Séminaire BOURBAKI
62ème année, 2009-2010, no 1013

Novembre 2009

p-ADIC FAMILIES OF MODULAR FORMS[after Hida, Coleman, and Mazur]by Matthew EMERTONINTRODUCTION
The theory of p-adic families of modular forms grew out of two highly related traditions in the arithmetic theory of modular forms: the theory of congruences of modular
forms (which dates back to work of Ramanujan) and the (more recent) theory of Galois
representations attached to modular forms. The first example of a p-adic family of
modular forms was the Eisenstein family, considered by Serre in [37]. This is a familyof q-expansions, parametrized by the weight k, whose coefficients are p-adically continuous functions of k. Serre’s immediate goal in studying this family was to obtain anunderstanding of the possible congruences between the q-expansion coefficients of modular forms in different weights, especially of the constant terms, since such congruences
lead to congruences between special values of ζ-functions.
The papers [22, 23] led to a decisive shift in the theory, placing it at the centre of thearithmetic theory of modular forms. In these papers, Hida constructed p-adic familiesof cuspforms, varying continuously with the weight k, which were also simultaneous
eigenforms for the Hecke operators. Thus, in light of the known construction of Galois representations attached to Hecke eigenforms, one found that associated to thesep-adic families of cuspidal eigenforms there were corresponding p-adic families of p-adic
Galois representations. The existence of such families led Mazur to develop his general
theory of deformations of Galois representations [31], which in turn inspired further
developments [45, 43].
Hida’s constructions had a certain limitation: if f is a Hecke eigenform of weightk ≥ 1 and level N prime to p, then f appears in a Hida family if and only if (at least)
one of the roots of the pth Hecke polynomial of f is of slope zero (i.e. a p-adic unit).
This restriction was removed by the work of Coleman and Mazur [10], who constructedp-adic analytic (more precisely, rigid analytic) curves of eigenforms containing any suchform f , whether or not its pth Hecke polynomial admits a unit root; these are theso-called eigencurves.
The eigencurves are fundamentally analytic objects. One can also ask whether there
is an algebraic family (or more precisely, a scheme) that parametrizes all the f as
above, regardless of the slopes of the roots of the pth Hecke polynomial. Indeed, there
is such an object; all the eigenforms f (of arbitrary weight but some fixed level N ) are

1013–02

parametrized by the Zp points of Spec T(N ), where T(N ) is the p-adic Hecke algebraof level N . These points are no longer parametrized by weight; indeed, Spec T(N ) is
(at least conjecturally) of relative dimension three over Spec Zp . It is conjectured thatevery continuous, two-dimensional, semi-simple odd p-adic Galois representation of GQ
that is unramified outside finitely many primes corresponds to a point of Spec T(N ) forsome appropriate value of N . This is one of the main motivations for the study of thefamilies Spec T(N ), and the related p-adic families of eigenforms constructed by Hida
and Coleman–Mazur.
In Section 1 of this exposé we recall the basic theory of modular forms, Hecke operators, and the Galois representations associated to Hecke eigenforms. In Section 2, weoutline the definitions and basic results and conjectures regarding the p-adic Hecke algebras T(N ), and the families of Hida and Coleman–Mazur. We focus more on systemsof Hecke eigenvalues attached to eigenforms, rather than on the eigenforms themselves.
This is in keeping with our focus on the relationship with Galois representations (although it takes us somewhat far in spirit from the concrete viewpoint of [37]).
Acknowledgments. I would like to thank J-P. Serre for his helpful comments on an
earlier version of this article.
0.1. Notation
As usual Q, R, and C denote the fields of rational, real, and complex numbers, andZ denotes the ring of integers. For any prime p, we let Zp denote the ring of p-adic
integers, and Qp denote the field of p-adic numbers.
We let Q denote the algebraic closure of Q in C, and let Z denote the integral closureof Z in Q. For each prime p, we fix an algebraic closure Qp of Qp , and let Zp denote theintegral closure of Zp in Qp . We also fix an embedding ıp : Q ,→ Qp . This restricts toan embedding Z ,→ Zp . We write Fp to denote the residue field of Zp . It is an algebraic
closure of the field Fp of p elements. We let ordp : Qp → Z ∪ {∞} denote the p-adic
valuation, normalized so that ordp (p) = 1. If x ∈ Qp , then ordp (x) is also called theslope of x. (Thus x has finite slope if and only if x 6= 0, while x has slope zero if and×
only if x ∈ Zp .)1. MODULAR FORMS, HECKE ALGEBRAS, AND GALOISREPRESENTATIONS
1.1. Modular formsLet
H = τ ∈ C | =(τ ) > 0
denote the complex upper half-plane. The group SL2 (Z) acts on H in the usual way:aτ + ba bτ=.
c dcτ + d1013–03

Let O(H) denote the space of holomorphic functions on H. If k is an integer, then we
define the weight k-action of SL2 (Z) on O(H) as follows:(f |k γ)(τ ) := (cτ + d)−k f (γτ ),for f ∈ O(H) and γ = ( ac db ) ∈ SL2 (Z); as the notation indicates, this is a right action.
If N ≥ 1, defineΓ1 (N ) := γ ∈ SL2 (Z) | γ ≡ ( 10 1∗ ) mod N .
Definition 1.1. — A modular form (resp. cuspform) of weight k and level N is aholomorphic function f ∈ O(H) that is invariant under the weight k-action of Γ1 (N ),and for which(1)lim (f |k γ)(iy)y→∞exists and is finite (resp. vanishes) for each γ ∈ SL2 (Z). We let Mk (N ) (resp. Sk (N ))denote the space of modular forms (resp. cuspforms)of weight k and level N .
Remark 1.2. — If f ∈ O(H) is invariant under the weight k-action of Γ1 (N ), then, inorder to check if f is a modular form or a cuspform, it suffices to study the limit (1) for
finitely many γ ∈ SL2 (Z) (namely, for a set of coset representatives for Γ1 (N )\SL2 (Z)).
Remark 1.3. — If f is a modular form of weight k and level N , then, applying theinvariance property of f to the matrix ( 10 11 ) ∈ Γ1 (N ), one finds that f (τ + 1) = f (τ ).
We may thus expand the function f (τ ) as a Fourier seriesf (τ ) :=∞X

cn (f )q n ,n=−∞where q := exp(2πiτ ). Condition (1), with γ = 1, then shows that cn (f ) = 0 for n < 0(resp. for n ≤ 0 if f is a cuspform). We refer to this Fourier series as the q-expansionof f .
Clearly Mk (N ) and Sk (N ) are vector subspaces of O(H). In fact they are also finite
dimensional. (See [39] for a discussion of this and other basic facts concerning modular
forms.)
Example 1.4. — If k < 0, then Mk (N ) = 0. When k = 0, the space M0 (N ) consists
simply of the constant functions on H (and so S0 (N ) = 0). To avoid these trivial cases,
we will typically assume that k ≥ 1 in all that follows. As k increases, the dimensionsof both Mk (N ) and Sk (N ) grow essentially linearly in k (with the exception that
Mk (N ) = 0 if N = 1 or 2 and k is odd).
Example 1.5. — The simplest examples of modular forms of positive weight are theEisenstein series Ek ∈ Mk (1). These are defined for even k ≥ 4. (It is easily shown1013–04

that Mk 1) vanishes if k is odd or 0 < k < 4.) The q-expansion of Ek is given by thefollowing formula:∞−Bk X X k−1 n+(
Ek (τ ) =d )q ,
2kn=1d|nwhere Bk is the kth Bernoulli number.
There is a direct sum decomposition
Mk (1) = CEk ⊕ Sk (1).
More generally, for any N , we may decompose Mk (N ) into the direct sum of a space ofEisenstein series (typically of dimension greater than one when N > 1) and the spaceof cuspforms. (See Example 1.18 below.)
1.2. Hecke operators
Fix integers k ≥ 1 and N ≥ 1. WriteΓ0 (N ) := γ ∈ SL2 (Z) | γ ≡ ( ∗0 ∗∗ ) mod N .
Note that Γ0 (N ) contains Γ1 (N ) as a normal subgroup, and that the map
a b
7→ d mod N
c d
induces an isomorphism(2)∼Γ0 (N )/Γ1 (N ) −→ (Z/N Z)× .

