VANISHING FOURIER COEFFICIENTS OF HECKEEIGENFORMSFRANK CALEGARI AND NASER TALEBIZADEH SARDARI

Abstract. We prove that, for fixed level (N, p) = 1 and p > 2, there are only
finitely many Hecke eigenforms f of level Γ1 (N ) and even weight with ap (f ) =0 which are not CM.

1. Introduction
Lehmer [Leh47] raised the question of whether τ (n) = 0 forQany of the non∞trivialFourier coefficients of Ramanujan’s Delta function ∆ = q n=1 (1 − q n )24 =Pτ (n)q n . He proved that if τ (n) = 0 for some n, then necessarily τ (p) = 0 fora prime p|n. Lehmer’s problem remains open, as does the analogous question for
any cuspidal eigenform f of level one. If one weakens the hypothesis further andassumes only that f has level N for some N prime to p, then there are a numberof ways in which ap (f ) = 0, including the following:(1) If f is a modular form with CM arising from an imaginary quadratic
field F/Q in which p is inert, then ap (f ) = 0.
(2) If f is a weight two modular form arising from an elliptic curve E/Q withgood supersingular reduction at p, and p ≥ 5, then ap (f ) = 0.
In this paper, we examine a vertical analogue of Lehmer’s conjecture where p isfixed and we vary the weight. Our main theorem is as follows:
Theorem 1.0.1. Fix a prime p > 2 and an integer (N, p) = 1. Then there are only
finitely many non-CM Hecke eigenforms of level N and even weight with ap (f ) = 0.
We shall deduce from this the following:
Corollary 1.0.2. Fix a prime p. There are only finitely many eigenforms of level 1
with ap (f ) = 0.
Our arguments are not effective. The existence of non-CM (modular) elliptic
curves E/Q which are supersingular at p shows that some exceptions must be
included. The assumption p > 2 and the assumptions on the weight are not intrinsic
to our method, but rather reflect the absence of certain R = T theorems either
when p = 2 or when the residual representation ρf |G(Q(ζp )) is reducible. The weight
condition can be weakened to requiring either that the weight n is even or n − 1 is
not divisible by (p + 1)/2.
We now explain two further motivations for considering this problem (in addition
to the analogy with Lehmer’s question).
The first author was supported in part by NSF Grants DMS-1701703 and DMS-2001097. The
second author was supported in part by NSF Grant DMS-2015305.
1

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F. CALEGARI AND N.TALEBIZADEH SARDARI(∞)

1.1. Analogies with counting Maass forms. Let AQ

denote the finite adeles,(∞)
let G = GL(2)/Q, and let U ⊆ G(AQ ) denote a compact open subgroup.

The
problem of counting spaces of cuspidal modular forms of level U and weight n ≥ 2amounts to computing the sumX(1)
dim π Uas π ⊂ L2cusp (G(Q)\G(AQ ), χ) ranges over all cuspidal automorphic representations (with a fixed suitably chosen central character χ) such that π∞ corresponds
to a discrete series representation Dn of weight n. In contrast, the problem ofcounting spaces of algebraic Maass forms with eigenvalue λ = 1/4 amounts to thesame sum (1) except now where π∞ corresponds to a particular principal series representation. The philosophical explanation for why the first sum can be estimated
precisely using the trace formula while the latter can not is that discrete series representations have positive measure in the Plancherel measure of the unitary dualof PGL2 (R) whereas any fixed principal series does not. (For this perspective on
counting automorphic forms, see [Shi12].) Finally, consider the problem of counting
modular forms of weight n ≥ 2 and level prime to p with ap = 0 (the subject of
this paper). This amounts to computing the same sum (1) where once more π∞corresponds to the discrete series Dn , but now one additionally requires that πp
is the spherical representation of GL2 (Qp ) with given central character and with
Satake parameters α and β satisfying α + β = 0. The obstacle in computing thissum is the same problem as for Maass forms except now the difficulties have moved
from the place ∞ to the place p, namely, the representation πp up to twist has
zero measure in the Plancherel measure of the unitary dual of PGL2 (Qp ). From ananalytic point of view, these difficulties are quite similar. Using the trace formula,
one can try to estimate (1) where now π∞ or πp respectively are now allowed torange over some class of unitary representations of positive measure (amounting
to allowing the Laplace eigenvalue λ or the Hecke eigenvalue ap to vary in an εball around λ = 1/4 or ap = 0 respectively) and then try to control the error as εbecomes small. The (upper) bounds one obtains in this way typically (see the discussion before Theorem 1 in [Duk95]) have the shape O(V / log V ) where V is thetrivial bound (which in the case of eigenforms with ap = 0 amounts to V  n).
Thus one is led to ask whether the extra arithmetic structure present when considering the latter question allows one to improve upon this analytic estimate (forwhich the answer is clearly yes).
1.2. Analogies with a question of Greenberg. Consider an irreducible modular Galois representation
ρf : GQ → GL2 (Qp )of weight n ≥ 2. If f has CM by an imaginary quadratic field F/Q in which p
splits, then the restriction of ρf to GQp splits into a direct sum of characters. A
well-known open question (attributed to Ralph Greenberg [GV04]) asks whether
the converse holds:
Question 1.2.1. Suppose that ρf splits after restriction to GQp . Does f necessarily
have CM?
Equivalently, is any local splitting of ρf due to a global splitting over some finite
extension of Q? The condition that ap = 0 for a modular form of level prime

