A GENERALIZATION OF THE COLEMAN MAP FOR HIDA DEFORMATIONS

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1 A GENERALIZATION OF THE COLEMAN MAP FOR HIDA DEFORMATIONS TADASHI OCHIAI Abstract. In this paper, we give a Coleman/Perrin-Riou type map for an ordinary type deformation and construct a two variable p-adic L-function for a Hida family from the Beilinson-Kato elements. Contents 1. Introduction 1 2. Λ-adic Forms and Galois Representations 5 3. The main result and its application to Hida s Galois deformation 9 4. Calculation of local Iwasawa modules Proof of Theorem References Introduction Fix a prime p 3. We denote by Qµ p ) the extension of the rational number field Q obtained by adjoining all p-power roots of unity. We fix a complex embedding Q C and a p-adic embedding Q Q p of an algebraic close Q of Q throughout the paper, where C is the field of complex numbers and Q p is an algebraic close of the field Q p of p-adic numbers. Let G be the Galois group of Qµ p )/Q and let D be the group of diamond operators for the tower of modular cves {Y 1 p t )} t 1 see 2). We have the canonical character χ : G Z p resp. κ : D 1 + pzp Z p ). A character η : G Q p resp. η : D Q p ) is called an arithmetic character of weight wη) Z resp. wη ) Z) if η resp. η ) coincides with the character χ wη) resp. κ wη ) ) on a sufficiently small open subgroup U resp. U ) of G resp. D ). For any arithmetic character η resp. η ) of G resp. D ), let O η,η = η η )Z p [[G D ]]) the finite flat extension of Z p obtained by adjoining the values of the character η η. We fix a positive integer N prime to p. In his celebrated paper [H2], Hida associates a continuous representation T of GalQ/Q) which is free of rank two over the complete group algebra Z p [[G D ]] to a Λ-adic cusp form F of level Np. The representation T has the following properties see also 2 for more detailed explanation on Hida s theory): The author is partially supported by JSPS. 1

2 1. Let η : G Q p resp. η : D Q p ) be a character. Assume that η is an arithmetic character of weight w 0. We denote by T η,η the specialization T Zp [[G D ]] O η,η of T. Then there exists a cusp form f η = a n f η )q n of 0<n< weight w +2 and level Np such that T η,η is isomorphic to T fη η where T fη is the Galois representation associated to f η in the sense of Deligne [De1] and η is the twist by the one dimensional Galois representation associated to η. In this sense, T is a family of modular representations when the cusp form and the character twist vary. 2. As a representation of the decomposition group G Qp at p, the representation T has a filtration: 0 F + T T F T 0 such that F + T and F T are free Z p [[G D ]]-modules of rank one. Let χ : G Q G Z p [[G ]] be the universal cyclotomic character and let α : G Qp Z p [[D ]] be the unramified character such that η αfrob p )) = a p f η ) for each arithmetic character η : D Q p of non-negative weight, where Frob p is the geometric Frobenius element at p. Then F + T is isomorphic to Z p [[G ]] χ) Z p [[D ]] α), where Z p [[G ]] χ) resp. Z p [[D ]] α)) is the free rank one Z p [[G ]]-module resp. Z p [[D ]]-module) on which G Qp acts via χ resp. α). In this sense, F + T interpolates the p-th Foier coefficient a p f η ) of the cusp form f η when η varies. We will construct the Coleman map for this deformation T. Before stating the main results, we prepare some notations. We define the Z p [[D ]]-module D to be Z p [[D ]] α) Zp Ẑ p ) G Qp, where Zp is the formal tensor product over Z p and Ẑ p is the p-adic completion of the maximal unramified extension of Z p. Let Z p 1) = s µ p s, where µ p s is the group of p s -th roots of unity. The absolute Galois group G Q of Q acts natally on Z p 1). We denote by T the Kummer dual Hom Zp [[G D ]]T, Z p [[G D ]]) Z p 1) of T. Let T η,η resp. T η,η ) be the specialization of T resp. T ) at a character η η of G D and let V η,η resp. V η,η ) be the extension T η,η Zp Q p resp. T η,η Zp Q p ). The module D defined above has the following properties see 3 for the proof): 1. The module D is free of rank one over Z p [[D ]]. 2. Let η resp. η ) be an arithmetic character of G resp. D ) satisfying 0 wη) 1 wη ). We denote by K η resp. K η ) be the finite extension of Q p obtained by adjoining the values of the character η resp. η ) to Q p. Then D η OK D η K η dr K η η)) is natally identified with Fontaine s filtered module D dr V η,η )/Fil 0 D dr V η,η ), where D η is the specialization of D at η, K η η) is the one dimensional K η -vector space on which G Q acts via η and O Kη K η is the ring of integers of K η K η. For a free Z p -module T with continuous G Qp -action, Bloch-Kato cf. [BK, 3]) defines a subgroup Hf 1Q p, T ) of H 1 Q p, T ) called the finite part. We denote the quotient 2

3 H 1 Q p, T ) H 1 f Q p, T ) by H1 /f Q p, T ) for short. Kato defines a map: H 1 /f Q p, T ) exp Fil 0 D dr T Zp Q p ) called the dual exponential map see Definition 3.9). Let w, w ) be a pair of integers such that 0 w 1 w. We denote by T w,w ) the quotient T /Φ w,w ) T, where Φ w,w ) is the height two ideal of Z p [[G D ]] defined to be the kernel of the homomorphism χ wps κ w p t : Z p [[G D ]] Q p. The projective it H/f 1 Q p, T w,w ) ) does not depend on the choice of w, w ) by Corollary We denote H/f 1 Q p, T w,w ) ) by H/f 1 Q p, T ). Let be the largest finite subgroup of G and let ω : Z/pZ) Z p be the Teichmuller character. We define an idempotent 1 e i Z p [ ] to be ω i g)g for each 0 i p 2. For a Z p [[G D ]]-module M, p 1 g we have the decomposition M = 0 i p 2 e i M), where each e i M) is natally regarded as a Z p [[G / ) D ]]-module. The module H/f 1 Q p, T ) defined above has the following properties see 4 for the proof): 1. For each 0 i p 2, e i H 1 /f Q p, T )) is a torsion free Z p [[G / ) D ]]-module note that Z p [[G / ) D ]] is an integral domain). 2. Let FracZ p [[G D ]]) be the total quotient ring of Z p [[G D ]]. Then H 1 /f Q p, T ) Zp[[G D ]]FracZ p [[G D ]]) is a free FracZ p [[G D ]])-module of rank one. 3. Let η resp. η ) be an arithmetic character of G resp. D ) such that 0 wη) 1 wη ). Then we have the specialization map H 1 /f Q p, T ) H 1 /f Q p, T η,η ) at η η whose cokernel is finite. From now on throughout the paper, we fix a norm compatible system {ζ p s} s 1 of primitive p s -th roots of unity. Let δ Qp 1) be the inverse image of 1 Q p via the isomorphism D dr Q p 1)) Q p determined by {ζ p s} s 1. Let us fix a basis d of the Z p [[D ]]-module D. Denote by K η,η the fraction field of O η,η. By the properties of D stated above, we have a basis d η,η of the one dimensional K η,η -vector space D dr V η,η )/Fil 0 D dr V η,η ) induced by the fixed data {ζ p s} s 1 and d see Definition 3.12 for the precise definition of d η,η ) for each η resp. η ) satisfying 0 wη) 1 wη ). O main result is to construct an interpolation of the dual exponential maps when the character η η of G D varies. The result is as follows: Theorem Theorem 3.13). Let us fix a basis d of D. Then, we have a Z p [[G D ]]- linear homomorphism Ξ d : H 1 /f Q p, T ) Z p [[G D ]] satisfying the following properties : 3

