THE EISENSTEIN IDEAL IN HIDA’S
ORDINARY HECKE ALGEBRA

Matthew Emerton
University of Michigan
Abstract. For p ≥ 5 a prime and i 6≡ 0 (mod p − 1) let Tord
denote Hida’spar,(i)ordinary Hecke algebra acting on the space of p-ordinary cusp forms of tame levelone, arbitrary weight k, and tame nebentypus equal to ω (i−k) (where ω is the Teichmuller character). Let I0Eis denote the Eisenstein ideal in Tord
, that is, thepar,(i)
ideal generated by the elements Tl − 1 − hlil−1 (for l a prime distinct from p) andthe element Up − 1. Let Lp (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) /Lp (ω i , 1 − s) = Tord
/I0
(a result originally established in [10]) and usepar,(i) Eis
this isomorphism to give a direct proof of the main result of [9]. Our proof of theisomorphism 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 (asconstructed in [5,6,7,8]) and let T denote the ring of Hecke operators acting on V×(as constructed in [1,2]). If Λ := Zp [[Z×p ]] denotes the completed group ring of Zp
over Zp then T is naturally a Λ algebra.
×
The Teichmuller character ω : F×p → Zp is uniquely determined as the inverse××of the reduction map Zp → Fp . If we let Γ denote the kernel of the reduction map,××and identify F×
p with its image under ω, then we
Qmay write Zp = Γ × Fp . The ring
Λ is semi-local, and is equal to the product Λ = i (mod p−1) Λ(i) , where Λ(i) is that(i)×local factor of Λ on which F×p acts via ω . If we forget the Fp action, we find that
Λ(i) is isomorphic to the completed group ring Zp [[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 Tord(i) of T(i) . There is a→Λ
obtainedbyconsideringtheaction of the Hecke
natural morphism Tord(i)(i)
operators on the Eisenstein series, whose kernel we denote by IEis . Thus IEis is
generated by the elements Tl − hlil−1 (where l runs over all primes other than p)
together with the element Up − 1. We let Tpar,(i) denote the quotient of T(i) whichacts on the subspace Vpar,(i) of V(i) consisting of cuspidal generalized modular
functions, and let I0Eis denote the image of IEis in Tordpar,(i) . Following the terminology of [10], we call I0Eis the Eisenstein ideal in Tordpar,(i) .
The main result of this note is the following Theorem:
1

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MATTHEW EMERTON

Theorem. Suppose that i 6≡ 0 (mod p − 1). Then the pullback of I0Eis via thenatural map Λ(i) → Tordpar,(i) is equal to the principal ideal of Λ(i) generated by the
Kubota-Leopoldt p-adic L-function Lp (ω i , 1 − s), and the consequent injection0Λ(i) /Lp (ω i , 1 − s) → Tordpar,(i) /IEis

is an isomorphism.
This has as a corollary the following result:Corollary. If i 6≡ 0 (mod p − 1) then the element NormTordpar,(i)/Λ(i) (Up − 1) of Λ(i)i

is divisible by Lp (ω , 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-Langtheory of modular units, and it occurs in conjunction with a subtle and sophisticatedanalysis of the arithmetic geometry of Jacobians of modular curves. The proof wewill give is much simpler by comparison.
The Corollary was first proved in [9], where the proof again involves an analysisof 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 A0
in T(i) for each i 6≡ 0 (mod p − 1) with the property thata0 (f ) = a1 (f |A0 )for any generalized modular function f . (Here an (f ) denotes the nth 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 thenotation that should be used in this theory, and so we take some care to relate ournotation 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 constructedin [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 Mk denoteL
the space of modular forms of level one and weight k defined overQp and let M = k Mk denote the graded ring of modular forms defined over Qp .
The q-expansion map allows us to consider M as a subring of Qp [[q]]. We have thefollowing elementary description of V [2,5]: V is equal to the p-adic completion of
the intersection M ∩ Zp [[q]], where this intersection is taken in Qp [[q]].
×
We write Λ := Zp [[Z×p ]] for the completed group ring of Zp with coefficients in×××
Zp . The Teichmuller mapping ω : Fp ,→ Zp induces the direct sum decompositionΛ=Yi (mod p−1)

