On the local behaviour of ordinary modular Galois representations

Eknath Ghate

Abstract. We show that if the restriction to the decomposition group at p of the p-adic Galois representation attached to one member of a Hida family of elliptic modular cusp forms is non-split then it is non-split for all but finitely many members of this family. We explain the relevance of this result to a question of Greenberg on the local splitting behaviour of ordinary modular Galois representations.

1. Greenberg’s question

In this paper we investigate a question of R. Greenberg concerning the local splitting behaviour of the Galois representation attached to an ordinary primitive elliptic modular cusp form.

Let us state Greenberg’s question. Let f = ∑ a(n, f) q^{n} denote a primitive elliptic modular cusp form of weight k ≥ 2, level N ≥ 1 and nebentypus χ : (ℤ/N)^{×} → ℂ^{×}. Here primitive means that f is a normalized newform that is a common eigenform of all the Hecke operators. Let ℚ̄ denote an algebraic closure of ℚ and fix an embedding ι_{∞} of ℚ̄ into ℂ. Let K_{f} denote the number field generated by the Fourier coefficients a(n, f) of f via the embedding ι_{∞}. It is well known that K_{f} is either a totally real field or a CM field [16, Page 185].

Let p be a prime. Fix an embedding ι_{p} of ℚ̄ into ℚ̄_{p} and let ℘ be the prime of ℚ̄ induced by this embedding. We write again ℘ for the restriction of ℘ to any subfield of ℚ̄. In particular ℘ is a prime of K_{f}. Let K_{f,℘} denote the completion of K_{f} at ℘. Eichler, Shimura and Deligne attach a Galois representation to f

ρ_{f} : Gal(ℚ̄/ℚ) → GL_{2}(K_{f,℘})

which has the property that for all primes ℓ outside Np

tr ρ_{f}(Frob_{ℓ}) = a(ℓ, f) and det ρ_{f}(Frob_{ℓ}) = χ(ℓ)ℓ^{k−1}.

We write ρ_{f,℘} instead of ρ_{f} when we wish to emphasize that ρ_{f} takes values in K_{f,℘}.

Let G_{p} denote the decomposition group at p determined by the embedding ι_{p}. We say that f is ordinary at ℘ (with respect to the embedding ι_{∞}) if a(p, f) is a ℘-adic unit. If f is ordinary at ℘ then Deligne, Mazur-Wiles [12] and Wiles [20, Theorem 2] have shown that the restriction of ρ_{f} to G_{p} is ‘upper triangular’. More precisely let V be a model for ρ_{f}|_{G_{p}}; this is a two dimensional vector space over K_{f,℘}. Let K̄_{f,℘} = ℚ̄_{p} be the algebraic closure of K_{f,℘}. Then there exists a basis of V̄ := V ⊗ K̄_{f,℘} with respect to which ρ_{f}|_{G_{p}} has the following shape:

2000 Mathematics Subject Classification. Primary 11F80, Secondary 11G10, 11R23.

ρ_{f}|_{G_{p}} ∼ (\begin{matrix}δ & u \\ 0 & ϵ \end{matrix}) (1.1)

where ϵ, δ : G_{p} → K̄_{f,℘}^{×} are characters with ϵ unramified, and u : G_{p} → K̄_{f,℘} is a continuous function that is not necessarily a homomorphism. Moreover ϵ(Frob_{p}) is equal to the unique ℘-adic unit root of the polynomial x^{2} − a(p, f)x + p^{k−1}χ(p), where we take χ(p) = 0 if p | N.

The map u has a cohomological interpretation. Let c : G_{p} → K̄_{f,℘} be the map defined by

c(g) = ϵ^{−1}(g) · u(g)

for all g ∈ G_{p}. Write K̄_{f,℘}(δϵ^{−1}) for K̄_{f,℘} thought of as a G_{p}-module via the character δϵ^{−1}. Then it is easy to check that c ∈ Z^{1}(G_{p}, K̄_{f,℘}(δϵ^{−1})) is a 1-cocycle. Thus c represents a cohomology class

[c] ∈ H^{1}(G_{p}, K̄_{f,℘}(δϵ^{−1})).

Let V_{1} be a G_{p}-stable line in V̄. A line V_{2} ⊂ V̄ is said to be complementary to V_{1} if V_{1} ⊕ V_{2} = V̄. Let us say that ρ_{f}|_{G_{p}} splits, or that ρ_{f} is split at p, if V̄ has a G_{p}-stable line and a G_{p}-stable complementary line to it. The following easily proved lemma characterizes when ρ_{f}|_{G_{p}} splits in terms of the vanishing of the cohomology class [c] defined above.

Lemma 1. ρ_{f}|_{G_{p}} splits if and only if [c] = 0 in H^{1}(G_{p}, K̄_{f,℘}(δϵ^{−1})).

Greenberg has asked the following question concerning the splitting behaviour of the restriction of ρ_{f} to G_{p}.

Question 1. Let f be a primitive cusp form that is ordinary at ℘. When is the representation ρ_{f}|_{G_{p}} split? That is, when is the cocycle c that is defined in terms of the upper-shoulder map u in (1.1) a coboundary?

The purpose of this paper is to provide some answers to Question 1. It is not difficult to see that if f has complex multiplication, or CM for short, then ρ_{f} splits at p (see Propositions 3 and 5 below). A natural guess for an answer to Question 1 is therefore:

(1.2) (??) ρ_{f}|_{G_{p}} splits if and only if f has CM.

As far as we are aware almost all evidence in support of the ‘only if’ direction in guess (1.2) is when f has weight 2 (more on this below). For this reason the following theorem concerning cusp forms of higher weight is of interest. To state it we need some notation. Assume for simplicity that p is prime to N. Let f_{l} for l ≥ 1 denote the members of the Hida family containing f (see [9], [10] and section 3 below). The form f_{l} has weight l, level dividing Np, and character χω_{p}^{k−l} where ω_{p} : (ℤ/p)^{×} → μ_{p−1} is the Teichmüller character. Moreover, each f_{l} is ordinary at ℘, and, if l = k then ρ_{f_{k}} ∼ ρ_{f}. Now, following a suggestion of R. Greenberg, we prove (see Theorem 3 below):

Theorem 1. If ρ_{f_{l_{0}}} |_{G_{p}} is not split for one weight l_{0} ≥ 2 then ρ_{f_{l}}|_{G_{p}} splits only for finitely many weights l ≥ 2.

It is a consequence of Hida’s theory that the forms f_{l} for l ≥ 2 either all have CM, or no f_{l} for l ≥ 2 has CM (see, for instance, [5]). In particular if f does not have CM then f_{2} does not have CM. In view of the weight 2 results to be described shortly one may now take l_{0} = 2 and specify conditions under which the hypothesis of Theorem 1 holds. Then the theorem shows that under these conditions the ‘only if’ direction of (1.2) holds for all but finitely many of the f_{l} for l ≥ 2.

While Theorem 1 does not answer Greenberg’s question for the cusp form f we started with, it is nevertheless possible in some cases to specify the finite set of exceptional weights l for which ρ_{f_{l}}|G_{p} is split: these are the weights l for which the quantity T = (1 + p)^{l} − 1 is a zero of a certain power series g(T). In Section 5 we illustrate this circle of ideas for f = Δ, the unique primitive cusp form of level 1 and weight 12. Under a suitable hypothesis of total ordinariness on Δ_{2}, the weight 2 member of the family containing Δ, we obtain a criterion for the splitting behaviour of ρ_{Δ}|_{G_{p}} in terms of the vanishing of a power series (see Theorem 4 below).