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

3

to p turns out to imply (see Theorem 2.1.1 below) that the representation ρf is
induced after restriction to GQp . If f has CM by an imaginary quadratic field F/Q
in which p is inert, then ρf is globally induced from F and the restriction of ρf
to GQp is induced from the unramified quadratic extension K/Qp . Hence theproblem we are considering is the analogue of Question 1.2.1 when split is replaced
by induced. The analogy here is not perfect, however, since (as noted above) nonCM elliptic curves with supersingular reduction provide a negative answer to thisquestion in general whereas the answer to Greenberg’s question is expected toalways be positive. Note that in addition to supersingular elliptic curves there areother isolated counterexamples, including
q + 4q 3 − 8q 4 − 5q 5 − 22q 6 − 11q 9 + . . . ∈ S4 (Γ0 (95), Q),
q − 2q 2 + 4q 4 + 2q 5 − 7q 7 − 8q 8 − 27q 9 + . . . ∈ S4 (Γ0 (154), Q)
with a2 = 0 and a3 = 0 respectively. The listed forms are identified by their labels 95.4.a.a and 154.4.a.b in the L-functions and modular forms database [LMF20].
1.3. Some preliminaries on Group representations. We recall some standard
facts about Galois representations. Letρ : Γ → GLn (Qp )be any continuous homomorphism of a compact group Γ (such as a local or global
Galois group with its natural topology). As explained in [Ski09, §2], ρ takes values in GLn (E) for some finite extension E/Qp which comes with a fixed embedding E ,→ Qp , and thus also fixed embeddings OE ,→ Zp and kE = OE /πE ,→ Fp .
Since Γ is compact, the image of ρ preserves a lattice Λ ⊂ E n and thus gives rise
to a representationρΛ : Γ → Aut(Λ) ' GLn (OE ).
We obtain from this a corresponding residual representationρΛ : Γ → GLn (kE ) ⊂ GLn (Fp ).
The representation ρΛ may depend on Λ. On the other hand, by the Brauer–
Nesbitt theorem, the semisimplification of ρΛ does not depend on any choices [Böc13,§3.2]. We denote the correspond semisimple representation Γ → GLn (Fp ) by ρ.
If k ⊂ Fp is any field such that ρ is valued in GLn (k), we say that k is a coefficientfield for ρ. By abuse of notation, we shall also let ρ denote the representation of Γ
to GLn (k) for any coefficient field k.
We now prove a few straightforward group theory lemmas to be used in thesequel.
Lemma 1.3.1. If H is an index 2 subgroup of G and χ is a character of H,
then IndG
H χ is reducible if and only if χ extends to a character of G.
Proof. If V = IndG
H χ is reducible, then V ' χ1 ⊕ χ2 for two characters χi of G.
Restricting to H, we deduce that χi |H = χ for at least one i, and hence χ extends
to a character of G. Conversely, if χ extends to a character of G (which by abuse ofnotation we also denote by χ), then V ' χ ⊕ (χ ⊗ ηG/H ) where ηG/H is the the nontrivial quadratic character of G with kernel H. In particular, V is reducible.
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F. CALEGARI AND N.TALEBIZADEH SARDARI

Lemma 1.3.2. Suppose G acts irreducibly on a 2-dimensional vector space V
over Fp , but this action becomes reducible after restriction to a normal subgroup H
such that G/H is cyclic of order prime to p. Then the projective image of G is dihedral, and the representation V is induced from an index two subgroup. Moreover,either the index two subgroup is unique and contains H, or the projective imageof G is D4 ' (Z/2Z)2 and V is induced from exactly three index two subgroupsof G, precisely one of which contains H.
Proof. We first claim that V |H is completely decomposable. Let V 0 ⊂ V be a
one-dimensional H-stable submodule. Since V is irreducible as a G-module, there
exists g ∈ G such that gV 0 6= V 0 . But since H is normal in G, it follows that gV 0
is also an H-stable submodule of V and hence there is a decomposition V ' V 0 ⊕gV 0 as H-modules. We deduce that the representation V restricted to H is the
direct sum of two characters, which implies that the projective representation of H
associated to V has cyclic image (given by the image of the ratio of the characters)
and has order prime to p. The assumption that G/H is cyclic of order prime to pthen implies that the image of G in PGL2 (Fp ) is metacyclic of order prime to p.
The finite subgroups of PGL2 (Fp ) of order prime to p consist of the groups Z/r,
D2r , A4 , S4 , and A5 (see for example [Ser72, §2.5]). The latter three groups are
not metacyclic. If the projective image of G is cyclic, then the image Γ of G on V
has the property that Γ/Z(Γ) is cyclic. But Γ/Z(Γ) is cyclic only if it is trivial,which implies that Γ is abelian and acts reducibly on V . Thus the projective imageof G is the dihedral group D2r for some r ≥ 2. If V is induced from more than
one index two subgroup, then the projective image of G must contain at least two
cyclic subgroups of index two, and for D2r and r ≥ 2 this happens only for r = 2,
when there are precisely three such subgroups. Taken together, this proves thelemma.

Lemma 1.3.3. Let (A, m) be a local Artinian ring with residue field k. Let VA
be an A[G]-module which is free of rank 2 over A. Assume that there exists a
decomposition VA /m =: Vk ' Uk ⊕ Uk0 of G-modules where dim Uk = dim Uk0 = 1
and Uk is not isomorphic to Uk0 . Then any G-equivariant decomposition:
VA ' UA ⊕ UA0
is unique, and — possibly swapping the factors — UA /m = Uk and UA0 /m0 = Uk0 .
Proof. If UA is any A[G]-module which is free of rank one as an A-module, then
all the Jordan–Hölder factors of UA are isomorphic to UA /m. Assume that there
exists two decompositions VA ' UA ⊕ UA0 and VA ' TA ⊕ TA0 . Since Uk is notisomorphic to Uk0 , the decomposition Vk = Uk ⊕ Uk0 is unique and thus (up toreordering) we may assume that UA /m = Uk and also TA /m = Uk . If TA 6= UA ,then I := (TA + UA ) ∩ UA0 must be non-trivial. Viewing I as a subspace of TA + UA
(which is a quotient of TA ⊕ UA ) and UA0 respectively, we deduce that all the
Jordan–Hölder factors of I are all both isomorphic to Uk and isomorphic to Uk0 , acontradiction if I is non-trivial. Hence TA = UA and similarly TA0 = UA0 .