Λ(i) ,

THE EISENSTEIN IDEAL IN HIDA’S ORDINARY HECKE ALGEBRA

3

where Λ(i) denotes a copy of the completed group ring Zp [[Γ]] (Γ = 1 + pZp ) madei
into a Λ-algebra by having F×p act via ω . For any Λ-module M we write
M(i) := M ⊗Λ Λ(i) .
If a ∈ Z×
p then we use the notation hai to denote the corresponding element of thegroup 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
Mk via the formula f |hγi = γ k f . One can show that with respect to this action
×M ∩ Zp [[q]] is a Z×
p submodule of M [5]. Thus there is induced an action of Zp 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 theΛ-subalgebra generated by the Hecke operators Tl (l 6= p) and Up . (In this choiceof notation we follow [1]. This Hecke algebra is denoted by a script ‘H’ in [2]. Notethat in the referenceQ[11] the symbol T is used to denote a Hecke algebra which
we would denote by i6≡2 (mod p−1) Tordpar,(i) in the notation to be introduced below.
See [1, §II.1.2, §III.1.2] for an explanation of why the compact open topology isthe appropriate topology to consider.) It will sometimes be convenient to write
Tp := Up . If n is any positive integer, then the usual formulae allow us to write the
Hecke operator Tn as a polynomial in the Tl (l ranging over all primes dividing n)with coefficients in Λ. To be precise:(1) T1 = the identity endomorphism; if m and n are coprime then Tmn = Tm Tn ;(2) if l is a prime distinct from p then for any integer k ≥ 2
Tlk = Tl Tlk−1 − hlil−1 Tlk−2 ;(3) for any positive integer k, Tpk = Tpk .
Note that the particular case Tl2 = Tl2 + hlil−1 of (3) yields the alternativedescription of T as the completion of the Zp -subalgebra of End(V) generated by
all the Hecke operators Tn .
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 = Tn forsome integer n.
We define the subspace Vpar of V by working with the spaces Sk of cusp formsof level one and weight k in place of the spaces Mk . It is easily seen that f ∈ V isan element of the subspace Vpar if and only if for every a ∈ Z×
p the element f |haiof V has vanishing constant term in its q-expansion. The subspace Vpar of V is
T-invariant, and we denote by Tpar that quotient of T which acts faithfully on
Vpar . (Note that this quotient is denoted by T0 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 6≡ 0 (mod p − 1), and an (f ) ∈ Zp forevery n ≥ 1, then a0 (f ) ∈ Zp , and thus f ∈ V.
Proof. Suppose that a0 (f ) ∈/ Zp . Let ν > 0 be the p-adic valuation of 1/a0 (f ), sothat pν f ∈ V(i) and pν a0 (f ) ∈ Z×
p . Thenpν f − pν a0 (f ) ≡ 0(mod pν )4

MATTHEW EMERTON

by virtue of our assumption about an (f ) for n > 0.
×ν
Let a be a generator of the cyclic group F×
p ,→ Zp . Since f ∈ V(i) , p f |hai =
i ν
ν
ν
ν
a p f , while since p a0 (f ) is a constant, p a0 (f )|hai = p a0 (f ). Thus
(1 − ai )pν a0 (f ) ≡ pν a0 (f ) − ai pν f ≡ (pν a0 (f ) − pν f )|hai ≡ 0(mod pν ).

Since i 6≡ 0 (mod p − 1) our choice of a guarantees that (1 − ai ) 6≡ 0 (mod p).
This implies that pν a0 (f ) ≡ 0 (mod pν ), a contradiction to our assumption thata0 (f ) ∈/ Zp . Thus we conclude that in fact a0 (f ) ∈ Zp , 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 Zp -algebra R we have an isomorphismof continuous R-modules, functorial in R:∼ˆ Zp R −→V(i) ⊗
Homcont (T(i) , R),

given via the pairing(f, T ) → a1 (f |T ),provided i 6≡ 0 (mod p − 1). (Here T(i) has its compact-open topology, R has its
p-adic topology, and Homcont (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 ofV(i) don’t cause any trouble. (See [1, §III.1.3] for a discussion of this trouble in thecase when one allows i ≡ 0 (mod p − 1).) 
The morphism f → a0 (f ) yields a morphism
ˆ → R,
V(i) ⊗R
functorial in p-adically complete Zp -algebras R. Thus we get a morphism
Homcont (T(i) , R) → R,functorial in R. Since T(i) , as a topological ring, is the inverse limit of p-adicallycomplete 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 A0,(i) ∈ T(i) ,the “universal constant term” of the elements of V(i) . By construction, if f ∈ V(i) ,then a1 (f |A0,(i) ) = A0,(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 ∈ Vpar,(i) .
Proof. If f |T A(i) = 0 for all T ∈ T(i) then in particular for any a ∈ Z×
p we computethat
a0 (f |hai) = a1 (f |haiA0,(i) ) = 0,
and thus f ∈ Vpar,(i) . Hence (i) implies (ii).

