Coleman Power Series for K_{2}

and p-Adic Zeta Functions

of Modular Forms

dedicated to Professor Kazuya Kato

Takako Fukaya^{1}

Received: November 30, 2002 Revised: July 3, 2003

Abstract. For a usual local field of mixed characteristic (0, p), we have the theory of Coleman power series [Co]. By applying this theory to the norm compatible system of cyclotomic elements, we obtain the p-adic Riemann zeta function of Kubota-Leopoldt [KL]. This application is very important in cyclotomic Iwasawa theory.

In [Fu1], the author defined and studied Coleman power series for K_{2} for certain class of local fields. The aim of this paper is following the analogy with the above classical case, to obtain p-adic zeta functions of various cusp forms (both in one variable attached to cusp forms, and in two variables attached to ordinary families of cusp forms) by Amice-Vélu, Vishik, Greenberg-Stevens, and Kitagawa,... by applying the K_{2} Coleman power series to the norm compatible system of Beilinson elements defined by Kato [Ka2] in the projective limit of K_{2} of modular curves.

2000 Mathematics Subject Classification: Primary 11F85; Secondary 11G55, 19F27 Keywords and Phrases: p-adic zeta function, modular form, algebraic K-theory, Euler system.

Contents

1. Introduction 388 2. Review of Coleman power series for K_{2} 391 3. Review of p-adic modular forms 393 4. Universal zeta modular form 396 5. Review of p-adic zeta functions of modular forms 412 6. The result on one-variable p-adic zeta function 416 7. The result on two-variable p-adic zeta function 433 References 441

1Supported by the JSPS Research Fellowship for Young Scientists

Documenta Mathematica · Extra Volume Kato (2003) 387–442

1. Introduction

1.1. Let p be any prime number. For a complete discrete valuation field H of mixed characteristic (0, p), with perfect residue field, we have the theory of Coleman power series, as reviewed in section 2. One of the important applications of this theory is the construction of the p-adic Riemann zeta function of Kubota and Leopoldt [KL], by applying the theory to the norm compatible system of cyclotomic elements. In the paper [Fu1], we have obtained “K_{2}-version of Coleman power series” for a certain class of local fields. Following the analogy with the case of the usual Coleman power series above, the aim of the present paper is to show that by applying the theory of K_{2} Coleman power series to the norm compatible system of Beilinson elements in the projective limit of K_{2} of modular curves defined by Kato [Ka2], we obtain p-adic zeta functions of various cusp forms, both in one variable attached to cusp forms (cf.Amice-Vélu [AV], Vishik [Vi]), and two variables attached to universal family of ordinary cusp forms (cf. Greenberg-Stevens [GS], Kitagawa [Ki]).

1.2. We describe our result briefly reviewing the classical result on the p-adic Riemann zeta function. Let denote by ζ_{p^{n}} ∈ \overline{ℚ_{p}} a primitive p^{n}-th root of unity and assume ζ_{p^{n+1}}^{p} = ζ_{p^{n}} for all n ≥ 1. We write Q(ℤ_{p}[[G_{∞}]]) for the total quotient ring of the completed group ring ℤ_{p}[[G_{∞}]] = lim_{\overleftarrow{n}} ℤ_{p}[(ℤ/p^{n}ℤ)^{×}] and G_{∞} = ℤ_{p}^{×} is regarded as the

Galois group Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}) associated to the cyclotomic p extension of ℚ_{p} via the cyclotomic character. Iwasawa [Iw] discovered a relationship between the norm compatibles system of cyclotomic elements (1 − ζ_{p^{n}})_{n} ∈ lim_{\overleftarrow{n}} ℚ_{p}(ζ_{p^{n}})^{×} and the p-adic Riemann zeta

function ζ_{p-adic} ∈ Q(ℤ_{p}[[G_{∞}]]) of Kubota and Leopoldt [KL]. The relation of these two appears in the theory of the usual Coleman power series as follows. The theory of Coleman power series for the multiplicative group induces a map C and C sends (1 − ζ_{p^{n}})_{n} ∈ lim_{\overleftarrow{n}} ℚ_{p}(ζ_{p^{n}})^{×} to ζ_{p-adic} ∈ Q(ℤ_{p}[[G_{∞}]]):

C : lim_{\overleftarrow{n}} ℚ_{p}(ζ_{p^{n}})^{×} ⟶^{via Coleman power series} Q(ℤ_{p}[[G_{∞}]]),

C((1 − ζ_{p^{n}})_{n}) = ζ_{p-adic}.

Coleman Power Series for K_{2} 389

The purpose of this paper is, by pursuing the analogy with this work, to obtain p-adic zeta functions in one variable attached to cusp forms, and in two variables attached to ordinary families of modular forms, whose existences are already known (for one-variable zeta functions cf. Amice-Vélu [AV], Vishik [Vi],..., and two-variable zeta functions associated to ordinary families of cusp forms cf. Greenberg-Stevens [GS], Kitagawa [Ki],...). Let

H = (lim_{\overleftarrow{n}}(ℤ/p^{n}ℤ[[q]][\frac{1}{q}]))[\frac{1}{p}],

where q is an indeterminate. This is a complete discrete valuation field of mixed characteristic (0, p) whose residue field k is an imperfect field satisfying [k : k^{p}] = p. As reviewed in section 2, for H, we have a theory of K_{2} Coleman power series [Fu1]. Let N be a positive integer which is prime to p. Let us denote by Y (Np^{n}, p^{n})

the modular curve corresponding to the subgroup Γ(Np^{n}, p^{n}) = \{(\begin{matrix}α & β \\ γ & δ \end{matrix}) ∈

SL_{2}(ℤ) ; α ≡ 1(Np^{n}), β ≡ 0(Np^{n}), γ ≡ 0(p^{n}), δ ≡ 1(p^{n})} whose total constant field is ℚ(ζ_{p^{n}}). In his paper [Ka2], Kato discovered a norm compatible system of Beilinson elements belonging to lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})). We study the image of this Beilinson-

Kato system under a map C_{N} below which is defined by using our K_{2} Coleman power series following the analogy with the classical map C. We call this image a universal zeta modular form (see section 4).

C_{N} : lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})) ⟶^{via K_{2} Coleman power series} Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]])

C_{N}(Beilinson-Kato system) = the universal zeta modular form.

(Precisely we will define the universal zeta modular form as an element obtained from this image with a suitable modification, cf. section 4.) Here G_{∞}^{(1)} ≅ G_{∞}^{(2)} ≅ G_{∞}, G_{∞}^{(1)} is a group of diamond operators acting on the space of padic modular forms (refer to sections 3 and 4), and G_{∞}^{(2)} = Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}). Further Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]) denotes the total quotient ring of the completed group ring O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]. Theorem 6.2 which is one of our two main results will state that the above universal zeta modular form produces p-adic zeta functions (in one variable) of eigen cusp forms which are not necessarily ordinary. Theorem 7.3 which is the other main theorem asserts roughly the following.

Theorem 1.3 (cf. Theorem 7.3). We assume p ≥ 5. Let 𝔥_{Np^{∞}}^{ord} be the ordinary part of the ring of Hecke operators of level Np^{∞} acting on the space of the padic cusp forms of level Np^{∞} (cf. section 3). The universal zeta modular form above produces, by the method in section 7, a p-adic zeta function in two variables

L_{p-adic}^{ord,univ} ∈ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{a}]

which displays property (1.1) below. Here I_{Np^{∞}}^{ord} ⊂ 𝔥_{Np^{∞}}^{ord} is a certain ideal (see3.7 in section 3), and a ∈ 𝔥_{Np^{∞}}^{ord}[[G_{∞}^{(2)}]] is a certain non-zerodivisor.

Let f be an ordinary p-stabilized newform of tame conductor N (for the definition of an ordinary p-stabilized newform, see 7.2.1 in section 7) of weight k ≥ 2. Attached to f, we have a ring homomorphism κ_{f} :

κ_{f} : 𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⟶ \overline{ℤ_{p}} ; T (n) ↦ a_{n}(f) (n ≥ 1),

with a_{n}(f) such that T (n)f = a_{n}(f)f. Suppose κ_{f} satisfies some “suitable” condition. Then κ_{f} induces a homomorphism which is also denoted by κ_{f} :

κ_{f} : (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}]][\frac{1}{a}] ⟶ Q(\overline{ℤ_{p}}[[G_{∞}^{(2)}]]),

and concerning the image L_{p-adic}^{ord,univ}(κ_{f}) of L_{p-adic}^{ord,univ} under this homomorphism κ_{f}, we have

L_{p-adic}^{ord,univ}(κ_{f}) = p-adic zeta function of f ∈ (O_{M}[[G_{∞}^{(2)}]]) ⊗_{O_{M}} M. (1.1)

Here M is the finite extension ℚ_{p}(a_{n}(f); n ≥ 1) of ℚ_{p}.

For the precise statement, see Theorem 7.3 in section 7.

The above L_{p-adic}^{ord,univ} is essentially the two-variable p-adic zeta function associated to ordinary families of cusp forms which has been already given by Greenberg-Stevens [GS], Kitagawa [Ki],..., by another method. The significance of our p-adic zeta function is that the coefficients in the p-adic zeta function belong to the ring of Hecke operators as above. Hence our L_{p-adic}^{ord,univ} is a p-adic zeta function associated with the universal family of ordinary cusp forms. By another method, Ochiai ([Oc]) has also constructed this kind of two-variable p-adic zeta function. The author found that Panchishkin [Pa1], [Pa2] gave a new way of the construction of p-adic zeta functions of modular forms by using something similar to our universal zeta modular form at almost the same time as the author gave talks in the conferences in the autumn of 2000 as described in the proceedings [Fu2], [Fu3] in Japanese. Our aim is to obtain p-adic zeta functions of modular forms by applying K_{2} Coleman power series to the norm compatible system of Beilinson elements in K_{2} of modular curves.

1.4. The organization of this paper is as follows. In section 2, we review the theory of Coleman power series both in the classical case and in the case for K_{2} [Fu1]. In section 3, we review the theory of p-adic modular forms (cf. [Hi1]). In section 4, we define and study a “universal zeta modular form” which is obtained from the image of Beilinson-Kato system under C_{N} appearing in 1.2. In our construction, p-adic properties of p-adic zeta functions are deduced from the p-adic properties of the universal zeta modular form and the relation between the universal zeta modular form and special values of zeta functions of modular forms. In section 5, we review the theory of p-adic zeta functions of modular forms.

In section 6, we prove our theorem (Theorem 6.2) on the construction of onevariable p-adic zeta functions of eigen cusp forms which are not necessarily ordinary. In section 7, we prove our theorem (Theorem 7.3) on the construction of padic zeta functions in two variables, which are attached to universal families of ordinary cusp forms.

The author would like to express her sincere gratitude to Professor Kazuya Kato for his guidance and constant encouragement. She has learned a great deal from him throughout her graduate course. She is very happy to dedicate this paper to his celebrated 50th birthday.

She is very thankful to Professor John Coates who gave her comments and constant encouragement for this study.

In this paper, for a complete discrete valuation field L, O_{L} denotes the ring of integers of L. For a ring R, Q(R) denotes the total quotient ring of R. We also fix once and for all an embedding of \overline{ℚ} into \overline{ℚ_{p}}.

2. Review of Coleman power series for K_{2}

In this section we give a brief review of the theory of Coleman power series both in the usual case (in 2.1 – 2.2) and our K_{2}-version case (in 2.3 – 2.5).

2.1. We review the classical case of the usual Coleman power series. The existence of Coleman power series were discovered by Coates and Wiles [CW] and almost immediately, Coleman [Co] generalized their approach by an alternative method. The theory of Coleman power series has been obtained for general Lubin-Tate groups, but here we review the theory only for the formal multiplicative group. Let H be a complete discrete valuation field of mixed characteristic (0, p) with perfect residue field k. We assume that H is absolutely unramified, i.e. p is a prime element of O_{H}. We denote by O_{H}[[ε − 1]] = lim_{\overleftarrow{n}} O_{H}[ε^{±1}]/(ε − 1)^{n}

the coordinate ring of the formal completion of the multiplicative group over O_{H}, and by O_{H}((ε − 1)) the Laurent series ring O_{H}[[ε − 1]][1/(ε − 1)]. Let σ denote the Frobenius automorphism of O_{H}. We extend σ to an endomorphism of O_{H}((ε − 1)) by putting σ(ε) = ε. We define a ring homomorphism

φ : O_{H}((ε − 1)) ⟶ O_{H}[[ε − 1]][\frac{1}{ε^{p} − 1}]

by φ(f)(ε) = (σf)(ε^{p}), and

N : O_{H}((ε − 1))^{×} ⟶ O_{H}((ε − 1))^{×}

to be the norm operator induced by the homomorphism φ. We write (O_{H}((ε − 1))^{×})^{N=1} for the group of all units f in O_{H}((ε − 1)) which satisfy N(f) = f. Now let ζ_{p^{n}} denote a primitive p^{n}-th root of unity, and assume ζ_{p^{n+1}}^{p} = ζ_{p^{n}} for all n ≥ 1. Put H_{n} = H(ζ_{p^{n}}). The aim in this case is to study lim_{\overleftarrow{n}} H_{n}^{×},

where the projective limit is taken with respect to the norm maps in the tower of fields H_{n} (n = 1, 2, 3, …).

Theorem 2.2 (Coleman [Co]). We have an isomorphism

Ψ : (O_{H}((ε − 1))^{×})^{N=1} ⟶^{≅} lim_{\overleftarrow{n}} H_{n}^{×}

given by Ψ(f(ε)) = ((σ^{−n}f)(ζ_{p^{n}}))_{n=1,2,3,...}.

2.3. Now we review our case of K_{2}-version of Theorem 2.2. (See [Fu1] for more details.) Let H be a complete discrete valuation field of characteristic 0, whose residue field k is an imperfect field of characteristic p satisfying [k : k^{p}] = p. We assume that H is absolutely unramified. We fix once and for all a p-base b of k, and a lifting q of b to H (all of our subsequent constructions depend on these choices). We define σ : O_{H} ⟶ O_{H} to be the unique ring homomorphism satisfying σ(q) = q^{p}, and the action of σ on k is given by raising to the p-th power. For simplicity, let us write

S = O_{H}[[ε − 1]], S′ = O_{H}((ε − 1)) = S[\frac{1}{ε − 1}].

We extend σ to an endomorphism of S′ by putting σ(ε) = ε. We then define a ring homomorphism

φ : S′ ⟶ S[\frac{1}{ε^{p} − 1}]

by φ(f)(ε) = (σf)(ε^{p}). For any ring A, let K_{2}(A) denote Quillen’s K_{2} group of A ([Qu]). Since S[1/(ε^{p} − 1)] is a free S′-module of rank p^{2} via φ, we have the K_{2} norm map (see [Qu], §4, Transfer maps)

K_{2}(S[\frac{1}{ε^{p} − 1}]) ⟶ K_{2}(S′).

The composition of this with

K_{2}(S′) ⟶ K_{2}(S[\frac{1}{ε^{p} − 1}])

induced by the inclusion map S′ ↪ S[1/(ε^{p} − 1)], gives rise to a K_{2} norm map

N : K_{2}(S′) ⟶ K_{2}(S′).

We consider the following tower of fields above H. We take a p^{n}-th root q^{1/p^{n}} of q in a fixed algebraic closure \overline{H} of H and assume that (q^{1/p^{n+1}})^{p} = q^{1/p^{n}} for all n ≥ 1. We define

H_{n} = H(ζ_{p^{n}}, q^{1/p^{n}}).

Moreover, we define a ring homomorphism

θ_{n} : O_{H} ⟶ O_{H(q^{1/p^{n}})}

by specifying that θ_{n}(q) = q^{1/p^{n}}, and that the induced map k ⟶ k(b^{1/p^{n}}) on the residue fields is the isomorphism x ↦ x^{1/p^{n}}. For n ≥ 1, θ_{n} induces a ring homomorphism

h_{n} : S′ ⟶ H_{n}

given by h_{n}(∑_{m=−r}^{∞} a_{m}(ε − 1)^{m}) = ∑_{m=−r}^{∞} θ_{n}(a_{m})(ζ_{p^{n}} − 1)^{m}. Thus we obtain a map

Ψ_{n} : K_{2}(S′) ⟶ K_{2}(H_{n}) (n ≥ 1)

which is induced by h_{n}. In order to state our theorem, we need to introduce certain completions K̂_{2} of our K_{2}-groups (see 2.5 below for the definition). All of the above homomorphisms give rise to corresponding maps between the completed K_{2}-groups, which we can check easily from the definition of the completions in 2.5 below and which we denote by the same symbol. We also write K̂_{2}(S′)^{N=1} for the subgroup of elements f in K̂_{2}(S′) which satisfy N(f) = f. Instead of lim_{\overleftarrow{n}} H_{n}^{×}, we study the projective limit lim_{\overleftarrow{n}} K̂_{2}(H_{n}) with respect to

the norm maps in the tower of fields H_{n} (n = 1, 2, 3, …).

Theorem 2.4 ([Fu1], Theorem 1.5). We have an isomorphism

Ψ : K̂_{2}(S′)^{N=1} ⟶_{≅} lim_{\overleftarrow{n}} K̂_{2}(H_{n})

given by Ψ(f) = (Ψ_{n}(f) : n = 1, 2, 3, …).

2.5. We describe the completions of the K_{2} groups appearing in Theorem 2.4. We introduce the completions K̂_{2}(A) in the following two cases by which the completions in Theorem 2.4 follows.

(i) A = S′. (ii)A is a complete discrete valuation field L. Let r ≥ 1. In the case of (i), let U^{(r)} = 1 + (p, (ε − 1))^{r}S, a subgroup of S^{×}, where (p, (ε − 1)) is the ideal of S generated by the elements in ( ). In the case of (ii), let U^{(r)} = U_{L}^{(r)}, where U_{L}^{(r)} is the r-th unit group of L, i.e. 1 + m_{L}^{r} ⊂ O_{L}^{×} for the maximal ideal m_{L} of L. We define a subgroup U^{(r)}K_{2}(A) of K_{2}(A) for a ring A of the type (i) or (ii), as the one which is generated by {a, A^{×}} for all a ∈ U^{(r)} ⊂ A^{×}, and define

K̂_{2}(A) = lim_{\overleftarrow{r}} K_{2}(A)/U^{(r)}K_{2}(A).

3. Review of p-adic modular forms

In this section, we briefly review the necessary facts for us on the theory of p-adic modular forms. We follow Hida [Hi1] to which we refer for more details. (See also Katz [Katz1], [Katz2], etc.)

3.1. Let L be a finite extension of ℚ_{p} in \overline{ℚ_{p}}, and we take a finite extension L_{0} of ℚ which is dense in L under the p-adic topology. For k ≥ 0 and for M ≥ 1, let M_{k}(X_{1}(M); L_{0}) be the space of modular forms for Γ_{1}(M) of weight k with Fourier coefficients in L_{0}. We define

M_{k}(X_{1}(M); L) = M_{k}(X_{1}(M); L_{0}) ⊗_{L_{0}} L.

Similarly we define the space of cusp forms S_{k}(X_{1}(M); L_{0}) and S_{k}(X_{1}(M); L). As in [Hi1], §1, it has been known that the spaces M_{k}(X_{1}(M); L), S_{k}(X_{1}(M); L) are independent of the choice of a subfield L_{0} in the evident sense. Now for j ≥ 0 we put

M^{j}(X_{1}(M); L) = ⨁_{k=0}^{j} M_{k}(X_{1}(M); L),

S^{j}(X_{1}(M); L) = ⨁_{k=1}^{j} S_{k}(X_{1}(M); L),

which are embedded in L[[q]] via the summation of q-expansions, and

M^{j}(X_{1}(M); O_{L}) = M^{j}(X_{1}(M); L) ∩ O_{L}[[q]],
S^{j}(X_{1}(M); O_{L}) = S^{j}(X_{1}(M); L) ∩ O_{L}[[q]].

We define \overline{M}(X_{1}(M); O_{L}) as the closure of ⋃_{j≥1} M^{j}(X_{1}(M); O_{L}) in O_{L}[[q]] for the p-adic topology, and \overline{S}(X_{1}(M); O_{L}) to be the closure of ⋃_{j≥1} S^{j}(X_{1}(M); O_{L}) in O_{L}[[q]] for the p-adic topology. For an integer N ≥ 1 which is prime to p, it has been proven that \overline{M}(X_{1}(Np^{t}); O_{L}) and \overline{S}(X_{1}(Np^{t}); O_{L}) are independent of the choice of t ≥ 1, as in Hida [Hi1], §1, Cor. 1.2 (i), and (1.19a), respectively. For simplicity we put

\overline{M}_{Np^{∞}} = \overline{M}(X_{1}(Np^{t}); ℤ_{p}), \overline{S}_{Np^{∞}} = \overline{S}(X_{1}(Np^{t}); ℤ_{p})

for any t ≥ 1. We introduced the above notation in a general situation for our later use, however in the rest of this section, we always take L = ℚ_{p}.

3.2. We review the definition of the rings of Hecke operators H_{Np^{∞}} and 𝔥_{Np^{∞}} acting on \overline{M}_{Np^{∞}} and \overline{S}_{Np^{∞}}, respectively. For t ≥ 1 and j ≥ 1, let H^{j}(X_{1}(Np^{t}); ℤ_{p}) (resp. 𝔥^{j}(X_{1}(Np^{t}); ℤ_{p})) be the ℤ_{p}-subalgebra of ℤ_{p}-endomorphism ring of M^{j}(X_{1}(Np^{t}); ℤ_{p}) (resp. S^{j}(X_{1}(Np^{t}); ℤ_{p})) generated over ℤ_{p} by T (n) (n ≥ 1). We put

H(X_{1}(Np^{t}); ℤ_{p}) = lim_{\overleftarrow{j}} H^{j}(X_{1}(Np^{t}); ℤ_{p}),

𝔥(X_{1}(Np^{t}); ℤ_{p}) = lim_{\overleftarrow{j}} 𝔥^{j}(X_{1}(Np^{t}); ℤ_{p}),

where the inverse limits are taken by natural homomorphisms given by the restriction of operators. As in Hida [Hi1], §1, (1.15a) and (1.19a), respectively, H(X_{1}(Np^{t}); ℤ_{p}) and 𝔥(X_{1}(Np^{t}); ℤ_{p}) do not depend on t ≥ 1. For simplicity we put

H_{Np^{∞}} = H(X_{1}(Np^{t}); ℤ_{p}), 𝔥_{Np^{∞}} = 𝔥(X_{1}(Np^{t}); ℤ_{p})

for any t ≥ 1. The rings H_{Np^{∞}} and 𝔥_{Np^{∞}} act on \overline{M}_{Np^{∞}} and \overline{S}_{Np^{∞}}, respectively. The ring 𝔥_{Np^{∞}} is, in fact, a quotient of H_{Np^{∞}} by the annihilator in H_{Np^{∞}} of \overline{S}_{Np^{∞}}.