1.4. Preliminaries on Galois deformations. If k is a finite field, we let W (k)denote the Witt vectors of k, andh·i : k × → W (k)×

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

5

the Teichmüller map. Let C denote the category of local Artinian W (k)-algebras (A, m)
with a fixed identification A/m = k.
Throughout this paper, we shall often consider 2-dimensional representations V =Vk of a group Γ over a field k.
Definition 1.4.1. Assume that Vk is absolutely irreducible as a representationof Γ. A deformation VA of Vk to A ∈ C consists of an A[Γ]-module VA free over A
such that VA ⊗A A/m ' Vk . Two deformations are equivalent if they are isomorphic
as A[Γ]-modules.
We now explain why this notion of deformation coincides with the alternate
description in terms of matrix representations in [Maz89]. A fixed choice of basis
for Vk gives rise to a representation: ρ : Γ → GL2 (k). A choice of basis for VA
lifting the given choices of basis for Vk gives rise to a representationρ : Γ → GL2 (A)such that the corresponding residual representation is isomorphic to ρ, and any
such ρ gives rise to a module VA . If ρ and ρ0 are two representations such that theunderlying modules VA and VA0 are isomorphic, then they are conjugate by somematrix M . Since we are assuming Vk is absolutely irreducible, it follows by Schur’slemma that this matrix must be scalar modulo m. Hence (after scaling) we may
assume M ∈ I +mM2 (A), which is the usual notion of strict equivalence in [Maz89].
The deformation functors D : C → Sets we consider will all be pro-represented
by complete local Noetherian rings. We shall assume basic familiarity with thefurther theory of Galois deformations as contained in [Maz89, Gou01].
2. The Argument
2.1. Local consequences of the condition ap (f ) = 0. Suppose we have a cuspidal modular eigenform f ∈ Sn (Γ1 (N ), Zp ), and let
ρf : Gal(Q/Q) → GL2 (Qp )denote the corresponding Galois representation. By the main theorem of [Sch90,
Sai97], the p-adic representation ρf |GQp is crystalline, and the characteristic polynomial of crystalline Frobenius is x2 − ap (f )x + pn−1 χ(p), where χ is the Nebentypus character of f . For irreducible 2-dimensional crystalline representationsof Gal(Qp /Qp ), the characteristic polynomial of crystalline Frobenius is enoughto determine the representation uniquely by [Bre03, Prop.3.1.1]. When ap (f ) = 0,
there is a very simple description of the corresponding local Galois representation
which we now describe. Let K/Qp denote the unique unramified quadratic exten×
sion. By local class field theory, there is a unique character K × → Gab
K → K×which sends p to 1 and z ∈ OK to z. With respect to the two embeddings of K×into Qp , this gives two characters ε2 , ε02 from GK to Qp which are permuted by
the action of Gal(K/Qp ). We have the following result which follows from [Bre03,
Prop 3.1.2]:
Theorem 2.1.1 (Breuil). Suppose that ap (f ) = 0. Then GQ
ρf |GQ = IndGKp εn−1⊗ψ2pfor some unramified character ψ with ψ 2 = χ|GQp .

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F. CALEGARI AND N.TALEBIZADEH SARDARI

Note that ψ is a priori only uniquely defined up to the unramified quadratic
character ηK/Qp , but since ρf |GQp ⊗ ηK/Qp ' ρf |GQp , either choice is correct.
×

Following §1.3, the characters ε2 and ε02 from GK to Fp correspond by local class××
field theory to the maps K × → Gab
K → kK sending p to 1 and z ∈ OK to z mod pcomposed with the two embeddings of kK ' Fp2 into Fp . Moreover, we also havea corresponding identification
 G

Q
(2)
ρf G = IndGKp εn−1⊗ψ2
QpFor any ρf , we fix a coefficient field k = kE ⊂ Fp as above such that ρf is valued
in GL2 (k). For the remainder of §2 (with the exception of of §2.6) we assume
that p > 2. We now show that ρf is absolutely irreducible after restriction to GQp .
Lemma 2.1.2. Assume that ap (f ) = 0.
(1) The representation ρf |GQp is absolutely irreducible if n − 1 is not divisibleby (p + 1), and in particular absolutely irreducible whenever n is even.
(2) The representation ρf |GQ(ζp ) is absolutely irreducible if n − 1 is not divisibleby (p+1)/2. In particular, if ρf |GQ(ζp ) is reducible and n is even, then p ≡ 1
mod 4 and (n − 1) ≡ (p + 1)/2 mod (p + 1).
Proof. The representation ρf |GQp is induced from a character of an index 2 subgroup. Thus, by Lemma 1.3.1, to prove part (1) it suffices (via the descriptionof ρf |GQp in equation (2)) to show that the character εn−1
of GK does not extend
2
n−1
to GQp . Assume otherwise. Then ε2coincides with its Gal(K/Qp ) conjugate.
By local class field theory, the action of Gal(K/Qp ) on Gab
K coincides with theaction of Gal(K/Qp ) on K × under the Artin map. Since the non-trivial elementof Gal(K/Qp ) acts on the residue field k of K by Frobenius, it follows that thep(n−1)
conjugate of ε2 is ε02 = εp2 . Thus we may assume that εn−1
= ε2and hence2(p−1)(n−1)

ε2

= 1.

Since ε2 has order |k × | = p2 − 1, this forces (n − 1) to be divisible by (p + 1). This
proves part (1).
Now suppose that ρf is irreducible as a representation of G = GQp but becomes
reducible over H = GQp (ζp ) . Since G/H = Gal(Qp (ζp )/Qp ) is cyclic of order prime
to p, it follows by Lemma 1.3.2 that ρf |GQp is induced from an index two subgroupof GQp containing GQp (ζp ) . In particular, it must be induced from GL where L/Qp
is the ramified quadratic extension inside Qp (ζp ). On the other hand, ρf |GQp is
also induced from the unramified extension K/Qp , and so ρ|GQp is irreducible andinduced from at least two distinct quadratic fields. By Lemma 1.3.2, it follows thatthe projective image of GQp is isomorphic to (Z/2Z)2 , and that the projective image
restricted to GK has order 2. Since the representation ρf restricted to GK is (up top(n−1)

twist) the direct sum εn−1
⊕ ε2, it follows that the ratio of the characters ε2n−1
2p(n−1)and ε2is quadratic, or equivalently that2(p−1)(n−1)

ε2

= 1.

