THE EISENSTEIN IDEAL IN HIDA’S

ORDINARY HECKE ALGEBRA

Matthew Emerton

University of Michigan

Abstract. For p ≥ 5 a prime and i ≢ 0 (mod p − 1) let T_{par,(i)}^{ord} denote Hida’s ordinary Hecke algebra acting on the space of p-ordinary cusp forms of tame level one, arbitrary weight k, and tame nebentypus equal to ω^{(i−k)} (where ω is the Teichmuller character). Let ℐ′_{Eis} denote the Eisenstein ideal in T_{par,(i)}^{ord}, that is, the ideal generated by the elements T_{l} − 1 − ⟨l⟩l^{−1} (for l a prime distinct from p) and the element U_{p} − 1. Let L_{p}(ω^{i}, 1 − s) denote the Kubota-Leopoldt p-adic L-function. In this note we give a simple proof of the fact that there is a canonical isomorphism Λ_{(i)}/L_{p}(ω^{i}, 1 − s) = T_{par,(i)}^{ord}/ℐ′_{Eis} (a result originally established in [10]) and use this isomorphism to give a direct proof of the main result of [9]. Our proof of the isomorphism depends on constructing a certain element in the full Hecke algebra T_{(i)} which one can think of as being the “universal constant term” of a (not necessarily cuspidal) p-adic modular form.

Introduction

Let us fix a prime number p ≥ 5. The object of this note is to study the relation between p-adic ordinary Eisenstein series and p-adic ordinary cusp forms.

Let V denote Katz’s ring of generalized p-adic modular functions of level one (as constructed in [5,6,7,8]) and let T denote the ring of Hecke operators acting on V (as constructed in [1,2]). If Λ := Z_{p}[[Z_{p}^{×}]] denotes the completed group ring of Z_{p}^{×} over Z_{p} then T is naturally a Λ algebra.

The Teichmuller character ω : F_{p}^{×} → Z_{p}^{×} is uniquely determined as the inverse of the reduction map Z_{p}^{×} → F_{p}^{×}. If we let Γ denote the kernel of the reduction map, and identify F_{p}^{×} with its image under ω, then we may write Z_{p}^{×} = Γ × F_{p}^{×}. The ring Λ is semi-local, and is equal to the product Λ = ∏_{i (mod p−1)} Λ_{(i)}, where Λ_{(i)} is that local factor of Λ on which F_{p}^{×} acts via ω^{(i)}. If we forget the F_{p}^{×} action, we find that Λ_{(i)} is isomorphic to the completed group ring Z_{p}[[Γ]]. We write V_{(i)} = V ⊗_{Λ} Λ_{(i)} and T_{(i)} = T ⊗_{Λ} Λ_{(i)}, so that T_{(i)} is the ring of Hecke operators on V_{(i)}.

Now follow [2,3] in cutting down to the ordinary part T_{(i)}^{ord} of T_{(i)}. There is a natural morphism T_{(i)}^{ord} → Λ_{(i)} obtained by considering the action of the Hecke operators on the Eisenstein series, whose kernel we denote by ℐ_{Eis}. Thus ℐ_{Eis} is generated by the elements T_{l} − ⟨l⟩l^{−1} (where l runs over all primes other than p) together with the element U_{p} − 1. We let T_{par,(i)} denote the quotient of T_{(i)} which acts on the subspace V_{par,(i)} of V_{(i)} consisting of cuspidal generalized modular functions, and let ℐ′_{Eis} denote the image of ℐ_{Eis} in T_{par,(i)}^{ord}. Following the terminology of [10], we call ℐ′_{Eis} the Eisenstein ideal in T_{par,(i)}^{ord}.

The main result of this note is the following Theorem:

Theorem. Suppose that i ≢ 0 (mod p − 1). Then the pullback of ℐ′_{Eis} via the natural map Λ_{(i)} → T_{par,(i)}^{ord} is equal to the principal ideal of Λ_{(i)} generated by the Kubota-Leopoldt p-adic L-function L_{p}(ω^{i}, 1 − s), and the consequent injection

Λ_{(i)}/L_{p}(ω^{i}, 1 − s) → T_{par,(i)}^{ord}/ℐ′_{Eis}

is an isomorphism.