Let us now describe what can be said about the ‘only if’ direction in guess (1.2) when f has weight k = 2. Some of the results we describe below are explicitly in the literature and others are simple extensions of these results and so may be known to the expert. Assume first that p does not divide N. If K_{f} = ℚ, so that f corresponds to an elliptic curve with good ordinary reduction at p, then Serre and Tate have proved that (1.2) holds (see [14]). It is possible to generalize this result as follows. We shall say that f is totally ordinary at p if f is ordinary at ℘ for each prime ℘|p of K_{f}. In addition we say that ρ_{f} is totally split at p if ρ_{f,℘} splits at p for all such ℘. Then we prove (see Proposition 4 below):

Proposition 1. Suppose f has weight 2 and that p | N̸. Suppose that f is totally ordinary at p and that ρ_{f} is totally split at p. Then f has CM.

The proof of Proposition 1 is based on a study of the endomorphism algebra of the abelian variety A_{f} associated to f by Shimura and uses the theory of p-divisible groups. It builds on the above mentioned work of Serre and Tate [14]. Proposition 1 was proved independently by Emerton [4] and, when p splits completely in K_{f}, by Ribet [13].

Let π denote the automorphic representation corresponding to f and let π_{p} denote the local component of π at p. If p | N̸ then π_{p} is in the unramified principal series. A statement similar to the proposition above also holds when p|N and the power of p dividing N is the same as the power of p dividing the conductor of χ in which case π_{p} is in the ramified principal series. One only needs to modify the notions ‘totally ordinary’ and ‘totally split’ appropriately. Since in this case

a(p, f)\overline{a(p, f)} = p^{k−1} = p,

the sum of the ℘-adic valuation and the \overline{℘}-adic valuation of a(p, f) is non-zero for any prime ℘|p of K_{f}. We shall say that f is totally ordinary at p if for each such prime ℘ either a(p, f) is a ℘-adic unit or it is a \overline{℘}-adic unit. We shall further say that ρ_{f} is totally split at p if in addition either ρ_{f,℘}|_{G_{p}} splits, when a(p, f) is a ℘-adic unit, or ρ_{f,\overline{℘}}|_{G_{p}} splits, when a(p, f) is a \overline{℘}-adic unit. We then have the following result (see Proposition 6 below):

Proposition 2. Suppose that f has weight 2 and that p|N. Assume that the power of p dividing N is the same as the power of p dividing the conductor of χ. Suppose that f is totally ordinary at p and ρ_{f} is totally split at p. Then f has CM.

If p||N and p does not divide the conductor of χ (in this case π_{p} is a twist of the Steinberg representation) then ρ_{f} is ordinary at a prime ℘ of ℚ̄ lying over p exactly when k = 2. Moreover in this case f does not have CM and it is a result of Ribet [13] that ρ_{f,℘} is not split at p for each prime ℘ of K_{f} lying over p. Finally if the exponent of the power of p dividing N is larger than one and unequal to the exponent of the power of p dividing the conductor of χ (this includes the cases where π_{p} is supercuspidal) and k ≥ 2 then a_{p} = 0. So f is not ordinary at ℘ for any prime ℘ of ℚ̄ and Greenberg’s question does not arise in this case.

The above discussion shows that much is known about Question 1 in the weight 2 case. However we stress that although all the above mentioned results support (1.2) it is remarkable that the ‘only if’ direction in (1.2) appears still to be open even in weight 2. For instance say f is a non-CM form of weight 2 such that A_{f} is an abelian surface, i.e. K_{f} is a quadratic field. Say that p | N̸ and p = ℘℘′ splits in K_{f}. Suppose f is ordinary at ℘ but not ordinary at ℘′. Then after (1.2) one might expect that ρ_{f,℘} is not split at p, but this does not appear to be known.

In forthcoming joint work with V. Vatsal [5] we study the weight 1 specializations of Hida families. As a consequence we show that in a non-CM family, all but finitely many of the weight l ≥ 2 specializations have non-split local Galois representation. This is a stronger result than Theorem 1 since we make no non-splitting assumption on any one member of the family. However, we do assume that p is odd, and work under some conditions on the mod p representation attached to the family. See [5] for further details.

We end this paper by giving an equivalent formulation of Question 1 in the case that N is prime to p in the setting of filtered modules. This was posed by Breuil during his lectures [1] at the CRM in Barcelona in the summer of 2001. As a final remark let us mention that much work has been done on the analog of Greenberg’s question in the mod p setting. Here we refer the reader to the papers of Gross [8] and Coleman-Voloch [2].

2. Abelian varieties and weight 2

The main objective of this section is to investigate Greenberg’s question when the cusp form f has weight k = 2. However since some of the results we present are valid for k ≥ 2 we do not make any blanket assumption on the weight k.

Some of the results we present below occur explicitly in the literature and others may be well known to the experts. We collect together here these results with proofs for our as well as for the reader’s convenience.

It is natural to break the problem up into two cases: p | N̸ and p|N.

2.1. Case p | N̸ Consider the first case. We have the following proposition.

Proposition 3. Suppose that f is a CM form of arbitrary weight k ≥ 2. Suppose that p | N̸ and that f is ordinary at ℘. Then ρ_{f,℘} splits at p.

Proof. Since f has CM, there is a quadratic Dirichlet character ψ such that a(q, f) = a(q, f)ψ(q) for all but finitely many primes q. This means that

(2.1)  ρ_{f} ∼ ρ_{f} ⊗ ψ

where ψ is thought of as a character of Gal(ℚ̄/ℚ). Let F denote the quadratic extension of ℚ cut out by ψ and let H = Gal(ℚ̄/F). Then (2.1) implies that

ρ_{f} ∼ Ind_{F}^{ℚ}(ϕ)

for some Hecke character ϕ : H → ℚ̄_{p}^{×}. Let σ denote the non-trivial element of Gal(F/ℚ) and let ϕ^{σ} denote the conjugate character of H. Then

(2.2)  ρ_{f}|_{H} ∼ (\begin{matrix}ϕ & 0 \\ 0 & ϕ^{σ} \end{matrix}) .

Comparing determinants on H we see that

ϕϕ^{σ} = χν_{p}^{k−1} (2.3)

where ν_{p} : Gal(ℚ̄/ℚ) → ℤ_{p}^{×} is the p-adic cyclotomic character.

Let 𝔭 be the prime of F determined by the embedding ι_{p}. Let H_{𝔭} denote a decomposition subgroup of H at 𝔭. After (1.1) at most one of ϕ or ϕ^{σ} is ramified at 𝔭. On the other hand, since p | N̸, χ is unramified at 𝔭. Since ν_{p}^{k−1} is ramified at 𝔭 at least one of ϕ or ϕ^{σ} is ramified at 𝔭. Thus exactly one of ϕ or ϕ^{σ}, say ϕ, is ramified at 𝔭. But this means that p splits in F. Indeed if it were inert, then ϕ would be unramified at 𝔭^{σ} = 𝔭 which is a contradiction. Likewise p cannot ramify in F. Thus p = 𝔭𝔭^{σ} splits in F and we have G_{p} = H_{𝔭} ⊂ H so that comparing (1.1) and (2.2) we see that ρ_{f,℘} splits at p.

The following simpler argument was suggested by the referee. Since we are working over a field of characteristic 0, to show ρ_{f}|_{G_{p}} is semisimple it suffices to show that ρ_{f} is semisimple on an open subgroup of G_{p}. Now since f has CM, ρ_{f} is semisimple and abelian on H. But an abelian semisimple representation remains semisimple after arbitrary restriction. The proposition follows.

