A p-ADIC VARIATIONAL HODGE CONJECTURE AND
MODULAR FORMS WITH COMPLEX MULTIPLICATION

Matthew Emerton
University of Chicago

If p is a prime number, N a positive integer prime to p, and k an integer, thenwe let Sk† (N ) denote the space of overconvergent p-adic cuspforms with coefficientsin Cp (of some radius of overconvergence) of weight k on Γ1 (N ) (see [10, chap. I]
for the definition of overconvergent modular forms). In [4, prop. 4.3], Coleman
k−1

d
k−1
induces a
proves that if k ≥ 2, then the differential operator θ= qdq†transformation from S−(k−2)
to Sk† . If f is a weight k normalized Hecke eigenformlying in the image of θk−1 , then one sees that the Up eigenvalue of f is divisible bypk−1 , and so f has slope at least k − 1 ([4, lem. 6.3]).
Suppose that f is actually a normalized classical eigenform on Γ1 (N ) ∩ Γ0 (p).
It follows by Atkin-Lehner theory [1] and the fact that f has slope at least k − 1that f has slope exactly k − 1, and that f is one of the twin oldforms attached toan ordinary classical form g on Γ1 (N ).
Conversely, suppose that g is a classical normalized Hecke eigenform of weight k
on Γ1 (N ), defined over Qp , which is ordinary at p. Then g gives rise to two oldforms(a pair of twins) on Γ1 (N )∩Γ0 (p), one of slope zero and one of slope k −1, which we
may think of as overconvergent Hecke eigenforms on Γ1 (N ). Denote the slope k − 1twin by f . In [4, §7], Coleman discusses the question as to whether or not f is in
the image of θk−1 . He shows that if g is a CM modular form, then f is in the imageof θk−1 . In fact, if g is attached to the (k − 1)st power of a Grössencharacter ψ,then f is obtained by applying θk−1 to a slope zero p-adic modular form of weight−(k − 2) attached to the (1 − k)th power of ψ [4, prop. 7.1]. Coleman also asks
whether there are any non-CM forms in the image of θk−1 [4, remark 2, p. 232].
Continue to suppose that g is a a normalized classical eigenform. The Fourier
coefficients span a finite extension E of Q. Let ℘ range over the primes of E
lying over p. Each prime ℘ gives an embedding of E into Qp (well-defined up to
conjugation in Qp ). Let g℘ denote g regarded as a modular form over Qp (welldefined up to algebraic conjugacy in Qp ) in this way. If g℘ is ordinary, then as
above we may form its slope k − 1 twin, which we denote f℘ .
The aim of this article is to state a certain analogue in p-adic Hodge theory of
the variational Hodge conjecture [12], and to explain how it implies the following
conjecture.
Conjecture (0.1). Let g be a classical Hecke eigenform on Γ1 (N ), let E be thefinite extension of Q generated by the Fourier coefficients of g, and let ℘ run over
1

