aa r X i v : . [ m a t h . N T ] J a n A generalization of Colmez-Greenberg-Stevens formula
Bingyong Xie ∗ Department of Mathematics, East China Normal University, Shanghai, [email protected]
Abstract
In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tateweights for families of Galois representations with triangulations. We give a generalization ofthe Fontaine-Mazur L -invariant and use it to build a formula which is a generalization of theColmez-Greenberg-Stevens formula. Key words: Frobenius, Hodge-Tate weight, L -invariant.MSC(2010) classification: 11F80, 11F85. In their remarkable paper [10], Mazur, Tate and Teitelbaum proposed a conjectural formula for thederivative at s = 1 of the p -adic L -function of an elliptic curve E over Q when p is a prime ofsplit multiplicative reduction. An important quantity in this formula is the so called L -invariant,namely L ( E ) = log p ( q E ) /v p ( q E ) where q E ∈ Q × p is the Tate period for E . This conjectural formulawas proved by Greenberg and Stevens [8] using Hida’s families. Indeed, for the weight 2 newform f attached to E , there exists a family of p -adic ordinary Hecke eigenforms containing f . A keyformula they proved is L ( E ) = − α ′ ( f ) α ( f ) (1.1)where α is the function of U p -eigenvalues of the eigenforms in the Hida family. On the otherhand, they showed that − α ′ ( f ) α ( f ) is equal to L ′ p ( f, L ( f, . Combining these two facts they obtained theconjectural formula.In this paper we will focus on (1.1) which was later generalized by Colmez [6] to the non-ordinarysetting. We state Colmez’s result below. Theorem 1.1. ( [6] ) Suppose that, at each closed point z of Max( S ) one of the Hodge-Tate weight of V z is , and there exists α ∈ S such that ( B ϕ = α cris ,S b ⊗ S V ) G Q p is locally free of rank over S . Suppose z is a closed point of Max( S ) such that V z is semistable with Hodge-Tate weights and k ≥ . ∗ This paper is supported by Science and Technology Commission of Shanghai Municipality (grant no.13dz2260400) and the National Natural Science Foundation of China (grant no. 11671137). In this paper, the Hodge-Tate weights are defined to be minus the generalized eigenvalues of Sen’s operators. Inparticular the Hodge-Tate weight of the cyclotomic character χ cyc is − hen the differential d αα − L d κ + 12 d δ is zero at z , where L is the Fontaine-Mazur L -invariant of V z . See [6] for the precise meanings of κ and δ . Roughly speaking, d δ is the derivative of Frobenius,and d κ is the derivative of Hodge-Tate weights.The condition that “( B ϕ = α cris ,S b ⊗ S V ) G Q p is locally free of rank 1 over S ” in Theorem 1.1 is equivalentto that V admits a triangulation [5]. So, Theorem 1.1 means that the derivatives of Frobenius andthe derivatives of Hodge-Tate weights of a family of 2-dimensional representations of G Q p with atriangulation satisfy a non-trivial relation at each semistable (but non-crystalline) point.Colmez’s theorem was generalized by Zhang [14] for families of 2-dimensional Galois represen-tations of G K ( K a finite extension of Q p ) and Pottharst [12] who considered families of (notnecessarily ´etale) ( ϕ, Γ)-modules of rank 2 instead of families of 2-dimensional Galois representa-tions.In this paper we give a generalization of Colmez’s theorem which includes the above generaliza-tions as special cases.Fix a finite extension K of Q p . What we work with is a family of K - B -pair (called S - B -pair inour context) that is locally triangulable. We will provide conditions for Fontaine-Mazur L -invariantto be defined. Note that, the L -invariant is now a vector with component number equal to [ K : Q p ]. Theorem 1.2.
Let W be an S - B -pair that is semistable at a point z ∈ Max( S ) . Suppose that W islocally triangulable at z with the local triangulation parameters ( δ , · · · , δ n ) . Assume that for D z ,the filtered E - ( ϕ, N ) -module attached to W z , the Fontaine-Mazur L -invariant ~ L s,t ( see Definition6.5 ) can be defined for s, t ∈ { , , · · · , n } . Then K : Q p ] (cid:18) d δ t ( p ) δ t ( p ) − d δ s ( p ) δ s ( p ) (cid:19) + ~ L s,t · (d ~w ( δ t ) − d ~w ( δ s )) = 0 . Here, ~w ( δ i ) is the Hodge-Tate weight of the character δ i . In [13] we proved Theorem 1.2 for a special case, where we consider the case of K = Q p anddemand that the Frobenius is simisimple at z . The motivation and some potential applications ofour theorem was also discussed in [13].Our paper is orginized as follows. In Section 2 we recall the theory of B -pairs built by Berger.Then in Section 3 we extend a part of this theory to families of B -pairs, and discuss the relationbetween triangulations of semistable B -pairs and refinements of their associated filtered ( ϕ, N )-modules. In Section 4 we compare cohomology groups of ( ϕ, Γ)-modules and those of B -pairs, andthen attach a 1-cocycle to each infinitesimal deformation of a B -pair. In Section 5 we use thereciprocity law to build an auxiliary formula for L -invariants. The L -invariant is defined in Section6. In Section 7 we prove a formula called “projection vanishing property” for the above 1-cocycle.Finally in Section 8 we use the auxiliary formula in Section 5 and the projection vanishing propertyto deduce Theorem 1.2. Notations
Let K be a finite extension of Q p , G K the absolute Galois group Gal( K/K ). Let K be the maximalabsolutely unramified subfield of K . Let G ab K denote the maximal abelian quotient of G K .2et χ cyc be the cyclotomic character of G K , H K the kernel of χ cyc and Γ K the quotient G K /H K .Then χ cyc induces an isomorphism from Γ K onto an open subgroup of Z × p .Let E be a finite extension of K such that all embeddings of K into an algebraic closure of E are contained in E , Emb( K, E ) the set of embeddings of K into E . We consider E as a coefficientfield and let G K acts trivially on E .Let rec K be the reciprocity map of local class field theory such that rec K ( π K ) is a lifting of theinverse of q th power Frobenius of k , where π K is a uniformizing element of K and k is the residuefield of K with cardinal number q . Note that the image of rec K coincides with the image of theWeil group W K ⊂ G K by the quotient map G K → G ab K . Let rec − K : W K → K × be the conversemap of rec K . ( ϕ, Γ K ) -modules and B -pairs We recall the construction of Fontaine’s period rings. Please consult [7, 2] for more details.Let C p be a completed algebraic closure of Q p with valuation subring o C p and p -adic valuation v p normalized such that v p ( p ) = 1.Let e E be { ( x ( i ) ) i ≥ | x ( i ) ∈ C p , ( x ( i +1) ) p = x ( i ) ∀ i ∈ N } , and let e E + be the subset of e E suchthat x (0) ∈ o C p . If x, y ∈ e E , we define x + y and xy by( x + y ) ( i ) = lim j →∞ ( x ( i + j ) + y ( i + j ) ) p j , ( xy ) ( i ) = x ( i ) y ( i ) . Then e E is a field of characteristic p . Define a function v E : e E → R ∪ { + ∞} by putting v E (( x ( n ) )) = v p ( x (0) ). This is a valuation for which e E is complete and e E + is the ring of integersin e E . If we let ε = ( ε ( n ) ) be an element of e E + with ǫ (0) = 1 and ǫ (1) = 1, then e E is a completedalgebraic closure of F p (( ε − ω = [ ε ] −