It follows that (n − 1) is divisible by (p + 1)/2, which proves part (2) after notingthat (p + 1)/2 is even if p ≡ 3 mod 4.
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7

The proof of Lemma 2.1.2 is the only place where we use the assumption thatthe weight n is even. The proof otherwise proceeds without any further changes
required under the weaker assumption that (n − 1) is not divisible by (p + 1)
and (n − 1) is divisible by (p + 1)/2 only if p ≡ 1 mod 4.
2.2. Some reductions. By a theorem of Jochnowitz [Joc82], there are only finitely
many irreducible modular residual representations of level N . By class field theory,×
there are only finitely many Nebentypus characters χ : (Z/N Z)× → Zp of level N .
Moreover, if
×hχi : (Z/N Z)× → k × → W (k)× → Zp
is the Teichmüller lift of the mod mZp reduction of χ, then χ/hχi is valued in 1 +mZp . Any such character is the square of a unique character valued in 1 + mZp ,√because 1 + x = 1 + x/2 + . . . converges in Zp for p > 2. Thus, after a finite
global twist, we may assume that the Nebentypus characters of f are fixed andvalued in W (k)× .
In particular, to prove Theorem 1.0.1, we may assume the following:
Assumption 2.2.1. There exist an infinite number of cuspidal eigenforms f oflevel Γ1 (N ) and even weight n with ap (f ) = 0 satisfying the following further
assumptions:(1) All such f have the same fixed residual representationρ = ρf : GQ → GL2 (k).
(2) There is an isomorphism
 G
Q
ρ|GQ = IndGKp εn−1⊗ψ2
pfor fixed n and fixed ψ : GQp → k × .
(3) If ψ denotes the character hψi : GQp → W (k)× , then the Nebentypus
character χ of f restricted to GQp is equal to ψ 2 . By Theorem 2.1.1, thisevaluated at p
is equivalent to saying that the representation det(ρf )|Gab
Q
p

ab
2
considered as an element of Q×
p → GQp via the Artin map is ψ (p).

2.3. Local deformation rings. In this section, we define some local deformation
rings associated to ρ.
Recall that we have fixed a coefficient field k for ρ. After increasing k if necessary,
we may assume that the eigenvalue of any element in the image of ρ lands in k, andmoreover that OK ⊆ W (k), where W (k) is the Witt vectors of k.
Associated to ρ|GQp is a local universal deformation ring Rloc which is a complete
local Noetherian W (k)-algebra (it represents the functor D recalled below). We now
construct a quotient of this ring corresponding to deformations which are “induced”
from K. Let Vk denote the underlying representation of ρ over k. After restricting
to GK , there is a canonical splitting Vk = Uk ⊕ Uk0 such that GK acts on Uk and Uk0
by εn−1⊗ ψ and (ε02 )n−1 ⊗ ψ respectively, and Uk is not isomorphic to Uk0 .
2
Definition 2.3.1 (Locally induced deformations). For a local Artinian W (k)algebra (A, m) ∈ C , let Dloc (A) denote the deformations VA of Vk to A. Let Dind (A)denote the subset of deformations VA which admit a splitting VA = UA ⊕ UA0 into
free A-modules of rank one such that UA and UA0 are GK -modules which are GK deformations of Uk and Uk0 respectively.

8

F. CALEGARI AND N.TALEBIZADEH SARDARILet Dind,ψ (A) denote the subset of Dind (A) such that the action of p ∈ K × →×
Gab
K on UA is given by ψ(p), where ψ : GQp → W (k) is the Teichmüller lift of ψas in Assumption 2.2.1(3).
Finally, let DU denote the deformations UA of Uk .
Let OK (p) = 1 + mK denote the units in K × which are 1 (mod p). Since p > 2,
the exponential map gives an isomorphism exp : mK → OK (p). The group mK
may be further identified as a topological group with (Zp )2 via an arbitrary choiceof a basis for mK over Zp .
Lemma 2.3.2. Dind is pro-represented by a complete W (k)-algebra Rloc,ind which
is isomorphic to the universal deformation ring of
εn−1⊗ ψ : GK → k × .
2
In particular, Rloc,ind ' W (k)JOK (p) ⊕ Zp K is smooth of relative dimension 3over W (k). Dind,ψ is pro-represented by the quotient Rloc,ind,ψ ' W (k)JOK (p)Kof Rloc,ind .
Proof. It suffices to identify the functors Dind and DU . Given an element of Dind ,there exists a corresponding free A-submodule UA ⊂ VA deforming Uk by definition,
and a decomposition VA = UA ⊕UA0 . This decomposition is unique by Lemma 1.3.3.
GQ
Conversely, given a deformation UA of Uk , then VA = IndGKp (UA ) gives an elementof Dind (A), and this gives the desired identification (one easily checks that thesemaps are mutual inverses). Hence the two deformation functors coincide.
Let ΓK denote the Galois group of the maximal pro-p abelian extension of K. By
×d×b
local class field theory, we have Gab
K ' K ' OK (p)⊕kK ⊕ Z and then (since p > 2)
we have an isomorphism ΓK ' OK (p) ⊕ Zp . It follows that the 1-dimensional
deformation ring associated to any character χ : GK → k × is isomorphic to
W (k)JΓK K ' W (k)JOK (p) ⊕ Zp K,

where the Zp factor on the right hand side is topologically generated by p ∈ K × .
The corresponding Galois representation
×K × → Gab
K → W (k)JΓK K

is given explicitly as follows:(3)

x 7→ hχ(x)i[ϕ(x)],where ϕ(x) ∈ ΓK is the image of x under the natural map K × → Gab
K → ΓK . In
particular, when χ = εn−1
ψ,
we
have
the
desiredisomorphism(notingthat ε2 (p) =2
1 and ψ is the Teichmüller lift of ψ)
Rloc,ind,ψ ' Rloc,ind /(hψ(p)i[ϕ(p)] − ψ(p)) ' Rloc,ind /([ϕ(p)] − 1) ' W (k)JOK (p)K.

We now define local deformation rings Rloc,unr and Rloc,split .
Definition 2.3.3 (Unramified and split local deformation rings). Let M/Qp denotethe fixed field of ker(ρf |GQp ). Let M unr denote the maximal unramified extensionof M . We define subfunctors Dunr and Dsplit of D as follows:(1) Dunr (A) ⊂ Dloc (A) consists of deformations VA such that the action of GQp
on VA factors through Gal(M unr /Qp ).