4 1. The map Ξ d is injective and the cokernel of Ξ d is a pseudo-null Z p [[D G ]]- module. 2. Let C be an element of H/f 1 Q p, T ) and let c η,η H/f 1 Q p, T η,η ) be the specialization of C for each arithmetic character η resp. η ) of G resp. D ). Assume the inequality 0 w 1 w for w = wη) and w = wη ). Then the specialization Ξ d C) η,η Q p of Ξ d C) at the character η η is given by ) 1) w 1 ap f η ) s ) w 1)! p w 1 1 pw 1 ϕp) 1 a ) pf η )ϕp) 1 a p f η ) p w exp c η,η ), d η,η, where, is the de Rham pairing : Fil 0 D dr V η,η ) D dr V η,η )/Fil 0 D dr V η,η ) D dr K η,η 1)) = K η,η, ϕ is the finite order character η χ w of G and s is the p-order of the conductor of ϕ. Assume that the residual representation G Q GLT /I, p)t ) = GL 2 F p ) is irreducible, where I is the augmentation ideal of Z p [[G D ]]. By a result of Kato [Ka3], we have an element Z H/f 1 Q p, T ) called the Beilinson-Kato element see 3). The specialization z η,η of Z is related to a L-value of the modular form f η. If we fix a basis d of D, z η,η has the property that exp z η,η ), d η,η / C p,η,d = Gϕ 1, ζ p s)2π 1) wη ) wη)+1 L p) f η, ϕ, wη)) C 1)wη ) wη)+1 ϕ 1),η where L p) f η, ϕ, s) is the Hecke L-function for the ϕ-twist of f η with its p-factor removed, C p,η,d resp. C ±,η ) is a p-adic resp. complex) period see 3 for the definition of these periods) and Gϕ 1, ζ p s) is the Gauss sum for ϕ 1. The following corollary shows that the image Ξ d Z) Z p [[G D ]] of the Beilinson-Kato element Z gives a two variable p-adic L-function for the Hida deformation corresponding to Z. Corollary Theorem 3.17). Let us fix a basis d of D. Assume that the residual representation G Q GLT /I, p)t ) = GL 2 F p ) is irreducible. Then Ξ d Z) Z p [[G D ]] has the following interpolation properties for each arithmetic character η resp. η ) of G resp. D ) satisfying the inequality 0 w 1 w for w = wη) and w = wη ) : Ξ d Z) η,η / Cp,η,d = 1) w 1 w 1)! Gϕ 1, ζ p s)2π 1) w w+1 ap f η ) C 1)w w+1 ϕ 1),η p w 1 ) s ) 1 ϕp)pw 1 Lf η, ϕ, w). a p f η ) Greenberg-Stevens, Kitagawa and Ohta also construct a two-variable p-adic L-function for an ordinary Λ-adic cusp form independently. Their method is to construct a Λ-adic interpolation of modular symbols and their definition of a p-adic period is an error term on the Betti side. In o case, the definition of a p-adic period is an error term on the de Rham side. The relation between o p-adic L-function and those of Greenberg-Stevens, Kitagawa and Ohta is not clear at present. One of the advantage of o construction of the p-adic L-function from Euler system is that it is useful to investigate the relation 4

5 between a p-adic L-function and a Selmer group Iwasawa Main conjecte). We give an application of the result in this paper to a two-variable Iwasawa Main conjecte for a Λ-adic cusp form in the paper [O], where we show one of the inequality predicted by the Main conjecte. Plan. The plan of this paper is as follows. In 2, we recall necessary facts from Hida theory. In 3, we state o main result for nearly ordinary Galois deformations not necessarily ited to Hida deformation. We deduce the result stated above in the case of two variable deformation coming from Hida theory from o main result. In 4, we give the calculation of the it of local cohomology groups, which is used to show the injectivity of the interpolation map of the main theorem. In 5, we give the proof of the main result. Notation. For a field K, we denote GalK/K) by G K where K is the separable close of K. Given a finite prime v of a number field F, we denote by Frob v the geometric Frobenius at v. For a commutative ring S, we denote by S the group of invertible elements in S. We denote by Sρ) the free S-module of rank one on which G F acts via a character ρ : G F S. Throughout the paper, we assume that the fixed integer p is an odd prime number. Acknowledgements. The author expresses his gratitude to Prof. Takeshi Saito for encoagement and discussion. He is grateful to Prof. Kazuya Kato for showing him the manuscript [Ka3] and for useful advice and is also grateful to Prof. Kazuhiro Fujiwara for useful advice and encoagement. He thanks Yoshitaka Hachimori and Kazuo Matsuno for encoagement and fruitful discussion. 2. Λ-adic Forms and Galois Representations In this section, we review some fundamental results on Λ-adic cusp forms and their Galois representations. We keep the notation of the previous section. Let O Q p be a commutative ring which is finite flat over Z p and let ψ be a Dirichlet character modulo Np r. We denote by M k Γ 1 Np r ), ψ; O) resp. S k Γ 1 Np r ), ψ; O)) the space of modular resp. cusp) forms of weight k, Neben character ψ and Foier coefficients in O for the group Γ 1 Np r ). For each integer t 1, the affine modular cve Y 1 p t ) /Q is the fine moduli of pairs E /S, e /S ) of an elliptic cve E over a Q-scheme S and an S-section e /S of order p t. Recall that the diamond operator a on Y 1 p t ) is the automorphism on Y 1 p t ) which sends a pair E /S, e /S ) to the pair E /S, ae /S ) for each a Z/p t Z). We denote the p-sylow subgroup of the group of diamond operators on Y 1 p t+1 ) by D t. D t is canonically isomorphic to the group 1 + pz/p t+1 Z) Z/p t+1 Z) ). We define the pro-p group D to be the projective it t D t. Definition 2.1. Let G and χ be as in the previous section and let D be as above. Let κ : D 1 + pzp Q p be the canonical character. 1) A character η resp. η ) of G resp. D ) is called an arithmetic character of weight wη) resp. wη )) if there exists an integer wη) resp. wη )) such that η resp. η ) coincides with χ wη) resp. κ wη ) ) on a sufficiently small open subgroup U 5