2

MATTHEW EMERTON

all the prime ideals of E lying over p. Suppose that each g℘ is ordinary, and thatf℘ lies in the image of θk−1 for every prime ℘. Then g is a CM eigenform.
Whether or not it is reasonable to weaken conjecture (0.1) by asking only thatg℘ be ordinary and f℘ be in the image of θk−1 for one prime ℘ is something on
which the author is undecided for the moment.
In section 1, after making some preliminary observations valid for all weights,
we prove conjecture (0.1) in the case that the weight k equals two. In this case,
we appeal directly to the Serre-Tate theory of deformations of abelian varieties incharacteristic p [15, 19] and Tate’s theorem on morphisms of p-divisible groups inmixed characteristic [24, thm. 4].
In section 2, we state our p-adic variational Hodge conjecture (conjecture (2.2))and elaborate on some aspects of it. In particular, proposition (2.7) gives a usefulalternative formulation of the conjecture, in terms of motives with good reduction
modulo p.
In section 3 we explain how conjecture (2.2) implies conjecture (0.1) for arbitrary
weights k ≥ 2. For this we will depend heavily on the existence of the motive
attached to a Hecke eigenform [21].
That conjecture (0.1) would be implied by an appropriate conjecture on algebraic
cycles was noticed independently by B. Mazur. The author first observed therelation between conjectures (0.1) and (2.2) some time in 1997. He would like tothank M. Kisin for encouraging him to write up this account of his observation, as
well as B. Conrad and S. Bloch for helpful conversations.
1. preliminaries, and the case of weight two eigenforms
Let us fix a choice of an algebraic closure Qp of Qp , and let Cp denote the p-adic
completion of Qp . If E is any finite extension of Q then Gal(Qp /Qp ) acts on theset of embeddings of E into Qp , and the orbits correspond to primes ℘ of E lying
over p.
If g is a classical normalized Hecke eigenform of weight at least two on Γ1 (N )and if E is the finite extension of Q in Q generated by the q-expansion coefficientsof g, then there is attached to g a representation ρg : Gal(Q/Q) → GL2 (E ⊗Q Qp )∼ Q[6]. Write E ⊗Q Qp −→ ℘|p E℘ , where the product ranges over the primes ℘ of E
lying over p, and E℘ denotes the ℘-adic completion of E. There is a corresponding∼ Q
factorization ρg −→ ℘|p ρg,℘ . The representation ρg,℘ reflects the properties of gregarded as a p-adic modular form via the embedding E → Qp corresponding to℘. As in the introduction, we let g℘ denote g regarded as a p-adic modular form inthis way.
If f is any p-adic normalized Hecke eigenform of some weight on Γ1 (N ) withcoefficients in Cp , we may attach to f a Galois representation ρf : Gal(Q/Q) →GL2 (Cp ). (See [10, §III.5] and [14, thm. II] for the construction of these representations. The first reference treats the case p > 7, and the second, the case p > 2 (inthe more general context of p-adic Hilbert modular forms). We will only need theexistence of these representations in the ordinary case, for p-adic modular forms
on GL2/Q , and in this case a modification of the methods of [13] can also be used
to construct the necessary representations when p = 2. One can also modify the
set-up of the argument of [14, §1] so that it treats the case p = 2. If the reader
prefers, they may assume that p is odd for the remainder of the paper.)p-ADIC HODGE THEORY AND MODULAR FORMS

3

Now suppose as above that g is a classical eigenform on Γ1 (N ), that ℘ is aprime lying over p in the coefficient field E of g, and that g℘ is ordinary. As in
the introduction let f℘ denote the slope k − 1 oldform on Γ0 (p) ∩ Γ1 (N ) attachedto g℘ . Then f℘ is a p-adic Hecke eigenform, and we have an equality of Galois
representations
ρf℘ = ρg,℘ .
Suppose that f℘ lies in the image of θk−1 . Let F℘ be the weight −(k − 2) overconvergent form satisfying θk−1 F℘ = f℘ . Then since f℘ has slope k − 1, we see thatF℘ must be ordinary.
The Galois representations ρf,℘ and ρF,℘ are related by the equation(1.1)

ρf,℘ = χk−1 ρF,℘

(where χ denotes the p-adic cyclotomic character).
The following proposition was also observed independently by Gouvea and Kisin.
It is the characteristic zero analogue of Serre’s observation concerning mod p Galois
representations and companion forms [11, thm. 13.8].
Proposition (1.2). Suppose that the classical eigenform g on Γ1 (N ) satisfies the
hypothesis of conjecture (0.1). Then the Galois representation ρg , when restricted
to an inertia group at p in Gal(Q/Q), splits as the direct sum of the trivial characterand the character χk−1 .
Proof. It suffices to prove this for each factor ρg,℘ = ρf℘ of ρg . Since ρf℘ is the
Galois representation attached to an ordinary p-adic modular form of weight k,when restricted to inertia at p it becomes a reducible representation, which is anextension of the trivial character by the character χk−1 (as was proved for classicalforms by Deligne in an unpublished letter to Serre; see [14, prop. 2.3] for a publishedproof, in the more general context of p-adic Hilbert modular forms). Similarly, ρF℘
becomes an extension of the trivial character by χ−(k−1) , when restricted to aninertia group at p. Equation (1.1) shows that ρf℘ thus also restricts to an extensionof the character χk−1 by the trivial character. Since k 6= 1, so that χk−1 is distinct
from the trivial character, these two descriptions of ρf℘ as an extension prove theproposition. 
We now prove conjecture (0.1) in the case of weight two.
Theorem (1.3). Let N be a positive integer, p a prime not dividing N , g a classicalnormalized Hecke eigenform of weight two on Γ1 (N ), and E the finite extension of
Q generated by the coefficients of g. Suppose that for every prime ℘ of E lying
over p, the form g℘ is ordinary, and the slope one form f℘ lies in the image ofθ : S0† (N ) → S2† (N ). Then g is a CM form on Γ1 (N ).
Proof. Since we are in the case k = 2, the modular form g determines an abelianvariety factor A of the Jacobian of X1 (N ) defined over Q, of dimension equal tothe degree of E over Q, and equipped with an embedding E → End(A) ⊗Z Q.
Let Tp (A) denote the p-adic Tate module of A, and write Vp (A) = Tp (A) ⊗Zp Qp .
Then Vp (A) is a naturally a two-dimensional E-vector space, and as a representation of Gal(Q/Q), is isomorphic to ρg . Proposition (1.2) implies that we have theisomorphism
(1.4)