A simple computation, using the normality of Γ1 (N ) in Γ0 (N ), shows that the weightk-action of Γ0 (N ) preserves Mk (N ) and Sk (N ). When restricted to these spaces,
this action obviously factors through the quotient Γ0 (N )/Γ1 (N ), and hence, via theisomorphism (2), we obtain an action of the group (Z/N Z)× on Mk (N ) and Sk (N ).
If d ∈ (Z/N Z)× , then we denote the corresponding automorphism of Mk (N ) by hdi.
(These operators are sometimes referred to as the diamond operators.)
Remark 1.6. — We note a simple but important identity for the action of the diamond
operator h−1i, namely(3)h−1if = (−1)k f,

for any f ∈ M
 k (N ). This is easily verified by considering the weight k-action of the−1 0
matrix 0 −1 ∈ Γ0 (N ) on f .
Definition 1.7. — If ` is a prime not dividing N , then we define the automorphism
S` of Mk (N ) via the formulaS` = h`i`k−2 .

1013–05

Since the diamond operators leave Sk (N ) invariant, so do the operators S` . In fact,
although it is traditional to single out the operators S` as defined above, it is theoperators `S` = h`i`k−1 that will be more important for us, as we see already in thenext definition.
Definition 1.8. — If ` is a prime not dividing N , then we define the endomorphism
T` of Mk (N ) via the formula
∞
∞
X
X
n
(4)
(T` f )(τ ) =cn` (f )q +`cn (S` f )q n` .
n=0

n=0

Remark 1.9. — It is not immediately obvious that T` , which we have defined simply
by its effect on q-expansions, actually preserves the space Mk (N ). In fact T` canbe thought of as a certain double coset operator, corresponding to the double coset
GL2 (Z` ) ( 0` 10 ) GL2 (Z` ) (see e.g. [39, Ch. 3]). From this point of view, it is easy to verify
that it preserves the space Mk (N ), as well as the subspace Sk (N ) of cuspforms.
The operator S` also has a double coset interpretation; it corresponds to the double
coset GL2 (Z` ) ( 0` 0` ) GL2 (Z` ). This is one reason to consider S` as a primary object,rather than the diamond operator h`i.
Definition 1.10. — We let Tk (N ), or simply Tk when the level N is understood,
denote the Z-subalgebra of End Mk (N ) generated by the operators `S` and T` as `ranges over the primes not dividing N . The algebra Tk (N ) is called the Hecke algebra(for the given weight k and level N ).
Remark 1.11. — Following [39, Ch. 3], one can extend Definition 1.8 and define Hecke
operators Tm acting on Mk (N ) for any positive integer m prime to N . The algebra
Tk (N ) defined above then coincides with the Z-algebra of endomorphisms of Mk (N )generated by the collection of these operators Tm .
The following result encapsulates the basic properties of the algebra Tk , and of its
action on Mk (N ).
Proposition 1.12. — The algebra Tk is commutative, reduced, and free of finite rankover Z. Furthermore, the tensor product C ⊗Z Tk acts faithfully on Mk (N ).
Remark 1.13. — The commutativity part of the statement is not difficult to verify; forexample, it is easily checked using the description of the Hecke operators in terms of
double cosets. The additional properties of Tk are then equivalent to the following
statements about the eigenspaces and eigenvalues of the Hecke operators:
1. Every eigenvalue of any of the Hecke operators is an algebraic integer. (Here one
sees the importance, when k = 1, of taking `S` rather than S` in the definition of
Tk , so as to avoid introducing denominators.)
2. The systems of simultaneous eigenvalues for the action of the Hecke operators on
Mk (N ) (which are collections of algebraic integers, by 1) are closed under theaction of Gal(Q/Q).

1013–06

3. The space Mk (N ) decomposes as a direct sum of simultaneous eigenspaces for the
Hecke operators.
Definition 1.14. — We say that f ∈ Mk (N ) is a Hecke eigenform if it is a simultaneous eigenvector for the Hecke operators `S` and T` (where ` ranges over all primes
not dividing N ), or equivalently, if there is a ring homomorphism λ : Tk → C such thatT f = λ(T )f for all T ∈ Tk .
We refer to a homomorphism λ : Tk → C as a system of Hecke eigenvalues. (Anysuch λ is the system of Hecke eigenvalues attached to some Hecke eigenform. Also,
according to the preceding remark, any such λ factors through the ring of algebraic
integers Z in Q.) If λ is a system of Hecke eigenvalues, then we write Mk (N )[λ] todenote the corresponding subspace of Hecke eigenforms.
As already noted in the preceding remark, the space Mk (N ) admits the direct sum
decompositionM
Mk (N )[λ],Mk (N ) =λwhere the direct sum is taken over all systems of Hecke eigenvalues.
Remark 1.15. — The formula (4) shows that if λ is a system of Hecke eigenvalues,
then the q-expansion of a Hecke eigenform f ∈ Mk (N )[λ], and hence the eigenformf itself, is to a large extent determined by the system of Hecke eigenvalues λ. For
example, if N = 1, then the group of diamond operators is trivial, and so `S` f = `k−1 f .
Formula (4) then shows that
λ(T` )cn (f ) = cn` (f ) + cn/` (f )`k−1
for every prime number ` (where we set cn/` = 0 if ` - n). Thus the Fourier coefficientscn (f ) (n ≥ 1) are determined recursively by the single coefficient c1 (f ), and so f isdetermined up to a scalar by its associated system of Hecke eigenvalues. In particular,
the λ-eigenspace in Mk (1) is one-dimensional.(1) If N > 1, then we find that f is determined by λ, together with the Fourier coefficients cm (f ), for those positive integers mdivisible only by primes dividing N . Thus f need not be uniquely determined (up toa scalar) by λ, and the λ-eigenspace in Mk (N ) can be of dimension greater than one.
However, the structure of this eigenspace is well-understood, either using the theory ofso-called oldforms and newforms as in [1], or in terms of the action of GL2 (A) on thespace of modular forms of weight k [25, 6]. We do not recall the details here, since
they will not be important for us. As we will explain in the following subsection, our(1)

A slight amount of caution is required here, because c0 (f ) is not directly determined by the cn (f )for n ≥ 1. However, since k ≥ 1, then in fact c0 (f ) is so determined, as one easily sees, since a constantfunction cannot be modular of weight k > 0. As Serre notes [37, Rem. 2), p. 221], one can directly
determine c0 (f ) from the cn (f ) for n ≥ 1 as follows: −c0 (f ) is the value at s = 0 of the meromorphicP∞function defined by analytic continuation of the Dirichlet series n=1 cn (f )n−s .

1013–07

attention will be focussed on the systems of eigenvalues λ themselves, rather than on
the associated Hecke eigenforms.
Remark 1.16. — Given a system of Hecke eigenvalues λ appearing in Mk (N ), it fol×lows from the definition of the operators S` that there is a Q -valued character ε of
(Z/N Z)× such that λ(`S` ) = ε(`)`k−1 . Thus we may recover the value of the weight kfrom the system of eigenvalues λ. Indeed, if ` is any prime not dividing N , thenk = (log` |λ(`S` )|) + 1.
Example 1.17. — If k ≥ 4 is even, then the Eisenstein series Ek ∈ Mk (1) is a Hecke
eigenform. The corresponding system of Hecke eigenvalues λ is given by
λ(`S` ) = `k−1 ,λ(T` ) = 1 + `k−1 .

(Here ` is an arbitrary prime, since we are in the case N = 1).
Example 1.18. — Let ψ1 : (Z/M1 Z)× → C× and ψ2 : (Z/M2 Z)× → C× be characters,
and let k ≥ 1 (unless M1 = M2 = 1, in which case we require that k ≥ 4) be chosen sothat ψ1 (−1)ψ2 (−1)(−1)k = 1. Then the following system of Hecke eigenvalues, which
we denote by λψ1 ,ψ2 ,k , appears in Mk (M1 M2 ):λψ1 ,ψ2 ,k (`S` ) = ψ1 (`)ψ2 (`)`k−1 ,λψ1 ,ψ2 ,k (T` ) = ψ1 (`) + ψ2 (`)`k−1 .