3.3. By the action of H_{Np^{∞}} on \overline{M}_{Np^{∞}} in 3.2, we have a canonical map

𝔦 : \overline{M}_{Np^{∞}} ⟶ Hom_{ℤ_{p}}(H_{Np^{∞}}, ℤ_{p}) ; f ↦ (T (n) ↦ a_{1}(T (n)f))

(f ∈ \overline{M}_{Np^{∞}}) where a_{1}(T (n)f) is the coefficient of q in the q-expansion of T (n)f ∈ \overline{M}_{Np^{∞}}. This map will play an important role later in the construction of our twovariable p-adic zeta function L_{p-adic}^{ord,univ}.

3.4. For j ≥ 1 let e^{j} = lim_{n⟶∞} T (p)^{n!} in H^{j}(X_{1}(Np^{t}); ℤ_{p}) or 𝔥^{j}(X_{1}(Np^{t}); ℤ_{p}), and e = lim_{\overleftarrow{j}} e^{j}. Then e^{2} = e.

We denote by H_{Np^{∞}}^{ord} the ordinary part e · H_{Np^{∞}} of H_{Np^{∞}} and by 𝔥_{Np^{∞}}^{ord} the ordinary part e · 𝔥_{Np^{∞}} of 𝔥_{Np^{∞}}. Let P_{Np^{∞}}^{ord} ⊂ H_{Np^{∞}}^{ord} (resp. p_{Np^{∞}}^{ord} ⊂ 𝔥_{Np^{∞}}^{ord}) be the annihilator of the old forms (resp. old cusp forms) of level N′p^{t} for all N′ such that N′|N and N′ < N. For the precise definition of P_{Np^{∞}}^{ord} (resp. p_{Np^{∞}}^{ord}), see [Hi1], §3. In the case N = 1, we have

P_{p^{∞}}^{ord} = H_{p^{∞}}^{ord}, p_{p^{∞}}^{ord} = 𝔥_{p^{∞}}^{ord}.

On P_{Np^{∞}}^{ord} and p_{Np^{∞}}^{ord}, Hida showed Proposition 3.6 below which is important for us. Preceding it, we set up notation.

3.5. We define a group G_{∞}^{(1)} which is endowed with an isomorphism to ℤ_{p}^{×} and which acts on the space \overline{M}_{Np^{∞}} in the following way.

Firstly for x ∈ (ℤ/Np^{t}ℤ)^{×} we denote by ⟨x⟩ the endomorphism (\begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix})^{∗}

on M_{k}(X_{1}(Np^{t}); ℚ) induced by the action of (\begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix}) ∈ GL_{2}(ℤ/Np^{t}ℤ) on

X(Np^{t}, Np^{t}). (The action of GL_{2}(ℤ/Np^{t}ℤ) on X(Np^{t}, Np^{t}) induces an endomorphism on M_{k}(X_{1}(Np^{t}); ℚ) by the fact that M_{k}(X_{1}(Np^{t}); ℚ) may be

regarded as the fixed part of M_{k}(X(Np^{t}, Np^{t})) by the group \{(\begin{matrix}u & v \\ w & x \end{matrix}) ∈

GL_{2}(ℤ/Np^{t}ℤ) ; u ≡ 1(Np^{t}), w ≡ 0(Np^{t})}, where M_{k}(X(Np^{t}, Np^{t})) is the space of modular forms on X(Np^{t}, Np^{t}) of weight k as, for example, in [Ka2], §3 (3.3.1), and §4.)

We also use the same notation ⟨x⟩ for the endomorphism on M_{k}(X_{1}(Np^{t}); ℚ_{p}) induced by the above. For a ∈ ℤ_{p}^{×}, let g_{a}^{(1)} ∈ G_{∞}^{(1)} denote the element corresponding to a under the given isomorphism. For f = ∑_{k} f_{k} ∈ ⋃_{j} M^{j}(X_{1}(Np^{t}); ℚ_{p}) with f_{k} ∈

M_{k}(X_{1}(Np^{t}); ℚ_{p}) and t ≥ 1, we define the action of g_{a}^{(1)} ∈ G_{∞}^{(1)} as

g_{a}^{(1)} · f = ∑_{k} a^{k−2}⟨a′⟩f_{k},

where a′ ∈ (ℤ/Np^{t}ℤ)^{×} is the element such that a′ ≡ a(p^{t}) and a′ ≡ 1(N). This action of G_{∞}^{(1)} may be extended to \overline{M}_{Np^{∞}}. We remark that the relation between this action of G_{∞}^{(1)} and the action of ℤ_{p}^{×} in Hida [Hi1], §3, (3.1) is

f | a = a^{2}g_{a}^{(1)} · f,

where f | a denotes the image of f under the action of a in the meaning of Hida. We put Λ = ℤ_{p}[[G_{∞}^{(1)}]]. By the above action of G_{∞}^{(1)} on \overline{M}_{Np^{∞}}, we have a ring homomorphism

Λ ⟶ H_{Np^{∞}}.

We see that via this homomorphism 𝔥_{Np^{∞}}, H_{Np^{∞}}^{ord}, and 𝔥_{Np^{∞}}^{ord} become also Λ-algebras, and P_{Np^{∞}}^{ord}, and p_{Np^{∞}}^{ord} are Λ-modules.

Proposition 3.6 (Hida [Hi1] Corollary 3.3). We assume p ≥ 5.

(1) The rings H_{Np^{∞}}^{ord} and 𝔥_{Np^{∞}}^{ord} are finitely generated projective modules over Λ.

(2) The ideal P_{Np^{∞}}^{ord} (resp. p_{Np^{∞}}^{ord}) is a finitely generated projective module over Λ. Moreover the intersection of P_{Np^{∞}}^{ord} (resp. p_{Np^{∞}}^{ord}) and the nilradical of H_{Np^{∞}}^{ord} (resp. 𝔥_{Np^{∞}}^{ord}) is null.

Proof. For the proof, see [Hi1]. □

3.7. We define an ideal

I_{Np^{∞}}^{ord} ⊂ 𝔥_{Np^{∞}}^{ord}

to be the annihilator of p_{Np^{∞}}^{ord} ⊂ 𝔥_{Np^{∞}}^{ord}. Then the natural map

p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ⟶ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ)

is an isomorphism, where the both hands sides are semisimple algebras over Q(Λ) (cf. [Hi1], §3).

4. Universal zeta modular form

In this section, we define and study a“universal zeta modular form” which is obtained from the norm compatible system of Beilinson elements defined by Kato [Ka2], via K_{2} Coleman power series. The p-adic properties of padic zeta functions of modular forms are deduced from the p-adic property of the universal zeta modular form and the relations between the universal zeta modular form and zeta values. In 4.1, we define a map

C_{N} : lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})) ⟶ Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]])

(see 4.1 for the details) by using K_{2} Coleman power series, and Proposition 4.4 shows that

C_{N} : Beilinson-Kato system ↦ the universal zeta modular form
(reviewed in 4.2)  (defined in 4.3).

In 4.5, we explain the properties of the universal zeta modular form concerning the relation with special values of zeta functions of cusp forms. In what follows,

H = (lim_{\overleftarrow{n}}(ℤ/p^{n}ℤ[[q]][\frac{1}{q}]))[\frac{1}{p}].

We fix a system (ζ_{p^{n}})_{n≥1} of primitive p^{n}-th roots of unity which satisfy ζ_{p^{n+1}}^{p} = ζ_{p^{n}} for all n ≥ 1. For q ∈ H and n ≥ 1, we fix p^{n}-th roots q^{1/p^{n}} of q in \overline{H} which satisfy (q^{1/p^{n+1}})^{p} = q^{1/p^{n}} for all n ≥ 1. Let N denote a positive integer such that (N, p) = 1.

4.1. By using K_{2} Coleman power series, we define a map

C_{N} : lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})) ⟶ Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]),

where the left hand side is the inverse limit of K_{2} of modular curves (cf. 4.1.1) taken with respect to the norm maps, and on the right hand side, the group G_{∞}^{(1)} is as in section 2 (we will review this in 4.3) and the group G_{∞}^{(2)} is the Galois group Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}). We put S = O_{H}[[ε − 1]] and S′ = O_{H}[[ε − 1]][1/(ε − 1)]. Let G_{∞} = ℤ_{p}^{×}. The map C_{N} is defined as the following composition:

C_{N} : lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n}))

⟶^{E_{N}} lim_{\overleftarrow{n}} K̂_{2}(H_{n})[[G_{∞}]]

⟶_{≅}^{Col} K̂_{2}(S′)^{N=1}[[G_{∞}]]

⟶^{d log} Ω_{S}^{2}(log)[[G_{∞}]] = \frac{S}{ε − 1} · d log(q) ∧ d log(ε)[[G_{∞}]]

⟶_{≅} \frac{S}{ε − 1}[[G_{∞}]] ⟶ Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]).

(4.1)

We explain each term and each arrow in the composition (4.1).

4.1.1. For M_{1}, M_{2} ≥ 1 such that M_{1} + M_{2} ≥ 5, let Y (M_{1}, M_{2}) be the modular curve over ℚ, which represents the functor

S ↦(the set of isomorphism classes of pairs (E, ι)

where E is an elliptic curve over S and ι is an injective homomorphism

ℤ/M_{1}ℤ × ℤ/M_{2}ℤ ⟶ E of group schemes over S).

For M ≥ 3 such that M_{1}|M and M_{2}|M, we have

Y (M_{1}, M_{2}) = G\Y (M, M),  (4.2)

where G is the group \{(\begin{matrix}u & v \\ w & x \end{matrix}) ∈ GL_{2}(ℤ/Mℤ) ; u ≡ 1(M_{1}), v ≡ 0(M_{1}), w ≡

0(M_{2}), x ≡ 1(M_{2})}. We define Y (M_{1}, M_{2}) for M_{1}, M_{2} ≥ 1, M_{1} + M_{2} < 5, by (4.2).

4.1.2. We explain A[[G_{∞}]] for an abelian group A. For a set J, ℤ[J] denotes a free ℤ-module on the set J. We define G_{n} = (ℤ/p^{n}ℤ)^{×}, A[G_{n}] = A ⊗_{ℤ} ℤ[G_{n}],

and A[[G_{∞}]] = lim_{\overleftarrow{n}} A[G_{n}].

4.1.3. Let

Ω_{S/ℤ}^{1}(log) = (Ω_{S/ℤ}^{1} ⊕ S ⊗_{ℤ} S^{′×})/N ,

where Ω_{S/ℤ}^{1} is the module of the absolute differential forms and N is the Ssubmodule of the direct sum which is generated by elements (−da, a ⊗ a) for a ∈ S ∩S^{′×}. In Ω_{S/ℤ}^{1}(log), we denote the class (0, 1⊗a) for a ∈ S^{′×} by d log(a). For r ≥ 1, let Ω_{S/ℤ}^{r}(log) = ⋀_{S}^{r} Ω_{S/ℤ}^{1}(log), and define

Ω_{S}^{r}(log) = lim_{\overleftarrow{n}} Ω_{S/ℤ}^{r}(log)/p^{n}Ω_{S/ℤ}^{r}(log).

Then we have Ω_{S}^{1}(log) is a free S-module and

Ω_{S}^{1}(log) = S · d log(q) ⊕ S · d log(ε − 1), Ω_{S}^{2}(log) = S · d log(q) ∧ d log(ε − 1),

Ω_{S}^{r}(log) = 0 for r ≥ 3.

4.1.4. We explain the definition of the map E_{N} in (4.1) (cf. [Fu1], §6). This map is induced by the following map for n ≥ 1 satisfying Np^{n} + p^{n} ≥ 5

K_{2}(Y (Np^{n}, p^{n})) ⟶ K_{2}(H_{n})[G_{n}] ; x ↦ ∑_{u∈G_{n}}( ∑_{w∈ℤ/p^{n}ℤ} x_{(u,w)})g_{u}.

Here x_{(u,w)} ∈ K_{2}(H_{n}) is the pull-back of x under the following composition:

Spec(H_{n}) ⟶ Spec(H_{n}(q^{1/N})) ⟶ Y (Np^{n}, p^{n}),  (4.3)

where q^{1/N} ∈ \overline{H} is a N-th root of q.

The first map in (4.3) is given by the homomorphism

H_{n}(q^{1/N}) ⟶ H_{n} ; ∑_{i=−∞}^{∞} a_{i}q^{i/Np^{n}} ↦ ∑_{i=−∞}^{∞} a_{i}q^{i/p^{n}}.

We define the second map of (4.3). Let 𝔈_{q} be the elliptic curve over O_{H} which is obtained from the Tate curve over ℤ[[q]][1/q] with q-invariant q. For each m ≥ 1, we have _{m}𝔈_{q}(O\overline{H}) = \{q^{a/m}ζ_{m}^{b} mod q^{ℤ} ; a, b ∈ ℤ\}, where _{m}𝔈_{q} = Ker(m : 𝔈_{q} ⟶ 𝔈_{q}). Now we define the second map of (4.3) by the open immersion corresponding to

(𝔈_{q} ⊗_{O_{H}} H_{n}(q^{1/N}), q^{u′/Np^{n}} mod q^{ℤ}, q^{w/p^{n}}ζ_{p^{n}} mod q^{ℤ}) over H_{n}(q^{1/N}).

Here u′ ∈ (ℤ/Np^{n}ℤ)^{×} is the element such that u′ ≡ u(p^{n}) and u′ ≡ 1(N).

4.1.5. The second arrow of (4.1), which is an isomorphism, is by Theorem 2.4 in section 2 on K_{2} Coleman power series.

4.1.6. We explain the map d log in (4.1). It is the map induced by the map d log : K̂_{2}(S′) ⟶ Ω_{S}^{2}(log) characterized by {α_{1}, α_{2}} ↦ d log(α_{1}) ∧ d log(α_{2}), where α_{1}, α_{2} ∈ S^{′×} and {α_{1}, α_{2}} ∈ K̂_{2}(S′) is the symbol. (The group K̂_{2}(S′) is topologically generated by the symbols. Refer to [Fu1], §4, 4.11.)

4.1.7. We explain the last arrow in (4.1). We firstly define a map

S[[G_{∞}]] ⟶ O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]  (4.4)

to be the O_{H}-homomorphism associated to

ε^{a}g_{u} ↦ \begin{cases}u^{−1}g_{u}^{(1)}g_{u^{−1}a}^{(2)} & if (a, p) = 1 \\ 0 & if (a, p) ≠ 1, \end{cases}

for a ∈ ℤ and u ∈ ℤ_{p}^{×}. Here for u ∈ ℤ_{p}^{×}, g_{u}^{(1)} ∈ G_{∞}^{(1)} denotes the corresponding element, and g_{u}^{(2)} ∈ G_{∞}^{(2)} denotes the corresponding element to u via the cyclotomic character χ_{cyclo} : G_{∞}^{(2)} ⟶^{≅} ℤ_{p}^{×}. Next for an integer d′ which is prime to p, let ν_{d′} : \frac{S}{ε−1} ⟶ \frac{S}{ε−1} be the O_{H}-homomorphism given by sending f(ε)/(ε − 1) (f(ε) ∈ S) to f(ε^{d′})/(ε^{d′} − 1). It follows from the definition that the image (1 − d′ν_{d′})(\frac{S}{ε−1}) is contained in S. Now the last map in (4.1) is defined as the composition

\begin{aligned}\frac{S}{ε − 1}[[G_{∞}]] & ⟶^{1−d′ν_{d′}} S[[G_{∞}]] ⟶^{(4.4)} O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] \\ & ⟶^{·(1−d′g_{d′}^{(2)})^{−1}} Q(O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]), \end{aligned}

where 1 − d′ν_{d′} is applied only for the coefficient \frac{S}{ε−1} of G_{∞}. It is easily seen that the map C_{N} is independent of the choice of d′.

4.2. Let c and d be integers satisfying (c, 6Np) = 1 and (d, 6p) = 1. We review the norm compatible system of Beilinson elements defined by Kato in [Ka2]

(_{c,d}z_{Np^{n},p^{n}})_{n} ∈ lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})),  (4.5)

where the projective limit is taken with respect to the norm maps, in the form which is enough for us here (the details are found in [Ka2]). For n ≥ 1 satisfying Np^{n} + p^{n} ≥ 5, the element (4.5) is given as

c,dz_{Np^{n},p^{n}} = {_{c}g_{Np^{n},0},_{d}g_{0,p^{n}}},

where _{c}g_{Np^{n},0} ∈ O(Y (Np^{n}, 1))^{×} and _{d}g_{0,p^{n}} ∈ O(Y (1, p^{n}))^{×} are Siegel units, and we introduce their properties which are necessary for us here (see, for example, [Ka2] for the details). For integers M and c such that M ≥ 3 and (c, 6M) = 1, we have an element cθ_{E} of O(E \_{c}E)^{×} which has the following property. Here E is the universal elliptic curve over Y (M, M), _{c}E = Ker(c : E ⟶ E), and O(E \ _{c}E)^{×} is the affine ring. For τ ∈ ℌ and z ∈ ℂ \ c^{−1}(ℤτ + ℤ), the value at z of _{c}θ_{E}, on the elliptic curve ℂ/(ℤτ + ℤ), is

q^{(1/12)(c^{2}−1)}(−t)^{(1/2)(c−c^{2})} · γ_{q}(t)^{c^{2}}γ_{q}(t^{c})^{−1},

where q = exp(2πiτ), t = exp(2πiz), and

γ_{q}(t) = ∏_{j≥0}(1 − q^{j}t) ∏_{j≥1}(1 − q^{j}t^{−1}).

Now the Siegel unit _{c}g_{α,β} for (α, β) = (a/M, b/M) ∈ (((1/M)ℤ)/ℤ)^{2} \ {(0, 0)} (a, b ∈ ℤ) may be defined by

cg_{α,β} = ι_{α,β}^{∗}(_{c}θ_{E}) ∈ O(Y (M, M))^{×}.

Here

ι_{α,β} = ae_{1} + be_{2} : Y (M, M) ⟶ E \_{c}E

with the canonical basis (e_{1}, e_{2}) of E over Y (M, M). In the case α = 0, cg_{0,β} ∈ O(Y (1, M))^{×} and in the case β = 0, _{c}g_{α,0} ∈ O(Y (M, 1))^{×}.) In [Ka2] (cf. [Sc]), it was shown that _{c,d}z_{Np^{n},p^{n}} (n ≥ 1) form a projective system with respect to the norm maps.

In the paper [Ka2], Kato always used norm compatible systems (_{c,d}z_{Mp^{n},M′p^{n}})_{n} ∈ lim_{\overleftarrow{n}} K_{2}(Y (Mp^{n}, M′p^{n})) (M, M′ ≥ 1, (M + M′)p^{n} ≥ 5) satisfying the con-

dition that M|M′ in application. However clearly the system (_{c,d}z_{Np^{n},p^{n}})_{n} ∈ lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})) which we use does not satisfy this condition. When Kato

used a system, for example, to construct a p-adic zeta function of an eigen cusp form f, he considered the “f^{∗}-component” of the system, where f^{∗} is the dual cusp form of f (see 6.5.1 in section 6 for the definition of the dual cusp form, and for the meaning of “component”, refer to section 6). But our method needs to study the “f-component” of the system. So we must slightly modify his system in application.

4.3. We define a“universal zeta modular form”

z_{Np^{∞}}^{univ} ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][\frac{1}{g}](⊂ O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][\frac{1}{g}])

which yields special values at s = r (r ∈ ℤ, 1 ≤ r ≤ k − 1) of the zeta functions of modular forms of weight k ≥ 2 and level Np^{t} for t ≥ 0. Here \overline{M}_{Np^{∞}} is as in section 3. Moreover G_{∞}^{(1)} ≅ G_{∞}^{(2)} ≅ G_{∞} = ℤ_{p}^{×}, and G_{∞}^{(1)} is , as before, the group acting on the space of p-adic modular forms \overline{M}_{Np^{∞}} whose action is characterized by g_{a}^{(1)}f = a^{k−2}⟨a′⟩f for f ∈ M_{k}(X_{1}(Np^{t}); ℚ_{p}), a ∈ ℤ_{p}^{×}, and a′ ∈ (ℤ/Np^{t}ℤ)^{×} such that a′ ≡ a(p^{t}) and a′ ≡ 1(N). Here ⟨a′⟩ is as in 3.5. The group G_{∞}^{(2)} is the Galois group Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}). We define z_{Np^{∞}}^{univ} as an element of O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g] and in 4.5.5, we will prove that it belongs to the subspace \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g]. Firstly we define F_{N,1}, F_{N,2} ∈ H[[G_{∞}]] = lim_{\overleftarrow{n}} H[G_{n}] to be

F_{N,1} = ( ∑_{\begin{matrix}i≥1 \\ (i,p)=1 \end{matrix}} ∑_{j≥1} q^{Nij})(g_{i} − g_{−i}) + lim_{\overleftarrow{n}}( ∑_{a∈(ℤ/p^{n}ℤ)^{×}} ζ_{≡a(p^{n})}(0) · g_{a}),

\begin{aligned}F_{N,2} & = ( ∑_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} q^{ij} · g_{i} − ∑_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} q^{ij} · g_{−i}) \\ & + lim_{\overleftarrow{n}}( ∑_{\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(Np^{n})}(0) · g_{a}). \end{aligned}

Here for a ∈ ℤ_{p}^{×} or a ∈ (ℤ/p^{n}ℤ)^{×}, g_{a} represents the corresponding element of G_{∞} or G_{n}, respectively. For M, m ∈ ℤ, M ≥ 1, and a ∈ ℤ/Mℤ, ζ_{≡a(M)}(m) is the evaluation at s = m of the partial Riemann zeta function

ζ_{≡a(M)}(s) = ∑_{\begin{matrix}j≥1 \\ j≡a mod M \end{matrix}} j^{−s},

and ∑\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix} ζ_{≡a(Np^{n})}(0) · g_{a} belongs to H[G_{n}].

We define the product F_{N,1} · F_{N,2} ∈ H[[G_{∞} × G_{∞}]] naturally (by the rule xg_{a} · yg_{b} ↦ xyg_{a,1}g_{b,2} with x, y ∈ H, a, b ∈ ℤ_{p}^{×}, where g_{a,1} (resp, g_{b,2}) means the corresponding element of the first (resp. the second) G_{∞}). Now we define the universal zeta modular form z_{Np^{∞}}^{univ} to be the image of F_{N,1} · F_{N,2} under the isomorphism of rings over H

H[[G_{∞} × G_{∞}]] ⟶ H[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ;  (4.6)
xg_{a,1}g_{b,2} ↦ xg_{b}^{(1)}g_{ab^{−1}}^{(2)} (x ∈ H, a, b ∈ ℤ_{p}^{×})
(F_{N,1} · F_{N,2} ↦ z_{Np^{∞}}^{univ}).

For integers c, d′ such that (c, p) = 1 and (d′, p) = 1, we have

(1 − c^{−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d′g_{d′}^{(2)})z_{Np^{∞}}^{univ} ∈ O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]].  (4.7)

This follows from the fact that

(1 − c^{−1}g_{c^{−1}}) · lim_{\overleftarrow{n}} ∑_{\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(Np^{n})}(0) · g_{a} ∈ O_{H}[[G_{∞}]].