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

9(2) Dsplit (A) ⊂ Dunr (A) ⊂ Dloc (A) consists of deformations VA such that theaction of GQp on VA factors through Gal(M/Qp ).
Let Dunr,ψ (A) and Dsplit,ψ (A) denote the subsets of Dunr and Dsplit respectively
22such that the action of p ∈ K × → Gab
K on ∧ VA is given by ψ (p). (Equivalently,
2
the determinant character of VA evaluated on p is ψ (p).)
Lemma 2.3.4. The functor Dunr is a subfunctor of Dind , and the functor Dunr,ψ
is a subfunctor of Dind,ψ . The functors Dunr , Dsplit and Dunr,ψ , Dsplit,ψ are prorepresentable by quotients of Rloc,ind and Rloc,ind,ψ respectively. There are isomorphisms and surjections as follows:
W (k)JOK (p) ⊕ Zp K ' Rloc,ind

Rloc,ind,ψ ' W (k)JOK (p)KW (k)JZp K ' Rloc,unr

Rloc,unr,ψ ' W (k)W (k) ' Rloc,split

Rloc,split,ψ ' W (k)where the maps in the first column send all elements of OK (p) to 1 for the firstmap and all elements of Zp to 1 for the second.
Proof. We start by proving that Dunr (A) ⊂ Dind (A). Let G = Gal(M unr /Qp ),let H = Gal(M unr /K), and let I ⊂ H ⊂ G denote the inertia group. There is
an isomorphism M unr = M.Qunr
p . We deduce that I is cyclic of order prime to p
and H is abelian. It follows that the action of I on VA is diagonalizable. Since theaction of I on Vk decomposes as the direct sum Uk ⊕ Uk0 of distinct characters, itfollows that the action of I on VA decomposes as UA ⊕UA0 where UA and UA0 are freerank one A-modules which reduce to Uk and Uk0 modulo m respectively. Since H
is abelian and contains I, it follows that hUA for h ∈ H is also a free rank one Amodule which is preserved by I, and thus VA = hUA ⊕ hUA0 . By Lemma 1.3.3, wededuce that hUA = UA or UA0 , and the former is ruled out by noting that hUk = Uk .
Hence the decomposition VA ' UA ⊕ UA0 extends to a decomposition of H-modules.
Since the image of GK in G is H, this says precisely that VA ∈ Dind (A).
Having shown that Rloc,unr is a quotient of Rloc,ind , we can reinterpret thefunctors Dunr , Dsplit in terms of the deformations of the one dimensional character εn−1
⊗ ψ : GK → k × . The unramified lifts correspond to deformations of2
the unramified character times the Teichmüller lift of the ramified character, andthe split lift corresponds to the Teichmüller lift of the entire representation. Since
the action of Gal(K/Qp ) on K × fixes p, the action of p ∈ K × → Gab
K on UA for
any element of Dind (A) coincides with the action of p on UA0 , and thus the subsets Dunr,ψ (A) ⊂ Dunr (A) and Dsplit,ψ (A) ⊂ Dsplit (A) are given by imposing thecondition that the image of p of any deformation of our character acts by ψ(p). The
explicit descriptions of these rings can then be read off from the explicit descriptionof Rind in Lemma 2.3.2 and from equation (3).

2.4. Global deformation rings. We now consider some global deformation rings
associated to ρ : GQ → GL2 (k).

10

F. CALEGARI AND N.TALEBIZADEH SARDARI

Definition 2.4.1. Let D(A) denote the deformations of Vk which are unramified
outside N p.
The functor D(A) is represented by the universal global deformation ring Rglob .
The ring Rglob is an Rloc -algebra by Yoneda’s lemma. This allows us to define thekey deformation ring R of interest, together with auxiliary rings Runr and Rsplit .
Definition 2.4.2. Let R = Rglob ⊗Rloc Rloc,ind,ψ , let Runr = Rglob ⊗Rloc Rloc,unr ,and let Rsplit ' Rglob ⊗Rloc Rloc,split .
The ring R represents the functor Dglob,ind,ψ (A) of deformations VA unramified
outside N p such that VA |GQp ∈ Dind,ψ (A). Similarly (following Lemma 2.3.4) therings Runr and Rsplit represent the subfunctor of Dglob,ind (A) for which the corresponding representations locally factor through Gal(M unr /Qp ) and Gal(M/Qp )respectively, where M is the fixed field of ρ|GQp . Because Rsplit,unr,ψ ' Rsplit,ψ byLemma 2.3.4, the ring Rsplit is a quotient of R. On the other hand, Runr need not
be a quotient of R. The construction of R guarantees that all of our eigenforms(satisfying Assumption 2.2.1) give rise to Qp -valued points of R.
The last global deformation ring we consider parametrizes deformations of ρwhich are globally induced from a quadratic field F . If ρ is induced from GF , thenthere is a decomposition Vk ' Uk ⊕ Uk0 as GF -modules. Since GK ⊂ GF , thisdecomposition must coincide with the unique such decomposition of GK -modules
for the unramified quadratic extension K/Qp .
Definition 2.4.3 (CM deformation rings). Suppose that ρ : GQ → GL2 (k) is
induced from a quadratic field F/Q. Let DCM,F (A) denote the subset of deformations D(A) of VA which admit a splitting VA = UA ⊕ UA0 into free A-modules ofrank one such that UA and UA0 are GF -modules which are GF -deformations of Uk
ab2
and Uk0 respectively, and such that the image of p ∈ Q×
p → GQp on ∧ VA is given2by ψ (p).
Lemma 2.4.4. There exist at most three quadratic fields F/Q such that ρ : GQ →GL2 (k) is induced from F . For any such F , the functor DCM,F (A) ⊂ D(A) is
pro-representable by a ring RFCM which is a quotient of R.
Proof. The first claim follows from Lemma 1.3.2. The second claim follows fromthe fact that any element of DCM,F (A) gives an element of Dind,ψ (A) by restriction
to GK ⊂ GF .

Remark 2.4.5. If ρ is induced from a quadratic extension F/Q, the field F/Q
is either real or imaginary. There is some abuse of notation to call RFCM a CM
deformation ring when F is real. This will not cause any issues since we use only
the fact that all CM deformations of ρ of level N lie on RFCM for some F (which will
be imaginary). When F/Q is a real quadratic field, the maximal pro-p extensionof F unramified outside N p is the compositum of the Zp -cyclotomic extension
with a finite extension of F , and so RFCM (Qp ) consists of a finite number of Artin
representations up to twist.
2.5. The ring R is small. We have now constructed a deformation ring R which
captures the eigenforms F with ap (f ) = 0. The first key step is establish a finitenessresult for this ring.
Lemma 2.5.1. R is finite as a module over Rloc,ind,ψ = W (k)JOK (p)K.

