Several Variables p-Adic L-Functions

for Hida Families of Hilbert Modular Forms

Tadashi Ochiai^{1}

Received: May 8, 2011 Revised: June 4, 2012

Communicated by Peter Schneider

Abstract. After formulating Conjecture A for p-adic L-functions defined over ordinary Hilbert modular Hida deformations on a totally real field F of degree d, we construct two p-adic L-functions of d + 1-variable depending on the parity of weight as a partial result on Conjecture A. We will also state Conjecture B which is a corollary of Conjecture A but is important by itself. Main issues of the construction are the study of Hida theory of Hilbert modular forms by using Hilbert modular varieties (without using Shimura curves), the study of higher dimensional modular symbols on Hilbert modular varieties and delicate treatments on archimedean and p-adic periods.

2010 Mathematics Subject Classification: 11R23, 11F41, 14G35, 11F67 Keywords and Phrases: p-adic L-function, Hilbert modular forms, Hida theory, Iwasawa theory, Modular symbol

Contents

1. Introduction  808
1.1. General overview.  808
1.2. Main results and technical difficulty on the work  815
1.3. List of notations.  818
2. Mellin transform of Hilbert modular forms and Hecke L-functions. 819
2.1. Automorphic forms on GL_{2}.  819
2.2. Hecke operators.  820
2.3. Nearly ordinary modular forms.  820
2.4. Hilbert modular varieties and standard local systems  821
2.5. Standard q-expansion and L-function.  823

1The author is partially supported by Grant-in-Aid for Young Scientists (B), No.20740013, Japan Society for the Promotion of Science.

2.6. Eichler-Shimura map  825
2.7. Mellin transform  825
3. Modular symbol cycle  826
3.1. Cohomological interpretation of special values  828
4. Hida families of Hilbert modular forms.  830
4.1. Various level structures.  831
4.2. Control theorem for nearly ordinary cohomology.  834
4.3. Freeness result for nearly ordinary cohomology.  835
4.4. p-adic periods  837
5. p-adic L-function  840
5.1. Statement of the main theorem (Theorem A)  840
5.2. Construction of L_{p}(R; b^{even}) and L_{p}(R; b^{odd})  841
5.3. Proof of the distribution property of Theorem A  841
5.4. Proof of the interpolation property of Theorem A  845
References  847

1. Introduction

1.1. General overview. Let p be an odd prime number fixed throughout the paper. In the spirit of Iwasawa theory, our interest is in constructing and studying p-adic analytic L-functions which interpolate special values of Hecke L-functions of various automorphic forms (or Hasse-Weil L-functions of various motives). According to the philosophy of Iwasawa Main Conjecture, the padic analytic L-function is expected to coincide with its algebraic counterpart encoding the behaviour of generalized class groups or Selmer groups. In the classical situation, we fix an automorphic form f for a certain algebraic group G on a number field F and we study the p-adic analytic L-function L_{p}(f) interpolating special values of the Hecke L-function L(f, ϕ, s) twisted by Hecke characters ϕ on F of p-power order and p-power conductor. Note that, by the class field theory, Hecke characters as above are identified with characters of the Galois group Gal(F_{{p}}/F) where F_{{p}} is the maximal abelian pro-p extension of F unramified outside primes above p. The most fundamental case is the case of a totally real number field F, in which case Leopoldt conjecture predicts that F_{{p}} is almost equal to the cyclotomic ℤ_{p}-extension F_{∞} of F. Thus, L_{p}(f) is an element of the cyclotomic Iwasawa algebra O[[Gal(F_{∞}/F)]] over the ring of integers O of a finite extension of ℚ_{p}, which is non-canonically isomorphic to a power series in one variable O[[T]]. When F = ℚ and G = 𝔾_{m}, the p-adic L-function L_{p}(f) ∈ O[[Gal(ℚ_{∞}/ ℚ)]] of an automorphic form on G (that is a Hecke character of F) is constructed by Kubota-Leopoldt, Iwasawa and Coleman. When F = ℚ and G = GL_{2}, L_{p}(f) ∈ O[[Gal(ℚ_{∞} / ℚ)]] is constructed by Mazur-Tate-Teitelbaum [MTT]. There are some known work of p-adic L-functions over the cyclotomic Iwasawa algebra O[[Gal(F_{∞}/F)]] for other algebraic groups G of higher rank. We do not try to give a list of previous work on such constructions.

Several Variables p-Adic L-Functions … 809

We are interested in a vast generalization of Iwasawa theory from the theory over the cyclotomic Iwasawa algebra to the theory over the whole algebra Rn.o ; \overline{ρ} of nearly ordinary Galois deformations (with suitable local conditions) of a given mod p representation \overline{ρ} (see [G] and [Oc2] for the project of an Iwasawa theory over deformation algebras). We are particularly interested in constructing p-adic analytic L-functions L_{p}(\overline{ρ}) ∈ Rn.o ; \overline{ρ}. Let ρ_{f,p} be a modular p-adic representation lifting \overline{ρ}. By the universal property, Rn.o ; \overline{ρ} parameterizes in particular twists of ρ_{f,p} by Hecke characters on F of p-power order and ppower conductor. Hence, there is a surjection Rn.o ; \overline{ρ} ↠ O[[Gal(F_{∞}/F)]] and we expect that the specialization of L_{p}(\overline{ρ}) ∈ Rn.o ; \overline{ρ} to be the classical cyclotomic p-adic L-function L_{p}(f) ∈ O[[Gal(F_{∞}/F)]]. In this way, our project is really a generalization which contains previous work. When a given mod p representation \overline{ρ} of Gal(\overline{F} /F) has rank one, Rn.o ; \overline{ρ} is isomorphic to O[[Gal(F_{∞}/F)]]. Hence, in this case, the theory is the same as the classical cyclotomic theory. The first important new case arises when rank(\overline{ρ}) = 2. In this case, the Krull dimension of Rn.o ; \overline{ρ} is greater than that of O[[Gal(F_{∞}/F)]]. We have a natural surjection from Rn.o ; \overline{ρ} to the (\overline{ρ}-component of) Hida’s nearly ordinary Hecke algebra 𝕋n.o ; \overline{ρ} and this surjection is conjectured to be an isomorphism. Up to now, we took the viewpoint of Galois deformation rings since we believe that this is the appropriate framework for construction of more general p-adic L-functions. However, from now on, we will try to define the p-adic analytic L-function L_{p}(\overline{ρ}) in 𝕋n.o ; \overline{ρ} rather than in Rn.o ; \overline{ρ} since 𝕋n.o ; \overline{ρ} is more closely related to the L-values. For the rest of the introduction, we fix a rank-two mod p representation \overline{ρ} of Gal(\overline{F} /F) which is modular and nearly ordinary. When F = ℚ, there is a canonical isomorphism:

𝕋n.o ; \overline{ρ} ≅ 𝕋ord ; \overline{ρ} ˆ⊗_{O}O[[Gal(ℚ_{∞} / ℚ)]] = 𝕋ord ; \overline{ρ}[[Gal(ℚ_{∞} / ℚ)]]

where 𝕋ord ; \overline{ρ} is (the \overline{ρ}-component of) nearly ordinary Hecke algebra, which is finite and flat over O[[1 + p ℤ_{p}]]. The algebras 𝕋ord ; \overline{ρ} and 𝕋n.o ; \overline{ρ} depend on a certain tame conductor N. However, if there is no confusion, we will omit N in the notations. By Hida’s theory, for every p-stabilized elliptic (nearly) ordinary eigen cuspform f of weight k ≥ 2 and conductor Np^{∗} (∗ ∈ ℕ) such that the residual representation \overline{ρ}_{f} associated to ρ_{f} is isomorphic to \overline{ρ}, there exists a unique κ = κ_{f} : 𝕋ord ; \overline{ρ} ⟶ \overline{ℚ}_{p} such that the q-expansion of f equals ∑_{n}κ_{f}(T_{n})q^{n}, where T_{n} is the n-th Hecke operator in 𝕋ord ; \overline{ρ}. Hence, for every

pair (f, ϕ) of ordinary p-stabilized eigen cuspform f as above and a character ϕ of Gal(ℚ_{∞} / ℚ), there is an unique κ = κ_{f,ϕ} : 𝕋n.o ; \overline{ρ} ⟶ \overline{ℚ}_{p} such that the q-expansion of f ⊗ ϕ equals ∑_{n}κ_{f,ϕ}(T_{n})q^{n}. The algebra 𝕋ord ; \overline{ρ} is a local algebra

which might contain zero divisors in general. A quotient R = 𝕋ord ; \overline{ρ}/𝔄 by a prime ideal 𝔄 of height zero is called a branch of 𝕋ord ; \overline{ρ}. A branch of 𝕋n.o ; \overline{ρ} is defined exactly in the same way. For a branch R of 𝕋ord ; \overline{ρ}, Rˆ⊗_{O}O[[Gal(ℚ_{∞} / ℚ)]] is a branch of 𝕋n.o ; \overline{ρ} and this correspondence gives a bijection between the set of branches of 𝕋ord ; \overline{ρ} and the set of branches of 𝕋n.o ; \overline{ρ}. The p-adic L-function L_{p}(f) ∈ O[[Gal(ℚ_{∞} / ℚ)]] of an elliptic modular form f has been constructed in the 70’s and extensively studied by the method of modular symbols. Kitagawa refined Mazur’s method of Λ-adic modular symbols which is a family of modular symbols associated to Hida family. We recall the following theorem of [Ki].

Theorem 1.1 (Kitagawa-Mazur). Let R^{ord} be a branch of 𝕋ord ; \overline{ρ}. Assume that R^{ord} is a Gorenstein ring. Then, there exists L_{p}(R^{ord}) ∈ R^{ord}[[Gal(ℚ_{∞} / ℚ)]] which is characterized by the interpolation property:

\frac{κ_{f,χ^{j−1}ϕ}(L_{p}(R^{ord}))}{C_{f,p}} = (j − 1)!G(ϕ, j)A_{p}(f)\frac{L(f, ϕω^{1−j}, j)}{(−2π\sqrt{−1})^{j−1}C_{f,∞}}

for every pair (f, χ^{j−1}ϕ) as follows:

• The form f is a p-stabilized ordinary eigen cuspform f ∈ S_{k}(Γ_{1}(Np^{∗})) satisfying \overline{ρ}_{f} ≅ \overline{ρ} such that κ_{f} : 𝕋ord ; \overline{ρ} ⟶ \overline{ℚ}_{p} factors through R.

• The character χ^{j−1}ϕ consists of a finite character ϕ of Gal(ℚ_{∞} / ℚ) and an integer j such that 1 ≤ j ≤ k − 1.

Here, the notation in the above equation is as follows. The element C_{p,f} ∈ \overline{ℤ}_{p}^{×}/\overline{ℤ}_{(p)}^{×} is a p-adic period for f and C_{f,∞} ∈ ℂ^{×}/\overline{ℤ}_{(p)}^{×} is a complex period for f where \overline{ℤ}_{p} (resp. \overline{ℤ}) is the ring of integers of \overline{ℚ}_{p} (resp. \overline{ℚ}) and \overline{ℤ}_{(p)} is the localization of \overline{ℤ} at the valuation induced by a fixed embedding \overline{ℚ} ↪ \overline{ℚ}_{p}. The symbol G(ϕ, j) is the Gauss sum for the Dirichlet character (ϕω^{a+1−j})^{−1} where a is an integer such that \overline{ρ} ⊗ ω^{−a} is ordinary. The term A_{p}(f) is given by

A_{p}(f) = \begin{cases}(1 − \frac{p^{j−1}}{a_{p}(f ⊗ ω^{−a})}) & if ϕω^{a+1−j} is trivial, \\ (\frac{p^{j−1}}{a_{p}(f ⊗ ω^{−a})})^{c(ϕ,j)} & if ϕω^{a+1−j} has conductor p^{c(ϕ,j)}. \end{cases}

By abuse of notation, a_{p}(f ⊗ω^{−a}) is the Hecke eigenvalue at p of the p-stabilized newform associated to f ⊗ ω^{−a}.

Remark 1.2. Kitagawa [Ki] constructs the p-adic L-functions on each branch 𝕋ord ; \overline{ρ}/𝔄 assuming that 𝕋ord ; \overline{ρ}/𝔄 is Gorenstein. As explained above, we believe that the construction on the whole 𝕋ord ; \overline{ρ} is more universal. It seems to us that such a universal construction will be possible if we assume that 𝕋ord ; \overline{ρ} is Gorenstein as well as some conditions.

For other results related to Theorem 1.1, we give the following remark:

Remark 1.3. (1) Results similar to that of Theorem 1.1 are obtained using a similar method of Λ-adic modular symbol by Greenberg-Stevens ([GS]) and by Ohta (unpublished). Later was found another construction based on the method of Rankin-Selberg and it has been generalized to obtain results similar to Kitagawa’s result by Fukaya[Fu], Ochiai[Oc1] and Panchishkin[P2]. (Panchishkin’s construction is done for a p-adic family of positive slope, but the same construction works for a Hida family)

(2) The p-adic L-functions obtained at [GS], [Fu], [Oc1] and [P2] are weaker than the one by Kitagawa since they are sometimes defined only at a localization of a branch R but not on the whole of R. In [Fu], [Oc1] and [P2], the complex periods C_{f,∞} cannot be optimally normalized and are defined at ℂ^{×}/\overline{ℚ}^{×}. In [GS], the analogue of C_{f,p} is not a padic unit as in the Kitagawa’s one and we only know that C_{f,p} are non zero except finitely many f’s.

We fix a totally real number field F with d := [F : ℚ] > 1 throughout the paper. We would like to introduce the nearly ordinary Hecke algebra 𝕋n.o ; \overline{ρ} over a general totally real field F of degree d > 1 and construct a p-adic Lfunction which generalizes Theorem 1.1 (Everything which will be discussed in this paper works for d = 1 except that we omit the condition of holomorphy at infinity in Definition 2.1 which is always valid for d > 1 thanks to the Kocher princicple).

Remark 1.4. In order to work in the setting of Hilbert modular forms, it is more efficient to switch to the terminology of nearly ordinary modular forms rather than ordinary modular forms twisted by characters. Let I_{F} be the set of embeddings of F into ℝ. It will also be more efficient to cinsider modular forms with double-digit weight w = (w_{1}, w_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}] as explained in [H6]. When F = ℚ, a cusp form of weight k in the usual sense is of weight (0, k − 1). For an ordinary elliptic cusp form of weight (0, k − 1), the twist by a Hecke character of weight j is a nearly ordinary cuspform of weight (−j, k − 1 − j). Conversely, every elliptic cuspform of weight (−j, k − 1 − j) which is nearly ordinary at p is obtained as a twist of an ordinary cuspform of weight (0, k −1). For general totally real fields F, the situation is a bit more complicated. Since all global Hecke character on totally real fields coincide with an integral power of the Norm character multiplied by a finite order global Hecke character, a cuspform of weight (w_{1}, w_{2}) and a cuspform of weight (w′_{1}, w′_{2}) nearly ordinary at p are a twist of the other only when there exists an integer j such that w_{1} − w′_{1} = w_{2} − w′_{2} = jt. From now on, we will switch to the notation of nearly ordinary forms with double-digit weight.

We denote by F̃_{∞} the composition of all ℤ_{p}-extensions of F. Note that Leopoldt’s conjecture predicts that F̃_{∞} = F_{∞}. We fix a residual modular Galois representation \overline{ρ} and an integral ideal 𝔫 of F which is prime to p. We note that, over a general totally real field F of degree d > 1, there are two different approaches to introduce the (\overline{ρ}-component of) nearly ordinary Hecke algebra 𝕋^{n.o} of tame conductor 𝔫. The original approach due to Hida ([H1], [H2], [H3]) uses a quarternion algebra B over F which is unramified at all finite primes of F. He chooses a quarternion algebra B such that Shimura varieties Y_{K_{1}(𝔫)∩K_{11}(p^{n})}^{B} associated to K_{1}(𝔫) ∩ K_{11}(p^{n}) (see Section 2.1 for the definition of K_{1}(𝔫) and K_{11}(p^{n})) are of dimension e = 1 (resp. e = 0) when [F : ℚ] is odd (resp. even). The nearly ordinary part H_{n.o}^{B}(w_{1}, w_{2}) of the limit \underleftarrow{lim}_{n} H^{e}(Y_{K_{1}(𝔫)∩K_{11}(p^{n})}^{B}; L(w_{1}, w_{2}; O)) for a certain local system L(w_{1}, w_{2}; O) is a finitely generated free module over a certain Iwasawa algebra Λ_{n.o} (see Definition 4.2). The nearly ordinary Hecke algebra in this context 𝕋_{B}^{n.o} is defined to be a Λ_{n.o}-subalgebra in End_{Λ_{n.o}}(H_{n.o}^{B}(w_{1}, w_{2})) generated by Hecke operators. Let \overline{ρ} be a mod p representation of F. In general, for a local domain R whose residue field R/𝔐_{R} is a finite field of characteristic p and an R-module M with R-linear action of Hecke correspondence, the \overline{ρ}-part M\overline{ρ} is defined by

(1) M\overline{ρ} = \{x ∈ M | T_{λ}x ≡ Tr(\overline{ρ}(Frob_{λ}))x mod 𝔐_{R}M for every λ ∤ 𝔫 p\},

where T_{λ} is the Hecke correspondence at the prime λ and \overline{ρ}(Frob_{λ}) means the action on the inertia fixed part when λ divides 𝔫. Then, \overline{ρ}-component (𝕋_{B}^{n.o})\overline{ρ} is defined to be a Λ_{n.o}-subalgebra in End_{Λ_{n.o}}(H_{n.o}^{B}(w_{1}, w_{2})\overline{ρ}) generated by Hecke operators. The algebra (𝕋_{B}^{n.o})\overline{ρ} is one of the local components of the semi-local Λ_{n.o}-algebra 𝕋_{B}^{n.o}. The second approach to introduce the (\overline{ρ}-component of) nearly ordinary Hecke algebra, which is essential to our work, is the one which uses Hilbert modular varieties. Taking the matrix algebra M_{2}(F) over F in place of B, the Shimura varieties Y_{K_{1}(𝔫)∩K_{11}(p^{n})}^{M_{2}(F)} are so called Hilbert modular vari-

eties, which are of dimension d. The nearly ordinary part H_{n.o}^{M_{2}(F)}(w_{1}, w_{2}) of the limit \underleftarrow{lim}_{n} H^{d}(Y_{K_{1}(𝔫)∩K_{11}(p^{n})}^{M_{2}(F)}; L(w_{1}, w_{2}; O)) for the standard local system L(w_{1}, w_{2}; O) (cf. §2.4) is not known to be a finitely generated free module over Λ_{n.o} (see Definition 4.2). However, we will prove later in this article, that H_{n.o}^{M_{2}(F)}(w_{1}, w_{2})\overline{ρ} for a suitable mod p representation \overline{ρ} is a finitely generated free module over Λ_{n.o}. Hence, (𝕋_{M_{2}(F)}^{n.o})\overline{ρ} is also defined to be a Λ_{n.o}-subalgebra

in End_{Λ_{n.o}}(H_{n.o}^{M_{2}(F)}(w_{1}, w_{2})\overline{ρ}) generated by Hecke operators. See Section 4 for the precise result.