By (4.7), we obtain that z_{Np^{∞}}^{univ} ∈ O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g] for a non-zero divisor g ∈ ℤ[[G_{∞}^{(1)} × G_{∞}^{(2)}]]. Thus the universal zeta modular form is something like a product of two “Λadic Eisenstein series” in the sense of Hida in [Hi3], Chapter 7, §7.1.

The following proposition describes the relation between the norm compatible system of Beilinson elements and the universal zeta modular form.

Proposition 4.4 . Let c and d be as in 4.2. We further assume that c ≡ 1 mod N. Then we have

C_{N}((_{c,d}z_{Np^{n},p^{n}})_{n}) = (c^{2} − cg_{c^{−1}}^{(1)}g_{c}^{(2)})(d^{2} − dg_{d}^{(2)}) · z_{Np^{∞}}^{univ}.

Proof. Firstly we consider a composition C′_{N,d′} determined by the relation that

(1 − d′g_{d′}^{(2)}) · C_{N} = (4.6) ∘ C′_{N,d′}, where d′ is, as in 4.1.7, an integer which is prime to p. Namely, C′_{N,d′} is as follows:

\begin{aligned}C′_{N,d′} : lim_{\overleftarrow{n}} K_{2}(Y (Np^{n}, p^{n})) & ⟶ \frac{S}{ε − 1}[[G_{∞}]] \\ & ⟶^{s} O_{H}[[G_{∞}]][[G_{∞}]], \end{aligned}

where the first arrow is the composition of the first four maps in (4.1), and the map s is defined by the composition

s : \frac{S}{ε − 1}[[G_{∞}]] ⟶^{1−d′ν_{d′}} S[[G_{∞}]] ⟶ O_{H}[[G_{∞}]][[G_{∞}]].

Here the second map is the O_{H}-homomorphism associated to

ε^{a}g_{u} ↦ \begin{cases}u^{−1}g_{a,1}g_{u,2} & if (a, p) = 1 \\ 0 & if (a, p) ≠ 1, \end{cases}

for a ∈ ℤ and u ∈ ℤ_{p}^{×}. For the proof of Proposition 4.4, by the definition of z_{Np^{∞}}^{univ}, our task is to show that under the map C′_{N,d′}, the norm compatible system of Beilinson elements (_{c,d}z_{Np^{n},p^{n}})_{n} is sent to (c^{2} − cg_{c^{−1},2})(d^{2} − dg_{d,1})(1 − d′g_{d′,1}) · F_{N,1} · F_{N,2}, where F_{N,1}, F_{N,2} are as in the definition of z_{Np^{∞}}^{univ}. We prove the above assertion by showing the result of the computation of the image of (_{c,d}z_{Np^{n},p^{n}})_{n} under each step in the composition defining C′_{N,d′}. Step 1. Firstly we compute the image of _{c,d}z_{Np^{n},p^{n}} = {_{c}g_{Np^{n},0},_{d}g_{0,p^{n}}} under K_{2}(Y (Np^{n}, p^{n})) ⟶ K_{2}(H_{n}(q^{1/N})) which is given by the pull-back by the latter map of (4.3). Directly from the definition, we have that the image is {A_{1}, B_{1}} ∈ K_{2}(H_{n}(q^{1/N})). Here the element A_{1} ∈ H_{n}(q^{1/N})^{×} is obtained from _{c}θ_{E}, where E is the universal elliptic curve over Y (Np^{n}, Np^{n}), by putting t = q^{u′/Np^{n}} with u′ ∈ (ℤ/Np^{n}ℤ)^{×} such that u′ ≡ u(p^{n}) and u′ ≡ 1(N). The element B_{1} ∈ H_{n}(q^{1/N})^{×} is obtained from _{d}θ_{E}, where E is the universal elliptic curve over Y (p^{n}, p^{n}) by putting t = q^{w/p^{n}}ζ_{p^{n}}. From this, it is easy to have the following result:

E_{N}((_{c,d}z_{Np^{n},p^{n}})_{n})

= ({ ∏_{u_{n}∈(ℤ/p^{n}ℤ)^{×}}q^{(N/12)(c^{2}−1)}(−q^{u′_{n}/p^{n}})^{(1/2)(c−c^{2})}γ_{q^{N}} (q^{u′_{n}/p^{n}})^{c^{2}}γ_{q^{N}} (q^{cu′_{n}/p^{n}})^{−1},

∏_{w_{n}∈ℤ/p^{n}ℤ} q^{(N/12)(d^{2}−1)}(−q^{Nw′_{n}/p^{n}}ζ_{p^{n}})^{(1/2)(d−d^{2})}

γ_{q^{N}} (q^{Nw′_{n}/p^{n}}ζ_{p^{n}})^{d^{2}}γ_{q^{N}} (q^{dNw′_{n}/p^{n}}ζ_{p^{n}}^{d})^{−1}}g_{u_{n}})_{n}

∈ lim_{\overleftarrow{n}} K̂_{2}(H_{n})[G_{n}] = (lim_{\overleftarrow{n}} K̂_{2}(H_{n}))[[G_{∞}]],

where for u_{n} ∈ (ℤ/p^{n}ℤ)^{×}, u′_{n} is an integer such that u′_{n} ≡ 1(N) and u′_{n} ≡ u_{n}(p^{n}), and for w_{n} ∈ ℤ/p^{n}ℤ, w′_{n} is an integer such that w′_{n} ≡ w_{n}(p^{n}). Step 2. From the definition, we obtain that under the notation in the composition (4.1)

Col ∘ E_{N}((_{c,d}z_{Np^{n},p^{n}})_{n}) ∈ K̂_{2}(S′)^{N=1}[[G_{∞}]]

coincides with the image of (A_{2}, B_{2}) with A_{2} ∈ O_{H}^{×}[[G_{∞}]] and B_{2} ∈ S^{′×} given below under the natural map

O_{H}^{×}[[G_{∞}]] × S^{′×} ⟶ K̂_{2}(S′)[[G_{∞}]] ; (x_{u}g_{u}, y) ↦ {x_{u}, y}g_{u}.

The elements A_{2}, B_{2} are as follows:

A_{2} = lim_{\overleftarrow{n}}( ∏_{u_{n}∈(ℤ/p^{n}ℤ)^{×}}(−q^{(−c^{2}ζ_{≡u′_{n}(Np^{n})}(−1)+ζ_{≡cu′_{n}(Np^{n})}(−1))})g_{u_{n}})

· ∏_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}}(1 − q^{i})^{c^{2}}g_{i} · ∏_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}}(1 − q^{i})^{c^{2}}g_{−i}

· ∏_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}}(1 − q^{i})^{−1}g_{c^{−1}i} · ∏_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}}(1 − q^{i})^{−1}g_{−c^{−1}i},

where u′_{n} is as before.

B_{2} = lim_{\overleftarrow{n}}( ∏_{w_{n}∈ℤ/p^{n}ℤ}(−q^{N(−d^{2}ζ_{≡wn(p^{n})}(−1)+ζ_{≡dwn(p^{n})(−1)})})

· ∏_{i≥1}(1 − q^{Ni}ε)^{d^{2}} · ∏_{i≥1}(1 − q^{Ni}ε^{−1})^{d^{2}}

· ∏_{i≥1}(1 − q^{Ni}ε^{d})^{−1} · ∏_{i≥1}(1 − q^{Ni}ε^{−d})^{−1}

· (1 − ε)^{d^{2}}(1 − ε^{d})^{−1}ε^{(1/2)(d−d^{2})}.

Step 3. By the definition of the map d log, we find easily that

d log ∘Col ∘ E_{N}((_{c,d}z_{Np^{n},p^{n}})_{n}) ∈ Ω_{S}^{2}(log)[[G_{∞}]]

coincides with the image of (A_{3}, B_{3}) with A_{3} ∈ O_{H}[[G_{∞}]] and B_{3} ∈ \frac{S}{ε−1} given below under the map

O_{H}[[G_{∞}]] × \frac{S}{ε − 1} ⟶ Ω_{S}^{2}(log)[[G_{∞}]] ;

(xg_{u}, y) ↦ xy · d log(q) ∧ d log(ε) · g_{u}

for x ∈ O_{H}, u ∈ ℤ_{p}^{×}, y ∈ \frac{S}{ε−1}. The elements A_{3}, B_{3} are as follows:

A_{3} = ( ∑_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} iq^{ij})(c^{2} · g_{i} − g_{c^{−1}i})

+ ( ∑_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} iq^{ij})(c^{2} · g_{−i} − g_{−c^{−1}i})

+ lim_{\overleftarrow{n}}( ∑_{\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(Np^{n})}(−1)(c^{2} · g_{a} − g_{c^{−1}a})).

B_{3} = d^{2}(∑_{i≥1} ∑_{j≥1} q^{Nij}ε^{j} − ∑_{i≥1} ∑_{j≥1} q^{Nij}ε^{−j})

− d(∑_{i≥1} ∑_{j≥1} q^{Nij}ε^{dj} − ∑_{i≥1} ∑_{j≥1} q^{Nij}ε^{−dj})

+ d^{2} \frac{ε}{1 − ε} − d\frac{ε^{d}}{1 − ε^{d}} + \frac{1}{2}(d − d^{2}).

Step 4. By the definition of the map s, we see that

s ∘ d log ∘Col ∘ E_{N}((_{c,d}z_{Np^{n},p^{n}})_{n}) ∈ O_{H}[[G_{∞}]][[G_{∞}]] coincides with the image of ((1 − d′g_{d′})B_{4}, A_{4}) with A_{4} ∈ O_{H}[[G_{∞}]] and B_{4} ∈ Q(O_{H}[[G_{∞}]]) ((1 − d′g_{d′})B_{4} ∈ O_{H}[[G_{∞}]]) given below under the natural O_{H}- homomorphism

O_{H}[[G_{∞}]] × O_{H}[[G_{∞}]] ⟶ O_{H}[[G_{∞}]][[G_{∞}]] ;
(xg_{a}, yg_{b}) ↦ xyg_{a,1}g_{b,2} (a, b ∈ ℤ_{p}^{×}).

The elements A_{4}, B_{4} are as follows:

A_{4} = ( ∑_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} q^{ij})(c^{2} · g_{i} − c · g_{c^{−1}i})

− ( ∑_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} q^{ij})(c^{2} · g_{−i} − c · g_{−c^{−1}i})

+ lim_{\overleftarrow{n}}( ∑_{\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(Np^{n})}(0)(c^{2} · g_{a} − c · g_{c^{−1}a})).

B_{4} = d^{2}(∑_{i≥1} ∑_{\begin{matrix}j≥1 \\ (j,p)=1 \end{matrix}} q^{Nij}g_{j} − ∑_{i≥1} ∑_{\begin{matrix}j≥1 \\ (j,p)=1 \end{matrix}} q^{Nij}g_{−j})

− d(∑_{i≥1} ∑_{\begin{matrix}j≥1 \\ (j,p)=1 \end{matrix}} q^{Nij}g_{dj} − ∑_{i≥1} ∑_{\begin{matrix}j≥1 \\ (j,p)=1 \end{matrix}} q^{Nij}g_{−dj})

+ d^{2} lim_{\overleftarrow{n}}( ∑_{a_{n}∈(ℤ/p^{n}ℤ)^{×}} ζ_{≡a_{n}(p^{n})}(0) · g_{a_{n}})

− d lim_{\overleftarrow{n}}( ∑_{a_{n}∈(ℤ/p^{n}ℤ)^{×}} ζ_{≡a_{n}(p^{n})}(0) · g_{da_{n}}).

By comparing A_{4} with F_{1} and B_{4} with F_{2}, we obtain the assertion of Proposition 4.4. □

4.5. We prove that z_{Np^{∞}}^{univ} which has been defined as an element of O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][[1/g], in fact, belongs to the subspace \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g]. We further show the relation between z_{Np^{∞}}^{univ} and special values of zeta functions of cusp forms. Preceding this, in 4.5.1 – 4.5.4, we review the zeta modular forms in [Ka2], which were defined basing on the works of Shimura [Sh], and whose period integrals yield special values of the zeta functions of cusp forms. In 4.5.5, we show that z_{Np^{∞}}^{univ} is contained in the subspace \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g], and then in 4.5.6, we describe the relation between z_{Np^{∞}}^{univ} and the zeta modular forms reviewed in 4.5.1 – 4.5.4.

4.5.1. We review some Eisenstein series appearing, for example, in [Ka2], §3. For M_{1}, M_{2} ≥ 1 such that M_{1} + M_{2} ≥ 5, as before, let M_{j}(X(M_{1}, M_{2})) be the space of modular forms on X(M_{1}, M_{2}) of weight j ≥ 1.

Let M ≥ 3, and x, y ∈ ((1/M)ℤ)/ℤ. We review the q-expansions of Eisenstein series

F_{x,y}^{(j)} ((j, x, y) ≠ (2, 0, 0)), E_{x,y}^{(j)} (j ≠ 2), Ẽ_{x,y}^{(2)} ∈ M_{j}(X(M, M)),

following Kato [Ka2], §4. (In the case x = 0, these modular forms are, in fact, elements of M_{j}(X(1, M)) and in the case y = 0, they are, in fact, elements of M_{j}(X(M, 1)).) For γ ∈ ℚ/ℤ, we define

ζ(γ, s) = ∑_{\begin{matrix}m∈ℚ,m>0 \\ m mod ℤ ≡ γ \end{matrix}} m^{−s}, ζ^{∗}(γ, s) = ∑_{m=1}^{∞} exp(2πiγm) · m^{−s}.

For (i) F = F_{x,y}^{(j)} ((j, x, y) ≠ (2, 0, 0)), (ii) F = E_{x,y}^{(j)} (j ≠ 2), or (iii) F = Ẽ_{x,y}^{(2)}, we write F = ∑\begin{matrix}m∈ℚ \\ m≥0 \end{matrix} a_{m}q^{m} (q = exp(2πiτ)).

In the case of (i), we assume that (j, x, y) ≠ (2, 0, 0). Then a_{m} for m > 0 can be obtained from the equation

∑_{\begin{matrix}m∈ℚ \\ m>0 \end{matrix}} a_{m}m^{−s} = ζ(x, s − j + 1)ζ^{∗}(y, s) + (−1)^{j}ζ(−x, s − j + 1)ζ^{∗}(−y, s).

In the case j ≠ 1, a_{0} = ζ(x, 1 − j). In the case j = 1, a_{0} = ζ(x, 0) if x ≠ 0, and a_{0} = (1/2)(ζ^{∗}(y, 0) − ζ^{∗}(−y, 0)) if x = 0.

In the case of (ii), we assume that j ≠ 2. Then a_{m} for m > 0 can be obtained from the equation

∑_{\begin{matrix}m∈ℚ \\ m>0 \end{matrix}} a_{m}m^{−s} = ζ(x, s)ζ^{∗}(y, s − j + 1) + (−1)^{j}ζ(−x, s)ζ^{∗}(−y, s − j + 1).

In the case j ≠ 1, a_{0} = 0 if x ≠ 0, and a_{0} = ζ^{∗}(y, 1 − j) if x = 0. In the case j = 1, a_{0} = ζ(x, 0) if x ≠ 0, and a_{0} = (1/2)(ζ^{∗}(y, 0) − ζ^{∗}(−y, 0)) if x = 0.

In the case of (iii), the a_{m} for m > 0 can be obtained from the equation

∑_{\begin{matrix}m∈ℚ \\ m>0 \end{matrix}} a_{m}m^{−s} = ζ(x, s)ζ^{∗}(y, s − 1) + ζ(−x, s)ζ^{∗}(−y, s − 1) − 2ζ(s)ζ(s − 1).

If x ≠ 0, a_{0} = 0, and if x = 0, a_{0} = ζ^{∗}(y, −1) − ζ(−1).

4.5.2. We review the zeta modular forms in [Ka2], §§4 and 5, which were defined basing on the works of Shimura [Sh]. These zeta modular forms yield special values of zeta functions of modular forms by period integrals (concerning this, refer to [Ka2], §5). Let k, r, m, n be integers such that k ≥ 2, 1 ≤ r ≤ k − 1, 1 ≤ m ≤ n, and N(p^{n} + p^{m}) ≥ 5. Further for an integer M, let prime(M) denote the set of all of the prime divisors of M. In the case r ≠ 2, the zeta modular forms are as follows:

\begin{matrix}z_{Np^{n},Np^{n}}^{(k,r)}(k, r, k − 1) \\ = (r − 1)!^{−1} · (Np^{n})^{k−r−2}(Np^{n})^{−r} · F_{1/Np^{n},0}^{(k−r)} · E_{0,1/Np^{n}}^{(r)} ∈ M_{k}(X(Np^{n}, Np^{n})), \end{matrix}

z_{1,Np^{m},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np))

= Tr_{Np^{m}}(z_{Np^{n},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np))) ∈ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})).

We remark that M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})) may be regarded as the fixed part

of M_{k}(X(Np^{n}, Np^{n})) by the group \{(\begin{matrix}u & v \\ w & x \end{matrix}) ∈ GL_{2}(ℤ/Np^{n}ℤ) ; u ≡

1(Np^{m}), w ≡ 0(Np^{m}), ux − vw ≡ 1(Np^{n})}. In the above

Tr_{Np^{m}} : M_{k}(X(Np^{n}, Np^{n})) ⟶ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}}))

denotes the trace map. Let c and d be integers such that (c, Np) = 1 and (d, p) = 1. In the case r = 2, the zeta modular forms are as follows:

c,dz_{Np^{n},Np^{n}}^{(k,2)}(k, 2, k − 1)
=(Np^{n})^{k−4}(Np^{n})^{−2}c^{2}d^{2}
· (F_{1/Np^{n},0}^{(k−2)} − c^{2−k} · F_{c/Np^{n},0}^{(k−2)}) · (Ẽ_{0,1/Np^{n}}^{(2)} − Ẽ_{0,d/Np^{n}}^{(2)})
∈ M_{k}(X(Np^{n}, Np^{n}))),

c,dz_{1,Np^{m},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np))
= Tr_{Np^{m}}(_{c,d}z_{Np^{n},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np)))
∈ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})).

The above zeta modular forms provide the value at s = r of the zeta functions of modular forms of weight k by period integrals. In our method, modular forms whose q-expansions belong to ℤ_{(p)}[[q]] or ℚ[[q]] are important. So we analyze zeta modular forms from this viewpoint. Firstly for j ∈ ℤ, j ≥ 1, and a ∈ ((1/M)ℤ)/ℤ satisfying (j, a) ≠ (2, 0), directly from the definition we have

∑_{x∈((1/M)ℤ)/ℤ} F_{a,x}^{(j)} ∈ M_{j}(X_{1}(M); ℚ).

Lemma 4.5.3 . We assume that r ≠ 2.

408 Takako Fukaya

(1) In M_{k}(X_{1}(Np^{n}); ℚ(ζ_{Np^{n}})), we have

z_{1,Np^{n},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np))

= (r − 1)!^{−1} · (Np^{n})^{k−r−2}(Np^{n})^{−r} · ∑_{x∈((1/Np^{n})ℤ)/ℤ} F_{1/Np^{n},x}^{(k−r)} · E_{0,1/Np^{n}}^{(r)}

= (r − 1)!^{−1} · (Np^{n})^{k−r−2}(Np^{n})^{−2}

· ( ∑_{a∈ℤ/Np^{n}ℤ}( ∑_{x,y∈((1/Np^{n})ℤ)/ℤ} F_{1/Np^{n},x}^{(k−r)} · F_{a/Np^{n},y}^{(r)}) · ζ_{Np^{n}}^{a}.

(2) Let Tr_{Np^{n},Np^{m}} : M_{k}(X_{1}(Np^{n}); ℚ(ζ_{Np^{n}})) ⟶ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}}))

be the trace map. In M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})), we have

z_{1,Np^{m},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np)) = Tr_{Np^{n},Np^{m}}(z_{1,Np^{n},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np))) = (r − 1)!^{−1} · (Np^{m})^{k−r−2}(Np^{n})^{−2}

· ( ∑_{a∈ℤ/Np^{n}ℤ} T (p)^{n−m}

( ∑_{x∈((1/Np^{m})ℤ)/ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ}(F_{1/Np^{m},x}^{(k−r)} · F_{a/Np^{n},y}^{(r)})) · ζ_{Np^{n}}^{a}).

Here T (p) = U(p) is the Hecke operator on the space M_{k}(X_{1}(Np^{n}); ℚ).

(3) Let tr_{Np^{m},p^{n}} : M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})) ⟶ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{p^{n}}))

be the trace map. In M_{k}(X_{1}(Np^{m}); ℚ(ζ_{p^{n}})), we have

tr_{Np^{n},p^{n}}(z_{1,Np^{m},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np))) = (r − 1)!^{−1} · (Np^{m})^{k−r−2}(Np^{n})^{−2} · N

· ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − l^{−r}T (l)ν_{l^{−1}})

( ∑_{a∈ℤ/p^{n}ℤ} T (p)^{n−m}

(  ∑_{x∈((1/Np^{m})ℤ)/ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ}(F_{1/Np^{m},x}^{(k−r)} · F_{Na/Np^{n},y}^{(r)})) · ζ_{p^{n}}^{a}),

(4.8)

where for x ∈ (ℤ/p^{n}ℤ)^{×}, ν_{x} is the corresponding element of Gal(ℚ(ζ_{p^{n}})/ℚ) via the cyclotomic character.

Documenta Mathematica · Extra Volume Kato (2003) 387–442

Coleman Power Series for K_{2} 409

Proof. (1) The first equality is direct from the definition. The equality (r ≠ 2)

E_{0,1/Np^{n}}^{(r)} = (Np^{n})^{r−2} · ∑_{a∈ℤ/Np^{n}ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ} F_{a/Np^{n},y}^{(r)} · ζ_{Np^{n}}^{a}

which can be obtained by computation, shows the second equality.

(2) (3) The results are immediate from the definitions.

□

The following Lemma 4.5.4 describes the case r = 2. Lemma 4.5.4 . We use the same notation as in Lemma 4.5.3.

(1) In M_{k}(X_{1}(Np^{n}); ℚ(ζ_{Np^{n}})), we have

c,dz_{1,Np^{n},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np)) = (Np^{n})^{k−4}(Np^{n})^{−2}c^{2}d^{2}

· ∑_{\begin{matrix}a∈ℤ/Np^{n}ℤ \\ a≠0 \end{matrix}} ∑_{x,y∈((1/Np^{n})ℤ)/ℤ}(F_{1/Np^{n},x}^{(k−2)} − c^{2−k} · F_{c/Np^{n},x}^{(k−2)})

· (F_{a/Np^{n},y}^{(2)} · ζ_{Np^{n}}^{a} − F_{a/Np^{n},y}^{(2)} · ζ_{Np^{n}}^{da}).