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

11

Proof. Let mRloc,ind,ψ denote the maximal ideal of Rloc,ind,ψ . By Nakayama’s lemma,
to show that R is finite over Rloc,ind,ψ it suffices to show that R/mRloc,ind,ψ is finite
over Rloc,ind,ψ /mRloc,ind,ψ ' k. Now by Lemma 2.3.4, we have isomorphisms
k = Rloc,ind,ψ /mRloc,ind,ψ ' Rloc,split /mRloc,split ,and thus
R/mRloc,ind,ψ ' Rsplit /mRloc,split .
If Rsplit is finite over Rloc,split ' W (k), then Rsplit /mRloc,split is certainly finite
over Rloc,split /mRloc,split ' k, and we would be done. The finiteness of Rsplit follows
immediately from [AC14, Thm.1(2)] under the additional Taylor–Wiles hypothesisthat ρ|GQ(ζp ) is absolutely irreducible. Indeed, that reference proves the stronger
claim that Runr is finite over W (k), and Rsplit is a quotient of Runr .
It suffices to consider the remaining case when the Taylor–Wiles hypothesis fails,
or equivalently that ρ|GQ(ζp ) is reducible. This certainly implies that ρ|GQp (ζp ) is
reducible, and hence, by Lemma 2.1.2, we may assume that n − 1 ≡ (p + 1)/2
mod (p + 1) and that p ≡ 1 mod 4. But now we may invoke Theorem A.0.1 of theappendix.
Remark 2.5.2. An alternative approach to proving finiteness is to specialize toa height one prime ideal p of Rloc,ind,ψ corresponding to a representation of the
GQp k−1form εm)⊗ψ, where 2 ≤ k ≤ p−1 and ε1 is the cyclotomic character.
1 ⊗(IndGK ε2
GQIn order for the corresponding residual representation to agree with (IndGKp ε2n−1 )⊗(p+1)

ψ, it suffices (using the identity ε1 = ε2
congruence is satisfied:) to chose m and k such that the followingm(p + 1) + k − 1 ≡ (n − 1) or (n − 1)pmod (p2 − 1).

We may take k ≡ n mod (p + 1) unless n ≡ 0 mod (p + 1), and we may take k ≡(p + 3 − n) ≡ 2 − n mod (p + 1) unless n ≡ 2 mod (p + 1), and so m and k exist
as long as p ≥ 5. The corresponding deformation ring Rloc,ind,ψ /p is a quotient of
the crystalline local deformation ring with Hodge–Tate weights [m, m + k − 1], andis thus a twist of a crystalline deformation ring of weight [0, k − 1] which is in the
Fontaine–Laffaille range. Since one expects to be able to prove R = T theorems
in this context (exploiting the fact that the corresponding local deformation rings
are Cohen–Macaulay), this leads to explicit bounds on dimQp (R/p)[1/p] in termsof dimensions of spaces of modular forms of weight at most p − 1, although in thisapproach one would also need to deal separately with the case when ρ|G(Q(ζp )) is
reducible.
We now turn to the study of the finite Λ := W (k)JOK (p)K-module R. The
eigenforms with ap (f ) = 0 captured by R give rise to a map from R to Qp andthus a prime of R. Any such prime is contained inside a minimal prime of R, andsince R is Noetherian, there are only finitely many minimal prime ideals, and thus
we may assume that there are infinitely many non-CM points which lie inside afixed minimal prime P, or equivalently lie on a fixed irreducible component R/Pof R.
Lemma 2.5.3. Suppose that there are infinitely many modular Galois representations ρf of level dividing N giving rise to points of R/P. Then the support of R/P
is all of Λ.

12

F. CALEGARI AND N.TALEBIZADEH SARDARI

Proof. The ring R/P is a finite Λ module by Lemma 2.5.1. The support of a finite
module is closed, and thus it suffices to show that the support includes a Zariski
dense subset of Λ. Since there are only finitely many Galois representations of any
fixed weight, we may assume that R/P has points ρf for modular eigenforms f of
infinitely many different weights. If f has weight n, then we may explicitly write
down the corresponding point of Λ. We make the explicit identification:
OK (p) ' (1 + p)Zp ⊕ (1 + p)η Zp ,√where η = u for any fixed non-quadratic residue u. We may then take X =[1 + p] − 1 and Y = [(1 + p)η ] − 1. The classical modular forms we are considering
all correspond to specializations where z ∈ O× maps to z n−1 , or equivalently to
X 7→ (1 + p)n−1 − 1, Y 7→ (1 + p)η(n−1) − 1.

(4)

It suffices to show that any infinite collection of these points are Zariski dense in Λ.
The problem is that the Zariski closure is trying to be given by the equation
H = η log(1 + X) − log(1 + Y ) = 0,

(5)

but this is not an element of Λ⊗Qp because the denominators grow without bound.
(Note that we have chosen the field k so that η ∈ OK ⊆ W (k).) Alternatively, theZariski closure wants to be (1 + X)η − (1 + Y ) = 0, although the corresponding
formal power series (unlike H) doesn’t even converge for all |X|, |Y | < 1 as we shallsee shortly in Sublemma 1 below.
Suppose the Zariski closure of these points is given by the vanishing set of F (X, Y )in W (k)JX, Y K. Choose a primitive pm th root of unity ζm for each m and let πm =1 − ζm . There is an inclusion
F (X, Y ) ∈ W (k)JX, Y K ⊂ W (k)[πm ]JX, Y K ⊂ W (k)[πm ]JX/πm , Y /πm K,

which amounts to considering the restriction of functions bounded by 1 on the
open unit ball B(1) to functions bounded by 1 on the open ball B(πm ). (Here B(r)denotes the open ball centered at the origin with radius |r|.)
Sublemma 1. Suppose that v(η) = 0 but η ∈/ Zp . Thenm−1

Hm := (1 + X)ηp

(6)

− (1 + Y )pm−1.

