡ n′_{a′,b⃗′,B′_{2}} mod ℓ^{2f} − 1,

where a′ ≡ ℓ^{−r}a mod ℓ^{f} − 1, b′_{i} = b_{i+r mod f} for i ∈ {0, …, f − 1} and B′_{ν} is such that J_{B′_{ν}} = J_{B_{ν}} ∘ Frob_{ℓ}^{−r} for ν = 1, 2. □

3.2. The reducible case. Now suppose ρ|_{D} is reducible, write ρ|_{D} ∼ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix})

and let c_{ρ} denote the corresponding class in H^{1}(K_{𝔭}, χ_{1}χ_{2}^{−1}). Consider now the set of pairs

W′(χ_{1}, χ_{2}) = { (V_{a⃗,b⃗}, J) | J ⊂ S, χ_{1}|_{I} = ∏_{τ∈S} ω_{τ}^{a_{τ}} ∏_{τ∈J} ω_{τ}^{b_{τ}} , χ_{2}|_{I} = ∏_{τ∈S} ω_{τ}^{a_{τ}} ∏_{τ∉J} ω_{τ}^{b_{τ}} },

with π_{1} and π_{2} denoting the projections (V_{a⃗,b⃗}, J) ↦ V_{a⃗,b⃗} and (V_{a⃗,b⃗}, J) ↦ J. Note that interchanging χ_{1} and χ_{2} replaces J by its complement. We shall see that the projection map π_{2} : W′(χ_{1}, χ_{2}) → {J ⊂ S} is typically a bijection, so that typically |W′(χ_{1}, χ_{2})| = 2^{f}. However W_{𝔭}(ρ) will be defined below as a subset of π_{1}(W′(χ_{1}, χ_{2})) depending on c_{ρ}.

We now analyse the set W′(χ_{1}, χ_{2}) in a manner analogous to the irreducible case. We write χ_{ν} = ω^{n_{ν}} with n_{ν} ∈ Z/(ℓ^{f} − 1)Z for ν = 1, 2, and we let n = n_{1} − n_{2}. If a ∈ Z/(ℓ^{f} −1)Z, b⃗ = (b_{0}, …, b_{f−1}) with each b_{i} ∈ {1, …, ℓ} and B ⊂ {0, …, f −1}, let

n_{a,b⃗,B} = a + ∑_{i∈B} b_{i}ℓ^{i} mod ℓ^{f} − 1.

Then W′(χ_{1}, χ_{2}) is in bijection with the set of triples (a, b⃗, B) as above with n_{1} ≡ n_{a,b⃗,B} mod ℓ^{f} − 1 and n_{2} ≡ n_{a,b⃗,\overline{B}} mod ℓ^{f} − 1 where \overline{B} is the complement of B in {0, …, f − 1}. Note that for each B ⊂ {0, …, f − 1} there is a unique such triple for each solution of

n ≡ ∑_{i∈B} b_{i}ℓ^{i} − ∑_{i∉B} b_{i}ℓ^{i} mod ℓ^{f} − 1.

with b_{0}, …, b_{f−1} ∈ {1, …, ℓ}. But the values of ∑_{i∈B} b_{i}ℓ^{i} − ∑_{i∉B} b_{i}ℓ^{i} are the ℓ^{f} consecutive integers from n_{B} + 1 − ℓ^{f} to n_{B} where

n_{B} = ∑_{i∈B} ℓ^{i+1} − ∑_{i∉B} ℓ^{i},

so there is a unique solution if n ≢ n_{B} mod ℓ^{f} − 1 and two solutions if n ≡ n_{B} mod ℓ^{f} − 1. In particular the projection W′(χ_{1}, χ_{2}) → {J ⊂ S} is surjective and |W′(χ_{1}, χ_{2})| = 2^{f} + |{ B | n ≡ n_{B} mod ℓ^{f} − 1}|.

We now show that if f is odd, then the 2^{f} values of n_{B} mod ℓ^{f} − 1 are distinct, unless ℓ = 2 or 3, in which case n_{{0,...,f−1}} ≡ n_{∅} mod ℓ^{f} −1 and the rest are distinct. First note that

n_{B} ≡ −1 + (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i} mod ℓ^{f} − 1,

where B^{∗} = ({0, 2, …, f−1}∩B)∪({1, 3, …, f−2}\B). Thus n_{B_{1}} ≡ n_{B_{2}} mod ℓ^{f} −1 if and only if

∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} ≡ ∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} mod (ℓ^{f} − 1)/d,

where d = gcd(ℓ+1, ℓ^{f} −1). If ℓ > 3, then d = 2 and the two sums differ by at most (ℓ^{f} − 1)/(ℓ − 1) < (ℓ^{f} − 1)/d, so the above congruence holds if and only if equality holds, in which case B_{1}^{∗} = B_{2}^{∗}, so B_{1} = B_{2}. If ℓ = 2 or 3, then d = ℓ − 1, but the two sums differ by (ℓ^{f} − 1)/(ℓ − 1) only when one of B_{1} or B_{2} is {0, …, f − 1} and the other is empty.

Now consider the case where f is even and ℓ > 3. We then have

n_{B} ≡ (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i} mod ℓ^{f} − 1,

where B^{∗} = ({0, 2, …, f−2}∩B)∪({1, 3, …, f−1}\B). Thus n_{B_{1}} ≡ n_{B_{2}} mod ℓ^{f} −1 if and only if

∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} ≡ ∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} mod (ℓ^{f} − 1)/(ℓ + 1).

But these two sums differ by at most (ℓ^{f} − 1)/(ℓ − 1), which is less than 2(ℓ^{f} − 1)/(ℓ + 1). So if the above congruence holds then either equality holds, in which case B_{1} = B_{2}, or

∑_{i∈B_{2}^{∗}}(−1)^{i}ℓ^{i} = ∑_{i∈B_{1}^{∗}}(−1)^{i}ℓ^{i} + ∑_{i=0}^{f−1}(−1)^{i}ℓ^{i},

exchanging B_{1} and B_{2} if necessary. Solving modℓ^{r} inductively on r, we see that the only possibility is that B_{1}^{∗} = ∅ and B_{2}^{∗} = {0, 1, …, f − 1}, in which case n_{B_{ν}} ≡ 0 mod ℓ^{f} − 1.

If f is even and ℓ = 3, then the situation is the same, except that we have (ℓ^{f} − 1)/(ℓ − 1) = 2(ℓ^{f} − 1)/(ℓ + 1), so in addition to the possibilities that arose for ℓ > 3, we have

n_{{0,...,f−1}} ≡ n_{∅} ≡ (ℓ^{f} − 1)/2 mod ℓ^{f} − 1

as for odd f.

Finally suppose that f is even and ℓ = 2. In this case we have

n_{B} ≡ 3 ∑_{i∈B^{∗}}(−1)^{i}2^{i} mod 2^{f} − 1,

where B^{∗} = ({0, 2, …, f − 2} ∩ B) ∪ ({1, 3, …, f − 1} \ B). In particular n_{B} ≡ 0 mod 3. Moreover we have |n_{B_{1}} − n_{B_{2}}| ≤ 3(2^{f} − 1) with equality possible only when one of B_{1} or B_{2} is {0, …, f − 1} and the other is empty, in which case n_{B_{ν}} ≡ 0 mod 2^{f} − 1. Thus each non-zero congruence class mod 2^{f} − 1 arises as n_{B} for at most 3 values of B, while 0 arises for at most 4. It follows that each of the (2^{f} − 4)/3 non-zero multiples of 3 mod 2^{f} − 1 arises as n_{B} for exactly three values of B, while 0 arises for exactly four values.

We have thus proved the following propositions:

Proposition 3.4. Suppose that ℓ > 3. If f is odd, then the congruence classes mod ℓ^{f} − 1 of the form

−1 + (ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i}

are distinct and non-zero as B^{∗} runs through all subsets of {0, 1, …, f − 1}. If f is even, then the congruence classes mod ℓ^{f} − 1 of the form

(ℓ + 1) ∑_{i∈B^{∗}}(−1)^{i}ℓ^{i}

are distinct and non-zero as B^{∗} runs through all non-empty proper subsets of {0, 1, …, f − 1}. Letting A denote the set of such classes in each case, we have

|W′(χ_{1}, χ_{2})| = \{ \begin{matrix}2^{f} + 2, & if n ≡ 0 mod ℓ^{f} − 1 and f is even, \\ 2^{f} + 1, & if n ∈ A, \\ 2^{f}, & otherwise. \end{matrix}

Proposition 3.5. Suppose that ℓ = 3. If f is odd, then the congruence classes mod 3^{f} − 1 of the form

−1 + 4 ∑_{i∈B^{∗}}(−1)^{i}3^{i}

are distinct and non-zero mod (3^{f} − 1)/2 as B^{∗} runs through all the subsets of {0, 1, …, f − 1} other than {0, 2, …, f − 1} and {1, 3, …, f − 2}. If f is even, then the congruence classes mod 3^{f} − 1 of the form

4 ∑_{i∈B^{∗}}(−1)^{i}3^{i}

are distinct and non-zero mod (3^{f} − 1)/2 as B^{∗} runs through all non-empty proper subsets of {0, 1, …, f − 1} other than {0, 2, …, f − 2} and {1, 3, …, f − 1}. Letting A denote the set of such classes in each case, we have

|W′(χ_{1}, χ_{2})| = \{ \begin{matrix}2^{f} + 2, & if n ≡ 0 mod ℓ^{f} − 1 and f is even, or n ≡ (ℓ^{f} − 1)/2 \\ 2^{f} + 1, & if n ∈ A, \\ 2^{f}, & otherwise. \end{matrix}

Proposition 3.6. Suppose that ℓ = 2. Then (all congruences on n below being modulo 2^{f} − 1) we have

|W′(χ_{1}, χ_{2})| = \{ \begin{matrix}2^{f} + 4, & if n ≡ 0 and f is even, \\ 2^{f} + 3 & if n ≢ 0, 3 | n and f is even, \\ 2^{f} + 2, & if n ≡ 0 and f is odd, \\ 2^{f} + 1, & if n ≢ 0 and f is odd, \\ 2^{f}, & if 3 ∤ n and f is even. \end{matrix}

Note that “3 | n” is well-defined when f is even (as then n is an integer modulo 4^{f/2} − 1 which is zero mod 3). As in the irreducible case, multiple B can occur with the same (a, b⃗).

Proposition 3.7. The projection map from W′(χ_{1}, χ_{2}) onto its first component fails to be injective if and only if ℓ^{r}n ≡ m mod ℓ^{f} − 1 for some integers r, m with |m| ≤ max{0, ℓ(ℓ^{f−2} − 1)/(ℓ − 1)}.

Proof. The statement that the projection map from W′(χ_{1}, χ_{2}) to its first component is not injective is equivalent to the statement that for some a, b⃗ and B_{1} ≠ B_{2}, we have

n_{1} ≡ n_{a,b⃗,B_{1}} ≡ n_{a,b⃗,B_{2}} mod ℓ^{f} − 1

and

n_{2} ≡ n_{a,b⃗,\overline{B}_{1}} ≡ n_{a,b⃗,\overline{B}_{2}} mod ℓ^{f} − 1,

where \overline{B} denotes the complement of B in \{0, 1, 2, …, f − 1\}.