In the case when M1 = M2 = 1, we obtain the systems of Hecke eigenvalues associatedto the Eisenstein series Ek , as considered in the preceding example. In general, we referto such a system of Hecke eigenvalues as an Eisenstein system of eigenvalues.
If we writeM
Mk (N )[λ],Ek (N ) =λ Eisenstein

where the sum ranges over all Eisenstein systems of Hecke eigenvalues for which
M1 M2 = N, then we refer to modular forms f ∈ Ek (N ) as Eisenstein series. There is adirect sum decomposition
Mk (N ) = Ek (N ) ⊕ Sk (N ).
Unlike the Eisenstein systems of eigenvalues considered in Example 1.18, the systems of eigenvalues appearing in the spaces of cuspforms do not admit an elementary
description. As we will see in the following subsection, they correspond to certain Galois
representations.
Example 1.19. — We close this subsection with a careful presentation of the preceding
concepts in the case N = 1 and k = 12. In this case
M12 (1) = E12 (1) ⊕ S12 (1),where E12 (1) is one-dimensional, spanned by
∞ X
X
691
691
+
(
d11 )q n =+ q + 2049q 2 + 177148q 3 + · · · ,E12 =32760 n=1
32760
d|n

1013–08

and S12 (1) is also one-dimensional, spanned by Ramanujan’s famous cuspform∆(τ ) = q

∞
Y

n 24

(1 − q )

n=1

=

∞
X

τ (n)q n = q − 24q 2 + 252q 3 + · · · .

n=1(Here τ (n) = cn (∆) is by definition the nth Fourier coefficient of ∆.) Each of thesemodular forms is a Hecke eigenform, and correspondingly T12 admits two systems of
Hecke eigenvalues.
If we write λ1 (resp. λ2 ) to denote the system of Hecke eigenvalues attached to E12(resp. ∆), thenλ1 × λ2 : T12 ,→ Z × Z.

(5)Note that since each of these eigenforms has been normalized so that c1 = 1, we mayread off the corresponding systems of Hecke eigenvalues from the Fourier coefficients, as
in Remark 1.15. The product λ1 × λ2 provides an embedding λ1 × λ2 : T12 ,→ Z × Z. It
was first observed by Ramanujan that this embedding is not an isomorphism. Indeed,
Ramanujan showed thatX
τ (n) ≡
d11 mod 691
d|n

for every natural number n, or equivalently,
λ1 (T` ) ≡ λ2 (T` ) mod 691
for each prime `. On the other hand, it is easily verified (just by considering the cases
when ` = 2 and 3) that no such congruence holds modulo any higher power of 691, nor
modulo any other prime. Thus (5) induces an isomorphism∼

T12 −→ {(u, v) ∈ Z × Z | u ≡ v mod 691}.
∼

If p is a prime and p 6= 691, then Zp ⊗Z T12 −→ Zp × Zp ; this reflects the fact thatthe distinct systems of eigenvalues λ1 and λ2 remain distinct when reduced modulo p.
On the other hand, the tensor product Z691 ⊗Z T12 does not factor as a product inany non-trivial way; rather, it is a local ring, reflecting the congruence of λ1 and λ2
modulo 691.
1.3. Galois representations
As in the preceding section, fix integers k ≥ 1 and N ≥ 1. From a certain point ofview, it is the systems of Hecke eigenvalues appearing in Mk (N ) that are of the greatest
interest, rather than the modular forms, or even the Hecke eigenforms, themselves. Thisis because they give rise to Galois representations, as we now recall.
Choose a prime number p. If λ is a system of Hecke eigenvalues appearing in Mk (N ),then since λ takes values in the ring Z of algebraic integers, we may compose it withour chosen embedding ıp : Q ,→ Qp , and so regard λ as taking values in Zp . For
the remainder of this subsection, we regard all systems of Hecke eigenvalues as being
Zp -valued. If λ : Tk → Zp is a system of Hecke eigenvalues, then we let λ : Tk → Fp be1013–09

the homomorphism obtained by composing λ with the map Zp → Fp given by reducing
modulo the maximal ideal of Zp .
Let Σ denote the (finite) set of primes dividing N p, let QΣ denote the maximal
algebraic extension of Q in Q that is unramified outside of the primes in Σ, and write
GQ,Σ := Gal(QΣ /Q). Recall that if ` is a prime not in Σ, then attached to ` is a Frobenius
element Frob` ∈ GQ,Σ , well-defined up to conjugacy, with the property that there is aprime ideal l lying over ` in the ring of algebraic integers in QΣ that is preserved by
Frob` , such that for any algebraic integer x ∈ QΣ , Frob` (x) ≡ x` mod l. The Čebotarev
density theorem furthermore implies that the union of these conjugacy classes is densein GQ,Σ .
If M is any integer divisible only by primes dividing N p, and if ζM denotes a primitive
M th root of unity, then ζM ∈ QΣ , and so there is a group homomorphism
χM : GQ,Σ → (Z/M Z)×describing the action of the elements of GQ,Σ on ζM , namely, for any σ ∈ GQ,Σ , we haveχσ(ζM ) = ζMM(σ).

We refer to χM as the mod M cyclotomic character. It can also be characterized by
the formulaχM (Frob` ) ≡ ` mod M,
for any prime ` not dividing N p. Also, if c ∈ GQ,Σ denotes complex conjugation, thenχM (c) = −1.
We also define the p-adic cyclotomic character χ : GQ,Σ → Z×p to be the projectivenlimit over n of the mod p -cyclotomic characters χpn . Again, the character χ is characterized by the formula χ(Frob` ) = ` for any ` not dividing N p, and we also have that
χ(c) = −1.
The various cyclotomic characters give the basic examples of characters (i.e. onedimensional representations) of the group GQ,Σ . The following theorem shows that
Hecke eigenforms are a source of two-dimensional representations of this group.
Theorem 1.20. — There is a continuous, semi-simple representation
ρλ : GQ,Σ → GL2 (Qp ),uniquely determined (up to equivalence) by the condition that for each prime ` - N p, thematrix ρλ (Frob` ) has characteristic polynomial equal to X 2 − λ(T` )X + λ(`S` ).
Remarks on the proof. — The uniqueness statement of the theorem is easily proved.
Indeed, if ρ1 and ρ2 are two representations both satisfying the conditions of the theorem, then by assumption their characteristic polynomials agree on the set of elements
Frob` , which by Čebotarev density are dense in GQ,Σ . Since they are continuous, theircharacteristic polynomials then agree on all elements of GQ,Σ . It follows that ρ1 andρ2 are equivalent, as claimed, since a semi-simple finite-dimensional representation of agroup is uniquely determined, up to equivalence, by its characteristic polynomials.

1013–10

In the case when λ is an Eisenstein system of Hecke eigenvalues, the existence of ρλis also easily proved; see Example 1.24 below. On the other hand, if λ is a system of
eigenvalues attached to a cuspform, then the construction of ρλ is much less trivial. Itsconstruction is due to Eichler, Shimura, and Igusa [13, 38, 24] (in the case k = 2), to
Deligne [11] (for k > 2), and to Deligne and Serre [12] (for k = 1).
It is useful to give a name to the characteristic polynomials appearing in Theorem 1.20.
Definition 1.21. — If λ : Tk (N ) → C is a system of Hecke eigenvalues, then foreach prime ` - N , we define the `th Hecke polynomial of λ to be the polynomial
X 2 − λ(T` )X + λ(`S` ).
Remark 1.22. — Since GQ,Σ is profinite, the representation ρλ may be conjugated so as
to take values in GL2 (Zp ), and we let ρ◦λ denote such a GL2 (Zp )-valued representation
underlying ρλ . The GL2 (Zp )-valued representation ρ◦λ is not always uniquely determined
up to equivalence by λ. However, if we let ρλ denote the semi-simplification of therepresentation GQ,Σ → GL2 (Fp ) obtained by reducing ρ◦λ modulo the maximal idealof Zp , then ρλ is uniquely determined, up to equivalence, by λ, and in fact, even by
λ (as the notation suggests). Indeed, ρλ is uniquely characterized, up to equivalence,by the condition that for each prime ` - N p, the matrix ρλ (Frob` ) has characteristic
polynomial equal to X 2 − λ(T` )X + λ(`S` ). (The proof of the uniqueness is identical tothat given in the proof of Theorem 1.20.)
×Remark 1.23. — As in Remark 1.16, write λ(`S` ) = ε(`)`k−1 for some Q -valued
character ε of (Z/N Z)× . We may compose ε with the mod N cyclotomic character××
χN to obtain a Q -valued character of GQ,Σ , which we regard as being Qp -valued via
our chosen embedding ıp : Q ,→ Qp . It then follows from the condition on the determinant of ρ(Frob` ) in the statement of Theorem 1.20, together with Čebotarev density
and the given relationship between λ(`S` ) and ε(`), thatdet ρλ := (ε ◦ χN )χk−1 ,where as above χ denotes the p-adic cyclotomic character.
In particular, if c ∈ GQ,Σ denotes complex conjugation, then one computes thatdet ρλ (c) = ε(−1)(−1)k−1 = −1
(the last equality following from (3)). One says that ρλ is odd. Similarly, the representation ρλ is odd.
Example 1.24. — If λψ1 ,ψ2 ,k is an Eisenstein system of Hecke eigenvalues attached tocharacters ψi : (Z/Mi Z)× → C× and the weight k, as in Example 1.18, then it is easy
to write down a corresponding Galois representation ρλψ1 ,ψ2 ,k satisfying the conditionsof Theorem 1.20; namely, we can takeρλψ1 ,ψ2 ,k = (ψ1 ◦ χM1 ) ⊕ (ψ2 ◦ χM2 )χk−1 .