THE EISENSTEIN IDEAL IN HIDA’S ORDINARY HECKE ALGEBRA

5

Conversely, if f is an element of Vpar,(i) , so is f |Tn for every n, and soan (f |A0,(i) ) = a1 (f |A0,(i) |Tn ) = a1 (f |Tn |A0,(i) ) = a0 (f |Tn ) = 0
for every n ≥ 1, showing that f |A0 = 0. Since Vpar,(i) is closed under the actionof T(i) we conclude more generally that f |T A0,(i) = 0 for any T ∈ T(i) . Hence (ii)
implies (i). 
The preceding Lemma shows that the principal ideal generated by A0,(i) is in
the kernel of the surjection T(i) → Tpar,(i) , and so there is an induced surjection
T(i) /A0,(i) → Tpar,(i) .
Proposition 4. The morphism T(i) /A0,(i) → Tpar,(i) is an isomorphism.
Proof. Consider the following commutative diagram, for any p-adically complete
Zp -algebra R:
Homcont (Tpar,(i) , R) .
5
kkk
∼kkkk
k
kkk
kkk

ˆ Zp R ∼ / Homcont (T(i) /A0,(i) , R)
Vpar,(i) ⊗ˆ Zp R
V(i) ⊗∼

/ Homcont (T(i) , R)

In this diagram the left hand vertical arrow is the completion of the tensor productof the inclusion of Vpar,(i) in V(i) with the identity morphism of R and the righthand vertical arrows are the injections determined by the surjections
T(i) → T(i) /A0,(i) → Tpar,(i) .
The lower horizontal arrow is the isomorphism of Corollary 2 and the diagonalarrow is the isomorphism of [1, Corollary III.1.3]. Lemma 3 shows that the upperhorizontal arrow is also an isomorphism. Thus the morphism
Homcont (Tpar,(i) , R) → Homcont (T(i) /A0,(i) , R)
is an isomorphism for every p-adically complete Zp -algebra R. Since any principal
ideal is closed in T(i) we conclude that the morphism
T(i) /A0,(i) → Tpar,(i)is an isomorphism. §3. Ordinary parts
In this section we cut down to the ordinary part Tord of T, as in [2,3,9,11]. (Notethat 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 asubscript 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) isQ
used to denotethe ordinary part of the Hecke algebra.) The factorization T = i T(i) yields a

6

MATTHEW EMERTON

Q
factorization Tord = i Tord(i) . If T is any element of T(i) we will use the same
symbol to denote its image in the ring Tord(i) . We will apply this convention inparticular to the operators Tn (n an integer) and A0,(i) .
For any i (mod p − 1) we denote by IEis,(i) the ideal in Tord(i) generated by theelements Tl −1−hlil−1 (where l ranges over all primes distinct from p) together withthe element Up − 1. The natural morphism Λ(i) → Tord(i) induces an isomorphism∼

Λ(i) → T/IEis,(i) .
The known formulae for the constant terms of Eisenstein series of level one andweight k show that if i 6≡ 0 (mod p − 1) then the image of A0,(i) in T/IEis,(i) = Λ(i)is equal to the Kubota-Leopoldt p-adic L-function Lp (ω i , 1 − s). (For the construction of this L-function, see [4,6]. For the connection with Eisenstein series from thep-adic point of view, see [6,12].)We let Tordpar,(i) denote the ordinary part of Tpar,(i) . Proposition 4 shows that
ord
ord
Tpar,(i) = T(i) /A0,(i) . We let I0Eis,(i) denote the image of the ideal IEis in Tordpar,(i) .
Theorem 5. Suppose that i 6≡ 0 (mod p − 1). Then the pullback of I0Eis,(i) via thenatural map Λ(i) → Tordpar,(i) is equal to the principal ideal of Λ(i) generated by the
Kubota-Leopoldt p-adic L-function Lp (ω i , 1 − s), and the consequent injection0Λ(i) /Lp (ω i , 1 − s) → Tordpar,(i) /IEis,(i)is an isomorphism.
Proof. This follows from a consideration of the following commutative diagram:ord