∼

Vp (A) −→ E ⊗Q Qp ⊕ E ⊗Q Qp (1)4

MATTHEW EMERTON

after restricting to an inertia group at p. Replacing A by an isogenous abelianvariety if necessary, we may assume that the ring of integers OE of E acts on A as aring of endomorphisms, and that the isomorphism (1.4) arises from an isomorphism
(1.5)

∼

Tp (A) −→ OE ⊗Z Zp ⊕ OE ⊗Z Zp (1)after restricting to an inertia group at p.
Let A[p∞ ] denote the p-divisible group of A. Then the isomorphism (1.5) yields
a splitting
(1.6)

∼

A[p∞ ] −→ OE ⊗Z (Qp /Zp ) ⊕ OE ⊗Z Ĝmafter restricting the p-divisible group A[p∞ ] to the maximal unramified extensionof Qp .
The abelian variety A has good reduction at p, since p is prime to N . Let A
denote the Néron model of A over W (Fp ). Tate’s theorem [24, thm. 4] implies thatthe factorization (1.6) induces an analogous splitting of the the p-divisible group
A[p∞ ] of A:∼
A[p∞ ] −→ OE ⊗Z (Qp /Zp ) ⊕ OE ⊗Z Ĝm .
Thus A is the canonical lift of its special fibre A0 [19, proof of thm. V.3.3], and isendowed with an OE -linear endomorphism F lifting the Frobenius endomorphismof the special fibre.
The morphism F is not given by an element of OE , since its kernel is entirelycontained in (indeed, is equal to the p-torsion subgroup of) Ĝm ⊗Z OE . Thus
OE [F ] is a commutative Z-algebra of endomorphisms of A of rank equal to twice
the dimension of A.
If OE [F ] is an integral domain, then A is endowed with endomorphisms by anorder in a field of twice its dimension, and so is a CM abelian variety. Consequently[23, prop. 1.6] implies that g is a CM modular form.
If OE [F ] is not an integral domain, then we may use it to cut out a proper
abelian subvariety B of A which is closed under the action of OE . Then this abeliansubvariety has CM (since [E : Q] = dim(A) > dim(B)), and again we conclude thatg is a CM form [23, prop. 1.6]. This completes the proof of the theorem. 
2. the p-adic variational hodge conjecture
In order to state our conjecture in p-adic Hodge theory, we introduce somenotation. Let k be a perfect field in characteristic p, let W denote the Witt ring ofk, and let K denote the fraction field of W .
Let X be a smooth proper W -scheme, and let Z0 be a codimension n-cycle inthe special fibre X0 of X . (Let us remark that we will always take our cycles
to have coefficients in Q, thus avoiding any problems with torsion.) Correspond2n
ing to Z0 is a cycle class c0 in the crystalline cohomology group Hcrys(X0 , K) :=2n
Hcrys (X0 , OX0 /W ) ⊗W K. Let X denote the generic fibre of X , which is a smoothand proper K-scheme. The comparison between crystalline and de Rham cohomology [2] yields an isomorphism
(2.1)