6 resp. U ) of G resp. D ). Let sη) resp. sη )) be the p-order of the conductor of the finite order character η χ wη) resp. η κ wη ) ). We denote the set of arithmetic characters of G resp. D ) by X arith G ) resp. X arith D )). 2) Let R be a local domain finite flat over Z p [[D ]]. We denote by XR) the set of non-trivial continuous algebra homomorphisms R Q p. We define a subset X arith R) of XR) by: X arith R) = {p XR) the character p D : D Q p is an arithmetic character}. We call an element p X arith R) an arithmetic point of R. The weight of p D X arith D ) is called the weight of the arithmetic point p and is denoted by wp). We also denote by ψ p) resp. sp)) the finite order character p D κ wp) of D resp. the p-order of the conductor of ψ p) ). Consider a formal q-expansion F = A n F)q n, where each A n F) is an element 0 n< of a fixed algebraic close of FracZ p [[D ]]). For F as above, we define the subring H of the algebraic close of FracZ p [[D ]]) to be the algebra generated by all A n F) s over Z p [[D ]]. We assume that H is finite flat over Z p [[D ]]. For each p X arith H), we denote by f p the formal q-expansion a n f p )q n, where a n f p ) Q p is the image of A n F) under the map p : H Q p. 0 n< Definition 2.2. Let ψ 0 be a character defined modulo Np. We call F a Λ-adic form of level Np with Dirichlet character ψ 0 if f p is the q-expansion of a modular form in M wp)+2 Γ 1 Np sp) ), ψ 0 ψ p) ω wp) ; Q p ) for each p X arith H) with wp) 0, where ω is the Teichmuller character. Definition 2.3. Let r 1. Let ψ be a character defined modulo Np r. Then a p-adic cusp form f S k Γ 1 Np r ), ψ; O) is called a p-stabilized newform of tame conductor N with Dirichlet character ψ if 1. f is an eigenform of S k Γ 1 Np r ), ψ; O) for all Hecke operators T l l Np) and U l l Np) which belong to EndS k Γ 1 Np r ), ψ; O)). 2. The newform associated to f has level Np r 0 for some r 0, 0 r 0 r. 3. The eigenvalue a p of f for U p EndS k Γ 1 Np r ), ψ; O)) is a p-adic unit. Let f be a newform in S k Γ 1 N), ψ; O). Assume that the eigenvalue a p of f for the Hecke operator T p is a p-adic unit. We define f S k Γ 1 Np), ψ; O) by f = fq) βfq p ), where β is the unique root of x 2 a p x + ψp)p k 1 with p-adic absolute value β < 1. Then f is a p-stabilized newform of tame conductor N whose n-th Foier coefficients equal to that of f for each natal number n prime to p. We call this f the p-stabilized newform associated to f. Definition 2.4. A normalized Λ-adic cusp form F is a Λ-adic newform of tame conductor N with Dirichlet character ψ 0 if f p is a weight wp) + 2 p-stabilized newform of tame conductor N, level Np sp) with character ψ 0 ψ p) ω wp) for each p X arith H) with wp) 0. 6

7 Remark 2.5. Let F be a Λ-adic newform of the tame conductor N with character ψ 0 modulo Np. Then it is known that the specialized p-stabilized newform f p is a newform that is, f p is also new at p) if and only if sp) > 1 or ψ 0 ψ p) ω wp) restricted to the subgroup = Z/pZ) is non-trivial. Theorem 2.6 Hida [H1] Corollary 3.2, Corollary 3.7). Let N be an integer prime to p and let ψ be a Dirichlet character defined modulo Np. Let f S k Γ 1 Np), ψ; O) be a p-stabilized newform of tame conductor N. Then there exist a Λ-adic newform F of tame conductor N with Dirichlet character ψω k 2 and p X arith H) with wp) = k 2 such that f p is equal to f. Recall the definition of continuity for Galois representations over the field of fractions K of H see [H3, 7.5], for example). Definition 2.7. The representation ρ : G Q GL 2 K) is continuous if there exists a finitely generated H-module T H 2 which is stable under G Q -action such that ρ : G Q AutT) is continuous with respect to the topology of T defined by the maximal ideal of H and that T H K = K 2. Hida associates a continuous Galois representation over H to a Λ-adic newform F as follows: Theorem 2.8 Hida [H2]). Let F be a Λ-adic newform with Dirichlet character ψ 0 modulo Np. Then, there exists a continuous irreducible representation ρ F : G Q GL 2 K) satisfying the following properties: 1. ρ F is unramified outside Np. 2. For the geometric Frobenius element Frob l at l Np, we have: Trace ρ F Frob l ) ) = A l F), det ρ F Frob l ) ) = ψ 0 l) κ 1 l)l 1, where l l is the projection Z p = Z/pZ) 1 + pz p ) 1 + pz p ) and κ is the tautological character 1 + pz p D Z p [[D ]] K. In order to study the Iwasawa theory for Hida deformations, it is convenient to assume the following condition: Int) The representation ρ F : G Q GL 2 K) has a G Q -stable lattice T which is isomorphic to H 2. The condition Int) is satisfied in fairly general situations. Before stating some sufficient conditions for Int) to be satisfied Proposition 2.10), we introduce some necessary notations. Definition 2.9. Let F be the residue field of H modulo the maximal ideal M of H. A semi-simple representation ρ : G Q GL 2 F) is called the residual representation associated to F if ρ is unramified outside Np and the characteristic polynomial of the geometric Frobenius Frob l for each prime l Np is congruent to X 2 A l F)X ψ 0 l) κ 1 l)l 1 modulo M. 7

8 Such a residual representation is always known to exist without assuming the condition Int) and it is unique up to isomorphism. The following is a list of cases where the condition Int) is known to be true. Proposition 2.10 Maz-Wiles, Tilouine, Maz-Tilouine). The condition Int) holds if one of the following conditions is satisfied: 1) The ring H is regular. 2) The tame conductor N of F is equal to 1 and the ring H is Gorenstein Maz-Wiles [MW2] 9). 3) Let a be a number such that ψ 0 Z/pZ) = ω a. Then a 0, 1 modulo p 1 and the ring H is Gorenstein Tilouine [Ti] Theorem 4.4). 4) The residual representation is irreducible see Maz-Tilouine [MT] 2, Corollary 6). We denote this condition by Ir). From now on throughout the paper, we will assume the condition Int). We have the following local property of ρ F due to Maz and Wiles: Proposition 2.11 [Wi] Theorem 2.2.2). The restriction ρ F GQp group G Qp of ρ F has the filtration: 0 F + T T F T 0 to the decomposition such that F + T and F T are free H-modules of rank one. Fther, G Qp acts on F + T via the unramified character α such that αfrob p ) = A p F) for the geometric Frobenius element Frob p. Remark Note that the normalization of the above proposition is dual to that of the paper [Wi]. In [Wi], the Frobenius element is normalized to be the arithmetic one. We normalize the Frobenius element to be the geometric one throughout this paper. For each p X arith H) with wp) 0, the specialization T p = T H ph) of T is isomorphic to the p-adic Galois representation T fp associated to f p by Deligne [De1]. In this sense, ρ F is a family of modular Galois representations when the weight of the modular form varies. Let χ : G Q G Z p [[G ]] be the universal cyclotomic character. We denote by Z p [[G ]] χ) the free Z p [[G ]]-module of rank one on which G Q acts via the character χ. The nearly ordinary deformation T associated to T is defined to be the formal tensor product T Zp Z p [[G ]] χ), where the action of G Q on T is given by the diagonal one. The representation T has the following properties: 1. T is free of rank two over R = H Zp Z p [[G ]] = H[[G ]]. 2. As a G Qp -module, we have the filtration: 0 F + T T F T 0, where F + T resp. F T ) is F + T Zp Z p [[G ]] χ) resp. F T Zp Z p [[G ]] χ)). 3. Let T η,p be the specialization of T at η, p) X arith G ) X arith H). Assume that wp) is non-negative. Then there exists a cusp form f p of weight wp) + 2 and T η,p is isomorphic to T fp η, where η is the twist by the one dimensional Galois representation corresponding to η. 8