Remark 1.5. In this way, for a mod p representation \overline{ρ} satisfying suitable conditions, there are two different definitions of nearly ordinary Hecke algebras (𝕋_{B}^{n.o})\overline{ρ} and (𝕋_{M_{2}(F)}^{n.o})\overline{ρ} which are both local rings finite and torsion free over Λ_{n.o}. There should be a comparison between these two different definitions, but it seems unknown at the moment. However, in order to discuss modular symbol method, it is necessary to work on Hilbert modular varieties Y_{K_{1}(𝔫)∩K_{11}(p^{n})}^{M_{2}(F)} which are non-compact and have cusps at infinity. Thus, we only take the second approach in this paper.

From now on through the introduction, we assume that our nearly oridinary modular mod p representation \overline{ρ} satisfies a certain condition so that (𝕋_{B}^{ord})\overline{ρ} and (𝕋_{M_{2}(F)}^{n.o})\overline{ρ} are defined and (the \overline{ρ}-component of) the nearly ordinary (resp. ordinary) Hecke algebra 𝕋n.o ; \overline{ρ} (resp. 𝕋ord ; \overline{ρ}) means always the one obtained by the second approach. As is shown in Theorem 4.15 later, for every p-stabilized Hilbert nearly ordinary eigen cuspform φ of cohomological weight (w_{1}, w_{2}), level 𝔫 p^{∗} satisfying \overline{ρ}_{φ} ≅ \overline{ρ}, there exists a unique algebra homomorphism κ = κ_{φ} : 𝕋n.o ; \overline{ρ} ⟶ \overline{ℚ}_{p} such that the q-expansion of φ equals to ∑_{𝔞}κ_{φ}(T_{𝔞})q^{𝔞} where T_{𝔞} is the 𝔞-th Hecke operator

in 𝕋n.o ; \overline{ρ} for each integral ideal 𝔞 of F. The algebra 𝕋n.o ; \overline{ρ} is finite and torsion free over O[[(1 + p ℤ_{p})^{d+1+δ_{F,p}}]], where δ_{F,p} is the Leopoldt defect (conjectured to be zero by Leopoldt conjecture). For each prime ideal 𝔄 of height zero on 𝕋n.o ; \overline{ρ}, R = 𝕋n.o ; \overline{ρ}/𝔄 is called a branch of 𝕋n.o ; \overline{ρ} as is the previous page. Every arithmetic specialization κ : 𝕋n.o ; \overline{ρ} ⟶ \overline{ℚ}_{p} factors through a unique branch R in which case we say that the arithmetic specialization κ is on the branch R of 𝕋n.o ; \overline{ρ}. A double-digit weight (w_{1}, w_{2}) is called critical if we have the inequality w_{1,τ} < 0 ≤ w_{2,τ} for every τ ∈ I_{F}, where w_{i,τ} ∈ ℤ is the coefficient given by w_{i} =

∑_{τ∈I_{F}} w_{i,τ} τ. Put t = ∑_{τ∈I_{F}} τ ∈ ℤ[I_{F}] where I_{F} = {τ : F ↪ ℝ} is the set of

embeddings. A double-digit weight (w_{1}, w_{2}) is called a cohomological doubledigit weight if w_{2} − w_{1} ≥ t and w_{1} + w_{2} ∈ ℤt. As a dictionary in this paper, we recall that k = w_{2} − w_{1} + t plays a role of the weight of modular form in the classical sense. We propose the following conjecture:

Conjecture A . Let R be a branch of 𝕋n.o ; \overline{ρ}. There exists an element L_{p}(R) ∈ R which satisfies the interpolation property:

\frac{κ_{φ}(L_{p}(R))}{C_{φ,p}} = ∏_{𝔭 |(p)} A_{𝔭}(φ)\frac{L(φ, 0)}{C_{φ,∞}},

for every p-stabilized nearly ordinary eigen cuspform φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔫p^{∗})) of critical cohomological double-digit weight w = (w_{1}, w_{2}) on R, where the term A_{𝔭}(φ) is defined as follows:

(2)  A_{𝔭}(φ) = \begin{cases}(1 − \frac{1}{N_{F/ℚ}(𝔭)a_{𝔭}(φ)}) & if a_{𝔭}(φ) ≠ 0, \\ (\frac{1}{N_{F} (𝔭)a_{𝔭}(φ^{0})})^{ord_{𝔭} Cond(ϕ_{0})} & if a_{𝔭}(φ) = 0, \end{cases}

where φ^{0} is the nearly ordinary form of weight (w_{1}, w_{2}) which is of minimal conductor among twists of φ by finite order Hecke characters of F with pprimary conductor and ϕ_{0} is the unique finite order character of Gal(F̃_{∞}/F) such that φ = φ^{0} ⊗ ϕ_{0}. The number N_{F} (𝔭) is the absolute norm of the prime ideal 𝔭. Further, C_{φ,p} ∈ \overline{ℤ}_{p}^{×}/\overline{ℤ}_{(p)}^{×} and C_{φ,∞} ∈ ℂ^{×}/\overline{ℤ}_{(p)}^{×} are a p-adic period and a complex period for φ (cf. Remark 1.6 (1)). For any ordinary eigen cuspform φ of critical double-digit weight (w_{1}, w_{2}) and for any nearly ordinary eigen cuspform φ′ of critical double-digit weight (w_{1} − r, w_{2} − r) respectively satisfying φ′ = φ ⊗ N_{F}^{r} ϕ with the norm character N_{F} of F and a finite Hecke character ϕ, we have

C_{φ,p,v} = C_{φ′,p,v′} (3)

C_{φ,∞,v} = C_{φ′,∞,v′} \frac{G(ϕ^{−1})}{(−2π\sqrt{−1})^{dr}} ∏_{τ∈I_{F}} \frac{Γ(−w′_{1,τ})}{Γ(−w_{1,τ})} (4)

where G(ϕ) is the Gauss sum for ϕ defined in Definition 3.6 and Γ(s) is the Gamma function.

Remark 1.6. (1) The complex period C_{φ,∞} is a usual motivic complex period defined via the comparison isomorphism between the de Rham realization and Betti realization. On the other hand, the p-adic period C_{φ,p} which appears here is not expected to be a motivic p-adic period obtained via the comparison theorem. Though it is not motivic, we call C_{φ,p} a p-adic period following Greenberg. Since motivic p-adic periods will probably transcendental over ℚ_{p} there should be some modifications if we state Conjecture A using motivic p-adic periods. The author has some speculations on these possible different formulations of Conjectures depending the choices of p-adic periods, but it will be discussed elsewhere.

(2) Though we do not have a canonical lift of C_{φ,p} (resp. C_{φ,∞}) to \overline{ℤ}_{p}^{×} (resp. ℂ^{×}), the ratio “C_{φ,p}/C_{φ,∞}” should be well-defined. (cf. Definition 4.16, Remark 4.17 )

(3) When F = ℚ and when R is a Gorenstein ring, Conjecture A is equivalent to Theorem 1.1.

(4) We expect that a more general conjecture by replacing R by the whole local component of the Hecke algebra 𝕋n.o ; \overline{ρ} should be true.

The ordinary Hecke algebra 𝕋ord ; \overline{ρ} is finite and torsion-free over O[[(1 + p ℤ_{p})^{1+δ_{F,p}}]]. Since d + 1 + δ_{F,p} > 2 + δ_{F,p}, the natural surjection 𝕋n.o ; \overline{ρ} ↠ 𝕋ord ; \overline{ρ}[[Cl_{F}^{+}(p^{∞})_{p}]] cannot be an isomorphism by looking at the Krull dimensions. As is also shown in Theorem 4.15 later, for every p-stabilized Hilbert ordinary eigen cuspform φ of parallel weight k ≥ 2 and of level Np^{∗} satisfying \overline{ρ}_{φ} ≅ \overline{ρ}, there exists an unique algebra homomorphism κ = κ_{φ} : 𝕋ord ; \overline{ρ} ⟶ \overline{ℚ}_{p} such that the q-expansion of φ equals ∑_{𝔞}κ_{φ}(T_{𝔞})q^{𝔞}. We propose another conjecture:

Conjecture B . Let R^{ord} be a branch of 𝕋ord ; \overline{ρ}. There exists L_{p}(R^{ord}) ∈ R^{ord}[[Cl_{F}^{+}(p^{∞})_{p}]] which has the interpolation property:

\frac{κ_{φ}(L_{p}(R^{ord}))}{C_{φ,p}} = ∏_{𝔭 |(p)} A_{𝔭}(φ)\frac{L(φ, 0)}{C_{φ,∞}}

Documenta Mathematica 17 (2012) 807–849

for every p-stabilized ordinary eigen cuspform of critical parallel weight k ≥ 2 and of level 𝔫p^{∗} on R^{ord}. Here, C_{φ,p} ∈ \overline{ℤ}_{p}^{×}/\overline{ℤ}_{(p)}^{×} and C_{φ,∞} ∈ ℂ^{×}/\overline{ℤ}_{(p)}^{×} are a p-adic period and a complex period for φ satisfying the same relation as (2) and (3) of Conjecture A.

Since, for every R^{ord}, R^{ord}[[Gal(F_{∞}/F)]] is a quotient of some branch R of 𝕋n.o ; \overline{ρ}, it is clear that Conjecture B is an immediate corollary of Conjecture A. We remark that Mok [Mo] constructs a two-variable p-adic L-function related to Conjecture B by the method of Rankin-Selberg and he obtains an application to the problem of trivial zeros of the one-variable p-adic L-functions of Hilbert modular forms. However, his construction is slightly weaker than the p-adic L-function of Conjecture B since his complex periods C_{φ,∞} are Rankin-Selberg type which are different from p-optimal modular symbol type periods required in Conjecture B. In fact, with Rankin-Selberg type period, the p-valuation of the special values which appear in the interpolation formula might not match with the value expected by generalized Birch and Swinneton-Dyer conjecture. It seems difficult to improve this point of the method of Rankin-Selberg (see Remark 1.3. 3. for a similar problem). Also, it is not clear if Mok’s method works for Conjecture A.

1.2. Main results and technical difficulty on the work. In 1976, Manin [M2] generalized the method of modular symbol on modular curves in the setting of Hilbert modular varieties and constructed the cyclotomic p-adic L-function L_{p}(φ) ∈ O[[Gal(F_{∞}/F)]] of a Hilbert modular form φ. In this paper, we prove a weaker version of Conjecture A by generalizing the method of Λ-adic modular symbols on Hilbert modular varieties (see Theorem 5.1 for the precise statement of Theorem A below). In order to state the result, we introduce the following conditions (Van\overline{ρ}) and (Ir\overline{ρ}) for our fixed \overline{ρ} and 𝔑. (Van\overline{ρ}) The module H^{i}(Y_{K_{1}(𝔫)∩K_{11}(p^{n})}, L(w_{1}, w_{2}, O))\overline{ρ} vanishes for any i ≠ d,

any n ∈ ℕ and any cohomological double-digit weight (w_{1}, w_{2}), where Y_{K_{1}(𝔫)∩K_{11}(p^{n})} is a Hilbert modular variety of level K_{1}(𝔫) ∩ K_{11}(p^{n}).

The groups K_{1}(𝔫) and K_{11}(p^{n}) above are defined at §2.1. See (1) for the definition of \overline{ρ}-part H^{i}(Y_{K_{1}(𝔫)∩K_{11}(p^{n})}, L(w_{1}, w_{2}, O))\overline{ρ}.

(Ir\overline{ρ}) The [F : ℚ]-th power mod p Galois representation \overline{ρ}^{⊗[F:ℚ]} of Gal(\overline{F} /F)

is irreducible. The main result of this article is as follows:

Theorem A . Assume that the conditions (Van\overline{ρ}) and (Ir\overline{ρ}) are satisfied for our fixed \overline{ρ} and 𝔫. Let R be a branch of 𝕋n.o ; \overline{ρ}. Assume that R is a Gorenstein algebra. Then, there exists a p-adic L-function L_{p}^{even}(R) ∈ R (resp. L_{p}^{odd}(R) ∈ R) which satisfies the interpolation property

\frac{κ_{φ}(L_{p}^{∗}(R))}{C_{φ,p}} = ∏_{𝔭 |(p)} A_{𝔭}(φ)\frac{L(φ, 0)}{C_{φ,∞}} (∗ = even (resp. odd)),

for every p-stabilized nearly ordinary eigen cuspform φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔫 p^{∗})) of critical cohomological weight w = (w_{1}, w_{2}) on R such that w_{1} + w_{2} = rt are multiples of t = ∑_{τ∈I_{F}} τ by even (resp. odd) integers r. Here, all terms in the

above interpolation are the same as explained in Conjecture A.

Remark 1.7. (1) Since w_{1} + w_{2} is a multiple of t for every critical cohomological weight w = (w_{1}, w_{2}), each of L_{p}^{even}(R) and L_{p}^{odd}(R) satisfy the half of the desired interpolation property. Hence, Conjecture A is true if L_{p}^{even}(R) is equal to L_{p}^{odd}(R) as an element of R. Though we prove only the half of Conjecture A, a phenomenon of such a partial interpolation and such a theorem were quite new as far as we know. We believe that our work shed a new light on this area.

(2) The assumption that R is Gorenstein might not be satisfied in general. When F = ℚ, there are some known explicit Eisenstein local components which are not Gorenstein. If R is not Gorenstein, we have some technical difficulties to construct the p-adic L-function in R. Probably, we can construct L_{p}(R) only as an element in a localization of R and we have a similar interpolation only by means p-adic periods C_{φ,p} which are not necessarily p-adic units.

Theorem A immediately implies a corollary which gives two p-adic L-functions for the interpolation predicted by Conjecture B depending on the parity of weight. However, we will prove in [DO] the following full interpolation result which is stronger than the corollary of Theorem A.

Theorem B . Assume that the conditions (Van\overline{ρ}) and (Ir\overline{ρ}) are satisfied for our fixed \overline{ρ} and 𝔫. Let R^{ord} be a branch of 𝕋ord ; \overline{ρ}. Assume that R^{ord} is a Gorenstein algebra. Then, there exists a p-adic L-function L_{p}(R^{ord}) ∈ R^{ord}[[Cl_{F}^{+}(p^{∞})_{p}]] which satisfies the same interpolation property as stated in Conjecture B.

Remark 1.8. We remark that the p-adic L-function in Theorem B interpolates not only the half of the desired specializations depending on the parity but all desired specializations. This is because we can simply use the space of ordinary modular symbols for the proof of Theorem B and we do not have to use the space of modular symbols for the level structure ZK_{11} which is necessary for the proof of Theorem A. As is remarked in the previous section, Mok [Mo] proves a result which is quite similar as Theorem B.

From the look of the statement of our main theorem (Theorem A), this work might seem to be done by a routine translation of the method of Mazur-Kitagawa form the method of modular symbols over modular curves into the method of modular symbols over Hilbert modular varieties. However, if one tries to establish this kind of result, one immediately finds a lot of difficulties including the ones coming from complicated natures of the generalization of Hida theory to Hilbert modular forms. This is why there has not been a result analogous to Theorem A long time after [Ki] and [M2] and why it took a long time for us to fill the detail of the work. So, it is important for us to explain difficulties of the work and ingredients of the paper.

(1) On the process of Mellin transform and the theory of modular symbols on Hilbert modular varieties, we often have the problem of the action of units of the totally real field F which did not exist in the classical situation of modular curves and where we considered the field of rationals ℚ.

(2) We have to control the torsion of the étale cohomology of Hilbert modular varieties so that our p-adic L-function interpolates special values of Hecke L-function (cf. the condition (Van\overline{ρ}) stated before Main Theorem A). Such problem as well as the freeness of the Hecke algebra was studied by Dimitrov[Di1] and by Lan-Suh[LS].

(3) By Shimura’s work, the special value L(φ, 0) of the Hecke L-function of a Hilbert modular form φ of critical weight is equal to a complex period Ω_{φ,∞} modulo multiplication by elements in \overline{ℚ}^{×}. The periods Ω_{φ,∞} are invariant modulo multiplication by elements in \overline{ℚ}^{×} under twists by Hecke characters. Our p-adic L-function should satisfy an interpolation property which matches well with Shimura’s results (and Deligne’s conjecture). Hence, in our nearly ordinary Hida deformations (consisting of d+1 variables), we have to separate d variables related to the weights of modular forms from the variable related to the twist by Hecke characters. Though the separation of the variables is not difficult for F = ℚ in Kitagawa’s work, the direct analogue of this construction for a general totally real field gives us only a result for Conjecture B. To have a positive result for Conjecture A, we need a delicate choice of the level (ℤ_{p} ⊗𝔬)^{×}K_{11}(p^{n}) which is between the usual level structures K_{0}(p^{n}) and K_{11}(p^{n}). The use of (ℤ_{p} ⊗𝔬)^{×}K_{11}(p^{n}) already appears in a work of Hida [H5] but, to our best knowledge, its relation to the construction of a p-adic L-function has not been pointed out anywhere. Roughly speaking, when F = ℚ, the level structure (ℤ_{p} ⊗𝔬)^{×}K_{11}(p^{n}) yields a one-variable Hida family characterized by j = \frac{k}{2} in the twovariable Hida family obtained by K_{11}(p^{n}) in which the weight k of modular forms and Tate twist j vary freely.

At the end of this introduction, we recall that the algebraic counterpart of the p-adic L-functions on branches R of 𝕋n.o ; \overline{ρ} is given by the Selmer group Sel_{T} of the universal Galois deformation T over R. The group Sel_{T} is defined to be a subgroup of the Galois cohomology H^{1}(F, T ⊗_{R} R^{∨}) and its Pontrjagin dual (Sel_{T})^{∨} is conjectured to be a finitely generated torsion module over R. This conjecture is proposed in [Oc3] and is partially proved in [FO]. We finish our paper by stating the d + 1-variable Iwasawa Main Conjecture with our p-adic L-functions constructed in this paper.

d + 1-variable Iwasawa Main Conjecture (cf. [Oc3])

Under certain conditions (see the above articles for the exact statements), we conjecture that the principal ideals of a d + 1 + δ_{F,p}-variable nearly ordinary Hecke algebra R generated by L_{p}^{even}(R) and L_{p}^{odd}(R) constructed in Theorem A are equal and they coincide with the characteristic ideal of (Sel_{T})^{∨}.