This has as a corollary the following result:

Corollary. If i ≢ 0 (mod p − 1) then the element Norm_{T_{par,(i)}^{ord}/Λ_{(i)}}(U_{p} − 1) of Λ_{(i)}

is divisible by L_{p}(ω^{i}, 1 − s).

This Theorem is proved (in a slightly different language) in [10]. The proof there is however rather indirect, depending on a “modulo p” version of the Kubota-Lang theory of modular units, and it occurs in conjunction with a subtle and sophisticated analysis of the arithmetic geometry of Jacobians of modular curves. The proof we will give is much simpler by comparison.

The Corollary was first proved in [9], where the proof again involves an analysis of the geometry of Jacobians of modular curves. In this note we point out that the Corollary is a direct consequence of the Theorem.

The proof of the Theorem that we give depends on constructing an element A_{0} in T_{(i)} for each i ≢ 0 (mod p − 1) with the property that

a_{0}(f) = a_{1}(f|A_{0})

for any generalized modular function f. (Here a_{n}(f) denotes the n^{th} q-expansion coefficient of the generalized modular function f.) This construction generalizes [12, Théorème 9].

§1. The set-up

In this section we establish our notation and recall some results on p-adic modular forms. Unfortunately there seems to be no consensus in the literature as to the notation that should be used in this theory, and so we take some care to relate our notation to that of the various references we cite.

Fix a prime p ≥ 5. We denote by V the ring of generalized p-adic modular functions of tame level one which are holomorphic at the cusps. This ring is constructed in [5,6,7,8] (see in particular [5, §4] for a discussion of the case of tame level one) in which it is denoted by V_{∞,∞}.

Let ℳ_{k} denote the space of modular forms of level one and weight k defined over Q_{p} and let ℳ = ⨁_{k} ℳ_{k} denote the graded ring of modular forms defined over Q_{p}. The q-expansion map allows us to consider ℳ as a subring of Q_{p}[[q]]. We have the following elementary description of V [2,5]: V is equal to the p-adic completion of the intersection ℳ ∩ Z_{p}[[q]], where this intersection is taken in Q_{p}[[q]].

We write Λ := Z_{p}[[Z_{p}^{×}]] for the completed group ring of Z_{p}^{×} with coefficients in Z_{p}^{×}. The Teichmuller mapping ω : F_{p}^{×} ↪ Z_{p}^{×} induces the direct sum decomposition

Λ = ∏_{i (mod p−1)} Λ_{(i)},

where Λ_{(i)} denotes a copy of the completed group ring Z_{p}[[Γ]] (Γ = 1 + pZ_{p}) made into a Λ-algebra by having F_{p}^{×} act via ω^{i}. For any Λ-module M we write

M_{(i)} := M ⊗_{Λ} Λ_{(i)}.

If a ∈ Z_{p}^{×} then we use the notation ⟨a⟩ to denote the corresponding element of the group ring Λ, as well as its projection to any of the factors Λ_{(i)} of Λ.

We define an action of Z_{p}^{×} on ℳ by defining the action on each direct summand ℳ_{k} via the formula f|⟨γ⟩ = γ^{k}f. One can show that with respect to this action ℳ ∩ Z_{p}[[q]] is a Z_{p}^{×} submodule of ℳ [5]. Thus there is induced an action of Z_{p}^{×} on V [5] via which we may regard V as a Λ-module and End(V) as a Λ-algebra.

We let T denote the completion in the compact-open topology of End(V) of the Λ-subalgebra generated by the Hecke operators T_{l} (l ≠ p) and U_{p}. (In this choice of notation we follow [1]. This Hecke algebra is denoted by a script ‘H’ in [2]. Note that in the reference [11] the symbol T is used to denote a Hecke algebra which we would denote by ∏_{i≢2 (mod p−1)} T_{par,(i)}^{ord} in the notation to be introduced below. See [1, §II.1.2, §III.1.2] for an explanation of why the compact open topology is the appropriate topology to consider.) It will sometimes be convenient to write T_{p} := U_{p}. If n is any positive integer, then the usual formulae allow us to write the Hecke operator T_{n} as a polynomial in the T_{l} (l ranging over all primes dividing n) with coefficients in Λ. To be precise:

(1) T_{1} = the identity endomorphism; if m and n are coprime then T_{mn} = T_{m}T_{n};

(2) if l is a prime distinct from p then for any integer k ≥ 2

T_{l^{k}} = T_{l}T_{l^{k−1}} − ⟨l⟩l^{−1}T_{l^{k−2}};

(3) for any positive integer k, T_{p^{k}} = T_{p}^{k}.