9 3. The main result and its application to Hida s Galois deformation In this section, we state o main results for general nearly ordinary deformations not necessarily ited to the Hida deformation introduced in 2. We give an application to the Hida family in the latter half of this section. Throughout the first half of the section, let D be a pro-p group which has the canonical isomorphism κ : D 1 + pzp we do not necessarily assume that D is the group of diamond operators as in 2). We fix a commutative ring H which is finite flat over Z p [[D ]]. Definition 3.1. Let T be a free H-module of rank two with continuous G Q -action. The representation T is called an ordinary deformation if the following conditions are satisfied: 1. The representation T has a filtration as a G Qp -module: 0 F + T T F T 0 such that F + T and F T are free rank one H-modules and that the action of G Qp on F + T is given by an unramified character α of G Qp. 2. There exists a Dirichlet character ψ 0 : G Q Z p [ψ 0 ] H such that G Q acts on the the determinant representation dett) = T 2 via the character ψ 0 χ 1 κ 1, where we regard the tautological character κ : 1 + pz p D Z p [[D ]] H as a character of G Q through the canonical character χ : G Q G 1 + pzp. Let R = H Zp Z p [[G ]] = H[[G ]], which is finite flat over Z p [[G D ]]. We denote by T the Galois representation T Zp Z p [[G ]] χ). We call T the nearly ordinary deformation associated to T. A nearly ordinary deformation T defined above is free of rank two over R. The Kummer dual T of T is defined to be Hom R T, R) Zp Z p 1). T has a rank one filtration F + T T defined by F + T = Hom R F T, R) Zp Z p 1). Let w, w ) Z Z and let s, t) Z 0 Z 0. We denote by T w,w ) resp. F + T w,w ), F T w,w ), T w,w ), F + T w,w ), F T w,w ) ) the specialization of T resp. F + T, F T, T, F + T, F T ) obtained by applying R R/Φ w,w ), where Φ w,w ) is the height two ideal defined in 1. These representations are free Z p -modules of finite rank on which G Q acts continuously. We denote by V w,w ) resp. F + V w,w ), F V w,w ), V w,w ), F + V w,w ), F V w,w ) ) the extension of T w,w ), F T w,w ) ) by applying Zp Q p. For each resp. F + T w,w ), F T w,w ), T w,w ), F + T w,w ) η, p) X arith G ) X arith H), we denote by T η,p resp. F + T η,p, F T η,p, T η,p, F + T η,p, F T η,p ) the specialization of T resp. F + T, F T, T, F + T, F T ) via R η pr), where η pr) is finite flat over Z p. Similarly, we define V η,p resp. F + V η,p, F V η,p, V η,p, F + V η,p, F V η,p ) by applying Zp Q p to the above representations. For later use in this section and the next, we summarize basic facts on these specializations: 1. For η, p) X arith G ) X arith H), T η,p resp. T η,p ) is a quotient of T w,w ) sη),sp) resp. T w,w ) sη),sp) ), where w = wη) and w = wp). 2. T η,p is the Kummer dual Hom Zp T η,p, Z p ) Zp Z p 1) of T η,p. 9

10 3. V w,w ) resp. V w,w ) ) is isomorphic to V η,p resp. V η,p ) where η resp. p) runs η,p η,p arithmetic characters of G resp. arithmetic points of H) satisfying wη) = w and sη) s resp. wp) = w and sp) t). Fontaine defines the rings of p-adic periods B crys B dr. The rings B crys and B dr have continuous G Qp -action and B dr is a complete discrete valuation field. We denote by B + dr the valuation ring of B dr and denote by Fil i B dr for each i Z the decreasing filtration of B dr defined by u i B + dr where u is a uniformizer of B+ dr. For a p-adic representation V of G Qp, we denote by D crys V ) resp. D dr V )) the Fontaine s module defined by V B crys ) G Qp resp. V B dr ) G Qp ). The module D crys V ) resp. D dr V )) is a finite dimensional Q p -vector space such that dim Qp D crys V ) dim Qp V resp. dim Qp D dr V ) dim Qp V ). A p-adic representation V is called a crystalline representation resp. de Rham representation) if dim Qp D crys V ) = dim Qp V resp. dim Qp D dr V ) = dim Qp V ). The module D dr V ) has a separated and exhausted decreasing filtration Fil i D dr V ) := V Fil i B dr ) G Qp. We refer the reader to [Bu] for Fontaine s theory of p-adic representations. For each η X arith G ) resp. p X arith H)), we denote by K η resp. K p, K η,p ) the fraction field of ηz p [[G ]]) resp. ph), η ph[[g ]])), which is a finite extension of Q p. We have the following lemma: Lemma 3.2. Let η, p) X arith G ) X arith H) satisfying 0 wη) 1 wp). Then, V η,p is a de Rham representation of G Qp such that D dr F + V η,p ) is canonically isomorphic to D dr V η,p )/Fil 0 D dr V η,p ). V η,p is also a de Rham representation such that D dr F V η,p ) is canonically isomorphic to Fil 0 D dr V η,p ). Proof. By the definition of T, there exists a finite extention K of Q p such that V η,p is an ordinary representation of G K. By a result of Perrin-Riou [P3], an ordinary representation is semi-stable in the sense of Fontaine. Especially, V η,p is a de Rham representation of G K. Since a potentially de Rham representation is a de Rham representation cf. [Bu]), V η,p is a de Rham representation of G Qp. We have Fil 0 D dr F + V η,p ) = F + V η,p B + dr )G Qp = 0 since the action of a sufficiently small open subgroup of the inertia subgroup I p on F + V η,p is given by χ w with w > 0. Thus we have a K η,p -linear injection D dr F + V η,p ) D dr V η,p )/Fil 0 D dr V η,p ). Similarly we have a K η,p -linear sjection Fil 0 D dr V η,p ) Fil 0 D dr F V η,p ) and dim Kη,p Fil 0 D dr V η,p /F + V η,p )) = 1. We have dim Kη,p D dr V η,p ) = 2 since V η,p is a de Rham representation and we have dim Kη,p D dr F + V η,p ) = 1 since a sub-representation of a de Rham representation is also a de Rham representation. In conclusion, we have dim Kη,p D dr V η,p )/Fil 0 D dr V η,p ) 1 and the above mentioned injection D dr F + V η,p ) D dr V η,p )/Fil 0 D dr V η,p ) must be an isomorphism. The assertion for V η,p is shown in the same way. Lemma 3.3. Let M be a free H-module of finite rank e endowed with unramified G Qp - action. Then M Zp Ẑ p ) G Qp is a free H-module of finite rank e. Proof. Let I be a height two ideal of H. Then we have M Zp Ẑ p ) G Qp = n M/I n ) Zp Ẑ p ) G Qp 10 = n M/I n Zp Ẑ p ) G Qp.