(2) In M_{k}(X_{1}(Np^{m}); ℚ(ζ_{Np^{n}})), we have c,dz_{1,Np^{m},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np))

= (Np^{m})^{k−4}(Np^{n})^{−2}c^{2}d^{2}

· T (p)^{n−m}( ∑_{\begin{matrix}a∈ℤ/Np^{n}ℤ \\ a≠0 \end{matrix}} ∑_{x∈((1/Np^{m})ℤ)/ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ}

((F_{1/Np^{m},x}^{(k−2)} − c^{2−k} · F_{c/Np^{m},x}^{(k−2)}) · (F_{a/Np^{n},y}^{(2)} · ζ_{Np^{n}}^{a} − F_{a/Np^{n},y}^{(2)} · ζ_{Np^{n}}^{da}))).

(3) In M_{k}(X_{1}(Np^{m}); ℚ(ζ_{p^{n}})), we have

tr_{Np^{n},p^{n}}(_{c,d}z_{1,Np^{m},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np)))

= (Np^{m})^{k−4}(Np^{n})^{−2}c^{2}d^{2} · N

· ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − l^{−r}T (l)ν_{l^{−1}})

(T (p)^{n−m}( ∑_{\begin{matrix}a∈ℤ/p^{n}ℤ \\ a≠0 \end{matrix}} ∑_{x∈((1/Np^{m})ℤ)/ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ}

((F_{1/Np^{m},x}^{(k−2)} − c^{2−k} · F_{c/Np^{m},x}^{(k−2)}) · (F_{Na/Np^{n},y}^{(2)} · ζ_{p^{n}}^{a} − F_{Na/Np^{n},y}^{(2)} · ζ_{p^{n}}^{da})))).

(4.9)

410 Takako Fukaya Proof. (1) The equality follows from the equality

Ẽ_{0,1/Np^{n}}^{(2)} −Ẽ_{0,d/Np^{n}}^{(2)} = ∑_{\begin{matrix}a∈ℤ/Np^{n}ℤ \\ a≠0 \end{matrix}} ∑_{y∈((1/Np^{n})ℤ)/ℤ}(F_{a/Np^{n},y}^{(2)} ·ζ_{Np^{n}}^{a} −F_{a/Np^{n},y}^{(2)} ·ζ_{Np^{n}}^{da})

which can be obtained by computation.

(2) (3) The results are immediate from the definitions.

□

4.5.5. We prove that z_{Np^{∞}}^{univ} ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g]. For any j ∈ ℤ, we define an isomorphism of rings

χ^{j} : ℤ_{p}[[G_{∞}]] ⟶_{≅} ℤ_{p}[[G_{∞}]] ; g_{a} ↦ a^{j}g_{a} (a ∈ ℤ_{p}^{×}),

and for a_{1}, a_{2} ∈ ℤ, we define an isomorphism of rings over O_{H}

O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ⟶^{(χ^{a_{1}} ,χ^{a_{2}} )} O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ;

x · g_{b_{1}}^{(1)} · g_{b_{2}}^{(2)} ↦ xb_{1}^{a_{1}}b_{2}^{a_{2}} · g_{b_{1}}^{(1)} · g_{b_{2}}^{(2)}

for x ∈ O_{H}, b_{1}, b_{2} ∈ ℤ_{p}^{×}. Let c and d′ be integers which are prime to p. We put

c,d′z_{Np^{∞}}^{univ} = (1 − c^{−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d′g_{d′}^{(2)})z_{Np^{∞}}^{univ} ∈ O_{H}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]. Let a_{1} and a_{2} be integers such that 0 ≤ a_{2} ≤ a_{1}. We regard ((1 − c^{a_{2}−a_{1}−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d^{′a_{2}+1}g_{d′}^{(2)}))^{−1} · _{c,d′}z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}) as an element of H[[G_{∞}^{(1)} × G_{∞}^{(2)}]] and write z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}) for it. Then directly from the definitions, z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}) coincides with the image of the product

(( ∑_{\begin{matrix}i≥1 \\ (i,p)=1 \end{matrix}} ∑_{j≥1} i^{a_{2}}q^{Nij})(g_{i,1} − (−1)^{a_{2}}g_{−i,1}) + lim_{\overleftarrow{n}}( ∑_{a∈(ℤ/p^{n}ℤ)^{×}} ζ_{≡a(p^{n})}(−a_{2}) · g_{a,1}))

· (( ∑_{\begin{matrix}i≥1,i≡1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} i^{a_{1}−a_{2}}q^{ij} · g_{i,2} − (−1)^{a_{1}−a_{2}} ∑_{\begin{matrix}i≥1,i≡−1(N) \\ (i,p)=1 \end{matrix}} ∑_{j≥1} i^{a_{1}−a_{2}}q^{ij} · g_{−i,2})

+ lim_{\overleftarrow{n}}( ∑_{\begin{matrix}a∈(ℤ/Np^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(Np^{n})}(−a_{1} + a_{2}) · g_{a,2}))

under the map (4.6). Hence concerning the image z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} of z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}) under the projection H[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ⟶ H[G_{n}^{(1)} × G_{n}^{(2)}], we find that

Documenta Mathematica · Extra Volume Kato (2003) 387–442

\begin{matrix}z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} = (Np^{n})^{a_{1}−a_{2}−1}N^{−1}(p^{n})^{a_{2}−1} \\ ∑_{\begin{matrix}b_{1}∈(ℤ/Np^{n}ℤ)^{×} \\ b_{1}≡1(N) \end{matrix}} ∑_{b_{2}∈(ℤ/p^{n}ℤ)^{×}}( ∑_{x,y((1/Np^{n})ℤ)/ℤ} F_{b_{1}/Np^{n},x}^{(a_{1}−a_{2}+1)} · F_{Nb_{2}/Np^{n},y}^{(a_{2}+1)}) · g_{b_{1}}^{(1)} · g_{b_{1}^{−1}b_{2}}^{(2)} \end{matrix} (4.10)

with F_{b_{1}/Np^{n},x}^{(a_{1}−a_{2}+1)} and F_{Nb_{2}/Np^{n},y}^{(a_{2}+1)} in 4.5.1. From this, we obtain that

z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} ∈ M_{a_{1}+2}(X_{1}(Np^{n}); ℚ)[G_{n}^{(1)} × G_{n}^{(2)}](⊂ H[G_{n}^{(1)} × G_{n}^{(2)}])

and hence

c,d′z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)}

∈ M_{a_{1}+2}(X_{1}(Np^{n}); ℤ_{(p)})[G_{n}^{(1)} × G_{n}^{(2)}](⊂ O_{H}[G_{n}^{(1)} × G_{n}^{(2)}]).

Here M_{a_{1}+2}(X_{1}(Np^{n}); ℤ_{(p)}) = M_{a_{1}+2}(X_{1}(Np^{n}); ℚ) ∩ ℤ_{(p)}[[q]]. The latter fact implies that z_{Np^{∞}}^{univ} ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][1/g].

4.5.6. We see the relation between z_{Np^{∞}}^{univ} and special values of zeta functions of cusp forms. This relation is described by the relation between the universal zeta modular form z_{Np^{∞}}^{univ} and the zeta modular forms reviewed in 4.5.2 – 4.5.4. This relation will play an important role in sections 6 and 7. Let

·ζ_{p^{n}} : ℚ[G_{n}^{(2)}] ⟶ ℚ(ζ_{p^{n}})  (4.11)

be the ℚ-linear map given by the action of G_{n}^{(2)} on ζ_{p^{n}} such that g_{a}^{(2)} ↦ ζ_{p^{n}}^{a} (a ∈ (ℤ/p^{n}ℤ)^{×}). We consider the image of z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} under the map

·ζ_{p^{n}} : M_{a_{1}+2}(X_{1}(Np^{n}); ℚ)[G_{n}^{(1)} × G_{n}^{(2)}] ⟶ M_{a_{1}+2}(X_{1}(Np^{n}); ℚ(ζ_{p^{n}}))[G_{n}^{(1)}] (4.12)

induced by the map (4.11). By the caluculation until now, we see that the above image is

\begin{matrix}(Np^{n})^{a_{1}−a_{2}−1}N^{−1}(p^{n})^{a_{2}−1} \\ ∑_{\begin{matrix}b_{1}∈(ℤ/Np^{n}ℤ)^{×} \\ b_{1}≡1(N) \end{matrix}} & ∑_{b_{2}∈(ℤ/p^{n}ℤ)^{×}} \\ (  ∑_{x,y∈((1/Np^{n})ℤ)/ℤ} F_{b_{1}/Np^{n},x}^{(a_{1}−a_{2}+1)} · F_{Nb_{2}/Np^{n},y}^{(a_{2}+1)}) · ζ_{p^{n}}^{b_{1}^{−1}b_{2}} · g_{b_{1}}^{(1)}. \end{matrix}  (4.13)

We assume a_{2} ≠ 1. Putting a_{1} = k − 2, a_{2} = r − 1 in (4.13), m = n in (4.8) in Lemma 4.5.3 (3), and comparing (4.13) with (4.8), we see that the element (4.13) is closely related to the element (4.8) in Lemma 4.5.3 (3). Roughly speaking, for an eigen cusp form f, “f-component” of ∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x)ν_{x} · (4.8),

with a character ψ : (ℤ/p^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} (n ≥ 0), yields L_{(Np)}(f, ψ, r) by period integrals. Here L_{(Np)}(f, ψ, s) denotes the function obtained from L(f, ψ, s) = ∑_{i=1}^{∞} a_{i}(f)ψ(i)i^{−s}, where a_{i}(f) is given by T (i)f = a_{i}(f)f, by removing prime(Np) factors of L(f, ψ, s). We will see in section 6 that ∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x)ν_{x} · (4.13) yields L_{(p)}(f, ψ, r). For the details, see section 6. In the case a_{2} = 1, the above statements must be modified as follows. The image of (1−c^{−a_{1}}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1−g_{d}^{(2)})z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ)|_{(n,n)} under the map (4.12) is closely related to the element in Lemma 4.5.4 (3).

5. Review of p-adic zeta functions of modular forms

In this section, we review the result of Amice-Vélu [AV] and Vishik [Vi] (Theorem 5.5) which concerns the existence and the characterizing properties of p-adic zeta functions of modular forms.

As referred above, in the rest of this paper, N denotes a positive integer which is prime to p.

5.1. Let

f = ∑_{n≥1} a_{n}(f)q^{n} ∈ M_{k}(X(1, Np^{t})) ⊗ ℂ

be a normalized eigen cusp form of weight k ≥ 2 of level Np^{t} for some t ≥ 0. We assume that the conductor of f is divisible by N. We further assume that t is the smallest integer ≥ 0 such that f ∈ M_{k}(X(1, Np^{t})) ⊗ ℂ. Set K = ℚ(a_{n}(f); n ≥ 1). We take a prime λ of K which is above p, and let K_{λ} be the completion of K by λ. Suppose that there exists an element α ∈ \overline{K_{λ}}^{×} satisfying v_{p}(α) < k − 1 for the additive valuation v_{p} of \overline{K_{λ}} normalized by v_{p}(p) = 1, and

1 − αp^{−s} | (p-factor of L(f, s))^{−1} in \overline{ℚ_{p}}[p^{−s}],

where L(f, s) = ∑_{n≥1} a_{n}(f)n^{−s} is the complex zeta function of f. Then the padic zeta function of f may be defined for each α satisfying the above conditions. We fix such α and suppress α in the notation of p-adic zeta functions. We will review the characterizing properties of a p-adic zeta function in 5.5.

5.2. We give a review of the space H_{K_{λ},k−1} to which the p-adic zeta function of f belongs. We first set up the notation. For the natural decomposition ℤ_{p}^{×} = 𝔽_{p}^{×} × (1 + pℤ_{p}) in the case p ≠ 2 (resp. ℤ_{2}^{×} = {±1} × (1 + 4ℤ_{2})), let u be a topological generator of the second component 1+pℤ_{p} (resp. 1+4ℤ_{2}). We denote 𝔽_{p}^{×} (resp. {±}) by Δ. As before G_{∞}^{(2)} is the Galois group Gal(ℚ_{p}(ζ_{p^{∞}})/ℚ_{p}) which is endowed with an isomorphism to ℤ_{p}^{×} via the cyclotomic character. Now for a finite extension L of ℚ_{p} and for a positive integer d, we define

H_{L,d} = \{∑_{\begin{matrix}n≥0 \\ a∈Δ \end{matrix}} c_{n,a} · g_{a}^{(2)} · (g_{u}^{(2)} − 1)^{n} ∈ L[Δ][[g_{u}^{(2)} − 1]] ;

lim_{n⟶∞} |c_{n,a}|_{p}n^{−d} = 0 for all a ∈ Δ}.

Here | |_{p} is the multiplicative valuation of L normalized by |p|_{p} = 1/p. The space H_{L,d} is independent of the choice of u in the evident sense. We have

O_{L}[[G_{∞}^{(2)}]] ⊗_{O_{L}} L ⊂ H_{L,1}, and H_{L,i} ⊂ H_{L,j} for 1 ≤ i ≤ j.

Here the first inclusion is given by the natural map. We put

H_{L,∞} = ⋃_{d≥1} H_{L,d},

then H_{L,∞} is a ring. For any positive integer d and for any subset U of ℤ, we define a map

i_{U} : H_{L,d} ⟶ ∏_{j∈U} L[[G_{∞}^{(2)}]] = ∏_{j∈U} lim_{\overleftarrow{n}} L[G_{n}^{(2)}]

∑_{\begin{matrix}n≥0 \\ a∈Δ \end{matrix}} c_{n,a} · g_{a}^{(2)} · (g_{u}^{(2)} − 1)^{n} ↦ (∑_{\begin{matrix}n≥0 \\ a∈Δ \end{matrix}} c_{n,a} · a^{j}g_{a}^{(2)} · (u^{j}g_{u}^{(2)} − 1)^{n})_{j}.

It is known that for any d ≥ 1, the map i_{ℤ} is injective. Moreover for any different d integers r_{1}, …, r_{d}, the map i_{{r_{1},...,r_{d}}} is already injective:

i_{{r_{1},...,r_{d}}} : H_{L,d} ↪ ∏_{j∈{r_{1},...,r_{d}}} L[[G_{∞}^{(2)}]](⊂ ∏_{j∈ℤ} L[[G_{∞}^{(2)}]]).

Concerning the above injection, we have a proposition (cf. [AV]).

Proposition 5.3 . Let

ℍ_{L,d} ⊂ ∏_{j∈\{1,...,d\}} lim_{\overleftarrow{n}} L[G_{n}^{(2)}] = ∏_{j∈\{1,...,d\}} L[[G_{∞}^{(2)}]]

be the subspace consisting of elements

μ = (μ_{j})_{j} = ((μ_{j,n})_{n})_{j} ∈ ∏_{j∈\{1,...,d\}} lim_{\overleftarrow{n}} L[G_{n}^{(2)}]

satisfying conditions (i) and (ii) below. Then the map

i_{{1,...,d}} : H_{L,d} ↪ ∏_{j∈{1,...,d}} L[[G_{∞}^{(2)}]]

induces a bijection from H_{L,d} onto ℍ_{L,d}.

(i) For any j = 1, …, d,

lim_{n⟶∞} p^{dn}μ_{j,n} = 0.

(ii) For n ≥ 1, let ϕ_{n} : (ℤ/p^{n}ℤ)^{×} ⟶ ℤ_{p}^{×} be a lifting, namely it is a map such

that the composition (ℤ/p^{n}ℤ)^{×} ⟶^{ϕ_{n}} ℤ_{p}^{×} ⟶^{proj.} (ℤ/p^{n}ℤ)^{×} coincides with the identity map. For j ∈ ℤ, let X_{ϕ_{n}}^{j} : L[G_{n}^{(2)}] ⟶ L[G_{n}^{(2)}] be the L-homomorphism induced by g_{a}^{(2)} ↦ ϕ_{n}(a)^{j}g_{a}^{(2)} for any a ∈ (ℤ/p^{n}ℤ)^{×}. Now for 1 ≤ i ≤ d and for any ϕ_{n} satisfying the above condition,

lim_{n⟶∞} p^{(d−i+1)n} ∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) μ_{j+1,n}(X_{ϕ_{n}}^{−j}) = 0.

Here μ_{j,n}(X_{ϕ_{n}}^{−j}) ∈ L[G_{n}^{(2)}] is the image of μ_{j,n} under X_{ϕ_{n}}^{−j}.

Proof. For the proof, see [AV]. □

In section 6, we will construct the p-adic zeta function of f as an element of ∏_{1≤j≤k−1} K_{λ}(α)[[G_{∞}^{(2)}]], and we will prove that it is contained in ℍ_{K_{λ}(α),k−1} by using Proposition 5.3.

5.4. In this subsection we give a preliminary discussion to introduce Theorem 5.5 concerning the existence and the characterizing properties of p-adic zeta functions. In the rest of this section, we assume that a_{p}(f) ≠ 0.

5.4.1. As in [Ka2], §6, we define S(f) to be

S(f) = (M_{k}(X_{1}(Np^{t}); ℚ) ⊗_{ℚ} K)/(T (n) ⊗ 1 − 1 ⊗ a_{n}(f) ; n ≥ 1),

which is the quotient of M_{k}(X_{1}(Np^{t}); ℚ) ⊗_{ℚ} K by the K-subspace generated by T (n) ⊗ 1 − 1 ⊗ a_{n}(f) for n ≥ 1. This S(f) is a one dimensional K-vector space.

5.4.2. We define V_{K}(f) to be the quotient of H^{1}(Y (1, Np^{t})(ℂ), Sym_{ℤ}^{k−2}(R^{1}λ_{∗} (ℤ)) ⊗_{ℤ} K) by the K-subspace generated by the images of T (n) ⊗ 1 − 1 ⊗ a_{n}(f) for n ≥ 1. Here λ : E ⟶ Y (1, Np^{t}) is the universal elliptic curve. This V_{K}(f) is a two dimensional K-vector space.

5.4.3. We put V_{ℂ}(f) = V_{K}(f) ⊗_{K} ℂ, and let

per_{f} : S(f) ⟶ V_{ℂ}(f)

be the one induced by the period map (cf. for example, [Ka2], §5, 5.4)

per_{1,Np^{t}} : M_{k}(X(1, Np^{t})) ⊗ ℂ ⟶ H^{1}(Y (1, Np^{t})(ℂ), Sym_{ℤ}^{k−2}(R^{1}λ_{∗}(ℤ)) ⊗_{ℤ} ℂ).

5.4.4. For the ℂ-linear map

ι : V_{ℂ}(f) ⟶ V_{ℂ}(f)

induced by the complex conjugation on Y (1, Np^{t})(ℂ) and E(ℂ), and for x ∈ V_{ℂ}(f), we define

x^{+} = \frac{1}{2}(1 + ι)(x), x^{−} = \frac{1}{2}(1 − ι)(x).

Now we take an element γ ∈ V_{K}(f) such that γ^{+} ≠ 0, γ^{−} ≠ 0. For ω ∈ S(f) and for the above γ ∈ V_{K}(f), we define Ω(ω, γ)_{+}, Ω(ω, γ)_{−} ∈ ℂ as

per_{f}(ω) = Ω(ω, γ)_{+} · γ^{+} + Ω(ω, γ)_{−} · γ^{−}.

5.4.5. For x ∈ (ℤ/Np^{t}ℤ)^{×}, let ⟨x⟩ be as in 3.5 in section 3. Let

ϵ_{f} : (ℤ/Np^{t}ℤ)^{×} ⟶ \overline{ℚ}^{×}

be the character defined by ⟨x⟩f = ϵ_{f}(x) · f for x ∈ (ℤ/Np^{t}ℤ)^{×}.

5.4.6. As before, let χ_{cyclo} : G_{∞}^{(2)} ⟶^{≅} ℤ_{p}^{×} be the cyclotomic character. For j ∈ ℤ, we regard

χ_{cyclo}^{j} : G_{∞}^{(2)} ⟶ ℤ_{p}^{×} ; g_{a}^{(2)} ↦ a^{j} (a ∈ ℤ_{p}^{×})

also as a character ℤ_{p}^{×} ⟶ ℤ_{p}^{×} ; a ↦ a^{j}. For μ = ∑\begin{matrix}n≥0 \\ a∈Δ \end{matrix} c_{n,a} · g_{a}^{(2)} · (g_{u} − 1)^{n} ∈ H_{L,d} and a continuous character ψ :

ℤ_{p}^{×} ⟶ \overline{ℚ_{p}}^{×}, we define

μ(ψ) = ∑_{\begin{matrix}n≥0 \\ a∈Δ \end{matrix}} c_{n,a} · ψ(a) · (ψ(u) − 1)^{n}.

Let α be as in 5.1.

Theorem 5.5 ([AV],[Vi]). Let h = min{n ∈ ℤ ; n ≥ 1, v_{p}(α) < n}(≤ k − 1). For γ ∈ V_{K}(f) such that γ^{+} ≠ 0, γ^{−} ≠ 0, and for a non-zero ω ∈ S(f), we have a function

L_{p-adic}(f)_{ω,γ} ∈ H_{K_{λ},h} ⊂ H_{K_{λ},k−1},

characterized by the properties (i) and (ii) below. In particular, if v_{p}(α) = 0, L_{p-adic}(f) belongs to the subspace

L_{p-adic}(f)_{ω,γ} ∈ O_{K_{λ}}[[G_{∞}^{(2)}]] ⊗_{O_{K_{λ}}} K_{λ} ⊂ H_{K_{λ},1}.

(i) Let ψ : (ℤ/p^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} be a character with conductor p^{n} (n ≥ 1). We put ± = (−1)^{k−r−1}ψ(−1)ϵ_{f}(−1). Then for any integer r such that 1 ≤ r ≤ k − 1, we have

L_{p-adic}(f)_{ω,γ}(χ_{cyclo}^{r}ψ^{−1})

= (r − 1)! · p^{nr}α^{−n} · G(ψ, ζ_{p^{n}})^{−1} · (2πi)^{k−r−1} · \frac{1}{Ω(ω, γ)_{±}} · L(f, ψ, r),

where G(ψ, ζ_{p^{n}}) denotes the Gauss sum ∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x)ζ_{p^{n}}^{x} and L(f, ψ, r) is the evaluation at s = r of the function L(f, ψ, s) = ∑_{i=1}^{∞} a_{i}(f)ψ(i)i^{−s}.

(ii) We put ± = (−1)^{k−r−1}ϵ_{f}(−1). For any integer r such that 1 ≤ r ≤ k − 1, we have

L_{p-adic}(f)_{ω,γ}(χ_{cyclo}^{r})

= (r − 1)! · (2πi)^{k−r−1} · \frac{1}{Ω(ω, γ)_{±}}

· (1 − p^{r−1}α^{−1})(1 − ϵ_{f}(p)p^{k−r−1}α^{−1}) · L(f, r).

Remark 5.5.1 . (1) In fact, both hands sides of the above equality belong to \overline{ℚ}.

(2) The function in Theorem 5.5 can be characterized by only property (i).

The function L_{p-adic}(f)_{ω,γ} in Theorem 5.5 is called the p-adic zeta function of f.

5.6. We have a canonical isomorphism

H_{K_{λ},d} ≅ H_{K_{λ},d}/(g_{−1}^{(2)} − 1) × H_{K_{λ},d}/(g_{−1}^{(2)} + 1).