We first deal with the special case n ≡ 0 mod ℓ^{f} − 1. In this case the map is never injective; take b⃗ = (ℓ − 1, ℓ − 1, …, ℓ − 1), B_{1} = {0, 1, …, f − 1} and B_{2} = ∅ (with the appropriate value of a).

So let us now assume that n ≢ 0 mod ℓ^{f} − 1. Suppose first that the projection map is not injective. Because n ≢ 0 we check that B_{2} cannot be the complement of B_{1} (note that this finishes the proof in the case f = 1). We can then assume f − 1 ∈ B_{1} \ B_{2} and 0 ∉ (B_{1} \ B_{2}) ∪ (B_{2} \ B_{1}) after exchanging B_{1} and B_{2} and multiplying n by a power of ℓ if necessary, and complete the argument as in the irreducible case. □

For each pair α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}) we shall define below a subspace L_{α} ⊂ H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})) of dimension |J| + δ, where δ = 0 except in certain cases where χ_{1}χ_{2}^{−1} is trivial or cyclotomic. We then define W_{𝔭}(ρ) by the following rule:

V_{a⃗,b⃗} ∈ W_{𝔭}(ρ) if and only if c_{ρ} ∈ L_{α} for some α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}).

Note in particular that W_{𝔭}(ρ) ⊂ π_{1}(W′(χ_{1}, χ_{2})), with equality if c_{ρ} = 0, that is, if ρ|D is split.

Before defining the subspace L_{α}, we recall some facts about crystalline representations. Recall that a character ψ : D → \overline{Q}_{ℓ}^{×} is crystalline if and only the filtered ϕ-module D_{crys}(ψ) = (B_{crys} ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}(ψ))^{D} is free of rank one over K_{𝔭} ⊗_{Q_{ℓ}} \overline{Q}_{ℓ}.

For each τ ∈ S, let e_{τ} : K_{𝔭} ⊗_{Q_{ℓ}} \overline{Q}_{ℓ} → \overline{Q}_{ℓ} denote the projection defined by a⊗b ↦ τ̃(a)b where τ̃ is the embedding K_{𝔭} → \overline{Q}_{ℓ} reducing to τ, and let e_{τ} D_{crys}(ψ) denote the filtered \overline{Q}_{ℓ}-vector space D_{crys}(ψ) ⊗_{K_{𝔭}⊗\overline{Q}_{ℓ},e_{τ}} \overline{Q}_{ℓ}.

Lemma 3.8. Suppose that ψ is a crystalline character and for each τ ∈ S, m_{τ} is the integer such that gr^{−m_{τ}} e_{τ} D_{crys}(ψ) ≠ 0. Then \overline{ψ}|_{I} = ∏_{τ∈S} ω_{τ}^{m_{τ}}.

Proof. Crystalline characters satisfying the first condition correspond to (weakly) admissible filtered ϕ-modules with the specified filtration. These exist, and any two such differ by an unramified twist. Taking tensor products, the lemma reduces to the case where m_{τ} = 1 if τ = τ_{0} and m_{τ} = 0 otherwise. The result in this case follows for example from Theorems 5.3 and 8.4 of [29]. □

Recall that α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}) if and only if

χ_{1}|_{I} = ∏_{τ∈S} ω_{τ}^{a_{τ}} ∏_{τ∈J} ω_{τ}^{b_{τ}} , χ_{2}|_{I} = ∏_{τ∈S} ω_{τ}^{a_{τ}} ∏_{τ∉J} ω_{τ}^{b_{τ}} .

Lemma 3.9. Suppose that α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}). Let m_{τ,α} = b_{τ} if τ ∈ J and m_{τ,α} = −b_{τ} if τ ∉ J. Then there is a unique lift χ_{α} of χ_{1}χ_{2}^{−1} with the following properties:

• χ_{α} is crystalline with gr^{−m_{τ}} e_{τ} D_{crys}(χ_{α}) ≠ 0 for each τ ∈ S;

• if g ∈ D^{ab} corresponds via local class field theory to ℓ, then χ_{α}(g) is the Teichmüller lift of χ_{1}χ_{2}^{−1}(g).

Proof. Let ψ be a character satisfying the first condition. The preceding lemma shows that the reduction of ψ is an unramified twist of χ_{1}χ_{2}^{−1}. Let χ_{α} = δψ, where δ is the unramified character with δ(g) = χ̃_{1}χ̃_{2}^{−1}(g)ψ^{−1}(g) (and the tildes denote Teichmüller lifts). □

Recall that if ψ : D → \overline{Q}_{ℓ}^{×} is a crystalline representation, then Bloch and Kato define a subspace H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(ψ)) corresponding to those extensions of topological \overline{Q}_{ℓ}D-modules

0 → \overline{Q}_{ℓ}(ψ) → E → \overline{Q}_{ℓ} → 0

which are crystalline. By Corollary 3.8.4 of [5] for example, we have

dim H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(ψ)) = dim H^{0}(K_{𝔭}, \overline{Q}_{ℓ}(ψ)) + dim D_{crys}(ψ) − dim Fil^{0}D_{crys}(ψ)

where the dimensions are over \overline{Q}_{ℓ}. Applying this for ψ = χ_{α} for α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}), we see that dim H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α})) = |J|. We then define H_{f}^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) as the preimage of H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α})) under the natural map

H^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) → H^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α}))

and L′_{α} as the image of H_{f}^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) under the natural map

H^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) → H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})).

We then let L_{α} = L′_{α} except in two cases:

• if χ_{1}χ_{2}^{−1} is the cyclotomic character, b⃗ = (ℓ, …, ℓ) and J = S, then we let

L_{α} = H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1}));

• if χ_{1}χ_{2}^{−1} is the trivial character and J ≠ S, then we let L_{α} = L′_{α} + L_{ur}, where L_{ur} is the one-dimensional space of unramified classes in H^{1}(K_{𝔭}, \overline{F}_{ℓ}).

Remark 3.10. Recall that in defining χ_{α} we chose a lift (namely the Teichmüller lift) of χ_{1}χ_{2}^{−1}(g). If b_{τ} < ℓ for all τ ∈ S, then one can show using Fontaine-Laffaille theory [29] that the space L′_{α} is independent of the choice of lift. We expect this to be true even if some b_{τ} = ℓ, except in the two cases where we accordingly modified the definition of L_{α}. This independence of lift is proved for f = 2 by Chang and one of the authors in [14], Remark 7.13.

Remark 3.11. If χ_{1}χ_{2}^{−1} is the cyclotomic character, b⃗ = (1, …, 1) and J = S, then χ_{α} is the ℓ-adic cyclotomic character. In that case we have the isomorphism H^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α})) ≅ (K_{𝔭}^{×})ˆ ⊗_{Z_{ℓ}} \overline{Q}_{ℓ} by Kummer theory (where ˆ· denotes ℓ-adic completion), and one knows that H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α})) ≅ (O_{K_{𝔭}}^{×})ˆ ⊗_{Z_{ℓ}} \overline{Q}_{ℓ} (these being the extensions arising from ℓ-divisible groups). It follows that L_{α} corresponds to O_{K_{𝔭}}^{×} ⊗\overline{F}_{ℓ} under the isomorphism H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})) ≅ K_{𝔭}^{×}⊗\overline{F}_{ℓ}, hence consists of the classes which are peu ramifiées in the sense of Serre [63].

Lemma 3.12. If α = (V_{a⃗,b⃗}, J) ∈ W′(χ_{1}, χ_{2}), then dim L_{α} = |J| except in the following cases:

(1) if χ_{1}χ_{2}^{−1} is cyclotomic, b⃗ = (ℓ, …, ℓ), J = S and ℓ > 2, then dim L_{α} =

|J| + 1;

(2) if χ_{1}χ_{2}^{−1} is trivial, then dim L_{α} = |J| + 1 unless either L_{ur} ⊄ L′_{α} or b⃗ =

(ℓ, …, ℓ), in which case dim L_{α} = |J| + 2.

Proof. Note first that H_{f}^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) contains H^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α}))_{tor} and that the quotient is free of rank |J| = dim H_{f}^{1}(K_{𝔭}, \overline{Q}_{ℓ}(χ_{α})). Therefore the natural map

H_{f}^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) ⊗_{\overline{Z}_{ℓ}} \overline{F}_{ℓ} → H^{1}(K_{𝔭}, \overline{Z}_{ℓ}(χ_{α})) ⊗_{\overline{Z}_{ℓ}} \overline{F}_{ℓ} → H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1}))

is injective and its image L′_{α} has dimension |J|, unless χ_{1} = χ_{2}, in which case the dimension is |J| + 1. If χ_{1} ≠ χ_{2}, then the lemma is now immediate from the definition of L_{α} and, in the cyclotomic J = S case, the local Euler characteristic formula. If χ_{1} = χ_{2}, then one also needs to note the following:

• If J = S and b⃗ ≠ (ℓ, …, ℓ), then b⃗ = (ℓ − 1, …, ℓ − 1) and L_{ur} ⊂ L′_{α}, for dimension reasons if ℓ > 2 and by Remark 3.11 if ℓ = 2.

• If b⃗ = (ℓ, …, ℓ), then ℓ = 2 and J is either S or ∅. In the latter case, L′_{α} is spanned by the homomorphism G_{K_{𝔭}} → \overline{F}_{2} defined by the reduction of (ω^{2} − 1)/8, where ω is the 2-adic cyclotomic character. So in this case L_{ur} ⊄ L′_{α}.

□

If ρ|_{D} ∼ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}) and c_{ρ} is the corresponding class in H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})), we

now define

(3)  W_{𝔭}(ρ) = { V | ∃J with c_{ρ} ∈ L_{α} for α = (V, J) ∈ W′(χ_{1}, χ_{2}) }.

Note that if ρ|_{D} ∼ χ_{1} ⊕ χ_{2}, or equivalently c_{ρ} = 0, then W_{𝔭}(ρ) = π_{1}(W′(χ_{1}, χ_{2})) is independent of the choice of ordering of χ_{1} and χ_{2}. Note also in this case that |W_{𝔭}(ρ)| has size approximately 2^{f}, as in the irreducible case (1). Moreover if ρ is reducible, one knows by [7], Prop A.3 or [14], Thm. 7.8 that |W_{𝔭}(ρ)| = 2^{d} for some d ∈ {0, …, f} depending on c_{ρ}, provided χ_{1}χ_{2}^{−1} is generic in the sense that its restriction to I_{K_{𝔭}} is of the form ∏_{τ∈S} ω_{τ}^{b_{τ}} with b_{τ} ∈ {1, …, ℓ − 2} for all τ ∈ S and b⃗ ∉ {(1, …, 1), (ℓ − 2, …, ℓ − 2)}.

Finally we remark that it is shown in [21] that in the cases where ρ|_{D} is semisimple, the set W_{𝔭}(ρ) is related to the set of Jordan-Hölder constituents of the reduction of a corresponding irreducible characteristic zero representation of GL_{2}(k).