11 M/I n is a free module of rank one over the ring H/I n with finite number of elements. It suffices to show that M/I n Zp Ẑ p ) G Qp is free of rank one over H/I n. Clearly the proof follows from the following claim: Claim 3.4. Let R be a Z p -algebra with finite number of elements. For a free R-module M of finite rank e endowed with unramified G Qp -action, M Zp Ẑ p ) G Qp is free of finite rank e over R. We prove the claim in the rest. Let p m be the characteristic of the ring R. Then M Zp Ẑ p is isomorphic to M Z/p m Z W m F p ), where W m F p ) is the ring of Witt vectors of length m cf. [Se, Chap. II, 6]). Since M is finite, there is an open subgroup H of GalQ p /Q p ) such that H acts trivially on M. We have M Zp Ẑ p ) G Qp = M Z/p m Z W m F)) G, where F is the fixed field F H p and G = GalQ p /Q p )/H. Since M Z/p m ZW m F) is isomorphic to M Z/p m Z Z/p m Z[G] as an R[G]-module, M Z/p m Z W m F)) G is free of rank e over R. By Lemma 3.3, we give the following definition. Definition 3.5. Let T be an ordinary deformation. We define a free H-module D of rank one by D = F + T Zp Ẑ p ) G Qp where the Galois action of g G Qp on F + T Zp Ẑ p is the diagonal action g g. Lemma 3.6. Let T = T Zp Z p [[G ]] χ) be a nearly ordinary deformation and let η, p) X arith G ) X arith H) satisfying 0 wη) 1 wp). Then we have the canonical isomorphism: D dr K η η)) OK η Kp D p = D dr F + V η,p ), where D p is the specialization D H ph) of D and K η η) is the one dimensional Galois representation over K η on which G Qp acts via η. Proof. Let V p be the representation T H ph)) Zp Q p. Since F + V p is unramified, D dr F + V p ) is canonically isomorphic to F + V p Q p ) G Qp. Hence D H K p = F + V p Q p ) G Qp is canonically isomorphic to D dr F + V p ). Recall that F + V η,p = K η η) Kη K p F + V p. Since the functor D dr is compatible with a tensor product of two de Rham representations, we have the canonical isomorphism D dr K η η)) Kη K p D dr F + V p ) = D dr F + V η,p ). This completes the proof of the lemma. Before giving the main result, we prepare some general definitions. Let B st be the ring of p-adic periods for semi-stable Galois representations, which is a subring of B dr equipped with continuous G Qp -action cf. [Bu]). For a representation V of G Qp, we denote by D pst V ) the inductive it V B st ) J where J runs through open subgroups of the J I p inertia subgroup I p of G Qp. Let σ be an arithmetic Frobenius element in GalQ p /Q p ). The module D pst V ) is a finite dimensional Q p -vector space with the following properties: 1. We have the inequality dim Q D pst V ) dim Qp V ). p 2. D pst V ) is endowed with the monodromy operator N, which is a Q p -linear nilpotent endomorphism on D pst V ) and is induced from the monodromy operator of B st. 11

12 3. We have a σ-semilinear G Qp -action on D pst V ) and the action of I p factors through a finite quotient of I p. 4. The module D pst V ) has the Frobenius operator f, which is σ-semilinear and is induced from the Frobenius operator of B st. The restriction of the action of G Qp on D pst V ) to the Weil group W p G Qp gives us a σ-semilinear action of W p on D pst V ). We denote by u : W p W p /I p Z the natal map which sends σ to 1. By the twist of the W p -action which replaces the action of g W p with the action of g f ug), we obtain a Q p -linear action of W p on D pst V ). Since the inertia subgroup I p W p acts through a finite quotient of I p, the complex absolute values of the eigenvalues of the Frobenius element of W p are well-defined if they are algebraic numbers. Definition 3.7. For a nearly ordinary deformation T, we consider the following condition: MW) Every eigenvalue α of the action of a lift of a geometric Frobenius Frob p on D pst V η,p ) is an algebraic number whose complex absolute value is p w +1 w 2 if the monodromy N is zero on D pst V η,p ), p w 2 w+1 if N is non zero on D pst V η,p ) and the eigen vector of α is in CokerN), p w 2 w if N is non zero on D pst V η,p ) and the eigen vector of α is in KerN). for each η, p) X arith G ) X arith H) with wp) 0, where w = wη) and w = wp). Remark 3.8. Assume that V p is the p-adic realization of a certain pe motive of weight wp) + 1 for each p X arith H) with wp) 0. Then the above assertion MW) is conjected to be true. By a result of T. Saito [Sa] for the monodromy-weight conjecte for elliptic modular forms, the above assertion on the complex eigenvalues of the lift of the Frobenius on D pst V η,p ) is true if V p is the p-adic representation associated to an elliptic cusp form of weight wp) + 2. Hence MW) is true if T is associated to a certain Λ-adic cusp form. Let us recall the definition of the dual exponential map exp. We denote by logχ) H 1 χ Q p, Q p ) = HomG Qp, Q p ) the homomorphism G Qp Z log p Q p defined by the cyclotomic character χ. Let V be a p-adic representation of G Qp. Let us consider the map: H 0 Q p, V B + dr ) logχ) H 1 Q p, V B + dr ), obtained by the cup product with logχ) H 1 Q p, Q p ). By [Ka1] Chap. II, Proposition 1.2.3, the above map logχ) is an isomorphism. Definition 3.9. The dual exponential map exp is defined to be the composite: H 1 Q p, V ) H 1 Q p, V B + dr ) logχ)) 1 Fil 0 D dr V ) = H 0 Q p, V B + dr ). Bloch-Kato [BK] defines a subgroup H 1 f Q p, V ) H 1 Q p, V ) called the finite part as follows: H 1 f Q p, V ) = Ker [ H 1 Q p, V ) H 1 Q p, V B crys ) ] 12

13 Let T resp. A) be a G Qp -stable lattice of V resp. a discrete Galois module T Q p /Z p ). We have the following exact sequence: H 1 Q p, T ) i H 1 p Q p, V ) H 1 Q p, A). We define H 1 f Q p, T ) H 1 Q p, T ) resp. H 1 f Q p, A) H 1 Q p, A)) to be the pullback i 1 H 1 f Q p, V ) resp. the push-forward p H 1 f Q p, V ). The dual exponential map is known to factor as: H 1 Q p, V ) H 1 /f Q p, V ) exp Fil 0 D dr V ), where H/f 1 Q p, V ) is H1 Q p, V ) Hf 1Q p, V ). 4.13): We prove the following lemma in 4 see Corollary Lemma Assume that T satisfies the condition MW). For each pair w, w ) Z Z satisfying 0 w 1 w, H/f 1 Q p, T w,w ) ) is canonically isomorphic to H/f 1 Q p, T 1,0) ). Especially, w, w ). We denote the module H/f 1 Q p, T w,w ) ) is independent of the choice of H/f 1 Q p, T w,w ) ) by H/f 1 Q p, T ). Since R is finite flat over Z p [[G D ]], Hom Zp [[G D ]]R, Z p [[G D ]]) is finitely generated R-module by r f)x) = fr x) for r R and f Hom Zp [[G D ]]R, Z p [[G D ]]). Definition The Z p [[G D ]]-algebra R = H[[G ]] is called a Gorenstein ring if Hom Zp[[G D ]]R, Z p [[G D ]]) is free of rank one over R. Definition map induced by the G Qp -invariant of the map F + T Zp Ẑ p 1. For each p X arith H), we denote by Sp p : D D dr F + V p ) the F + T p Zp Ẑ p, where T p = T H ph) note that D dr F + V p ) = D crys F + V p ) = F + T p Zp Ẑ p ) G Qp Zp Q p since F + T p is unramified). 2. Let η X arith G ) and let ϕ be the finite order character ηχ wη). We denote by O a finite flat extension of Z p whose fraction field is K. We define Sp η : O[[G ]] D dr K K Kη K η )η)) to be the Z p -linear homomorphism: O[[G ]] D dr Kχ w )) Qp Q p µ p s) = D dr Kχ w ) Qp Q p [G s ]) D dr Kχ w ) Qp K η ϕ)) D dr Kχ w ) K Kη K η ϕ)) = D dr K K Kη K η )η)), where the first map is the O[[G ]]-linear map which sends g G to δ w Q p1) ζ g p s. The isomorphism in the upper line is nothing but the isomorphism Kχw ) B dr ) G Qpµ p s ) = Kχ w ) Q p [G s ] B dr ) G Qp 13 by Shapiro s lemma.