Note that the particular case T_{l^{2}} = T_{l}^{2} + ⟨l⟩l^{−1} of (3) yields the alternative description of T as the completion of the Z_{p}-subalgebra of End(V) generated by all the Hecke operators T_{n}.

For each i (mod p−1) the subspace V_{(i)} of V is T-invariant, and T acts on V_{(i)} through its quotient T_{(i)}. If T is any operator in T, we will denote by the same letter its image in T_{(i)} (for any (i)). This will apply in particular when T = T_{n} for some integer n.

We define the subspace V_{par} of V by working with the spaces 𝒮_{k} of cusp forms of level one and weight k in place of the spaces ℳ_{k}. It is easily seen that f ∈ V is an element of the subspace V_{par} if and only if for every a ∈ Z_{p}^{×} the element f|⟨a⟩ of V has vanishing constant term in its q-expansion. The subspace V_{par} of V is T-invariant, and we denote by T_{par} that quotient of T which acts faithfully on V_{par}. (Note that this quotient is denoted by T_{0} in [1] and by a script ‘h’ in [2,3].)

§2. The universal constant term

We need the following variant of Katz’s Key Lemma [6]:

Proposition 1. If f ∈ V_{(i)}[1/p] for some i ≢ 0 (mod p − 1), and a_{n}(f) ∈ Z_{p} for every n ≥ 1, then a_{0}(f) ∈ Z_{p}, and thus f ∈ V.

Proof. Suppose that a_{0}(f) ∉ Z_{p}. Let ν > 0 be the p-adic valuation of 1/a_{0}(f), so that p^{ν}f ∈ V_{(i)} and p^{ν}a_{0}(f) ∈ Z_{p}^{×}. Then

p^{ν}f − p^{ν}a_{0}(f) ≡ 0 (mod p^{ν})

by virtue of our assumption about a_{n}(f) for n > 0.

Let a be a generator of the cyclic group F_{p}^{×} ↪ Z_{p}^{×}. Since f ∈ V_{(i)}, p^{ν}f|⟨a⟩ = a^{i}p^{ν}f, while since p^{ν}a_{0}(f) is a constant, p^{ν}a_{0}(f)|⟨a⟩ = p^{ν}a_{0}(f). Thus

(1 − a^{i})p^{ν}a_{0}(f) ≡ p^{ν}a_{0}(f) − a^{i}p^{ν}f ≡ (p^{ν}a_{0}(f) − p^{ν}f)|⟨a⟩ ≡ 0 (mod p^{ν}).

Since i ≢ 0 (mod p − 1) our choice of a guarantees that (1 − a^{i}) ≢ 0 (mod p). This implies that p^{ν}a_{0}(f) ≡ 0 (mod p^{ν}), a contradiction to our assumption that a_{0}(f) ∉ Z_{p}. Thus we conclude that in fact a_{0}(f) ∈ Z_{p}, and the q-expansion principle now shows that f ∈ V_{(i)}. □

From this we deduce the following Corollary:

Corollary 2. For any p-adically complete Z_{p}-algebra R we have an isomorphism of continuous R-modules, functorial in R:

V_{(i)} ˆ⊗_{Z_{p}}R ⟶^{∼} Hom_{cont}(T_{(i)}, R),

given via the pairing

(f, T) → a_{1}(f|T),

provided i ≢ 0 (mod p − 1). (Here T_{(i)} has its compact-open topology, R has its p-adic topology, and Hom_{cont}(T_{(i)}, R) has the topology of uniform convergence.)

Proof. This result can be proved using the methods of [1, §III.1.2] (see also [2, §2]) since the preceding Corollary assures us that the constant terms of the elements of V_{(i)} don’t cause any trouble. (See [1, §III.1.3] for a discussion of this trouble in the case when one allows i ≡ 0 (mod p − 1).) □

The morphism f → a_{0}(f) yields a morphism

V_{(i)} ˆ⊗R → R,

functorial in p-adically complete Z_{p}-algebras R. Thus we get a morphism

Hom_{cont}(T_{(i)}, R) → R,

functorial in R. Since T_{(i)}, as a topological ring, is the inverse limit of p-adically complete rings (for example, the classical Hecke algebras at finite level and weight; see [1, §II.1.2] or [2, §2]) this morphism corresponds to an element A_{0,(i)} ∈ T_{(i)}, the “universal constant term” of the elements of V_{(i)}. By construction, if f ∈ V_{(i)}, then a_{1}(f|A_{0,(i)}) = A_{0,(i)}(f).