What about a converse to the above proposition? We offer the following result when f has weight 2. The proof was inspired by an argument of Serre contained at the back of his book [14, Chapter IV, A.2.3-4]. The result below was proved independently by Emerton [4, Theorem 1.3] and a slightly weaker result (requiring p to split completely in K_{f}) may be found in Ribet [13, page 785].

Proposition 4. Let f have weight 2. Suppose that p | N̸. If f is ordinary at each prime ℘ of ℚ̄ lying over p and ρ_{f,℘} is split at p for each such ℘, then f is a CM form.

Proof. Write A = A_{f} for the abelian variety attached to f by Shimura. Recall that A is defined over ℚ, is simple over ℚ, and has multiplication by an order in the ring of integers of K_{f} (making it an abelian variety of GL_{2}-type). Since p | N̸, A has good reduction at p, and the representation of Gal(ℚ̄/ℚ) on the ℓ-adic Tate module of A, where ℓ is a prime distinct from p, is unramified at p (cf. the criterion of Néron-Ogg-Shafarevich [15, Theorem 1]). Let Ã denote the reduction of A at p. It is an abelian variety over 𝔽_{p}. Let End(A) denote the ring of all the endomorphisms of A thought of as an abelian variety over ℚ̄, and likewise let End(Ã) denote the ring of all the endomorphisms of Ã thought of as an abelian variety over 𝔽̄_{p}. There is a natural map

End(A) → End(Ã) (2.4)

which is well known to be an injection. We claim that, under the hypotheses of the proposition, the map (2.4) is also a surjection after possibly replacing A by an isogenous abelian variety.

Let A(p) denote the p-divisible group attached to A and likewise let Ã(p) denote the p-divisible group attached to Ã. These are p-divisible groups over ℚ and 𝔽_{p} respectively. Let f̃ ∈ End(Ã). We wish to show that f̃ lifts to an endomorphism f of A. To prove this it suffices to show that the induced morphism f̃(p) of Ã(p) lifts to a morphism of A(p). That is, it suffices to show that the induced map on p-divisible groups

End(A(p)) → End(Ã(p)) (2.5)

is a surjection (it is known to be an injection). In the setting of [14, Chapter IV, A.2.3-4] (where A is defined over ℤ_{p} to start with) the fact that (2.5) is surjective implies that (2.4) is surjective is attributed to an unpublished result of Tate. We now show, following a suggestion of Conrad, how the surjectivity of (2.4) follows from that of (2.5) in our setting where A is defined over ℚ. Let ℓ denote an arbitrary prime and let A(ℓ) and Ã(ℓ) denote the associated ℓ-divisible groups over ℚ and 𝔽_{p} respectively. If (2.5) is an isomorphism then the natural map End(A(ℓ)) → End(Ã(ℓ)) is an isomorphism for every prime ℓ since for ℓ ≠ p it is automatically an isomorphism because both the ℓ-divisible groups involved are étale. On the other hand by the Tate conjecture for abelian varieties over number fields (due to Ribet in the case at hand, namely modular abelian varieties defined over ℚ, but in general due to Faltings) we have

End(A) ⊗ ℤ_{ℓ} = End_{G_{L}}(T_{ℓ}(A)) = End(A(ℓ))

where L is a number field over which all the endomorphisms of A are defined and G_{L} = Gal(ℚ̄/L). Similarly (this time by a result of Tate himself) we have

End(Ã) ⊗ ℤ_{ℓ} = End_{H}(T_{ℓ}(Ã)) = End(Ã(ℓ))

where H is a finite index subgroup of Gal(𝔽̄_{p}/𝔽_{p}). Putting things together we see that if (2.5) is an isomorphism then (2.4) is an isomorphism when tensored with ℤ_{ℓ} for all primes ℓ and therefore that (2.4) itself is an isomorphism.

We now prove that (2.5) is an isomorphism after possibly replacing A by an isogenous abelian variety. To do so we may replace A by the Néron model of A defined over ℤ_{p} which we continue to write as A.

Let O_{f} denote the ring of integers of K_{f}. By replacing A by an isogenous variety if necessary we may assume that O_{f} ⊂ End(A). Consider the decomposition of p into prime ideals given by pO_{f} = ∏_{℘|p} ℘^{e_{℘}} where e_{℘} is the ramification index of ℘|p. For each prime ℘ of K_{f} lying over p let A(℘) denote the p-divisible group formed out of the commutative finite flat group schemes A[℘^{e_{℘}n}] for n = 1, 2, . . .. Then A(℘) has height 2e_{℘}f_{℘} where f_{℘} is the residue degree of ℘|p. Further the decomposition of the group scheme A[p^{n}] = ∏_{℘|p} A[℘^{e_{℘}n}] for each n induces the decomposition

A(p) = ∏_{℘|p} A(℘)

of the p-divisible group A(p).

By hypothesis each ρ_{f,℘} splits at p. This means that A(℘), thought of as a p-divisible group over ℚ_{p}, splits into two p-divisible groups over ℚ_{p} of height e_{℘}f_{℘} each. By a theorem of Tate [19, Theorem 4] we deduce that A(℘) splits over ℤ_{p} itself:

A(℘) = H_{1}(℘) × H_{2}(℘)

where the H_{i}(℘) are p-divisible groups over ℤ_{p} of height e_{℘}f_{℘}. In particular Ã(℘) = H̃_{1}(℘) × H̃_{2}(℘).

Since f is ℘-ordinary the characteristic polynomial of the crystalline Frobenius on the ℚ_{p}-crystal attached to Ã(℘) namely

N_{ℚ_{p}}^{K_{f,℘}} (x^{2} − a(p, f)x + χ(p)p^{k−1}) (2.6)

has an ordinary Newton polygon. Since in the isogeny category of p-divisible groups over 𝔽_{p} the H̃_{i}(℘) are K_{f,℘}-modules, one of the H̃_{i}(℘) must be étale and the other is therefore multiplicative. Say H̃_{1}(℘) is étale and H̃_{2}(℘) is multiplicative. If follows that H_{1}(℘) is étale and that H_{2}(℘) is multiplicative. In particular A(p) = H_{1} ×H_{2} where H_{1} = ∏ H_{1}(℘) is étale and H_{2} = ∏ H_{2}(℘) is multiplicative.

Since there are no maps between an étale p-divisible group and a multiplicative p-divisible group we are reduced to showing that if G is a ‘pure’ (that is either an étale or a multiplicative) p-divisible group over ℤ_{p} then

End(G) = End(G̃). (2.7)

But by duality it suffices to prove (2.7) when G is étale in which case it is known to be true [14, Lemma, Chapter IV, A.2.3]. Summarizing we have shown that after a possible replacement of A by an isogenous abelian variety End(A(p)) = End(Ã(p)) and the map (2.4) is an isomorphism.

Recall that an abelian variety X defined over an arbitrary field is said to be of CM-type if End(X) ⊗ ℚ contains a semi-simple commutative ℚ-subalgebra of dimension 2 dim(X) over ℚ. Since Ã is defined over a finite field it is an abelian variety of CM-type. After tensoring with ℚ, (2.4) is an isomorphism for A = A_{f}, so A_{f} is of CM-type as well. By a result of Shimura [18, Proposition 1.6], f is a CM form.