14 3. Let η, p) X arith G ) X arith H). Then we denote by Sp η,p : D Zp Z p [[G ]] D dr F + V η,p ) the composite: D Zp Z p [[G ]] Sp p 1 D dr F + V p ) O p [[G ]] 1 Sp η D dr F + V p ) Kp K η D dr K η η)) = D dr F + V η,p ) Let be the largest finite subgroup of G and let R be R Zp [ ] Z p. The ring R is an integral domain and we have an isomorphism R = R. O main result in this paper is the following theorem: 1 i p 1 Theorem Let T be a nearly ordinary deformation in the sense of Definition 3.1. Assume that R is Gorenstein, R is a normal domain and that T satisfies the condition MW). Let us fix a basis d of the H-module D = F + T Zp Ẑ p ) G Qp. Then we have an R-linear homomorphism Ξ d : H/f 1 Q p, T ) R with the following properties: 1. The map Ξ d is an injective R-homomorphism whose cokernel is a pseudo-null R- module. 2. Let C be an element of H/f 1 Q p, T ) and let c η,p H/f 1 Q p, T η,p ) be the specialization of C at η, p) X arith G ) X arith H). Assume that 0 w 1 w for w = wη) and w = wp). Then, Ξ d C) η,p is given by : 1) w 1 w 1)! ap p w 1 ) s 1 pw 1 ϕp) a p ) 1 a pϕp) p w ) 1 exp c η,p), d η,p, where a p is the value of the action of the geometric Frobenius Frob p on F + V p,, is the pairing : Fil 0 D dr V η,p ) D dr V η,p )/Fil 0 D dr V η,p ) D dr K η,p 1)) = K η,p, ϕ is the finite order character ηχ w of G and s is the p-order of the conductor of ϕ. Now, we apply o main theorem to a two-variable modular deformation explained in 2. From now on throughout the section, we take T to be the nearly ordinary deformation associated to a certain Λ-adic new form F. In order to introduce Beilinson-Kato elements, we prepare notations. For each p X arith H) with wp) 0, we denote by f p = n>0a n f p ) c q n the dual modular form of f p = n>0a n f p )q n where c is the complex conjugate. The dual modular form f p is known to be a Hecke cuspidal eigen form of weight wp)+2 with Neben character dual of that of f p. We denote by Q f p the finite extension of Q obtained by adjoining Foier coefficients of f p to Q. For a Dirichlet character ϕ of p-power conductor, let Q f p,ϕ be the finite extension of Q f obtained by adjoining the values of ϕ and let V p dr ϕ) be the de Rham realization of the Dirichlet motive for ϕ, which is a one dimensional vector space over 14

15 K ϕ. We associate the de Rham representation V dr f p ) to f p. The de Rham realization V dr f p ) has the following properties: 1. V dr f p ) is a two dimensional vector space over Q f p and is equipped with a de Rham filtration Fil i V dr f p ) V dr f p ), which is a decreasing filtration of Q f p - vector spaces. 2. We have Fil 0 V dr f p ) = V dr f p ) and Fil wp)+2 V dr f p ) = {0}. For each w such that 0 w 1 wp), Fil w V dr f p ) is natally identified with the one-dimensional Q f p -vector space Q f p f p. 3. For a Dirichlet character ϕ of p-power conductor, we define the ϕ-twist V dr f p )ϕ) of V dr f p ) to be V dr f p ) Qfp K ϕ V dr ϕ), which has dimension two over Q f p,ϕ. If ϕ is the finite order character ηχ wη) for η X arith G ) satisfying 1 wη) wp)+1, Fil wp) w+2 V dr f p )ϕ 1 ) K Qfp,ϕ η,p is natally identified with Fil 0 D dr V η,p ). We also associate the Betti representation V B f p ) to f p. The Betti realization V B f p ) has the following properties: 1. V B f p ) is a two dimensional vector space over Q f p and is equipped with the action of the complex conjugate c on V B f p ). 2. Each ±1-eigen space V B f p ) ± for the action of c is one dimensional over Q f p. λ Let E t Y1 Np t ) Y1 Np t ) be the universal elliptic cve and let ν : H Y 1 Np t )C) = H/Γ 1 Np t ) be the uniformization map, where H is the complex upper half plane. Consider the continuous map τ : 0, ) Y 1 Np t )C), y νy 1). Let H 1 be the higher direct image R 1 λ t, Z), which is a locally free sheaf on Y 1 Np t )C) of Z-rank two. We denote by H 1 the dual sheaf HomH 1, Z). The stalk of τ 1 H 1 ) at y 0, ) is identified with H 1 C/y 1Z + Z, Z) = y 1Z + Z. The sheaf τ 1 H 1 ) is a constant free sheaf of Z-rank two with basis e 1 = y 1, 0), e 2 = 0, 1). Definition Let p X arith H) be an arithmetic point of weight w. 1. For each integer w satisfying 0 w 1 w, we denote by δ dr p the Q f p -basis of Fil w w+2 V dr f p ) coming from f p. 2. We define a basis δ B,± p of V B f p ) ± to be the image of the class τ, e w 1 ) in the cohomology H 1 X 1 Np sp) ), X 1 Np sp) ) \ Y 1 Np sp) ); Sym w H 1 )) under the following composite map: H 1 X 1 Np sp) ), X 1 Np sp) ) \ Y 1 Np sp) ); Sym w H 1 )) H 1 X 1 Np sp) ), X 1 Np sp) ) \ Y 1 Np sp) ); Sym w H 1 )) We have the Eichler-Shima isomorphism: I ±,p : Fil w w+2 V dr f p )ϕ 1 ) Qfp,ϕ C H 1 Y 1 Np sp) ); Sym w H 1 )) V B f p ) ±. 15 V B f p ) ± 1)w w+1 ϕ 1) Qfp C.