Lemma 3. The following conditions are equivalent for any f ∈ V_{(i)}:

(i) f|T A_{(i)} = 0 for all T ∈ T_{(i)};

(ii) f ∈ V_{par,(i)}.

Proof. If f|T A_{(i)} = 0 for all T ∈ T_{(i)} then in particular for any a ∈ Z_{p}^{×} we compute that

a_{0}(f|⟨a⟩) = a_{1}(f|⟨a⟩A_{0,(i)}) = 0,

and thus f ∈ V_{par,(i)}. Hence (i) implies (ii).

Conversely, if f is an element of V_{par,(i)}, so is f|T_{n} for every n, and so

a_{n}(f|A_{0,(i)}) = a_{1}(f|A_{0,(i)}|T_{n}) = a_{1}(f|T_{n}|A_{0,(i)}) = a_{0}(f|T_{n}) = 0

for every n ≥ 1, showing that f|A_{0} = 0. Since V_{par,(i)} is closed under the action of T_{(i)} we conclude more generally that f|T A_{0,(i)} = 0 for any T ∈ T_{(i)}. Hence (ii) implies (i). □

The preceding Lemma shows that the principal ideal generated by A_{0,(i)} is in the kernel of the surjection T_{(i)} → T_{par,(i)}, and so there is an induced surjection T_{(i)}/A_{0,(i)} → T_{par,(i)}.

Proposition 4. The morphism T_{(i)}/A_{0,(i)} → T_{par,(i)} is an isomorphism.

Proof. Consider the following commutative diagram, for any p-adically complete Z_{p}-algebra R:

\xymatrix{ & Hom_{cont}(T_{par,(i)}, R) \ar[d] \\ V_{par,(i)} ˆ⊗_{Z_{p}}R \ar@{-->}[ur]^{∼} \ar[r]^{∼} \ar[d] & Hom_{cont}(T_{(i)}/A_{0,(i)}, R) \ar[d] \\ V_{(i)} ˆ⊗_{Z_{p}}R \ar[r]^{∼} & Hom_{cont}(T_{(i)}, R)} .

In this diagram the left hand vertical arrow is the completion of the tensor product of the inclusion of V_{par,(i)} in V_{(i)} with the identity morphism of R and the right hand vertical arrows are the injections determined by the surjections

T_{(i)} → T_{(i)}/A_{0,(i)} → T_{par,(i)}.

The lower horizontal arrow is the isomorphism of Corollary 2 and the diagonal arrow is the isomorphism of [1, Corollary III.1.3]. Lemma 3 shows that the upper horizontal arrow is also an isomorphism. Thus the morphism

Hom_{cont}(T_{par,(i)}, R) → Hom_{cont}(T_{(i)}/A_{0,(i)}, R)

is an isomorphism for every p-adically complete Z_{p}-algebra R. Since any principal ideal is closed in T_{(i)} we conclude that the morphism

T_{(i)}/A_{0,(i)} → T_{par,(i)}

is an isomorphism. □

§3. Ordinary parts

In this section we cut down to the ordinary part T^{ord} of T, as in [2,3,9,11]. (Note that we follow the conventions of [1] in using a superscript ord to denote taking the ordinary part of any T-module. The convention of [2,3] is to use either a subscript or a superscript 0, while the convention of [11] is to use a subscript U. Note however that in this last reference T (with no subscript) is used to denote the ordinary part of the Hecke algebra.) The factorization T = ∏_{i} T_{(i)} yields a factorization T^{ord} = ∏_{i} T_{(i)}^{ord}. If T is any element of T_{(i)} we will use the same symbol to denote its image in the ring T_{(i)}^{ord}. We will apply this convention in particular to the operators T_{n} (n an integer) and A_{0,(i)}.

For any i (mod p − 1) we denote by ℐ_{Eis,(i)} the ideal in T_{(i)}^{ord} generated by the elements T_{l}−1−⟨l⟩l^{−1} (where l ranges over all primes distinct from p) together with the element U_{p} − 1. The natural morphism Λ_{(i)} → T_{(i)}^{ord} induces an isomorphism

Λ_{(i)} →^{∼} T/ℐ_{Eis,(i)}.