1013–11

On the other hand, if λ arises from a cuspform, then ρλ does not admit a descriptionin terms of characters. Indeed, one has the following result [35, Thm. 2.3].
Proposition 1.25. — If the system of Hecke eigenvalues λ is attached to a cuspform,
then the representation ρλ associated to λ by Theorem 1.20 is irreducible.
(p)

For any prime p, write Tk to denote the subalgebra of Tk generated by the elements`S` and T` for ` not dividing N p (i.e. we omit the Hecke operators at p). If λ : Tk → Zp(p)
is a system of Hecke eigenvalues, we write λ(p) to denote the restriction of λ to Tk , andrefer to λ(p) as the p-deprived system of Hecke eigenvalues associated to λ. Similarly,(p)(p)
we let λ : Tk → Fp denote the restriction of λ. The conditions on the Galois
representation ρλ given in Theorem 1.20 evidently depend only on λ(p) , and in fact
we can use the existence of the Galois representations attached to λ to show that λ(p)already determines λ. Indeed, we have the following more general result.
Proposition 1.26. — If λ1 and λ2 are two systems of Hecke eigenvalues such that
λ1 (T` ) = λ2 (T` ) for all but finitely many primes ` not dividing N, then λ1 and λ2
coincide.
Proof. — This is proved by the same argument used to establish the uniqueness claimof Theorem 1.20. Let q be some fixed prime not dividing N , and choose p to be distinct
from q. Let ρλ1 and ρλ2 denote the Galois representations associated to λ1 and λ2 as in
Theorem 1.20, regarded as representations over Qp . Then ρλ1 and ρλ2 have the same
traces on the elements Frob` , for all but finitely many `. Čebotarev density implies
that the set of elements Frob` (where ` ranges over all but finitely many primes notdividing N p) is dense in GQ,Σ , and so, since ρλ1 and ρλ2 are continuous, we see that theirtraces coincide. Thus they have isomorphic semi-simplifications (since we are working
over the field Qp of characteristic zero), and so their characteristic polynomials coincide
on any element of GQ,Σ . Applying this to Frobq , we find that λ1 (Sq ) = λ2 (Sq ) andthat λ1 (Tq ) = λ2 (Tq ). Since q was an arbitrary prime not dividing N , the proposition
follows.
The preceding proposition has the following technical corollary.
(p)Corollary 1.27. — The ring Tk has finite index in Tk .
(p)∼

Proof. — Since Tk is finite over Z, it suffices to show that C ⊗Z Tk −→ C ⊗Z Tk .
Equivalently, we must show that distinct systems of Hecke eigenvalues remain distinct
after omitting the eigenvalues corresponding to the Hecke operators at p. This follows
from the proposition.
Remark 1.28. — The finite index of Corollary 1.27 can be greater than 1. For example,(2)
if N = 23, k = 2, and p = 2, then the index of T2 in T2 is equal to 2. (More precisely,
√
√
(2)
T2 ∼
= Z[ 5].)= Z[(1 + 5)/2], while T2 ∼

1013–12

Remark 1.29. — Corollary 1.27 (or better, its proof) shows that λ 7→ λ(p) induces abijection between the set of homomorphisms Tk → C and the set of homomorphisms(p)
Tk → C, and hence between the set of homomorphisms Tk → Zp and the set of(p)homomorphisms Tk → Zp .

2. p-ADIC FAMILIES OF SYSTEMS OF HECKE EIGENVALUES
2.1. The p-adic Hecke algebra
Let N be a positive integer, and fix a prime p not dividing N . If k ≥ 1 is a positive
integer, then for each prime ` not dividing N we define the operators S` and T` on theLdirect sum ki=1 Mi (N ) in the obvious way: S` and T` act on each summand via the
Hecke operator with the same name.
(p)

(p)

Definition 2.1. — We let T≤k (N ), or simply T≤k if the level N is understood, denoteLthe Z-algebra of endomorphisms of ki=1 Mi (N ) generated by the operators `S` and T` ,as ` ranges over all primes not dividing N p.
Since each operator S` and T` is determined by its action on each of the direct
summands, there is a natural injection(6)(p)
T≤k ,→

k
Y

Ti .

i=1

Remark 2.2. — We could consider the analogous algebra in which we included theoperators pSp and Tp . However, for our later purposes, it is important to omit these
operators from the algebra under consideration.
Q
Proposition 2.3. — The image of (6) has finite index in ki=0 Ti .
Proof. — Given that the source and target of (6) are both finite Z-algebras, it suffices toshow that (6) becomes an isomorphism after tensoring with C over Z. This follows fromthe fact that the p-deprived systems of eigenvalues appearing in Mk (N ) are distinct
for different values of k, by Remarks 1.16 and 1.29.
Example 2.4. — Take N = 1, p = 2, and k = 6. The spaces Mi (1) for 1 ≤ i ≤ 6vanish unless i = 4 or 6, in which case they are one-dimensional, spanned by E4 and∼∼E6 respectively. Thus T4 −→ Z and T6 −→ Z, and so (6) becomes in this case anembedding
(7)
Now

(2)

T≤6 ,→ Z × Z.
∞ X
X
1
E4 =
+
(
d3 )q n ,240 n=1d|n1013–13

while∞−1 X X 5 n
E6 =+(d )q .
504 n=1d|n

One immediately checks that
1 + `3 ≡ 1 + `5 mod 12,
for all ` 6= 2. Furthermore, no analogous congruence holds modulo any larger modulus,
and thus the embedding (7) induces an isomorphism(2)∼T≤6 −→ {(u, v) ∈ Z × Z | u ≡ v mod 12}.
A similar calculation shows that(3)∼T≤6 −→ {(u, v) ∈ Z × Z | u ≡ v mod 6}.
These examples exhibit congruences similar to those discussed in Example 1.19, butinvolving congruences between systems of Hecke eigenvalues in different weights.
If k 0 ≥ k, thenkM0

Mi (N ) ⊂

i=0

k
M

Mi (N ),

i=0

and so restriction induces a surjection
(p)

(p)

T≤k0 → T≤k .
Tensoring this with Zp over Z, we obtain a surjection
(8)

(p)

(p)

Zp ⊗Z T≤k0 → Zp ⊗Z T≤k .

Definition 2.5. — The p-adic Hecke algebra T(N ), or simply T if the level N isunderstood, is defined to be the projective limit
(9)

(p)

Zp ⊗Z T≤k ,T := lim←−kwhere the transition maps are the maps (8).
Remark 2.6. — Note that since any prime ` 6= p is invertible in Zp , the operator S` =(p)`−1 (`S` ) lies in each of the algebras Zp ⊗Z Tk , for each ` - N p, and so we may regard
each of these algebras as being generated by the elements S` and T` (` - N p), just as
well as by `S` . Also, since the transition maps (8) take the elements S` and T` in thesource to the elements S` and T` in the target, these elements give rise to well-definedelements S` and T` in the projective limit T, for any prime ` - N p.
From the various embeddings (6), we obtain an embeddingYT ,→Zp ⊗Z Tk .
k≥1

1013–14

The target of this embedding is a countable product of non-zero rings; in particular,
it is not Noetherian. On the other hand, we have the following result regarding thesource.
Theorem 2.7. — The ring T is a p-adically complete, Noetherian Zp -algebra, and isin fact the product of finitely many complete Noetherian local Zp -algebras.
(p)Remarks on the proof. — Each Zp ⊗Z Tk is a finite Zp -algebra, and so is a product offinitely many complete local finite Zp -algebras. It follows that T is p-adically complete,and is a product of a countable collection of complete local Zp -algebras. The fact that
this product involves only finitely many local algebras is not formal; it is equivalent
to a statement about Fp -valued systems of Hecke eigenvalues that is the subject of
Proposition 2.8 below. The fact that these local factors are Noetherian is also notformal; it is proved via a consideration of relation between the ring T and Galois
representations, as discussed in the following subsection. (More precisely, each local
component of T is canonically the quotient of a certain Galois pseudo-deformation ring,and hence is Noetherian; see [30, §1.4] for a discussion of the latter, and in particular
Lemma 1.4.2 for a proof of the Noetherianness of pseudo-deformation rings.(2) )Informally speaking, this theorem can be thought of as showing that the phenomenon
exhibited in Example 2.4 is typical: as k grows, the power of p dividing the index of theimage of (6) in its target grows progressively larger, reflecting the existence of many
congruences modulo powers of p between systems of eigenvalues appearing in various
weights.
We present one concrete manifestation of this abundance of congruences in the following proposition (due to Jochnowitz [26], generalizing an argument of Serre in thecase N = 1), which is an important ingredient in the proof of Theorem 2.7. Indeed, its
statement is a straightforward reformulation of the claim that T has only finitely many
maximal ideals. In order to state the proposition, we introduce additional notation.
(p)Suppose given a p-deprived system of Hecke eigenvalues λ(p) : Tk → Zp . We then(p)(p)
write λ : Tk → Fp to denote the reduction of λ(p) modulo the maximal ideal of Zp .
Proposition 2.8. — As λ(p) ranges over all p-deprived systems of eigenvalues of allweights k ≥ 0, there are only finitely many possibilities for the collection of eigenvalues

(p)
(p)
λ (`S` ), λ (T` ) `-N p .
One has the following precise conjecture regarding the Krull dimension of the ring T.
Conjecture 2.9. — The ring T is equidimensional of Krull dimension 4, i.e. each
irreducible component of Spec T is of dimension 4.
(2)

Technically, the results of [30, §1.4] only apply when p is odd; however, with the appropriate modifications, they should also apply in the case when p = 2.