∼

2n
2n
Hcrys
(X0 , K) −→ HdR(X, K).

p-ADIC HODGE THEORY AND MODULAR FORMS

5

Suppose that Z0 is the specialization of a cycle Z on X. The cycle class c of Z lies
2n
(X, K) (where the filtration on this vector space isin the nth filtered piece of HdRof course the Hodge filtration), and under the isomorphism (2.1), c0 is identified
with c. (See [9] for the construction of cycle classes in crystalline cohomology, anda proof of this last claim.) We conjecture that the converse is true.
Conjecture (2.2). Suppose that Z0 is a codimension n cycle on the special fibre
X0 of the smooth proper W -scheme X , such that under the isomorphism (2.1) the2nclass c0 lies in the nth filtered piece of HdR(X, K). Then there is a cycle Z on X
which specializes to Z0 .
This conjecture is an obvious p-adic analogue of the variational Hodge conjecture
discussed in [12].
Remark (2.3) (a) Conjecture (2.2) is known in the case when n = 1 [3, thm. 3.8].
The method of proof is analogous to that used by Kodaira and Spencer to provethe classical Lefschetz (1, 1)-theorem [17].
(b) Conjecture (2.2) is also known in the special case when X is the product ofabelian varieties A and B, and Z0 is the graph of a morphism between the special
fibres A0 and B0 [3, thm. 3.15]. (As the proof of this result makes clear, thisis a crystalline incarnation of the Serre-Tate deformation theory of morphisms ofabelian varieties.)
Remark (2.4) We will elaborate on conjecture (2.2) in the particular case when
Z0 is a correspondence between the special fibres X0 and Y0 of two smooth proper
W -schemes X and Y. That is, Z0 is a cycle on the product X0 × Y0 , of codimension
n + dim(X0 ), say.
The Künneth formula and Poincaré duality together show that2(dimX0 +n)
Hcrys(X0 × Y0 , K)2(dimX0 +n)−ii= ⊕i Hcrys(X0 , K) ⊗ Hcrys(Y0 , K)i−2ni
= ⊕i Hom(Hcrys(X0 , K), Hcrys(Y0 , K)),and the cycle class c(Z0 ) decomposes as a direct sum c(Z0 ) = ⊕i ci (Z0 ), where eachi−2nici (Z0 ) ∈ Hom(Hcrys(X0 , K), Hcrys(Y0 , K))is the map on cohomology induced by the correspondence Z0 . In particular, if Z0
is the graph Γf of a morphism f : Y0 → X0 of k-schemes (so that n = 0), then eachci (Γf ) is the map on degree i cohomology induced by f .
Let X and Y denote the generic fibres of X and Y respectively. The Künneth
formula and Poincaré duality are equally valid for the de Rham cohomology of X×Y(regarded as filtered K-vector spaces), and are respected by the isomorphism (2.1).
Hence we see that c(Z0 ) lies in the (dimX + n)th piece of the Hodge filtration on2(dimX+n)
HdR(X × Y, K) if and only if each ci (Z0 ) is of degree n with respect to the
Hodge filtration on its source and target. In particular, if Z0 = Γf , we see that the
hypothesis of conjecture (2.2) is satisfied for c(Γf ) if and only if the maps induced
by f on cohomology respect the Hodge filtration on their source and target.
It is useful (indeed, crucial for the applications to conjecture (0.1)) to observethat conjecture (2.2) has a reformulation as a statement generalizing remark (2.4) to6