For x ∈ H_{K_{λ},d}, we call its image in H_{K_{λ},d}/(g_{−1}^{(2)} − 1) (resp. H_{K_{λ},d}/(g_{−1}^{(2)} + 1)) under the above isomorphism the +-part (resp. −-part) of x. Moreover we call H_{K_{λ},d}/(g_{−1}^{(2)} − 1) (resp. H_{K_{λ},d}/(g_{−1}^{(2)} + 1)) the +-part (resp. −-part) of H_{K_{λ},d}.

5.7. As in [Ka2], §6, we define δ(f, k − 1, 0(1)) ∈ V_{K}(f) to be the image of δ(k, k − 1, 0(1)) ∈ H^{1}(Y (1, Np^{t})(ℂ), Sym_{ℤ}^{k−2}(R^{1}λ_{∗}(ℤ)) ⊗_{ℤ} ℚ) in [Ka2], §5, 5.4. It is known that δ(f, k −1, 0(1))^{+} = 0, and if L(f, k −1) ≠ 0, δ(f, k −1, 0(1)) = δ(f, k − 1, 0(1))^{−} ≠ 0. In what follows, in the case L(f, k − 1) ≠ 0, we take δ(f, k − 1, 0(1)) ∈ V_{K}(f) as γ ∈ V_{K}(f), we consider (−1)^{k} · ϵ_{f}(−1)-part of the p-adic zeta function of f, and we suppress γ in the notation of p-adic zeta functions.

6. The result on one-variable p-adic zeta function

Let the notation and the setting be as in section 5. Suppose f is an eigen cusp form of weight k ≥ 2 and of level Np^{t} with t ≥ 1 which satisfies the condition in 5.1. In 6.1, for a certain subspace A of \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ⊗_{ℤ_{p}[[G_{∞}^{(1)}×G_{∞}^{(2)}]]}

Q(ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]) to which the universal zeta modular form z_{Np^{∞}}^{univ} belongs, we define a map “to take f-component”

𝔏_{f,{k−2},{0,...,k−2},t} : A ⟶ ∏_{r∈{1,...,k−1}} K_{λ}[[G_{∞}^{(2)}]][G_{t}^{(1)}].

The main theorem (Theorem 6.2) of this section is, roughly speaking, that if L(f, k − 1) ≠ 0,

𝔏_{f,{k−2},{0,...,k−2},t} : z_{Np^{∞}}^{univ} ↦ p-adic zeta function of f

(see Theorem 6.2 for the precise statement).

6.1. We define the subspace A and the map 𝔏_{f,{k−2},{0,...,k−2},t} in a more general forms M[[G_{∞}^{(1)} ×G_{∞}^{(2)}]]_{I_{1},I_{2}} (see 6.1.4) and 𝔏_{f,I_{1},I_{2},i} (see 6.1.6), respectively. (For simplicity we assumed that t ≥ 1 in the above, but in fact, we can treat also the case t = 0, as seen below.) We begin with some preliminaries. Let f, K, and the other setting be as in 5.1 in section 5.

6.1.1. As in 5.1, we fix α under the notation there. We first consider the case that t = 0. We denote the p-factor of L(f, s) by (1−αp^{−s})^{−1}(1−βp^{−s})^{−1} with β ∈ \overline{K_{λ}}^{×}.

Let

f_{α} = f − β · φ_{q}(f) ∈ M_{k}(X(1, Np)) ⊗ ℂ,

where φ_{q}(f) = ∑_{n≥1} a_{n}(f)q^{pn} with f = ∑_{n≥1} a_{n}(f)q^{n}. We have

T (p)f_{α} = α · f_{α},
L(f_{α}, s) = (1 − βp^{−s})L(f, s),

and ϵ_{f_{α}} : (ℤ/Npℤ)^{×} ⟶ \overline{ℚ}^{×} is the one induced by (ℤ/Nℤ)^{×} ⟶^{ϵ_{f}} \overline{ℚ}^{×}. This f_{α} is an eigen cusp form but is not a newform.

6.1.2. For f in 5.1, we put 𝔣 = f in the case that t ≥ 1 with t in 5.1, and 𝔣 = f_{α} in the case t = 0, where f_{α} is as in 6.1.1. Moreover let Np^{m} denote the level of 𝔣. (Namely m = t in the case that 𝔣 = f and m = 1 in the case that 𝔣 = f_{α} as above.) Further put

L = \begin{cases}K_{λ} & t ≥ 1 \\ K_{λ}(α) & t = 0. \end{cases}

6.1.3. Let j, s ∈ ℤ, and let M^{j}(X_{1}(Np^{s}); ℚ) be as in section 3. The space (∪_{j≥2,s≥1}M^{j}(X_{1}(Np^{s}); ℚ) ⊗_{ℚ} L)/(T (n) ⊗ 1 − 1 ⊗ a_{n}(𝔣) ; n ≥ 1) is a one dimensional L-vector space in which the class of f is a base. We define a map pr_{𝔣} by the following composition

\begin{aligned}pr_{𝔣} : & ⋃_{j≥2 s≥1} M^{j}(X_{1}(Np^{s}); ℚ) \\ & ⟶ ( ⋃_{j≥2,s≥1} M^{j}(X_{1}(Np^{s}); ℚ) ⊗_{ℚ} L)/(T (n) ⊗ 1 − 1 ⊗ a_{n}(𝔣) ; n ≥ 1) \\ & ⟶ L, \end{aligned} (6.1)

where the first map is the natural projection and the second map is by sending the class of f to 1.

6.1.4. Let I_{1}, I_{2} ⊂ ℤ be subsets. We denote by M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}} the ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]-submodule of \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] defined in the following way:

M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}}
:= \{x ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ;

x(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} ∈ ( ⋃_{j≥2, s≥1} M^{j}(X_{1}(Np^{s}); ℤ_{p}))[G_{n}^{(1)} × G_{n}^{(2)}]

for any (a_{1}, a_{2}) ∈ I_{1} × I_{2} and n ≥ 1}.

6.1.5. Let c, d′ be integers which are prime to p. By the result in 4.5.5, we find that for any k ≥ 2,

c,d′z_{Np^{∞}}^{univ} =(1 − c^{−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d′g_{d′}^{(2)})z_{Np^{∞}}^{univ}
∈ M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{{k−2},{0,...,k−2}}.

(6.2)

6.1.6. Let i be a positive integer. We define a map “to take 𝔣-component”

𝔏_{f,I_{1},I_{2},i} : M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}} ⟶ ∏_{(a_{1},a_{2})∈I_{1}×I_{2}} L[[G_{∞}^{(2)}]][G_{i}^{(1)}]

as follows:

𝔏_{f,I_{1},I_{2},i} = ∏_{(a_{1},a_{2})∈I_{1}×I_{2}} 𝔏_{f,a_{1},a_{2},i}

for

𝔏_{f,a_{1},a_{2},i} : M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}} ⟶ L[[G_{∞}^{(2)}]][G_{i}^{(1)}].

The map 𝔏_{f,a_{1},a_{2},i} is defined as the composition

𝔏_{f,a_{1},a_{2},i} : M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}} ⟶^{(χ^{a_{1}} ,χ^{a_{2}} )} M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{{0},{0}}

⟶^{proj.} M[[G_{∞}^{(2)}]][G_{i}^{(1)}]_{{0},{0}}

⟶^{pr_{𝔣}} L[[G_{∞}^{(2)}]][G_{i}^{(1)}],

where M[[G_{∞}^{(2)}]][G_{i}^{(1)}]_{{0},{0}} denotes the image of M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{{0},{0}} under the natural projection \overline{M}_{Np^{∞}}[[G_{∞}^{(1)}×G_{∞}^{(2)}]] ⟶ \overline{M}_{Np^{∞}}[[G_{∞}^{(2)}]][G_{i}^{(1)}], and the last map is given by pr_{𝔣} (6.1) for each coefficients of G_{n}^{(2)} and by taking lim_{\overleftarrow{n}}.

For x ∈ M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}} ⊗_{ℤ_{p}[[G_{∞}^{(1)}×G_{∞}^{(2)}]]} Q(ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]), if there is

a non-zerodivisor g ∈ ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] such that g(χ^{a_{1}}, χ^{a_{2}}) is invertible in ℚ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] for all (a_{1}, a_{2}) ∈ I_{1} × I_{2} and gx ∈ M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{I_{1},I_{2}}, then we define

\begin{aligned}𝔏_{f,I_{1},I_{2},i}(x) = 𝔏_{f,I_{1},I_{2},i}(gx) & · ∏_{(a_{1},a_{2})∈I_{1}×I_{2}} g(a_{1}, a_{2})^{−1} \\ & ∈ ∏_{(a_{1},a_{2})∈I_{1}×I_{2}} L[[G_{∞}^{(2)}]][G_{i}^{(1)}]. \end{aligned}

6.1.7. By (6.2), we find that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) ∈ ∏_{a_{2}∈{0,...,k−2}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}] can be defined in the sense at the end of 6.1.6.

In what follows, by putting a_{2} = r − 1, we write ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}]

instead of ∏_{a_{2}∈{0,...,k−2}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}]. Furthermore for L[[G_{∞}^{(2)}]], we define + and − parts in the same way as in 5.6, and we define the ∗-part of ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}] with ∗ = + or ∗ = − by ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]]^{∗·(−1)^{r}}[G_{m}^{(1)}], where L[[G_{∞}^{(2)}]]^{∗·(−1)^{r}} is the ∗ · (−1)^{r}-part

of L[[G_{∞}^{(2)}]].

We state our main theorem in this section. In the situation of Theorem 5.5, we take the class of f as ω ∈ S(𝔣), and suppress the ω in the notation of p-adic zeta functions appearing below. Since we will assume L(𝔣, k −1) ≠ 0, we take δ(𝔣, k −1, 0(1)) ∈ V_{K}(𝔣)\{0} as γ, and suppress γ in the notation of p-adic zeta functions, as referred in 5.7.

Theorem 6.2 . Put ± = (−1)^{k}ϵ_{f}(−1). Let h = min{n ∈ ℤ ; n ≥ 1, v_{p}(α) < n}(≤ k − 1), as in Theorem 5.5. Then we have

𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} ∈ ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}]  (6.3)

is contained in the subspace

i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}].

Here 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} represents the ±-part. Moreover if v_{p}(α) = 0, (6.3) belongs to i_{{1,...,k−1}}(O_{L}[[G_{∞}^{(2)}]] ⊗_{O_{L}} L)[G_{m}^{(1)}]. Concerning the relation with p-adic zeta function, we have the following result. Suppose L(𝔣, k − 1) ≠ 0. In the rest of this theorem, we identify an element of H_{L,h} with its image under i_{{1,...,k−1}} : H_{L,h} ↪ ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]].

(1) In the case 𝔣 = f, we have

𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±}

= α^{m} ∑_{a∈(ℤ/p^{m}ℤ)^{×}} L_{p-adic}(f)^{±}ϵ_{f}(a^{′−1}) · g_{a}^{(1)} ∈ ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}].

Here a′ ∈ (ℤ/Np^{m}ℤ)^{×} is the element such that a′ ≡ 1(N) and a′ ≡ a(p^{m}).

(2) In the case 𝔣 = f_{α}, we have

𝔏_{f,{k−2},{0,...,k−2},1}(z_{Np^{∞}}^{univ})^{±}

= α ∑_{a∈(ℤ/pℤ)^{×}} L_{p-adic}(f_{α})^{±} · g_{a}^{(1)} ∈ ∏_{r∈{1,...,k−1}} L[[G_{∞}^{(2)}]][G_{1}^{(1)}].

(3) In the above, we only considered the (−1)^{k}ϵ_{f}(−1)-parts of L_{p-adic}(𝔣), and we put the assumption that L(𝔣, k − 1) ≠ 0 which always holds in the case k ≥ 3. However we can obtain by the method in 6.7 below, the whole L_{p-adic}(𝔣), including the (−1)^{k−1}ϵ_{f}(−1)-part, without the assumption that L(𝔣, k − 1) ≠ 0.

Remark 6.2.1 . In Theorem 6.2 (2), we present the p-adic zeta function of f_{α} instead of the p-adic zeta function of f. By the characterizing property of p-adic zeta functions in Theorem 5.5, their p-adic zeta functions are a multiple of the other by a non-zero constant.

The rest of this section is devoted to the proof of Theorem 6.2. In 6.3, we prove that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} is contained in i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}]. Then in 6.4 – 6.7, we prove that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) displays the characterizing property (i) of the p-adic zeta function in Theorem 5.5. (Cf. Remark 5.5.1 in section 5.)

6.3. We show that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} belongs to i_{{1,...,k−1}}(H_{L,h}) [G_{m}^{(1)}]. First, we give a preliminary discussion which will be important also for the proof on the characterizing properties of p-adic zeta functions.

6.3.1. We define a homomorphism

φ_{q} : H ⟶ H

by φ_{q}(∑_{i=−∞}^{∞} a_{i}q^{i}) = ∑_{i=−∞}^{∞} a_{i}q^{pi} (a_{i} ∈ ℚ_{p}, the valuation of a_{i} is bounded below, and a_{i} → 0 when i → −∞). Let

Tr_{q} : H ⟶ H

be the trace map associated to φ_{q}. We use the same symbol Tr_{q}

H[G_{a_{1}}^{(1)} × G_{a_{2}}^{(2)}] ⟶ H[G_{a_{1}}^{(1)} × G_{a_{2}}^{(2)}] (a_{1}, a_{2} ≥ 1)

for the map induced by Tr_{q} on the coefficients. For an element x of L[[G_{∞}^{(1)} × G_{∞}^{(2)}]] and for positive integers a_{1}, a_{2}, we denote by x|_{(a_{1},a_{2})} the image of x under the natural projection

L[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ⟶ L[G_{a_{1}}^{(1)} × G_{a_{2}}^{(2)}].

Proposition 6.3.2 . Let a_{1}, a_{2} be integers such that 0 ≤ a_{2} ≤ a_{1}. Then for all positive integers n and m such that n ≥ m, we have

Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(m,n)}) ∈ M_{a_{1}+2}(X_{1}(Np^{m}); ℚ)[G_{m}^{(1)} × G_{n}^{(2)}],

Tr_{q}^{n−m}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(m,n)}) ∈ M_{a_{1}+2}(X_{1}(Np^{m}); ℤ_{(p)})[G_{m}^{(1)} × G_{n}^{(2)}],

where z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}) is as in 4.5.5.

Proof. Clearly ∗|_{(m,n)} is the image of ∗|_{(n,n)} under the projection H[G_{n}^{(1)} × G_{n}^{(2)}] ⟶ H[G_{m}^{(1)} × G_{n}^{(2)}]. By (4.10), we know z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} which is an element of M_{a_{1}+2}(X_{1}(Np^{n}); ℚ)[G_{n}^{(1)} × G_{n}^{(2)}] precisely, and hence we can calculate Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}))|_{(m,n)}. By this calculation, we find that the element Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{a_{1}}, χ^{a_{2}}))|_{(m,n)} belongs to M_{a_{1}+2}(X_{1}(Np^{m}); ℚ)[G_{m}^{(1)} × G_{n}^{(2)}]. The assertion for _{c,d′}z_{Np^{∞}}^{univ} follows from this. □

In 6.3.3 – 6.3.6, we show that the element 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) belongs to i_{{1,...,k−1}}(H_{L,h}⊗_{ℤ_{p}[[G_{∞}^{(2)}]]}ℤ_{p}[[G_{∞}^{(2)}]][1/a])[G_{m}^{(1)}] = i_{{1,...,k−1}}(H_{L,h}[1/a])[G_{m}^{(1)}] (⊂ (i_{{1,...,k−1}}(H_{L,h}) ⊗_{ℤ_{p}[[G_{∞}^{(2)}]]} Q(ℤ_{p}[[G_{∞}^{(2)}]]))[G_{m}^{(1)}]) with a certain non-

zerodivisor a ∈ ℤ_{p}[[G_{∞}^{(2)}]] which satisfies that a(χ^{r}) (r = 1, …, k−1) are invertible in ℚ_{p}[[G_{∞}^{(2)}]]. In 6.3.7 – 6.3.12, we prove that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} is, in fact, contained in i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}].

6.3.3. We prove that

𝔏_{f,{k−2},{0,...,k−2},m}(_{c,d′}z_{Np^{∞}}^{univ}) ∈ i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}], (6.4)

which means that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) ∈ i_{{1,...,k−1}}(H_{L,h}[1/a])[G_{m}^{(1)}] with a non-zerodivisor a ∈ ℤ_{p}[[G_{∞}^{(2)}]] (which satisfies that a(χ^{r}) (r = 1, …, k − 1) are invertible in ℚ_{p}[[G_{∞}^{(2)}]]). It follows directly from the definition that the projection M_{k}(X_{1}(Np^{n}); ℚ) ⟶ S(𝔣) (n ≥ m) commutes with the Hecke operator T (p) = U(p) = Tr_{q}. By this and by the fact that the action of T (p) on S(𝔣) coincides with the multiplication by α, we obtain

𝔏_{f,{k−2},{r−1},m}(_{c,d′}z_{Np^{∞}}^{univ})|_{(m,n)}

= α^{m−n} · pr_{𝔣}(Tr_{q}^{n−m}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)})). (6.5)

Therefore for the proof of (6.4) it is enough to show that

∏_{r∈\{1,...,k−1\}} lim_{\overleftarrow{n}}(α^{−n} · pr_{𝔣}(Tr_{q}^{n−m}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)})))

∈ i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}]. (6.6)

We denote the r-component (r ∈ {1, …, k − 1}) of the left hand side of (6.6) by

μ_{r}(𝔣) = (μ_{r,n}(𝔣))_{n} ∈ lim_{\overleftarrow{n}} L[G_{n}^{(2)}][G_{m}^{(1)}] = L[[G_{∞}^{(1)}]][G_{m}^{(1)}].

In order to prove the assertion (6.6), by Proposition 5.3 in section 5, it is sufficient to show the following two assertions: One is to show that

lim_{n⟶∞} p^{hn}μ_{r,n}(𝔣) = 0 for all r ∈ {1, …, k − 1}.  (6.7)

The other is to prove that for any d ∈ ℤ such that h ≤ d ≤ k −1, (μ_{r}(𝔣))_{r=1,...,d} satisfy the condition (ii) in Proposition 5.3.

6.3.4. We check that (μ_{r}(𝔣))_{r=1,...,k−1} satisfy (6.7) above. By Proposition 6.3.2 which shows

Tr_{q}^{n−m}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)}) ∈ M_{k}(X_{1}(Np^{m}); ℤ_{(p)})[G_{m}^{(1)}][G_{n}^{(2)}]

for all n ≥ m and by the fact that M_{k}(X_{1}(Np^{m}); ℤ_{(p)}) is a finitely generated ℤ_{(p)}-module, we have that the image of M_{k}(X_{1}(Np^{m}); ℤ_{(p)}) under pr_{𝔣} : M_{k}(X_{1}(Np^{m}); ℚ) ⟶ L is contained in a · O_{L} for some a ∈ L^{×}.

From this, p^{nh} times μ_{r,n}(𝔣) is contained in p^{nh} · α^{−n}a · O_{L}[G_{m}^{(1)}][G_{n}^{(2)}]. As v_{p}(α) < h, we obtain lim_{n→∞} p^{nh} · α^{−n}a · O_{L} = 0 which implies lim_{n→∞} p^{nh}μ_{r,n}(𝔣) = 0 as desired.

6.3.5. We show that (μ_{r}(𝔣))_{r=1,...,d} satisfy the condition (ii) of Proposition 5.3 for any d ∈ ℤ such that h ≤ d ≤ k − 1. We use the notation in Proposition 5.3 (ii). We write c,d′z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{j}X_{ϕ_{n}}^{−j})|_{(m,n)} for (_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{j})|_{(m,n)})(id, X_{ϕ_{n}}^{−j}) ∈

M_{k}(X_{1}(Np^{n}); ℤ_{(p)})[G_{m}^{(1)}][G_{n}^{(2)}]. We can prove that

∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) μ(𝔣)_{j+1,n}(X_{ϕ_{n}}^{−j})

= ∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) · α^{−n} · pr_{𝔣}(Tr_{q}^{n−m}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{j}X_{ϕ_{n}}^{−j})|_{(m,n)}))

in Proposition 5.3 coincides with

α^{−n} · pr_{𝔣}(Tr_{q}^{n−m}(∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) · (_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{j}X_{ϕ_{n}}^{−j})|_{(m,n)}))).

(6.8)

Moreover we have

\begin{matrix}∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) · (_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{j}X_{ϕ_{n}}^{−j})|_{(m,n)}) \\ ∈ M_{k}(X_{1}(Np^{n}); p^{n(i−1)}ℤ_{(p)})[G_{m}^{(1)}][G_{n}^{(2)}] \end{matrix}

which follows from the general argument that

∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) · x(χ^{j}X_{ϕ_{n}}^{−j})|_{n} ∈ p^{n(i−1)}ℤ_{(p)}[G_{n}]

for any x ∈ ℤ_{(p)}[[G_{∞}]]. Here x(χ^{j}X_{ϕ_{n}}^{−j})|_{n} represents x(χ^{j})|_{n}(X_{ϕ_{n}}^{−j}) for the image x(χ^{j})|_{n} ∈ ℤ_{(p)}[G_{n}] of x(χ^{j}) ∈ ℤ_{(p)}[[G_{∞}]] under the projection. By this, we find (6.8) is contained in α^{−n} ·a·p^{n(i−1)}O_{L}[G_{m}^{(1)}][G_{n}^{(2)}] with a ∈ L^{×} in 6.3.4. As v_{p}(α) < h ≤ d, we obtain

lim_{n⟶∞} p^{(d−i+1)n} ∑_{j=0}^{i−1}(−1)^{i−j−1} (\begin{matrix}i − 1 \\ j \end{matrix}) μ(𝔣)_{j+1,n}(X_{ϕ_{n}}^{−j}) = 0,

which says that (μ_{r}(𝔣))_{r=1,...,d} (h ≤ d ≤ k − 1) satisfy the condition (ii) of Proposition 5.3, as desired.

The above arguments conclude our claim (6.4) in 6.3.3.

6.3.6. The argument in 6.3.4 shows that if v_{p}(α) = 0,

μ_{r}(𝔣) ∈ (O_{L}[[G_{∞}^{(2)}]] ⊗_{O_{L}} L)[G_{m}^{(1)}].

This and the argument in 6.3.5 show that in this case, 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) is contained in i_{{1,...,k−1}}((O_{L}[[G_{∞}^{(2)}]] ⊗_{O_{L}} L)[1/a])[G_{m}^{(1)}].