2.2. Case p | N Let us turn our attention to the second case, namely, let us assume that p|N. Write N_{p} for the exponent of the exact power of p that divides N. Let C denote the conductor of χ and let C_{p} denote the exponent of the exact power of p that divides C. Then we have that either

1. (Principal series) N_{p} = C_{p} in which case |a(p, f)| = p^{(k−1)/2}, or,

2. (Steinberg) N_{p} = 1 and C_{p} = 0 in which case a(p, f)^{2} = χ(p)p^{k−2}, or,

3. (Other) N_{p} ≥ 2 and N_{p} ≠ C_{p} in which case a(p, f) = 0.

The last case contains the cases where π_{p}, the local factor at p in the automorphic representation π attached to f, is supercuspidal.

We now discuss each of the above cases in turn, mostly under the additional hypothesis that k = 2.

2.2.1. Principal series case

Proposition 5. Suppose that f is a CM form of weight k ≥ 2. Say p|N and N_{p} = C_{p}. If f is ℘-ordinary then ρ_{f,℘} splits at p.

Proof. The proof is similar to the proof of Proposition 3. The only difference worth noting is that the nebentypus character χ is now ramified at 𝔭. Nonetheless the equation (2.3) shows that at least one of ϕ or ϕ^{σ} is ramified at 𝔭 and the proof proceeds as before. Note also that the proof of Proposition 3 suggested by the referee applies verbatim in the present case.

Again we can offer a converse similar to Proposition 4 when the weight of f is 2. Recall that complex conjugation induces an automorphism of K_{f}. When k = 2 we have a(p, f)\overline{a(p, f)} = p. In particular the sum of the ℘-adic valuation of a(p, f) and the \overline{℘}-adic valuation of a(p, f) is non-zero.

Proposition 6. Suppose that f has weight 2, and that p|N with N_{p} = C_{p}. Suppose that for each prime ℘|p of K_{f} either a(p, f) is a ℘-adic unit and ρ_{f,℘} splits at p, or a(p, f) is a \overline{℘}-adic unit and ρ_{f,\overline{℘}} splits at p. Then f is a CM form.

Proof. The proof is similar to that of Proposition 4 and so we only outline the differences here. This time the abelian variety A = A_{f} has potentially good reduction at p. This may be seen by studying the ℓ-adic Galois representation ρ : Gal(ℚ̄/ℚ) → Aut(T_{ℓ}(A)) on the ℓ-adic Tate module of A for a prime ℓ ≠ p. It turns out that ρ(I_{p}) has finite image, so that ρ is unramified when restricted to a finite extension F of ℚ. Further, the number field F is independent of ℓ and may be taken to be the field cut out by the p-part of the nebentypus character χ. By [15, Theorem 2], A has good reduction at 𝔭, the unique prime of F lying over p.

Let B = A ×_{ℚ} F and let B̃ denote the abelian variety over O_{F} /𝔭 which is the reduction of B at the prime 𝔭. The proof of Proposition 4 now goes through as before if we replace A by B. Indeed for each prime ℘|p of K_{f} we must have ℘ ≠ \overline{℘} since the sum of the ℘-adic valuation of a(p, f) and \overline{℘}-adic valuation of a(p, f) is non-zero whereas exactly one of these two valuations is zero by hypothesis. Thus, replacing B by an isogenous abelian variety if necessary, we may decompose the p-divisible group B(p) as follows:

B(p) = ∏_{℘|p} B(℘) = G_{1} × G_{2}

where G_{1} is the product of all the B(℘) with a(p, f) a ℘-adic unit and G_{2} = Ḡ_{1} is the product of the remaining B(℘).

Now by hypothesis ρ_{f,℘} is split when restricted to H_{𝔭}, the decomposition group of F at 𝔭, for all primes ℘ of K_{f} such that B(℘) occurs in G_{1}. So for each factor B(℘) occurring in G_{1} we may write B(℘) = H_{1}(℘) × H_{2}(℘) as before. On the other hand the characteristic polynomial of the crystalline Frobenius on the ℚ_{p}-crystal attached to B̃(℘) is

N_{ℚ_{p}}^{K_{f,℘}} ((x − a(p, f))(x − χ′(p)\overline{a(p, f)}))

where χ′ is the prime to p-part of χ (compare with (2.6)). Thus as before B̃(℘) is ordinary and is a product of an étale and a multiplicative p-divisible group. We conclude as before that say H_{1}(℘) is étale and H_{2}(℘) is multiplicative. In particular G_{1} is a product of an étale and a multiplicative p-divisible group. Since G_{2} = Ḡ_{1} we see that the same is true for G_{2} and therefore also for G. The rest of the proof proceeds as before showing that (replacing B by an isogenous variety if necessary) End(B(p)) = End(B̃(p)) and B is an abelian variety of CM-type. We conclude that A is an abelian variety of CM-type and that f is a CM form.

2.2.2. Steinberg case The following result of Ribet [13, Theorem 3.6.2 a)] completely answers Greenberg’s question in the Steinberg case when k = 2. For completeness we will give the proof.

Theorem 2. Suppose that f has weight 2 and that the p-part of χ is trivial. Suppose that p||N. Then f is ordinary at each prime ℘ of ℚ̄ and ρ_{f,℘} is not split at p for all such ℘.

Proof. For the first statement note that f is ordinary at every prime ℘ of ℚ̄ simply because a(p, f) is a root of unity. For the second, let A = A_{f} be the abelian variety attached to f. First assume that A = E is an elliptic curve with split multiplicative reduction at p (so a(p, f) = 1 and not −1). Then it is well known that the Galois representation attached to p is non-split when restricted to G_{p}. This follows from the theory of the Tate curve. Indeed, under the above hypotheses, E_{/ℚ_{p}} is isomorphic to a Tate curve defined over ℚ_{p}, so that

E(ℚ̄_{p}) = \frac{ℚ̄_{p}^{×}}{q^{ℤ}}

for the Tate period q ∈ ℚ_{p} of E. A set of q^{ℤ}-coset representatives of the p^{n}-torsion points of E is therefore given by ζ^{l}α^{m} for 0 ≤ l, m ≤ n − 1 where ζ is a primitive p^{n^{th}}-root of 1 and α is a p^{n^{th}}-root of q. As a module for G_{p} = Gal(ℚ̄_{p}/ℚ_{p}) we see that E[p^{n}](ℚ̄_{p}) fits in an exact sequence

1 → μ_{p^{n}} → E[p^{n}] → ℤ/p^{n} → 1,

where the map E[p^{n}] → ℤ/p^{n} is induced by ζ^{l}α^{m} ↦ m. Thus the above sequence is not split as a sequence of ℤ[G_{p}]-modules. If V_{p}(E) is the Tate module of E tensored with ℚ_{p} we obtain a non-split exact sequence of ℚ_{p}[G_{p}]-modules:

1 → ℚ_{p}(1) → V_{p}(E) → ℚ_{p} → 1

and thus ρ_{f} does not split at p in this case. If A = E has non-split multiplicative reduction at p (so a(p, f) = −1), then E acquires split multiplicative reduction over an unramified quadratic extension K of ℚ_{p}, and E_{/K} is isomorphic to a Tate curve defined over K. Now the above argument goes through if one works over K, showing that ρ_{f} is not split on Gal(ℚ̄_{p}/K) ⊂ G_{p}. It follows that ρ_{f} does not split at p.