Note that we also have 2-variable and 1-variable Iwasawa Main Conjectures corresponding respectively to Theorem B and Theorem C as special cases of the above Iwasawa Main Conjecture which are formulated in the same way.

1.3. List of notations. At the last page, we will list the symbols which appear in the article. Here, we list some of the most basic notations in the article.

• Let 𝔬 be the ring of integers of our fixed totally real field F and let 𝔬_{+}^{×} be the group of totally positive units of 𝔬.

• We denote by 𝔸_{F} the ring of adèles of F and by 𝔸_{F}^{fin} the subring of finite adèles F ⊗ ℤ̂. We put 𝔬̂ = 𝔬 ⊗_{ℤ}ℤ̂, where ℤ̂ is the profinite completion of ℤ.

• p ≥ 5 is a fixed odd prime number unramified in F.

• ℌ is the upper half-plane {z ∈ ℂ|Imz > 0} and ℌ^{±} = ℂ − ℝ.

• Let O be a discrete valuation ring finite flat over ℤ_{p} which contains all conjugates of 𝔬. We denote by K the field of fractions of O.

• For any n ∈ ℤ[I_{F}], we put n_{max} = max{n_{τ} |τ ∈ I_{F} }, n_{min} = min{n_{τ} |τ ∈ I_{F} }.

• We denote by T the p-Sylow subgroup of the maximal torus in GL_{2}(𝔬⊗_{ℤ} ℤ_{p}). We refer to 4.1 for various tori T_{n.o}, T_{Z}, T_{ord}, T^{arith}, T_{n.o}^{arith}, T_{Z}^{arith} related to T.

• For a (pro-)abelian group A, we denote by A_{p} the p-Sylow subgroup of A.

• For a locally compact abelian group A, we denote by A^{PD} the Pontrjagin dual of A.

• Suppose that the ideal (p) ⊂ ℤ decomposes into the product 𝔭_{1} ⋯ 𝔭_{s} in 𝔬. In this paper, if there is no possibility of confusion, we often take the multi-index notation p^{α} which means 𝔭_{1}^{α_{1}} ⋯ 𝔭_{s}^{α_{s}}.

Acknowledgments. This project was started at the beginning of 2006. The author expresses his sincere gratitude to the University Paris 13 for their hospitality where he stayed from February 2006 to February 2008 and where some preparatory steps of this work were done. He is also grateful to Jacques Tilouine who was the host during his stay at the University Paris 13. The author also thanks Mladen Dimitrov with whom he made a collaboration on the subject of modular symbols and Hida theory of Hilbert modular forms for some periods since the middle of 2007. This article, the article [Di2] by Dimitrov and a joint article [DO] in preparation all grew up after our collaboration.

2. Mellin transform of Hilbert modular forms and Hecke L-functions.

2.1. Automorphic forms on GL_{2}. We define the ℂ-vector space of Hilbert automorphic forms as in [H1], except that our normalization is cohomological.

Definition 2.1. Let K be an open compact subgroup of GL_{2}(𝔸_{F}). Let us take a cohomological double-digit weight w = (w_{1}, w_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}] (as defined in §1.3). The space M_{w_{1},w_{2}}(K; ℂ) of adelic Hilbert modular forms of weight (w_{1}, w_{2}), level K is the ℂ-vector space of functions φ : GL_{2}(𝔸_{F}) → ℂ satisfying the followings three conditions:

(i) φ(γgy) = φ(g) for all γ ∈ GL_{2}(F), y ∈ K and g ∈ GL_{2}(𝔸_{F}). (ii) φ(gu) = det(u)^{w_{1}−t} exp(−\sqrt{−1} ∑_{τ∈I_{F}} k_{τ} θ_{τ})φ(g), for all u ∈ GL_{2}(F ⊗_{ℚ} ℝ)

and g ∈ GL_{2}(𝔸_{F}), where θ_{τ} ∈ ℝ is such that u_{τ} ∈ ℝ^{×} · ( \begin{matrix}cos(θ_{τ}) & − sin(θ_{τ}) \\ sin(θ_{τ}) & cos(θ_{τ}) \end{matrix} ) and k ∈ ℤ[I_{F}] is defined to be k = w_{2} − w_{1} + t.

(iii) For all δ ∈ GL_{2}(𝔸_{F}^{fin}),

φ_{δ} : z = (z_{τ}) ∈ ℌ^{I_{F}} ↦ φ (δ ( \begin{matrix}1 & Re z \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\sqrt{Im z} & 0 \\ 0 & \sqrt{Im z}^{−1} \end{matrix} ))

is holomorphic at z_{τ} ∈ ℌ for every τ ∈ I_{F}. The space S_{w_{1},w_{2}}(K) of adelic Hilbert automorphic cuspforms is the subspace of M_{w_{1},w_{2}}(K) consisting of functions satisfying the following additional condition: (iv) ∫_{𝔸_{F} /F} φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) dx = 0 for all g ∈ GL_{2}(𝔸_{F}), where dx denotes an additive Haar measure.

For an ideal 𝔩 of 𝔬, we consider the following open compact subgroups of GL_{2}(𝔸_{F}^{fin}) :

K_{0}(𝔩) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(𝔬̂)| c ∈ 𝔩 𝔬̂\} ,

K_{1}(𝔩) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{0}(𝔩)| d − 1 ∈ 𝔩 𝔬̂\} ,

K_{11}(𝔩) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{1}(𝔩)| a − 1 ∈ 𝔩 𝔬̂\} .

Let Z be the p-Sylow subgroup of (𝔬 ⊗ ℤ_{p})^{×} viewed as a subgroup of the center of GL_{2}(𝔸_{F}^{fin}). Then, ZK_{11}(𝔩) is another important example of an open compact subgroups of GL_{2}(𝔸_{F}^{fin}). We have the following inclusions for every prime 𝔩 of F:

ZK_{11}(𝔩) ⊂ K_{0}(𝔩)
∪  ∪
K_{11}(𝔩) ⊂ K_{1}(𝔩),

For z = (z_{τ}) ∈ ℌ^{I_{F}} and γ = (\begin{matrix}a_{τ} & b_{τ} \\ c_{τ} & d_{τ} \end{matrix} ) ∈ G(F ⊗_{ℚ}ℝ) we put j(γ, z) = c_{τ} z_{τ} +d_{τ}.

2.2. Hecke operators. For an open compact subgroup K of GL_{2}(𝔸_{F}^{fin}) and an element δ ∈ GL_{2}(𝔸_{F}^{fin}), the Hecke operator [KδK] acts on the left on φ ∈ S_{k}(K) as follows:

φ|_{[KδK]}(x) = ∑_{i} φ(xδ_{i}) , where [KδK] = ∐_{i} δ_{i}K.

Assume that K is factorizable as ∏_{λ} K_{λ} where K_{λ} is an open compact subgroup of GL_{2}(K_{λ}) (λ running over the set of all finite places of F) and K_{λ} is an open compact subgroup of GL_{2}(𝔬_{λ}) for almost all λ with 𝔬_{λ} the completion 𝔬 of at

λ. Let ϖ_{λ} be a uniformizer of 𝔬_{λ}. Then, the Hecke operator [K (\begin{matrix}ϖ_{λ} & 0 \\ 0 & 1 \end{matrix}) K]

is denoted by T_{λ}, if K_{λ} = GL_{2}(𝔬_{λ}), and by U_{λ}, otherwise. Here, we regard

(\begin{matrix}ϖ_{λ} & 0 \\ 0 & 1 \end{matrix}) ∈ GL_{2}(K_{λ}) as an element in GL_{2}(𝔸_{F}^{fin}) which is 1 in GL_{2}(K_{λ′}) at all

other finite primes λ′. It is important to observe that U_{λ} might depend on the choice of ϖ_{λ} (for example when K ⊂ K_{11}(λ)). In order to show the dependence

on the choice of a uniformizer, we sometimes denote [K (\begin{matrix}ϖ_{λ} & 0 \\ 0 & 1 \end{matrix}) K] by U_{ϖ_{λ}}.

The group {±1}^{I_{F}} acts on the space of Hilbert modular forms by the action of

the element ι = (ι_{τ})_{τ∈I_{F}} ∈ GL_{2}(F ⊗ ℝ) with ι_{τ} = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) at every infinite

place τ. This action naturally commutes with the Hecke action.

2.3. Nearly ordinary modular forms. For an element m = ∑_{τ∈I_{F}} m_{τ} τ ∈

ℤ[I_{F}], we denote by ℚ(m) the smallest subfield of F which contains x^{m} for every x ∈ F. Let A ⊂ ℂ be an algebra satisfying the following properties:

(1) The algebra A contains the ring of integers of ℚ(m).

(2) For every element δ ∈ (𝔸_{F}^{fin})^{×}, the fractional ideal δ^{m}A is a principal

ideal. In fact, for every non-archimedean prime λ of F, we fix a generator {ϖ_{λ}^{m}} ∈ A, which determines a generator {δ^{m}} ∈ A for any δ ∈ (𝔸_{F}^{fin})^{×}. We refer to [H1, Sect.3] for the proof of the existence of such algebra A. Let us define modified Hecke operators T_{0,λ} and U_{0,λ} for non-archimedean prime λ of F when the weight k is not parallel acting on the space S_{w_{1},w_{2}}(K; A). On the space S_{w_{1},w_{2}}(K; A), we define:

T_{0,λ} = {ϖ_{λ}^{w_{1}}}T_{λ}, U_{0,λ} = {ϖ_{λ}^{w_{1}}}U_{λ},

T_{0}(p) = ∏_{𝔭 |p} T_{0,𝔭}, U_{0}(p) = ∏_{𝔭 |p} U_{0,𝔭}.

By using these modified Hecke operators, we define nearly ordinary modular forms as follows:

Definition 2.2. A Hilbert modular eigenform φ ∈ M_{w_{1},w_{2}}(K; \overline{ℚ}) is nearly ordinary at p if the eigenvalue of φ with respect to T_{0,𝔭} (or U_{0,𝔭}) is a p-adic unit for all primes 𝔭 of F dividing p.

2.4. Hilbert modular varieties and standard local systems. For an open compact subgroup K of GL_{2}(𝔸_{F}^{fin}), we denote by Y_{K} the Hilbert modular variety of level K with complex points

Y_{K}(ℂ) = GL_{2}(F)\GL_{2}(𝔸_{F})/KK_{∞}^{+}

where K_{∞}^{+} is the group (SO_{2}(ℝ)ℝ_{+}^{×})^{d}. We will consider the Hilbert modular varieties as analytic varieties. By the strong approximation theorem for GL_{2}, Y_{K} has h_{K} := |F^{×}\𝔸_{F}^{×} /det(K)(F ⊗ ℝ)_{+}^{×}| connected components, each isomorphic to a quotient of ℌ^{I_{F}} by a congruence subgroup of GL_{2}(F) of the form GL_{2}(F) ∩ ( \begin{matrix}𝔠 & 0 \\ 0 & 1 \end{matrix} )^{−1}K GL_{2}(F ⊗_{ℚ} ℝ)_{+}( \begin{matrix}𝔠 & 0 \\ 0 & 1 \end{matrix} ) for each 𝔠 which represents a class of F^{×}\𝔸_{F}^{×} /det(K)(F ⊗ℝ)_{+}^{×}. Let us fix a set of representative {𝔠_{i} ∈ 𝔸_{F}^{×}}_{1≤i≤h_{F}} of the narrow class group Cl_{F}^{+} := F^{×}\𝔸_{F}^{×} /𝔬̂^{×}(F ⊗ ℝ)_{+}^{×} of F where h_{F} = ♯ Cl_{F}^{+} is the narrow class number of F. Throughout the paper, we assume the condition:

(5)  𝔠_{i,p} = 1 for every i,

where 𝔠_{i,p} is the p-component of 𝔠_{i}. Note that the number of connected components of Y_{K_{1}(𝔪)} equals to h_{F} and

Y_{K_{1}(𝔪)} ≅ ∐_{1≤i≤h_{F}} Γ_{K_{1}(𝔪)}(𝔠_{i})\ℌ^{I_{F}} ,

where Γ_{K_{1}(𝔪)}(𝔠_{i}) = GL_{2}(F) ∩ ( \begin{matrix}𝔠_{i} & 0 \\ 0 & 1 \end{matrix} )^{−1}K_{1}(𝔪) GL_{2}(F ⊗_{ℚ} ℝ)_{+}( \begin{matrix}𝔠_{i} & 0 \\ 0 & 1 \end{matrix} ) for each

representative 𝔠_{i} ∈ 𝔸_{F}^{×} of the narrow class group of F. For each 𝔠_{i}, we have the following description of the group Γ_{K_{1}(𝔪)}(𝔠_{i}):

Γ_{K_{1}(𝔪)}(𝔠_{i}) =

= \{ (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(F)∩(\begin{matrix}𝔬 & 𝔠_{i}^{−1} 𝔡^{−1} \\ 𝔠_{i}𝔡𝔪 & 𝔬 \end{matrix}) | ad−bc ∈ 𝔬_{+}^{×}, a ≡ 1 (mod 𝔪)\}

where 𝔡 is the different of 𝔬. We fixed an integral ideal 𝔫 ⊂ 𝔬 prime to p. We denote by Y_{11}(𝔫p^{α}) (resp. Y_{Z}(𝔫p^{α}), Y_{1}(𝔫p^{α})) the Hilbert modular variety Y_{K} for K = K_{1}(𝔫) ∩ K_{11}(p^{α}) (resp. ZK_{1}(𝔫) ∩ K_{11}(p^{α}), K_{1}(𝔫p^{α})). Similar analytic descriptions as above holds for these varieties, but we omit them since these are more or less parallel to the above description.

Definition 2.3. Let A be a ℤ-algebra and let (w_{1}, w_{2}) be an element of ℤ[I_{F}]× ℤ[I_{F}].

(1) For each τ ∈ I_{F}, we define a module L(w_{1,τ} , w_{2,τ}; A) to be

L(w_{1,τ} , w_{2,τ}; A) = ⨁_{\begin{matrix}0≤i,j≤w_{2,τ} −w_{1,τ} −1 \\ i+j=w_{2,τ} −w_{1,τ} −1 \end{matrix}} AX_{τ}^{i} Y_{τ}^{j}.

Put (\begin{matrix}X_{τ} \\ Y_{τ} \end{matrix} )^{w_{2,τ} −w_{1,τ} −1} =

= ^{t}(X_{τ}^{w_{2,τ} −w_{1,τ} −1}, X_{τ}^{w_{2,τ} −w_{1,τ} −2}Y_{τ} , …, Y_{τ}^{w_{2,τ} −w_{1,τ} −1}) .

On the module L(w_{1,τ} , w_{2,τ}; A), (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ GL_{2}(A) acts

from the left on the way which sends (\begin{matrix}X_{τ} \\ Y_{τ} \end{matrix} )^{w_{2,τ} −w_{1,τ} −1} to

(ad − bc)^{w_{1,τ}} (\begin{matrix}aX_{τ} + bY_{τ} \\ cX_{τ} + dY_{τ} \end{matrix} )^{w_{2,τ} −w_{1,τ} −1}.

(2) We define a module L(w_{1}, w_{2}; A) to be ⊗_{τ∈I_{F}} L(w_{1,τ} , w_{2,τ}; A) where the

tensor product is taken over A.

Let K be an open compact subgroup of GL_{2}(𝔸_{F}^{fin}). Similarly as the case of K = K_{1}(𝔪), Y_{K} is presented as follows:

Y_{K} = GL_{2}(F)\GL_{2}(𝔸_{F})/KK_{∞}^{+}

= GL_{2}(F)\(GL_{2}(𝔸_{F}^{fin}) × GL_{2}(F ⊗_{ℚ} ℝ))/KK_{∞}^{+}

= GL_{2}(F)\(GL_{2}(𝔸_{F}^{fin})/K × GL_{2}(F ⊗_{ℚ} ℝ)/K_{∞}^{+})

= GL_{2}(F)\(GL_{2}(𝔸_{F}^{fin})/K × (ℌ^{±})^{I_{F}} )

≅ ∏_{1≤i≤h_{F}} Γ_{K}(𝔠_{i})\ℌ^{I_{F}}

where the discrete group Γ_{K}(𝔠_{i}) is defined to be:

Γ_{K}(𝔠_{i}) = GL_{2}(F) ∩ ( \begin{matrix}𝔠_{i} & 0 \\ 0 & 1 \end{matrix} )^{−1}K GL_{2}(F ⊗_{ℚ} ℝ)_{+}( \begin{matrix}𝔠_{i} & 0 \\ 0 & 1 \end{matrix} )

for each representative 𝔠_{i} ∈ 𝔸_{F}^{×} of the ray class group F^{×}\𝔸_{F}^{×} /det(K)(F ⊗ℝ)_{+}^{×}. Each component Γ_{K}(𝔠_{i})\(ℌ^{+})^{I_{F}} is an affine algebraic variety of dimension d.

Definition 2.4. Let A be a subring of ℂ or \overline{ℚ}_{p} which contains the Galois closure of 𝔬[𝔡^{−1}𝔠_{i}^{−1}] for each 1 ≤ i ≤ h_{K}. We define the local system L(w_{1}, w_{2}; A) on Y_{K} to be:

(6)  GL_{2}(F)\GL_{2}(𝔸_{F}) × L(w_{1}, w_{2}; A)/KK_{∞}^{+}.

Here, the group KK_{∞}^{+} acts on L(w_{1}, w_{2}; A) trivially and (6) is presented as

∏_{i=1}^{h_{K}} Γ_{K}(𝔠_{i})\(ℌ^{I_{F}} × L(w_{1}, w_{2}; A)),

where Γ_{K}(𝔠_{i}) acts on (ℌ^{+})^{I_{F}} × L(w_{1}, w_{2}; A) diagonally for each conjugate corresponding to τ ∈ I_{F}.

Note that for K′ ⊂ K, there is a natural projection pr : Y_{K′} → Y_{K}. For δ ∈ GL_{2}(𝔸_{F}^{fin}), we define the Hecke correspondence [KδK] on Y_{K} to be (pr_{1})_{∗} ∘ (·δ)^{∗} ∘ (pr_{2})^{∗} from the diagram below:

\xymatrix{ & Y_{K∩δ^{−1}Kδ} \ar@{-->}[dl]_{_{pr_{1}}} \ar[r]^{·δ} & Y_{δKδ^{−1}∩K} \ar@{-->}[dr]^{_{pr_{2}}} \\ Y_{K} &  &  & Y_{K},}

where ( )^{∗} means the pull-back and ( )_{∗} means the trace map.