1. Let ˜ p be an element of e E such that ˜ p (0) = p .Let e A + be the ring W ( e E + ) of Witt vectors with coefficients in e E + , e A the ring of Witt vectors W ( e E ), and e B + = e A [1 /p ]. The map θ : e B + → C p , X n ≫−∞ p k [ x k ] X n ≫−∞ p k x (0) k is surjective. Let B +dR be the ker( θ )-adic completion of e B + . Then t cyc = log[ ε ] is an element of B +dR ,and put B dR = B +dR [1 /t cyc ]. There is a filtration Fil • on B dR such that Fil i B dR = L j ≥ i B +dR t j cyc .Let B +max be the subring of e B + consisting of elements of the form P n ≥ b n ([˜ p ] /p ) n , where b n ∈ e B + and b n → n → + ∞ . Put B max = B +max [1 /t cyc ]; B max is equipped with a ϕ -action.Put B log = B max [log[˜ p ]]; B log is equipped with a ϕ -action and a monodromy N ; B N =0log = B max ; B log is a subring of B dR . Put B e = B ϕ =1max . We have the following fundamental exact sequence0 / / Q p / / B e / / B dR / B +dR / / . If r and s are two elements in N [1 /p ] ∪ { + ∞} , we put e A [ r,s ] = e A + { p [¯ ω r ] , [¯ ω s ] p } and e B [ r,s ] = e A [ r,s ] [1 /p ] with the convention that p/ [¯ ω + ∞ ] = 1 / [¯ ω ] and [¯ ω + ∞ ] /p = 0. We equip these rings with3he p -adic topology. There are natural continuous G K -actions on e A [ r,s ] and e B [ r,s ] . Frobenius inducesisomorphisms ϕ : e A [ r,s ] ∼ −→ e A [ pr,ps ] and ϕ : e B [ r,s ] ∼ −→ e B [ pr,ps ] . If r ≤ r ≤ s ≤ s , then we have the G K -equivariant injective natural map e A [ r,s ] ֒ → e A [ r ,s ] . For r > e B † ,r rig = T s ∈ [ r, + ∞ ) e B [ r,s ] (equipped with certain Frechet topology) and e B † rig = ∪ r> e B † ,r rig (equipped with the inductive limittopology). Frobenius induces isomorphisms ϕ : e B † ,r rig ∼ −→ e B † ,pr rig and ϕ : e B † rig ∼ −→ e B † rig .Put A K ′ = { + ∞ X k ≥−∞ a k ω k | a k ∈ o K ′ , a k → k → −∞ ) } and B K ′ = A K ′ [1 /p ]. Here K ′ is the maximal absolutely unramified subfield of K ∞ = K ( µ p ∞ ).Then A K ′ is a complete discrete valuation ring with p as a prime element, and B K ′ is the fractionalfield of A K ′ . The G K -action and ϕ preserve A K ′ : ϕ ( ω ) = (1 + ω ) p − g ( ω ) = (1 + ω ) χ cyc ( g ) − A be the p -adic completion of the maximal unramified extension of A K ′ in e A , B its fractionalfield. Then ϕ and the G K -action preserve A and B .We put B K = B H K and B † ,rK = B K ∩ e B † ,r . Let B † ,r rig ,K be the Frechet completion of B † ,rK for thetopology induced from that on e B † ,r rig , and put B † rig ,K = ∪ r> B † ,r rig ,K equipped with the inductive limittopology. Frobunius induces injections B † ,r rig ,K ֒ → B † ,pr rig ,K and B † rig ,K ֒ → B † rig ,K ; there are continuousΓ K -actions on B † ,r rig ,K and B † rig ,K .We end this subsection by the definition of E -( ϕ, Γ K )-modules [11]. Definition 2.1. An E -( ϕ, Γ K )-module is a finite B † rig ,K ⊗ Q p E -module M equipped with a Frobe-nius semilinear action ϕ M and a comtinuous semilinear Γ K -action such that M is free as a B † rig ,K -module, that id B † rig ,K ⊗ ϕ M : B † rig ,K N ϕ, B † rig ,K M → M is an isomorphism, and that ϕ M and theΓ K -action commute with each other.By [11, Lemma 1.30] if M is an E -( ϕ, Γ K )-module, then M is free over B † rig ,K ⊗ Q p E . B -pairs We recall the theory of E - B -pairs [3, 11].Put B e,E = B e ⊗ Q p E , B +dR ,E = B +dR ⊗ Q p E and B dR ,E = B dR ⊗ Q p E . We extend the G K -actions E -linearly to these rings. Definition 2.2. An E - B -pair of G K is a couple W = ( W e , W +dR ) such that • W e is a finite B e,E -module with a continuous semilinear action G K -action which is free as a B e -module. • W +dR ⊂ W dR = B dR ⊗ B e W e is a G K -stable B +dR ,E -lattice.By [11, Remark 1.3] W e is free over B e,E and W +dR is free over B +dR ,E .If V is an E -representation of G K , then W ( V ) = ( B e,E ⊗ E V, B +dR ,E ⊗ E V ) is an E - B -pair,called the E - B -pair attached to V .If S is a Banach E -algebra, we can define S - B -pairs similarly; to each S -representation V of G K is associated an S - B -pair W ( V ) = ( B e,E ⊗ E V, B +dR ,E ⊗ E V ).4f W = ( W ,e , W +1 , dR ) and W = ( W ,e , W +2 , dR ) are two E - B -pairs, we define W N W to be( W ,e O B e,E W ,e , W +1 , dR O B +dR ,E W +2 , dR ) . Here, W ,e N B e,E W ,e is equipped with the diagonal G K -action, and W +1 , dR ⊗ B +dR ,E W +2 , dR is naturallyconsidered as a G K -stable B +dR ,E -lattice of B dR ⊗ B e ( W ,e O B e,E W ,e ) = W , dR O B dR ,E W , dR , where W , dR = B dR ⊗ B e W ,e and W , dR = B dR ⊗ B e W ,e .If W = ( W e , W +dR ) is an E - B -pair with W dR = B dR ⊗ B e W e , we define the dual of W to be W ∗ = ( W ∗ e , W ∗ , +dR ), where W ∗ e is Hom B e ( W, B e ) equipped with the natural G K -action, and W ∗ , +dR is the G K -stable lattice of B dR ⊗ B e W ∗ e ∼ = Hom B dR ( W dR , B dR ) defined by { ℓ ∈ Hom B dR ( W dR , B dR ) : ℓ ( x ) ∈ B +dR for all x ∈ W +dR } . The relation between ( ϕ, Γ K )-modules and B -pairs is built by Berger [3]. We recall Berger’sconstruction below.Let M be a ( ϕ, Γ K )-module of rank d over the Robba ring B † rig ,K . Berger [3] showed that W e ( M ) := ( e B † rig [1 /t ] ⊗ B † rig ,K M ) ϕ =1 is a free B e -module of rank d and equipped with a continuous semilinear G K -action.For sufficiently large r > K -stable finite free B † ,r rig ,K -submodule M r ⊂ M such that B † rig ,K ⊗ B † ,r rig ,K M r = M and id B † ,pr rig ,K ⊗ ϕ M : B † ,pr rig ,K ⊗ B † ,r rig ,K M r ∼ −→ M pr for any r ≥ r . Berger [3] showed that the B +dR -module W +dR ( M ) := B +dR ⊗ i n , B † , ( p − pn − ,K M ( p − p n − is independent of any n such that ( p − p n − ≥ r , and showed that there is a canonical G K -equivariant isomorphism B dR ⊗ B e W e ( M ) ∼ −→ B dR ⊗ B +dR W +dR ( M ).Put W ( M ) = ( W e ( M ) , W +dR ( M )). This is an E - B -pair of rank d = rank B † rig ,K M .The following is a variant version of Berger’s result [3, Theorem 2.2.7]. Proposition 2.3. [11, Theorem 1.36] The functor M W ( M ) is an exact functor and thisgives an equivalence of categories between the category of E - ( ϕ, Γ K ) -modules and the category of E - B -pairs of G K . Proposition 2.4.