is an element of W (k)[πm , Y ]JX/πm K ⊂ W (k)[πm ]JX/πm , Y /πm K, but is not anelement of W (k)[πm+1 ]JX/πm+1 , Y /πm+1 K ⊗ Qp .
Proof. It suffices to analyze the growth of the coefficient of X n as n varies. The
coefficient is explicitly given by a binomial coefficient, and hence its valuation is!
 m−1 
n−1
Y
ηp
m−1
v
=v
(ηp
− i) − v(n!)
ni=0Our assumptions on η imply that the valuation of ηpm−1 −i is v(i) when v(i) ≤ m−1
and m − 1 otherwise. Hence the valuation of the X n coefficient is equal to

 X

m−1
∞ 
X
n−1
n
(7)
1+
−
.
pk
pk
k=1

k=1

A lower bound for this expression is given by


m−1
∞ 
∞ 
∞
X n  X
X
X
n
n
n
n
−
=
−
≥
−
= − m−1
,
pk
pk
pk
pk
p
(p − 1)
k=1

k=1

k=m

k=m

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

13

whereas an upper bound when n = pr and r ≥ m − 1 is given by
 r
 X


m−1
∞  r
r
X
X
n
p −1
1
p
pr
1+
−
=
−
=
−
1
−
.
pk
pk
pk
(p − 1)pm−1
pr−m−1
k=1

k=1

k=m

m−1

Since v(πm ) = 1/(p(p−1)), the lower bound implies that Hm ∈ W (k)[πm , Y ]JX/πm K.
On the other hand, the upper bound shows that this cannot be improved.

Now Hm also vanishes at all the weights z 7→ z n−1 . We claim that Hm isirreducible in W (k)[πm ]JX/πm , Y /πm K. Viewing Hm as an element inside the larger
ring W (k)[πm ]JX/π1 , Y /π1 K, we do have the factorization
Hm =

m−1
pY


i
1 + Y − (1 + X)η ζm−1.

i=1

Certainly any factorization over the smaller ring is thus promoted to a factorization
over W (k)[πm , Y ]JX/πm K. But if there exists a factor with r < pm−1 terms, thenthe constant term considered as a polynomial in (Y + 1) will be a non-zero multipleof (1 + X)ηr , which does not lie in this ring by Sublemma 1, a contradiction.
Because Hm ∈ W (k)[πm ]JX/πm , Y /πm K is irreducible, we deduce that F (X, Y )must vanish at all points in B(πm ) where Hm vanishes. But then F (X, Y ) mustvanish at the finitely many pairs (ξ1 −1, ξ2 −1) of p-power roots of unity with v(ξi −1) > v(πm ). But repeating this with m arbitrarily large implies that F (X, Y )vanishes at all such pairs of p-power roots of unity, which is impossible because
they are Zariski dense in Λ.
We note in passing that (formally) limm→∞Hm (X, Y )= H(X, Y ).
pm−1

Lemma 2.5.4. If a component R/P has infinitely many points which correspondto modular Galois representations which are not CM, then the support of CM points
on R/P is either empty or lies on a proper closed subscheme of Λ.
Proof. If there are no CM points the result is immediate. If R/P has a single point
which admits CM by F/Q, then certainly ρ is induced from F . By Lemma 2.4.4,
there are at most three such fields F , and it suffices to prove the lemma for any
given F . In particular, the points with CM by F all give rise to points on RFCM andhence on the intersection R/P⊗R RFCM . Either this is all of R/P, which contradicts
the assumption, or, because R/P is irreducible, it has positive co-dimension. But
since R is finite over Λ, the dimension of R and any of its quotients coincides withthe dimension of its support, and hence we are done.
To complete the proof of Theorem 1.0.1, it suffices to show that if the supportof R/P is all of Λ, then R/P contains a Zariski dense set of points which are
CM. To this end, we use a variation of the idea of Ghate–Vatsal [GV04] to prove
local indecomposability of non-CM Hida families by specializations in weight one.
×
Consider points in Λ corresponding to maps OK (p) → Qp with finite image andsuch that the ratio of any such map to its Gal(K/Qp )-conjugate has order greaterthan 60. These points are clearly Zariski dense in Λ. The assumption that R/P
has full support means that for any such specialization we obtain corresponding
global Galois representations:ρ : GQ → GL2 (Qp )14

F. CALEGARI AND N.TALEBIZADEH SARDARI

which locally at p has finite image on inertia at p. Thus, by [PS16, Thm.0.2]
when ρ|G(Q(ζp )) is irreducible and by [Sas, Thm.1] otherwise (noting that we may
assume that p ≡ 1 mod 4 and hence ρ is induced from a real quadratic extension),
it follows that ρ is modular of weight one. By construction, the image of inertia
in the corresponding projective representation has order greater than 60 (given our
choice of point of Λ), and hence the global projective representation also has order
greater than 60. This ensures that the projective image is not of exceptional type
(A4 , S4 , or A5 ). It follows that ρ must be of dihedral type. Exactly as in [GV04],
all but finitely many of these forms must additionally be of CM type. But then we
have produced a Zariski dense set of CM points on R/P, a contradiction, and weare done.
Remark 2.5.5. The fact that any infinite set of characters of the form z 7→ z n
×are Zariski dense in some irreducible component of Λ = Spec(Zp JOKK) can beviewed as a special case of a local p-adic analogue of Lang’s conjecture [Lan83](See also [Ser18]). The classical analogue of our example is the statement that any
infinite set of points (exp(ηx), exp(x)) are Zariski dense in (C× )2 whenever η 6∈ Q.
2.6. Proof of Corollary 1.0.2. If f is a form of level 1 which is CM, then thecorresponding automorphic representation is induced from a character on someimaginary quadratic field K/Q. But then the level of f will be divisible by any
prime dividing the discriminant of K, a contradiction. Hence Corollary 1.0.2 follows
immediately for all p > 2. For p = 2, we prove directly that if a2 (f ) = 0 then ρf isdihedral, and so f is CM, from which the result follows by the argument above. Notethat for N = 1 and p = 2 the representation ρ will have trivial semi-simplification(see [Che08, Lemme 1.7]), and it follows that the image of ρf factors through the
maximal pro-2 extension of Q unramified outside 2. By [Che08, Prop 1.8], it follows
that the image of ρf is isomorphic to the image of ρf restricted to inertia at 2. But
the assumption that a2 (f ) = 0 implies that ρf is locally induced, which now implies
it is also globally induced, and thus CM.
3. Acknowledgments
This paper owes its genesis to two conversations between the authors during theirrespective number theory seminars, one in Wisconsin in February of 2018, and thesecond in Chicago in January of 2020. The first author would like to thank George
Boxer from whom he learned the surprising fact that the characters z → z n are×
Zariski dense in Zp JOK
K, and both authors would like to thank Patrick Allen andToby Gee for comments. The second author would like to thank Professors Nigel
Boston, Jordan Ellenberg, Lue Pan and Richard Taylor for their comments on theearlier versions of this work and also he would like to thank the hospitality of thedepartment of mathematics of University of Chicago.
Appendix A. Finiteness of unramified deformation rings
Let ρ : GQ → GL2 (k) be an absolutely irreducible odd Galois representation ofG
the form IndGQ
χ, where L/Q is the quadratic subfield L ⊂ Q(ζp ). Suppose that,L
up to unramified twist:
p+1
GQ
(8)
ρ|GQ = IndGKp εn−1
, n−1≡
mod p + 1.
2
p2
The main theorem of this section is the following.