16 Kato constructs elements in the K 2 of modular cves [Ka3]. By using his elements, we have the following system of elements in Galois cohomology. Proposition [Ka3] Let T be the nearly ordinary representation associated to a certain Λ-adic newform F satisfying the condition Ir) stated in 2 for Ir). There exists an element Z H 1 /f Q p, T ) satisfying the following properties: 1. Let z η,p H 1 /f Q p, T η,p ) be the specialization of Z at η, p) X arith G ) X arith H) satisfying 0 w 1 w for w = wη) and w = wp). Then exp z η,p ) Fil 0 D dr V η,p ) is contained in Fil w w+2 V dr f p )ϕ 1 ) Fil 0 D dr V η,p ). 2. The image of exp z η,p ) Fil w w+2 V dr f p )ϕ 1 ) under the map I +,p : Fil w w+2 V dr f p )ϕ 1 ) V B f p ) 1)w w+1 ϕ 1) Qfp C is equal to Gϕ 1, ζ p s)2π 1) w w+1 L p) f p, ϕ, w) δ B, 1)w w+1 ϕ 1) p. We define a complex period and a p-adic period at each arithmetic point p as follows: Definition Let the notations be as defined in Definition A complex period C ±,p is the complex number given by I,p δ dr p ) = C ±,pδ B,± p. 2. Fix a basis d of the H-module D. For each arithmetic point p : H Q p of weight w and each integer w such that 0 w 1 w, we denote by d η,p the basis of rank one K η,p -vector space D dr V η,p )/Fil 0 D dr V η,p ) defined as the image of d 1 via the map Sp η,p : D Z p [[G ]] D dr V η,p )/Fil 0 D dr V η,p ) of Definition We define a p-adic period C p,p = C p,p,d depending on the choices of d) to be the value δ dr p, d η,p where, is the pairing : Fil 0 D dr V η,p ) D dr V η,p )/Fil 0 D dr V η,p ) D dr K η,p 1)) = K η,p. The p-adic period C p,p,d does not depend on w and depends only on d. We fix a basis d of the H-module D from now on. Let η, p) X arith G ) X arith H) with 0 w 1 w for w = wη), w = wp). Then exp z η,p ) Fil w w+2 V dr f p )ϕ 1 ) is equal to 2π 1) w w+1 L p) f p, ϕ, w) δ dr p, where ϕ = ηχ w. Hence exp z η,p )/ C p,p,d C 1)w w+1 ϕ 1),p Fil 0 D dr V η,p ) Kη,p Q p is sent to the L-value Gϕ 1, ζ p s) 2π 1) w w+1 L p) f p, ϕ, w) C 1)w w+1 ϕ 1),p Q under the pairing Fil 0 D dr V η,p ) Kη,p Q p,d η,p Q p. As stated in Remark 3.8, a Hida deformation T satisfies the condition MW). Hence we have the interpolation map Ξ d by Theorem By the interpolation property of the 16

17 / map Ξ d, Ξ d Z) η,p Cp,p,d is given by: 1) w 1 ap f p ) w 1)! = 1) w 1 w 1)! p w 1 ap f p ) p w 1 ) s 1 pw 1 ϕp) a p f p ) ) s 1 pw 1 ϕp) a p f p ) ) 1 a pf p )ϕp) p w ) 1 exp z η,p ), d η,p C p,p,d ) 2π 1) w w+1 L p) f p, ϕ, w). Gϕ 1, ζ p s) 1 C 1)w w+1 ϕ 1),p From the above argument, we obtain the following theorem by applying Theorem 3.13: Theorem Assume the condition Ir) for F. Assume that R is Gorenstein and / integrally closed in FracR). Let us fix a basis d of the H-module D. Then Ξ d Z) η,p Cp,p,d is given by : 1) w 1 w 1)!Gϕ 1, ζ p s)2π 1) w w+1 C 1)w w+1 ϕ 1),p ap f p ) p w 1 ) s ) 1 ϕp)pw 1 Lf p, ϕ, w), a p f p ) at each η, p) X arith G ) X arith H) satisfying 0 w 1 w with w = wη) and w = wp), where ϕ is the finite order character ηχ w of G and s is the p-order of the conductor of ϕ. Remark A two-variable p-adic L-function for a Hida deformation is also constructed by Greenberg-Stevens, Kitagawa and Ohta independently. The main ingredient of their work is a construction of the H-adic modular symbol B ±, which is a free H-module of rank one. The module B ± has an interpolation property that B ± /pb ± is canonically identified with the p-adic completion of the Betti realization H B f p ) ±wp) for each arithmetic point p of H with wp) 0. They define their p-adic period C ± Q p,p,b ± p to be the error term δp B,±wp) = C ± b ± p,p,b ± p where b ± p is a basis of B ± /pb ± coming from a fixed H-basis b ± of B ±. On the other hand, o p-adic period C p,p,d is defined to be the error term on the de Rham side. Fukaya [Fu] announces another construction of the p-adic L-functions as an application of her theory of K 2 -version of the theory of Coleman power series. 4. Calculation of local Iwasawa modules In this section, we calculate projective its of various local Galois cohomology groups. The local calculation given in this section immediately implies the coincidence of twovariable Selmer groups of Greenberg type and of Bloch-Kato type for Hida s nearly ordinary deformations T see [O]). For the proof of the main theorem Theorem 3.13), we need only Corollary For a p-adic representation V of G Qp, a subspace Hg 1 Q p, V ) resp. He 1 Q p, V )) of H 1 Q p, V ) is defined as follows see [BK, 3]): Hg 1 Q p, V ) = Ker [ H 1 Q p, V ) H 1 Q p, V Qp B dr ) ], [ ] He 1 Q p, V ) = Ker H 1 Q p, V ) H 1 Q p, V Qp Bcrys f=1 ). 17

18 We have H 1 e Q p, V ) H 1 f Q p, V ) H 1 g Q p, V ) H 1 Q p, V ) by definition. Let T be a nearly ordinary deformation. For a pair w, w ) of integers and a pair s, t) of non-negative integers, we define the specialization T w,w ) resp. V w,w ) ) of T and the specialization T w,w ) resp. V w,w ) ) of T as in the beginning of 3. We define a subspace H 1 Q Gr p, V w,w ) ) of H 1 Q p, V w,w ) ) to be: [ ] H 1 Gr Q p, V w,w ) ) = Ker H 1 Q p, V w,w ) ) H 1 Q p, F V w,w ) ) By a result of Flach [Fl], we have the following lemma: Lemma 4.1. For integers w, w such that 0 w 1 w and integers s, t 0, the subspace Hg 1 Q p, V w,w ) ) of H 1 Q p, V w,w ) ) is equal to H 1 Q Gr p, V w,w ) ). We prove the following lemma: Lemma 4.2. Assume that the nearly ordinary deformation T satisfies the condition MW) cf. Definition 3.7). Then we have the equality Hf 1Q p, V w,w ) ) = Hg 1 Q p, V w,w ) ) for integers w, w such that 0 w 1 w and integers s, t 0. Proof. By Proposition 3.8 and Corollary of [BK], we see: H 1 g Q p, V w,w ) ) H 1 f Q p, V w,w ) ) = = ) Hf 1 Q p, V w,w ) )/He 1 Q p, V w,w ) ) D crys V w,w ) )/1 f)d crys V w,w ) )), where ) means a Q p -linear dual. Since a slope of D crys V w,w ) ) is w w 2 or w 1, this implies that Hf 1Q p, V w,w ) ) is equal to Hg 1 Q p, V w,w ) ) when w 1 Note that w w 2 can not be zero by the assumption of the lemma). Let us discuss the case w = 1 in the rest. To see that D crys V 1,w ) )/1 f)d crys V 1,w ) ) is zero in this case, we study the complex absolutes of the eigenvalues of the Frobenius f on D crys V 1,w ) ) = Hom Qp D pst V 0,w ) ) N, Q p ) G Qp, where D pst V 0,w ) ) N is the cokernel of the monodromy operator N acting on D pst V 0,w ) ). The set of eigenvalues of f on D crys V 1,w ) ) is equal to the set of the eigenvalues of the inverse of f on D pst V 0,w ) ) N and hence is equal to the set of the eigenvalues of the inverse of the geometric Frobenius element Frob p on D pst V 0,w ) ) N. By the assumption MW), the complex absolute values of the eigenvalues of Frob p on D pst V 0,w ) ) are equal to p w +1 2 or p w Thus, the eigenvalues of f on D crys V 1,w ) ) can not be trivial for any w 0. This completes the proof. For each pair s, t) of non-negative integers and each pair w, w ) of integers such that 0 w 1 w, we define HGr 1 Q p, A w,w ) ) to be: [ ] HGrQ 1 p, A w,w ) ) = Ker H 1 Q p, A w,w ) ) H 1 Q p, F A w,w ) ), 18