1013–15

Since T is a torsion free and p-adically complete Zp -algebra, this is equivalent toSpec T having relative dimension 3 over Spec Zp . This conjecture is motivated by the
known and conjectured relations between the ring T and Galois representations. (SeeRemark 2.14 below.) We will prove in Corollary 2.28 below that each irreducible component of Spec T has Krull dimension at least 4.
2.2. Galois representations again
As in the preceding subsections, we regard all systems of Hecke eigenvalues as taking
values in Zp .
Definition 2.10. — A p-adic system of Hecke eigenvalues is a homomorphism of
Zp -algebras ξ : T → Zp .
(p)

Suppose that λ(p) : Tk → Zp is a p-deprived system of Hecke eigenvalues. Since
the target of λ(p) is a Zp -algebra, this homomorphism extends to a homomorphism(p)λ(p) : Zp ⊗Z Tk → Zp . Composing this homomorphism with the natural surjection T →(p)
Tk , we obtain a homomorphism ξ : T → Zp . We refer to p-adic systems of Hecke
eigenvalues arising in this way as classical.
Theorem 2.11. — If ξ : T → Zp is any p-adic system of Hecke eigenvalues, then there
is a continuous, semi-simple representationρξ : GQ,Σ → GL2 (Qp ),uniquely determined (up to equivalence) by the condition that for each prime ` - N p, thematrix ρξ (Frob` ) has characteristic polynomial equal to X 2 − ξ(T` )X + ξ(`S` ).
Sketch of proof. — The uniqueness proof is identical to that given in the proof of
Theorem 1.20. As for existence, if ξ is a classical system, arising from the p-deprived
system of eigenvalues λ(p) , then we can clearly set ρξ := ρλ . To construct ρξ in general,
one uses the fact that the classical ξ are dense in the set of all ξ (in a suitable sense),
and then constructs ρξ by an interpolation argument.
Just as in the case of Theorem 1.20, one shows that if ξ is a p-adic system of Hecke
eigenvalues, then ρξ is odd, and so we see that T parametrizes a family of odd twodimensional p-adic Galois representations. Furthermore, one has the following fundamental conjecture to the effect that all odd two-dimensional Galois representations
should be of this form. (See e.g. the conjecture on p. 108 of [20].)
Conjecture 2.12. — If Σ is any finite set of primes containing p, and if
ρ : GQ,Σ → GL2 (Qp )is continuous, semi-simple, and odd, then ρ = ρξ for some p-adic system of Hecke
eigenvalues of some level N divisible only by primes in Σ distinct from p.
Remark 2.13. — In fact, one expects to be able to take N to be the tame (i.e. primeto-p) Artin conductor of ρ.

1013–16

Remark 2.14. — One can use techniques from Galois cohomology to show that if ρ isan odd, irreducible, continuous two-dimensional p-adic representation of GQ,Σ (for somefixed finite set of primes Σ), then the expected dimension of a neighbourhood of ρ inthe space of all such representations is three. (See Corollary 3 on p. 405 of [31]. Thisreference treats the case of mod p Galois representations, but is easily adapted to thecontext of p-adic Galois representations.) Taken together with Conjecture 2.12, thismotivates Conjecture 2.9.
Building on ideas of Gouvêa and Mazur [21] (in particular, the infinite fern, as considered in Subsection 2.5 below), together with the techniques of Wiles [45] and Taylor–Wiles [43], Böckle [3] has proved a strong result in the direction of Conjectures 2.12
and 2.9. Since the statement of his result is a little technical, we do not recall it here.
However, by appealing to a result of Kisin, one can improve the part of Böckle’s theorem
that pertains to Conjecture 2.9, as follows.
Before stating it, we recall that Q(ζp3 ) contains a unique quadratic extension of Q
when p is odd, and three such extensions when p = 2. For any such quadratic
L ⊂ Q(ζp3 ), we write GL,Σ := Gal(QΣ /L).
Theorem 2.15. — Let ξ : T → Zp be classical, and suppose that ρξ |GL,Σ is irreducible,
for each quadratic extension L ⊂ Q(ζp3 ). Then Spec T has dimension 3 in a neighbourhood of ξ.
Proof. — It follows from [29, Thm., p. 277] that this dimension is at most 3, while
Corollary 2.28 below establishes the opposite inequality. This proves the result.
2.3. Families parametrized by weight: the Eisenstein family
Since Spec T is (at least conjecturally) of relative dimension 3 over Spec Zp , one canthink of the set of all p-adic systems of Hecke eigenvalues ξ as depending on threeparameters. Unfortunately, even in those cases when Conjecture 2.9 is known, there
is no particularly canonical choice of these three parameters. A little more formally,
if Spec T has Krull dimension 4, then Noether normalization allows one to construct afinite map Spec T → Spec Zp [[T1 , T2 , T3 ]]. However, there is no canonical choice for sucha map.
On the other hand, there is a canonical map Spec T → Spec Zp [[T ]], as we now
explain. Write q = p if p is odd, and q = 4 if p = 2, and set Γ = 1 + qZp . Let
L := {` prime | ` ≡ 1 mod N q}.
We regard L as a subset of Γ. Dirichlet’s theorem on primes in arithmetic progression
shows that L is in fact dense in Γ.
Lemma 2.16. — The map L → T given by ` 7→ S` extends uniquely to a continuous
homomorphism of groups Γ → T× .

1013–17

Proof. — If ` ∈ L, and if λ is any system of Hecke eigenvalues of weight k, thenλ(S` ) = `k−2 . (Because ` ≡ 1 mod N , the diamond operator h`i is trivial.) The function(p)x 7→ xk−2 is continuous on Γ, and so the map ` 7→ S` from L → Zp ⊗Z Tk extends(p)to a continuous homomorphism Γ → (Zp ⊗Z Tk )× , for any weight k. The lemma now
follows by an easy passage to the limit.
nZp [Γ/Γp ]. This is the so-called completed group ring of Γ over Zp ;Write Zp [[Γ]] := lim←−nthere is an evident embedding of the usual group ring Zp [Γ] ,→ Zp [[Γ]]. If x ∈ Γ, we write[x] to denote the corresponding element of Zp [[Γ]] (so as to avoid confusion with the sameelement x regarded as belonging to the ring of coefficients Zp ). There is an isomorphism∼of Zp -algebras Zp [[T ]] −→ Zp [[Γ], determined by the condition T 7→ [1 + q] − 1.
The continuous map Γ → T× of the preceding lemma extends uniquely to a homomorphism of Zp -algebras(10)w : Zp [[Γ]] → T,

which we may equally well regard as a map Zp [[T ]] → T. Passing to Specs, we get the
canonical map(11)∼Spec T → Spec Zp [[Γ]] −→ Spec Zp [[T ]]referred to above. What is the meaning of this map?
Well, giving a Zp -valued point of Spec Zp [[Γ]] is the same as giving a continuous×
character κ : Γ → Zp . Thus Spec Zp [[Γ]] is the space of characters of Γ. (The iso∼morphism Spec Zp [[Γ]] −→ Spec Zp [[T ]] is then given by mapping a character κ to thevalue T = κ(1 + q) − 1; in this way, the space of continuous characters of Γ is identified
with the maximal ideal of Zp , or, in more geometric terms, the open unit disk around×
the origin of Qp .) If k is an integer, then we may define a character κk : Γ → Zp via
the formula κk (x) = xk−2 . These points are Zariski dense in Spec Zp [[Γ]] (in fact, any
infinite collection of them is Zariski dense), and so we regard Spec Zp [[Γ]] as a certain
kind of interpolation of the set of integers, and refer to it as weight space. In particular,
the Zp -valued point κk is said to be the point of weight k.
Now suppose that ξ : T → Zp is classical, arising from the system of Hecke eigenvalues
λ : Tk → Zp . One computes that the composite ξ ◦ w is equal to κk , the point ofweight k. Thus we may think of the w as mapping a system of Hecke eigenvalues to
its corresponding weight (which explains our choice of notation). From this, we also
see that w is injective (since there exist systems of Hecke eigenvalues of arbitrarily highweight), and hence that (11) is dominant.
Now the weight is a very natural parameter to consider, and so it is reasonable toask whether we can find families of systems of Hecke eigenvalues, and hence families ofGalois representations, that are parametrized by the weight. Somewhat more precisely,
we can ask whether we can find a closed subscheme Z ,→ Spec T such that the composite