Now say that A = A_{f} has dimension g ≥ 1. It is known that A has purely multiplicative (not necessarily split) reduction at p (see [3]). There is a general theory of p-adic uniformization of abelian varieties that are defined over finite extensions of ℚ_{p} and have purely multiplicative reduction at p. This is due to Tate, McCabe, Mumford and Raynaud (for references see [13]). This theory was used in [13] to show that ρ_{f,℘} restricted to G_{p} does not split at p for each prime ℘ of K_{f} with ℘|p. Let us recall the argument. It basically extends the argument for the case g = 1 outlined above. First assume that A has split multiplicative reduction at p. Then, working over ℚ_{p}, one has

A(ℚ̄_{p}) = \frac{(ℚ̄_{p}^{×})^{g}}{L}

for a certain free subgroup L =< q_{1}, q_{2}, …, q_{g} > of rank g generated by periods q_{i} ∈ (ℚ_{p}^{×})^{g}. Write q_{i} = (q_{i1}, q_{i2}, … q_{ig}) for q_{ij} ∈ ℚ_{p}^{×}. Let α_{ij} denote a p^{n^{th}}-root of q_{ij}. Then a set of L-coset representatives of the p^{n}-torsion points of A is

u(l_{r}, m_{s}) := (ζ^{l_{1}}α_{11}^{m_{1}}α_{21}^{m_{2}} ⋯ α_{g1}^{m_{g}} , ζ^{l_{2}}α_{12}^{m_{1}}α_{22}^{m_{2}} ⋯ α_{g2}^{m_{g}} , …, ζ^{l_{g}} α_{1g}^{m_{1}}α_{2g}^{m_{2}} ⋯ α_{gg}^{m_{g}})

where 0 ≤ l_{r}, m_{s} ≤ p^{n} − 1 for 1 ≤ r, s ≤ g. As a module for G_{p} = Gal(ℚ̄_{p}/ℚ_{p}) we see that A[p^{n}](ℚ̄_{p}) fits in the exact sequence

1 → μ_{p^{n}}^{g} → A[p^{n}] → (ℤ/p^{n})^{g} → 1,

where the map A[p^{n}] → (ℤ/p^{n})^{g} is induced by

u(l_{r}, m_{s}) ↦ (m_{1}, m_{2}, …, m_{g}).

Again the above sequence is not split as a sequence of ℤ[G_{p}]-modules. If V_{p}(A) is the Tate module of A tensored with ℚ_{p} we obtain a non-split exact sequence of ℚ_{p}[G_{p}]-modules:

1 → ℚ_{p}(1)^{g} → V_{p}(A) → ℚ_{p}^{g} → 1. (2.8)

Now V_{p}(A) is a free K_{f} ⊗ ℚ_{p} module of rank 2. Note that

K_{f} ⊗ ℚ_{p} = ⊕_{℘|p}K_{f,℘}.

Since the action of K_{f} commutes with the action of G_{p} the module ℚ_{p}(1)^{g} above is a K_{f} ⊗ ℚ_{p}-module which in fact is free of rank 1. Thus, if ℘|p, we may take ℘-eigenspaces in the exact sequence (2.8) above, to obtain the exact sequence of ℤ[G_{p}]-modules:

1 → K_{f,℘}(1) → V_{℘}(A) → K_{f,℘} → 1. (2.9)

Here V_{p}(A) = ⊕_{℘|p}V_{℘}(A) and V_{℘}(A) = K_{f,℘}^{2} as a vector space is the ℘-eigenspace of V_{p}(A). An easy argument of Ribet now shows that the sequence (2.9) is still non-split. We conclude that ρ_{f,℘} restricted to G_{p} is non-split. If A does not have split multiplicative reduction at p, then, as in the elliptic curve case, we may extend scalars to a finite unramified extension K of ℚ_{p} over which A has split multiplicative reduction, and deduce that ρ_{f,℘} is non-split on Gal(ℚ̄_{p}/K). Hence ρ_{f,℘} is non-split on G_{p}.

It is a consequence of the above theorem that if f has weight 2 and is Steinberg at p, that is N_{p} = 1 and C_{p} = 0, then f does not have CM. This can also be seen using ℓ-adic representations for ℓ ≠ p. Indeed, the inertia group I_{p} at p has finite image under the ℓ-adic representation attached to a CM form. On the other hand, it is a result of Langlands [11] that I_{p} has infinite image under the ℓ-adic representation attached to a form that is Steinberg at p.

2.2.3. Other cases If N_{p} ≥ 2 and N_{p} ≠ C_{p} then a(p, f) = 0 and the cusp form f is not ordinary at any prime ℘ of ℚ̄. In particular Greenberg’s question does not arise in this case.

3. Hida families

In the previous section we gave some partial answers to Greenberg’s question (Question 1) under the assumption that f has weight k = 2. In this and the next section we deal with the case k > 2.

Ideally one should approach Question 1 in the higher weight case by studying the fine properties of the motive M_{f} attached to f since it is the étale cohomology of M_{f} which gives rise to the Galois representation ρ_{f}. However this approach has not yet borne fruit.

Instead we use Hida families and some Iwasawa theoretic arguments to study Question 1 in the higher weight case. While these tools do not allow us to answer Question 1 for any particular ℘-ordinary form f of weight k > 2, we are able to prove a result of the following form: all but finitely many of the Galois representations attached to the members of the Hida family containing f are non-split at p under a suitable non-splitting hypothesis on one member of the family.

In this section we recall some properties of Hida families and leave the statements of our results (and their proofs) to the next section. General references for the material presented below are [9] and [10].

Let f ∈ S_{k}(N, χ) be a primitive form of arbitrary weight k ≥ 2. Fix an embedding ι_{p} of ℚ̄ into ℚ̄_{p}, equivalently, fix a prime ℘ of ℚ̄. Suppose that f is ordinary at ℘ (with respect to the embedding i_{∞} of ℚ̄ into ℂ).

Assume for simplicity that

(3.1) p is prime to N.

Let {f_{l} | l ≥ 1} denote the Hida family to which f ‘belongs’. It is a consequence of Hida’s theory (more precisely an étaleness statement for the new part of Hida’s ordinary Hecke algebra localized at a suitable height one prime of the Iwasawa algebra) that the forms f_{l} are uniquely determined by f. Now

f_{l} ∈ S_{l}^{ord}(Np, χω_{p}^{k−l})

where ω_{p} : (ℤ/p)^{×} → μ_{p−1} is the Teichmüller character. The superscript ord means that for each l ≥ 1, the cusp form f_{l} is ordinary at ℘ with respect to the embeddings ι_{∞} and ι_{p}. Moreover f_{k}, the weight k member of this family, is related to f by the following process of ‘p-stabilization’. Let α denote the unique ℘-adic unit root of x^{2} −a(p, f)x+χ(p)p^{k−1} and let β be defined by αβ = χ(p)p^{k−1}. Then f(z) and f(pz) ∈ S_{k}(Np, χ) as old forms and f_{k} = f(z) − βf(pz). Note that f_{k} is an oldform that is a common eigenform of all the Hecke operators T_{q} for q prime to N with eigenvalue a(q, f) and for the Hecke operators U_{q} for all primes q|N with eigenvalue a(q, f). It is also an eigenvector of U_{p} with eigenvalue α.

A reflection of the fact that f ‘lives’ in a family is that there is a ‘large’ Galois representation that interpolates the Galois representations ρ_{f_{l},℘} attached to f_{ℓ} by Deligne for l ≥ 2. Let us explain this in more detail. Let Λ = ℤ_{p}[[T]]. This is a regular local domain of Krull dimension 2. Consider for each l ≥ 1 the homomorphism ϕ_{l} of Λ given by

ϕ_{l} : Λ → ℤ_{p}
T ↦ (1 + p)^{l} − 1.