The functor M W ( M ) respects the tensor products and duals. roof. Let M and M be two E -( ϕ, Γ K )-modules. By taking ϕ -invariants, the isomorphism( e B † rig [1 /t ] ⊗ e B † ,r rig ,K M ) ⊗ e B † rig ⊗ Q p E [1 /t ] ( e B † rig [1 /t ] ⊗ e B † ,r rig ,K M ) ∼ −→ e B † rig [1 /t ] ⊗ e B † rig ,K ( M ⊗ M )induces a G K -equivariant injective map W e ( M ) ⊗ B e,E W e ( M ) → W e ( M ⊗ M ) . Here, M ⊗ M denotes the E -( ϕ, Γ K )-module M ⊗ B † rig ,K ⊗ Q p E M . Comparing dimensions andusing [11, Lemma 1.10] we see that this map is in fact an isomorphism. From the above Berger’sconstruction we see that the natural map W +dR ( M ) ⊗ B +dR ⊗ Q p E W +dR ( M ) → W +dR ( M ⊗ M )is an isomorphism. This proves that the functor M W ( M ) respects tensor products. The proofof that it respects duals is similar. E - B -pairs Definition 2.5. An E - ( ϕ, N ) -module over K is a K ⊗ Q p E -module D with a ϕ ⊗ ϕ D : D → D , and a K ⊗ Q p E -linear map N D : D → D such that N D ϕ D = pϕ D N D .A filtered E - ( ϕ, N ) -module over K is an E -( ϕ, N )-module with an exhaustive Z -indexed descendingfiltration Fil • on K ⊗ K D .We have an isomorphism of rings K ⊗ Q p E ∼ −→ M τ ∈ Emb(
K,E ) E τ , a ⊗ b ( τ ( a ) b ) τ , (2.1)where E τ is a copy of E for each τ ∈ Emb(
K, E ). Let e τ be the unity of E τ . Then 1 = P τ e τ . Put D τ = e τ ( K ⊗ K D ). Then K ⊗ K D = L τ ∈ Emb(
K,E ) D τ . Let Fil τ denote the induced filtration on D τ . Definition 2.6.
Let W = ( W e , W +dR ) be an E - B -pair. We define D cris ( W ) = ( B max ⊗ B e W e ) G K , D st ( W ) = ( B log ⊗ B e W e ) G K and D dR ( W ) = ( B dR ⊗ B e W e ) G K . Then we have dim K ( D ? ( W )) ≤ rank B e W e for ? = cris , st, and dim K ( D dR ( W )) ≤ rank B e W e . We say that W is crystalline (resp. semistable ) if dim K ( D ? ( W )) := rank B e W e for ? = cris (resp. st).If W is a semistable E - B -pair, we attach to W a filtered E -( ϕ, N )-module as follows. Theunderlying E -( ϕ, N )-module is D st ( W ); the filtration on D dR ( W ) = K ⊗ K D st ( W ) is given byFil i D dR ( W ) = t i W +dR ∩ D dR ( W ). Proposition 2.7. (a)
The functor W D st ( W ) realizes an equivalence of categories betweenthe category of semistable E - B -pairs of G K and the category of filtered E - ( ϕ, N ) -modules over K . (b) If W and W are semistable, then so is W ⊗ W . (c) The functor W D st ( W ) respects the tensor products and duals. If / / W / / W / / W / / is a short exact sequence of E - B -pairs, and W is semistable, then W and W are semistable. (e) The functor W D st ( W ) is exact.Proof. Assertion (a) follows from [3, Proposition 2.3.4]. See also [11, Theorem 1.18 (2)].Let W and W be two E - B -pairs. The isomorphism( B log ⊗ B e W ) ⊗ B log ⊗ Q p E ( B log ⊗ B e W ) ∼ −→ B log ⊗ B e ( W ⊗ W )induces an injective map D st ( W ) ⊗ K ⊗ Q p E D st ( W ) → D st ( W ⊗ W ) . (2.2)When W and W are semistable, the dimension of the source over K is rank B e W rank B e W [ E : Q p ] . Thedimension of the target over K is always equal to or less than rank B e ( W ⊗ W ) = rank B e W rank B e W [ E : Q p ] .Hence, (2.2) is an isomorphism, and so W ⊗ W is semistable. This proves (b). Similarly, theisomorphism ( B dR ⊗ B e W ) ⊗ B dR ⊗ Q p E ( B dR ⊗ B e W ) ∼ −→ B dR ⊗ B e ( W ⊗ W ) (2.3)induces an isomorphism D dR ( W ) ⊗ K ⊗ Q p E D dR ( W ) → D dR ( W ⊗ W ) . Via the isomorphism (2.3) the filtration on ( B dR ⊗ B e W ) ⊗ B dR ⊗ Q p E ( B dR ⊗ B e W ) coincideswith that on B dR ⊗ B e ( W ⊗ W ). Therefore, the filtration on D dR ( W ) ⊗ K ⊗ Q p E D dR ( W )and that on D dR ( W ⊗ W ) coincide. Indeed, they are the restrictions of the filtrations on( B dR ⊗ B e W ) ⊗ B dR ⊗ Q p E ( B dR ⊗ B e W ) and B dR ⊗ B e ( W ⊗ W ) respectively. Similarly we canshow that W D st ( W ) respects duals. This proves (c).For (d) we have the following exact sequence0 / / D st ( W ) / / D st ( W ) / / D st ( W ) . (2.4)So (d) follows from a dimension argument. Furthermore, when W is semistable, D st ( W ) → D st ( W ) is surjective. For any i ∈ Z we write d i ( W ) for dim K Fil i D st ( W ). As the maps inthe exact sequence (2.4) respect filtrations, we have d i ( W ) ≤ d i ( W ) + d i ( W ). Similarly, wehave d − i ( W ∗ ) ≤ d − i ( W ∗ ) + d − i ( W ∗ ). As W D st ( W ) respects duals, we have d i ( W ) =dim K ( D dR ( W )) − d − i ( W ∗ ). Then d i ( W ) = dim K ( D dR ( W )) − d − i ( W ∗ ) ≥ (dim K ( D dR ( W )) − d − i ( W ∗ )) + dim K ( D dR ( W )) − d − i ( W ∗ )= d i ( W ) + d i ( W ) . Thus we must have d i ( W ) = d i ( W ) + d i ( W ) for all i ∈ Z . In other words, the maps in (2.4) arestrict for the filtrations, which shows (e). 7y [3, Proposition 2.3.4] the quasi-inverse of the functor D st is given by D B ( D ) = (( B log ⊗ K D ) ϕ =1 ,N =0 , Fil ( B dR ⊗ K D )) . (2.5)For a filtered E -( ϕ, N )-module D we put X log ( D ) = ( B log ⊗ K D ) ϕ =1 ,N =0 and X dR ( D ) = B dR ⊗ K D/ Fil ( B dR ⊗ K D ) . If D B ( D ) = ( W e , W +dR ), then X log ( D ) = W e and X dR ( D ) = ( B dR ⊗ B e W e ) /W +dR . S - B -pairs of rank and triangulations S - B -pairs of rank Let S be a Banach E -algebra.For any a ∈ S × we define a filtered S - ϕ -module D a as follows. As a K ⊗ Q p S -module, D a = K ⊗ Q p S = ⊕ τ : K ֒ → E Se τ ;the ϕ ⊗ ϕ on D a satisfies ϕ ( e id ) = e ϕ − , ϕ ( e ϕ − ) = e ϕ − , · · · , ϕ ( e ϕ − f ) = ae id ;the descending filtration on D a,K = K ⊗ Q p S is given by Fil D a,K = D a,K and Fil D a,K = 0. Lemma 3.1. If a ∈ S satisfies that a − is topologically nilpotent, then there exists a unit u ∈ B max b ⊗ K S such that ϕ [ K : Q p ] ( u ) = au . Consequently { x ∈ B max b ⊗ K S : ϕ [ K : Q p ] ( x ) = ax } = ( B e,K b ⊗ K S ) u . Proof.