We define the standard Hecke cohomological operators T_{λ} = [K (\begin{matrix}ϖ_{λ} & 0 \\ 0 & 1 \end{matrix}) K],

S_{λ} = [K (\begin{matrix}ϖ_{λ} & 0 \\ 0 & ϖ_{λ} \end{matrix}) K], for λ ∉ Σ_{K}, and U_{λ} = [K (\begin{matrix}ϖ_{λ} & 0 \\ 0 & 1 \end{matrix}) K], for

λ ∈ Σ_{K}, where ϖ_{λ} is a uniformizer of F_{λ}. The Betti cohomology groups H^{•}(Y_{K}, L(w_{1}, w_{2}; A)) admit a natural action of all Hecke correspondences.

The action of the group {±1}^{I_{F}} on (Y_{K}, L(w_{1}, w_{2}; A)) is induced by the ac-

tion of (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) on the τ-component. Note that, for each τ ∈ I_{F}, the

action of \{±1\}^{I_{F}} on Y_{K} is the one induced by sending z_{τ} to −\overline{z}_{τ} on the upper-half plane ℌ^{I_{F}} corresponding to the τ-component of Y_{τ}. The action of {±1}^{I_{F}} on (Y_{K}, L(w_{1}, w_{2}; A)) induces the action of {±1}^{I_{F}} on the cohomology H^{•}(Y_{K}, L(w_{1}, w_{2}; A)). This action commutes with the Hecke action. For a character ϵ : {±1}^{I_{F}} → {±1} we denote by H^{•}(Y_{K}, L(w_{1}, w_{2}; A))[ϵ] the ϵ-isotypic part for this action.

2.5. Standard q-expansion and L-function. In this paragraph we consider forms of level K_{1}(𝔪). We recall the relation between the adelic definition of the modular form and modular forms over Hilbert upper half planes defined as follows:

Definition 2.5. For each i with 1 ≤ i ≤ h_{K}, the space M_{w_{1},w_{2}}(Γ_{K_{1}(𝔪)}(𝔠_{i}); ℂ) of Hilbert modular forms of cohomological weight (w_{1}, w_{2}), level Γ_{K_{1}(𝔪)}(𝔠_{i}) is the ℂ-vector space of the functions f_{i} : ℌ^{I_{F}} → ℂ which are holomorphic at z_{τ} for every τ ∈ I_{F} and such that, for every γ ∈ Γ, we have f_{i}(γ(z)) = det(γ)^{w_{1}}j(γ, z)^{k}f_{i}(z) where k ∈ ℤ[I_{F}] is w_{2} − w_{1} + t. The space S_{w_{1},w_{2}}(Γ_{K_{1}(𝔪)}(𝔠_{i}); ℂ) of Hilbert modular cuspforms is the subspace of M_{w_{1},w_{2}}(Γ_{K_{1}(𝔪)}(𝔠_{i}); ℂ), consisting of functions vanishing at all cusps.

Lemma 2.6. We have an isomorphism:

(7) S_{w_{1},w_{2}}(K_{1}(𝔪); ℂ) ≅ ⨁_{1≤i≤h_{F}} S_{w_{1},w_{2}}(Γ_{K_{1}(𝔪)}(𝔠_{i}); ℂ), φ ↦ (f_{i}(z))_{1≤i≤h_{F}} ,

whose definition is explained in the proof below.

Proof. In order to explain the correspondence of the map (7), we recall the notion of Hilbert automorphic functions on GL_{2}(F ⊗_{ℚ} ℝ) of weight (w_{1}, w_{2}), level Γ_{K_{1}(𝔪)}(𝔠_{i}) which are defined to be functions φ_{ℝ} satisfying the following conditions:

(i) φ_{ℝ}(γg) = φ_{ℝ}(g) for all γ ∈ Γ_{K_{1}(𝔪)}(𝔠_{i}) and g ∈ GL_{2}(F ⊗_{ℚ} ℝ). (ii) φ_{ℝ}(gu) = det(u)^{w_{1}} exp(−\sqrt{−1} ∑_{τ∈I_{F}} k_{τ} θ_{τ})φ_{ℝ}(g), for all u ∈ GL_{2}(F ⊗_{ℚ} ℝ)

and g ∈ GL_{2}(F ⊗_{ℚ} ℝ), where θ_{τ} ∈ ℝ is given by u_{τ} ∈ ℝ^{×} · ( \begin{matrix}cos(θ_{τ}) & − sin(θ_{τ}) \\ sin(θ_{τ}) & cos(θ_{τ}) \end{matrix} )

and k_{τ} = w_{2,τ} − w_{1,τ} + 1.

(iii) z ↦ φ_{ℝ} (( \begin{matrix}1 & Re z \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\sqrt{Im z} & 0 \\ 0 & \sqrt{Im z}^{−1} \end{matrix} )) is holomorphic at z_{τ} ∈ ℌ for every τ ∈ I_{F}.

Note that, since ℌ^{I_{F}} is isomorphic to SL_{2}(F ⊗_{ℚ} ℝ)/SO_{2}(ℝ)^{d}, for modular forms f_{i}(z) of weight (w_{1}, w_{2}) and level Γ_{K_{1}(𝔪)}(𝔠_{i}) in the sense of Definition 2.5, (cz+d)^{k}f_{i}(z) corresponds naturally to a Hilbert automorphic function φ_{ℝ,i} on GL_{2}(F ⊗_{ℚ} ℝ) of weight (w_{1}, w_{2}), level Γ_{K_{1}(𝔪)}(𝔠_{i}) in the above sense. In fact, this correspondence gives a bijection between two spaces. On the other hand, let us consider an element φ of M_{w_{1},w_{2}}(K_{1}(𝔪)) as introduced in Definition 2.1 and Definition 2.2. For each 1 ≤ i ≤ h_{F}, we denote by φ_{ℝ,i} the function φ(( \begin{matrix}t_{i} & 0 \\ 0 & 1 \end{matrix} ) g) restricted to GL_{2}(F ⊗_{ℚ} ℝ) ⊂ GL_{2}(𝔸_{F}^{fin}) where we choose t_{i} ∈ 𝔸_{F}^{fin} such that 𝔠_{i} 𝔡 = (t_{i}). φ_{ℝ,i} is naturally a Hilbert automorphic function φ_{ℝ,i} on GL_{2}(F ⊗_{ℚ} ℝ) of weight k, level Γ_{K_{1}(𝔪)}(𝔠_{i}) in the above sense. In fact, this correspondence gives a bijection between M_{w_{1},w_{2}}(K_{1}(𝔪)) and the direct sum for 1 ≤ i ≤ h_{F} of the spaces of Hilbert automorphic functions on GL_{2}(F ⊗_{ℚ} ℝ) of weight k, level Γ_{K_{1}(𝔪)}(𝔠_{i}). Now, the desired isomorphism is obtained by taking the composition of two correspondences explained above. □

Since f_{i} is invariant with respect to the action of the matrix (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) with

x ∈ (𝔠_{i} 𝔡)^{−1}, f_{i} has a q-expansion of the form ∑_{ξ∈(𝔠_{i})_{+}} a(f_{i}, ξ)q^{ξ}, where q^{ξ} = exp(2π\sqrt{−1} Tr_{F/ ℚ}(ξz)). Note the important relation:

(8)  a(f_{i}, ϵξ) = ϵ^{w_{1}}a(f_{i}, ξ) , for all ϵ ∈ 𝔬_{+}^{×} .

We will describe the relation between Fourier coefficients and Hecke eigenvalues.

Lemma 2.7. For each 𝔠_{i} and for an element ξ ∈ (𝔠_{i})_{+}, the quantity

a(f_{i}, ξ)ξ^{−w_{1}}

depends only on the ideal (ξ) 𝔠_{i} ⊂ 𝔬.

Definition 2.8. For an integral ideal 𝔞 of 𝔬, there is a unique i such that 𝔞 belongs to the same narrow ideal class as 𝔠_{i}^{−1}, that is 𝔞 = ξ 𝔠_{i}^{−1} with ξ ∈ (𝔠_{i})_{+}. We put then:

C(φ, 𝔞) := a(f_{i}, ξ)ξ^{−w_{1}}.

Lemma 2.9. ([H1, Prop.4.1,Thm.5.2]) Let φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔪)) be an eigen cuspform. Then, for every integral ideal 𝔞 of 𝔬, the eigenvalue of T_{𝔞} on φ is C(φ, 𝔞).

Definition 2.10. We define the standard L-function of φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔪)) to be:

L(φ, s) := ∑_{𝔞⊂𝔬} \frac{C(φ, 𝔞)}{N_{F/ ℚ}(𝔞)^{s}} .

Similarly, for any Hecke character ϕ of finite order, we define the standard L-function of φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔪)) twisted by ϕ to be:

L(φ ⊗ ϕ, s) := ∑_{𝔞⊂𝔬} \frac{C(φ, 𝔞)ϕ(𝔞)}{N_{F/ ℚ}(𝔞)^{s}}.

2.6. Eichler-Shimura map. Let φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔪), ℂ) be an eigen cuspform. Recall that, by (7), φ corresponds to ∑_{1≤i≤h_{F}} f_{i}(z) where f_{i}(z) ∈

S_{w_{1},w_{2}}(Γ_{K_{1}(𝔪)}(𝔠_{i}); ℂ) for each i. Then, the integration of the vector-valued d-form

∑_{1≤i≤h_{F}} f_{i}(z_{τ})_{∏τ∈I_{F}} (X_{τ} + z_{τ} Y_{τ})^{w_{2,τ} −w_{1,τ} −1} ∧_{τ∈I_{F}} dz_{τ}

on h_{F} -copies of ℌ^{I_{F}} defines an element of

H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ)) ≅ ⨁_{1≤i≤h_{F}} H^{d}(Γ_{K_{1}(𝔪)}(𝔠_{i})\ℌ^{I_{F}}; L(w_{1}, w_{2}; ℂ)).

We call this the Eichler-Shimura class of φ and we denote it by [φ] ∈ H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ)). Let H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}] (resp. H_{c}^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}]) be the component on which the Hecke algebra for M_{w_{1},w_{2}}(K_{1}(𝔪); ℂ) acts by the Hecke eigenvalues of φ. The class [φ] ∈ H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ)) naturally falls in H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}]. By the isomorphism H^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}] ≅ H_{c}^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}], we also have the Eichler-Shimura class in H_{c}^{d}(Y_{1}(𝔪); L(w_{1}, w_{2}; ℂ))[λ_{φ}].

2.7. Mellin transform. In this section, we compute L(φ, s) for an eigen cuspform φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔪)) by using a Mellin transform.

∑_{1≤i≤h_{F}} N_{F/ ℚ}(𝔠_{i})^{s} ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}/ 𝔬_{+}^{×}} f_{i}(z)z^{s−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

= ∑_{1≤i≤h_{F}} N_{F/ ℚ}(𝔠_{i})^{s} ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}/ 𝔬_{+}^{×}} ∑_{ξ∈(𝔠_{i})_{+}} a(f_{i}, ξ) exp(2π\sqrt{−1} Tr_{F/ ℚ}(ξz))z^{s−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

= ∑_{1≤i≤h_{F}} N_{F/ ℚ}(𝔠_{i})^{s}×

× ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}/ 𝔬_{+}^{×}} ∑_{ξ∈(𝔠_{i})_{+}} a \frac{a(f_{i}, ξ)ξ^{w_{1}}}{N_{F/ ℚ}(ξ)^{s}} exp(2π\sqrt{−1} Tr_{F/ ℚ}(ξz))(ξz)^{s−t−w_{1}} ∧_{τ∈I_{F}} d(ξz_{τ})

By using the invariance of the integrant with respect to the action of units, the last term is equal to:

∑_{1≤i≤h_{F}} N_{F/ ℚ}(𝔠_{i})^{s} ∑_{ξ∈(𝔠_{i})_{+}/ 𝔬_{+}^{×}} \frac{a(f_{i}, ξ)ξ^{w_{1}}}{N_{F/ ℚ}(ξ)^{s}} ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}} exp(2π\sqrt{−1} Tr_{F/ ℚ}(z))z^{s−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

= ∑_{1≤i≤h_{F}} ∑_{ξ∈(𝔠_{i})_{+}/ 𝔬_{+}^{×}} \frac{a(f_{i}, ξ)ξ^{w_{1}}}{N_{F/ ℚ}(ξ 𝔠_{i}^{−1})^{s}} ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}} exp(2π\sqrt{−1} Tr_{F/ ℚ}(z))z^{s−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

= (∑_{𝔞⊂𝔬} \frac{C(φ, 𝔞)}{N_{F/ ℚ}(𝔞)^{s}} ) × ∏_{τ∈I_{F}} ∫_{0}^{\sqrt{−1}∞}exp(2π\sqrt{−1}y_{τ})y_{τ}^{s−1−w_{1,τ}} dy_{τ}

= (∑_{𝔞⊂𝔬} \frac{C(φ, 𝔞)}{N_{F/ ℚ}(𝔞)^{s}} ) × ∏_{τ∈I_{F}} ∫_{0}^{∞}exp(−2πy_{τ})(\sqrt{−1}y_{τ})^{s−1−w_{1,τ}} d(\sqrt{−1}y_{τ})

= (∑_{𝔞⊂𝔬} \frac{C(φ, 𝔞)}{N_{F/ ℚ}(𝔞)^{s}} ) × ∏_{τ∈I_{F}} \frac{(\sqrt{−1})^{s−w_{1,τ}}}{(2π)^{s−w_{1,τ}}} ∫_{0}^{∞}exp(−2πy_{τ})(2πy_{τ})^{s−1−w_{1,τ}} d(2πy_{τ})

= L(φ, s) ∏_{τ∈I_{F}} \frac{Γ(s − w_{1,τ})}{(−2π\sqrt{−1})^{s−w_{1,τ}}} .

3. Modular symbol cycle

The image of (F ⊗ ℝ)_{+}^{×}/ 𝔬_{+}^{×} in the Hilbert modular variety Y_{K} of level K is a closed d-cycle, which is a generalization to the Hilbert modular case, of the classical modular symbol linking the cusps 0 and ∞ on a modular curve. Recall that the number of components of Y_{K} is h_{K} = |F^{×}\𝔸_{F}^{×} /det(K)(F ⊗ ℝ)_{+}^{×}|. Thus Y_{K} is a quotient of ℌ^{I_{F}} ∐ ⋯ ∐ ℌ^{I_{F}} (h_{K} copies). For each i satisfying 1 ≤ i ≤ h_{K}, we fix an element u_{i} ∈ F of the cusp ℙ^{1}(F) of i-th component ℌ^{I_{F}}. We define H_{u_{i}} ⊂ ℌ^{I_{F}} to be

H_{u_{i}} = \{u_{i} + \sqrt{−1}y | y ∈ (F ⊗ ℝ)_{+}^{×}\}.

For each u = {u_{i} ∈ F }_{1≤i≤h_{K}}, we denote by H_{u} ⊂ ℌ^{I_{F}} ∐ ⋯ ∐ ℌ^{I_{F}} (h_{K} copies) the disjoint union ∐_{i=1}^{h_{K}} H_{u_{i}}. Let 𝔪 be an integral ideal of F such that 𝔪u_{i} is contained in 𝔬 for every i. Let E be a finite index subgroup of 𝔬_{+}^{×} consisting of totally positive units e congruent to 1 modulo 𝔪. For such a sufficiently small subgroup E ⊂ 𝔬_{+}^{×}, we have:

G_{K,E,u} : H_{u}/E ⟶ Y_{K}.

Note that each component of H_{u}/E is a finite cover of (F ⊗_{ℚ} ℝ)_{+}^{×}/ 𝔬_{+}^{×} and it is homeomorphic to ℝ ×(𝕊^{1})^{d−1}. In Section 2.4, we introduced a local system L(w_{1}, w_{2}; A) on Y_{K} for any subring A of ℂ or \overline{ℚ}_{p} which contains the Galois closure of 𝔬[𝔡^{−1}𝔠_{i}^{−1}] for each 1 ≤ i ≤ h_{K}. By definition, the pull-back of the local system L(w_{1}, w_{2}; A) to H_{u}/E is the etale space

(H_{u} × L(w_{1}, w_{2}; A))/E

where e ∈ E acts on H_{u} by multiplication and acts on L(w_{1}, w_{2}; A) via the

matrix (\begin{matrix}e & 0 \\ 0 & 1 \end{matrix}).

Definition 3.1. A cohomological double-digit weight (w_{1}, w_{2}) ∈ ℤ[I_{F}]×ℤ[I_{F}] is said to be critical if w_{1,τ} < 0 ≤ w_{2,τ} holds for every τ ∈ I_{F}.

For each critical (w_{1}, w_{2}), we consider the map:

(9)  L(w_{1}, w_{2}; A) ⟶ A

∑ a_{w}X^{w_{2}−w_{1}−t−w}Y^{w} ↦ the coefficient of (\begin{matrix}w_{2} − w_{1} − t \\ −w_{1} − t \end{matrix}) X^{w_{2}}Y^{−w_{1}−t} , (10)

where all the indeterminates are of multi-index and w runs through elements in ℤ[I_{F}] satisfying 0 ≤ w ≤ w_{2} −w_{1} −t. Note that, by definition, a_{w} is divisible

by (\begin{matrix}w_{2} − w_{1} − t \\ −w_{1} − t \end{matrix}). Hence, (9) induces a map of sheaves

(11)  L(w_{1}, w_{2}; A) ⟶ A

on H_{u}/E.

Definition 3.2. Let A be a subring of ℂ or \overline{ℚ}_{p} which contains the Galois closure of 𝔬[𝔡^{−1}𝔠_{i}^{−1}] for each 1 ≤ i ≤ h_{K} and let (w_{1}, w_{2}) be a critical cohomological double-digit weight. For each u = {u_{i} ∈ F }_{1≤i≤h_{K}}, we have the Gysin map

(12)  H_{c}^{d}(Y_{K}; L(w_{1}, w_{2}; A)) ⟶ H_{c}^{d}(H_{u}/E; L(w_{1}, w_{2}; A))

for the map G_{K,E,u}. We define the evaluation map

(13)  ev_{K,u}^{(w_{1},w_{2})} : H_{c}^{d}(Y_{K}; L(w_{1}, w_{2}; A)) ⟶ A ⊗_{ℤ} ℚ

to be the composition of (12) and the map (14) defined as follows: (14)

H_{c}^{d}(H_{u}/E; L(w_{1}, w_{2}; A)) ⟶ H_{c}^{d}(H_{u}/E; A) ≅ A^{⊕h_{K}} ⟶ A ⟶^{×_{\frac{1}{[𝔬{× \atop +}:E]}}} A ⊗_{ℤ} ℚ.