The known formulae for the constant terms of Eisenstein series of level one and weight k show that if i ≢ 0 (mod p−1) then the image of A_{0,(i)} in T/ℐ_{Eis,(i)} = Λ_{(i)} is equal to the Kubota-Leopoldt p-adic L-function L_{p}(ω^{i}, 1 − s). (For the construction of this L-function, see [4,6]. For the connection with Eisenstein series from the p-adic point of view, see [6,12].)

We let T_{par,(i)}^{ord} denote the ordinary part of T_{par,(i)}. Proposition 4 shows that T_{par,(i)}^{ord} = T_{(i)}^{ord}/A_{0,(i)}. We let ℐ′_{Eis,(i)} denote the image of the ideal ℐ_{Eis} in T_{par,(i)}^{ord}.

Theorem 5. Suppose that i ≢ 0 (mod p − 1). Then the pullback of ℐ′_{Eis,(i)} via the natural map Λ_{(i)} → T_{par,(i)}^{ord} is equal to the principal ideal of Λ_{(i)} generated by the Kubota-Leopoldt p-adic L-function L_{p}(ω^{i}, 1 − s), and the consequent injection

Λ_{(i)}/L_{p}(ω^{i}, 1 − s) → T_{par,(i)}^{ord}/ℐ′_{Eis,(i)}

is an isomorphism.

Proof. This follows from a consideration of the following commutative diagram:

\xymatrix{ & T_{(i)}^{ord} \ar[d] \ar[dr] \\ Λ_{(i)} \ar[ur] \ar[r]^{∼} \ar[d] & T_{(i)}^{ord}/ℐ_{Eis,(i)} \ar[d] & T_{(i)}^{ord}/A_{0,(i)} = T_{par,(i)}^{ord} \ar[dl] \\ Λ_{(i)}/L_{p}(ω^{i}, 1 − s) \ar[r]^{∼} & T_{(i)}^{ord}/(ℐ_{Eis}, A_{0,(i)}) = T_{par,(i)}^{ord}/ℐ′_{Eis,(i)}.}

□

In [2] it is shown that T_{par,(i)}^{ord} is finite flat over Λ_{(i)} for every i (mod p − 1). (An alternative proof is given in [9] for the case i ≠ 2.) Thus we can define the norm map

Norm_{T_{(i)}^{par}/Λ_{(i)}} : T_{par,(i)}^{ord} → Λ_{(i)}.

Corollary 6. If i ≢ 0 (mod p − 1) then the element Norm_{T_{par,(i)}^{ord}/Λ_{(i)}}(U_{p} − 1) of

Λ_{(i)} is divisible by L_{p}(ω^{i}, 1 − s).

Proof. If T_{par,(i)}^{ord} = 0 there is nothing to prove. Thus we suppose that T_{par,(i)}^{ord} ≠ 0, so that we may regard Λ as a subring of T_{par,(i)}^{ord}.

Write N := Norm_{T_{(i)}^{par}/Λ_{(i)}}(U_{p} − 1). Then we have an inclusion of ideals

NΛ_{(i)} ⊂ (U_{p} − 1)T_{par,(i)}^{ord} ∩ Λ_{(i)} ⊂ ℐ′_{Eis,(i)} ∩ Λ_{(i)} = L_{p}(ω^{i}, 1 − s)Λ_{(i)}

(where the first inclusion follows immediately from the definition of the norm and the final equality is the result of Theorem 5). □

We can improve the result of the Corollary a little: we may complete T_{par,(i)}^{ord} at

the ideal ℐ′_{Eis,(i)} to obtain a local ring T_{par,(i)}^{ord,Eis}. There is a consequent factorization of T_{par,(i)}^{ord} as a product

T_{par,(i)}^{ord} = T_{par,(i)}^{ord,Eis} × T_{par,(i)}^{ord,0}.

Then one sees immediately that the above corollary remains true with T_{par,(i)}^{ord,Eis} in place of T_{par,(i)}^{ord} :

Corollary 7. If i ≢ 0 (mod p − 1) then the element Norm_{T_{par,(i)}^{ord}/Λ_{(i)}}(U_{p} − 1) of

Λ_{(i)} is divisible by L_{p}(ω^{i}, 1 − s).