Let Q ur p be the completed unramified extension of Q p . Then there exists an inclusion Q ur p ֒ → B max that is compatible with ϕ .As ϕ [ K : Q p ] − Q ur p , there exists a sequence c = 1 , c , · · · of elements in Q ur p such that ( ϕ [ K : Q p ] − c i = c i − for i ≥
1. The image of c i by the map Q ur p ֒ → B max → B max b ⊗ K S is again denoted by c i . Put u = ∞ X i =0 c i ( a − i . Then u is a unit and we have ϕ [ K : Q p ] u = au . Proposition 3.2. If a ∈ S satisfies that a − is topologically nilpotent, then D B ( D a ) is an S - B -pairof rank . Here D B is the functor defined by ( ) . roof. For each z ∈ B max b ⊗ Q p D a we write z = P c τ e τ with c τ ∈ B max b ⊗ K ,τ S . Then ϕ ( z ) = z ifand only if ϕ ( c ϕ i ) = c ϕ i − ( i = 1 , · · · , [ K : Q p ]) and ϕ [ K : Q p ] ( c id ) = ac id . Our assertion followsfrom Lemma 3.1.For any a ∈ S × , let δ a : K × → S × denote the character such that δ a ( π K ) = a and δ a | o × K = 1. Remark . In the case of S = E , for any u ∈ E × , D B ( D u ) coincides with the E - B -pair W ( δ u )defined in [11] (see [11, § W ( δ u ) from E to S is again denotedby W ( δ u ).Let δ : K × → S × be a continuous character such that δ ( π K ) is of the form δ ( π K ) = au , where u ∈ E × and a ∈ S satisfies that a − a goodcharacter . Let W a be the resulting S - B -pair in Proposition 3.2. Let δ ′ be the unitary continuouscharacter K × → E × such that δ ′ | o × K = δ | o × K and δ ′ ( π K ) = 1. By local class field theory, this inducesa continuous character e δ ′ : G K → S × such that e δ ′ ◦ rec K = δ ′ . Then we put W ( δ ) = W ( S ( e δ ′ )) ⊗ W ( δ u ) ⊗ W a , where W ( S ( e δ ′ )) is the S - B -pair attached to the Galois representation S ( e δ ′ ).If δ is a continuous character δ : K × → S × , we write log( δ ) for the logarithmic of δ | o × K , whichis a Z p -linear homomorphism log( δ ) : K → S .For any τ ∈ Emb(
K, E ) we use the same notation τ to denote the composition of τ : K ֒ → E and E ֒ → S . Then { τ : K ֒ → S } is a basis of Hom Z p ( E, S ) over S . Write log( δ ) = P τ k τ τ , k τ ∈ S .We call ( k τ ) τ the weight vector of δ and denote it by ~w ( δ ). We use w τ ( δ ) to denote k τ . Remark . Let S be an affinoid algebra over E . For any continuous character δ : K × → S × andany point z of Max( S ), there exists an affinoid neighborhood U = Max( S ′ ) of z in Max( S ) suchthat the restriction of δ to U is good. Lemma 3.5.
Let δ be a character of K × with values in S = E [ Z ] / ( Z ) , ¯ δ the character of K × with values in E obtained from δ modulo ( Z ) . Write δ = ¯ δ S (1 + Zǫ ) , where ¯ δ S is the character K × ¯ δ −→ E × ֒ → S × . Let ǫ ′ be the additive character of G K such that ǫ ′ ◦ rec K ( p ) = 0 and ǫ ′ ◦ rec K | o × K = ǫ | o × K .Assume that W (¯ δ ) is crystalline and ϕ [ K : Q p ] acts on D cris ( W (¯ δ )) by α . Then there is a nonzeroelement x ∈ ( B max ,E ⊗ B e,E W ( δ ) e ) ϕ [ K Q p ] = α (1+ Zv p ( π K ) ǫ ( p )) ,G K =(1+ Zǫ ′ ) whose reduction modulo Z is a basis of D st ( W (¯ δ )) over K ⊗ Q p E .Proof. This follows from the fact that W ( δ ) = W (¯ δ S ) ⊗ W δ Zvp ( πK ) ǫ ( p ) ⊗ W (1 + Zǫ ′ ) . Now let S be an affinoid algebra over E . For any open affinoid subset U of S and any S - B -pair W let W U denote the restriction to U of W . Definition 3.6.
Let W be an S - B -pair of rank n , z a point of Max( S ). If there is9 an affinoid neighborhood U = Max( S U ) of z , • a strictly increasing filtration { } = Fil W U ⊂ Fil W U ⊂ · · · ⊂ Fil n W U = W U of saturated free sub- S U - B -pairs, and • n good continuous characters δ i : Q × p → S × U such that for any i = 1 , · · · , n , Fil i W U / Fil i − W U ≃ W ( δ i ) , we say that W is locally triangulable at z ; we call Fil • a local triangulation of W at z , and call( δ , · · · , δ n ) the local triangulation parameters attached to Fil • .Please consult [6, 4] for more knowledge on triangulations.To discuss the relation between triangulations and refinements, we restrict ourselves to the caseof S = E .Let D be a filtered E -( ϕ, N )-module of rank n . The operator ϕ [ K : Q p ] on D is K ⊗ Q p E -linear.We assume that the eigenvalues of ϕ [ K : Q p ] : D → D are all in K ⊗ Q p E , i.e. there exists a basisof D over K ⊗ Q p E such the matrix of ϕ [ K : Q p ] with respect to this basis is upper-triangular.Following Mazur [9] we define a refinement of D to be a filtration on D F D ⊂ F D ⊂ · · · ⊂ F n D = D by E -subspaces stable by ϕ D and N D , such that each factor gr F i D = F i D/ F i − D ( i = 1 , · · · , n ) isof rank 1 over K ⊗ Q p E . Any refinement fixes an ordering α , · · · , α n of eigenvalues of ϕ [ K : Q p ] and an ordering ~k , · · · , ~k n of Hodge-Tate weights of K ⊗ K D taken with multiplicities such thatthe eigenvalue of ϕ [ K : Q p ] on gr F i D is α i and the Hodge-Tate weight of gr F i D is ~k i .We have the following analogue of [1, Proposition 1.3.2]. Proposition 3.7.
Let W be a semistable E - B -pair, D = D st ( W ) . (a) The equivalence of categories between the category of semistable E - B -pairs and the categoryof filtered E - ( ϕ, N ) -modules induces a bijection between the set of triangulations on W andthe set of refinements on D . (b) If (Fil i W ) is a triangulation of W with triangulation parameters ( δ , · · · , δ n ) that correspondto a refinement F • D of D with the ordering of Hodge-Tate weights being ~k , · · · , ~k n , then δ i = ˜ δ i Q τ ∈ Emb(
K,E ) τ ( x ) k i,τ , where ˜ δ i is a smooth character.Proof. Assertion (a) follows from the fact that D st is an exact. Assertion (b) follows from [11,Lemma 4.1]. ( ϕ, Γ K ) -modules and cohomology of B -pairs Let M be a ( ϕ, Γ K )-module. Assume that Γ K has a topological generator γ . Define the cohomology H • ΦΓ ( M ) by the complex C • ( M ) defined by C ( M ) = M ( γ − ,ϕ − −−−−−−−→ C ( M ) = M ⊕ M → C ( M ) = M, C ( M ) → C ( M ) is given by ( x, y ) ( ϕ − x − ( γ − y . Denote the kernel of C ( M ) → C ( M ) by Z ( M ).There is a one-to-one correspondence between H ( M ) and the set of extensions of M by M in the category of ( ϕ, Γ K )-modules, where M = B † rig ,K e is the trivial ( ϕ, Γ K )-module with ϕ ( e ) = γ ( e ) = e . Let ˜ M be an extension of M by M , and let ˜ e be any lifting of e in ˜ M . Thenthe element in H ( M ) corresponding to the extension ˜ M is the class of (( γ − e, ( ϕ − e ) ∈ Z ( M ).In [11] Nakamura introduced a cohomology for B -pairs and use it to compute the cohomologyof ( ϕ, Γ K )-modules.If W = ( W e , W +dR ) is an E - B -pair, let C • ( W ) be the complex of G K -modules defined by C ( W ) := W e → C ( W ) := W dR /W +dR . Here, W e → W dR /W +dR is the natural map. Definition 4.1.