(the first arrow of (14) is induced by (11) and the second one is the trace map.)

Remark 3.3. Later, in §5, we use the evaluation maps to define a certain measure on \underleftarrow{lim}_{β} Cl_{F}^{+}(p^{β}). Since K varies in K_{1}(𝔫) ∩ K_{11}(p^{α}) where p^{α} ⊂ 𝔬 becomes arbitrary small, it might seem that the denominator _{\frac{1}{[𝔬_{+}^{×}:E]}} is out of

control. However, by distribution property with respect to α proved at Lemma 5.3 and Corollary 5.7, it will follow that the measure we obtain is integral.

We recall some basic facts and fix notations related to cusps and evaluation maps. For a multi-index notation p^{β} = 𝔭_{1}^{β_{1}} ⋯ 𝔭_{s}^{β_{s}}, we have the extension as follows:

(15)  1 ⟶ (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×} ⟶ Cl_{F}^{+}(p^{β}) ⟶ Cl_{F}^{+} ⟶ 1.

When K = K_{1}(𝔫p^{α}) for fixed tame conductor 𝔫, the number of components of Y_{K} is h_{F} = |Cl_{F}^{+}|. We fix a splitting of the sequence (15) in a manner which is compatible with respect to β. Every element of u ∈ Cl_{F}^{+}(p^{β}) is identified with {u_{i} ∈ (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×}}_{1≤i≤h_{F}}. Here, we note that p^{−β}/ 𝔬 ⊂ F and we fix representatives of (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×} in p^{−β}/ 𝔬 in a manner which is compatible with respect to β = (β_{1}, ⋯, β_{s}). Under this situation, we denote ev_{K,u}^{(w_{1},w_{2})} with

K = K_{1}(𝔫p^{α}) and u ∈ Cl_{F}^{+}(p^{β}) by ev_{u,α,β}^{(w_{1},w_{2})}. In §5, we need a similar construction for the level K_{Z}(p^{α}) = ZK_{1}(𝔫) ∩ K_{11}(p^{α}) for fixed tame conductor 𝔫 where Z is the p-Sylow subgroup of (𝔬 ⊗ ℤ_{p})^{×} viewed as a subgroup of the center of GL_{2}(𝔸_{F}^{fin}) as introduced at the end of the introduction. The number of components h_{K_{Z}(p^{α})} = |F^{×}\𝔸_{F}^{×} /det(K_{Z}(p^{α}))(F ⊗ ℝ)_{+}^{×}| of Y_{K_{Z}(p^{α})} is 2^{t}h_{F} independently of α, where t satisfies 1 ≤ t ≤ s. Since p is not equal to 2, we have

(16) 1 ⟶ ((𝔬 /p^{β})^{×}/ 𝔬_{+}^{×})_{p} ⟶ Cl_{F}^{+}(p^{β})_{p} ⟶ (F^{×}\𝔸_{F}^{×} /det(K_{Z}(p^{α}))(F ⊗ ℝ)_{+}^{×})_{p} ⟶ 1.

As in the previous case, we fix a splitting of the sequence (16) in a manner which is compatible with respect to β = (β_{1}, ⋯, β_{s}). If we consider the situation where the coefficient ring is p-adic and the cohomology H_{c}^{d}(Y_{K_{Z}(p^{α})}; L(w_{1}, w_{2}; A)) is localized at a maximal ideal of the p-adic Hecke algebra acting on the cohomology (we denote the maximal ideal by \overline{ρ}), we have the cusp associated to u = {u_{i} ∈ ((𝔬 /p^{β})^{×}/ 𝔬_{+}^{×})_{p}} ∈ Cl_{F}^{+}(p^{β})_{p} and an evaluation map:

(17)  ev_{K_{Z}(p^{α}),u}^{(w_{1},w_{2})} : H_{c}^{d}(Y_{K_{Z}(p^{α})}; L(w_{1}, w_{2}; A))\overline{ρ} ⟶ A ⊗_{ℤ} ℚ.

We also denote ev_{K_{Z}(p^{α}),u}^{(w_{1},w_{2})} for u ∈ Cl_{F}^{+}(p^{β})_{p} by ev_{u,α,β}^{(w_{1},w_{2})}

3.1. Cohomological interpretation of special values. Now we consider the paring between the modular symbol cycle and the cohomology class of φ under Eichler-Shimura map and relate it to the special value with help of the subsection 2.7. By the definition of Eichler-Shimura class, we have the following proposition:

Theorem 3.4. Let φ be an eigen cuspform of critical double-digit weight (w_{1}, w_{2}) and level K_{1}(𝔫p^{α}). We denote by [φ] ∈ H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; ℂ))[λ_{φ}] the Eichler-Shimura class of a cuspform φ. Let β = (β_{1}, ⋯, β_{s}) be a multiindex as above. Then, for each u = {u_{i} ∈ (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×}} ∈ Cl_{F}^{+}(p^{β}), we have

ev_{u,α,β}^{(w_{1},w_{2})}([φ]) = \frac{1}{[𝔬_{+}^{×} : E]} ∑_{1≤i≤h_{F}} ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}/E} f_{i}(z + u_{i})z^{−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

via the correspondence from φ to {f_{i}(z)} obtained in Lemma 2.6 where E is a sufficiently small subgroup of 𝔬_{+}^{×} with finite index introduced at the beginning of Section 3.

Let 𝔬_{φ} be the ring of integers of the Hecke field which is obtained by adjoining the eigenvalues of Hecke operators to ℚ and let 𝔬_{φ,(p)} be the localization of 𝔬_{φ} at the prime ideal above p which is specified from the fixed embedding \overline{ℚ} ↪ \overline{ℚ}_{p}.

Definition 3.5. Let φ be an eigen cuspform of critical double-digit weight (w_{1}, w_{2}) and level K_{1}(𝔫p^{α}). Let v be an 𝔬_{φ,(p)}-basis of the free 𝔬_{φ,(p)}-module of rank one obtained by H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; 𝔬_{φ,(p)}))[λ_{φ}, ϵ] modulo 𝔬_{φ,(p)}- torsion part. By abuse of notation, we denote the image of the above v via the map

H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; 𝔬_{φ,(p)}))[λ_{φ}, ϵ]/(𝔬_{φ,(p)}-torsion) ↪

↪ H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; ℂ))[λ_{φ}, ϵ]

by the same letter v. We define the complex period C_{φ,∞,v}^{ϵ} ∈ ℂ^{×} to be the complex number satisfying v = C_{φ,∞,v}^{ϵ} · [φ].

Definition 3.6. Let ϕ be a finite order character of Cl_{F}^{+}(p^{β}). As in (15), (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×} is canonically identified with a subgroup of Cl_{F}^{+}(p^{β}). Thus we have a homomorphism:

(p^{−β}/ 𝔬)^{×} →^{∼} (𝔬 /p^{β})^{×} ↠ (𝔬 /p^{β})^{×}/ 𝔬_{+}^{×} ↪ Cl_{F}^{+}(p^{β})

where the first term (p^{−β}/ 𝔬)^{×} is defined to be a subset of p^{−β}/ 𝔬 consisting of elements whose annihirater is p^{β}. Then, the Gauss sum G(ϕ) for ϕ is defined as follows:

G(ϕ) = ∑_{x∈(p^{−β}/ 𝔬)^{×}} ϕ(x)exp(2π\sqrt{−1}Tr_{F/ℚ}(x)).

By a similar calculation as §2.7, we have

\frac{1}{[𝔬_{+}^{×} : E]} ∑_{1≤i≤h_{F}} N_{F/ ℚ}(𝔠_{i})^{s} ∑_{u_{j}∈((p^{−β}/ 𝔬)^{×}/ 𝔬_{+}^{×})_{i}} ϕ_{i}^{−1}(u_{j}) ∫_{\sqrt{−1}(F ⊗ℝ)_{+}^{×}/E} f_{i}(z+u_{j})z^{s−t−w_{1}} ∧_{τ∈I_{F}} dz_{τ}

= L(φ ⊗ ϕ, s)G(ϕ^{−1}) ∏_{τ∈I_{F}} \frac{Γ(s − w_{1,τ})}{(−2π\sqrt{−1})^{s−w_{1,τ}}} ,

for any finite order character ϕ of Cl_{F}^{+}(p^{β}), where ((p^{−β}/ 𝔬)^{×}/ 𝔬_{+}^{×})_{i} is the inverse image of each 𝔠_{i} ∈ Cl_{F}^{+} in (15) which is isomorphic to (p^{−β}/ 𝔬)^{×}/ 𝔬_{+}^{×} and ϕ_{i} is a map on each component ((p^{−β}/ 𝔬)^{×}/ 𝔬_{+}^{×})_{i} induced by ϕ. In the integral of the above equation, we choose a lift u_{i} ∈ (p^{−β}/ 𝔬)^{×} of u_{i} ∈ (p^{−β}/ 𝔬)^{×}/ 𝔬_{+}^{×} (we use the same symbol by abuse of notation). It is not difficult to see that the integral does not depend on the choice of such a lift. Using this, we have the following corollary:

Corollary 3.7. Let φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔫 p^{α})) be a normalized eigen cusp form of weight (w_{1}, w_{2}) and of level K_{1}(𝔫 p^{α}). Let v be an 𝔬_{φ,(p)}-basis of H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; 𝔬_{φ,(p)}))[λ_{φ}, ϵ] modulo 𝔬_{φ,(p)}-torsion part. Then we have

∑_{u∈Cl_{F}^{+}(p^{β})} ϕ^{−1}(u)ev_{u,α,β}^{(w_{1},w_{2})}(v) = G(ϕ^{−1})\frac{L(φ ⊗ ϕ, 0)}{C_{φ,∞,v}^{ϵ}}.

for any Herke character of finite order, where ϵ is the equal to ϕ restricted to (F ⊗ ℝ)^{×} identifying (F ⊗ ℝ)^{×}/(F ⊗ ℝ)_{+}^{×} with {±1}^{I_{F}}.

The following proposition is very important for the property of “invariance” of periods with respect to character twists:

Proposition 3.8. Let φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔐)) be an eigen cuspform which is critical in the sense of Definition 3.1. Assume that φ is not a twist of another eigen cuspform by a Hecke character of a finite order. We take a twist φ′ = φ ⊗ N_{F}^{r} ϕ with the Norm character N_{F} and a Hecke character of finite order ϕ, which is of weight (w_{1} − rt, w_{2} − rt). We assume that (w_{1} − rt, w_{2} − rt) is also critical. Let v be an 𝔬_{φ,(p)}-basis of

H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1}, w_{2}; 𝔬_{φ,(p)}))[λ_{φ}, ϵ]/(𝔬_{φ,(p)}-torsion)

and let v′ be the twist v[r] ⊗ ϕ which is an 𝔬_{φ,(p)}-basis of

H_{c}^{d}(Y_{1}(𝔫p^{α}); L(w_{1} − rt, w_{2} − rt; 𝔬_{φ′,(p)}[ϕ]))[λ_{φ′}, ϵ]/(𝔬_{φ′,(p)}-torsion)

where v[r] is a shift of weight of v by r. Then we have:

C_{φ,∞,v}^{ϵ} = C_{φ′,∞,v′}^{ϵ} \frac{G(ϕ^{−1})}{(−2π\sqrt{−1})^{dr}} ∏_{τ∈I_{F}} \frac{Γ(−w_{1,τ} + r)}{Γ(−w_{1,τ})}.

If there are not so much confusion, we sometimes omit the dependence on v of the period and denote it by C_{φ,∞}. (see Remark 4.17 (2) for our idea for eliminating the dependence on the choice of v)

4. Hida families of Hilbert modular forms.

In the first half of this section we review Hida’s theory of nearly ordinary Hilbert modular forms as developed in [H1] and [H2]. In particular, we recall Hida’s control theorem for nearly ordinary p-adic Hecke algebras. Hida’s proof relies, via the Jacquet-Langlands correspondence, on control theorems for cohomology groups coming from quaternion algebras over F which are totally definite, or indefinite but yield Shimura curves. The introduction of [H1] ends up with the hope that those results would be extended to general quaternion algebras yielding varieties of higher dimension. In the second half of this section we extend Hida’s theory to the case of M_{2}(F), corresponding to the d-dimensional Hilbert modular variety. Using the results of [Di1], we prove a control theorem 4.9 as well as a freeness result over the universal nearly ordinary p-adic Hecke algebra (see theorem 4.15).

The universal nearly ordinary p-adic Hecke algebra introduced by Hida is naturally an algebra over the Iwasawa algebra in d + 1 + δ_{F,p} variables (δ_{F,p} being the Leopoldt defect), and it is very important in our application in the following sections to separate the first d variables corresponding to the weight, from the other 1 + δ_{F,p} variables corresponding to twists by Hecke characters. We do that by a careful choice of level structure at primes dividing p.

Since we assumed that p is unramified in F, we denote by 𝔭_{1},...,𝔭_{s} the ideals of F above p. We fix an ideal 𝔫 of F prime to p.

4.1. Various level structures. In Section 2.4, we introduced for α ≥ 0 the Hilbert modular varieties Y_{11}(𝔫p^{α}), Y_{Z}(p^{α}) and Y_{1}(p^{α}) of level K_{1}(𝔫)∩K_{11}(p^{α}), K_{Z}(p^{α}) = ZK_{1}(𝔫) ∩ K_{11}(p^{α}) and K_{1}(𝔫 p^{α}), as well as the sheaves L(w_{1}, w_{2}; A)

on them. Here Z is a group of elements (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) with a ∈ (𝔬 ⊗ℤ_{p})^{×}. Using the

multi-index notation of 1.3, we have:

(18)  K_{0}(p^{α})/K_{11}(p^{α}) ≅ (𝔬 /p^{α})^{×} × (𝔬 /p^{α})^{×},

where the first factor is represented by the group of elements

\{(\begin{matrix}a & 0 \\ 0 & 1 \end{matrix})| a ∈ (𝔬 / ∏ 𝔭_{i}^{α_{i}})^{×}\} and the second factor is represented by the

group of elements \{(\begin{matrix}1 & 0 \\ 0 & d \end{matrix})| d ∈ (𝔬 / ∏ 𝔭_{i}^{α_{i}})^{×}\}. Hence, we have

(19)  K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×} ≅ ((𝔬 /p^{α})^{×} × (𝔬 /p^{α})^{×}) / 𝔬^{×}

where we divide by the group generated by the elements \{(\begin{matrix}d & 0 \\ 0 & d \end{matrix})\} where d

runs through the image of 𝔬^{×} in (𝔬 /p^{α})^{×} which is also denoted by 𝔬^{×} by abuse of notation. We introduce several tori as follows:

Definition 4.1. Let us define T (resp. T_{n.o}, T_{Z}, T_{ord}) to be the p-Sylow subgroup of \underleftarrow{lim}_{α}K_{0}(p^{α})/K_{11}(p^{α}) (resp. \underleftarrow{lim}_{α}K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×},

\underleftarrow{lim}_{α}K_{0}(p^{α})/ZK_{11}(p^{α}), \underleftarrow{lim}_{α}K_{0}(p^{α})/K_{1}(p^{α}) 𝔬^{×}).

By definition, we have the following relation:

(20)  \begin{aligned}& & T \\ & & ↠ \\ T_{ord} & ↞ & T_{n.o} & ↠ & T_{Z}. \end{aligned}

Definition 4.2. We define an Iwasawa algebra Λ (resp. Λ_{n.o}, Λ_{Z}, Λ_{ord}) to be O[[T]] (resp. O[[T_{n.o}]], O[[T_{Z}]], O[[T_{ord}]]).

The algebra Λ (resp. Λ_{Z}) is isomorphic to O[[(1 + p ℤ_{p})^{2d}]] (resp. O[[(1 + p ℤ_{p})^{d}]]). The algebra Λ_{n.o} (resp. Λ_{ord}) is isomorphic to O[[(1 + p ℤ_{p})^{d+1+δ_{F,p}}]] (resp. O[[(1 + p ℤ_{p})^{1+δ_{F,p}}]]), where δ_{F,p} is the Leopoldt defect for F and p which is expected to be zero by the Leopoldt conjecture.

We have the following relation:

(21)

Λ ↠

Λ_{ord} ↞ Λ_{n.o} ↠ Λ_{Z}

where two vertical arrows are surjective. We remark that a Λ_{n.o}-module (resp. Λ_{Z}-module, Λ_{ord}-module) M is naturally regarded as a Λ-module. Let the module H_{n.o}^{•}(Y_{11}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) be the largest O-direct summand of the module H^{•}(Y_{11}(p^{α}), L(w_{1}, w_{2}; K/ O)) on which the Hecke operator T_{0}(p) in Definition 2.2 is invertible. The Pontryagin dual H_{n.o}^{•}(w_{1}, w_{2}) of the limit \underrightarrow{lim}_{α}H_{n.o}^{•}(Y_{11}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) has a natural action of T_{n.o} and it is

a module over the algebra Λ_{n.o}. Let the module H_{n.o}^{•}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) be the largest O-direct summand of the module H^{•}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) on which the Hecke operator T_{0}(p) of Definition 2.2 is invertible. The Pontryagin dual H_{Z}^{•}(w_{1}, w_{2}) of the limit \underrightarrow{lim}_{α}H_{n.o}^{•}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) has a natural action of T_{Z} and it

is a module over the algebra Λ_{Z}. Let the module H_{ord}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) be the largest O-direct summand of the module H^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) on which the Hecke operator T_{0}(p) of Definition 2.2 is invertible. The Pontryagin dual H_{ord}^{•}(w_{1}, w_{2}) of the limit

\underrightarrow{lim}_{α}H_{ord}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))

has a natural action of T_{ord} and it is a module over O[[T_{ord}]] ≅ O[[(1 + p ℤ_{p})^{1+δ_{F,p}}]].

Remark 4.3. Let \overline{ρ} be a nearly ordinary residual Galois representation of G_{F} which is modular of level K_{1}(𝔫) ∩ K_{11}(p^{α}) and nearly ordinary at p. For any cohomological weight w = (w_{1}, w_{2}), we have

H_{n.o}^{•}(Y_{11}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} ≠ 0,
H_{n.o}^{•}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} ≠ 0,

where ( )\overline{ρ} means the \overline{ρ}-part as explained at the equation (1). On the other hand, H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} can be zero for certain (w_{1}, w_{2}). However, there exists a double-digit weight (w_{1}, w_{2}) such that H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} ≠ 0.