3.3. Basic properties of the definition. The set W_{𝔭}(ρ) was defined in terms of the restriction of ρ to G_{K_{𝔭}}. We now check that it is in fact non-empty and depends only on the restriction to inertia.

Proposition 3.13. If ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd, then W_{𝔭}(ρ) is non-empty and depends only on ρ|_{I_{K𝔭}}.

Proof. We first prove that W_{𝔭}(ρ) ≠ ∅. If ρ|_{G_{K𝔭}} is irreducible, then it is induced from a character ξ, and Propositions 3.1 and 3.2 show that W′(ξ) is non-empty, and hence so is W_{𝔭}(ρ) (note that if ℓ^{f} = 2, then n is not divisible by 3). If

ρ|_{G_{K𝔭}} is reducible, then it is of the form (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}), and we showed that the

projection map W′(χ_{1}, χ_{2}) → {J ⊂ S} is surjective. In particular, there is an element α = (V_{a⃗,b⃗}, S) ∈ W′(χ_{1}, χ_{2}). Moreover if χ_{1}χ_{2}^{−1} is cyclotomic, we may choose b⃗ = (ℓ, …, ℓ), so that in all cases L_{α} = H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})) by Lemma 3.12 and the local Euler characteristic formula. It follows that c_{ρ} ∈ L_{α} and V_{a⃗,b⃗} ∈ W_{𝔭}(ρ).

For the dependence only on inertia, first note that the irreducibility of ρ|_{G_{K𝔭}} is determined by ρ|_{I_{K𝔭}} (see for example section 2.4 of [24]). In the case that ρ|_{G_{K𝔭}} is irreducible, W_{𝔭}(ρ) is determined by W′(ξ), which depends only on ξ|_{I_{K𝔭}}, which in turn depends only on ρ|_{I_{K𝔭}}.

Suppose now that ρ|_{G_{K𝔭}} is reducible and ρ′ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is such that

ρ|_{I_{K𝔭}} ∼ ρ′|_{I_{K𝔭}}. Changing bases, we may assume ρ|_{G_{K𝔭}} = (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}) and ρ|_{I_{K𝔭}} =

ρ′|_{I_{K𝔭}}. Note that the function G_{K_{𝔭}}/I_{K_{𝔭}} → GL_{2}(\overline{F}_{ℓ}) defined by g ↦ ρ′(g)ρ(g)^{−1} takes values in Z(ρ(I_{K_{𝔭}})). We divide the proof into cases according to the possible centralisers.

Suppose first that χ_{1}χ_{2}^{−1} is ramified and c_{ρ} has non-trivial restriction to I_{K_{𝔭}}. In this case ρ|_{I_{K𝔭}} is indecomposable and has centraliser consisting only of the scalar

matrices. It follows that ρ′ = ψρ for some unramified character ψ : G_{K_{𝔭}} → \overline{F}_{ℓ}^{×}. Since W′(χ_{1}, χ_{2}) depends only on the restriction of χ_{1} and χ_{2} to I_{K_{𝔭}}, we have W′(ψχ_{1}, ψχ_{2}) = W′(χ_{1}, χ_{2}). Moreover the subspaces L_{α} of H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})) do not change if ρ is replaced by an unramified twist, nor does the class c_{ρ}. It follows that W_{𝔭}(ρ′) = W_{𝔭}(ρ).

Next suppose that χ_{1}χ_{2}^{−1} is ramified and c_{ρ} has trivial restriction to I_{K_{𝔭}}. Then

in fact c_{ρ} = 0, so we may assume ρ|_{G_{K𝔭}} = (\begin{matrix}χ_{1} & 0 \\ 0 & χ_{2} \end{matrix}). In this case the centraliser of

ρ(I_{K_{𝔭}}) consists of the diagonal matrices. It follows that ρ′|_{G_{K𝔭}} = (\begin{matrix}ψ_{1}χ_{1} & 0 \\ 0 & ψ_{2}χ_{2} \end{matrix}) for some unramified characters ψ_{1} and ψ_{2}. Hence c_{ρ′} = 0 and W′(ψ_{1}χ_{1}, ψ_{2}χ_{2}) = W′(χ_{1}, χ_{2}), so W_{𝔭}(ρ′) = W_{𝔭}(ρ).

Next suppose that χ_{1}χ_{2}^{−1} is unramified and c_{ρ} has non-trivial restriction to I_{K_{𝔭}}. In this case we have

Z(ρ(I_{K_{𝔭}})) = \{ (\begin{matrix}x & xy \\ 0 & x \end{matrix}) | x ∈ \overline{F}_{ℓ}^{×}, y ∈ \overline{F}_{ℓ} \} ,

so if g ∈ G_{K_{𝔭}}, then

ρ′(g) = (\begin{matrix}1 & μ(g) \\ 0 & 1 \end{matrix}) ψ(g)ρ(g)

for some unramified character ψ and cocycle μ : G_{K_{𝔭}}/I_{K_{𝔭}} → \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1}). In partic-

ular, ρ′ ∼ (\begin{matrix}χ′_{1} & ∗ \\ 0 & χ′_{2} \end{matrix}) with χ′_{1}(χ′_{2})^{−1} = χ_{1}χ_{2}^{−1}. Moreover if χ_{1} ≠ χ_{2}, then c_{ρ′} = c_{ρ}

in H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})), and if χ_{1} = χ_{2}, then c_{ρ′} − c_{ρ} ∈ L_{ur}. Since the spaces L_{α} are the same for ρ and ρ′ and contain L_{ur} if χ_{1} = χ_{2}, we conclude that W_{𝔭}(ρ′) = W_{𝔭}(ρ).

Finally suppose that χ_{1}χ_{2}^{−1} is unramified and c_{ρ} has trivial restriction to I_{K_{𝔭}}, so

ρ′(g) = ρ(g) = (\begin{matrix}χ_{1}(g) & 0 \\ 0 & χ_{1}(g) \end{matrix})

for g ∈ I_{K_{𝔭}}. Note that c_{ρ} = 0 unless χ_{1} = χ_{2}, in which case c_{ρ} ∈ L_{ur}, and similarly for c_{ρ′}. It follows that W_{𝔭}(ρ′) = π_{1}(W′(χ′_{1}, χ′_{2})) = π_{1}(W′(χ_{1}, χ_{2})) = W_{𝔭}(ρ). □

We are now ready to state the weight conjecture. Recall that W (ρ) is defined as the set of representations of ∏_{𝔭|ℓ} GL_{2}(k_{𝔭}) of the form ⊗_{\overline{F}_{ℓ}}V_{𝔭} with each V_{𝔭} ∈ W_{𝔭}(ρ). By the preceding proposition W (ρ) is non-empty and depends only on the restrictions of ρ to inertia groups at primes over ℓ.

Conjecture 3.14. If ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is modular, then

W (ρ) = { V | ρ is modular of weight V }.

We now check compatibility of the conjectural weight set with twists and determinants.

Proposition 3.15. Suppose that ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd.

(1) Let χ : G_{K} → \overline{F}_{ℓ}^{×} be such that χ|_{I_{K𝔭}} = ∏_{τ∈S_{𝔭}} ω_{τ}^{c_{τ}} for each 𝔭|ℓ. Then

V ∈ W (ρ) if and only V ⊗ V_{χ} ∈ W (χρ), where
V_{χ} = ⊗_{𝔭|ℓ} ⊗_{τ∈S_{𝔭}} det^{c_{τ}} ⊗_{τ} \overline{F}_{ℓ}
is an \overline{F}_{ℓ}-valued character of G.

(2) If V ∈ W (ρ) and V has central character ⊗_{𝔭|ℓ} ∏_{τ∈S_{𝔭}} τ^{c_{τ}}, then det ρ|_{I_{K𝔭}} =

∏_{τ∈S_{𝔭}} ω_{τ}^{c_{τ} +1} for each 𝔭|ℓ.

Proof. To prove the first assertion it suffices to show that V ∈ W_{𝔭}(ρ) if and only if V ⊗ V_{χ_{𝔭}} ∈ W_{𝔭}(χρ) where χ_{𝔭} = χ|_{G_{K𝔭}} and V_{χ_{𝔭}} = ⊗_{τ∈S_{𝔭}} det^{c_{τ}} ⊗_{τ} \overline{F}_{ℓ}. If ρ|_{G_{K𝔭}} ∼ Ind_{D′}^{D}ξ is irreducible, then (V, J) ∈ W′(ξ) if and only (V ⊗ V_{χ_{𝔭}}, J) ∈

W′(ξχ_{𝔭}), yielding the assertion in this case. If ρ|_{G_{K𝔭}} ∼ (\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}) is reducible,

then α = (V, J) ∈ W′(χ_{1}, χ_{2}) if and only α′ = (V ⊗ V_{χ_{𝔭}}, J) ∈ W′(χ_{1}χ_{𝔭}, χ_{2}χ_{𝔭}).

Moreover, since V_{a⃗,b⃗} ⊗ V_{χ_{𝔭}} is of the form V_{a⃗′,b⃗}, we see that χ_{α} = χ_{α′}, so L_{α} = L_{α′} ⊂ H^{1}(K_{𝔭}, \overline{F}_{ℓ}(χ_{1}χ_{2}^{−1})). Since also c_{ρ} = c_{χρ}, we get the assertion in this case as well.

To prove the second assertion, we can again work locally at primes 𝔭|ℓ. Writing V = ⊗_{𝔭|ℓ}V_{𝔭}, we have V_{𝔭} ∈ W_{𝔭}(ρ) for each 𝔭|ℓ. If V_{𝔭} = V_{a⃗,b⃗}, this gives det ρ|_{I_{K𝔭}} = ∏_{τ∈S_{𝔭}} ω_{τ}^{2a_{τ} +b_{τ}}. Since V_{a⃗,b⃗} has central character ∏_{τ∈S_{𝔭}} τ^{2a_{τ} +b_{τ} −1}, we have

∑_{i=0}^{f_{𝔭}−1} c_{τ_{i}}ℓ^{i} ≡ ∑_{i=0}^{f_{𝔭}−1}(2a_{τ_{i}} + b_{τ_{i}} − 1)ℓ^{i} mod (ℓ^{f_{𝔭}} − 1).

Adding ∑_{i=0}^{f_{𝔭}−1} ℓ^{i} to each side of the congruence, we deduce that ∏_{τ∈S_{𝔭}} ω_{τ}^{c_{τ} +1} = ∏_{τ∈S_{𝔭}} ω_{τ}^{2a_{τ} +b_{τ}} . □

Combining the first part of the proposition with Corollary 2.11(b), we deduce the following:

Corollary 3.16. Suppose that ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) is continuous, irreducible and totally odd and χ : G_{K} → \overline{F}_{ℓ}^{×} is a character. Then Conjecture 3.14 holds for ρ if and only if it holds for χρ.

In the case K = Q, Conjecture 3.14 follows from known results on Serre’s Conjecture (see [48], Thm. 5 and [2], §2 for similar statements along these lines).

Theorem 3.17. Conjecture 3.14 holds if K = Q.