Let W = ( W e , W +dR ) be an E - B -pair. We define the Galois cohomology of W by H iB ( W ) := H i ( G K , C • ( W )).By definition there is a long exact sequence · · · → H iB ( W ) → H i ( G K , W e ) → H i ( G K , W dR /W +dR ) → · · · . (4.1)For a G K -module M put C ( M ) = M and let C i ( M ) be the space of continuous functionsfrom ( G K ) × i to M . Let δ : C ( M ) → C ( M ) be the map x ( g g ( x ) − x ) and let δ : C ( M ) → C ( M ) be the map f (( g , g ) f ( g g ) − f ( g ) − g f ( g )).Nakamura [11] showed that H B ( W ) is isomorphic to ker(˜ δ ) / im(˜ δ ), where ˜ δ and ˜ δ are definedby ˜ δ : C ( W e ) ⊕ C ( W +dR ) → C ( W e ) ⊕ C ( W +dR ) ⊕ C ( W dR ) :( x, y ) ( δ ( x ) , δ ( y ) , x − y ) , ˜ δ : C ( W e ) ⊕ C ( W +dR ) ⊕ C ( W dR ) → C ( W e ) ⊕ C ( W +dR ) ⊕ C ( W dR ) :( f , f , x ) ( δ ( f ) , δ ( f ) , f − f − δ ( x )) . The map H B ( W ) → H ( G K , W e ) is induced by the forgetful map C ( W e ) ⊕ C ( W +dR ) ⊕ C ( W dR ) → C ( W e ) . There is a one-to-one correspondence between H ( G K , W ) and the set of extensions of W by W in the category of E - B -pairs. Here, W = ( B e ⊗ Q p E, B +dR ⊗ Q p E ) is the trivial E - B -pair. Let ˜ W = ( ˜ W e , ˜ W +dR ) be an extension of W by W . Let ( ˜ w e , ˜ w +dR ) be a lifting in ˜ W of(1 , ∈ W . Then the element in H B ( W ) corresponding to the extension ˜ W is just the class of(( σ ( σ −
1) ˜ w e ) , ( σ ( σ −
1) ˜ w +dR ) , ˜ w e − ˜ w +dR ) ∈ ker(˜ δ ).By Proposition 2.3 there is a one-to-one correspondence between Ext( M , M ) and Ext( W , W ( M )).It induces a natrual isomorphism i M : H ( M ) → H B ( W ( M )) . .2 -cocycles from infinitesimal deformations Let S be the E -algebra E [ Z ] / ( Z ), ˜ M an S -( ϕ, Γ K )-module. Let { e , · · · , e n } be an S -basis of˜ M , { e ∗ , · · · , e ∗ n } the dual basis of ˜ M ∗ . Put M = ˜ M ⊗ S E and M ∗ = ˜ M ∗ ⊗ S E . Let e i,z denote e i mod Z , and e ∗ j,z denote e ∗ j mod Z . Then { e ,z , · · · , e n,z } is an E -basis of M , and { e ∗ ,z , · · · , e ∗ n,z } is an E -basis of M ∗ .The matrices of ϕ and γ with respect to { e , · · · , e n } are denote by ˜ A ϕ and ˜ A γ respectively,so that ϕ ( e j ) = P i ( ˜ A ϕ ) ij e i and γ ( e j ) = P i ( ˜ A γ ) ij e i . Write ˜ A ϕ = ( I n + ZU ϕ ) A ϕ and ˜ A γ =( I n + ZU γ ) A γ . Put c ΦΓ ( ˜ M ) = ( X i,j ( U ϕ ) i,j e ∗ j,z ⊗ e i,z , X i,j ( U γ ) i,j e ∗ j,z ⊗ e i,z ) . Write D B ( ˜ M ) = ( ˜ W e , ˜ W +dR ), D B ( M ) = W and D B ( M ∗ ) = W ∗ .Let f , · · · , f n be a basis of ˜ W e over B e,E , and let g , · · · , g n be a basis of ˜ W +dR over B +dR ,E . Wewrite the matrix of σ ∈ G K with respect to the basis { f , · · · , f n } by ( I n + ZU e,σ ) A e,σ , and thematrix of σ with respect to the basis { g , · · · , g n } by ( I n + ZU +dR ,σ ) A +dR ,σ . Here, U e,σ ∈ M n ( B e,E ) , U +dR ,σ ∈ M n ( B +dR ,E ) , A e,σ ∈ GL n ( B e,E ) , and A +dR ,σ ∈ GL n ( B +dR ,E ) . Write ( f , · · · , f n ) = ( g , · · · , g n )( I n + ZU dR ) A dR and put c B ( ˜ M ) = (cid:16) ( σ X i,j ( U e,σ ) ij f ∗ j,z ⊗ f i,z ) , ( σ X i,j ( U +dR ,σ ) ij g ∗ j,z ⊗ g i,z ) , X i,j ( U dR ) ij g ∗ j,z ⊗ g i,z (cid:17) . Proposition 4.2. (a) c ΦΓ ( ˜ M ) is in Z ( M ∗ ⊗ M ) . (b) c B ( ˜ M ) is in ker(˜ δ ,W ∗ ⊗ W ) . (c) We have i M ([ c ΦΓ ( ˜ M )]) = [ c B ( ˜ M )] .Proof. It is easy to verify (a) and (b).Put M ∗ S = M ∗ ⊗ E S . We consider M ∗ S ⊗ S ˜ M as an extension of M ∗ ⊗ E M by itself, and formthe following commutative diagram M (cid:15) (cid:15) / / M ∗ ⊗ E M / / M ∗ S ⊗ S ˜ M / / M ∗ ⊗ E M / / , where the vertical map M → M ∗ ⊗ E M is given by 1 P ni =1 e ∗ i,z ⊗ e i,z , which does not dependon the choice of the basis { e , · · · , e n } . Pulling back M ∗ S ⊗ S ˜ M via M → M ∗ ⊗ E M we obtain anextension of M by M ∗ ⊗ E M . Let M denote the resulting extension. Then M is a sub- E - B -pairof M ∗ S ⊗ S ˜ M . Put D B ( M ) = ( W e , W +dR ). 12 lifting of 1 in W e is P j f ∗ j,z ⊗ f j , and a lifting of 1 in W +dR is P j g ∗ j,z ⊗ g j . We have( σ − X j f ∗ j,z ⊗ f j = σ ( f ∗ ,z , · · · , f ∗ n,z ) ⊗ σ f f ... f n − ( f ∗ ,z , · · · , f ∗ n,z ) ⊗ f f ... f n = ( f ∗ ,z , · · · , f ∗ n,z )( A te,σ ) − ⊗ A te,σ (1 + zU te,σ )= ( f ∗ ,z , · · · , f ∗ n,z ) ⊗ U te,σ z f f ... f n . Similarly, ( σ − X j g ∗ j,z ⊗ g j = ( g ∗ ,z , · · · , g ∗ n,z ) ⊗ ( U +dR ,σ ) t z g g ... g n , and X j f ∗ j,z ⊗ f j − X j g ∗ j,z ⊗ g j = ( g ∗ ,z , · · · , g ∗ n,z ) ⊗ U t dR z g g ... g n . Hence the element in H B ( D B ( M ∗ ⊗ E M )) attached to the extension D B ( M ) is [ c B ( ˜ M )].A similar computation shows that the element in H ( M ∗ ⊗ E M ) attached to the extension M is [ c ΦΓ ( ˜ M )]. Now (c) follows. In [14, Section 2] using local class field theory Zhang precisely described the perfect pairing H ( G K , E ) × H ( G K , E (1)) → H ( G K , E (1)) . We recall it below.The Kummer theory gives us a canonical isomorphism so called the Kummer maplim ←−− n ( K × / ( K × ) p n ) ⊗ Z p E → H ( G K , E (1)) X i α i ⊗ a i X i a i [( α i )] . α ) is the 1-cocycle such that g ( pn √ α ) α = ε ( α g ) n for α ∈ K × and g ∈ G K , where ( pn +1 √ α ) p = pn √ α . Combining the Kummer map and the exponentmap exp : p o K → K × and extending it by linearity we obtain an embedding from K ⊗ Q p E to H ( G K , E (1)), againdenoted by exp. Then we have H ( G K , E (1)) = exp( K ⊗ Q p E ) ⊕ E · [( p )] . Let Hom( G K , E ) be the group of additive characters of G K with values in E . As the action of G K on E is trivial, H ( G K , E ) is naturally isomorphic to Hom( G K , E ). Let ψ : G K → E be theadditive character that vanishes on the inertial subgroup of G K and maps the geometrical Frobeniusto [ K : Q p ]. For any τ ∈ Emb(
K, E ) let ψ τ be the composition τ ◦ log ◦ rec − K , where log isnormalized such that log( p ) = 0. Then { ψ , ψ τ : τ ∈ Emb(
K, E ) } is an E -basis of H ( G K , E ). Lemma 5.1. [14, Proposition 2.1] The cup product of a ψ + P τ ∈ Emb(
K,E ) a τ ψ τ ( a , a τ ∈ E ) and b [( p )] + exp( b ) ( b ∈ E, b ∈ K ⊗ Q p E ) is (cid:16) a b − tr K/ Q p (( a τ ) τ · b ) (cid:17) ( ψ ∪ [( p )]) . Here, ( a τ ) τ is considered as an element in K ⊗ Q p E via the isomorphism (2 . . Lemma 5.2.