We say that a residual representation \overline{ρ} is realized in

H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))

when we have H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} ≠ 0. We recall the following lemma, which is straightforward from the construction of Hida family.

Lemma 4.4. Suppose that \overline{ρ} : G_{F} ⟶ \overline{𝔽}_{p} is a nearly ordinary residual Galois representation of level K_{1}(𝔫p) and double-digit weight (a_{1}, a_{2}). Then \overline{ρ} is realized in the cohomology H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) if and only if the maps (𝔬 /p)^{×} ⟶ (𝔬 /p)^{×}, u ↦ u^{w_{1}−a_{1}} and (𝔬 /p)^{×} ⟶ (𝔬 /p)^{×}, u ↦ u^{w_{2}−a_{2}} annihilate the image of 𝔬_{+}^{×} in (𝔬 /p)^{×}.

When \overline{ρ} is not realized in

H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)),

H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ} is defined to be zero. We denote by H_{n.o}^{•}(w_{1}, w_{2})\overline{ρ} the \overline{ρ}-part of H_{n.o}^{•}(w_{1}, w_{2}), which is defined to be the Pontrjagin dual of \underrightarrow{lim}_{α}H_{n.o}^{•}(Y_{11}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))\overline{ρ}.

Let (r_{1}, r_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}]. For a Λ-module M, we denote by tw^{(r_{1},r_{2})}(M) the module M whose Λ-module structure is twisted by

(22)  Λ → Λ , [a, d] ↦ a^{r_{1}}d^{r_{2}}[a, d],

where we identify (𝔬 ⊗ ℤ_{p})^{×} × (𝔬 ⊗ ℤ_{p})^{×} with T by

(a, d) ↦ (\begin{matrix}a & 0 \\ 0 & d \end{matrix}) .

By Hida, H_{n.o}^{•}(w_{1}, w_{2})\overline{ρ} is independent of (w_{1}, w_{2}) in the following sense:

Theorem 4.5. [H2, (3.3)] The Λ-modules H_{n.o}^{•}(w_{1}, w_{2})\overline{ρ} and tw^{(w_{1}−w′_{1},w_{2}−w′_{2})}(H_{n.o}^{•}(w′_{1}, w′_{2})\overline{ρ}) are isomorphic to each other for (w_{1}, w_{2}), (w′_{1}, w′_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}] cohomological double-digit weights (cf. §2.4).

Corollary 4.6. Let (w_{1}, w_{2}), (w′_{1}, w′_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}] be cohomological weights (cf. §2.4).

(i) The Λ_{n.o}-module H_{n.o}^{•}(w_{1}, w_{2})\overline{ρ} is always isomorphic to H_{n.o}^{•}(w′_{1}, w′_{2})\overline{ρ} with Λ_{n.o}-structure is twisted by the above twists. (ii) The Λ_{ord}-module H_{ord}^{•}(w_{1}, w_{2})\overline{ρ} is isomorphic to

tw^{(w_{1}−w′_{1},w_{2}−w′_{2})}(H_{ord}^{•}(w′_{1}, w′_{2})\overline{ρ})

with Λ_{ord}-structure twisted by (22) if and only if we have w_{1} = w′_{1} and w_{2} − w′_{2} ∈ ℤ t. (iii) The Λ_{Z}-module H_{Z}^{•}(w_{1}, w_{2})\overline{ρ} is isomorphic to H_{Z}^{•}(w′_{1}, w′_{2})\overline{ρ} with Λ_{Z}-structure is twisted by (22) if and only if w_{1} + w_{2} = w′_{1} + w′_{2}.

Definition 4.7. Let (w_{1}, w_{2}) be a cohomological double-digit weight. We denote by 𝕋n.o ; \overline{ρ}(w_{1}, w_{2}) (resp. 𝕋Z ; \overline{ρ}(w_{1}, w_{2}), 𝕋ord ; \overline{ρ}(w_{1}, w_{2})) the Hecke algebra which is defined to be the subalgebra in the Λ_{n.o}-linear (resp. Λ_{Z}-linear, Λ_{ord}-linear) endomorphism ring of H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} (resp. H_{Z}^{d}(w_{1}, w_{2})\overline{ρ}, H_{ord}^{d}(w_{1}, w_{2})\overline{ρ}) generated by all Hecke operators.

Suppose that \overline{ρ} is realized in H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O)) and it is also realized in H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(w′_{1}, w′_{2}; K/ O)). By Corollary 4.6, 𝕋n.o ; \overline{ρ}(w_{1}, w_{2}) (resp. 𝕋Z ; \overline{ρ}(w_{1}, w_{2}), 𝕋ord ; \overline{ρ}(w_{1}, w_{2})) is isomorphic to 𝕋n.o ; \overline{ρ}(w′_{1}, w′_{2}) (resp. 𝕋Z ; \overline{ρ}(w′_{1}, w′_{2}), 𝕋ord ; \overline{ρ}(w′_{1}, w′_{2})) without any condition (resp. if w_{1} + w_{2} = w′_{1} + w′_{2}, if w_{1} = w′_{1} and w_{2} − w′_{2} ∈ ℤ[t]).

4.2. Control theorem for nearly ordinary cohomology. As explained in the introduction, the condition (Van\overline{ρ}) is essential for the construction of our p-adic L-function. Before starting the construction, we will list some of known sufficient conditions which ensure the condition (Van\overline{ρ}) stated at the beginning of §1.2.

Remark 4.8. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫) ∩ K_{11}(p).

(1) The paper [Di1] by Dimitrov shows that (Van\overline{ρ}) holds if the conditions

(⋆) and (⋆⋆) below are true:

(⋆) \overline{ρ} is realized in H_{n.o}^{d}(Y_{K_{1}(𝔫)}, L(a_{1}, a_{2}, K/ O)) of double-digit weight a = (a_{1}, a_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}] satisfying

∑_{τ∈I_{F}} (a_{2,τ} − a_{1,τ}) < p − 1.

(⋆⋆) The representation ⊗_{τ∈I_{F}} \overline{ρ}(τ^{−1} · τ) of the absolute Galois group

GF̃ is irreducible of order divisible by p, where F̃/ℚ denotes the Galois closure of F/ℚ in \overline{ℚ}.

We also remark that, if F is Galois over ℚ and if φ is a (parabolic) newform which is not a theta series, nor a twist of a base change, then for almost all ordinary primes p, \overline{ρ} = ρ_{φ} mod p satisfies (⋆) and (⋆⋆).

(2) The paper [LS] by Lan and Suh shows that the condition (Van\overline{ρ}) is true if the weight of \overline{ρ} is regular.

The general formalism of control theorems established in [H1, §10] and [H4] and the condition (Van\overline{ρ}) immediately implies the following theorem.

Theorem 4.9. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫)∩K_{11}(p) satisfying the condition (Van\overline{ρ}) stated at the beginning of §1.2. Then we have,

(i) (H_{n.o}^{q})\overline{ρ} is a free Λ_{n.o}-module of finite rank and vanishes if q ≠ d. (ii) For any cohomological double-digit weight (w_{1}, w_{2}), we have the following isomorphism:

tw^{(w_{1},w_{2})}((H_{n.o}^{d})\overline{ρ}) ⊗_{Λ_{n.o}} O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}]

≅ (H_{n.o}^{d}(w_{1}, w_{2}))\overline{ρ} ⊗_{Λ_{n.o}} O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}]

≅ H_{n.o}^{d}(Y_{11}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ}.

Corollary 4.10. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫)∩K_{11}(p) satisfying the condition (Van\overline{ρ}).

(i) H_{ord}^{q}(w_{1}, w_{2})\overline{ρ} (resp. H_{Z}^{q}(w_{1}, w_{2})\overline{ρ}) is a free Λ_{ord}-module (resp. Λ_{Z}- module) of finite rank and vanishes if q ≠ d.

(ii) We have the following isomorphisms:

H_{ord}^{d}(w_{1}, w_{2})\overline{ρ}⊗_{Λ_{ord}} O [(K_{0}(p^{α})/K_{1}(p^{α}) 𝔬^{×})_{p}] ≅ H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ}
H_{Z}^{d}(w_{1}, w_{2})\overline{ρ}⊗_{Λ_{Z}} O [(K_{0}(p^{α})/ZK_{11}(p^{α}))_{p}] ≅ H_{n.o}^{d}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ}

Theorem 4.9 (ii) and Corollary 4.10 (ii) ensure the existence of Hida families in our setting (cf. Remark 1.5 for the relation between our construction and Hida’s construction [H1], [H2], [H3]). Now, we recall that we have another group of global units:

(23)  \{(\begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix})| d ∈ 𝔬^{×}\} .

We have the following definition by using the above group.

Definition 4.11. (1) Let w = (w_{1}, w_{2}) be a cohomological pair and let α = (α_{1}, ⋯, α_{s}) be an element of ℤ^{s}. We define an ideal P_{w;α}^{n.o} ∈ Λ_{n.o} (resp. P_{w;α}^{Z} ∈ Λ_{Z} ) to be the ideal such that

tw^{−w_{1},−w_{2}}(P_{w;α}^{n.o}) = Ker [Λ_{n.o} ⟶ O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}]]

tw^{−w_{1},−w_{2}}(P_{w;α}^{Z}) = Ker [Λ_{Z} ⟶ O [(K_{0}(p^{α})/ZK_{11}(p^{α}))_{p}]]

(2) Let O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}]^{arith} (resp. O [(K_{0}(p^{α})/ZK_{11}(p^{α}))_{p}]^{arith}) the quotient ring of

O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}] (resp. O [(K_{0}(p^{α})/ZK_{11}(p^{α}))_{p}])

by the global units action of (23). We define an ideal P_{w;α}^{n.o,arith} ∈ Λ_{n.o} (resp. P_{w;α}^{Z,arith} ∈ Λ_{Z} ) to be the ideal such that

tw^{−w_{1},−w_{2}}(P_{w;α}^{n.o,arith}) = Ker [Λ_{n.o} ⟶ O [(K_{0}(p^{α})/K_{11}(p^{α}) 𝔬^{×})_{p}]^{arith}]
tw^{−w_{1},−w_{2}}(P_{w;α}^{Z,arith}) = Ker [Λ_{Z} ⟶ O [(K_{0}(p^{α})/ZK_{11}(p^{α}))_{p}]^{arith}]

Note that, P_{w;α}^{n.o} and P_{w;α}^{n.o,arith} are defined for any w = (w_{1}, w_{2}), while P_{w;α}^{Z} and P_{w;α}^{Z,arith} are defined only for w = (w_{1}, w_{2}) such that w_{1} +w_{2} = 0. By definition, we have P_{w;α}^{n.o,arith} ⊃ P_{w;α}^{n.o} (resp. P_{w;α}^{Z,arith} ⊃ P_{w;α}^{Z}).

4.3. Freeness result for nearly ordinary cohomology. Let us recall the following results:

Theorem 4.12. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫) ∩ K_{11}(p) satisfying the condition (Van\overline{ρ}). Let us fix a cohomological double-digit weight a = (a_{1}, a_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}]. Then, the Hecke algebra 𝕋n.o ; \overline{ρ}(a_{1}, a_{2}) (resp. 𝕋Z ; \overline{ρ}(a_{1}, a_{2})) is free over Λ_{n.o} (resp. Λ_{Z}) when it is realized in H_{n.o}^{•}(Y_{1}(𝔫p^{α}), L(a_{1}, a_{2}; K/ O)) in the sense given before Lemma 4.4. For any cohomological double-digit weight w = (w_{1}, w_{2}), we have exact control:

𝕋n.o ; \overline{ρ}(a_{1}, a_{2}) ⊗_{Λ_{n.o}} Λ_{n.o}/P_{w−a,α}^{n.o,arith} ≅ (𝔥_{w;α}^{n.o})\overline{ρ},

𝕋Z ; \overline{ρ}(a_{1}, a_{2}) ⊗_{Λ_{Z}} (Λ_{Z}/P_{w−a;α}^{Z,arith}) ≅ (𝔥_{w,α}^{Z})\overline{ρ}

where 𝔥_{w,α}^{n.o} (resp. 𝔥_{w,α}^{Z}) is the Hecke algebra acting on

H_{n.o}^{d}(Y_{11}(p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ} (resp. H_{n.o}^{d}(Y_{Z}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ}).

Proof. By Theorem 4.9 (ii) and by the assumption (Van\overline{ρ}), we have

H_{n.o}^{d}(a_{1}, a_{2})\overline{ρ} ⊗ Λ_{n.o}/P_{w−a;α}^{n.o,arith} ≅ (H^{d}(Y_{K_{1}(𝔫)∩K_{11}(p^{α})}, L(w_{1}, w_{2}; K/ O))\overline{ρ})^{PD}

at each cohomological double-digit weight w = (w_{1}, w_{2}), By specializing at P_{w−a;α}^{n.o,arith}, we identify

𝕋n.o ; \overline{ρ}(a_{1}, a_{2}) ⊗ Λ_{n.o}/P_{w−a;α}^{n.o,arith} ⊂ End_{O}(H_{n.o}^{d}(a_{1}, a_{2})\overline{ρ} ⊗ Λ_{n.o}/P_{w−a;α}^{n.o,arith})

with 𝔥_{w;α}^{n.o}. The statement for 𝔥_{w;α}^{Z} is done in the same way. Hence, we omit the proof for 𝔥_{w;α}^{Z}. □

Definition 4.13. Let w = (w_{1}, w_{2}) be a cohomological weight and let α be an element of ℕ^{s}. A ring homomorphism κ : Λ^{n.o} ⟶ \overline{ℚ}_{p} is called an arithmetic specialization of weight w and level p^{α} if the kernel of κ contains P_{w;α}^{n.o,arith}. For any finite Λ^{n.o}-algebra R, a ring homomorphism κ : R ⟶ \overline{ℚ}_{p} is called an arithmetic specialization of weight w and level p^{α} if Λ^{n.o} ⟶ R ⟶^{κ} \overline{ℚ}_{p} is an arithmetic specialization of weight w and level p^{α} in the above sense. The arithmetic specialization of Λ^{Z} or a finite Λ^{Z}-algebra is defined similarly.

Remark 4.14. By Theorem 4.12, we have a one-to-one correspondence under the condition (Van\overline{ρ}):

normalized Hilbert modular eigen cuspforms φ of double-digit weight (w_{1}, w_{2}), level K_{1}(𝔫 p^{α}) which are nearly ordinary at p and whose residual Galois representations are isomorphic to \overline{ρ}

⟶^{∼} \begin{cases}arithmetic specializations \\ κ : 𝕋\overline{ρ}⟶\overline{ℚ}_{p} \\ which have weight (w_{1}, w_{2}) and level p^{α} \end{cases}\}

We denote by κ_{φ} : 𝕋\overline{ρ}⟶\overline{ℚ}_{p} the arithmetic specialization associated to an eigen cuspform φ via the above correspondence.

As is explained after Definition 4.7, 𝕋Z ; \overline{ρ}(w_{1}, w_{2}) depends only on w_{1} + w_{2}. Hence, we denote 𝕋Z ; \overline{ρ}(w_{1}, w_{2}) by 𝕋Z,w_{1}+w_{2} ; \overline{ρ} . Note that, once we fix a branch R

of 𝕋n.o ; \overline{ρ}(w_{1}, w_{2}), a branch R^{Z,w_{1}+w_{2}} := R ⊗_{𝕋n.o ; \overline{ρ}(w_{1},w_{2})} 𝕋Z,w_{1}+w_{2} ; \overline{ρ} of 𝕋Z,w_{1}+w_{2} ; \overline{ρ} is

induced for each cohomological double-digit weight (w_{1}, w_{2}). We will present the following refinement of Theorem 4.9 which takes into counts the action of the whole Hecke algebra.

Theorem 4.15. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫) ∩ K_{11}(p) satisfying the condition (Van\overline{ρ}). Let w = (w_{1}, w_{2}) be a cohomological double-digit weight and let R be a branch of

𝕋n.o ; \overline{ρ}(w_{1}, w_{2}). Then,

(1) The R-module M(w_{1}, w_{2})_{R} := H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} ⊗_{𝕋}n.o ; \overline{ρ} R is free of rank one over R. Similarly, The R^{Z,w_{1}+w_{2}}-module M^{Z}(w_{1}, w_{2})_{R^{Z,w_{1}+w_{2}}} :=

M(w_{1}, w_{2})_{R} ⊗_{R} R^{Z,w_{1}+w_{2}} is free of rank 2^{d} over R^{Z,w_{1}+w_{2}}.

(2) For any character ϵ of the {±1}^{I_{F}}, M(w_{1}, w_{2})_{R}[ϵ] (resp. M^{Z}(w_{1}, w_{2})_{R^{Z,w_{1}+w_{2}}} [ϵ]) is free of rank one over R (resp. R^{Z,w_{1}+w_{2}}).

Proof. We will write the proof only for M(w_{1}, w_{2})_{R} and M(w_{1}, w_{2})_{R}[ϵ]. We can apply exactly the same proof for M^{Z}(w_{1}, w_{2})_{R^{Z,w_{1}+w_{2}}} and M^{Z}(w_{1}, w_{2})_{R^{Z,w_{1}+w_{2}}} [ϵ]. Since p is not equal to 2 and the group {±1}^{I_{F}} is of 2-power order, we have a direct sum decomposition as 𝕋n.o ; \overline{ρ}(w_{1}, w_{2})-module with respect to the action of the group {±1}^{I_{F}} on the Betti cohomology as follows:

H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} = ⨁_{ϵ} H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ}[ϵ].

This induces the direct sum decomposition as R-module

M(w_{1}, w_{2})_{R} = ⨁_{ϵ} M(w_{1}, w_{2})_{R}[ϵ].