hh4 T(i) VVVVVVV
hhhh
VVVV
h
h
h
VVVV
hhh
h
VVVV
h
h
hh
VVVV
h
h
h
h

+
h
h
hh
h
∼
ord
/ TordΛ(i)Tord(i) /IEis,(i)(i) /A0,(i) = Tpar,(i)
h
h
hh
hhhh
hhhh
h
h
h

shhhh

∼ /
ord
ord
0
i
T
/(I
,
A
)
=
T
Λ(i) /Lp (ω , 1 − s)
Eis
0,(i)
(i)
par,(i) /IEis,(i) .
In [2] it is shown that Tordpar,(i) is finite flat over Λ(i) for every i (mod p − 1). (Analternative proof is given in [9] for the case i 6= 2.) Thus we can define the norm
map
ord
NormTpar/Λ(i) : Tpar,(i) → Λ(i) .
(i)Corollary 6. If i 6≡ 0 (mod p − 1) then the element NormTordpar,(i)/Λ(i) (Up − 1) ofi

Λ(i) is divisible by Lp (ω , 1 − s).
ordProof. If Tordpar,(i) = 0 there is nothing to prove. Thus we suppose that Tpar,(i) 6= 0,
ord
so that we may regard Λ as a subring of Tpar,(i) .

THE EISENSTEIN IDEAL IN HIDA’S ORDINARY HECKE ALGEBRA

7

Write N := NormTpar/Λ(i) (Up − 1). Then we have an inclusion of ideals(i)0i
N Λ(i) ⊂ (Up − 1)Tord
par,(i) ∩ Λ(i) ⊂ IEis,(i) ∩ Λ(i) = Lp (ω , 1 − s)Λ(i)(where the first inclusion follows immediately from the definition of the norm andthe final equality is the result of Theorem 5). We can improve the result of the Corollary a little: we may complete Tordpar,(i) at
the ideal I0Eis,(i) to obtain a local ring Tord,Eispar,(i) . There is a consequent factorizationof Tordasa
productpar,(i)ord,Eisord,0
Tord
par,(i) = Tpar,(i) × Tpar,(i) .

Then one sees immediately that the above corollary remains true with Tord,Eispar,(i) in
place of Tord:par,(i)Corollary 7. If i 6≡ 0 (mod p − 1) then the element NormTordpar,(i)/Λ(i) (Up − 1) of

Λ(i) is divisible by Lp (ω i , 1 − s).
We now explain the connection between this result and the result of [9]. In thatreference 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 thecompletion (returning to the notation of this paper) of the ring Tordpar,(i+2) at the
ord,U ≡1pprincipal ideal generated by (Up − 1). Let us write Tpar,(i+2)
for this completion.
Then the following diagram commutes:

Λ(i)

T(χ)

hai7→a−2 hai/ Λ(i+2)ord,U ≡1
Tpar,(i)p .

In this diagram the left vertical arrow gives the Λ-algebra structure on T(χ) whichis considered in [9], the right vertical arrow gives the Λ-algebra structure considered
ord,U ≡1
on Tpar,(i)p in this note, and the upper horizontal arrow is the isomorphism Λ(i) →Λ(i+2) given by the formula hai 7→ a−2 hai for elements a ∈ Γ. (The reason for theshifting 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 thisaction to be given by the formula f |hai := a2 f.)
On [9, p. 520] a local factor T(χ)Eis of T(χ) is defined, and the identification
par,Up ≡1
T(χ) = Tpar,(i+2)
induces an identification T(χ)Eis = Tord,Eispar,(i+2) . Also note that−2
under the isomorphism induced by hai 7→ a hai the element Lp (χω 2 , −1 − s) of
Λ(i) is taken to the element Lp (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 NormT(χ)/Λ(i) (Up − 1) of Λ(i) is divisble by Lp (χω 2 , −1 − s). In [9] thisnorm is denoted by Dp (χ, s)Eis . Thus we have provided a proof of the statement“ Lp (χω 2 , −1 − s) divides Dp (χ, s)Eis ” which occurs at the bottom of [9, p. 520].
(In [9] the further restriction i 6= 0 is imposed. However, this is no real restriction,
since Lp (ω 2 , −1 − s) is a unit in Zp [[Γ]] when p ≥ 5, as follows from the fact that
ζ(−1) = −1/12.)

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MATTHEW EMERTON

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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