Proof. Replacing ρ by a twist, we can assume ρ|_{I_{ℓ}} has the form (\begin{matrix}ω_{2}^{b} & 0 \\ 0 & ω_{2}^{ℓb} \end{matrix}) or

(\begin{matrix}ω^{b} & ∗ \\ 0 & 1 \end{matrix}) for some b with 1 ≤ b ≤ ℓ − 1. In the second case we write ρ|_{G_{Q_{ℓ}}} ∼

(\begin{matrix}χ_{1} & ∗ \\ 0 & χ_{2} \end{matrix}). We shall also use ω to denote the mod ℓ cyclotomic character on G_{Q}

or G_{Q_{ℓ}}.

In the first case we find that W (ρ) = {V_{0,b}, V_{b−1,ℓ+1−b}}. In the second case, we have the following possibilities:

W′(χ_{1}, χ_{2}) = \{ \begin{matrix}\{(V_{0,b}, S), (V_{b,ℓ−1−b}, ∅)\}, & if 1 < b < ℓ − 2, \\ \{(V_{0,ℓ−1}, S), (V_{0,ℓ−1}, ∅)\}, & if b = ℓ − 1 and ℓ > 2, \\ \{(V_{0,ℓ}, S), (V_{0,1}, S), (V_{1,ℓ−2}, ∅)\}, & if b = 1 and ℓ > 3, \\ \{(V_{0,ℓ−2}, S), (V_{ℓ−2,ℓ}, ∅), (V_{ℓ−2,1}, ∅)\}, & if b = ℓ − 2 and ℓ > 3, \\ \{(V_{0,ℓ}, S), (V_{0,1}, S), (V_{1,ℓ}, ∅), (V_{1,1}, ∅)\}, & if b = 1 and ℓ ≤ 3. \end{matrix}

Moreover, dimension considerations show that L_{(V,J)} = H^{1}(G_{Q_{ℓ}}, \overline{F}(χ_{1}χ_{2}^{−1})) whenever J = S, unless χ_{1}χ_{2}^{−1} = ω and V = V_{(0,1)}, in which case L_{(V,J)} consists of the peu ramifiées classes (see Remark 3.11). Considering dimensions in this case then gives that L_{(V,J)} consists precisely of the classes which are peu ramifiées. Note also that L_{(V,J)} = 0 whenever J = ∅ unless χ_{1}χ_{2}^{−1} is trivial, in which case all we need to know is that if ℓ = 2, then L_{(V_{0,1},∅)} is the set of peu ramifiées classes by a direct calculation. This gives

W (ρ) = \{ \begin{matrix}\{V_{0,b}\}, & if 1 < b < ℓ − 1 and ρ|_{G_{Q_{ℓ}}} is non-split, \\ \{V_{0,b}, V_{b,ℓ−1−b}\}, & if 1 < b < ℓ − 2 and ρ|_{G_{Q_{ℓ}}} is split, \\ \{V_{0,ℓ−2}, V_{ℓ−2,ℓ}, V_{ℓ−2,1}\}, & if b = ℓ − 2, ℓ > 3 and ρ|_{G_{Q_{ℓ}}} is split, \\ \{V_{0,ℓ−1}\}, & if b = ℓ − 1 and ℓ > 2, \\ \{V_{0,ℓ}\}, & if b = 1, χ_{1}χ_{2}^{−1} = ω and ρ|_{G_{Q_{ℓ}}} is très ramifiée, \\ \{V_{0,ℓ}, V_{0,1}, V_{1,ℓ−2}\}, & if b = 1, ℓ > 3 and ρ|_{G_{Q_{ℓ}}} is split, \\ \{V_{0,3}, V_{0,1}, V_{1,3}, V_{1,1}\}, & if b = 1, ℓ = 3 and ρ|_{G_{Q_{ℓ}}} is split, \\ \{V_{0,ℓ}, V_{0,1}\}, & otherwise. \end{matrix}

Propositions 2.5 and Corollary 2.11(2) show that ρ is modular of weight^{2} b + 1 and level prime to ℓ if and only if ω^{a}ρ is modular of weight V_{a,b}. If ρ is modular of weight 2 and level prime to ℓ, then ρ|_{G_{Q_{ℓ}}} arises from a finite flat group scheme over Z_{ℓ}, so it follows from results of Deligne and Fontaine ([24], Theorems 2.5, 2.6) and the explicit description of W (ρ) above that if ρ is modular of weight V, then V ∈ W (ρ).

To show that if V ∈ W (ρ), then ρ is modular of weight V, we combine the following four results. The first of these is a standard consequence of multiplication by the Hasse invariant (or Eisenstein series) of weight ℓ − 1.

Lemma 3.18. If ρ is modular of weight 2 and level prime to ℓ, then ρ is modular of weight ℓ + 1 and level prime to ℓ.

The theorem in the irreducible case is then a consequence of the following result of Edixhoven; see the second paragraph of [24, 4.5].

Lemma 3.19. Suppose that ρ is modular of weight b + 1 and level prime to ℓ with 2 ≤ b ≤ ℓ. If ρ|_{G_{Q_{ℓ}}} is irreducible, then ω^{1−b}ρ is modular of weight ℓ + 2 − b and level prime to ℓ.

To treat the reducible case, we first apply Mazur’s Principle [24, 2.8].

Theorem 3.20. Suppose that ρ is modular of weight ℓ + 1 and level prime to ℓ. If ρ is not modular of weight 2 and level prime to ℓ, then ρ|_{G_{Q_{ℓ}}} is a très ramifiée

representation of the form (\begin{matrix}ωχ_{2} & ∗ \\ 0 & χ_{2} \end{matrix}) for some unramified character χ_{2}.

The theorem is then a consequence of the companion forms theorem of Gross [40]^{3}:

Theorem 3.21. Suppose that ρ is modular of weight b + 1 and level prime to ℓ with 1 ≤ b ≤ ℓ − 2. If ρ|_{G_{Q_{ℓ}}} is reducible and split, then ω^{−b}ρ is modular of weight ℓ − b and level prime to ℓ.

Theorem 3.17 now follows. □

2Most of the literature on Serre’s Conjecture in the classical case uses arithmetic conventions, so for the purpose of this proof, we view ρ as “modular of weight k” if ρ ∼ \overline{ρ}_{π} for some cuspidal automorphic π with π_{∞} ≅ D_{k,k} in the notation of §2 and [13].

3Gross’s proof relies on the Hecke-equivariance of certain isomorphisms, later checked by Cais in [11]. In the meantime, other proofs were given by Coleman and Voloch [15] (but under slightly different hypotheses than we need), Faltings and Jordan [28], and Gee [35]

We end by remarking that Edixhoven’s refinement of Serre’s conjecture includes the statement that if ρ is unramified at ℓ then it should come from a mod ℓ modular form of weight 1. This refinement is not implied by our conjecture.

4. Mod ℓ Langlands correspondences

As Serre himself remarks in [63], his conjecture can be viewed as part of a “mod ℓ Langlands philosophy.” Indeed the weak conjecture can be viewed as asserting the existence of a global mod ℓ Langlands correspondence for GL_{2} /Q, and the refinement can be viewed as a local-global compatibility statement. This was made precise by Emerton in [26] as follows. Consider the representation

H = lim_{U^{→}} H_{ét}^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ})

of G_{Q} × GL_{2}(A_{f}). The weak form of Serre’s conjecture is the statement that if ρ : G_{Q} → GL_{2}(\overline{F}_{ℓ}) is continuous, odd and irreducible, then

π(ρ) := Hom_{\overline{F}_{ℓ}[G_{Q}]}(ρ, H)

is non-zero. Under some technical hypotheses on ρ|_{G_{Q_{ℓ}}}, Emerton shows that π(ρ) factors as a restricted tensor product of representations π_{p} where π_{p} is a representation of GL_{2}(Q_{p}) determined by ρ|_{G_{Qp}}. The “level part” of Serre’s refinement can then be recovered from the fact that if p ≠ ℓ and ρ|_{G_{Qp}} has Artin conductor p^{c_{p}},

then π_{p}^{U_{1}(p^{cp})} ≠ 0, and the “weight part” from the fact that

Hom_{GL_{2}(Z_{ℓ})}(det^{1−k} ⊗ Symm^{k−2} \overline{F}_{ℓ}^{2}, π_{ℓ}) ≠ 0

where k = k(ρ) ≥ 2 is the weight of ρ|_{G_{Q_{ℓ}}} as defined by Serre in [63]. In this section we formulate a conjectural extension of Emerton’s refinement to our setting, namely that of mod ℓ representations arising from a quaternion algebra D over a totally real field K.

Suppose now that

ρ : G_{K} → GL_{2}(\overline{F}_{ℓ})

is continuous, totally odd and irreducible. We shall associate to ρ a smooth representation π^{D}(ρ) of (D ⊗ Ẑ)^{×} over \overline{F}_{ℓ} and give a conjectural description for it as a product of local factors. Thus for each prime 𝔭 of K we would like to associate to ρ a smooth admissible representation of D_{𝔭}^{×} defined over \overline{F}_{ℓ}, ideally depending only on ρ|_{G_{K𝔭}}. We are able to achieve this for primes 𝔭 not dividing ℓ; indeed this was already done by Emerton if D is split at 𝔭 and the main new ingredient of this section is to treat the case where D_{𝔭} is a quaternion algebra. For 𝔭|ℓ, we are not yet able to give a conjectural description of the local factor, but the weight conjecture formulated in the preceding section can be interpreted as a description of the Jordan-Hölder factors of its socle under a maximal compact subgroup of D_{𝔭}^{×}. (We are grateful to Breuil and Emerton for this observation.)

We begin by recalling Emerton’s formulation of a mod ℓ local Langlands correspondence for GL_{2} over K_{𝔭} for 𝔭 not dividing ℓ; see [27]. Emerton’s construction is a modification of one provided by Vignéras in [70], on whose results it relies, a key difference being that [27] involves reducible representations of GL_{2}(K_{𝔭}) in order to prove local-global compatibility.

Fix for now a prime 𝔭 not dividing ℓ and let q = N(𝔭) = #(O_{K}/𝔭). For a continuous representation ρ̃ : G_{K_{𝔭}} → GL_{2}(\overline{Q}_{ℓ}), we let π(ρ̃) denote its local Langlands correspondent as modified in [25] or [27]. More precisely, π(ρ̃) is the usual^{4} irreducible admissible \overline{Q}_{ℓ}-representation of GL_{2}(K_{𝔭}) provided by the local Langlands correspondence, unless ρ̃ is the sum of two characters whose ratio is cyclotomic in which case π(ρ̃) is a generic representation of length two.

Theorem 4.1. (Emerton) There is a map ρ ↦ π(ρ) from the set of isomorphism classes of continuous representations G_{K_{𝔭}} → GL_{2}(\overline{F}_{ℓ}) to the set of isomorphism classes of finite-length smooth admissible \overline{F}_{ℓ}-representations of G = GL_{2}(K_{𝔭}), uniquely determined by the following properties:

(1) The representation π(ρ) has a unique irreducible subrepresentation π′, and π(ρ)/π′ is finite-dimensional.