19 where A w,w ) is the discrete Galois representation T w,w ) Zp Q p /Z p. The inductive it HGr 1 Q p, A w,w ) ) is equal to Ker [ H 1 Q p, A) H 1 Q p, F A) ]. Since it is indepen- dent of the choice of w, w ) with 0 w 1 w, we denote it by H 1 Gr Q p, A). We have the following proposition: Proposition 4.3. Assume the condition MW) for T. Let w, w ) be a pair of integers such that 0 w 1 w. Then the following statements hold: 1. The group H/f 1 Q p, T w,w ) ) is a quotient of the Pontryagin dual of the group HGr 1 Q p, A). 2. Assume that w 1. Then the group H/f 1 Q p, T w,w ) ) is the Pontryagin dual of HGr 1 Q p, A). Remark 4.4. We will einate the assumption w 1 later and prove that the group H/f 1 Q p, T w,w ) ) is the Pontryagin dual of HGr 1 Q p, A) for any pair of integers w, w ) such that 0 w 1 w see Corollary 4.13). First, we have the following lemma: Lemma 4.5. Let T = T Zp Z p [[G ]] χ) be a nearly ordinary deformation. We assume the condition MW) for T. Then the value αfrob p ) H at Frob p is not a root of unity for the unramified character α associated to the unramified representation F + T. Proof. The specialization αfrob p ) p Q p of αfrob p ) at an arithmetic point p is equal to the eigenvalue of the action of Frob p on F + V p Q Qp p = D crys F + V p ) Q Qp p = D pst F + V p ) D pst V p ). On the other hand, the eigenvalues of every lift of Frob p on D pst V p ) are algebraic integers with the complex eigenvalues p wp) 2 by the assumption MW). Hence αfrob p ) p is not a root of unity for each p X arith H) with wp) > 0. This completes the proof. For the proof of Proposition 4.3, we introduce other subgroups of H 1 Q p, A). Define H 1 Q Gr p, A w,w ) ) by: [ ] H 1 Gr Q p, A w,w ) ) = Ker H 1 Q p, A w,w ) ) H 1 Q p, F A w,w ) ). The inductive it H 1 Q Gr p, A w,w ) ) is equal to Ker [ H 1 Q p, A) H 1 Q p, F A) ]. Since it is also independent of the choice of w, w ) with 0 w 1 w, we denote it by H 1 Q Gr p, A). Let Hf 1Q p, A w,w ) ) be as given after Definition 3.9. By Lemma 4.1 and Lemma 4.2, Hf 1Q p, A w,w ) ) is the maximal divisible subgroup of H 1 Q Gr p, A w,w ) ). Taking inductive it with respect to s, t, we define Hf 1 Q p, A) w,w ) = Hf 1 Q p, A w,w ) ). By [BK, Proposition 3.8], the group H/f 1 Q p, T w,w ) ) is the Pontryagin dual of Hf 1Q p, A) w,w ). Hence Proposition 4.3 is equivalent to the following proposition: 19

20 Proposition 4.6. Assume the condition MW) for T. Let w, w ) be a pair of integers such that 0 w 1 w. Then the following statements hold: 1. The group H 1 f Q p, A) w,w ) is a subgroup of H 1 Gr Q p, A). 2. If fther w 1, H 1 f Q p, A) w,w ) is equal to H 1 Gr Q p, A). For a finitely generated R-module M, we denote the specialization M/Φ w,w ) M by M w,w ). For s, t ) s, t), we have a natal sjection M w,w ) s,t M w,w ). We define the augmentation map M w,w ) M w,w ) s,t by x gg x, where x is g G ps /G ps,g D pt /D pt a lift of x. The augmentation map is well-defined and is independent of the choice of a lift x. We have the following lemma: Lemma 4.7. Let M be a finitely generated torsion R-module whose ideal of support has height at least two. Let w, w ) be a pair of integers. For any t 0, assume that M w,w ) is a finite group whose order is bounded when s 0 varies. Then the it M w,w ) with respect to the augmentation maps above is equal to zero. Proof. It suffices to show that sm w,w ) = 0 for each t 0. By the assumption of the lemma, sm w,w ) is finite for any t 0. Hence there exists a sufficiently large natal number such that G s 0 acts trivially on sm w,w ). For s > s s 0, the augmentation map M w,w ) M w,w ) s,t is the multiplication by p s s. Hence, by taking s s greater than the p-order of M w,w ), M w,w ) M w,w ) s,t is the zero map. This completes the proof. Let Φ w) s resp. Ψ w ) t ) be the height one ideal of Z p [[G ]] resp. Z p [[D ]]) defined to be the kernel of the algebra homomorphism χ wps : Z p [[G ]] Q p resp. κ w p t : Z p [[D ]] Q p ). We also denote by Φ w) s and Ψ w ) t the height one ideals of Z p [[G D ]] through the inclusions Z p [[G ]] Z p [[G D ]] and Z p [[D ]] Z p [[G D ]]. Lemma 4.8. Assume the condition MW) for T. Let w, w ) be a pair of integers such that w > 0. Then, for any t 0, T w,w ) ) = T ) GQ p G Q p )w,w resp. F + T w,w ) ) = GQ p F + ) T GQ p )w,w ) is a finite group whose order is bounded when s 0 varies. Proof. As a G Qp -module, we have the following exact sequence see Definition 3.1): 0 H α) Zp Z p [[G ]] χ) T H α 1 χ 1 κ 1 ψ 0 ) Zp Z p [[G ]] χ) 0, where H α) resp. H α 1 χ 1 κ 1 ψ 0 )) is the free H-module of rank one on which G Qp acts via the character α resp. α 1 χ 1 κ 1 ψ 0 ). Hence we have the following exact sequence for each t 0: 0 H α) w Ψ ) Zp Z p [[G ]] χ) t T w Ψ ) H α 1 χ 1 κ 1 ψ 0 ) w t Ψ ) Zp Z p [[G ]] χ) 0, t 20