Let K be a sufficiently large field extension of finite degree of the quotient field of Λ. Let L be the integral closure of Λ in K. Let O be the ring of integers in a sufficiently large finite extension of ℚ_{p}. Let P_{l} : L → O denote an algebra homomorphism which extends the homomorphisms ϕ_{l} on Λ. Now Hida has constructed a Galois representation

ρ_{Λ} : Gal(ℚ̄/ℚ) → GL_{2}(K)

which has the following property: for each l ≥ 1 there is a height one prime of L, which is the kernel of some homomorphism P_{l} : L → O as above, such that the reduction of ρ_{Λ} at this prime yields a representation

ρ_{ℓ} : Gal(ℚ̄/ℚ) → GL_{2}(ℚ̄_{p})

which is equivalent to Deligne’s representation ρ_{f_{l},℘} when l ≥ 2. In particular the representation ρ_{k} is equivalent to ρ_{f}.

The representation ρ_{Λ} above satisfies two important additional properties. Firstly, ρ_{Λ} restricted to a decomposition group is also ‘upper triangular’: there are characters δ_{Λ} and ϵ_{Λ} of G_{p} with values in K^{×} with ϵ_{Λ} unramified such that

ρ_{Λ}|_{G_{p}} ∼ (\begin{matrix}δ_{Λ} & u_{Λ} \\ 0 & ϵ_{Λ} \end{matrix}) . (3.2)

Again u_{Λ} : G_{p} → K is just a map, and not necessarily a homomorphism.

Secondly, the reductions of the Galois representations ρ_{l} for l ≥ 1 are all mutually equivalent. More precisely let 𝔽 = L/𝔪_{L} denote the residue field of L and for l ≥ 1 let

ρ̄_{l}^{ss} : Gal(ℚ̄/ℚ) → GL_{2}(𝔽)

denote the semi-simplification of the reduction of ρ_{l}. Then

(3.3)  ρ̄_{l_{1}}^{ss} ∼ ρ̄_{l_{2}}^{ss}

for any two weights l_{1} ≥ 1 and l_{2} ≥ 1. It is this last condition that is the key defining property of the Hida family.

4. Iwasawa theoretic arguments

Keep the notation of the previous section. In particular f is a ℘-ordinary form (of prime to p level N). We are interested in the behaviour of ρ_{f} restricted to G_{p}. Recall that f_{l} for l ≥ 1 are the members of the Hida family containing f.

Theorem 3. Suppose that ρ_{f_{l_{0}}} is non-split when restricted to G_{p} for one integer l_{0} ≥ 1. Then for all but finitely many l ≥ 1 the representation ρ_{f_{l}} is non-split when restricted to G_{p}.

Proof. Let 𝔪_{L} denote the maximal ideal of L. By replacing L by a finite extension if necessary, we may assume that L contains a complete set of representatives of its residue field 𝔽 = L/𝔪_{L}. Then

L^{×} = μ_{q−1} × (1 + 𝔪_{L}) (4.1)

where q denotes the order of 𝔽. Note that μ_{q−1} has order prime to p and that 1 + 𝔪_{L} is a pro-p group.

For a character of ψ : G_{p} → L^{×} write ψ^{t}, respectively ψ^{w} for the projection of ψ onto the first, respectively second, factor in (4.1) above.

Let E^{t} denote the union of the (finitely many) tamely ramified extensions of ℚ_{p} of degree dividing q − 1. Let Δ = Gal(E^{t}/ℚ). Note that both δ_{Λ}^{t} and ϵ_{Λ}^{t} factor through Δ.

Let ℚ_{p}^{unr} denote the maximal unramified ℤ_{p}-extension of ℚ_{p} and let ℚ_{p}^{cycl} denote the cyclotomic ℤ_{p}-extension of ℚ_{p}. Let E^{w} be the maximal abelian pro-p extension of ℚ_{p}. It is well known that

E^{w} = ℚ_{p}^{unr} · ℚ_{p}^{cycl}

and that Γ := Gal(E^{w}/ℚ_{p}) ⟶^{∼} ℤ_{p}^{2}. Note that both δ_{Λ}^{w} and ϵ_{Λ}^{w} factor through Γ.

Let E = E^{t} · E^{w} denote the compositum of E^{t} and E^{w}. Note that E is a Galois extension of ℚ_{p} with Galois group G := Gal(E/ℚ_{p}) = Δ × Γ. After the discussion above the characters δ_{Λ} and ϵ_{Λ} factor through G.

Let H = Gal(ℚ̄_{p}/E). Since δ_{Λ} and ϵ_{Λ} are trivial on H we have

ρ_{Λ}|_{H} = (\begin{matrix}1 & λ \\ 0 & 1 \end{matrix})

for a genuine homomorphism λ : H → L.

We now claim that image(λ) ≠ 0. Indeed if λ = 0 then ρ_{Λ} is split when restricted to H. This means that the specializations ρ_{l} for all l ≥ 1 are all split when restricted to H. In particular ρ_{l_{0}} splits when restricted to H. In cohomological terms this means that the class of the cocycle c attached to ρ_{l_{0}}|_{G_{p}} as in Section 1 has trivial image under the restriction map

H^{1}(G_{p}, K_{0}(χ)) → H^{1}(H, K_{0}(χ)) (4.2)

where K_{0} = K̄_{f_{l_{0}} ,℘} = ℚ̄_{p} and χ = δ_{l_{0}}ϵ_{l_{0}}^{−1}. On the other hand the map (4.2) is injective. Accepting this for the moment we deduce that the class of c in H^{1}(G_{p}, K_{0}(χ)) is trivial. This contradicts the hypothesis of the theorem that ρ_{l_{0}} is not split when restricted to G_{p}, proving the claim.

It remains to show that (4.2) is injective. This can be seen as follows. By the inflation-restriction sequence the kernel of (4.2) is

H^{1}(G, K_{0}(χ)^{H}) = H^{1}(G, K_{0}(χ)) = H^{1}(Δ, K_{0}(χ)^{Γ}) ⊕ H^{1}(Γ, K_{0}(χ)^{Δ}).

The first cohomology group on the right hand side vanishes since Δ is a finite group. As for the second cohomology group it vanishes as well if χ is non-trivial on Δ, so we may assume that K_{0}(χ)^{Δ} = K_{0}(χ), in which case

H^{1}(G, K_{0}(χ)^{H}) = H^{1}(Γ, K_{0}(χ)).

Now Γ = Γ_{1} × Γ_{2} with Γ_{i} = ℤ_{p} for i = 1, 2. Consider the exact sequence

0 → H^{1}(Γ_{2}, K_{0}(χ)^{Γ_{1}}) → H^{1}(Γ, K_{0}(χ)) → H^{1}(Γ_{1}, K_{0}(χ)).

Note that χ ≠ 1 on Γ. Without loss of generality we may assume that χ restricts to a non-trivial character on Γ_{1}. Then K_{0}(χ)^{Γ_{1}} = 0 and

H^{1}(Γ_{1}, K_{0}(χ)) = K_{0}/(χ(γ) − 1)K_{0} = 0

where γ is a topological generator of Γ_{1}. This shows that H^{1}(Γ, K_{0}(χ)) = 0 and that the map (4.2) is injective.

Let I denote the (non-zero) ideal of L generated by the image of λ. Recall that P_{l} is an extension to L of the weight l specialization map ϕ_{l} of Λ. Let 𝔭_{l} := ker P_{l} ⊂ L. It is a height one prime ideal of L. Clearly ρ_{l} is split at G_{p} if and only if image(λ) ⊂ 𝔭_{l}, that is, if and only if I ⊂ 𝔭_{l}. Since the intersection of infinitely many height one primes in L is 0, the ideal I can only be contained in finitely many 𝔭_{l} and we conclude that ρ_{l} is split when restricted to G_{p} for finitely many weights l ≥ 1. This proves the theorem.