(2) If ρ̃: G_{K_{𝔭}} → GL_{2}(\overline{Q}_{ℓ}) is a continuous lift of ρ, then there is a G-equivariant \overline{Z}_{ℓ}-lattice in π(ρ̃) whose reduction admits a G-equivariant embedding into π(ρ).

(3) If π is a finite-length smooth admissible \overline{F}_{ℓ}-representation of G satisfying conditions (1) and (2), then there exists a G-equivariant embedding π ↪ π(ρ).

If D is split at 𝔭, we let π^{D_{𝔭}}(ρ) denote the representation of D_{𝔭}^{×} ≅ GL_{2}(K_{𝔭}) given by the theorem.

We would like an analogue of the theorem which associates to ρ a representation of D_{𝔭}^{×} when D_{𝔭} is a non-split quaternion algebra over K_{𝔭}. Again our construction is a modification of one already provided by Vignéras (this time in [69]) on whose results we rely. Since the irreducible smooth admissible \overline{F}_{ℓ}-representations of D_{𝔭}^{×} are finite-dimensional, one might expect a result like Theorem 4.1 but without condition (1). However results of Ribet [59] show this naive generalisation would not satisfy local-global compatibility, and there are further complications when ℓ = 2. On the other hand we can give a more explicit description of the desired representation π^{D_{𝔭}}(ρ).

Recall from Theorem 15.1 of [42] that the (local) Jacquet-Langlands correspondence establishes a bijection between isomorphism classes of irreducible admissible square-integrable \overline{Q}_{ℓ}-representations of GL_{2}(K_{𝔭}) and irreducible admissible \overline{Q}_{ℓ}- representations of D_{𝔭}^{×}. If ρ̃: G_{K_{𝔭}} → GL_{2}(\overline{Q}_{ℓ}) is a continuous representation, then π(ρ̃) is square-integrable if and only if ρ̃ is either irreducible or isomorphic to a twist of the non-split representation of the form ( \begin{matrix}1 & ∗ \\ 0 & ω^{−1} \end{matrix} ) where ω is the ℓ-adic cyclotomic character, and then we let π^{D_{𝔭}}(ρ̃) the representation of D_{𝔭}^{×} corresponding to π(ρ̃) via Jacquet-Langlands.

Now consider again a continuous representation ρ : G_{K_{𝔭}} → GL_{2}(\overline{F}_{ℓ}). It is straightforward to check that there exist continuous representations ρ̃ : G_{K_{𝔭}} → GL_{2}(\overline{Q}_{ℓ}) containing G_{K_{𝔭}}-stable lattices whose reduction is ρ. Moreover, if ρ̃ is irreducible, then so is ρ except in the following case: if q ≡ −1 mod ℓ, L is the quadratic unramified extension of K_{𝔭} and ρ̃ ∼ χ ⊗ Ind_{G_{L}}^{G_{K𝔭}} ξ for some characters χ of G_{K_{𝔭}} and ξ of G_{L} such that ξ is different from its Gal(L/K_{𝔭})-conjugate and has ℓ-power order, then ρ has semi-simplification isomorphic to \overline{χ}⊕\overline{χ}\overline{ω}^{−1} where \overline{ω} is the mod ℓ cyclotomic character. (Note that if q ≡ −1 mod ℓ, then \overline{ω} is the quadratic unramified character of G_{K_{𝔭}}, unless ℓ = 2 in which case \overline{ω} = 1.)

4Recall that we follow Carayol’s conventions in [13] with regard to the local Langlands correspondence; these differ from Emerton’s, but we make the same modification as in [25, 27].

For representations of D_{𝔭}^{×} we have the following result of Vignéras (Propositions 9 and 11 and Corollary 12 of [69]):

Proposition 4.2. (Vignéras) Suppose that ρ : G_{K_{𝔭}} → GL_{2}(\overline{F}_{ℓ}) is continuous and irreducible. If ρ̃ : G_{K_{𝔭}} → GL_{2}(\overline{Q}_{ℓ}) is a lift of ρ, then the reduction of π^{D_{𝔭}}(ρ̃) is irreducible and depends only on ρ.

If ρ : G_{K_{𝔭}} → GL_{2}(\overline{F}_{ℓ}) is irreducible, we define π^{D_{𝔭}}(ρ) to be the reduction of π^{D_{𝔭}}(ρ̃) for any lift ρ̃ of ρ; this is well-defined by the proposition.

Suppose now that ρ is reducible. If ρ is not a twist of a representation of the form ( \begin{matrix}1 & ∗ \\ 0 & \overline{ω}^{−1} \end{matrix} ) (where \overline{ω} is the mod ℓ cyclotomic character), then ρ has no lifts ρ̃ such that π(ρ̃) is square-integrable, and we define π^{D_{𝔭}}(ρ) = 0.

Suppose now that ρ ∼ ( \begin{matrix}χ & ∗ \\ 0 & χ\overline{ω}^{−1} \end{matrix} ) for some character χ : G_{K_{𝔭}} → \overline{F}_{ℓ}^{×} (i.e., ρ is any extension of χ\overline{ω}^{−1} by χ). If q ≢ −1 mod ℓ, then the only lifts ρ̃ of ρ for which π(ρ̃) is square-integrable are non-split representations of the form ( \begin{matrix}χ̃ & ∗ \\ 0 & χ̃ω^{−1} \end{matrix} ) where χ̃ : G_{K_{𝔭}} → O_{K_{𝔭}}^{×} lifts χ. In this case π^{D_{𝔭}}(ρ̃) = χ̃^{−1} ∘ det where det : D_{𝔭}^{×} → K_{𝔭}^{×} is the reduced norm (using χ̃ and χ also to denote the characters of K_{𝔭}^{×} to which they correspond via class field theory). We then define π^{D_{𝔭}}(ρ) to be χ^{−1} ∘ det, unless q ≡ 1 mod ℓ and ρ is the split representation χ ⊕ χ in which case we define π^{D_{𝔭}}(ρ) to be (χ^{−1} ∘ det) ⊕ (χ^{−1} ∘ det) (note that \overline{ω} is trivial).

Remark 4.3. Note that the reduction of π^{D_{𝔭}}(ρ̃) is χ^{−1} ∘ det, which then coincides with π^{D_{𝔭}}(ρ) unless we are in the exceptional case where q ≡ 1 mod ℓ and ρ is scalar. In this case our definition of π^{D_{𝔭}}(ρ) is motivated by results of Ribet [59] and Yang [73] on multiplicities of Galois representations in the cohomology of Shimura curves.

Suppose now that ρ ∼ ( \begin{matrix}χ & ∗ \\ 0 & χ\overline{ω}^{−1} \end{matrix} ) for some character χ : G_{K_{𝔭}} → \overline{F}_{ℓ}^{×} and that q ≡ −1 mod ℓ. Let c_{ρ} denote the extension class associated to ρ in

Ext_{\overline{F}_{ℓ}[G_{K𝔭}]}^{1}(χ\overline{ω}^{−1}, χ) ≅ H^{1}(G_{K_{𝔭}}, \overline{F}_{ℓ}(\overline{ω})).

Note that this space is 1-dimensional unless ℓ = 2 in which case it is 2-dimensional. Recall that such ρ have irreducible lifts ρ̃ which are twists of tamely ramified representations induced from G_{L} where L is the unramified quadratic extension of K_{𝔭}. For such ρ̃, π^{D_{𝔭}}(ρ̃) is a two-dimensional representation of D_{𝔭}^{×} whose reduction (which depends on a choice of lattice) has semi-simplification χ^{−1} ∘det ⊕(χ^{−1}\overline{ω}^{−1})∘ det (see [69]). We will define π^{D_{𝔭}}(ρ) as a certain extension of (χ^{−1}\overline{ω}^{−1}) ∘ det by χ^{−1} ∘ det depending on c_{ρ}. To this end we will first compute

Ext_{\overline{F}_{ℓ}[D_{𝔭}^{×}]}^{1}((χ^{−1}\overline{ω}^{−1}) ∘ det, χ^{−1} ∘ det) ≅ H^{1}(D_{𝔭}^{×}, \overline{F}_{ℓ}(\overline{ω} ∘ det)).

Although a unified treatment is possible (see Remark 4.4), it is simpler to consider separately the cases ℓ > 2 and ℓ = 2.

Suppose first that ℓ > 2. Let O_{D_{𝔭}} denote the maximal order in D_{𝔭} and Π a uniformizer, so val_{K_{𝔭}}(det Π) = 1 and O_{D_{𝔭}}/ΠO_{D_{𝔭}} ≅ F_{q^{2}}. Letting Γ = D_{𝔭}^{×}/(1 + ΠO_{D_{𝔭}}), we have an exact sequence

1 → F_{q^{2}}^{×} → Γ → Z → 0

where the map Γ → Z is val∘det and n ∈ Z acts on F_{q^{2}}^{×} by x ↦ x^{q^{n}}. Note that F_{q^{2}}^{×} acts trivially on \overline{F}_{ℓ}(\overline{ω} ∘ det) and the induced action of n ∈ Z is via the character μ(n) = (−1)^{n} = q^{n}. Since 1 + ΠO_{D_{𝔭}} is pro-p, we have that

H^{1}(D_{𝔭}^{×}, \overline{F}_{ℓ}(\overline{ω} ∘ det)) ≅ H^{1}(Γ, \overline{F}_{ℓ}(\overline{ω} ∘ det)).

Since H^{1}(Z, \overline{F}_{ℓ}(μ)) = H^{2}(Z, \overline{F}_{ℓ}(μ)) = 0, we have that

H^{1}(Γ, \overline{F}_{ℓ}(\overline{ω} ∘ det)) = Hom_{Z}(F_{q^{2}}^{×}, \overline{F}_{ℓ}(μ))

is one-dimensional. Hence there is a unique isomorphism class of \overline{F}_{ℓ}-representations of D_{𝔭}^{×} which are non-trivial extensions of (χ^{−1}\overline{ω}^{−1}) ∘ det by χ^{−1} ∘ det. We define π^{D_{𝔭}}(ρ) to be the extension which is split if and only if c_{ρ} is trivial.

Finally consider the case ℓ = 2. Then \overline{ω} is trivial and

H^{1}(D_{𝔭}^{×}, \overline{F}_{2}) = Hom(D_{𝔭}^{×}/(D_{𝔭}^{×})^{2}, \overline{F}_{2}),

and one checks easily that det induces an isomorphism

D_{𝔭}^{×}/(D_{𝔭}^{×})^{2}⟶^{∼}K_{𝔭}^{×}/(K_{𝔭}^{×})^{2}.

On the other hand local class field theory yields an isomorphism

H^{1}(G_{K_{𝔭}}, \overline{F}_{2}) ≅ Hom(K_{𝔭}^{×}/(K_{𝔭}^{×})^{2}, \overline{F}_{2}).

Putting these isomorphisms together yields

H^{1}(G_{K_{𝔭}}, \overline{F}_{2})⟶^{∼}H^{1}(D_{𝔭}^{×}, \overline{F}_{2}),

and we define π^{D_{𝔭}}(ρ) to be the extension obtained from the image of c_{ρ}.