For λ , λ τ ∈ E ( τ ∈ Emb(
K, E )) , the extension of E ( as a trivial G K -module ) by E corresponding to the cocycle λ ψ + P τ ∈ Emb(
K,E ) λ τ ψ τ is de Rham if and only if λ τ = 0 for each τ .Proof. By [11, Lemma 4.3], the subspace of extensions of E by E that are de Rham is 1-dimensional,and so consists of those corresponding to the cocycles λ ψ ( λ ∈ E ). Let ~ L = ( L σ ) σ : K֒ → E be a vector. We consider ~ L as an element of K ⊗ Q p E via the isomorphism(2.1).Let D be a filtered E -( ϕ, N )-module: the underlying E -( ϕ, N )-module D is a ( K ⊗ Q p E )-modulewith a basis { f , f , f } such that ϕ [ K : Q p ] f = p − [ K : Q p ] f , ϕ [ K : Q p ] f = f , ϕ [ K : Q p ] f = f , and N ( f ) = 0 , N ( f ) = − f , N ( f ) = f ;the filtration on K ⊗ K D = ( K ⊗ Q p E ) f ⊕ ( K ⊗ Q p E ) f ⊕ ( K ⊗ Q p E ) f Since the character ψ τ of the Weil group W K sends any lifting of the q th power Frobenius to 0, it can be extendedto a character of G K which is again denoted by ψ τ i D = (cid:26) ( K ⊗ Q p E )( f − ~ L f ) ⊕ ( K ⊗ Q p E )( f + ~ L f ) if i = 0 , i > . Let π i be the projection map X log ( D ) → B log ,E , X j =1 a j f j a i . Lemma 5.3.
Let c : G K → X log ( D ) be a -cocycle whose class in H ( G K , X log ( D )) belongs to ker( H ( G K , X log ( D )) → H ( G K , X dR ( D ))) . Then there exist γ , , γ ,τ , γ , , γ ,τ ∈ E ( τ ∈ Emb(
K, E )) such that π ( c ) = γ , ψ + X τ ∈ Emb(
K,E ) γ ,τ ψ τ and π ( c ) = γ , ψ + X τ ∈ Emb(
K,E ) γ ,τ ψ τ . Furthermore, γ , − γ , = X τ ∈ Emb(
K,E ) L τ ( γ ,τ − γ ,τ ) . In our proof of Lemma 5.3 we need the following
Lemma 5.4.
Let D be an E - ( ϕ, N ) -module. If Fil and Fil are two filtrations on K ⊗ K D suchthat Fil ( K ⊗ K D ) = Fil ( K ⊗ K D ) , then the kernel of H ( G K , X log ( D )) → H ( G K , X dR ( D, Fil )) coincides with the kernel of H ( G K , X log ( D )) → H ( G K , X dR ( D, Fil )) . Proof.
The proof is similar to that of [13, Proposition 2.5]
Proof of Lemma 5.3.
The argument is similar to the proof of [13, Lemma 5.1]. We only give asketch.Write c σ = λ ,σ f + λ ,σ f + λ ,σ f . As c takes values in X log ( D ), we have λ ,σ , λ ,σ ∈ E . Thisensures the existence of γ , , γ ,τ , γ , , γ ,τ .Let F il be the filtration on D such that F il − D = D and F il i D = Fil i D if i ≥
0. Then (
D, F il )is admissible. Let V be the semistable E -representation of G K attached to D V = ( D, F il ). ByLemma 5.4, [ c ] is in the kernel of H ( G K , X log ( D V )) → H ( G K , X dR ( D V )) and so there exists a1-cocycle c (1) : G K → V such that the image of [ c (1) ] by H ( G K , V ) → H ( G K , X log ( D V )) is [ c ].15e form the following commutative diagram V ′ (cid:15) (cid:15) / / V / / V / / π V,V (cid:15) (cid:15) T / / (cid:15) (cid:15) / / V / / V / / T / / V (resp. V ′ ) is the subrepresentation of V correspond-ing to the filtered E -( ϕ, N )-submodule of D V generated by f (resp. by f + f ) which is admissible.From (5.1) we obtain the following commutative diagram H ( G K , V ) / / π V,V (cid:15) (cid:15) H ( G K , T ) / / (cid:15) (cid:15) H ( G K , V ) H ( G K , V ) / / H ( G K , T ) / / H ( G K , V ) , where the horizontal lines are exact.Write c (2) for the 1-cocycle G K c (1) −−→ V → T → T . By a simple computation we obtain[ c (2) ] = [ (cid:16) ( γ , − γ , ) ψ + X τ ∈ Emb(
K,E ) ( γ ,τ − γ ,τ ) ψ τ (cid:17) ¯ f ] , where ¯ f is the image of f ∈ V in T . Note that T is isomorphic to E , and V is isomorphic to E (1).Being the image of [ π V,V ( c (1) )] in H ( T ), [ c (2) ] lies in the kernel of H ( G K , T ) → H ( G K , V ).By [14, Lemma 5.5], as an extension of E by E (1), V corresponds to the element [( p )] + exp( ~ L ).Now Lemma 5.1 yields our second assertion. L -invariants Let D be a filtered E -( ϕ, N )-module of rank n . Fix a refinement F of D . Then F fixes an ordering α , · · · , α n of the eigenvalues of ϕ [ K : Q p ] and an ordering ~k , · · · , ~k n of the Hodge-Tate weights. N F The operator ϕ induces a K ⊗ Q p E -semilinear operator ϕ F on gr F• D = n L i =1 F i D/ F i − D .We define a K ⊗ Q p E -linear operator N F on gr F• D . The definition is similar to the one definedin [13], so we omit some details.For any i ∈ { , · · · , n } , if N ( F i D ) = N ( F i − D ), we demand that N F maps gr F i D to zero.Now we assume that N ( F i D ) ) N ( F i − D ). Let j be the minimal integer such that N ( F i D ) ⊆ N ( F i − D ) + F j D. roposition 6.1. N ( F i − D ) ∩ F j D = N ( F i − D ) ∩ F j − D .Proof. Note that F j D , F j − D , N ( F i − D ) + F j D and N ( F i − D ) + F j − D are stable by ϕ . Thus( N ( F i − D ) + F j D ) / ( N ( F i − D ) + F j − D ) is a ϕ -module, and so must be free over K ⊗ Q p E .Hence the map F j D/ F j − D → ( N ( F i − D ) + F j D ) / ( N ( F i − D ) + F j − D ) (6.1)is an isomorphism. It follows that N ( F i − D ) ∩ F j D = N ( F i − D ) ∩ F j − D .The operator N induces a K ⊗ Q p E -linear map F i D/ F i − D → ( N ( F i − D ) + F j D ) / ( N ( F i − D ) + F j − D ) . We define the map N F : gr F i D → gr F j D to be the composition of this map and the inverse of (6.1).Finally we extend N F to the whole gr F• D by K ⊗ Q p E -linearity. Note that N F ϕ F = pϕ F N F .By definition, for any i we have either N (gr F i D ) = 0 or N (gr F i D ) = gr F j D for some j . Definition 6.2.
For j ∈ { , · · · , n − } we say that j is marked (or a marked index ) for F if thereis some i ∈ { , · · · , n } such that N F (gr F i D ) = gr F j D .Note that i and j in the above definition are determined by each other. We write i = t F ( j ) and j = s F ( i ). Proposition 6.3.