MATTHEW EMERTON

the context of motives (statement (ii) of proposition (2.7) below). We now explain
this.
We work in the setting of homological equivalence motives (also called Grothendieck motives) over K admitting good reduction over the ring W . It is not ourintention to attempt to expound a complete theory of such objects. We recall as
much as we need of this theory in order to explain the statement for motives that
we need in order to deduce conjecture (0.1) from conjecture (0.2). We refer to[16, 18, 22] for a more detailed discussions of motives, and to [9, rem. B.3.8] for a
discussion of motives over K with good reduction over W .
A homological equivalence motive M over K is a triple (X, π, n) consisting of asmooth proper K-scheme X, a correspondence π from X to itself, and an integer
n. The correspondence π is required to induce an idempotent endomorphism of
the total cohomology of X. (As to what cohomology theory we are using, see thefollowing remark.) The motive M = (X, π, n) is to be thought of as the nth Tate
twist of the image of π acting on the total cohomology of n. In particular, we define
the total cohomology of M to be equal to be the graded vector space obtained as
the nth Tate twist of the kernel of the total cohomology of X. We say that M has
good reduction over W if X admits an extension to a smooth proper W -scheme X ,Remark (2.5) We can use any of a number of cohomology theories and obtainthe same notion of motive: for example, Q` -adic étale cohomology of X for any
prime `, or the de Rham cohomology of X. To see that these do give the same
notion of motive, one uses the comparison isomorphisms afforded by the existenceof topological singular cohomology on X (together with the Lefschetz principle).
Note that these comparison isomorphisms are compatible with the action of correspondences.
We can also use cohomology theories attached to the special fibre of X ; forexample Q` -adic étale cohomology (for ` 6= p), or crystalline cohomology. To seethat these give the same notion of motive, one uses the proper base change and local
acyclicity of smooth morphisms to compare Q` -adic cohomology on the generic andspecial fibres of X (assuming ` 6= p), and the comparison isomorphism (2.1) between
the de Rham cohomology of the general fibre and the crystalline cohomology of thespecial fibre of X . Although it is redundant for the purposes of this remark, one
also has the p-adic comparison isomorphism between p-adic étale cohomology ofthe geometric generic fibre and the crystalline cohomology of the special fibre [7,thm. 5.6].
In the case that the cohomology theory that we use is computed on the special
fibre of X, we should also note that the correspondence π can be specialized to thisspecial fibre, and thus induces an endomorphism on cohomology in this case also.
Furthermore, the comparison isomorphisms between cohomology on the generic andspecial fibres of X are compatible with respect to specialization and the action ofcorrespondences.
If M = (X, π, n) is a homological equivalence motive over K, admitting good
reduction over W , then we define the special fibre M0 of M to be the homologicalequivalence motive over k given by the triple (X0 , π0 , n) (here π0 is the specializationof the correspondence π to the special fibre X0 of the smooth proper X which
extends X). Although X may not be uniquely determined, the motive M0 is welldefined up to isomorphism in the category of homological equivalence motives overk. (If X 0 is another smooth proper W -scheme extending X, then the graph of thep-ADIC HODGE THEORY AND MODULAR FORMS

7

identity in the generic fibre of X × X 0 specializes to a correspondence from X0 to
X00 which induces an isomorphism on cohomology.)
If M is a homological equivalence motive over K, admitting good reduction∗
over W , we define Hw.a.
(M, K) to be the crystalline cohomology of the special fi∗
bre Hcrys (M0 , K), equipped with the filtration induced by the Hodge filtration on∗
HdR(M, K) and the isomorphism (2.1). (The subscript w.a. is for weakly admis∗
sible, and is justified by the fact that Hw.a.
(M, K), equipped with its Frobenius
endomorphism and filtration, is a weakly admissible filtered φ-module in the senseof [8, §4.4]; this follows from the comparison theorem of [7, thm. 5.6], together with
[8, prop. 5.4.2 (i)].)Now suppose given a pair of homological equivalence motives M and M 0 over K,
both having good reduction over W . Let Z0 be a morphism from the special fibre
M0 of M to the special fibre M00 of the motive M 0 . Then Z0 induces a morphism∗∗c∗0 (Z0 ) : Hcrys(M0 ) → Hcrys(M00 ),and hence a morphism
(2.6)