Remark 4.4. To give a unified treatment for the cases ℓ = 2 and ℓ > 2 when q ≡ −1 mod ℓ and ρ as above, embed the unramified quadratic extension L of K_{𝔭} in D_{𝔭} and let N denote the normaliser of the image of L^{×} in D_{𝔭}^{×}. One can then check that restriction induces an isomorphism

H^{1}(D_{𝔭}^{×}, \overline{F}_{ℓ}(\overline{ω} ∘ det)) ≅ H^{1}(N, \overline{F}_{ℓ}(μ))

where μ(x) = 1 if x ∈ L^{×} and μ(x) = −1 otherwise. On the other hand one finds that N is isomorphic by local class field theory to the image of the Weil group of K_{𝔭} in Gal(L^{ab}/K_{𝔭}) and that the inflation and restriction maps induce isomorphisms

H^{1}(G_{K_{𝔭}}, \overline{F}_{ℓ}(\overline{ω})) ≅ H^{1}(Gal(L^{ab}/K_{𝔭}), \overline{F}_{ℓ}(\overline{ω})) ≅ H^{1}(N, \overline{F}_{ℓ}(μ)).

Remark 4.5. It is straightforward to check that if q ≡ −1 mod ℓ and ξ is a character of G_{L} of ℓ-power order and not equal to its Galois conjugate, then every representation ρ with semisimplification 1 ⊕ \overline{ω} is isomorphic to the reduction of a G_{K_{𝔭}}-stable lattice in Ind_{G_{L}}^{G_{K𝔭}} ξ. It follows that if ℓ = 2, then every representation with scalar semisimplification has the same set of lifts ρ̃ such that π(ρ̃) is squareintegrable, so π^{D_{𝔭}}(ρ) is not characterised by the set of π^{D_{𝔭}}(ρ̃). (Note that this is also the case for trivial reasons if q ≡ 1 mod ℓ and ℓ > 2.)

We now return to the global setting of a totally odd, continuous, irreducible

ρ : G_{K} → GL_{2}(\overline{F}_{ℓ})

and construct the \overline{F}_{ℓ}-representation of D_{f}^{×} = (D ⊗ Ẑ)^{×} whose local factors should be the π^{D_{𝔭}}(ρ). We first consider the case of a totally definite quaternion algebra D over K. Fix a maximal order O_{D} in D and isomorphisms O_{D,𝔭} ≅ M_{2}(O_{K,𝔭}) for each prime 𝔭 of K at which D is split.

For each open compact subgroup U of D_{f}^{×} we define

S^{D}(U) = \{ f : D^{×}\D_{f}^{×}/U → \overline{F}_{ℓ} \}.

The obvious projection maps for V ⊂ U induce inclusions S^{D}(U) → S^{D}(V) and we define S^{D} as the direct limit of the S^{D}(U). Thus S^{D} can equivalently be defined as the set of smooth functions f : D^{×}\D_{f}^{×} → \overline{F}_{ℓ}. Note that this \overline{F}_{ℓ}-vector space admits a natural left action of D_{f}^{×} by right translation, and S^{D}(U) = (S^{D})^{U}, the Uinvariant functions in S^{D}. Moreover for any g ∈ D_{f}^{×} and open compact U, V ⊂ D_{f}^{×} we have the double coset operator [UgV] : S^{D}(V) → S^{D}(U) defined in the usual way. In particular, for each prime 𝔭 at which D is split and GL_{2}(O_{K,𝔭}) ⊂ U, we have the endomorphisms

T_{𝔭} = [U( \begin{matrix}ϖ_{𝔭} & 0 \\ 0 & 1 \end{matrix} )U] and S_{𝔭} = [U( \begin{matrix}ϖ_{𝔭} & 0 \\ 0 & ϖ_{𝔭} \end{matrix} )U]

of S^{D}(U), where ϖ_{𝔭} is any uniformizer of O_{K,𝔭}. If Σ is a finite set of primes of K containing all those such that:

• D is ramified at 𝔭,

• GL_{2}(O_{K,𝔭}) ⊄ U,

• ρ is ramified at 𝔭, or

• 𝔭 divides ℓ, then we let T^{Σ}(U) denote the commutative \overline{F}_{ℓ}-subalgebra of End_{\overline{F}_{ℓ}}(S^{D}(U)) generated by the S_{𝔭} and T_{𝔭} for 𝔭 ∉ Σ. We let 𝔪_{ρ}^{Σ} = 𝔪_{ρ}^{Σ}(U) denote the ideal of T^{Σ}(U) generated by the operators

T_{𝔭} − S_{𝔭}tr(ρ(Frob_{𝔭})) and N(𝔭) − S_{𝔭} det(ρ(Frob_{𝔭}))

for all 𝔭 ∉ Σ. We let

S^{D}(U)[𝔪_{ρ}^{Σ}] = { f ∈ S^{D}(U) | T f = 0 for all T ∈ 𝔪_{ρ}^{Σ} }.

If ρ = \overline{ρ}_{π} for some (necessarily cuspidal) automorphic representation π of D^{×} with weight (2, 0) and π^{U} ≠ 0, then 𝔪_{ρ}^{Σ} is a maximal ideal of T^{Σ}(U) and S^{D}(U)[𝔪_{ρ}^{Σ}] ≠ 0; otherwise 𝔪_{ρ}^{Σ} = T^{Σ}(U) and S^{D}(U)[𝔪_{ρ}^{Σ}] = 0.

Lemma 4.6. Suppose that D, U, ρ and Σ are as above. Then

a) S^{D}(U)[𝔪_{ρ}^{Σ}] is independent of Σ (so we will denote it S^{D}(U)[𝔪_{ρ}]);

b) if g ∈ D_{f}^{×} and V is an open compact subgroup of D_{f}^{×} such that V ⊂ gUg^{−1}, then g sends S^{D}(U)[𝔪_{ρ}] to S^{D}(V)[𝔪_{ρ}].

Proof. a) We may assume Σ′ = Σ ∪ {𝔭} for some 𝔭 ∉ Σ and that 𝔪_{ρ}^{Σ′} ≠ T^{Σ′}(U). Since ρ is irreducible, a standard argument using the representations ρ_{π} lifting ρ gives a representation

ρ′ : G_{K} → GL_{2}(T^{Σ′}(U)_{𝔪_{ρ}^{Σ′}})

lifting ρ such that

T_{𝔭} = S_{𝔭}tr(ρ′(Frob_{𝔭})) and N(𝔭) = S_{𝔭} det(ρ′(Frob_{𝔭}))

as endomorphisms of S^{D}(U)_{𝔪_{ρ}^{Σ′}}. It follows that S^{D}(U)[𝔪_{ρ}^{Σ}] = S^{D}(U)[𝔪_{ρ}^{Σ′}].

b) Choosing Σ sufficiently large that GL_{2}(O_{K,𝔭}) ⊂ V and g_{𝔭} ∈ GL_{2}(O_{K,𝔭}) for all 𝔭 ∉ Σ, we see that g commutes with T_{𝔭} for all 𝔭 ∉ Σ. □

We can now consider the direct limit over U of the spaces S^{D}(U)[𝔪_{ρ}]; by the lemma, this makes sense and yields a representation S^{D}[𝔪_{ρ}] of D_{f}^{×}. We can now state an analogue of Emerton’s local-global compatibility conjecture [26].

Conjecture 4.7. Suppose that K is a totally real field,

ρ : G_{K} → GL_{2}(\overline{F}_{ℓ})

is a continuous, irreducible and totally odd representation, and D is a totally definite quaternion algebra over K. Then the \overline{F}_{ℓ}-representation S^{D}[𝔪_{ρ}] of D_{f}^{×} is isomorphic to a restricted tensor product ⊗′_{𝔭}π_{𝔭} where π_{𝔭} is a smooth admissible representation of D_{𝔭}^{×} such that

• if 𝔭 does not divide ℓ, then π_{𝔭} ≅ π^{D_{𝔭}}(ρ|_{G_{K𝔭}});

• if 𝔭|ℓ then π_{𝔭} ≠ 0; moreover if K and D are unramified at 𝔭, and σ is an irreducible \overline{F}_{ℓ}-representation of GL_{2}(O_{K,𝔭}), then Hom_{GL_{2}(O_{K,𝔭})}(σ, π_{𝔭}) ≠ 0 if and only if σ ∈ W_{𝔭}(ρ^{∨}|_{G_{K𝔭}}).

Remark 4.8. In the case K = Q and D = M_{2}(Q) and ρ|_{G_{Q_{ℓ}}} is not a twist of a representation of the form ( \begin{matrix}1 & ∗ \\ 0 & \overline{ω} \end{matrix} ), then Emerton predicts the precise form for π_{ℓ} in [26] as well and goes on to prove his conjecture under technical hypotheses (including the assumption that ℓ > 2). It is reasonable to expect that π_{𝔭} is of the form predicted there whenever D_{𝔭} ≅ M_{2}(Q_{ℓ}) and ρ|_{G_{K𝔭}} is as above.

Under the hypotheses that K_{𝔭} is an unramified extension of Q_{ℓ} and D is split at 𝔭, the relation with W_{𝔭} can be viewed as a description of the GL_{2}(O_{K,𝔭})-socle of π_{𝔭} (which in most cases is expected to be multiplicity-free). In many cases, Breuil and Paskunas [9] construct infinitely many (isomorphism classes of) representations with the desired socle. These representations are irreducible whenever ρ|_{G_{K𝔭}} is, and are expected to have finite length in general. However, the multitude of possibilities raises the question of whether one should still expect π_{𝔭} to be completely determined by ρ|_{G_{K𝔭}}.

Some work has also been done towards defining a conjectural set of weights W_{𝔭} when F_{𝔭} is a ramified extension of Q_{ℓ} and D_{𝔭} is split. In particular, Schein [62] gives a definition of W_{𝔭} when ρ|_{G_{K𝔭}} is tamely ramified, and Gee [37] gives a more general but less explicit definition than ours.

In the case where D_{𝔭} is not split, Gee and Savitt [39] have described the set of weights, again in a less explicit form in general. We remark however that one can show in this case that π_{𝔭}, if it exists, has infinite length.

Suppose now that D is split at exactly one infinite place. We exclude the case D = M_{2}(Q) already considered by Emerton. We now define S^{D}(U) = H^{1}(Y_{U,K̄}, \overline{F}_{ℓ}) and S^{D} = lim S^{D}(U), the limit taken over all open compact U ⊂ D_{f}^{×} with respect to the maps on cohomology induced by the natural projections Y_{V} → Y_{U} for U ⊂ V. If V ⊂ gUg^{−1}, then we have a K-morphism Y_{V} → Y_{U} corresponding to right multiplication by g on complex points, inducing a homomorphism S^{D}(U) → S^{D}(V) which we also denote by g, making S^{D} a G_{K} × D_{f}^{×}-module. However the natural map S^{D}(U) → (S^{D})^{U} is not necessarily an isomorphism.