The following two assertions are equivalent: (a) s is marked and t = t F ( s ) . (b) N F t − D ∩ F s D = N F t − D ∩ F s − D and N F t D ∩ F s D ) N F t D ∩ F s − D .Proof. We have already seen that, if (a) holds, then (b) holds. Conversely, we assume that (b)holds. Then N F t D ∩ F s D ) N F t − D ∩ F s D . Thus N F t D ) N F t − D .We show that N F t D N F t − D + F s − D . If it is not true, then there exists y ∈ F t D \F t − D which is a lifting of a basis of gr F t D over K ⊗ Q p E such that N ( y ) ∈ F s − D . For any z ∈ F t D ,write z = w + λy with w ∈ F t − D and λ ∈ K ⊗ Q p E . If N ( z ) is in F s D , then N ( w ) is alsoin F s D . But N F t − D ∩ F s D = N F t − D ∩ F s − D . Thus N ( w ) is in F s − D , which implies that N ( z ) = N ( w ) + λN ( y ) is also in F s − D . So, N F t D ∩ F s D = N F t D ∩ F s − D , a contradiction.From N F t D ∩ F s D ) N F t − D ∩ F s D we see that there is x ∈ F t D \F t − D such that N ( x ) ∈F s D . We must have N F t D ⊆ N F t − D + F s D . Otherwise, let j be the smallest integer such that N F t D ⊆ N F t − D + F j D and assume that j > s . Then N F ( x + F t − D ) = 0, which contradictsthe fact that N F : gr F t D → gr F j D is an isomorphism. L -invariants Assume that s is marked for F and t = t F ( s ). We consider the decompositions F t D/ F s − D = ( K ⊗ Q p E ) · ¯ e s ⊕ L ⊕ ( K ⊗ Q p E )¯ e t that satisfy the following conditions: • F ( F t D/ F s − D ) = ( K ⊗ Q p E )¯ e s and F t − s ( F t D/ F s − D ) = ( K ⊗ Q p E )¯ e s ⊕ L , where F isthe refinement on F t D/ F s − D induced by F . • Both L and ( K ⊗ Q p E )¯ e s ⊕ ( K ⊗ Q p E )¯ e t are stable by ϕ and N ; ϕ [ K : Q p ] (¯ e t ) = α t ¯ e t and N (¯ e t ) = ¯ e s .Such a decomposition is called an s -decomposition. emark . s -decompositions may be not exist. However, if ϕ is semisimple, then s -decompositionsalways exist (see [13]).Let dec denote an s -decomposition F t D/ F s − D = E ¯ e s ⊕ L ⊕ E ¯ e t . There is a natural isomorphism E ¯ e s ⊕ E ¯ e t → ( F t D/ F s − D ) /L of ( ϕ, N )-modules. Usually thefiltration on the filtered E -( ϕ, N )-submodule E ¯ e s ⊕ E ¯ e t and that on ( F t D/ F s − D ) /L are different.When these two filtrations satisfy certain compatible condition, we say the decomposition dec isperfect. Precisely, we say that dec is perfect if for any τ : K ֒ → E we have k s,τ < k t,τ , and if thereexist k ′ s,τ , k ′ t,τ and L dec ,τ ∈ E satisfying k s,τ ≤ k ′ s,τ < k ′ t,τ ≤ k t,τ such that the following conditionshold. • The filtration on the filtered E -( ϕ, N )-submodule E ¯ e s ⊕ E ¯ e t satisfiesFil iτ ( E ¯ e s ⊕ E ¯ e t ) = E ¯ e s,τ ⊕ E ¯ e t,τ if i ≤ k s,τ ,E (¯ e t,τ + L dec ,τ ¯ e s,τ ) if k s,τ < i ≤ k ′ t,τ , i > k ′ t,τ , • The filtration on the quotient of F t D/ F s − D by L satisfiesFil iτ F t D/ F s − D = E ¯ e s,τ ⊕ E ¯ e t,τ if i ≤ k ′ s,τ ,E (¯ e t + L dec ,τ ¯ e s ) if k ′ s,τ < i ≤ k t,τ , i > k t,τ , where the images of ¯ e s and ¯ e t in F t D/ F s − D are again denoted by ¯ e s and ¯ e t . Definition 6.5.
If there exists a perfect s -decomposition, we say that s is strongly marked (or a strongly marked index ). In this case we attached to each pair ( s, t ) with t = t F ( s ) an invariant ~ L F ,s,t = ( L dec ,τ ) τ , where dec is a perfect s -decomposition. Proposition 6.6 below tells us that ~ L F ,s,t is independent of the choice of perfect s -decompositions. We call ~ L F ,s,t the Fontaine-Mazur L -invariant associated to ( F , s, t ), and denote L dec ,τ by L F ,s,t,τ .In the case of t = s + 1, s is strongly marked if and only if k s,τ < k t,τ for all τ . Proposition 6.6. If dec and dec are two perfect s -decompositions, then L dec ,τ = L dec ,τ forany τ .Proof. The argument is similar to the proof of [13, Proposition 4.9].Let D ∗ be the filtered E -( ϕ, N )-module that is the dual of D . Let ˇ F be the refinement on D ∗ such that ˇ F i D ∗ := ( F n − i D ) ⊥ = { y ∈ D ∗ : h y, x i = 0 for all x ∈ F n − i D } . We call ˇ F the dual refinement of F .If L ⊂ M are submodules of D , then M ⊥ ⊂ L ⊥ . The pairing h· , ·i : L ⊥ × M induces a non-degenerate pairing on L ⊥ /M ⊥ × M/L , so that we can identify L ⊥ /M ⊥ with the dual of M/L naturally. In particular, gr ˇ F i D ∗ is naturally isomorphic to the dual of gr F n +1 − i D . Thus gr ˇ F• D ∗ isnaturally isomorphic to the dual of gr F• D . Proposition 6.7. (a) N ˇ F is dual to − N F . (b) s is marked for F if and only if n + 1 − t F ( s ) is marked for ˇ F . (c) s is strongly marked for F if and only if n + 1 − t F ( s ) is strongly marked for ˇ F .Proof. The proof of (a) is similar to that of [13, Proposition 4.14]. The proof of (b) is similar tothat of [13, Proposition 4.13]. The proof of (c) is similar to that of [13, Proposition 4.15 (a)].18
Projection vanishing property
Put S = E [ Z ] / ( Z ). Let z be the closed point defined by the maximal ideal ( Z ) of S .Let W = ( W e , W +dR ) be an S - B -pair. Let { w , · · · , w n } be a B e,S -basis of W e . Suppose that W admits a triangulation Fil • . Let ( δ , · · · , δ n ) be the corresponding triangulation parameters. Thenfor each i = 1 , · · · , n there exists a continuous additive character ǫ i of K × with values in E suchthat δ i = δ i,z (1 + Zǫ i ).Suppose that W z , the evaluation of W at z , is semistable, and let D z be the filtered E -( ϕ, N )-module attached to W z . Let F be the refinement of D z corresponding to the induced triangulationof W z , and let { e ,z , e ,z , · · · , e n,z } be a ( K ⊗ Q p E )-basis of D z that is compatible with F i.e. F i D = ( K ⊗ Q p E ) e ,z ⊕ · · · ⊕ ( K ⊗ Q p E ) e i,z . Let α i,z ∈ E be such that ϕ [ K : Q p ] ( e i,z ) = α i,z e i,z mod F i − .Let x ij ∈ B log ,E ( i, j = 1 , · · · , n ) be such that e i,z = x i w ,z + · · · + x ni w n,z . (7.1)Then X = ( x ij ) is in GL n ( B log ,E ). Write the matrix of σ ∈ G K with respect to the basis { w , · · · , w n } by ( I n + ZU e,σ ) A e,σ . As e ,z , · · · , e n,z are fixed by G K , we have X − A e,σ σ ( X ) = I n for all σ ∈ G K .For i = 1 , · · · , n put e i = x i w + · · · + x ni w n . Then { e , · · · , e n } is a basis of B log ,S ⊗ S W e over B log ,S . Lemma 7.1. If T is the matrix of ϕ D z for the basis { e ,z , · · · , e n,z } , then T is also the matrix of ϕ B log ,S ⊗ S W e for the basis { e , · · · , e n } .Proof. The assertion follows from the definition of { e , · · · , e n } and the fact that w ,z , · · · , w n,z , w , · · · , w n are fixed by ϕ .In Section 4.1 we attach to W an element c B ( W ) in H B ( W ∗ z ⊗ W z ). Consider the composition H B ( W ∗ z ⊗ W z ) → H ( G K , W ∗ e,z ⊗ B e,E W e,z ) → H ( G K , B log ,E ⊗ E ( D ∗ z ⊗ D z )) . As the matrix of σ ∈ G K for the basis { e , · · · , e n } is I n + ZX − U e,σ X , from the discussion inSection 4 we see that the image of c B in H ( G K , B log ,E ⊗ E ( D ∗ z ⊗ D z )) is the class of the 1-cocycle( U e,σ ) ij w ∗ j,z ⊗ w i,z = ( X − U e,σ X ) ij e ∗ j,z ⊗ e i,z . Let π hℓ be the projection B log ,E ⊗ E ( D ∗ z ⊗ D z ) → B log ,E , X j,i b ji e ∗ j,z ⊗ e i,z b hℓ . (7.2)For h = 1 , · · · , n , let ǫ ′ h be the additive character of G K such that ǫ ′ h ◦ rec K ( p ) = 0 and ǫ ′ h ◦ rec K | o × K = ǫ h | o × K . Theorem 7.2. (a)
For any pair of integers ( h, ℓ ) such that h < ℓ we have π hℓ ([ c ]) = 0 . (b) For any h = 1 , · · · , n , π h,h ([ c ]) coincides with the image of [ ǫ ′ h ] in H ( G K , B log ,E ) . We call (a) the projection vanishing property .19 roof.