VANISHING FOURIER COEFFICIENTS OF HECKE EIGENFORMS

15

Theorem A.0.1. Assume that p ≡ 1 mod 4, so L/Q is real. Let (N, p) = 1, and
let Rsplit denote the universal deformation ring of ρ consisting of representations
which are unramified outside N and totally split at p defined in Definition 2.4.2.
Then Rsplit is finite over W (k).
Before proving this, we begin with a preliminary lemma:
Lemma A.0.2. There exists a finite extension F/Q with the following properties(1) F is totally real.
(2) If v|N , then ρ|Fv is trivial.
(3) ρ|GF is absolutely irreducible.
(4) If v|p, then Fv ' K, the unique unramified quadratic extension of Qp .
Proof. The existence of F follows immediately from [Cal12, Proposition 3.2] (seealso [MB90]). For example, one may can choose G = GL2 (k), and then make thefollowing choices:(1) φv for v|N is the map ρ|Gv → G,(2) φv for v = p is any injective map Gal(K/Qp ) → G,(3) c∞ is trivial.
The irreducibility of ρ|GF is then guaranteed by choosing F/Q to be linearly disjoint
from the fixed field M of ker(ρ) by [Cal12, Lemma 3.2] (2).
Proof of Theorem A.0.1. We begin with some reductions. If F/E is any finite extension so that ρ|GF remains absolutely irreducible, and RF and RE denote theuniversal deformation rings of ρ|GF and ρ|GE respectively, then the map RF → RE
is always finite. As a consequence, to prove the finiteness of Rsplit , it suffices toreplace Q by any totally real field in which ρ remains absolutely irreducible. We
replace Q by the field F/Q constructed in Lemma A.0.2, so that ρ|Gv is trivial
for each v|N . By Lemma A.0.2, the field Fv for v|p is precisely the unramified
extension K/Qp for all v|p. In particular,p(n−1)

ρ|Fv ' εn−1
⊕ ε22

is reducible and p-distinguished. Make an arbitrary choice of one of these charactersfor each v|p, which gives a distinguished choice Uk of the decomposition Vk =Uk ⊕ Uk0 as GFv modules for each v|p.
We now recall the deformation ring RD defined in [SW01, §2] with respect to D =(W (k), Σ, ∅) where Σ is the set of primes dividing N . Note that ρ|GF is absolutely
irreducible and satisfies all the conditions of [SW01] by construction. The ring RD
is global deformation ring of ρ unramified outside N p subject to the following
condition: for all v|p, there exists a short exact sequence
0 → UA → VA → UA0 → 0of GFv -modules where UA and UA0 are free over A of rank one and UA /m = Uk .
After extending F if necessary we may assume that [F : Q] is even and thus
the hypothesis (Heven ) of [SW01, §3] holds. By [All14, Lemma 5.1.2], there exists
a lift ρ0 of ρ giving rise to a point on RD relative to our choices both over F andany totally real extension of F in which ρ|GF remains irreducible. This implies thatcondition (Hdef ) of [SW01, §3] holds for any such F . In this situation we have acorresponding Hecke ring TD as defined in [SW01, p.196], and a surjection RD →TD of Λ-modules where Λ is an Iwasawa algebra [SW01, p.191] (denoted ΛO ).

16

F. CALEGARI AND N.TALEBIZADEH SARDARI

There is a surjection RD → Rsplit . Any deformation coming from Rsplit locally has
the form UA ⊕ UA0 where the action of GK on UA and UA0 is via the Teichmüller
p(n−1)
lifts of εn−1
and ε2
respectively. In particular, the map
2
Λ → RD → Rsplit
factors through a quotient of the form W (k), as can be seen from the formulasin [SW01] on the last line of p.191 and the first line of p.192 respectively. A more
intrinsic way to see this is that the ring Λ represents weight space and all split
representations lie in the same fixed unramified weight. Hence to prove that Rsplit
is finite over W (k) it suffices to show that RD is finite over Λ.
By taking the compositum of F with a suitably large totally subfield of a cyclotomic extension (exactly as in the first two lines at the top of page 204 of [SW01])
we may further ensure that the pair (F, ρ0 ) is good in the sense of [SW01, §4]. It
follows from [SW01, Prop 4.1] and [SW01, Prop 8.2] that all primes of RD are promodular. If TD is the Hecke ring defined in [SW01, p.196], it follows that there is
an isomorphism RD /p ' TD /p for every prime p of RD , which implies immediatelythat (RD )red = (TD )red and hence (RD )red is finite over Λ (since TD is finite over Λ
by [SW01, Lemma 3.3]). Since RD is Noetherian, it follows that RD is also finite
over Λ. But RD surjects onto Rsplit /p, and the kernel contains the image of the
maximal ideal of Λ. It follows that Rsplit /p is finite over k and hence Rsplit is also
finite over W (k), as claimed.


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Email address: fcale@math.uchicago.edu
The University of Chicago, 5734 S University Ave, Chicago, IL 60637, USA
Email address: ntalebiz@ias.edu
The Institute For Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540
USA