For g ∈ D_{f}^{×}, and U,V open compact subgroups of D_{f}^{×}, we have the double coset operator [V gU] : S^{D}(U) → S^{D}(V) defined as the composite S^{D}(U) → S^{D}(V′) → S^{D}(V) where V′ = V ∩ gUg^{−1}, the first map is defined by g, and the second is the trace morphism times the integer [V : V′]/ deg(Y_{V′}/Y_{V}). We can thus define Hecke operators T_{𝔭} for 𝔭 ∉ Σ, algebras T^{Σ}(U) ⊂ End_{\overline{F}_{ℓ}}(S^{D}(U)), ideals 𝔪_{ρ}^{Σ} and subspaces S^{D}(U)[𝔪_{ρ}^{Σ}] ⊂ S^{D}(U), just as in the case of totally definite D. The analogue of Lemma 4.6 is proved in exactly the same way, now yielding a representation S^{D}[𝔪_{ρ}] of G_{K} × D_{f}^{×}.

Conjecture 4.9. Suppose that K is a totally real field,

ρ : G_{K} → GL_{2}(\overline{F}_{ℓ})

is a continuous, irreducible and totally odd representation, and D is a quaternion algebra over K split at exactly one real place. Then the \overline{F}_{ℓ}-representation S^{D}[𝔪_{ρ}] of G_{K}×D_{f}^{×} is isomorphic to ρ⊗(⊗′_{𝔭}π_{𝔭}) where π_{𝔭} is a smooth admissible representation of D_{𝔭}^{×} such that

• if 𝔭 does not divide ℓ, then π_{𝔭} ≅ π^{D_{𝔭}}(ρ|_{G_{K𝔭}});

• if 𝔭|ℓ then π_{𝔭} ≠ 0; moreover if K and D are unramified at 𝔭, and σ is an irreducible \overline{F}_{ℓ}-representation of GL_{2}(O_{K,𝔭}), then Hom_{GL_{2}(O_{K,𝔭})}(σ, π_{𝔭}) ≠ 0 if and only if σ ∈ W_{𝔭}(ρ^{∨}|_{G_{K𝔭}}).

By the following lemma, Conjecture 4.9 could be reformulated as saying that the representation Hom_{\overline{F}_{ℓ}[G_{K}]}(ρ, S^{D}) of D_{f}^{×} has the prescribed form, as in [26].

Lemma 4.10. The evaluation map ρ⊗_{\overline{F}_{ℓ}} Hom_{\overline{F}_{ℓ}[G_{K}]}(ρ, S^{D}) → S^{D} induces a G_{K} × D_{f}^{×}-linear isomorphism:

ρ ⊗_{\overline{F}_{ℓ}} Hom_{\overline{F}_{ℓ}[G_{K}]}(ρ, S^{D})⟶^{∼}S^{D}[𝔪_{ρ}].

Proof. It suffices to prove the lemma with S^{D} replaced by S^{D}(U) and take direct limits. Since ρ is irreducible, the evaluation map

ρ ⊗_{\overline{F}_{ℓ}} Hom_{\overline{F}_{ℓ}[G_{K}]}(ρ, S^{D}(U)) → S^{D}(U)

is injective by Schur’s Lemma. Using the Eichler-Shimura relations on Y_{U} (in particular, that Frob_{𝔭}^{2} −S_{𝔭}^{−1}T_{𝔭} Frob_{𝔭} +N(𝔭)S_{𝔭}^{−1} = 0 on H^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) for all 𝔭 ∉ Σ), one shows as in the proof of Prop. 3.2.3 of [8] that the image lies in S^{D}(U)[𝔪_{ρ}]. Finally, the main result of [6] shows that S^{D}(U)[𝔪^{ρ}] is isomorphic to a direct sum of copies of ρ, hence the map is surjective. □

Next we show how one can usually recover S^{D}(U)[𝔪_{ρ}] from S^{D}[𝔪_{ρ}]. The caveat (an observation going back to Ribet) is that it is not quite true that “all errors are Eisenstein”. Let us say that a representation ρ is badly dihedral if ρ is induced from a character of G_{K′} where K′ is a totally imaginary quadratic extension of K of the form K(δ) for some δ such that δ^{ℓ} ∈ K. For ℓ > 2 it is not difficult to check that if ρ is badly dihedral then K must contain the maximal real subfield Q(μ_{ℓ})^{+} of Q(μ_{ℓ}) and that K′ = K(μ_{ℓ}) (because the field K′ contains ζ := \overline{δ}/δ with \overline{δ} the Galois conjugate of δ, and ζ ≠ 1 is an ℓth root of unity). In particular if ℓ > 3 and ℓ is unramified in K, then there will be no badly dihedral representations at all. However for ℓ = 2 there may be more than one possibility for K′ (but only finitely many).

Lemma 4.11. The natural map S^{D}(U)[𝔪_{ρ}] → (S^{D}[𝔪_{ρ}])^{U} is injective; moreover it is an isomorphism if either

• Y_{U} has no elliptic points of order a multiple of ℓ or

• ρ is not badly dihedral.

Proof. It suffices to show that if V is any normal open compact subgroup of U then the natural map

H^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) → H^{1}(Y_{V,\overline{K}}, \overline{F}_{ℓ})^{U/V}

is injective after localising at 𝔪_{ρ}, and is an isomorphism under the additional hypotheses. Equivalently we must show that ρ does not appear in the \overline{F}_{ℓ}[G_{K}]- semisimplification of the kernel of the above map, and under the additional hypotheses it does not appear in the cokernel either.

Let Z_{U} denote the reduced closed subscheme of Y_{U} defined by its elliptic points and let Z_{V} = (Z_{U} ×_{Y_{U}} Y_{V})^{red}, W_{U} = Y_{U} − Z_{U} and W_{V} = Y_{V} − Z_{V}. Then the morphism W_{V} → W_{U} is étale with Galois group Γ (a quotient of U/V), and the Hochschild-Serre spectral sequence yields an exact sequence:

\begin{aligned}0 → H^{1}(Γ, H^{0}(W_{V,\overline{K}}, \overline{F}_{ℓ})) & → H^{1}(W_{U,\overline{K}}, \overline{F}_{ℓ}) \\ & → H^{1}(W_{V,\overline{K}}, \overline{F}_{ℓ})^{U} → H^{2}(Γ, H^{0}(W_{V,\overline{K}}, \overline{F}_{ℓ})). \end{aligned}

The inclusion W_{U} → Y_{U} yields an exact sequence:

\begin{matrix}H_{Z_{U,\overline{K}}}^{1} (Y_{U,\overline{K}}, \overline{F}_{ℓ}) → & H^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) \\ → & H^{1}(W_{U,\overline{K}}, \overline{F}_{ℓ}) → H_{Z_{U,\overline{K}}}^{2} (Y_{U,\overline{K}}, \overline{F}_{ℓ}) \end{matrix}

By the excision theorem, H_{Z_{U,\overline{K}}}^{i} (Y_{U,\overline{K}}, \overline{F}_{ℓ}) = ⨁_{z∈Z_{U} (K̄)} H_{\{z\}}^{i}(Y_{U,\overline{K}}, \overline{F}_{ℓ}). By the

Betti-étale comparison theorem for example, we see that each H_{\{z\}}^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) = 0 and that each H_{\{z\}}^{2}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) is one-dimensional; moreover if ψ : X′ → X is a non-constant morphism of smooth proper curves over \overline{K} with ψ(x′) = x, then the induced map H_{\{x\}}^{2}(X, \overline{F}_{ℓ}) → H_{\{x′\}}^{2}(X′, \overline{F}_{ℓ}) is trivial if the ramification degree e(x′/x) is divisible by ℓ and it is an isomorphism otherwise. In particular, if z ∈ Z_{U}(\overline{K}) is defined over L, then the morphism Y_{U,L} → P_{L}^{1} gotten from a uniformizer at z induces an isomorphism

\overline{F}_{ℓ}(1) = H_{\{0\}}^{2}(P1 ; \overline{L}, \overline{F}_{ℓ})⟶^{∼}H_{\{z\}}^{2}(Y_{U,\overline{L}}, \overline{F}_{ℓ})

of G_{L}-modules. It follows that H_{Z_{U,\overline{K}}}^{2} (Y_{U,\overline{K}}, \overline{F}_{ℓ}) ≅ ⨁_{P ∈Z_{U}} Ind_{G_{K(P)}}^{G_{K}} \overline{F}_{ℓ}(1) as G_{K}- modules. Combining this with the corresponding exact sequence arising from W_{V} → Y_{V} yields a commutative diagram

0 → H^{1}(Y_{U,\overline{K}}, \overline{F}_{ℓ}) → H^{1}(W_{U,\overline{K}}, \overline{F}_{ℓ}) → H_{Z_{U,\overline{K}}}^{2} (Y_{U,\overline{K}}, \overline{F}_{ℓ})
↓ ↓ ↓
0 → H^{1}(Y_{V,\overline{K}}, \overline{F}_{ℓ})^{Γ} → H^{1}(W_{V,\overline{K}}, \overline{F}_{ℓ})^{Γ} → H_{Z_{V,\overline{K}}}^{2} (Y_{V,\overline{K}}, \overline{F}_{ℓ})^{Γ}

such that the kernel of the rightmost vertical map is isomorphic to the direct sum of the Ind_{G_{K(P)}}^{G_{K}} \overline{F}_{ℓ}(1) over the P ∈ Y_{U} whose ramification degree in Y_{V} is divisible by ℓ. If P is such an elliptic point, then it is fixed by some δ ∈ D such that δ has order ℓ in D^{×}/K^{×}. The extension K′ := K[δ] is a commutative integral domain within D, and it is finite over K and hence a field; moreover it must be a quadratic extension of K, imaginary at our preferred infinite place τ_{0} since δ has isolated fixed points in ℌ^{±}, and at the other infinite places since K′ ⊂ D and hence K′ splits D. The elliptic point will then be a special point for the Shimura curve Y_{U} with respect to the torus Res_{K′/Q}(G_{m}) and by Lemma 3.11 of [17] the elliptic point will be defined over an abelian extension of K′. Now under the additional hypotheses of the lemma it follows that ρ does not appear in the \overline{F}_{ℓ}[G_{K}]-semisimplification of the direct sum of the Ind_{G_{K(P)}}^{G_{K}} \overline{F}_{ℓ}(1) as above.

Recall from Lemma 2.2 that the action of G_{K} on H^{0}(W_{V,\overline{K}}, \overline{F}_{ℓ}) = H^{0}(Y_{V,\overline{K}}, \overline{F}_{ℓ}), hence on the kernel and cokernel of the middle vertical map, factors through an abelian quotient. Finally we deduce from the snake lemma that ρ does not appear in the semisimplification of the kernel of the leftmost vertical map, nor that of the cokernel under the additional hypotheses. □

Finally we record some consequences of Conjectures 4.7 and 4.9. (Recall that Conjecture 4.9 is formulated for an arbitrary totally real K, but that Conjecture 3.14 assumes K is unramified at ℓ.)

Proposition 4.12. Conjecture 4.9 implies Conjecture 3.14.