We now try and get a better handle on the image of the homomorphism λ : H → L. This will allow us to describe the number of generators of the ideal I of L generated by this image. We keep the notation introduced in the proof of Theorem 3.

Let μ = δ_{Λ}ϵ_{Λ}^{−1} as a character of G with values in L^{×}. Let Λ_{G} = ℤ_{p}[[G]] be the completed group algebra of G. We write μ again for the induced homomorphism of algebras:

μ : Λ_{G} → L.

Write L(μ) for L considered as a Λ_{G}-module via μ.

Let M denote the maximal pro-p abelian extension of E and let

X := Gal(M/E).

Since X is both a ℤ_{p}-module and a G-module it is a module for Λ_{G}. Moreover since L is a pro-p group, the homomorphism λ : H → L factors through X. We have:

Lemma 2. The homomorphism λ : X → L(μ) is a morphism of Λ_{G}-modules.

Proof. This follows from the following identity

(\begin{matrix}δ_{Λ} & 0 \\ 0 & ϵ_{Λ} \end{matrix}) · (\begin{matrix}1 & λ \\ 0 & 1 \end{matrix}) · (\begin{matrix}δ_{Λ}^{−1} & 0 \\ 0 & ϵ_{Λ}^{−1} \end{matrix}) = (\begin{matrix}1 & μ · λ \\ 0 & 1 \end{matrix}) .

We may write Λ_{G} = ℤ_{p}[[G]] = ℤ_{p}[[Γ]][Δ]. Since Δ has order prime to p, X decomposes into eigenspaces for the action of Δ on X:

(4.3)  X = ⨁_{χ∈Δ̂} X^{χ}.

Each eigenspace is a ℤ_{p}[[Γ]]-module which is essentially free of rank 1. More precisely, the ℤ_{p}[[Γ]]-module structure of the eigenspaces is given by the following proposition. We shall assume for simplicity that

L = Λ so that q = p. (4.4)

Proposition 7. Under (4.4), X^{χ} is a projective ℤ_{p}[[Γ]]-module of rank 1. More precisely we have:

1. If χ ≠ ω_{p} then X^{χ} ⟶^{∼} ℤ_{p}[[Γ]].

2. If χ = ω_{p} then X^{χ} ⟶^{∼} 𝔞 where 𝔞 is the the augmentation ideal in ℤ_{p}[[Γ]], considered as a power series ring in two variables.

Proof. The proof follows from local class field theory. The argument is similar to the proof of [7, Theorem 3.11].

Consider μ^{t} as a character on Δ. By Lemma 2 we see that λ : X → L(μ) factors through X^{μ^{t}}. Let M^{λ} denote the fixed field of the kernel of λ and let

X(λ) := Gal(M^{λ}/E).

Then there are natural surjections X ↠ X^{μ^{t}} ↠ X(λ).

Let Λ_{Γ} = ℤ_{p}[[Γ]]. By the above proposition X^{μ^{t}} is isomorphic to Λ_{Γ} or to 𝔞 as a module over Λ_{Γ}. The character μ^{w} of Γ induces a homomorphism μ^{w} : Λ_{Γ} → R. Therefore, by restricting to 𝔞 if necessary, we may consider μ^{w} as a homomorphism

μ^{w} : X^{μ^{t}} → L.

Let M^{μ} denote the fixed field of ker(μ^{w}) ⊂ X^{μ^{t}} and let

X(μ) := Gal(M^{μ}/E).

Thus we have natural surjections X ↠ X^{μ^{t}} ↠ X(μ).

Lemma 3. We have M^{λ} ⊂ M^{μ}.

Proof. To prove the lemma it suffices to show that ker(μ^{w}) ⊂ ker(λ) as subgroups of X^{μ^{t}}. Suppose that r ∈ ker(μ^{w}). Then by Lemma 2 we have

λ(r · x) = μ^{w}(r) · λ(x) = 0

for all x ∈ X^{μ^{t}}. If X^{μ^{t}} = Λ_{Γ}, then taking x = 1 we see that λ(r) = 0 as desired. If X^{μ^{t}} = 𝔞 then we still have that

μ^{w}(x) · λ(r) = λ(x · r) = λ(r · x) = 0

for all x ∈ 𝔞. But L is an integral domain. Since μ^{w}(x) ≠ 0 for some x ∈ 𝔞 we see that λ(r) = 0, as desired.

By Lemma 3 we have X(μ) ↠ X(λ). In sum we see that there is a map of Λ_{Γ}-modules

X^{μ^{t}} ↠ image(λ).

Since X^{μ^{t}} is generated by at most two elements as a Λ_{Γ}-module (cf. Proposition 7) we see that image(λ) is generated by less than two elements as a Λ_{Γ}-module. In particular I, the ideal generated by image(λ) in L, is generated by at most two elements. More precisely we have:

Corollary 1. Assume that L = Λ. Then the ideal I ⊂ L is generated by at most two elements, and, if μ^{t} ≠ ω_{p} then it is generated by one element. In particular if μ^{t} ≠ ω_{p} then there is a power-series g(T) ∈ Λ = ℤ_{p}[[T]] such that the finite set of weights l ≥ 1 for which ρ_{l} is split are zeros of g(T) of the form T = (1 + p)^{l} − 1.

Proof. The first statement follows from the discussion above. For the second say that I is generated by the power series g(T) ∈ Λ. Recall that ρ_{l}|_{G_{p}} is split if and only if I ⊂ 𝔭_{l} and in the present context this translates into the condition that g(T) has a zero at T = (1 + p)^{l} − 1. This proves the corollary.

We remark that in the situation L = Λ and μ^{t} ≠ ω_{p} another way to see that the set of weights l ≥ 1 for which ρ_{l}|_{G_{p}} is split is finite is to note that the non-zero power series g(T) has only finitely many zeros by the Weierstrass preparation theorem.

5. The Δ-function

Let Δ = ∑ τ(n)q^{n} be the unique primitive cusp form of level N = 1 and weight k = 12. Let p be a prime which is ordinary for Δ and let ρ_{Δ} : Gal(ℚ̄/ℚ) → GL_{2}(ℚ_{p}) denote the associated Galois representation. In this section we illustrate how one can obtain some information towards Greenberg’s question for ρ_{Δ} using the results proved in this paper. The main thrust of the argument is to combine the weight 2 results in Section 2 with Theorem 3 taking l_{0} = 2. We obtain, under an assumption of total ordinariness, a criterion for whether ρ_{Δ}|_{G_{p}} is split in terms of the vanishing of certain power series.

Recall that Δ has Fourier coefficients τ(n) ∈ ℤ. Moreover τ(n) is coprime to n for all primes n smaller than 10^{6} except for n = 2, 3, 5, 7 and 2411 (see Gouvêa [6]). In particular p ≥ 11. Note Δ is automatically ordinary at every prime ℘ of ℚ̄ lying over p since τ(p) ∈ ℤ. Fix a prime ℘ of ℚ̄ lying over p. Let Δ_{l} ∈ S_{l}(p, ω_{p}^{12−l}) denote the weight l ≥ 1 member of the corresponding Hida family which contains Δ. Recall that here the word contains means that the level p form Δ_{12} is the p-stabilized modular form associated to (the level one form) Δ by the process described in Section 3.