1013–18

Z ,→ Spec T → Spec Zp [[T ]] is dominant with finite fibres; such a subscheme Z could
then be thought of as a family of Galois representations, parametrized by the weight.(3)Of course, if we impose no other conditions on Z, then such subschemes Z exist
for very general geometric reasons; on the other hand, a further natural condition to
impose is that Z contain a Zariski dense set of points corresponding to classical systemsof Hecke eigenvalues. The scheme Z could then be regarded as a one-dimensional family
of systems of Hecke eigenvalues, parametrized by weight, and interpolating a collectionof classical systems of Hecke eigenvalues.
Example 2.17. — The most basic example of a one-dimensional family of systems of
Hecke eigenvalues, parametrized by weight, is the Eisenstein family. This is the originalp-adic family of modular forms, introduced by Serre in [37]. We describe it here, in thelanguage of systems of Hecke eigenvalues that we have introduced.
For simplicity we take N = 1, and we fix an even residue class i mod p−1 if p is odd.
(p)Consider the p-deprived systems of Hecke eigenvalues λk associated to the Eisensteinseries Ek , for k ≥ 4 and congruent to i mod p − 1 if p is odd (resp. k ≥ 4 and even if
p = 2). Recall that these are given by(p)

λk (`S` ) = `k−1 ,(p)

λk (T` ) = 1 + `k−1 ,where ` ranges over all primes distinct from p.
We wish to rewrite these formulas slightly. Recall that Z×p = µp−1 × Γ (if p is odd)
or µ2 × Γ (if p = 2). In either case, let µ denote the first factor, and write ω : Z×
p → µ
(p)
to denote the corresponding projection. Then we may rewrite the formulas for λk as(p)

λk (`S` ) = `ω(`)i−2 `ω(`)−1

k−2

,

(p)

λk (T` ) = 1 + `ω(`)i−2 `ω(`)−1

k−2

,

where we set i = 0 if p = 2. We may evidently interpolate these formulas into a
Zp [[Γ]]-valued point of Spec T. Namely, there is a homomorphism E : T → Zp [[Γ]],defined byS` 7→ ω(`)i−2 [`ω(`)−1 ],T` 7→ 1 + `ω(`)i−2 [`ω(`)−1 ].

By construction, the composite κk ◦ E is equal to λk , for any k ≡ i mod p − 1 (or any
even k, if p = 2). Again by construction, E ◦ w is the identity on Zp [[Γ]]. Thus, in moregeometric terms, we have constructed a map Spec Zp [[Γ]] → Spec T which is a section
to the weight map w : Spec T → Spec Zp [[Γ]], namely a family of Eisenstein systems ofeigenvalues, parametrized by their weight.
(3)

We are using the word “parametrized” in a somewhat liberal sense, in that we are allowing ourfamily to be a multi-valued function of the weight, i.e. we are asking that Z → Spec Zp [[T ] have finite
fibres, but not that it necessarily be injective.

1013–19

2.4. Families parametrized by weight: Hida families and the eigencurve
In our discussion of p-adic systems of Hecke eigenvalues, we have systematicallyignored the Hecke operators Sp and Tp . This is important; for example, for the family
λk of the Example 2.17, we have λk (Sp ) = pk−2 and λk (Tp ) = 1 + pk−1 . These functions
do not interpolate well as p-adic functions of k. However, if we consider the pth Hecke
polynomial X 2 − λk (Tp )X + pλk (Sp ), we see that it has the form
X 2 − (1 + pk−1 )X + pk−1 = (X − 1)(X − pk−1 ).
One of the two roots of this polynomial is in fact constant in the family, and so interpolates without difficulty in the family. It is the second root which does not interpolate
well. This motivates the idea of changing our context slightly, and considering points
not just in Spec T, but in Spec T × Gm (here the fibre product is with Spec Z[T, T −1 ]over Spec Z, or equivalently, with Spec Zp [T, T −1 ] over Spec Zp ). To any system of Hecke
eigenvalues λ appearing in some Mk (N ), we can plot a pair of Qp -valued points in thisfibre product, whose first coordinate (for either point) is the associated classical p-adic
system of eigenvalues ξ, and whose second coordinates are the roots of the pth Hecke
polynomial of λ.
Definition 2.18. — Let X denote the set of Qp -valued points of Spec T × Gm consisting of pairs (ξ, α), where ξ : Spec T → Zp is classical, attached to some system of
Hecke eigenvalues λ : Tk → Zp with k ≥ 1, and α is a root of the pth Hecke polynomial
X 2 − λ(Tp )X + pλ(Sp ).
×

Let X ord denote the subset of X consisting of pairs (ξ, α) for which α ∈ Zp . (Thesuperscript ord is for ordinary.)
Remark 2.19. — The reason for singling out the subset X ord of X is that (since any
system of Hecke eigenvalues is a Zp -valued point of Spec T) these are precisely the pointsof X that consist of Zp -valued points of Spec T × Gm .
The following theorem, due to Hida [22, 23], describes the interpolation of the points
in X ord . (The map Spec T × Gm → Spec Zp [[Γ]] appearing in the statement of thetheorem is the one obtained by first projecting onto the factor Spec T, and then applying
the map w.)
Theorem 2.20. — The Zariski closure C ord of X ord in Spec T×Gm is one-dimensional.
The composite C ord → Spec T × Gm → Spec Zp [[Γ]] is finite, and is furthermore étalein the neighbourhood of those points of X ord that are attached to systems of Hecke
eigenvalues appearing in weight k ≥ 2.
Definition 2.21. — We refer to C ord as the Hida family, or ordinary family, of tamelevel N .
Remark 2.22. — We will see in Subsection 2.5 below that it is necessary to restrict toweights k ≥ 2 in the final statement of the theorem.

1013–20

The curve C ord is (almost) precisely a family of the type we envisaged in the previous
subsection. (We say “almost” because it lies in Spec T × Gm rather than in Spec T
itself.) On the other hand, not every classical system of eigenvalues appears in C ord ; itis certainly possible that if λ : Tk → Zp , then both roots of the pth Hecke polynomial
may have positive slope.
We thus turn to our next result, due to Coleman and Mazur [10], which deals with
the interpolation of the entire set X . In this case, taking the algebraic Zariski closure ofthese points in Spec T × Gm turns out to be too coarse of an operation, and we cannot
hope to construct an algebraic family of the type envisaged in the previous subsectionthat contains all the points of X . Rather, we will construct a rigid analytic family,
lying inside the associated rigid analytic space (Spec T × Gm )an .(4)Theorem 2.23. — The rigid analytic Zariski closure C of X in (Spec T × Gm )an is
one-dimensional. More precisely, the composite
C ,→ (Spec T × Gm )an → (Spec Zp [[Γ]])an(12)

is flat, and has discrete fibres. Furthermore, for any positive constant C, there are only
finitely many points (ξ, α) in any given fibre satisfying ordp (α) ≤ C. (In other words,
the slopes of the Gm -coordinates go to ∞ in each fibre.)
Definition 2.24. — The curve C is called the eigencurve of tame level N . The analytification of C ord is called the slope zero part, or the ordinary part, of the eigencurve.
It is a union of connected components of C.
Remark 2.25. — The map (12) is in fact étale in the neighbourhood of a point(ξ, α) ∈ X , unless α is a repeated root of the pth Hecke polynomial of the system of
eigenvalues λ giving rise to ξ. (Compare the discussion of [10, p. 5].) It is conjectured
that such repeated roots cannot occur when the weight k ≥ 2 [9].
Remark 2.26. — By construction, each of C ord and C contains a(n algebraic or rigid
analytic, as the case may be) Zariski dense set of points (ξ, α) for which ξ is a classicalsystem of eigenvalues. It is natural to ask whether the converse holds, namely, if (ξ, α)×is any Qp -valued point of C ord of C lying over the weight κk : Γ → Zp , for some positive
integer k, then is ξ classical?
The answer is no in general, for trivial reasons. One already sees this with the Eisenstein family of Example 2.17. Indeed, in the notation of that example (and assuming(4)

Concretely, if T =Qm

i=1 Zp [[T1 , . . . , Tri ]]/(fi,1 , . . . , fi,si ), thenmaSpec Zp [[T1 , . . . , Tri ]][T, T −1 ]/(fi,1 , . . . , fi,si ),Spec T × Gm =i=1an

and (Spec T × Gm )mais the rigid analytic space{ (T1 , . . . , Tri , T ) ||T1 |, . . . , |Tri | < 1, T 6= 0, fi,1 (T1 , . . . , Tri ) = · · · = fi,si (T1 , . . . , Tri ) = 0 }.

i=1

Also, (Spec Zp [[Γ]])an ∼= (Spec Zp [[T ]])an = {T | |T | < 1}, i.e. the open unit disk in Qp .