Proof. Since the conjecture is known for Q, we can assume K ≠ Q. Now ρ is modular of weight σ if and only if ρ(−1) is isomorphic to an \overline{F}_{ℓ}[G_{K}]-subquotient of Hom_{\overline{F}_{ℓ}[U]}(σ^{∨}, H^{1}(Y_{U′,\overline{K}}, \overline{F}_{ℓ})) for some D and U as in Definition 2.1. This is equivalent to saying that 𝔪_{ρ(−1)}^{Σ} (for any Σ at level U′) is (maximal and) in the support of Hom_{\overline{F}_{ℓ}[U]}(σ^{∨}, S^{D}(U′)), or equivalently that

Hom_{\overline{F}_{ℓ}[U]}(σ^{∨}, S^{D}(U′)[𝔪_{ρ(−1)}]) ≠ 0.

(Note in particular that by the proof of Lemma 4.10, “\overline{F}_{ℓ}G_{K}-subquotient” can be replaced with “\overline{F}_{ℓ}G_{K}-submodule” as claimed after Definition 2.1.)

So if ρ is modular of an irreducible weight σ, then Hom_{\overline{F}_{ℓ}[U]}(σ^{∨}, S^{D}[𝔪_{ρ(−1)}]) ≠ 0 by Lemma 4.11. If Conjecture 4.9 holds, then we may write S^{D}[𝔪_{ρ(−1)}] = ρ ⊗ (⊗′π_{𝔭}); moreover for each 𝔭|ℓ, we have Hom_{\overline{F}_{ℓ}[U_{𝔭}]}(σ_{𝔭}^{∨}, π_{𝔭}) ≠ 0 so that σ_{𝔭}^{∨} ∈ W_{𝔭}(ρ(−1)^{∨}), or equivalently that σ^{∨} ∈ W (ρ(−1)^{∨}). Since ρ ≅ det(ρ) ⊗ ρ^{∨} and σ ≅ ψσ^{∨} where ψ is the central character of σ, it follows easily from Proposition 3.15 that this is equivalent to σ ∈ W (ρ).

Conversely suppose that σ ∈ W (ρ). If [K : Q] is odd, then let D be a quaternion algebra over K ramified at all but one infinite places and split at all finite places. If [K : Q] is even, then let L denote the splitting field of ρ and choose a prime 𝔮 unramified in L(μ_{ℓ}) so that the conjugacy class of Frob_{𝔮} in Gal(L(μ_{ℓ})/K) is that of a complex conjugation. Let D be a quaternion algebra over K ramified at exactly 𝔮 and all but one infinite place. Then π^{D_{𝔭}}(ρ) ≠ 0 for all primes 𝔭, so we can choose U sufficiently small (of level prime to ℓ) so that Y_{U} has no elliptic points and π_{𝔭}^{U_{𝔭}} ≠ 0 for all 𝔭 not dividing ℓ. We can then reverse the above argument to conclude that ρ is modular of weight σ. □

Level-lowering theorems of Fujiwara [32], Rajaei [57] and the third author [44, 45] can be viewed as partial results in the direction of Conjectures 4.7 and 4.9, of which they are also consequences.

Proposition 4.13. Let ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) be continuous, irreducible and totally odd, let 𝔫 be the (prime to ℓ) conductor of ρ and let 𝔫′ = 𝔫 ∏_{𝔭|ℓ} 𝔭^{2}.

a) Suppose that Conjecture 4.7 holds, or that Conjecture 4.9 holds and ρ is not badly dihedral. Then ρ ∼ \overline{ρ}_{π} for some cuspidal automorphic representation π of GL_{2} /K of weight (2, 0) and conductor dividing 𝔫′.

b) If [K : Q] is even, suppose that Conjecture 4.7 holds for K; if [K : Q] is odd, suppose that Conjecture 4.9 holds for K and that ρ is not badly dihedral if ℓ = 2 or 3. If K is unramified at ℓ and ρ|_{G_{K𝔭}} arises from a finite flat group scheme over O_{K,𝔭} for each 𝔭|ℓ, then ρ ∼ \overline{ρ}_{π} for some cuspidal automorphic representation π of GL_{2} /K of weight (2, 2) and conductor 𝔫.

Proof. To prove (a) assuming Conjecture 4.7, let D be a quaternion algebra over K ramified at all infinite places and at most one prime 𝔭_{0} over ℓ, but no other finite places. Let U = ∏_{𝔮} U_{𝔮} be the open compact subgroup of D_{f}^{×} with

U_{𝔮} = \{ ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}(O_{K,𝔮}) | c ≡ d − 1 ≡ 0 mod 𝔫O_{K,𝔮} \}

for 𝔮 not dividing ℓ, and U_{𝔮} a pro-ℓ-Sylow subgroup of a maximal compact subgroup of D_{𝔮}^{×} for 𝔮|ℓ. It follows from Emerton’s characterisation of π_{𝔮} in Theorem 4.1 that π_{𝔮}^{U_{𝔮}} ≠ 0 for all 𝔮 not dividing ℓ. The same is true for 𝔮|ℓ since U_{𝔮} is pro-ℓ and π_{𝔮} ≠ 0. Therefore S^{D}(U)[𝔪_{ρ}] = S^{D}[𝔪_{ρ}]^{U} ≠ 0 and ρ ∼ \overline{ρ}_{π′} for some cuspidal automorphic representation π′ of D^{×} of weight (2, 0) with π′^{U} ≠ 0. Then the cuspidal automorphic representation π of GL_{2} /K corresponding to π via Jacquet-Langlands has conductor dividing 𝔫′. (Note that the form of U_{𝔮} for 𝔮|ℓ implies that π_{𝔮} has conductor dividing 𝔮^{2}.)

The proof of (a) assuming Conjecture 4.9 is similar, except that we use the assumption that ρ is not badly dihedral in order to apply Lemma 4.11.

The proof of (b) is also similar, using the fact that if ρ|_{G_{K𝔭}} arises from a finite flat group scheme, then W_{𝔭}(ρ) contains the trivial representation. (Recall that we are assuming ℓ to be unramified in K, so that badly dihedral representations only occur if ℓ = 2 or 3, as remarked above.) □

We remark that in fact the conclusions of Proposition 4.13 follow from weaker modularity conjectures than Conjectures 4.7 and 4.9, together with known level lowering results, at least if ℓ > 2 and ρ is not badly dihedral. Indeed, we will explain that weak modularity (Conjecture 1.1) implies Proposition 4.13(a), given these level lowering results. It seems that Proposition 4.13(b) is more subtle in that it requires more control over the level at ℓ, but these follow from Conjecture 1.1 and our weight conjecture (Conjecture 3.14), together with level lowering. (In fact, the only part of Conjecture 3.14 we need is the case where ρ is finite at 𝔭, so that W_{𝔭}(ρ) contains the trivial representation.)

To deduce (a) from the weak conjecture, we can assume that ρ ∼ \overline{ρ}_{π} for some automorphic representation π of GL_{2}/K of some weight and level, by weak modularity. By Corollary 2.12, we can assume that the weight is (2, 0), and that the level is 𝔪 ∏_{𝔭|ℓ} 𝔭^{a_{𝔭}} for some ideal 𝔪 and some integers a_{𝔭}. The same argument as in the proof of Proposition 4.13(a) above gives that a_{𝔭} ≤ 2. For primes 𝔮 ∤ ℓ, we may use existing level lowering results to deduce that we may take 𝔪 = 𝔫, so that ρ is modular of weight (2, 0) and level 𝔫′, as required. For [K : Q] odd, these are due to the third author and to Rajaei [44, 45, 57], under the technical hypothesis that ℓ > 2 and ρ is not badly dihedral. When [K : Q] is even, a similar argument applies, but in order to use the level lowering results mentioned above, one needs to begin by raising the level by adding a prime, using Taylor’s theorem [66]. One can then switch to an appropriate quaternion algebra to perform the level lowering, and finally remove the prime that we added, using Fujiwara’s unpublished version [32] of Mazur’s Principle in the case [K : Q] even.

Proposition 4.13(b) would work in the same way, given sufficiently strong level lowering statements for primes 𝔭|ℓ. However, these are not yet sufficient to deduce (b) from weak modularity and level lowering. But Conjecture 1.1 and Conjecture 3.14, together with level lowering (and the results of Taylor and Fujiwara when [K : Q] is even), is sufficient to deduce Proposition 4.13(b); one simply uses the observation made in the course of the proof above that the trivial representation lies in W_{𝔭}(ρ) if ρ is finite at 𝔭.

Corollary 4.14. If Conjecture 4.7 or 4.9 holds, then there are only finitely many continuous, irreducible, totally odd ρ : G_{K} → GL_{2}(\overline{F}_{ℓ}) of conductor dividing 𝔫.

Proof. Note that by class field theory, there are only finitely many badly dihedral ρ of a given conductor. We can therefore assume ρ is not badly dihedral, so by Proposition 4.13(a), either conjecture implies ρ is modular of weight (2, 0) and level ℓ^{2}.𝔫(ρ), where 𝔫(ρ) denotes the Artin conductor of ρ. Since there are only finitely many automorphic representations of weight (2, 0) and given bounded level, the result follows. □

Corollary 4.15. Suppose that Conjecture 4.7 holds if [K : Q] is even, and Conjecture 4.9 holds if [K : Q] is odd. If E is an elliptic curve over K, then E is modular.

Proof. Given E of conductor 𝔫, let ℓ run through all primes greater than 3 and unramified in K, such that E has good reduction at all 𝔭|ℓ. Then ρ_{E,ℓ} is finite at 𝔭, so Proposition 4.13(b) implies that ρ_{E,ℓ} is modular of weight (2, 2) and level equal to 𝔫(ρ_{E,ℓ}), which divides the conductor of E. So there is an automorphic representation π^{(ℓ)} of level U_{1}(𝔫) and weight (2, 2) whose mod ℓ representation agrees with ρ_{E,ℓ}, or equivalently one of weight (2, 0) giving rise to ρ_{E,ℓ}(−1). There are only finitely many such automorphic representations, so there is a π such that π = π^{(ℓ)} for infinitely many ℓ. It follows that for all 𝔭 not dividing 𝔫, the action of T_{𝔭} on π^{U_{1}(𝔫)} is by a_{𝔭}(E) and that of S_{𝔭} is trivial. Therefore ρ_{E,ℓ}(−1) ∼ ρ_{π} (for any ℓ), and hence L(E, s) = L(π, s). □

Remark 4.16. The remarks after Proposition 4.13 show that the conclusion of Corollary 4.14 for ℓ > 2 actually follows from Conjecture 1.1 and known level lowering results. Similarly one sees that the conclusion of Corollary 4.15 follows from Conjecture 1.1, Conjecture 3.14 and level lowering results. In fact the modularity of E even follows from Conjecture 1.1 using modularity lifting results of Skinner-Wiles [65], Fujiwara [31] or Taylor [67]. Furthermore, using the lifting results of Kisin [53] and Gee [34] one can show (unconditionally) that if ρ_{E,3} is irreducible and not badly dihedral, then E is modular. See also [64], [65] and [47] for additional cases where modularity of E is known.

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Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, UK

E-mail address: k.buzzard@imperial.ac.uk

Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK

E-mail address: fred.diamond@kcl.ac.uk

Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, UK

E-mail address: A.F.Jarvis@sheffield.ac.uk