Now we show that 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})^{±} ∈ i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}]. The main line of the proof is roughly as follows. Let M be an integer ≥ 1 which is prime to Np. By using _{c,d′}z_{M,Np^{∞}} which is similar with the element _{c,d′}z_{Np^{∞}}^{univ} in 4.5.5 and which is defined in 6.3.7 and 6.3.8, we will construct a function L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}) ∈ i_{{1,...,k−1}}(H_{L(ϕ),h})[G_{m}^{(1)}] in 6.3.9. Here ϕ denotes a character (ℤ/Mℤ)^{×} ⟶ \overline{ℚ}^{×}. Assertions in 6.3.11 and 6.3.12 say that L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}) is, in fact, a function which is a multiple of our 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}) by a non-zerodivisor in ℤ_{p}[[G_{∞}^{(2)}]] which is prime to a ∈ ℤ_{p}[[G_{∞}]] in 6.3.3.

6.3.7. Let M be a positive integer such that (M, Np) = 1. We define an element

z_{M,Np^{∞}} ∈ \overline{M}_{MNp^{∞}}[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]][\frac{1}{g′}]

with a certain non-zerodivisor

g′ ∈ ℤ[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]] ⊂ \overline{M}_{MNp^{∞}}[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]],

which is similar to the universal zeta modular form z_{Np^{∞}}^{univ}. In fact, in the case M = 1, we have z_{1,Np^{∞}} = z_{Np^{∞}}^{univ}. Here

G_{Mp^{∞}}^{(1)} ≅ G_{Mp^{∞}}^{(2)} ≅ G_{Mp^{∞}} = (ℤ/Mℤ)^{×} × ℤ_{p}^{×},

the group G_{Mp^{∞}}^{(1)} is the one acting on the space \overline{M}_{MNp^{∞}} in the following way. For a = (a_{1}, a_{2}) ∈ (ℤ/Mℤ)^{×} × ℤ_{p}^{×}, the action of the corresponding element 𝔤_{a}^{(1)} ∈ G_{Mp^{∞}}^{(1)} on f = ∑_{k} f_{k} ∈ ⋃_{j} M^{j}(X_{1}(MNp^{t}); ℚ_{p}) with f_{k} ∈ M_{k}(X_{1}(MNp^{t}); ℚ_{p}) and t ≥ 1 is given as

𝔤_{a}^{(1)} · f = ∑_{k} a_{2}^{k−2}⟨a′⟩f_{k},

where a′ ∈ (ℤ/MNp^{t}ℤ)^{×} is the element such that a′ ≡ a(Mp^{t}) and a′ ≡ 1(N). The group G_{Mp^{∞}}^{(2)} is the Galois group Gal(ℚ_{p}(ζ_{Mp^{∞}})/ℚ_{p}) which is endowed with an isomorphism to (ℤ/Mℤ)^{×} × ℤ_{p}^{×} via the cyclotomic character. The element z_{M,Np^{∞}} is the image of the product F′_{N,1} · F′_{N,2} ∈ H[[G_{Mp^{∞}}]][[G_{Mp^{∞}}]] with F′_{N,1}, F′_{N,2} ∈ H[[G_{Mp^{∞}}]] below under the isomorphism of rings over H

H[[G_{Mp^{∞}}]][[G_{Mp^{∞}}]] ⟶ H[[G_{Mp^{∞}}^{(1)}]][[G_{Mp^{∞}}^{(2)}]] ; x𝔤_{a,1}𝔤_{b,2} ↦ x𝔤_{b}^{(1)}𝔤_{ab^{−1}}^{(2)}

(x ∈ H, a, b ∈ ℤ_{p}^{×} ×(ℤ/Mℤ)^{×}), where 𝔤_{a} ∈ G_{Mp^{∞}} is the corresponding element to a. Here

F′_{N,1} = ( ∑_{\begin{matrix}i≥1 \\ (i,Mp)=1 \end{matrix}} ∑_{j≥1} q^{Nij})(𝔤_{i} − 𝔤_{−i}) + lim_{\overleftarrow{n}}( ∑_{a∈(ℤ/Mp^{n}ℤ)^{×}} ζ_{≡a(Mp^{n})}(0) · 𝔤_{a}),

\begin{aligned}F′_{N,2} & = ( ∑_{\begin{matrix}i≥1,i≡1(N) \\ (i,Mp)=1 \end{matrix}} ∑_{j≥1} q^{ij} · 𝔤_{i} − ∑_{\begin{matrix}i≥1,i≡−1(N) \\ (i,Mp)=1 \end{matrix}} ∑_{j≥1} q^{ij} · 𝔤_{−i}) \\ & + lim_{\overleftarrow{n}}( ∑_{\begin{matrix}a∈(ℤ/NMp^{n}ℤ)^{×} \\ a≡1(N) \end{matrix}} ζ_{≡a(NMp^{n})}(0) · 𝔤_{a}). \end{aligned}

Originally z_{M,Np^{∞}} belongs to H[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]], but we can prove that z_{M,Np^{∞}} ∈ \overline{M}_{MNp^{∞}}[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]][1/g′]. This follows from the following lemma which can be proven in the same way as for z_{Np^{∞}}^{univ}. For n ≥ 1, we write G_{Mp^{n}} for the group (ℤ/Mℤ)^{×} × G_{n}.

Lemma 6.3.8 . (1) Let c, d′ be integers which are prime to p. Then

c,d′z_{M,Np^{∞}} := (1−c^{−1}𝔤_{c^{−1}}^{(1)}𝔤_{c}^{(2)})(1−d′𝔤_{d′}^{(2)})z_{M,Np^{∞}} ∈ O_{H}[[G_{Mp^{∞}}^{(1)} ×G_{Mp^{∞}}^{(2)}]].

(2) For i ∈ ℤ, let us denote by χ^{i} : G_{Mp^{∞}} ⟶ G_{Mp^{∞}} the map induced by χ^{i} on the component ℤ_{p}^{×}. Let a_{1}, a_{2} be integers such that 0 ≤ a_{2} ≤ a_{1}. We define z_{M,Np^{∞}}(χ^{a_{1}}, χ^{a_{2}}) in the same way as for z_{Np^{∞}}^{univ} in section 4. Concerning the image of the natural projection, we have

\begin{matrix}z_{M,Np^{∞}}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} ∈ & M_{a_{1}+2}(X_{1}(MNp^{n}); ℚ)[G_{Mp^{n}}^{(1)} × G_{Mp^{n}}^{(2)}] \\ (⊂ H[G_{Mp^{n}}^{(1)} × G_{Mp^{n}}^{(2)}]), \end{matrix}

\begin{matrix}_{c,d′}z_{M,Np^{∞}}(χ^{a_{1}}, χ^{a_{2}})|_{(n,n)} ∈ & M_{a_{1}+2}(X_{1}(MNp^{n}); ℤ_{(p)})[G_{Mp^{n}}^{(1)} × G_{Mp^{n}}^{(2)}] \\ (⊂ O_{H}[G_{Mp^{n}}^{(1)} × G_{Mp^{n}}^{(2)}]). \end{matrix}

(3) Let a_{1} and a_{2} be as in (2). For any integers n, m such that n ≥ m ≥ 1, we have

Tr_{q}^{n−m}(z_{M,Np^{∞}}(χ^{a_{1}}, χ^{a_{2}})|_{(m,n)}) ∈ M_{a_{1}+2}(X_{1}(MNp^{m}); ℚ)[G_{Mp^{m}}^{(1)}][G_{Mp^{n}}^{(2)}].

Tr_{q}^{n−m}(_{c,d′}z_{M,Np^{∞}}(χ^{a_{1}}, χ^{a_{2}})|_{(m,n)}) ∈ M_{a_{1}+2}(X_{1}(MNp^{m}); ℤ_{(p)})[G_{Mp^{m}}^{(1)}][G_{Mp^{n}}^{(2)}].

6.3.9. Let ϕ : (ℤ/Mℤ)^{×} ⟶ \overline{ℚ}^{×} be a character whose conductor is M. Let 𝔣_{ϕ} be the modular form given by

𝔣_{ϕ} = ∑_{n≥1} a_{n}(𝔣)ϕ(n)q^{n}.

It is an eigen cusp form of level MNp^{m} with zeta function L(𝔣_{ϕ}, s) = L(𝔣, ϕ, s).

In the same manner as in 6.3.3 (6.5), we can show that

(α^{−n}ϕ(p)^{−n} · pr_{𝔣_{ϕ}}(Tr_{q}^{n−m}(_{c,d′}z_{M,Np^{∞}}(χ^{k−2}, χ^{r−1})|_{(m,n)})))_{n}  (6.9)

belongs to lim_{\overleftarrow{n}} L(ϕ)[G_{Mp^{n}}^{(2)}][G_{Mp^{m}}^{(1)}], where L(ϕ) is the field generated over L by

the values of ϕ. Here remark that a_{p}(𝔣_{ϕ}) = a_{p}(𝔣)ϕ(p) = αϕ(p). We consider the image of the element (6.9) under the L(ϕ)-homomorphism given by

L(ϕ)[[G_{Mp^{∞}}^{(2)}]][G_{Mp^{m}}^{(1)}] ⟶ L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}] ; x𝔤_{a}^{(2)}𝔤_{b}^{(1)} ↦ xϕ(ab^{2})g_{a}^{(2)}g_{b}^{(1)} (6.10)

(x ∈ L(ϕ), 𝔤_{∗} ∈ G_{Mp^{∞}}, g_{∗} ∈ G_{∞}). We write L_{ϕ,f,{k−2},{r−1},m}(_{c,d′}z_{M,Np^{∞}}) ∈ L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}] for this image, and we put

L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}) =

∏_{r∈{1,...,k−1}} L_{ϕ,f,{k−2},{r−1},m}(_{c,d′}z_{M,Np^{∞}}) ∈ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}].

Concerning L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}), we have the following propositions, which are crucial to our purpose.

Proposition 6.3.10 . The element L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}) is contained in the subspace i_{{1,...,k−1}}(H_{L(ϕ),h})[G_{m}^{(1)}].

Proof. We can prove Proposition 6.3.10 in the same manner as in 6.3.1 – 6.3.6. □

In the rest of 6.3, we identify an element of H_{L(ϕ),h} and its image under i_{{1,...,k−1}} : H_{L(ϕ),h} ↪ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]].

Proposition 6.3.11 . Assume L(𝔣_{ϕ}, k−1) = L(𝔣, ϕ, k−1) ≠ 0 and ϕ(−1) = 1. Then we have

L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}})^{±}

= x · ( ∏_{r∈\{1,...,k−1\}}( ∏_{\begin{matrix}l:prime \\ l|M \end{matrix}}(1 − a_{l}(𝔣)l^{−r}g_{l^{−1}}^{(2)} + ϵ_{𝔣}(l)l^{k−1−2r}g_{l^{−2}}^{(2)})

· (1 − c^{r−k}ϵ_{𝔣}(c^{−1})ϕ(c^{−1})g_{c}^{(2)})(1 − d^{′r}ϕ(d′)g_{d′}^{(2)}))
· α^{m} · 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ}))^{±}
∈ H_{L(ϕ),h}[G_{m}^{(1)}]

for some x ∈ L(ϕ)^{×}, where ± = (−1)^{k}ϵ_{𝔣}(−1).

Proof. An element of H_{L(ϕ),h}[1/e](⊂ H_{L(ϕ),h} ⊗_{ℤ_{p}[[G_{∞}^{(2)}]]} Q(ℤ_{p}[[G_{∞}^{(2)}]])) can be

characterized by specializations H_{L(ϕ),h}[1/e] ⟶ \overline{L} induced by ψ ∘ χ^{r_{i}} : G_{∞}^{(2)} ⟶ \overline{ℚ}^{×} for different h integers r_{i} (i = 1, …, h) and all but finitely many Dirichlet characters ψ. So we can prove Proposition 6.3.11 by comparing the images of both hands sides under such specializations. The image of the right hand side under specializations will be studied in 6.4 – 6.6 below. The image of the left hand side may be obtained in the same way as for the right hand side, but we omit the details. □

6.3.12. We finish proving that the elements in Theorem 6.2 are contained in i_{{1,...,k−1}}(H_{L,h})[G_{m}^{(1)}]. We have

𝔏_{f,{k−2},{0,...,k−2},m}(_{c,d′}z_{Np^{∞}}^{univ})

= ( ∏_{r∈{1,...,k−1}}(1 − c^{r−k}ϵ_{𝔣}(c^{−1})g_{c}^{(2)})(1 − d^{′r}g_{d′}^{(2)}))

· 𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})

∈ H_{L,h}[G_{m}^{(1)}](⊂ H_{L,h} ⊗_{ℤ_{p}[[G_{∞}^{(2)}]]} Q(ℤ_{p}[[G_{∞}^{(2)}]])[G_{m}^{(1)}]),

which follows directly from the definition. Since (1 − a_{l}(𝔣)g_{l^{−1}}^{(2)} + ϵ_{𝔣}(l)l^{k−1}g_{l^{−2}}^{(2)}) for all of the prime numbers l which are prime to Np do not have a common divisor, by Propositions 6.3.10 and 6.3.11, it is sufficient to show the following assertion. There exist c, d′, M, and ϕ which satisfy the above given conditions and the following condition. For some characters ψ_{1}, ψ_{2} : (ℤ/p^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} with conductor divisible by p, ϕ(c) = ψ_{1}(c) ≠ 1 and ϕ(d′) = ψ_{2}(d′) ≠ 1, and ϕ(−1) = 1 hold. We can take such elements, therefore we obtain the desired result.

6.4. We prove that our elements in Theorem 6.2 (1) and (2) satisfy the characterizing properties of L_{p-adic}(𝔣) in Theorem 5.5. In fact, the difference between Theorem 6.2 (1) and (2) comes from the fact that we take pr_{𝔣} instead of “pr_{f}”. We treat the cases Theorem 6.2 (1) and (2) together. We use the notation in Theorem 5.5.

6.4.1. As referred earlier, for L[[G_{∞}^{(2)}]], we define + and − parts in the same way as in 5.6, and for ∗ = + or −, we define the ∗-part of L[[G_{∞}^{(2)}]][G_{m}^{(1)}] by L[[G_{∞}^{(2)}]]^{∗}[G_{m}^{(1)}]. In both cases of Theorem 6.2 (1) and (2), we have

𝔏_{f,{k−2},{0,...,k−2},m}(z_{Np^{∞}}^{univ})

= ∏_{r∈\{1,...,k−1\}} lim_{\overleftarrow{n}}(α^{m−n} · pr_{𝔣}(Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)})))

∈ ∏_{r∈\{1,...,k−1\}} lim_{\overleftarrow{n}} L[G_{n}^{(2)}][G_{m}^{(1)}] = ∏_{r∈\{1,...,k−1\}} L[[G_{∞}^{(2)}]][G_{m}^{(1)}]. (6.11)

This follows from the equality (6.5). We will prove the assertion that the image of the coefficient belonging to lim_{\overleftarrow{n}} L[G_{n}^{(2)}] = L[[G_{∞}^{(2)}]] of g_{1}^{(1)} in

( ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − a_{l}(𝔣)l^{−r}g_{l^{−1}}^{(2)})· lim_{\overleftarrow{n}}(α^{−n} · pr_{𝔣}(Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)}))))^{±·(−1)^{r}}

∈ lim_{\overleftarrow{n}} L[G_{n}^{(2)}][G_{m}^{(1)}] = L[[G_{∞}^{(2)}]][G_{m}^{(1)}], (6.12)

where ± = (−1)^{k}ϵ_{𝔣}(−1) and ( )^{±·(−1)^{r}} represents the ± · (−1)^{r}-part of the element in ( ), under the map ψ^{−1} : L[[G_{∞}^{(2)}]] ⟶ \overline{L} coincides with the image of

( ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − a_{l}(𝔣)g_{l^{−1}}^{(2)}) · L_{p-adic}(𝔣))^{±}

under χ_{cyclo}^{r}ψ^{−1} : H_{L,h} ⟶ \overline{L}. By the following facts, this assertion deduces Theorem 6.2. Firstly we have the equality (6.11). Secondly \begin{matrix}l:prime \\ l|N \end{matrix}(1 − a_{l}(𝔣)g_{l^{−1}}^{(2)}) ∈ O_{L}[[G_{∞}^{(2)}]] ⊗_{O_{L}} L is a non-zerodivisor of H_{L,h}.

Finally by the results in 4.5.5, we have

\begin{matrix}(Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)}))_{n} = ∑_{a∈(ℤ/p^{m}ℤ)^{×}}⟨a^{′−1}⟩x · g_{a}^{(1)} \\ ∈ M_{k}(X_{1}(Np^{m}); ℚ)[G_{m}^{(1)}][[G_{∞}^{(2)}]], \end{matrix} (6.13)

where x is the coefficient of g_{1}^{(1)} in the left hand side, and a′ ∈ (ℤ/Np^{m}ℤ)^{×} is the element such that a′ ≡ a(p^{m}) and a′ ≡ 1(N). Hence the coefficient in lim_{\overleftarrow{n}} L[G_{n}^{(2)}] = L[[G_{∞}^{(2)}]] of g_{a}^{(1)} in (6.12) is ϵ_{𝔣}(a^{′−1}) times the coefficient of g_{1}^{(1)}

in (6.12). (Remark that in the case of Theorem 6.2 (2), ϵ_{𝔣}(a′) = 1 holds.) Thus we prove the assertion above.

6.4.2. We use the same notation as in Theorem 5.5. We consider the following composition

ψ_{ζ_{p^{n}}} : L[[G_{∞}^{(2)}]] ⟶^{proj.} L[G_{n}^{(2)}] ⟶ \overline{L},  (6.14)

where the second map is defined by

∑_{u∈(ℤ/p^{n}ℤ)^{×}} a_{u}g_{u}^{(2)} ↦ ∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x) ∑_{u∈(ℤ/p^{n}ℤ)^{×}} a_{u}ζ_{p^{n}}^{ux} (a_{u} ∈ L).

Lemma 6.4.3 . Let μ be an element of L[[G_{∞}^{(2)}]]. Then we have

μ(ψ^{−1}) = ψ_{ζ_{p^{n}}} (μ) · G(ψ, ζ_{p^{n}})^{−1} ∈ \overline{L}.

Proof. We can prove the lemma by direct computation. □

6.4.4. We assume ψ(−1) = (−1)^{k−r}ϵ_{𝔣}(−1). By Lemma 6.4.3, our task is showing that the image of the coefficient of g_{1}^{(1)} in (6.12) under the map (6.14) is

(r − 1)! · p^{nr} · α^{−n} · (2πi)^{k−r−1} · \frac{1}{Ω(f, δ(𝔣, k − 1, 1(0)))_{−}} · L_{(Np)}(𝔣, ψ, r).

6.5. In order to prove our claim in 6.4.4, we need to review the result in [Ka2].

6.5.1. As in [Ka2], let

z_{Np^{n}}(𝔣, r, k − 1, 0(1), prim(Np)) ∈ S(𝔣) ⊗ ℚ(ζ_{Np^{n}}) (r ≠ 2),
c,dz_{Np^{n}}(𝔣, 2, k − 1, 0(1), prim(Np)) ∈ S(𝔣) ⊗ ℚ(ζ_{Np^{n}})

be the images of z_{1,Np^{m},Np^{n}}^{(k,r)}(k, r, k − 1, 0(1), prim(Np)) in the case r ≠ 2,

and _{c,d}z_{1,Np^{m},Np^{n}}^{(k,2)}(k, 2, k − 1, 0(1), prim(Np)) in the case r = 2, both of which are elements of M_{k}(X_{1}(Np^{m}); ℚ) ⊗_{ℚ} ℚ(ζ_{Np^{n}}), respectively, in 4.5.2 under the projection

M_{k}(X_{1}(Np^{m}); ℚ) ⊗_{ℚ} ℚ(ζ_{Np^{n}}) ⟶ S(𝔣) ⊗_{ℚ} ℚ(ζ_{Np^{n}}).  (6.15)

Let 𝔣^{∗} = ∑_{n≥1} \overline{a_{n}}(𝔣)q^{n} denote the dual cusp form of 𝔣. Here \overline{a_{n}}(𝔣) are the complex conjugates of a_{n}(𝔣). This is also a normalized eigen cusp form. For n ≥ 1 such that (n, Np) = 1, it is known that

(T (n)⟨n^{−1}⟩)(𝔣^{∗}) = a_{n}(𝔣)𝔣^{∗}.  (6.16)

For x ∈ (ℤ/Np^{n}ℤ)^{×}, let us denote by ν_{x} the corresponding element to x of Gal(ℚ_{p}(ζ_{Np^{n}})/ℚ_{p}) via the cyclotomic character.

Proposition 6.5.2 . Assume r ≠ 2. Let ψ : (ℤ/Np^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} be a character. We put ± = (−1)^{k−r−1}ψ(−1). Then we have

\begin{matrix}∑_{x∈(ℤ/Np^{n}ℤ)^{×}} ψ(x) · per_{𝔣}((⟨x^{−1}⟩ ⊗ ν_{x})(z_{Np^{n}}(𝔣, r, k − 1, 0(1), prim(Np))))^{±} \\ = L_{(Np)}(𝔣^{∗}, ψ, r) · (2πi)^{k−r−1} · δ(𝔣, k − 1, 0(1))^{±}, \end{matrix}

where L_{(Np)}(𝔣^{∗}, ψ, s) denotes the function obtained from L(𝔣^{∗}, ψ, s) by removing prime(Np) factors.

Proof. Remark that the definitions of the actions of Galois group Gal(ℚ(ζ_{Np^{n}})/ℚ) are different between in [Ka2] and here: The action of σ_{x} in [Ka2], §6, Theorem 6.6 on S(𝔣) ⊗ ℚ(ζ_{Np^{n}}) is equal to the action of ⟨x^{−1}⟩ ⊗ ν_{x} in our notation. By this relation we see that the above equation is equivalent to 6.6 in [Ka2] which can be deduced from the work of Shimura [Sh]. □

In the case r = 2, Proposition 6.5.2 must be modified as

∑_{x∈(ℤ/Np^{n}ℤ)^{×}} ψ(x) · per_{𝔣}((⟨x^{−1}⟩ ⊗ ν_{x})(_{c,d}z_{Np^{n}}(𝔣, 2, k − 1, 0(1), prim(Np))))^{±}

= c^{2}d^{2} · (1 − c^{2−k}ψ(c)^{−1})(1 − ψ(d)^{−1}ϵ_{𝔣}(d))
· L_{(Np)}(𝔣^{∗}, ψ, 2) · (2πi)^{k−3} · δ(𝔣, k − 1, 0(1))^{±}.

The above proposition induces the following corollary.

Corollary 6.5.3 . Assume r ≠ 2. Let ψ be as in Proposition 6.5.2. We put ± = (−1)^{k−r−1}ψ(−1)ϵ_{𝔣}(−1). Then we have

\begin{matrix}∑_{x∈(ℤ/Np^{n}ℤ)^{×}} ψ(x) · per_{𝔣}((1 ⊗ ν_{x})(z_{Np^{n}}(𝔣, r, k − 1, 0(1), prim(Np))))^{±} \\ = L_{(Np)}(𝔣, ψ, r) · (2πi)^{k−r−1} · δ(𝔣, k − 1, 0(1))^{±}. \end{matrix}

Proof. By the definition of ϵ_{𝔣}, we have the equality

∑_{x∈(ℤ/Np^{n}ℤ)^{×}} ψ(x)

· per_{𝔣}((⟨x^{−1}⟩ ⊗ ν_{x})(z_{Np^{n}}(𝔣, r, k − 1, 0(1), prim(Np))))^{(−1)^{k−r−1}ψ(−1)}

= ∑_{x∈(ℤ/Np^{n}ℤ)^{×}} ψ(x)ϵ_{𝔣}(x^{−1})

· per_{𝔣}((1 ⊗ ν_{x})(z_{Np^{n}}(𝔣, r, k − 1, 0(1), prim(Np))))^{(−1)^{k−r−1}ψ(−1)}.