∗
∗
c∗0 (Z0 ) : Hw.a.
(M0 ) → Hw.a.
(M00 ),

which respects the Frobenius endomorphisms of these filtered φ-modules, but which
may not respect their filtrations.
If Z0 is the specialization of a morphism Z : M → M 0 , then (2.6) certainly
will respect the filtrations. The following proposition discusses the converse of thisstatement.
Proposition (2.7). Conjecture (2.2) is equivalent to the following statement:
If Z0 : M0 → M00 is a morphism between the special fibres of two homologicalequivalence motives M and M 0 over K, both of which admit good reduction overW , for which the morphism (2.6) respects the filtrations on its source and target,then there is a morphism of motives over K, Z : M → M 0 , which specializes to Z0 .
Proof. Let us suppose first that the given statement holds, and see that conjecture (2.2) follows. For this, it suffices to note that if X is a smooth proper variety
over W , then giving a codimension n cycle on the special (respectively, the generic)fibre of X is the same as giving a morphism of the special (respectively, the generic)fibre of the motive Q(−n) to the motive of X0 (respectively, the motive of X). (Recall that Q(−n) denotes the motive given by the triple (Spec K, id, −n).)
Now suppose that conjecture (2.2) holds. Let Z0 : M0 → M00 be a morphism of
the special fibres of the motives M and M 0 , and suppose that the morphism (2.6)
preserves the filtrations on its source and target. Suppose that M = (X, p, m) and
M 0 = (Y, q, n), and let X and Y be extensions of X and Y respectively to smoothproper schemes over W . By definition, there is a correspondence U0 on X0 × Y0 , ofcodimension n − m + dim(X0 ), such that q0 ◦ U0 and U0 ◦ p0 are both homologicallyequivalent to U0 , and such that Z0 is induced by the correspondence U0 . The
assumption that (2.6) preserves the filtrations on its source and target is equivalent(as the discussion of remark (2.4) shows) to the assumption that c(U0 ) lies in the2(n−m+dim(X0 ))(n − m + dim(X0 ))th piece of the Hodge filtration of HdR(X × Y, K).
Conjecture (2.2) implies that there is a cycle U on X × Y specializing to U0 . Now
take Z : M → M 0 to be the morphism induced by the correspondence U . 

8

MATTHEW EMERTON

Let us fix a motive M over K, admitting good reduction over W . Let σ denotethe canonical Frobenius automorphism of K. Then we may pull-back M along σ toobtain the motive M σ over K, which again admits good reduction over W (since σ
restricts to an automorphism of W ). There is a natural morphism from the special
fibre of M σ to the special fibre of M , given by the relative Frobenius morphism.
Lemma 2.8. Let M be a homological equivalence motive over K, which admitsiigood reduction over W . The morphism Hw.a.
(M σ , K) → Hw.a.
(M, K) induced by
the relative Frobenius map M0σ → M0 respects the filtrations on its source andtarget if and only if the same is true of the canonical Frobenius endomorphism ofi(X, K).
Hw.a.
Proof. This follows by the usual comparison between relative and absolute Frobenius. For notational simplicity we treat the case when M is the motive corresponding to the generic fibre of a smooth proper W scheme X . The general caseifollows from this. The Frobenius endomorphism on Hw.a.
(X, K) is determined byii∼
the isomorphism Hw.a. (X, K) = Hcrys (X0 , K), and the endomorphism of the lattercohomology group induced by the absolute Frobenius endomorphism of X0 . Thisabsolute Frobenius endomorphism can be factored as the composite
FX /k

(p)

0
X0 −→
X0

∼

−→ X0 ,∼

where the second arrow is the base-change of the isomorphism k −→ k given
by absolute Frobenius. Thus we can think of the Frobenius endomorphism φ ofi
(X, K) as being the composite of the map (of K-vector spaces; it will generally
Hw.a.
i
i
(X, K)(X (σ) , K) → Hw.a.
not respect the filtrations on its source and target) Hw.a.
iinduced by the relative Frobenius FX0 /k and the isomorphism Hw.a. (X, K) →∼i
(X (σ) , K) determined by the isomorphism X (σ) −→ X obtained by baseHw.a.
changing the automorphism σ of K, which is σ-linear and which does preserve thefiltrations. It follows that φ preserves Hodge filtrations if and only if the same istrue of the K-linear endomorphism induced by the relative Frobenius of X0 . The
lemma follows from this discussion. 
3. higher weight forms.
In this section we explain how conjecture (2.2) implies conjecture (0.1) for modular forms of arbitrary weight k ≥ 2.
Theorem (3.1). Let N be a positive integer, p a prime not dividing N , g a classicalnormalized Hecke eigenform of weight k ≥ 2 on Γ1 (N ), and E the finite extensionof Q generated by the coefficients of g. Suppose that for every prime ℘ of E lying
over p, the form g℘ is ordinary, and the slope k − 1 form f℘ lies in the image of†(N ) → Sk† (N ). Then conjecture (2.2) implies that g is a CM form on
θ : S−(k−2)Γ1 (N ).
Proof. Scholl [21] explains how to attach to g a homological equivalence motive
M defined over Q equipped with an action of E which makes it of rank two overE, having good reduction modulo p (when regarded as a motive over the fraction
field Qp of Zp = W (Fp )), and whose associated p-adic Galois representation isexactly ρg .