1013–21

p is odd for simplicity), if k 6≡ i mod p − 1 then the associated system of eigenvalues
is not associated to a modular form of level 1; rather, its values on `S` and T` (for(p)
` - N p) coincide with those of the system of eigenvalues λ1,ωi−k ,k (in the notation of
Example 1.18), corresponding to an Eisenstein series of level p.
Hida showed in general that if (ξ, α) is a Zp -valued (or equivalently, Qp -valued) pointof C ord lying over the character κk for k ≥ 2 (or, more generally, a character of theform ψκk , where ψ has finite order and k ≥ 2), then there is a system of eigenvalues
λ : Tk (N p) → Zp such that ξ(`S` ) = λ(`S` ) and ξ(T` ) = λ(T` ) for all ` - N p.
In the non-ordinary case, the situation is more complicated. The fibre of C over anyκk (or over any character ψκk , where ψ is of finite order) is typically infinite, and all
but finitely many of the points do not arise from a classical eigenform (of any level).
However, Coleman showed [7, 8] that if (ξ, α) is such a point, and if the slope of α
is less than k − 1, then just as in the ordinary case, there is a system of eigenvalues
λ : Tk (N p) → Zp such that ξ(`S` ) = λ(`S` ) and ξ(T` ) = λ(T` ) for all ` - N p. (One canshow, e.g. using Theorem 2.33 below, that, conversely, if such a λ exists, then the slopeof α is at most k − 1. Of course, all but finitely many of the points lying over κk haveslope > k − 1.)
Idea of proofs. — The first step in the proof of Theorems 2.20 and 2.23 is to define aspace of p-adic modular forms on which the p-adic Hecke algebra T acts. In fact one
can literally work with such a space, namely the space of generalized p-adic modular
functions of Katz (as defined in [27], see also [19] and [22]) — this is the approach takenin [22] for the ordinary case and in [10] for the general case — or with a surrogate,
constructed from the group cohomology of Γ1 (N ) and certain of its subgroups. The
cohomological approach to the ordinary case is developed in [23], and for the generalcase is developed in [42, 2]. There is another approach, via the p-adically completed
cohomology of modular curves [15, §4], which is somewhat different, and which we will
say a little about below. To simplify the exposition, from now on we will speak simply
of “the space of p-adic modular forms”, meaning either the space of generalized p-adic
modular functions, or one of the cohomological surrogates of [2, 23, 42].
The next step is to introduce an additional Hecke operator on this space, the so-called
Up -operator. In the context of p-adic modular forms, this operator has the following
effect on q-expansions:∞X
Up f =anp (f )q n .
n=0

We let T∗ denote the quotient of T[Up ] that acts faithfully on the space of p-adic modular
forms. Evidently, Spec T∗ ,→ Spec T × A1 .
Suppose for a moment that f is a modular form of weight k and level N, with p - N ,and let α and β be the roots of the pth Hecke polynomial. Then f (τ ) − βf (pτ ) isa modular form of level N p, which is a Up -eigenform with eigenvalue α. (This canbe checked directly on the level of q-expansions.) Thus Spec T∗ contains the set X ,and hence also the Zariski closure of this set. The technical difficulty that arises in1013–22

establishing the theorems is that Spec T∗ is much bigger than either C ord or C, roughly
speaking because Up has a huge kernel on the space of the p-adic modular forms, while
we are trying to construct curves lying in Spec T × Gm , i.e. systems of eigenvalues ofT∗ for which the associated Up -eigenvalue is non-zero.
It is at this point that the proofs of the two theorems diverge somewhat, with theproof of Theorem 2.20 being technically simpler than that of Theorem 2.23. The pointsof X ord correspond to eigenforms whose Up -eigenvalue is ordinary. If f is any eigenform
for T∗ whose Up -eigenvalue α is of positive slope, then limn Upn f = αn f → 0 as n → 0.
Thus by iterating Up on the space of p-adic modular forms and passing to a limit, we
can cut out the ordinary part of the space of p-adic modular forms, on which Up actswith only ordinary eigenvalues. (This process can be summarized by using the limitsof powers of Un to construct the so-called ordinary projector, which projects to theordinary part.) The quotient of T∗ acting faithfully on this ordinary part is denotedby Tord , and C ord = Spec Tord . The key fact, underlying the proof of Theorem 2.20,
is that Tord is finite over Zp [[Γ]]. This can be proved in various ways; either using
the theory of mod p modular forms, if one is working with generalized p-adic modular
functions (this is the approach taken in [22]), or by arguments with group cohomology(this is the approach of [23]).
As already indicated, the proof of Theorem 2.23 is more technical. The reason is asfollows: if m∗ is a maximal ideal of T∗ lying over a maximal ideal m of T, and if m∗ is
not ordinary (i.e. if Up ∈ m∗ ), then it follows from [19, Prop. II.3.14] thatT∗m∗ ∼= Tm [[Up ]].
(Here T∗m∗ and Tm denote completions, and Tm [[Up ]] is the formal power series ring in
Tm with variable Up .) Thus if ξ : Tm → Zp is a system of eigenvalues, we can extend
it to a system of eigenvalues of T∗m by assigning any positive-slope value of Up that
we choose; even if ξ is classical, attached to some system of eigenvalues λ : Tk → Zp ,the algebra T∗ has no way of distinguishing the positive slope roots of the pth Hecke
polynomial of λ from any other positive slope elements of Zp .
Thus one cannot reasonably interpolate the points X by algebra alone; it is necessary
to use some analysis. In the generalized p-adic modular functions setting, the key stepis to replace this space by a certain subspace of so-called overconvergent modular forms.
(This is the approach of [10].) In the cohomological framework, this step can be taken at
the beginning, by working with rigid analytic (rather than merely continuous) modular
symbols (as is done in [42, 2]). In these settings, the operator Up is a compact operator,and so has a reasonable spectral theory. One can then analyze, and obtain finitenessresults for, all of its non-zero eigenspaces, rather than just the ordinary eigenspaces.
The analysis of these eigenspaces is at the heart of the proof of Theorem 2.23.
As mentioned above, there is another approach to the proof of Theorem 2.23, via
p-adically completed cohomology [15]. In this setting, one does not directly have an
action of the Up -operator, but rather has an action of the entire group GL2 (Qp ), and the1013–23

introduction of the Up -operator and the passage to its non-zero eigenspaces is effected
in a single step, by applying the locally analytic Jacquet module functor of [14].
We make some further technical remarks. In the paper [10], the authors prove Theorem 2.23 only in the case when N = 1. The generalization to arbitrary N can be foundin [4, Part II], or [15, §4]. Also, in most of the papers cited, the authors work with
Hecke algebras that contain the Hecke operators U` for `|N, ` 6= p, as well as the operators S` and T` that we have considered. We have avoided any consideration of these
operators, since they are not essential for the consideration of Galois representations.
It is not difficult to deduce the results in the form that we have stated them from thecorresponding results cited, which perhaps involve these additional operators.
2.5. The infinite fern
The composition of the closed embedding C ord ,→ Spec T × Gm with the projection
onto the first factor gives a map(13)