We now explain the connection between this result and the result of [9]. In that reference a certain Hecke ring denoted T(χ) is considered, where χ is a power of the Teichmuller character. If χ = ω^{i} then the ring T(χ) of [9] is equal to the completion (returning to the notation of this paper) of the ring T_{par,(i+2)}^{ord} at the

principal ideal generated by (U_{p} − 1). Let us write T_{par,(i+2)}^{ord,U_{p}≡1} for this completion. Then the following diagram commutes:

[figure: Λ_{(i)}; ⟨a⟩↦a^{−2}⟨a⟩; Λ_{(i+2)}; T(χ); T_{par,(i)}^{ord,U_{p}≡1}.]

In this diagram the left vertical arrow gives the Λ-algebra structure on T(χ) which is considered in [9], the right vertical arrow gives the Λ-algebra structure considered on T_{par,(i)}^{ord,U_{p}≡1} in this note, and the upper horizontal arrow is the isomorphism Λ_{(i)} → Λ_{(i+2)} given by the formula ⟨a⟩ ↦ a^{−2}⟨a⟩ for elements a ∈ Γ. (The reason for the shifting by two is that in [9,11] all actions of Γ are considered with regard to weight two modular forms, so that the Γ-action on weight two modular forms with trivial nebentypus is taken to be trivial, whereas we have followed [2,5] in defining this action to be given by the formula f|⟨a⟩ := a^{2}f.)

On [9, p. 520] a local factor T(χ)^{Eis} of T(χ) is defined, and the identification T(χ) = T_{par,(i+2)}^{par,U_{p}≡1} induces an identification T(χ)^{Eis} = T_{par,(i+2)}^{ord,Eis}. Also note that under the isomorphism induced by ⟨a⟩ ↦ a^{−2}⟨a⟩ the element L_{p}(χω^{2}, −1 − s) of Λ_{(i)} is taken to the element L_{p}(ω^{i+2}, 1 − s) of Λ_{(i+2)}.

Thus Corollary 7 is equivalent to the following result (in the notation of [9]): the element Norm_{T(χ)/Λ_{(i)}}(U_{p} − 1) of Λ_{(i)} is divisble by L_{p}(χω^{2}, −1 − s). In [9] this norm is denoted by D_{p}(χ, s)^{Eis}. Thus we have provided a proof of the statement “ L_{p}(χω^{2}, −1 − s) divides D_{p}(χ, s)^{Eis} ” which occurs at the bottom of [9, p. 520]. (In [9] the further restriction i ≠ 0 is imposed. However, this is no real restriction, since L_{p}(ω^{2}, −1 − s) is a unit in Z_{p}[[Γ]] when p ≥ 5, as follows from the fact that ζ(−1) = −1/12.)

References

1. F. Gouvea, Arithmetic of p-adic modular forms, Lecture Notes in Math., vol 1304, Springer-Verlag, Berlin and New York, 1988.

2. H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. Ec. Norm. Sup. (4) 19, 231-273.

3. H. Hida, Galois representations into GL_{2}(Z_{p}[[X]]) attached to ordinary cusp forms, Inv. Math. 85, 545-613.

4. K. Iwasasa, Lectures on p-adic L functions, Annals of Mathematical Studies, vol. 89, Princeton University Press, Princeton, New Jersey, 1969.

5. N. Katz, High congruences between modular forms, Ann. of Math. 101, 332-367.

6. N. Katz, p-adic L functions via moduli, Proceedings of Symposia in Pure Mathematics, vol. 29, American Math Society, Providence, Rhode Island, 1975, pp. 479-506.

7. N. Katz, p-adic interpolation of real analytic Eisenstein series, Ann. of Math. 104, 459-571.

8. N. Katz, The Eisenstein measure and p-adic interpolation, Am. J. Math. 99, 238-311.

9. B. Mazur and A. Wiles, Analogies between function fields and number fields, Am. J. Math. 105, 507-521.

10. B. Mazur and A. Wiles, Class fields of abelian extensions of Q, Inv. Math. 76, 179-330.

11. B. Mazur and A. Wiles, On p-adic analytic families of Galois representations, Comp. Math. 59, 231-264.

12. J. -P. Serre, Formes modulaires et fonctions zêta p-adiques, Lecture Notes in Math., vol 350, Springer-Verlag, Berlin and New York, 1973, pp. 191-268.

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109

  E-mail address: emerton@math.lsa.umich.edu