Hence, the statement (2) implies the statement (1). We will prove the statement (2) in the rest of the proof. By Theorem 4.9 (ii) and by the condition (Van\overline{ρ}), we have

M(w_{1}, w_{2})_{R}[ϵ] ⊗_{R} R/𝔐 ≅ H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ}[ϵ] ⊗_{𝕋}n.o ; \overline{ρ} R/𝔐
≅ (H^{d}(Y_{K_{1}(𝔫)∩K_{0}(p)}, L(w_{1}, w_{2}; R/𝔐)[ϵ])^{PD}
≅ (H^{d}(Y_{K_{1}(𝔫)}, L(w_{1}, w_{2}; R/𝔐))[ϵ])^{PD}

where the second isomorphism holds thanks to the ordinarity. Note that the condition (Van\overline{ρ}) easily implies that H^{d}(Y_{K_{1}(𝔫)}, L(w_{1}, w_{2}; R/𝔐))[ϵ]) is of rank one over R/𝔐. By Nakayama’s lemma, M(w_{1}, w_{2})_{R}[ϵ] is a cyclic R-module. Thus, M(w_{1}, w_{2})_{R}[ϵ] is isomorphic to R/I for some ideal I of R. Since R is an integral domain which is integral over Λ_{n.o}, we have I ≠ 0 if and only if I ∩ Λ_{n.o} ≠ 0. Since M(w_{1}, w_{2})_{R}[ϵ] is a torsion-free Λ_{n.o}-module by Theorem 4.12, this implies I = 0. Thus M(w_{1}, w_{2})_{R}[ϵ] is a free R-module of rank one. □

4.4. p-adic periods. Let us consider the homomorphism

(24)  T^{n.o} ↠ ((𝔬 ⊗_{ℤ}ℤ_{p})^{×}/\overline{𝔬^{×}})_{p}.

induced from the map T = ((𝔬 ⊗_{ℤ}ℤ_{p})^{×}/\overline{𝔬^{×}})_{p} × ((𝔬 ⊗_{ℤ}ℤ_{p})^{×}/\overline{𝔬^{×}})_{p} ↠ ((𝔬 ⊗_{ℤ}ℤ_{p})^{×}/\overline{𝔬^{×}})_{p}, (a, d) ↦ ad^{−1}. Note that the kernel of (24) is equal to Z. By identifying ((𝔬 ⊗_{ℤ}ℤ_{p})^{×}/\overline{𝔬^{×}})_{p} canonically with Gal(F_{\{p\}}/F) where F_{\{p\}} is the maximal abelian pro-p extension of F unramified outside primes above p, we have the isomorphism

(25)  Λ ≅ Λ_{Z} ˆ⊗_{ℤ_{p}}ℤ_{p}[[Gal(F_{{p}}/F)]].

The isomorphism (25) induces the following isomorphism for any cohomological double-digit weight (w_{1}, w_{2}) ∈ ℤ[I_{F}] × ℤ[I_{F}]:

(26)  H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} ≅ H_{Z}^{d}(w_{1}, w_{2})\overline{ρ} ˆ⊗_{ℤ_{p}}ℤ_{p}[[Gal(F_{\{p\}}/F)]].

Let w = (w_{1}, w_{2}) ∈ ℤ[I_{F}]×ℤ[I_{F}] be a cohomological double-digit weight. Since w_{1} + w_{2} ∈ tℤ, there exists an integer r = r(w) such that w_{1} + w_{2} = rt, we denote by w^{Z} = (w_{1}^{Z}, w_{2}^{Z}) ∈ ℤ[I_{F}] × ℤ[I_{F}] by

w_{i}^{Z} = \begin{cases}w_{i} − (\frac{r}{2} + 1)t & if r is even, \\ w_{i} − \frac{(r+1)}{2}t & if r is odd, \end{cases}

for i = 1, 2. We also have the identification:

(27)  H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} ≅ H_{n.o}^{d}(w_{1}^{Z}, w_{2}^{Z})\overline{ρ}.

induced by the twist by χ_{cyc}^{\frac{r(w)}{2} +1} (resp. χcyc^{\frac{r(w)+1}{2}} ) when r(w) is even (reps. odd). The isomorphisms (26) and (27) induce the following isomorphism:

(28)  H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ} ≅ H_{Z}^{d}(w_{1}^{Z}, w_{2}^{Z})\overline{ρ} ˆ⊗_{ℤ_{p}}ℤ_{p}[[Gal(F_{\{p\}}/F)]].

Note that w_{1}^{Z}+w_{2}^{Z} is equal to −2t (resp. −t) when r(w) is even (resp. odd). For a branch R of 𝕋n.o ; \overline{ρ} ⊂ End_{Λ_{n.o}}(H_{n.o}^{d}(w_{1}, w_{2})\overline{ρ}), the identification (28) induces the following isomorphism:

(29)  R ≅ \begin{cases}R^{Z,−2t} ˆ⊗_{ℤ_{p}}ℤ_{p}[[Gal(F_{\{p\}}/F)]] & when r(w) is even, \\ R^{Z,−t} ˆ⊗_{ℤ_{p}}ℤ_{p}[[Gal(F_{\{p\}}/F)]] & when r(w) is odd, \end{cases}

where R^{Z,−2t} (resp. R^{Z,−t}) is a branch of 𝕋Z,−2t ; \overline{ρ} (resp. 𝕋Z,−t ; \overline{ρ}) when r(w) is even (resp. odd). Thus, we have the following one-to-one map:

(30) \begin{cases}arithmetic specializations κ of weight w = (w_{1}, w_{2}) \\ and level K_{1}(𝔫p^{α}) on R \end{cases}\}

⟶ \begin{cases}a pair (κ^{Z}, η_{κ}) which consists of \\ arithmetic specializations κ^{Z} on R^{Z,−2t} (resp. R^{Z,−t}) \\ of weight w^{Z} = (w_{1}^{Z}, w_{2}^{Z}) \\ η_{κ} : Gal(F_{\{p\}}/F) ⟶ \overline{ℚ}_{p}^{×} such that η_{κ}/χ_{cyc}^{\frac{r(w)}{2} +1} \\ (resp. η_{κ}/χcyc^{\frac{r(w)+1}{2}} ) is of finite order \end{cases}\} ,

when r(w) is even (resp. odd).

Let us fix an R^{Z,−2t}-basis element (resp. R^{Z,−t}-basis element) b^{even,ϵ} ∈ M^{Z}(−2t, 0)_{R^{Z,−2t}}[ϵ] (resp. b^{odd,ϵ} ∈ M^{Z}(−t, 0)_{R^{Z,−t}}[ϵ]). For any arithmetic specialization κ on R of cohomological double-digit weight w = (w_{1}, w_{2}) with w_{1} + w_{2}, we have

(31) κ^{Z}(M(w_{1}^{Z}, w_{2}^{Z})_{R^{Z,w^{Z}_{1} +w^{Z}_{2}}} [ϵ])

≅ (H_{n.o}^{d}(Y_{Z}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; K/O))PD ; \overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{Z})\overline{ρ}} κ(R)

≅ (H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; K/ O))PD ; \overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R)

Hence, if we denote by b_{κ^{Z}}^{even,ϵ} (resp. b_{κ^{Z}}^{odd,ϵ}) the image of b^{even,ϵ} (resp. b^{odd,ϵ}) via the specialization κ^{Z}, we have the canonical isomorphism

(32)  (H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; K/ O))PD ; \overline{ρ}[ϵ]) ⊗_{(𝔥_{w;α}^{n.o})\overline{ρ}} κ(R) ≅ b_{κ^{Z}}^{ϵ} · κ(R)

Definition 4.16. Assume the condition (Van\overline{ρ}) for \overline{ρ} associated to the fixed branch R. Let κ = κ_{φ} be an arithmetic specialization on R which corresponds to a nearly ordinary eigenform φ of level K_{1}(𝔫p^{α}) and of weight w = (w_{1}, w_{2}). Denote by 𝔬_{φ} the ring of integers of the Hecke field for φ and let be the localization at the prime over p induced from the fixed embedding 𝔬_{φ} ↪ \overline{ℤ}_{p}.

H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; 𝔬_{φ,(p)}))\overline{ρ}[ϵ]

⟶ (H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; O))\overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R)

⟶ (H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R)

where 𝔬_{φ,(p)} is the discrete valuation ring defined before Definition 3.5 and O contains the completion of 𝔬_{φ,(p)}. The image of the module H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}, w_{2}; 𝔬_{φ,(p)}))\overline{ρ}[ϵ] gives an 𝔬_{φ,(p)}-rational structure on the image H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[ϵ]⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R) which is a free κ(R)-module of rank one. Since 𝔬_{φ,(p)} is principal, the image of the above composite map is free of rank one over 𝔬_{φ,(p)}. Let v be an 𝔬_{φ,(p)}-basis of the image. On the other hand, we have the isomorphism

(H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R)

≅ (H_{n.o}^{d}(Y_{1}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; K/O))∨ ; \overline{ρ}[ϵ]) ⊗_{(𝔥_{w^{Z};α}^{n.o})\overline{ρ}} κ(R) ≅ b_{κ^{Z}}^{ϵ} · κ(R)

via the Poincare duality theorem. We define the p-adic period C_{φ,p}^{ϵ} = C_{φ,p,v}^{ϵ} in κ(R)^{×} by b_{κ^{Z}}^{ϵ} = C_{φ,p,v}^{ϵ} · v

Remark 4.17. (1) A p-adic period does not depend on the twist by a finite character. That is, if an arithmetic specialization κ on R corresponds to a modular form φ and if an arithmetic specialization κ′ on R corresponds to a modular form φ′ = φ ⊗ ϕ with a finite character ϕ of Gal(F_{∞}/F), we have C_{φ,p,v}^{ϵ} = C_{φ′,p,v}^{ϵ}.

(2) Note that the p-adic period C_{φ,p,v}^{ϵ} as well as the complex period C_{φ,∞,v}^{ϵ}

depend on the choice of v and they are well-defined only up to multiplication of an element in (𝔬_{φ,(p)})^{×}. Nevertheless, for two different choices of v, the p-adic and complex periods are incremented by the same amount. Hence, the pair (C_{φ,p}^{ϵ}, C_{φ,∞}^{ϵ}) (or the ratio “C_{φ,p}^{ϵ}/C_{φ,∞}^{ϵ}”) is independent of the choice of v. By this reason, later in the interpolation of the p-adic L-function, we will denote the periods by C_{φ,p}^{ϵ} and C_{φ,∞}^{ϵ} without specifying the choice of v.

5. p-adic L-function

5.1. Statement of the main theorem (Theorem A). Under the one-to-one correspondence between arithmetic specializations of R and modular forms φ on the Hida family over R, we sometimes denote an arithmetic specialization κ of R by κ_{φ} to specify the corresponding modular form φ (cf. Remark 4.14). As is explained after Definition 4.7, 𝕋n.o ; \overline{ρ}(w_{1}, w_{2}) is independent of the choice of a cohomological weight (w_{1}, w_{2}). Hence, we denote 𝕋n.o ; \overline{ρ}(w_{1}, w_{2}) by 𝕋n.o ; \overline{ρ}. We sometimes denote the complex and p-adic periods C_{φ,∞,v}^{ϵ} and C_{φ,p,v}^{ϵ} by C_{φ,∞,v} and C_{φ,p,v} when ϵ is the trivial character 1. We also denote b^{even,ϵ} (resp. b^{odd,ϵ}) by b^{even} (resp. b^{odd}) when ϵ is the trivial character 1. The theorem below is the main theorem of our paper which we denoted by Theorem A in the introduction.

Theorem 5.1. Let \overline{ρ} be a residual Galois representation of G_{F} which is nearly ordinary at p and of level K_{1}(𝔫) ∩ K_{11}(p) satisfying the conditions (Van\overline{ρ}) and (Ir\overline{ρ}) stated at the beginning of §1.2. Let R be a branch of 𝕋n.o ; \overline{ρ} and let us fix an R^{Z,−2t}-basis (resp. R^{Z,−t}-basis) element b^{even} (resp. b^{odd}) of M^{Z}(−2t, 0)_{R^{Z,−2t}}[1] (resp. M^{Z}(−t, 0)_{R^{Z,−t}}[1]) (see Theorem 4.15). Assume that R^{Z,−2t} (resp. R^{Z,−2t}) is a Gorenstein algebra. Then, there exists a p-adic L-function L_{p}(R; b^{even}) ∈ R (resp. L_{p}(R; b^{odd}) ∈ R) which satisfies the interpolation property

\frac{κ_{φ}(L_{p}(R; b^{even}))}{C_{φ,p}} = ∏_{𝔭 |(p)} A_{𝔭}(φ)\frac{L(φ, 0)}{C_{φ,∞}}

(resp. the same interpolation for L_{p}(R; b^{odd})),

for every p-stabilized nearly ordinary eigen cuspform φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔫p^{∗})) of critical cohomological weight w = (w_{1}, w_{2}) on R such that w_{1} + w_{2} = rt for an even (resp. odd) integer r where C_{φ,p}^{ϵ} is a p-adic period (cf. Definition 4.16) and A_{𝔭}(φ) is defined as follows:

A_{𝔭}(φ) = \begin{cases}(1 − \frac{1}{N_{F} (𝔭)a_{𝔭}(φ)}) & if 𝔭 ∤ Cond(ϕ_{0}ω^{−j}), \\ (\frac{1}{N_{F} (𝔭)a_{𝔭}(φ^{0})})^{ord_{𝔭} Cond(ϕ_{0}ω^{−j})} & if 𝔭 |Cond(ϕ_{0}ω^{−j}). \end{cases}

where φ^{0} (resp. ϕ_{0}) is a unique ordinary form of weight (w_{1}, w_{2}) (resp. a unique finite order character of Gal(F̃_{∞}/F)) which is not a twist of another nearly ordinary form of weight (k − 2, j − 1) by a finite order Hecke character of F with p-primary conductor such that φ = φ^{0} ⊗ ϕ_{0}ω^{−j}. The number N_{F} (𝔭) is the absolute norm of the prime ideal 𝔭.

5.2. Construction of L_{p}(R; b^{even}) and L_{p}(R; b^{odd}). The p-adic L-function L_{p}(R; b^{even}) (resp. L_{p}(R; b^{odd})) will be constructed as an element of R ≅ R^{Z,−2t}[[Gal(F_{{p}}/F)]] (resp. R ≅ R^{Z,−t}[[Gal(F_{{p}}/F)]]) interpolating special values of L-function of Hilbert modular forms twisted by Hecke characters. Since the construction of L_{p}(R; b^{odd}) is parallel to that of L_{p}(R; b^{even}), we will explain mainly the construction of L_{p}(R; b^{even}). Note that, for a fixed cohomological weight w^{Z} = (w_{1}^{Z}, w_{2}^{Z}) ∈ ℤ[I_{F}] × ℤ[I_{F}] with w_{1}^{Z} + w_{2}^{Z} = −2t, we have:

R^{Z,−2t}[[Gal(F_{\{p\}}/F)]] = \underleftarrow{lim}_{α,β} R_{w^{Z};α}^{Z}[Cl_{F}^{+}(p^{β})_{p}],

where R_{w^{Z},α}^{Z} is a local component of the full Hecke algebra 𝔥_{α}^{Z,(w_{1}^{Z},w_{2}^{Z})} acting on the cohomology H^{d}(Y_{Z}(𝔫p^{α}), L(w_{1}^{Z}, w_{2}^{Z}; K/ O)) generated by T_{λ} with λ prime to p 𝔫, U_{λ} with λ| 𝔫, and U_{ϖ_{i}} for ϖ_{i} running over all uniformizers of 𝔭_{i}. Recall that, it is an immediate consequence of Corollary 4.6 that \underleftarrow{lim}_{α} R_{w^{Z};α}^{Z} depends only on w_{1}^{Z} + w_{2}^{Z}. Hence the notation R^{Z,−2t} is justified (cf. notations given after Remark 4.14). Therefore, in order to construct L_{p}(R; b^{even}), it is enough to construct an inverse system

{x_{α,β} = x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ R_{w^{Z},α}^{Z}[Cl_{F}^{+}(p^{β})_{p}]}_{α,β}

with respect to multi-indexes α and β for each w^{Z} = (w_{1}^{Z}, w_{2}^{Z}) with w_{1}^{Z} + w_{2}^{Z} = −2t in ℤ[I_{F}]. For w^{Z} = (w_{1}^{Z}, w_{2}^{Z}) with w_{1}^{Z} + w_{2}^{Z} = −2t, we will define

L_{p}^{(w_{1}^{Z},w_{2}^{Z})}(R; b^{even}) ∈ R^{Z,−2t}[[Gal(F_{\{p\}}/F)]] to be \underleftarrow{lim}_{α,β}x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) and we

denote L_{p}^{(−2t,0)}(R; b^{even}) by L_{p}(R; b^{even}). Finally, by help of the canonical identification R = R^{Z,−2t}[[Gal(F_{{p}}/F)]], we regard L_{p}(R; b^{even}) as an element of R. The construction of x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) will occupy the next paragraph 5.3 and the desired interpolation property will be proved in the paragraph 5.4.

5.3. Proof of the distribution property of Theorem A. Recall that, for every u ∈ Cl_{F}^{+}(p^{β})_{p}, we constructed:

ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})} ∈ Hom_{O}(H^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1], K)

after the equation (17) of Section 3. Note that we can omit the compact support, since the localization at \overline{ρ} kills the boundary cohomology. The R^{Z,−2t}-basis tw^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ M^{Z}(w_{1}^{Z}, w_{2}^{Z})_{R^{Z,−2t}}[1] from Paragraph 4.4 specializes to an R_{w^{Z},α}^{Z}-basis b_{α}^{even} of H_{c}^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1]. Now, let us consider the composite map:

(33)

Hom_{O}(H^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1], K) ↠ Hom_{O}(R_{w^{Z} ,α}^{Z}, K) →^{∼} R_{w^{Z} ,α}^{Z} ⊗_{O} K,

where the first morphism of (33) is induced by R_{w^{Z},α}^{Z}-basis b_{α}^{even}.

Definition 5.2. We denote by ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even}) ∈ R_{w^{Z},α}^{Z} ⊗_{O} K the image of

ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})} by the map (33). Then, for each α = (α_{1}, ⋯, α_{s}), β = (β_{1}, ⋯, β_{s}) ∈

ℕ^{s} with α ≤ β and with β_{j} > 0 for all j, we define x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ (R_{w^{Z},α}^{Z}⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}] to be:

x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) = U_{0}(p)^{−β} · ( ∑_{u∈Cl_{F}^{+}(p^{β})_{p}} ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even}) · [u]) .

Here, U_{0}(p)^{β} is the modified Hecke operator ∏_{1≤i≤s}(U_{0,𝔭_{i}})^{β_{i}} associated to fixed

uniformizers ϖ_{𝔭_{i}} of 𝔭_{i} (see 2.3 for the definition).

By the definition of ordinary forms in 2.3, the operator U_{0}(p) defined as in Definition 5.2 is a unit in the nearly ordinary Hecke algebra R^{Z,−2t} and its quotients R_{w^{Z},α}^{Z}.

We will show that the set {x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ (R_{w^{Z},α}^{Z} ⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}]}_{α,β}

forms a projective system, by checking the compatibility with respect to α and β. Thus, our construction via the inverse limit of x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) for the p-adic L-function L_{p}(R; b^{even}) is an analogue of Iwasawa’s construction via Stickelberger elements for Kubota-Leopoldt p-adic L-function. The compatibility with respect to α is not so hard.

Lemma 5.3. For each β and for each pair α, α′ satisfying α ≤ α′, x_{α′,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) is mapped to x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) via the natural map (R_{w^{Z},α′}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β})_{p}] ↠ (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β})_{p}].