C ord → Spec T,

which is very close to being injective on Zp -points. Indeed, if (ξ1 , α) and (ξ2 , β) are twosuch points mapping to the same point of Spec T, then ξ1 = ξ2 = ξ (say), and we seethat ρξ | GQp admits unramified quotients on which Frobp acts by α and β respectively.
Thus if α 6= β, we see that ρξ | GQp is unramified. It is then conjectured (as a special caseof [18, Conj. 3c]), and is proved in most cases [5], that ξ is a classical system of Hecke
eigenvalues, arising from a weight 1 Hecke eigenform of level N . Hence (13) has (or atleast, is expected to have) at most finitely many double points, arising from classicalsystems of Hecke eigenvalues in weight one.
On the other hand, if we consider the analogous map(14)

C → (Spec T)an ,then every system of eigenvalues λ : Tk → Zp gives rise to a pair of points (ξ, α) and(ξ, β), where ξ is a p-adic system of Hecke eigenvalues associated to λ, andX 2 − λ(T` )X + λ(`S` ) = (X − α)(X − β).
Unless α = β (which, as we already noted, is expected to be impossible unless k = 1),
we see that the image of (14) admits a double point at ξ. Thus the image of (14) is avery complicated curve, with an infinite number of double points. It is known as theinfinite fern [32, 21]. The following theorem, due to Gouvêa and Mazur [21], showsthat it is a kind of “space-filling curve” in (Spec T)an .
Theorem 2.27. — Each component of the Zariski closure of the infinite fern in(Spec T)an is at least two-dimensional.
Sketch of proof. — Since C is defined to be the Zariski closure of X , we see that
the Zariski closure of the image of (14) is equal to the Zariski closure of the set ξ ofclassical p-adic systems of Hecke eigenvalues. Suppose that this Zariski closure contains

1013–24

a component Z that is one-dimensional. Since the singular locus of Z is a Zariski closed
proper subset of Z, we may find a classical ξ lying in the smooth locus of Z. Since Z
is one-dimensional, the map C → Z must be surjective in a neighbourhood of ξ. Let
λ : Tk → Zp , for some k ≥ 1, be the system of eigenvalues giving rise to ξ, and let α
and β be the two roots of the pth Hecke polynomial of λ. Then (unless α = β), theimage of C is branched at ξ, contradicting the fact that Z is smooth at ξ.
If α = β, then by appealing to the result of Coleman mentioned in Remark 2.26, it
is easy to see that we may find arbitrarily small perturbations ξ 0 of ξ, for which ξ 0 isclassical and such that the corresponding roots α0 and β 0 of the pth Hecke polynomial
are distinct. Then we may apply the above argument to ξ 0 instead, and again derive acontradiction.
The previous result has the following corollary on the Krull dimension of Spec T,
which is again due to Gouvêa and Mazur [21].
Corollary 2.28. — The Krull dimension of each component of Spec T is at least 4.
Sketch of proof. — It is equivalent to show that each component of the associatedrigid analytic space (Spec T)an is at least three-dimensional. Any such component Y
contains a component Z of the image of (14), which is two-dimensional. Twistingby characters of p-power conductor then provides a one-dimensional deformation of Z
inside Y, showing that the Y is at least three-dimensional.
2.6. Galois representations over Hida families and the eigencurve
The following result, due to Mazur–Wiles [34] and Wiles [44], gives a Galois-theoreticinterpretation of the points of C ord , and in particular, of the Gm -coordinate. Before stating it, we note that the chosen embedding ıp : Q ,→ Qp induces a map
GQp → GQ → GQ,Σ (where we have written GQp and GQ to denote Gal(Qp /Qp ) andGal(Q/Q) respectively, and where the second arrow is the natural surjection). For any
representation ρ of GQ,Σ , we write ρ| GQp to denote the restriction of ρ to a representation of GQp via this map. Recall that GQp contains a normal subgroup Ip (the inertia∼
subgroup), such that GQp /Ip −→ GFp , the absolute Galois group of Fp . This lattergroup is topologically generated by the Frobenius automorphism Frobp . We say thata representation of GQp is unramified if it is trivial when restricted to Ip ; any such
representation is then endowed with an action of Frobp .
Theorem 2.29. — If (ξ, α) is Zp -valued point of C ord , then ρξ | GQp admits a onedimensional unramified quotient on which Frobp acts with eigenvalue α.
One has the following conjecture, which is an analogue for C ord of Conjecture 2.12.
It was first made by Mazur and Tilouine [33].

1013–25

Conjecture 2.30. — If Σ is any finite set of primes containing p, and if
ρ : GQ,Σ → GL2 (Qp )is a continuous, semi-simple, and odd representation whose restriction to GQp admits
a one-dimensional unramified quotient on which Frobp acts through the eigenvalue α,then there is a Zp -valued point (ξ, α) in the Hida family for some level N divisible only
by primes in Σ distinct from p such that ρ = ρξ .
In their papers [40, 41], Skinner and Wiles have established this conjecture in a large
number of cases.
Theorem 2.31. — Let ρ and α be as in the statement of Conjecture 2.30, and let
ρ : GQ,Σ → GL2 (Fp ) be the representation obtained by descending ρ to Zp and thenreducing modulo the maximal ideal of Zp . If det ρ|Ip = ψχk−1 for some finite ordercharacter ψ and some integer k ≥ 2 (recall that χ denotes the p-adic cyclotomic character), and if the semi-simplification of ρ | GQp (which is necessarily the direct sum of two
characters, and which is well-defined independently of the choice of Zp -model of ρ giving
rise to ρ) is the direct sum of distinct characters, then Conjecture 2.30 holds for ρ.
Remark 2.32. — Suppose that ρ and α are as in the preceding result, and let (ξ, α)
be the point on the Hida family (of an appropriately chosen level), whose existence isgiven by the theorem, for which ρ = ρξ . The assumption on det ρ in the theorem (andin particular, the assumption that k ≥ 2) implies, by the result of Hida recalled above,that ξ is obtained from a system of eigenvalues λ : Tk (N p) → Zp . Thus, in the context
of this result, one concludes that ρ actually arises from the system of Hecke eigenvalues
attached to a classical modular form (of level possibly divisible by p).
The following result gives a Galois-theoretic interpretation of the points on C, analogous to Theorem 2.29. It is due to Kisin [28]. The statement requires the language+of Fontaine’s theory [17]. Recall that Fontaine has defined a ring Bcris, equipped with
commuting actions of the group GQp and of a “Frobenius” operator ϕ. If V is any++
representation of GQp over Qp , then Dcris(V ) := (Bcris⊗Qp V )GQp is a Qp -vector space ofdimension at most that of V , equipped with an operator ϕ induced by the operator ϕ+.
on Bcris
Theorem 2.33. — If (ξ, α) is a Qp -valued point of C, and if ρ∨ξ | GQp denotes the con+tragredient representation to ρξ | GQp , then Dcris(ρ∨ξ | GQp ) contains a one-dimensional subspace on which ϕ acts via α.
Sketch of proof. — If the p-adic system of Hecke eigenvalues ξ is classical, arising froma system of Hecke eigenvalues λ : Tk → Zp attached to some modular form of weightk ≥ 1, then the representation ρ∨ξ | GQp is in fact crystalline, with Hodge–Tate weights+
equal to 0 and 1 − k, and so Dcris(ρ∨ξ | GQp ) is two-dimensional over Qp . In this context, it
is known that the characteristic polynomial of ϕ is equal to the pth Hecke polynomial

1013–26

of λ [36]. Thus, if α and β are the two roots of this polynomial, then we see that
+
+
Dcris
(ρ∨ξ | GQp )ϕ=α and Dcris(ρ∨ξ | GQp )ϕ=β are both non-zero. The theorem is then provedby showing that these non-zero spaces interpolate over the curve C.
×+Remark 2.34. — In the context of Theorem 2.33, if α ∈ Zp , then Dcris(ρ∨ξ | GQp )ϕ=α is
non-zero if and only if ρξ | GQp contains an unramified quotient on which Frobp acts via α.
Thus Theorem 2.29 is a consequence of Theorem 2.33.

The following conjecture is analogous to Conjecture 2.30 in the ordinary case. (Seethe hope expressed in Remark (2) of [28, p. 450].)
Conjecture 2.35. — If Σ is any finite set of primes containing p, and if
ρ : GQ,Σ → GL2 (Qp )+is a continuous, semi-simple, and odd representation such that Dcris(ρ∨ξ | GQp )ϕ=α is non-zero, then there is a Qp -valued point (ξ, α) in C for some level N divisible only by primesin Σ distinct from p such that ρ = ρξ .
There has been recent progress on this conjecture (see the corollary on p. 3 of [30] as
well as the forthcoming paper [16]).

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Matthew EMERTON
Department of Mathematics
Northwestern University
2033 Sheridan Rd
EVANSTON, IL 60208-2730 – USA
E-mail : emerton@math.northwestern.edu