In this case we may take L = Λ = ℤ_{p}[[T]]. Moreover one may check that

det(ρ_{Λ}) = κ · ω_{p}^{k} · ν^{−1}

where κ : Gal(ℚ̄/ℚ) → Λ^{×} is the character induced by composing the projection Gal(ℚ̄/ℚ) ↠ Gal(ℚ_{∞}/ℚ) = 1 + pℤ_{p} where ℚ_{∞} is the cyclotomic ℤ_{p}-extension of ℚ with the character 1+pℤ_{p} → Λ^{×} which takes (1+p)^{s} for s ∈ ℤ_{p} to (1+T)^{s} ∈ Λ^{×} and ν : Gal(ℚ̄/ℚ) ↠ ℤ_{p}^{×} ↪ Λ^{×} is the p-adic cyclotomic character. Recall that μ = δ_{Λ}ϵ_{Λ}^{−1} = det(ρ_{Λ}) · ϵ_{Λ}^{−2}. Hence μ^{t} = ω_{p}^{k−1} · (ϵ_{Λ}^{t})^{−2}. Since ϵ_{Λ}^{t} is unramified it is not hard to see that if p > 11 then μ^{t} ≠ ω_{p}. Thus assuming p ≠ 11 puts us in the setting of Corollary 1 and we obtain a power series g(T) ∈ ℤ_{p}[[T]]. Applying Theorem 3 we get:

Proposition 8. Let p > 11 be ordinary for Δ. Assume that ρ_{l_{0}}|_{G_{p}} is not split for some weight l_{0} ≥ 1. Then ρ_{Δ}|_{G_{p}} splits if and only if g(T) vanishes at T = (1 + p)^{12} − 1.

We are interested in applying the above proposition when l_{0} = 2. The form Δ_{2} ∈ S_{2}(p, ω_{p}^{10}). The associated nebentypus ω_{p}^{10} has order \frac{p−1}{gcd(10,p−1)}. Thus if p > 11 then ω_{p}^{10} ≠ 1 and the component at p of the automorphic representation corresponding to Δ_{2} is in the (ramified) principal series. On the other hand we have:

Lemma 4. Let p ≥ 11. Then Δ_{2} does not have CM.

Proof. This follows from the general fact that the specializations of a non-CM Λ-adic modular form are non-CM forms in all weights l ≥ 2 (see, for instance, [5]). In this particular case the lemma can also be seen directly. If p = 11 then Δ_{2} is Steinberg at p and so cannot have CM. So assume that p > 11. Recall (cf. [17, Lemma 3]) that a general cusp form g has CM if g is of the form g = ∑_{𝔞} λ(𝔞)q^{N(𝔞)} where λ is a Hecke character of an imaginary quadratic field M and 𝔞 varies through all integral ideals of M. Further there is an integer ν ≥ 1 and an integral ideal 𝔠 of M such that λ((α)) = α^{ν} for all α ≡ 1 mod 𝔠. If M has discriminant D then g is a cusp form of level N(𝔠)|D|, weight ν + 1 and nebentypus λ|_{ℚ} · ω_{M/ℚ} where λ|_{ℚ} is the Dirichlet character of level N(𝔠) taking a to λ((a))/a^{ν} for all a ∈ ℤ with (a, N(𝔠)) = 1 and ω_{M/ℚ} is the quadratic character corresponding to M/ℚ. A necessary condition for λ as above to exist is

(5.1) ζ^{ν} = 1 for all roots of unity ζ ∈ K with ζ ≡ 1 mod 𝔠.

If Δ_{2} = g for some g as above then we must have ν = 1 and 𝔠 = 1 (and D = −p where p must satisfy p ≡ 3 mod 4 and p > 11). Taking ζ = −1 we see that the the condition (5.1) fails and so Δ_{2} does not have CM.

Let K_{2} denote the Hecke field of Δ_{2}. The lemma shows that K_{2} is a CM field when p > 11. Assume from now on that p > 11. As in the introduction we shall say that p is totally ordinary for Δ_{2} if for each prime ℘ of K_{2} lying above p either a(p, Δ_{2}) is a ℘-adic unit or it is a \overline{℘}-adic unit. We have the following theorem.

Theorem 4. Assume that p > 11 and that Δ_{2} is totally ordinary at p. Then ρ_{Δ} is split at p if and only if g(T) has a zero at T = (1 + p)^{12} − 1.

Proof. Since Δ_{2} does not have CM then by Proposition 6 there must be a place ℘ of ℚ̄ lying above p such that ρ_{2,℘}|_{G_{p}} is non-split. By conjugating with an element of Gal(ℚ̄/ℚ) if necessary we may assume that ℘ is the place of ℚ̄ that was earlier fixed in order to speak of the Hida family containing Δ. The theorem now follows from Proposition 8.

Note that the condition of total ordinariness on Δ_{2} is easy to check numerically. It is true if p = 13 for instance.

6. Filtered modules

We conclude with an equivalent formulation by Breuil of Greenberg’s question, in the case that N is prime to p, in terms of filtered modules.

Let f be a ℘-ordinary primitive form of level N prime to p. Fontaine has associated a filtered (ϕ, N)-module D = D_{st}(ρ_{f}|_{G_{p}}) to the restriction of ρ_{f} : Gal(ℚ̄/ℚ) → GL_{2}(K_{f,℘}) to the decomposition group at p. This is a finite dimensional ℚ_{p}-vector space which is equipped with a linear map ϕ : D → D, a decreasing exhaustive filtration Fil^{i}(D) of ℚ_{p}-subspaces of D, and a linear operator N : D → D called the monodromy operator, which, because of our assumption that p is prime to the level, is trivial. Moreover D has an action of K_{f,℘} which commutes with all the structures on D mentioned above and makes D into a two-dimensional K_{f,℘}-module.

Let μ_{1} and μ_{2} be defined by μ_{1}μ_{2} = χ(p) and p^{k−1}μ_{1} +μ_{2} = a(p, f). Let K′_{f,℘} be a sufficiently large finite extension of K′_{f,℘} and consider the following filtered (ϕ, N)-modules of dimension two over K′_{f,℘} with basis e_{1}, e_{2} defined by

D_{split} : \begin{cases}ϕ(e_{1}) = p^{k−1}μ_{1}e_{1} \\ ϕ(e_{2}) = μ_{2}e_{2} \\ Fil^{k−1}(D) = K′_{f,℘} · e_{1} \\ N = 0 \end{cases} D_{non−split} : \begin{cases}ϕ(e_{1}) = p^{k−1}(μ_{1}e_{1} + e_{2}) \\ ϕ(e_{2}) = μ_{2}e_{2} \\ Fil^{k−1}(D) = K′_{f,℘} · e_{1} \\ N = 0 \end{cases} .

Let D^{∗} denote the filtered (ϕ, N)-module corresponding to the representation which is dual to ρ_{f}|_{G_{p}}. It turns out that D^{∗} ∼ D_{split} if ρ_{f} is split at p and D^{∗} ∼ D_{non−split} if ρ_{f} is not split at p. Breuil has asked the following question (cf. [1]). It is the exact analogue of Greenberg’s question in the setting of filtered modules (in the case that p is prime to N).

Question 2. Suppose that f is a ℘-ordinary primitive modular cusp form of weight at least 2. Is the associated filtered module D^{∗} isomorphic to D_{split} or to D_{non−split}?

Acknowledgements. We wish to thank R. Greenberg for posing the question investigated in this paper during a visit to TIFR in January 2001, and for very helpful discussions related to it. We also thank C. Breuil, A. Brown, O. Bültel, B. Conrad, D. Prasad and the referee for some helpful remarks.

20 Eknath Ghate

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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India.

E-mail address: eghate@math.tifr.res.in