By Proposition 6.5.2 and by this, we see that the left hand side of the equation in Corollary 6.5.3 is equal to

L_{(Np)}(𝔣^{∗}, ψ · ϵ_{𝔣}, r) · (2πi)^{k−r−1} · δ(𝔣, k − 1, 0(1))^{±}.

By (6.16) and by the fact that ϵ_{𝔣}(n) = ϵ_{𝔣^{∗}}(n^{−1}) for n ∈ ℤ such that (n, Np) = 1, we have

L_{(Np)}(𝔣^{∗}, ψ · ϵ_{𝔣}, s) = L_{(Np)}(𝔣, ψ, s).

This shows the result. □

With a suitable modification, we have a similar result with Corollary 6.5.3, in the case r = 2.

As a corollary to Corollary 6.5.3, we obtain the following Corollary 6.5.4 which is important for the proof of our Theorem 6.2. For r ≠ 2, let A(k, r) be the element of M_{k}(X_{1}(Np^{m}); ℚ) ⊗_{ℚ} ℚ(ζ_{p^{n}}) obtained from the right hand side of (4.8) in Lemma 4.5.3 (3) in section 4 by replacing “a ∈ ℤ/p^{n}ℤ” in the right hand side of (4.8) by “a ∈ (ℤ/p^{n}ℤ)^{×}”. Namely,

A(k, r) =(r − 1)!^{−1} · (Np^{m})^{k−r−2}(Np^{n})^{−2} · N

· ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − l^{−r}T (l)ν_{l^{−1}})

( ∑_{a∈(ℤ/p^{n}ℤ)^{×}} T (p)^{n−m}

(  ∑_{x∈((1/Np^{m})ℤ)/ℤ} ∑_{y∈((1/Np^{n})ℤ)/ℤ}(F_{1/Np^{m},x}^{(k−r)} · F_{Na/Np^{n},y}^{(r)})) · ζ_{p^{n}}^{a}).

We define A(𝔣, r) to be the image of A(k, r) under the projection (6.15).

Corollary 6.5.4 . Assume r ≠ 2. Let ψ : (ℤ/p^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} be a character which does not factor through (ℤ/p^{n−1}ℤ)^{×} and which satisfies ψ(−1) = (−1)^{k−r}ϵ_{𝔣}(−1). Then we have

\begin{aligned}∑_{x∈(ℤ/p^{n}ℤ)^{×}} & ψ(x) · per_{𝔣}((1 ⊗ ν_{x})(A(𝔣, r)))^{−} \\ & = L_{(Np)}(𝔣, ψ, r) · (2πi)^{k−r−1} · δ(𝔣, k − 1, 0(1))^{−}. \end{aligned}

Proof. We take ψ as a character which factors through (ℤ/p^{n}ℤ)^{×} ⟶ \overline{ℚ}^{×} and does not factor through (ℤ/p^{n−1}ℤ)^{×}, and apply Corollary 6.5.3. Then the part of the sum ∑_{x∈(ℤ/Nℤ)^{×}} is equivalent to take the trace map tr_{Np^{n},p^{n}} in Lemma 4.5.3 (3). Concerning the problem of changing a, since ψ in Corollary 6.5.4 is primitive, the iterated sum over ∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x)ζ_{p^{n}}^{ax} for (a, p) > 1 become zero. □

In the case r = 2, Corollary 6.5.4 must be modified as follows. We take c,dA(k, 2) as the element which is obtained from (4.9) by replacing “a ∈ ℤ/p^{n}ℤ, a ≠ 0” in the right hand side of (4.9) by “a ∈ (ℤ/p^{n}ℤ)^{×}”. We also define _{c,d}A(𝔣, 2) to be the image of _{c,d}A(k, 2) under the projection (6.15). Then we have

∑_{x∈(ℤ/p^{n}ℤ)^{×}} ψ(x) · per_{𝔣}((1 ⊗ ν_{x})(_{c,d}A(𝔣, 2)))^{−}

= c^{2}d^{2} · ((1 − c^{2−k}ϵ_{𝔣}(c)^{−1}ψ(c)^{−1})(1 − ψ(d)^{−1}))

L_{(Np)}(𝔣, ψ, 2) · (2πi)^{k−3} · δ(𝔣, k − 1, 0(1))^{−}.

6.6. Now we prove our claim in 6.4.4. Since T (l)𝔣 = a_{l}(𝔣)𝔣, (6.12) coincides with

(lim_{\overleftarrow{n}}(α^{−n}

· pr_{𝔣}( ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − T (l)l^{−r}g_{l^{−1}}^{(2)})(Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)})))))^{±·(−1)^{r}}.

So to analyze (6.12), we consider the image of

∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − T (l)l^{−r}g_{l^{−1}}^{(2)})(Tr_{q}^{n−m}(z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)}))

∈ M_{k}(X_{1}(Np^{m}); ℚ)[G_{m}^{(1)} × G_{n}^{(2)}] (6.17)

under the map

·ζ_{p^{n}} : M_{k}(X_{1}(Np^{m}); ℚ)[G_{m}^{(1)} × G_{n}^{(2)}] ⟶ M_{k}(X_{1}(Np^{m}); ℚ(ζ_{p^{n}}))[G_{m}^{(1)}](6.18)

defined in the similar way as (4.12). In the case r ≠ 2, by comparing (4.8) in Lemma 4.5.3 (3) and equation (4.13) in 4.5.6 with a_{1} = k −2 and a_{2} = r −1, we see that the coefficient of g_{1}^{(1)} ∈ G_{m}^{(1)} in the image of (6.17) under the map (6.18) is

(r − 1)! · p^{nr} · A(k, r)

with A(k, r) in Corollary 6.5.4. Thus under the composition

\begin{aligned}L[[G_{∞}^{(2)}]][G_{m}^{(1)}] & ⟶^{proj.} L[G_{n}^{(2)}][G_{m}^{(1)}] ⟶^{·ζ_{p^{n}}} L(ζ_{p^{n}})[G_{m}^{(1)}] \\ & ⟶^{·f} (L(ζ_{p^{n}}) · f)[G_{m}^{(1)}], \end{aligned}  (6.19)

the coefficient of g_{1}^{(1)} in the element (6.12) is sent to

α^{−n} · (r − 1)! · p^{nr} · A(𝔣, r).

Hence by Corollary 6.5.4 and by comparing the definition of the map (6.14) and the map in Corollary 6.5.4, we obtain that under the assumption in 6.4.4, per_{𝔣}(f)^{−} times the image in question in 6.4.4 equals

α^{−n} · (r − 1)! · p^{nr} · L_{(Np)}(𝔣, ψ, r) · (2πi)^{k−r−1} · δ(𝔣, k − 1, 0(1))^{−}.

Since L(𝔣, k − 1) ≠ 0, we have δ(𝔣, k − 1, 0(1)) = δ(𝔣, k − 1, 0(1))^{−} ≠ 0. As per_{𝔣}(f)^{−} = Ω(f, δ(𝔣, k − 1, 1(0)))_{−} · δ(𝔣, k − 1, 0(1))^{−}, we find that the claim in 6.4.4 is true in the case r ≠ 2. In the case r = 2, in the same way as for r ≠ 2, we can show that the coefficient of g_{1}^{(1)} in the image of c^{2}d^{2} · (1 − c^{2−k}ϵ_{𝔣}(c^{−1})g_{c}^{(2)})(1 − g_{d}^{(2)}) times element (6.17) under the map (6.18) is p^{2n} ·_{c,d}A(k, 2) with _{c,d}A(k, 2) just after Corollary 6.5.4. Hence the image of the coefficient of g_{1}^{(1)} in c^{2}d^{2} · (1 − c^{2−k}ϵ_{𝔣}(c^{−1})g_{c}^{(2)})(1 − g_{d}^{(2)}) times (6.12) under the composition (6.19) is α^{−n} · p^{2n} ·_{c,d}A(𝔣, 2). From this, in the same way as for r ≠ 2, we can show that the claim in 6.4.4 is true for r = 2. The above arguments conclude our desired result that the left hands sides of the equations in Theorem 6.2 (1) and (2) satisfy the characterizing property of the right hands sides of them.

6.7. We explain that by our method, we can obtain the whole p-adic zeta function L_{p-adic}(𝔣) without assuming that L(𝔣, k − 1) ≠ 0.

Similar with z_{Np^{∞}}^{univ}, elements z_{M,Np^{∞}} in 6.3.7 may be obtained via K_{2} Coleman power series from the system of Beilinson elements (_{c,d}z_{MNp^{n},Mp^{n}})_{n} ∈ lim_{\overleftarrow{n}} K_{2}(Y (MNp^{n}, Mp^{n})), with c, d ∈ ℤ such that (c, 6MNp) = (d, 6Mp) = 1

and c ≡ 1(N). In this case, we take the field H(ζ_{M}) instead of H. We define L_{ϕ,f,{k−2},{0,...,k−2},m}(z_{M,Np^{∞}}) ∈ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}] by replacing _{c,d′}z_{M,Np^{∞}} by z_{M,Np^{∞}} in (6.9) in the construction of the function L_{ϕ,f,{k−2},{0,...,k−2},m}(_{c,d′}z_{M,Np^{∞}}). Then in the same way as for Theorem 6.2, one can show that if L(𝔣_{ϕ}, k − 1) = L(𝔣, ϕ, k − 1) ≠ 0,

L_{ϕ,f,{k−2},{0,...,k−2},m}(z_{M,Np^{∞}})^{±·ϕ(−1)}

= x · ∑_{a∈(ℤ/p^{m}ℤ)^{×}}(( ∏_{r∈\{1,...,k−1\}}( ∏_{\begin{matrix}l:prime \\ l|M \end{matrix}}(1 − a_{l}(𝔣)l^{−r}g_{l^{−1}}^{(2)} + ϵ_{𝔣}(l)l^{k−1−2r}g_{l^{−2}}^{(2)})))

· L_{p-adic}(𝔣))^{±·ϕ(−1)}ϵ_{𝔣}(a^{′−1})g_{a}^{(1)} ∈ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}]

with some x ∈ L(ϕ)^{×}, and ± = (−1)^{k}ϵ_{𝔣}(−1). Moreover by replacing the pair (M, Np) by (N, p), we define a function. Let ϕ : (ℤ/Nℤ)^{×} ⟶ \overline{ℚ}^{×} be a character whose conductor is N. Now we define L_{ϕ,f,{k−2},{0,...,k−2},m}(z_{N,p^{∞}}) ∈ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}] to be the image of

\begin{matrix}∏_{r∈\{1,...,k−1\}} & (α^{−n}ϕ(p)^{−n} · pr_{𝔣_{ϕ}}(Tr_{q}^{n−m}(_{c,d′}z_{N,p^{∞}}(χ^{k−2}, χ^{r−1})|_{(m,n)})))_{n} \\ ∈ ∏_{r∈\{1,...,k−1\}} L(ϕ)[[G_{Np^{∞}}]][G_{Np^{m}}^{(1)}] \end{matrix}

under an L(ϕ)-homomorphism given by

L(ϕ)[[G_{Np^{∞}}^{(2)}]][G_{Np^{m}}^{(1)}] ⟶ L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}] ; x𝔤_{a}^{(2)}𝔤_{b}^{(1)} ↦ xϕ(ab^{2})g_{a}^{(2)}g_{b}^{(1)}

(x ∈ L(ϕ), 𝔤_{∗} ∈ G_{Np^{∞}}, g_{∗} ∈ G_{∞}). One can show that if L(𝔣_{ϕ}, k − 1) = L(𝔣, ϕ, k − 1) ≠ 0,

L_{ϕ,f,{k−2},{0,...,k−2},m}(z_{N,p^{∞}})^{±·ϕ(−1)}

= x · ∑_{a∈(ℤ/p^{m}ℤ)^{×}}((( ∏_{r∈\{1,...,k−1\}}( ∏_{\begin{matrix}l:prime \\ l|N \end{matrix}}(1 − a_{l}(𝔣)l^{−r}g_{l^{−1}}^{(2)}))

· L_{p-adic}(𝔣))^{±·ϕ(−1)})ϵ_{𝔣}(a^{′−1})g_{a}^{(1)} ∈ ∏_{r∈{1,...,k−1}} L(ϕ)[[G_{∞}^{(2)}]][G_{m}^{(1)}]

with some x ∈ L(ϕ)^{×}, and ± = (−1)^{k}ϵ_{𝔣}(−1).

From the above equalities, we find that by taking various ϕ which satisfy L(𝔣, ϕ, k−1) ≠ 0, the whole (i.e. including the (−1)^{k−1}ϵ_{𝔣}(−1)-part of) L_{p-adic}(𝔣) can be obtained by our method without assuming L(𝔣, k − 1) ≠ 0.

7. The result on two-variable p-adic zeta function

For a module A over ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]], we put A_{Q} = A ⊗_{ℤ_{p}[[G_{∞}^{(1)}×G_{∞}^{(2)}]]}

Q(ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]). In 7.1, for a certain subspace B of \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{Q} to which the universal zeta modular form z_{Np^{∞}}^{univ} belongs, we define a map (in 7.1.3)

L_{N} : B ⟶ (((𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ))[[G_{∞}^{(2)}]])_{Q}.

The main theorem (Theorem 7.3) of this section is that

L_{N} : z_{Np^{∞}}^{univ} ↦ a “universal ordinary p-adic zeta function”,

where the universal ordinary p-adic zeta function is a p-adic zeta function in two variables associated to the universal family of ordinary cusp forms (see Theorem 7.3 for the details).

7.1. We define the subspace B and the map L_{N}.

7.1.1. We put Λ = ℤ_{p}[[G_{∞}^{(1)}]] as in 3.5. Let us define a subspace \overline{m}_{Λ} of \overline{M}_{Np^{∞}}[[G_{∞}^{(1)}]] in the following way:

\overline{m}_{Λ} = \{x ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)}]] ; g_{a^{−1}}^{(1)} · x_{n,b} = x_{n,ab} for all a, b ∈ (ℤ/p^{n}ℤ)^{×}\}.

Here x_{n,a} ∈ \overline{M}_{Np^{∞}} are defined by x|_{n} = ∑_{a∈(ℤ/p^{n}ℤ)^{×}} x_{n,a}g_{a}^{(1)} ∈ \overline{M}_{Np^{∞}}[G_{n}^{(1)}]

with the image x|_{n} of x under the projection \overline{M}_{Np^{∞}}[[G_{∞}^{(1)}]] ⟶ \overline{M}_{Np^{∞}}[G_{n}^{(1)}].

Proposition 7.1.2 . (1) We have

z_{Np^{∞}}^{univ} ∈ \overline{m}_{Λ}[[G_{∞}^{(2)}]][\frac{1}{g}](⊂ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][\frac{1}{g}]),

where g is as before.

(2) Let 𝔦 : \overline{M}_{Np^{∞}} ⟶ Hom_{ℤ_{p}}(H_{Np^{∞}}, ℤ_{p}) be as in 3.3. We denote by the same symbol 𝔦 the map

\begin{aligned}𝔦 : \overline{M}_{Np^{∞}}[[G_{∞}^{(1)}]] & ⟶ Hom_{ℤ_{p}}(H_{Np^{∞}}, ℤ_{p})[[G_{∞}^{(1)}]] \\ & = Hom_{ℤ_{p}}(H_{Np^{∞}}, Λ) \end{aligned}

induced by 𝔦 in 3.3. Then

𝔦(\overline{m}_{Λ}) ⊂ Hom_{Λ}(H_{Np^{∞}}, Λ).

Proof. (1)The results in 4.5.5 deduce the assertion.

(2) The result follows directly from the definition. □

7.1.3. We define a map

\overline{m}_{Λ} ⟶ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ)  (7.1)

which induces

L_{N} : \overline{m}_{Λ}[[G_{∞}^{(2)}]][\frac{1}{g}] ⟶ ((𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ))[[G_{∞}^{(2)}]][\frac{1}{g}]

for a non-zerodivisor g ∈ ℤ_{p}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]. The map (7.1) is defined as the following composition:

\overline{m}_{Λ} ⟶^{𝔦} Hom_{Λ}(H_{Np^{∞}}, Λ) ⟶^{proj.} Hom_{Λ}(H_{Np^{∞}}^{ord}, Λ)
⟶ Hom_{Q(Λ)}(P_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ), Q(Λ))
⟶ Hom_{Q(Λ)}(p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ), Q(Λ))
⟶ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ).

(7.2)

The second arrow is by the natural projection. The third arrow is the evident one. The fourth arrow is given as follows. By the definition of P_{Np^{∞}}^{ord} and p_{Np^{∞}}^{ord}, it follows that the natural map P_{Np^{∞}}^{ord} ⟶ p_{Np^{∞}}^{ord} is surjective. The fourth arrow is given as the unique section p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ⟶ P_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) as algebras over Q(Λ) of the surjective homomorphism between semisimple algebras P_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ⟶ p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) which is induced by the above surjective map. Finally the last arrow in (7.2) is defined in the following way. By Proposition 3.6, p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ≅ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ) are finitely generated semisimple algebras over Q(Λ). Hence we have an isomorphism

(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ) ≅ Hom_{Q(Λ)}((𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ), Q(Λ)) ;(7.3)

a ↦ (x ↦ Tr(a · x)),

where Tr is the trace map

Tr : 𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ⟶ Q(Λ).

This map gives the last map of (7.2).

7.1.4. We define a universal ordinary p-adic zeta function

L_{p-adic}^{ord,univ} ∈ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{hg}],

where h ∈ 𝔥_{Np^{∞}}^{ord} is a certain non-zerodivisor and g is as before, as

L_{p-adic}^{ord,univ} = L_{N}(z_{Np^{∞}}^{univ}).

From the definition, one can see that universal ordinary p-adic zeta function L_{p-adic}^{ord,univ} is an element of (1/h)(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][1/g], which is contained in

(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][1/hg].

7.2. We review basic facts and results of Hida about ordinary eigen cusp forms.

7.2.1. Let k ≥ 2 and m ≥ 1. Let f ∈ M_{k}(X(1, Np^{m})) ⊗ ℂ be a normalized eigen cusp form. We recall that f is called ordinary if the eigenvalue of T (p) on f is a p-adic unit. Further f is called an ordinary p-stabilized newform of tame conductor N when f is ordinary, the conductor of f is divisible by N, and m ≥ 1 (cf. [GS]). We call Np^{m} the level of an ordinary p-stabilized newform f of tame conductor N if m is the smallest (positive) integer such that f ∈ M_{k}(X(1, Np^{m})) ⊗ ℂ. An ordinary p-stabilized newform of tame conductor N of level Np^{m} is either it is already a newform of level Np^{m} or it is an ordinary eigen cusp form f_{α} obtained from a newform f of level N when α is a p-adic unit by the method in 6.1.1 (cf.[GS]).

7.2.2. We call a ring homomorphism κ : 𝔥_{Np^{∞}}^{ord} ⟶ \overline{ℤ_{p}} satisfying the following conditions an N-primitive arithmetic point (cf. [GS]). For an integer i such that i ≥ 0 and a Dirichlet character ψ of conductor p^{n} (n ≥ 0), let 𝔓_{i,ψ} be the kernel of the ℤ_{p}-homomorphism Λ ⟶ \overline{ℤ_{p}} induced by ψ ∘ χ^{i} : G_{∞}^{(1)} ⟶ \overline{ℤ_{p}}^{×}. The condition for an N-primitive arithmetic point is that it factors through

κ : 𝔥_{Np^{∞}}^{ord} ⟶ 𝔥_{Np^{∞}}^{ord}/(𝔓_{i,ψ} + I_{Np^{∞}}^{ord}) ⟶ \overline{ℤ_{p}}  (7.4)

for some i ≥ 0 and some ψ. Here I_{Np^{∞}}^{ord} is the ideal defined in 3.7 in section 3.

7.2.3. In the case p ≥ 5, Hida [Hi2], §1, Corollary 1.3 proved that for an Nprimitive arithmetic point κ which factors as (7.4), we have a unique ordinary p-stabilized newform f = ∑_{n≥0} a_{n}(f)q^{n} of weight i + 2 of tame conductor N such that κ(T (n)) = a_{n}(f).

Theorem 7.3 . We assume p ≥ 5. The universal ordinary p-adic zeta function

L_{p-adic}^{ord,univ} ∈ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{hg}]

defined in 7.1.4 displays property (7.6) below. Let

κ : 𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⟶ \overline{ℤ_{p}}

be an N-primitive arithmetic point. We write f_{κ} = ∑_{n≥1} a_{n}(f_{κ})q^{n} for the ordinary p-stabilized newform of tame conductor N attached in the sense of 7.2.3 with κ. We denote the weight and level of f_{κ} by k and Np^{m} (m ≥ 1), respectively. We always have that κ(g) ∈ \overline{ℤ_{p}}[[G_{∞}^{(2)}]] is a non-zero divisor. We assume that κ(h) ≠ 0. Then κ induces the following homomorphism which is also denoted by κ :

κ : (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{hg}] ⟶ Q(\overline{ℤ_{p}}[[G_{∞}^{(2)}]]).  (7.5)

Now if L(f_{κ}, k − 1) ≠ 0, then concerning the image L_{p-adic}^{ord,univ}(κ) of L_{p-adic}^{ord,univ} under (7.5), we have

L_{p-adic}^{ord,univ}(κ)^{±} = p^{m−1}(p − 1) ·κ(T (p))^{m} ·L_{p-adic}(f_{κ})^{±}(χ) ∈ O_{M}[[G_{∞}^{(2)}]] ⊗_{O_{M}} M. (7.6)

Here concerning (one-variable) p-adic zeta function L_{p-adic}(f_{κ}), we take the class of f_{κ} as ω ∈ S(f_{κ}) in the situation of Theorem 5.5. Moreover ± = (−1)^{k}κ(⟨−1⟩), L_{p-adic}^{ord,univ}(κ)^{±} and L_{p-adic}(f_{κ})^{±} are the ±-parts of L_{p-adic}^{ord,univ}(κ) and L_{p-adic}(f_{κ}), respectively, and M is the finite extension ℚ_{p}(a_{n}(f_{κ}); n ≥ 1) of ℚ_{p}.

Remark 7.3.1 . (1) Recall that as in Theorem 6.2, even for p = 2, 3, the p-adic zeta function L_{p-adic}(f) of each ordinary p-stabilized newform f was constructed from z_{Np^{∞}}^{univ}.