Proof. Note that {b_{α}^{even}}_{α≥0} is a compatible system by definition. Hence, it

follows that x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) is the image of x_{α′,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) by the natural homomorphism

(R_{w^{Z},α′}^{Z} ⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}] ↠ (R_{w^{Z},α}^{Z} ⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}] .

Note also that the image of U_{0}(p)^{β} in R_{w^{Z},α}^{Z} ⊗_{O} K is compatible with respect

to α once β is fixed. Thus, we have the compatibility of x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) and

x_{α′,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}). □

The compatibility with respect to β which is more delicate than that of α is proved below.

Lemma 5.4. For each α and for each pair β, β′ satisfying β ≤ β′, x_{α,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) is mapped to x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) via the natural map (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β′})_{p}] ↠ (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β})_{p}].

Proof. It suffices to show the equality:

(34)  U_{0}(p)^{β′−β}ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})} = ∑_{u′} ev_{u′,α,β′}^{(w_{1}^{Z},w_{2}^{Z})},

for any u ∈ Cl_{F}^{+}(p^{β})_{p} in the space Hom_{O}(H^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1], O), where u′ runs through elements in Cl_{F}^{+}(p^{β′})_{p} which are mapped to u. By an inductive argument, the proof of (34) is reduced to the case of β ≤ β′ where there exists j such that β′ = (β_{1}, ⋯, β_{j−1}, β_{j} + 1, β_{j+1}, ⋯, β_{s}). Then, the equation (34) is reduced to

(35)  U_{0,ϖ_{j}}ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})} = ∑_{u′} ev_{u′,α,β′}^{(w_{1}^{Z},w_{2}^{Z})},

where the sum runs through archimedean embeddings of F with which ϖ_{j}-adic topology is compatible with the induced p-adic embedding. For any α ≥ 0, we

have the coset decomposition of the double coset K_{Z}(p^{α}) (\begin{matrix}ϖ_{𝔭_{j}} & 0 \\ 0 & 1 \end{matrix}) K_{Z}(p^{α})

which induces the operator U_{𝔭_{j}} as follows:

(36)  K_{Z}(p^{α}) (\begin{matrix}ϖ_{𝔭_{j}} & 0 \\ 0 & 1 \end{matrix}) K_{Z}(p^{α}) = ∐_{a} (\begin{matrix}ϖ_{𝔭_{j}} & a \\ 0 & 1 \end{matrix}) K_{Z}(p^{α}).

where a in the right-hand side runs through a set of representatives of 𝔬_{+}^{×}-orbits in 𝔬 /(ϖ_{𝔭_{j}}) = 𝔬 /𝔭_{j}. By the sequence (16), u is identified with a set of certain numbers of elements {u_{i}} where each u_{i} is an (𝔬_{+}^{×})_{p}-orbits in 𝔬 /p^{β} = p^{−β}/ 𝔬.

Since we have p^{−β}ϖ_{𝔭_{j}}^{−1} = p^{−β′}, \frac{a + u_{i}}{ϖ_{𝔭_{j}}} runs through the set of representatives of

a fixed u_{i} ∈ p^{−β}/ 𝔬 in p^{−β′}/ 𝔬 when a varies the representatives of the elements which appeared in (36). Here, when we talk about these representatives, we consider p^{−β′}/ 𝔬 ↠ p^{−β}/ 𝔬 induced from 𝔬 /p^{β′} ↠ 𝔬 /p^{β} by fixed identifications p^{−β}/ 𝔬 = 𝔬 /p^{β} and p^{−β′}/ 𝔬 = 𝔬 /p^{β′}. This completes the proof. □

If there is an index j with β_{j} = 0, the element x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})})(b_{κ^{Z}}^{even}) is not defined at Definition 5.2. We extend the definition to general β as follows.

Definition 5.5. Let S be the subset of {1, …, s} such that β_{j} = 0 for j ∈

S and β_{j} > 0 for j ∉ S. Then, we define x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ (R_{w^{Z},α}^{Z} ⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}] to be:

x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) =

= ∏_{j∈S} (1 − \frac{1}{N_{F} (𝔭_{j})U_{0,𝔭_{j}}} ) · U_{0}(p)^{−β} · ( ∑_{u∈Cl_{F}^{+}(p^{β})_{p}} ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even}) · [u]) .

With this extended definition, we have the following proposition:

Proposition 5.6. For each pair α, α′ satisfying α ≤ α′ and for each pair β, β′ satisfying β ≤ β′, x_{α′,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) is mapped to x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) via the natural map (R_{w^{Z},α′}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β′})_{p}] ↠ (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β})_{p}].

Proof. Thanks to Lemma 5.3 and Lemma 5.4, the desired compatibility holds unless there is an index j with β_{j} = 0. Hence, we fix α and we assume that there is j such that β_{j} = 0. Let β′ = (β′_{1}, ⋯, β′_{s}) be the multi-index such

that β′_{j} = 1 and β′_{k} = β_{k} for k ≠ j. It suffices to prove that x_{α,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even})

is mapped to x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) via the natural map (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β′})_{p}] ↠ (R_{w^{Z},α}^{Z} ⊗_{O} K)[Cl_{F}^{+}(p^{β})_{p}]. Let us calculate the ratio:

(37)  ∑ ; ũ′ev_{ũ,α,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even})/ ∑ ; ũ ev_{ũ,α,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even})

where ũ runs through the element of 𝔬_{+}^{×}-orbits of ∏_{k≠j}(𝔬 /𝔭_{k}^{β_{k}})^{×} × 𝔬 /𝔭_{j} which are lifts of u in the summation of the denominator, and ũ runs through the element of 𝔬_{+}^{×}-orbits of ∏_{k≠j}(𝔬 /𝔭_{k}^{β_{k}})^{×} × (𝔬 /𝔭_{j})^{×} which are lifts of u in the

summation of the numerator. The ratio is equal to the value of 1− \frac{1}{N_{F} (𝔭_{j})U_{0,𝔭_{j}}}

acting on ev_{ũ,α,β′}^{(w_{1}^{Z},w_{2}^{Z})}(b_{α}^{even}). By modifying the definition counting this factor, we have the desired compatibility, which completes the proof. □

Finally, We have the following corollary.

Corollary 5.7. Assume the condition (Ir\overline{ρ}). For each α = (α_{1}, ⋯, α_{s}), β =

(β_{1}, ⋯, β_{s}) ∈ ℕ^{s} with α ≤ β, the element x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ (R_{w^{Z},α}^{Z} ⊗_{O} K) [Cl_{F}^{+}(p^{β})_{p}] introduced at Definition 5.5 lies in the subalgebra R_{w^{Z},α}^{Z} [Cl_{F}^{+}(p^{β})_{p}].

Proof. Let us consider the case of α = 0. In this case, since we consider the

set of all u in Cl_{F}^{+}(p^{β})_{p}, (∐_{u} H_{u})/ 𝔬_{+}^{×} is well-defined. (Note that (\begin{matrix}e & 0 \\ 0 & 1 \end{matrix}) is

contained in K_{1}(𝔫) ⊂ K_{11}(p^{0}) = K_{1}(𝔫) for any unit e ∈ 𝔬_{+}^{×}.) We recall that the Betti cohomology with compact support H_{c}^{d}(Y_{K_{1}(𝔫)}; L(w_{1}, w_{2}; K)) is isomorphic to the étale cohomology with compact support H_{ét,c}^{d}((Y_{K_{1}(𝔫)})\overline{ℚ}; L(w_{1}, w_{2}; K)) where (Y_{K_{1}(𝔫)})\overline{ℚ} is a model of Y_{K_{1}(𝔫)} over \overline{ℚ}. By the fact that Y_{K_{1}(𝔫)} has a model over ℚ, the étale cohomology H_{ét,c}^{d}((Y_{K_{1}(𝔫)})\overline{ℚ}; L(w_{1}, w_{2}; K)) has a natural Galois action of G_{ℚ}. By [BL], for each eigen cusp form φ in \overline{ρ}-component, it is known that we have

H_{ét,c}^{d}((Y_{K_{1}(𝔫)})\overline{ℚ}; L(w_{1}, w_{2}; K))[λ_{φ}] ≅ (ρ_{φ})^{⊗[F:ℚ]}

as a representation of G_{F} where ρ_{φ} is a Galois representation of φ on K which lifts our \overline{ρ}. We denote by A the Pontrjagin dual of the image of the integral cohomology H_{ét,c}^{d}((Y_{K_{1}(𝔫)})\overline{ℚ}; L(w_{1}, w_{2}; O))[λ_{φ}], in H_{ét,c}^{d}((Y_{K_{1}(𝔫)})\overline{ℚ}; L(w_{1}, w_{2}; K))[λ_{φ}]. The representation A is an O-module isomorphic to (K/O)^{⊕2^{d}} with continuous G_{ℚ}-action and A[ϖ] is isomorphic to the dual of \overline{ρ}^{⊗[F:ℚ]} as a representation of G_{F} for a uniformizer ϖ of O. Thus, the element

x_{0,β}^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) = ∏_{1≤j≤s} (1 − \frac{1}{N_{F} (𝔭_{j})U_{0,𝔭_{j}}} ) · ev_{u,0,β}^{(w_{1}^{Z},w_{2}^{Z})}(b_{0}^{even}) · [u] .

lies in R_{w^{Z},0}^{Z} [Cl_{F}^{+}(p^{β})_{p}] since we can choose a finite-index subgroup E of 𝔬_{+}^{×} whose index [𝔬_{+}^{×} : E] causes the denominator to be E = 𝔬_{+}^{×} (cf. Definition

3.2) and since the assumption that ev_{u,0,β}^{(w_{1}^{Z},w_{2}^{Z})}(b_{0}^{even}) is not integral would induce a G_{F} -stable line in A[ϖ], which contradicts to the result of [BL] and our condotion (Ir\overline{ρ}). By Lemma 5.3 and by the fact that R_{w^{Z},α}^{Z} is the preimage of R_{w^{Z},0}^{Z} ⊂ R_{w^{Z},0}^{Z}⊗_{O} K under the natural projection map R_{w^{Z},α}^{Z} ⊗_{O} K ⟶ R_{w^{Z},0}^{Z} ⊗_{O} K, we obtain the desired integrality for any α. □

The R^{Z,−2t}-basis tw^{(w_{1}^{Z},w_{2}^{Z})}(b^{even}) ∈ M^{Z}(w_{1}^{Z}, w_{2}^{Z})_{R^{Z,−2t}}[1] from Paragraph 4.4 specializes to an R_{w^{Z},α}^{Z}-basis b_{α}^{even} of H_{c}^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1]. Moreover, there exists an O-algebra isomorphism Hom_{O}(R_{w^{Z},α}^{Z}, O) ≅ R_{w^{Z},α}^{Z} since R_{w^{Z},α}^{Z} is Gorenstein by assumption. Now, let us consider the composite map: (38)

Hom_{O}(H^{d}(Y_{Z}(𝔫p^{α}); L(w_{1}^{Z}, w_{2}^{Z}; O))\overline{ρ}[1], O) ↠ Hom_{O}(R_{w^{Z},α}^{Z}, O) →^{∼} R_{w^{Z},α}^{Z},

where the first isomorphism of (38) is induced by R_{w^{Z},α}^{Z}-basis b_{α}^{even} and the second isomorphism of (38) is the one induced by the Gorenstein property.

5.4. Proof of the interpolation property of Theorem A. We will evaluate L_{p}(R; b^{even}) at an arithmetic specialization κ = κ_{φ} of R corresponding to a cuspform φ ∈ S_{w_{1},w_{2}}(K_{1}(𝔫p^{α})). Let r and η be as introduced in 4.4. We have the following commutative diagram with a unique arithmetic specialization κ_{φ}^{Z} on R:

R  ⟶^{κ_{φ}} \overline{ℚ}_{p}
↓  ∥
R^{Z,−2t}[[Gal(F_{\{p\}}/F)]]  ⟶_{η_{κ}∘κ^{Z}}  \overline{ℚ}_{p}.

Let φ^{Z} be the eigen cuspform corresponding to the arithmetic specialization κ_{φ}^{Z} and let (w_{1}^{0}, w_{2}^{0}) be the critical pair which is the largest among the shifts (w_{1}^{Z} + nt, w_{2}^{Z} + nt) by integer multiples nt of t which are critical. We denote by φ^{0} the unique twist of φ^{Z} of weight (w_{1}^{0}, w_{2}^{0}) which is not a twist of another nearly ordinary form of smaller level by a finite order Hecke character of F. (Since it will be a long and tedious description, we do not repeat what the twist by a finite order character of a modular form or the corresponding arithmetic specialization means.) Note that there exists a unique integer r_{0} (resp. r_{Z}) and a unique finite order character ϕ_{Z} (resp. ϕ_{0}) of Cl_{F}^{+}(p^{∞}) such that the form the newform associated with φ ⊗ N_{F}^{−r_{0}}ϕ_{0}^{−1} (resp. φ^{Z} ⊗ N_{F}^{−r_{Z}} ϕ_{Z}^{−1}) is equal to φ^{0}. By definition, we have:

κ_{φ}(L_{p}(R; b^{even})) = (η_{κ} ∘ κ_{φ}^{Z})(L_{p}^{(−2t,0)}(R; b^{even})).

The element κ_{φ}^{Z}(L_{p}^{(−2t,0)}(R; b^{even})) is an element in κ(R)[[Gal(F_{{p}}/F)]] ⊂ κ(R)[[Gal(F_{{p}}/F)]]. Recall that η_{κ} is equal to χ_{cyc}^{r_{Z}+1}ϕ_{Z} for some finite Hecke character ϕ_{Z}. By construction, κ_{φ}^{Z}(L_{p}^{(−2t,0)}(R; b^{even})) ⊗ ϕ_{Z}^{−1} is equal to

(\underleftarrow{lim}_{β} κ_{φ}^{Z}(x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})})(b_{κ^{Z}}^{even}))⊗ϕ_{Z}^{−1} = \underleftarrow{lim}_{β}(κ_{φ}^{Z}⊗ϕ_{Z}^{−1})(x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})}⊗ϕ_{Z}^{−1})(b_{κ^{Z}}^{even}⊗ϕ_{Z}^{−1}).

Let v^{0} be a basis of a free 𝔬_{φ^{0},(p)}-module H_{c}^{d}(Y_{1}(𝔫 p^{α}); L(w_{1}^{0}, w_{2}^{0}; 𝔬_{φ^{0},(p)}))[λ_{φ}, ϵ] of rank one. Then, by Definition 5.2, we have:

\underleftarrow{lim}_{β}(κ_{φ}^{Z} ⊗ ϕ_{Z}^{−1})(x_{α,β}^{(w_{1}^{Z} ,w_{2}^{Z})} ⊗ ϕ_{Z}^{−1})(b_{κ^{Z}}^{even} ⊗ ϕ_{Z}^{−1})

= \underleftarrow{lim}_{β} ( ∑_{u∈Cl_{F}^{+}(p^{β})_{p}}U_{0}(p)^{−β}ev_{u,α,β}^{(w_{1}^{Z} ,w_{2}^{Z})}(b_{κ^{Z}}^{even} ⊗ ϕ_{Z}^{−1})[u])

= C_{φ,p,v^{0}} · \underleftarrow{lim}_{β} ( ∑_{u∈Cl_{F}^{+}(p^{β})_{p}}U_{0}(p)^{−β}ev_{u,α,β}^{(w_{1}^{Z} ,w_{2}^{Z})}(v^{0}[r_{Z}])[u]) .

The evaluation of \underleftarrow{lim}_{β} ∑_{u∈Cl_{F}^{+}(p^{β})_{p}}U_{0}(p)^{−β}ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})}(v^{0}[r_{Z}])[u] at χ_{cyc}^{r_{0}−r_{Z}} ϕ_{0} is

equal to the evaluation of \underleftarrow{lim}_{β} ∑_{u∈Cl_{F}^{+}(p^{β})_{p}}U_{0}(p)^{−β}ev_{u,α,β}^{(w_{1},w_{2})}(v^{0}[r_{0}])[u] at ϕ_{0} by

the standard argument between Tate twists of critical values. Let ∏_{j=1}^{s} 𝔭_{j}^{β_{j}} be

the conductor of ϕ_{0}. If we have β_{j} > 0 for all j, we simply evaluate U_{0}(p)^{−β} ·

(∑_{u∈Cl_{F}^{+}(p^{β})_{p}} ev_{u,α,β}^{(w_{1}^{Z},w_{2}^{Z})}(v^{0}[r_{0}]) [u]) . via the character ϕ_{0}, whose value is equal to:

∏_{j=1}^{s} (\frac{1}{N_{F} (𝔭)a_{𝔭}(φ^{0})})^{β_{j}} ( ∑_{u∈Cl_{F}^{+}(p^{β})_{p}}ϕ_{0}^{−1}(u)ev_{u,α,β}^{(w_{1},w_{2})}(v^{0}[r_{0}]))

= ∏_{j=1}^{s} (\frac{1}{N_{F} (𝔭)a_{𝔭}(φ^{0})})^{β_{j}} G(ϕ_{0}^{−1})\frac{L(φ^{0} ⊗ N_{F}^{r_{0}}, 0)}{C_{φ,∞,v^{0}[r_{0}]}}

= ∏_{j=1}^{s} (\frac{1}{N_{F} (𝔭)a_{𝔭}(φ^{0})})^{β_{j}} \frac{L(φ^{0}, 0)}{C_{φ,∞,v}},

where the first equality follows from Corollary 3.7 and the second equality follows from Proposition 3.8. This completes the proof of the interpolation property for L_{p}(R; b^{even}) for κ_{φ} with a_{𝔭_{j}}(φ) = 0 for all j. If there are j with a_{𝔭_{j}}(φ) ≠ 0 or equivalently β_{j} = 0 for some component of the conductor ∏_{j=1}^{s} 𝔭_{j}^{β_{j}} of ϕ_{0}, the interpolation is clear from the extendes

definition of the element x_{α,β}^{(w_{1}^{Z},w_{2}^{Z})})(b_{κ^{Z}}^{ϵ}) at Definition 5.5 (see also Proposition 5.6) which complete the proof for L_{p}(R; b^{even}). The proof for L_{p}(R; b^{odd}) is done in the same way except that we replace the role of M^{Z}(−2t, 0)[1] by that of M^{Z}(−t, 0)[1].

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Tadashi Ochiai Graduate school of Science Osaka University 1-1 Machikaneyama Toyonaka Osaka 560-0043 Japan ochiai@math.sci.osaka-u.ac.jp 850

Documenta Mathematica 17 (2012)