p-ADIC HODGE THEORY AND MODULAR FORMS

9The crystalline comparison theorem [7, thm. 5.6] shows that the restriction ofρg to a decomposition group at p is a crystalline Galois representation (in the sense∗of [8, 5.1.4]), and that Hw.a.
(M ) is the crystalline Dieudonné-module of ρg . The∗
graded pieces of the Hodge filtration of Hw.a.
(M ) are supported in degrees zero andk − 1, and each is free of rank one over E ⊗Q Qp .
Proposition (1.2) shows that the restriction of ρg to an inertia group at p splits as
a direct sum of two distinct characters (the trivial character and χk−1 ). It follows∗
that Hw.a.
(M ) also splits as a direct sum∗
Hw.a.
= H0 ⊕ Hk−1 ,with each of H0 , Hk−1 free of rank one over E ⊗Q Qp , such that Hodge filtration of
H0 (respectively, of Hk−1 ) is supported in degree zero (respectively, degree k − 1).
∗
As a corollary, we see that the Frobenius endomorphism of Hw.a
(M ) must respect∗
the Hodge filtration of Hw.a. (M ). Lemma (2.8) then implies that the graph of∗
the Frobenius endomorphism of M0 preserves the Hodge filtration on Hw.a.
(M ).
Assuming that conjecture (2.2) holds, we find (by appealing to the equivalent formof it given by proposition (2.7)) that there is a morphism Z of M which lifts the
Frobenius endomorphism of M0 . This morphism (that is, the underlying cycle thatgives rise to it) is defined a priori only over Qp , but can in fact be assumed to be
defined over some finite extension L of Q, since M is defined over Q, and since
specializing cycles preserves their homological equivalence class.
To complete the argument, it is convenient to work with a field of coefficients(rather than the ring of coefficients E ⊗Q Qp ). Corresponding to the factorization
∼ Q
∼ Q
∼
∗
E ⊗Q Qp −→ ℘|p E℘ , we have factorizations ρg −→ ℘|p ρg,℘ and Hw.a.
(M ) −→Q∗∗℘|p Hw.a. (M )℘ . For each ℘, the weakly admissible module Hw.a. (M )℘ corresponds
via the crystalline comparison theorem to the restriction of ρg,℘ to the decompo∼∗(M )℘ −→
sition group at p, and for each ℘ we also have an isomorphism Hw.a.
H0,℘ ⊕ Hk−1,℘ .
Now the morphism Z of M over L induces an endomorphism of each ρg,℘ restricted to Gal(Q/L). We claim that this endomorphism is non-scalar. For if it werescalar, then (again by applying the crystalline comparison theorem) we would see∗that the Frobenius endomorphism of Hw.a.
(M )℘ would be scalar. But this is notpossible, because the Frobenius endomorphism must act on H0,℘ with slope zeroand on Hk−1,℘ with slope k − 1 (since each of these filtered φ-modules is weakly
admissible). (This is a particularly transparent case of [5, thm 3.1].)
We conclude that each ρg,℘ is not absolutely irreducible when restricted toGal(Q/L), since the endomorphism induced by Z will have at least one onedimensional eigenspace defined over E ℘ (since it is non-scalar) and this will be aGal(Q/L)-closed subspace of ρg,℘ . It follows from [20, prop. 4.4, thm. 4.5] thatg is a CM modular form. (An examination of the arguments shows that [20,prop. 4.4] remains true if one replaces “irreducible” with “absolutely irreducible”
in its statement.) This shows that conjecture (2.2) implies conjecture (0.1) for
arbitrary weights k ≥ 2. Note that the proof of theorem (1.3) is really just a specialization of this argument to the case of k = 2, taking advantage of remark (2.3) (b) to know thatthe relevant case of conjecture (2.2) is satisfied, and using the structure theory of
the p-divisible groups of abelian varieties and [24, thm. 4] to replace the generalcomparison theorem.