(2) By the argument in 4.5.5, we obtain _{c,d′}z_{Np^{∞}}^{univ} ∈ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} ×G_{∞}^{(2)}]]. From this and from the fact that κ((1 − c^{−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d′g_{d′}^{(2)})) ∈ \overline{ℤ_{p}}[[G_{∞}^{(2)}]] is a non-zerodivisor for any N-primitive arithmetic point κ, we find that κ(g) ∈ \overline{ℤ_{p}}[[G_{∞}^{(2)}]] is a non-zerodivisor for any N-primitive arithmetic point κ.

(3) In the above, we put the assumption that L(f_{κ}, k − 1) ≠ 0. (As referred before, L(f_{κ}, k−1) = 0 occurs only in the case k = 2.) Moreover we only considered the (−1)^{k}κ(⟨−1⟩)-parts of L_{p-adic}(f_{κ}). However as explained briefly in 7.7 later, by using z_{M,Np^{∞}}, z_{N,p^{∞}}, and ϕ in 6.7, with some device, we can construct a two-variable p-adic zeta function which can provide the (−1)^{k+1}κ(⟨−1⟩)-part of L_{p-adic}(f_{κ}) for κ satisfying some conditions (see 7.7 for this condition) even though L(f_{κ}, k − 1) = 0.

We prove Theorem 7.3 by using Theorem 6.2 and the argument in the proof of it. We use the notation in Theorem 7.3 and we assume κ(h) ≠ 0.

7.4. For integers c, d′ which are prime to p, we put

c,d′L_{p-adic}^{ord,univ} = (1 − c^{−1}g_{c^{−1}}^{(1)}g_{c}^{(2)})(1 − d′g_{d′}^{(2)})L_{p-adic}^{ord,univ} ∈ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[\frac{1}{h}][[G_{∞}]].

One can see that _{c,d′}L_{p-adic}^{ord,univ} = L_{N}(_{c,d′}z_{Np^{∞}}^{univ}). We also put

c,d′L_{p-adic}(f_{κ}) = ((1 − c^{−k}ϵ_{f_{κ}}(c^{−1})g_{c}^{(2)})(1 − g_{d′}^{(2)})) · L_{p-adic}(f_{κ}).

In 7.5 – 7.6, we will prove that under an N-primitive arithmetic point κ satisfying the condition in Theorem 7.3,

c,d′L_{p-adic}^{ord,univ}(κ) = p^{m−1}(p − 1) · κ(T (p))^{m} ·_{c,d′}L_{p-adic}(f_{κ})(χ). (7.7)

The result in Theorem 7.3 will follow from this.

7.5. We consider some consequences of Theorem 6.2 for the proof of (7.7).

7.5.1. Since f_{κ} is ordinary, L_{p-adic}(f_{κ}) may be characterized by the specialization property in Theorem 5.5 (i) for only one r among 1, …, k − 1 under the notation there. We have that S(f_{κ})_{M} := M_{k}(X_{1}(Np^{n}); M)/(T (n) − κ(T (n)) ; n ≥ 1) is a one dimensional M-vector space with a base f_{κ}. We define pr_{f_{κ}} : M_{k}(X_{1}(Np^{n}); M) ⟶ M as the composition M_{k}(X_{1}(Np^{n}); M) ⟶^{proj.} S(f_{κ})_{M} ⟶ M, where the second map is the one given by sending the class of f_{κ} to 1. (In the case that f_{κ} is a newform, the map pr_{f_{κ}} coincides with the map pr_{𝔣} in section 6. In the case f_{κ} = f_{α} for some newform f of conductor N and α as before, in section 6, we took f as the base of S(f_{κ})_{M}. So if we use pr_{f_{κ}} in Theorem 6.2 instead of pr_{𝔣}, we obtain the p-adic zeta function of f_{κ} which takes the class of f_{κ} as ω in the notation in Theorem 5.5.) By Theorem 6.2, we obtain

(lim_{\overleftarrow{n}} pr_{f_{κ}}(_{c,d′}z_{Np^{∞}}^{univ}(χ^{k−2}, χ^{r−1})|_{(m,n)}))^{±·(−1)^{r}}

= α^{m} · ∑_{a∈(ℤ/p^{m}ℤ)^{×}} i_{\{r\}}(_{c,d′}L_{p-adic}(f_{κ})^{±})ϵ_{f_{κ}}(a^{′−1})g_{a}^{(1)} ∈ lim_{\overleftarrow{n}} M[G_{n}^{(2)}][G_{m}^{(1)}], (7.8)

with ± = (−1)^{k}κ(⟨−1⟩), where a′ ∈ (ℤ/Np^{m}ℤ)^{×} is the element such that a′ ≡ 1(N) and a′ ≡ a(p^{m}) for each a.

7.5.2. Let ϵ_{p} : G_{∞}^{(1)} ⟶ ℤ_{p}^{×} denote the character such that the restriction of κ to G_{∞}^{(1)} coincides with ϵ_{p} ∘ χ^{k−2}. Then ϵ_{p}(g_{a}^{(1)}) = ϵ_{f_{κ}}(a′) for any a ∈ (ℤ/p^{m}ℤ)^{×}, and a′ ∈ (ℤ/Np^{m}ℤ)^{×} is the element such that a′ ≡ a(p^{m}) and a′ ≡ 1(N). Clearly the image of (7.8) under ϵ_{p} : M[[G_{∞}^{(2)}]][G_{m}^{(1)}] ⟶ M[[G_{∞}^{(2)}]] is

p^{m−1}(p − 1) · α^{m} · i_{{r}}(_{c,d′}L_{p-adic}(f_{κ})^{±}).

As f_{κ} is ordinary, by Theorem 6.2, we know L_{p-adic}(f_{κ}) ∈ O_{M}[[G_{∞}^{(2)}]] ⊗_{O_{M}} M, and hence we have

i_{{r}}(_{c,d′}L_{p-adic}(f_{κ})^{±}) = _{c,d′}L_{p-adic}(f_{κ})^{±}(χ^{r}) ∈ O_{M}[[G_{∞}^{(2)}]] ⊗_{O_{M}} M.

7.5.3. By the commutative diagram

M_{k}(X_{1}(Np^{n}); ℤ_{(p)})[G_{m}^{(1)}] ⟶^{ϵ_{p}} M_{k}(X_{1}(Np^{n}); ℤ_{(p)}[ϵ_{p}])
↓^{pr}^{fκ}  ↓^{pr}^{fκ}
M[G_{m}^{(1)}]  ⟶^{ϵ_{p}}  M,

and (7.8), it follows that

pr_{f_{κ}}(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, χ^{r−1}))^{±·(−1)^{r}}= p^{m−1}(p − 1) · α^{m} ·_{c,d′}L_{p-adic}(f_{κ})^{±}(χ^{r})

∈ O_{M}[[G_{∞}^{(2)}]] ⊗_{O_{M}} M. (7.9)

Here we denote by pr_{f_{κ}}(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, χ^{r−1}))^{±·(−1)^{r}} the inverse limit

(lim_{\overleftarrow{n}} pr_{f_{κ}}(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p}∘χ^{k−2}, χ^{r−1})|_{n}))^{±·(−1)^{r}} with_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p}∘χ^{k−2}, χ^{r−1})|_{n} ∈

M_{k}(X_{1}(Np^{n}); ℤ_{(p)}[ϵ_{p}])[G_{n}^{(2)}]. Furthermore by (6.13), we have

c,d′z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, χ^{r−1})|_{n} ∈ M_{k}(X_{1}(Np^{n}), ϵ_{p}; ℤ_{(p)}[ϵ_{p}])[G_{n}^{(2)}], where M_{k}(X_{1}(Np^{n}), ϵ_{p}; ℤ_{(p)}[ϵ_{p}]) is the sub ℤ_{(p)}[ϵ_{p}]-space of M_{k}(X_{1}(Np^{n}); ℤ_{(p)}[ϵ_{p}]) on which G_{∞}^{(1)} acts via ϵ_{p} ∘ χ^{k−2}.

7.6. Now we study _{c,d′}L_{p-adic}^{ord,univ}.

7.6.1. The following composition of Λ-homomorphisms

𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⟶ p_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ)
⟶ P_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ) ⟶ H_{Np^{∞}}^{ord} ⊗_{Λ} Q(Λ)

induces the Λ-homomorphism

Hom_{Λ}(H_{Np^{∞}}^{ord}, Λ) ⟶ Hom_{Λ}(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}, \frac{1}{h′} Λ)  (7.10)

with some non-zerodivisor h′ ∈ Λ. It is easy to see that the composition of (7.10) and

Hom_{Λ}(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}, \frac{1}{h′} Λ) ⟶ (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ),

which is given by (7.3), coincides with the composition in (7.2). From this, we see (h′)^{−1} ⊂ (h)^{−1}. The Λ-homomorphism \overline{m}_{Λ}[[G_{∞}^{(2)}]] ⟶^{𝔦} Hom_{Λ}(H_{Np^{∞}}, Λ)[[G_{∞}^{(2)}]] and (7.10) give a Λ-homomorphism

\overline{m}_{Λ}[[G_{∞}^{(2)}]] ⟶ Hom_{Λ}(𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}, Λ[\frac{1}{h′}])[[G_{∞}^{(2)}]],

and this Λ-homomorphism induces

\overline{m}_{Λ/𝔓_{k−2,ϵp}} [[G_{∞}^{(2)}]]

⟶ Hom_{Λ/𝔓_{k−2,ϵp}} (𝔥_{Np^{∞}}^{ord}/(𝔓_{k−2,ϵ_{p}} + I_{Np^{∞}}^{ord}), Λ[\frac{1}{h′}]/𝔓_{k−2,ϵ_{p}})[[G_{∞}^{(2)}]], (7.11)

where \overline{m}_{Λ/𝔓_{k−2,ϵp}} denotes the image of \overline{m}_{Λ} under the map

\overline{M}_{Np^{∞}}[[G_{∞}^{(1)}]] ⟶^{ϵ_{p}∘χ^{k−2}} \overline{M}_{Np^{∞}} ⊗_{ℤ_{p}} ℤ_{p}[ϵ_{p}]. The map (7.11) is well-defined as κ(h) ≠ 0. We put L = Λ[1/h′]/𝔓_{k−2,ϵ_{p}}. Now the result of Hida [Hi2], Corollary 1.3 affirms that Hom_{Λ/𝔓_{k−2,ϵp}} (𝔥_{Np^{∞}}^{ord}/(𝔓_{k−2,ϵ_{p}} + I_{Np^{∞}}^{ord}), Λ[1/h′]/𝔓_{k−2,ϵ_{p}}) is contained in S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L), where S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L) is the sub L-space of e · M_{k}(X_{1}(Np^{m}), ϵ_{p}; L) with e = lim_{\overleftarrow{n}} T (p)^{n!}, consisting of cusp forms with

conductor divisible by N. Hence (7.11) induces a homomorphism

\overline{m}_{Λ/𝔓_{k−2,ϵp}} [[G_{∞}^{(2)}]] ⟶ S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L)[[G_{∞}^{(2)}]].  (7.12)

7.6.2. We define m_{Λ}[[G_{∞}^{(2)}]] = \overline{m}_{Λ}[[G_{∞}^{(2)}]] ∩ M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{\{k−2\},\{0,...,k−2\}}(⊂ \overline{M}_{Np^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]]) with \overline{m}_{Λ} in 7.1.1 and M[[G_{∞}^{(1)} × G_{∞}^{(2)}]]_{\{k−2\},\{0,...,k−2\}} in 6.1.4. We see that _{c,d′}z_{Np^{∞}}^{univ} ∈ m_{Λ}[[G_{∞}^{(2)}]].

We define m_{Λ/𝔓_{k−2,ϵp}} [[G_{∞}^{(2)}]] to be the image of of m_{Λ}[[G_{∞}^{(2)}]] under the map

\overline{M}_{Np^{∞}}[[G_{∞}^{(1)}×G_{∞}^{(2)}]] ⟶^{(ϵ_{p}∘χ^{k−2},id)} \overline{M}_{Np^{∞}}⊗_{ℤ_{p}}ℤ_{p}[ϵ_{p}][[G_{∞}^{(2)}]]. Then the map (7.12)

induces a map

𝔧 : m_{Λ/𝔓_{k−2,ϵp}} [[G_{∞}^{(2)}]] ⟶ S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L)[[G_{∞}^{(2)}]].  (7.13)

By definition, the following diagram is commutative for each n ≥ 1:

m_{Λ/𝔓_{k−2,ϵp}} [G_{n}^{(2)}] ⟶^{𝔧} S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L)[G_{n}^{(2)}]
↓^{pr}^{fκ}  ↓^{pr}^{fκ}
M  ⟶^{id}  M.

(7.14)

We write pr_{f_{κ}}(𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id))) for the inverse limit lim_{\overleftarrow{n}}(pr_{f_{κ}}(𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, χ^{r−1}))|_{n})). By the commutative diagram

(7.14), we obtain

pr_{f_{κ}}(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id)) = pr_{f_{κ}}(𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id))).

Hence by this and by (7.9), if we prove the assertion in 7.6.3 below, Theorem 7.3 follows.

7.6.3. The assertion in question is as follows. We have

pr_{f_{κ}}(𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id))) = _{c,d′}L_{p-adic}^{ord,univ}(κ).

We prove this assertion. As an element of the right hand side of (7.11), 𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id)) is the element

x ↦ Tr_{k,ϵ_{p}}(_{c,d′}L_{p-adic}^{ord,univ} · x) (x ∈ 𝔥_{Np^{∞}}^{ord}/(𝔓_{k−2,ϵ_{p}} + I_{Np^{∞}}^{ord})),

where Tr_{k,ϵ_{p}} : Hom_{Λ/𝔓_{k−2,ϵp}} (𝔥_{Np^{∞}}^{ord}/(𝔓_{k−2,ϵ_{p}} + I_{Np^{∞}}^{ord}), Λ/𝔓_{k−2,ϵ_{p}}) is the trace map as Λ/𝔓_{k−2,ϵ_{p}}-modules. By the result of Hida [Hi2], Corollary 1.3, we find that

Tr_{k,ϵ_{p}} = ∑_{κ′} f_{κ′},

where f_{κ′} runs over all ordinary p-stabilized newform ∈ S_{k}^{ord,N}(X_{1}(Np^{m}), ϵ_{p}; L) of tame conductor N attached to κ′ : 𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⟶ \overline{ℤ_{p}} satisfying κ′|_{G_{∞}^{(1)}} = ϵ_{p} ∘ χ^{k−2}. By the definition of _{c,d′}L_{p-adic}^{ord,univ}, we obtain

𝔧(_{c,d′}z_{Np^{∞}}^{univ}(ϵ_{p} ∘ χ^{k−2}, id)) = _{c,d′}L_{p-adic}^{ord,univ}(∑_{κ′} f_{κ′}) = ∑_{κ′} _{c,d′}L_{p-adic}^{ord,univ}(κ′)f_{κ′},

where in ∑_{κ′}, κ′ runs as above. This concludes the assertion.

Therefore Theorem 7.3 is proven.

7.7. We briefly explain how to obtain a two-variable p-adic zeta function attached to the universal family of cusp forms which can provide (−1)^{k+1}κ(⟨−1⟩)- part of L_{p-adic}(f_{κ}) for κ satisfying a certain condition, even though L(f_{κ}, k − 1) = 0. We use the notation in 6.3.7 – 6.3.12 and in 6.7. As before, let M be a positive integer which is prime to Np. We consider the image of z_{M,Np^{∞}} under the \overline{M}_{MNp^{∞}}-homomorphism which is similar to (6.10) and which is given as follows

\overline{M}_{MNp^{∞}}[[G_{Mp^{∞}}^{(1)} × G_{Mp^{∞}}^{(2)}]] ⟶ \overline{M}_{MNp^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]] ; (7.15)

x𝔤_{a}^{(2)}𝔤_{b}^{(1)} ↦ xϕ(ab^{2})g_{a}^{(2)}g_{b}^{(1)} (x ∈ \overline{M}_{MNp^{∞}}, 𝔤_{∗} ∈ G_{Mp^{∞}}, g_{∗} ∈ G_{∞}).

We write z_{M,Np^{∞},ϕ} for the image of z_{M,Np^{∞}} under the above map. Now by replacing N by NM in the definition of \overline{m}_{Λ}, we define \overline{m}_{Λ}. Then for this \overline{m}_{Λ}, the following which is similar to the result in Proposition 7.1.2 (1) holds. Namely we have

z_{M,Np^{∞},ϕ} ∈ \overline{m}_{Λ}[[G_{∞}^{(2)}]][\frac{1}{g′′}](⊂ \overline{M}_{NMp^{∞}}[[G_{∞}^{(1)} × G_{∞}^{(2)}]][\frac{1}{g′′}]),

where g′′ is the image of g′ in 6.3.7 under the homomorphism (7.15). We study the image of z_{M,Np^{∞},ϕ} under the map

\overline{m}_{Λ}[[G_{∞}^{(2)}]][\frac{1}{g′′}] ⟶ ((𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord}) ⊗_{Λ} Q(Λ))[[G_{∞}^{(2)}]][\frac{1}{g′′}]  (7.16)

which is defined as L_{N}. We denote this image by

L_{p-adic,M,ϕ}^{ord} ∈ (𝔥_{NMp^{∞}}^{ord}/I_{NMp^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{h′g′′}],

where h′ ∈ 𝔥_{NMp^{∞}}^{ord} is a certain non-zerodivisor. This function L_{p-adic,M,ϕ}^{ord} displays the property (7.17) below. For an N-primitive arithmetic point κ : 𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord} ⟶ \overline{ℤ_{p}}, let κ_{ϕ} : 𝔥_{NMp^{∞}}^{ord}/I_{NMp^{∞}}^{ord} ⟶ \overline{ℤ_{p}} be the NM-primitive arithmetic point characterized by κ_{ϕ}(T (n)) = κ(T (n))ϕ(n). We always have that κ_{ϕ}(g′′) ∈ \overline{ℤ_{p}}[[G_{∞}^{(2)}]] is a non-zero divisor. We assume that κ_{ϕ}(h′) ≠ 0. Then κ_{ϕ} induces the following homomorphism which is also denoted by κ_{ϕ} :

κ_{ϕ} : (𝔥_{Np^{∞}}^{ord}/I_{Np^{∞}}^{ord})[[G_{∞}^{(2)}]][\frac{1}{h′g′′}] ⟶ Q(\overline{ℤ_{p}}[[G_{∞}^{(2)}]]).

Now if L(f_{κ}, ϕ, k − 1) ≠ 0, we have L_{p-adic,M,ϕ}^{ord}(κ_{ϕ})^{±·ϕ(−1)}

= x · (( ∏_{\begin{matrix}l:prime \\ l|M \end{matrix}}(1 − a_{l}(𝔣)g_{l^{−1}}^{(2)} + ϵ_{𝔣}(l)l^{k−1}g_{l^{−2}}^{(2)})) · L_{p-adic}(f_{κ}))^{±·ϕ(−1)}(χ)

∈ O_{M(ϕ)}[[G_{∞}^{(2)}]] ⊗_{O_{M(ϕ)}} M(ϕ)

(7.17)

with some x ∈ M(ϕ)^{×}, and ± = (−1)^{k}ϵ_{𝔣}(−1). By using the arguments in 6.3.7 – 6.3.12 and 6.7, we can prove the above assertion in the same way as for Theorem 6.2. Furthermore, by using z_{N,p^{∞}}, we can produce another two-variable p-adic zeta function attached to universal family of ordinary cusp forms in the same manner. In this way, by taking various ϕ, we can construct two-variable p-adic zeta functions which can provide not only (−1)^{k}κ(⟨−1⟩)-part but also (−1)^{k+1}κ(⟨−1⟩)- part of L_{p-adic}(f_{κ}) for κ satisfying the condition that κ_{ϕ}(h′) ≠ 0, without assuming L(f_{κ}, k − 1) ≠ 0.

References

[AV] Amice, Y. and Vélu, J., Distributions p-adiques associées aux séries de Hecke, Astérisque 24-25(1975) 119–131.

[Co] Coleman, R., Division values in local fields, Invent. Math. 53 (1979) 91–116.

[CW] Coates, J. and Wiles, A., On p-adic L-functions and Elliptic Units, J. Austral. Math. Soc (Series A) 26 (1978) 1–25.

[Fu1] Fukaya, T., The theory of Coleman power series for K_{2}, Journal of Alg. Geometry 12 (2003) 1–80.

[Fu2] Fukaya, T., The theory of Coleman power series for K_{2}, Proceedings from Tsudajuku University in Japanese 21 (2001) 75-85.

[Fu3] Fukaya, T., K_{2}-version of Coleman power series and p-adic zeta functions of modular forms, Shurikaisekikenkyusho Kokyuroku (in Japanese) from RIMS (Kyoto University) 1200 (2001) 48-59.

[GS] Greenberg, R. and Stevens, G., p-adic L functions and p-adic periods of modular forms, Invent. Math. 111 (1993) 407–447.

[Hi1] Hida, H., Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. 19 (1986) 231–273.

[Hi2] Hida, H., Galois representations into GL_{2}(ℤ_{p}[[X]]) attached to ordinary cusp forms, Invent. Math. 85 (1986) 545–613.

[Hi3] Hida, H., Elementary theory of L-functions and Eisenstein series, London Math. Soc. Student Texts 26, Cambridge Univ. Press (1993).

[Iw] Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964) 42–82.

[Ka1] Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil L-functions via B_{dR}, Lecture Notes in Math. 1553, Springer (1993) 50–163.

[Ka2] Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, preprint.

[Katz1] Katz N.M., p-adic interpolation of real analytic Eisenstein series, Ann. of Math., 104 (1976) 459–571.

[Katz2] Katz N.M., Higher congruences between modular forms, Ann. of Math., 101 (1975) 332–367.

[Ki] Kitagawa, K., On standard p-adic L-functions of families of elliptic cusp forms, Contemp. Math., 165 (1991) 81–110.

[KL] Kubota, T. and Leopoldt, H.W, Eine p-adische Theorie der Zetawerte, J. Reine Angew. Math. 214/215 (1964), 328–339.

[Mi] Milnor, J., Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970) 318–344.

[Oc] Ochiai, T., Doctoral Thesis, University of Tokyo (2001).

[Pa1] Panchishkin On the Siegel-Eisenstein measure and its applications, Israel Journal of Mathematics 120 B (2000), 467–509.

[Pa2] Panchishkin A new method of constructing p-adic L-functions associated with modular forms, Moscow Mathematical Journal 2 (2001) 1-16.

[Qu] Quillen, D., Higher algebraic K-theory I, Lecture Notes in Math. 342, Springer (1973) 179–198.

[Sc] Scholl, J., An introduction to Kato’s Euler systems, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press (1998) 379–460.

[Sh] Shimura, G. The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) 783–804.

[Vi] Vishik, M., M., Non-archimedean measures connected with Dirichlet series, Math. USSR, Sbornik 28 (1976) 216–228.

Takako Fukaya Keio University 4-1-1 Hiyoshi, Kohoku-ku Yokohama, 223-8521 Japan takakof@hc.cc.keio.ac.jp