10

MATTHEW EMERTON

References
1. Atkin, A. O. L., Lehner, J., Hecke operators on Γ0 (m), Math. Ann. 185 (1970), 134–160.
2. Berthelot, P., Ogus, A., Notes on crystalline cohomology. Math. Notes, vol. 21, Princeton
Univ. Press, Princeton, 1978.
3. Berthelot, P., Ogus, A., F -Isocrystals and de Rham cohomology. I, Invent. Math. 72 (1983),
159–199.
4. Coleman, R. F., Classical and overconvergent modular forms, Invent. Math. 124 (1996), 215–
241.
5. Coleman, R. F., Edixhoven, B., On the semi-simplicity of the Up -operator on modular forms,
Math. Ann. 310 (1998), 119-127.
6. Deligne, P., Formes modulaires et représentations `-adiques. Sém. Bourbaki, éxp. 355, SLN
179 (1969), 139–172.
7. Faltings, G., Crystalline Cohomology and p-adic Galois Representations, Algebraic Analysis,
Geometry and Number Theory (J. I. Igusa, ed.), John Hopkins Univ. Press, Baltimore, 1989,
pp. 25–80.
8. Fontaine, J. M., Représentations p-adiques semi-stables, Astérisque 223 (1994), 113–184.
9. Gillet, H., Messing, W., Cycle classes and Riemann-Roch for crystalline cohomology, Duke
Math. J. 55 (1987), 501–538.
10. Gouvea, F. Q., Arithmetic of p-adic modular forms, SLN 1304 (1988).
11. Gross, B. H., On a tameness criterion for Galois representations associated to modular forms(mod p), Duke Math. J. 61 (1990), 445–517.
12. Grothendieck, A., On the de Rham cohomology of algebraic varieties, Inst. Hautes Étud. Sci.
Publ. Math. 29 (1966), 95–103.
13. Hida, H., Galois representations into GL2 (Zp [[X]]) attached to ordinary cusp forms, Inv.
Math. 85 (1986), 545–613.
14. Hida, H., Nearly ordinary Hecke algebras and Galois representations of several variables, Algebraic Analysis, Geometry and Number Theory (J. I. Igusa, ed .), John Hopkins Univ. Press,
Baltimore, 1989, pp. 114–134.
15. Katz, N., Serre-Tate local moduli, SLN 868 (1982), 138–202.
16. Kleiman, S., Motives, Algebraic Geometry, Oslo, 1970 (F. Oort, ed.), Walters-Noordhoff,
Groningen, 1972, pp. 53–82.
17. Kodaira, K., Spencer, D. C., Groups of complex line bundles over compact Kähler varieties,
Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 868–872.
18. Manin, Yu. I., Correspondences, motifs and monoidal transformations, Math. USSR Sb. 6(1968), 439–470.
19. Messing, W., The crystals associated to Barsotti-Tate groups: with applications to abelianschemes, SLN 264 (1972).
20. Ribet, K. A., Galois representations attached to eigenforms with nebentypus, SLN 601 (1976),
17–36.
21. Scholl, A. J., Motives for modular forms, Invent. Math. 100 (1990), 419–430.
22. Scholl, A. J., Classical motives, Proc. Symp. Pure Math. 55 (1994), Part I, 163–187.
23. Shimura, G., Class fields over real quadratic fields and Hecke operators, Ann. of Math. 95(1972), 130–190.
24. Tate, J. T., p-divisible groups, Proceedings of a conference on local fields, NUFFIC Summer
school, Driedberger, Springer-Verlag, New York and Berlin, 1967, pp. 76–83.
University of Chicago
Department of Mathematics
5734 S. University Ave.
Chicago, IL 60637
E-mail address: emerton@math.uchicago.edu