The filtered E -( ϕ, N )-module attached to W z / Fil h − W z is D z / F h − D z . We denote theimage of e ℓ,z ( ℓ ≥ h ) in D z / F h − D z again by e ℓ,z .Let δ ′ h be the character of G K such that δ ′ h = 1 + Zǫ ′ h . By Lemma 3.5 there exists an element x ∈ ( B max ,E ⊗ B e,E ( W/ Fil h − W ) e ) G K = δ ′ h ,ϕ [ K Q p ] = α i,z (1+ Zv p ( π K ) ǫ h ( p )) whose image in D z / F h − D z is e h,z . Write x = e h + Z P ℓ ≥ h λ ℓ e ℓ with λ ℓ ∈ B log ,E .As the matrix of σ ∈ G K for the basis { e , · · · , e n } is I n + ZX − U e,σ X , we have[1 + Zǫ ′ h ( σ )] x = [1 + Zǫ ′ h ( σ )]( e h + Z X ℓ ≥ h λ ℓ e ℓ )= σ ( x ) = e h + Z X ℓ ≥ h ( X − U e,σ X ) ℓh e ℓ + Z X ℓ ≥ h σ ( λ ℓ ) e ℓ . For ℓ > h , comparing the coefficients of e ℓ we obtain( X − U e,σ X ) ℓh = (1 − σ ) λ ℓ , which shows (a). Similarly, comparing coefficients of e h we obtain( X − U e,σ X ) hh − ǫ ′ h ( σ ) = (1 − σ ) λ h , (7.3)which implies (b). We will need the following lemmas.
Lemma 8.1.
The inclusion
E ֒ → B e,E induces an isomorphism H ( G K , E ) ∼ −→ ker( N : H ( G K , B e,E ) → H ( G K , B log ,E )) . Proof.
The proof is identical to that of [13, Corollary 1.4].
Lemma 8.2.
The map N : B ϕ = p log ,E → B ϕ =1log ,E is surjective.Proof. The proof is identical to that of [13, Lemma 1.2].For the proof of Theorem 1.2 we may assume that S = E [ Z ] / ( Z ), and z is the closed pointdefined by the maximal ideal ( Z ). Let W be as in Theorem 1.2. Replacing W by the E - B -pair F t W/ F s − W and replacing F by the induced refinement on F t W/ F s − W , we may assume that s = 1 and t = n = rank B e,E ( W e ). Let e ,z , e ,z , · · · , e n,z be a K ⊗ Q p E -basis of D z such that( K ⊗ Q p E ) e ,z M L M ( K ⊗ Q p E ) e n,z (8.1)with L = ⊕ n − i =2 ( K ⊗ Q p E ) e i,z a perfect 1-decomposition of D z for F (see § e ∗ ,z , e ∗ ,z , · · · , e ∗ n,z be the dual basis of D ∗ z over K ⊗ Q p E .20et D be the quotient of D z by L , D ∗ the quotient of D ∗ z by ⊕ n − i =2 ( K ⊗ Q p E ) e ∗ i,z . Put D = D ∗ ⊗ D . The images of e ,z and e n,z in D are again denoted by e ,z and e n,z , and the imagesof e ∗ ,z and e ∗ n,z in D ∗ are again denoted by e ∗ ,z and e ∗ n,z respectively. So e ∗ ,z ⊗ e ,z , e ∗ ,z ⊗ e n,z , e ∗ n,z ⊗ e ,z , e ∗ n,z ⊗ e n,z form a K ⊗ Q p E -basis of D . Let D be the filtered E -( ϕ, N )-submodule of D with a K ⊗ Q p E -basis { e ∗ ,z ⊗ e ,z , e ∗ n,z ⊗ e ,z , e ∗ n,z ⊗ e n,z } . Let W = ( W e , W +dR ) (resp. W ) bethe E - B -pair attached to D (resp. D ). Note that ϕ [ K : Q p ] ( e ∗ ,z ⊗ e ,z ) = e ∗ ,z ⊗ e ,z , ϕ [ K : Q p ] ( e ∗ n,z ⊗ e n,z ) = e ∗ n,z ⊗ e n,z ,ϕ [ K : Q p ] ( e ∗ n,z ⊗ e ,z ) = p − [ K : Q p ] e ∗ n,z ⊗ e ,z , and − N ( e ∗ ,z ⊗ e ,z ) = N ( e ∗ n,z ⊗ e n,z ) = e ∗ n,z ⊗ e ,z , N ( e ∗ n,z ⊗ e ,z ) = 0Let ~ L F = ~ L F ,s,t be the L -invariant defined in Definition 6.5. As (8.1) is a prefect decomposition,we have Fil ( K ⊗ K D ) = Ee ∗ n,z ⊗ ( e n,z + ~ L F e ,z ) ⊕ E ( e ∗ ,z − ~ L F e ∗ n,z ) ⊗ e ,z ⊕ E ( e ∗ ,z − ~ L F e ∗ n,z ) ⊗ ( e n,z + ~ L F e ,z ) . and Fil ( K ⊗ K D ) = Ee ∗ n,z ⊗ ( e n,z + ~ L F e ,z ) ⊕ E ( e ∗ ,z − ~ L F e ∗ n,z ) ⊗ e ,z . Consider W as an infinitesimal deformation of W z . In Section 4.2 we attach to this infinitesimaldeformation an element c B ( W ) in H B ( W ∗ z ⊗ W z ). Let [ c ] be the image of c B ( W ) by the composition H B ( W ∗ z ⊗ W z ) → H ( G K , W ∗ e,z ⊗ B e,E W e,z ) → H ( G K , B log ,E ⊗ K ⊗ Q p E ( D ∗ z ⊗ D z )) , and choose a 1-cocyle c representing [ c ]. Write c in the form c = X j,i c j,i e ∗ j,z ⊗ e i,z with c i,j being a 1-cocycle of G K with values in B log ,E . By the projection vanishing property(Theorem 7.2 (a)) we have [ c ,n ] = 0. Lemma 8.3.
There exist ξ , ξ n ∈ B e,E and γ , , γ ,τ , γ n, , γ n,τ ( τ ∈ Emb(
K, E )) such that c , ( σ ) = ( σ − ξ + γ , ψ ( σ ) + X τ ∈ Emb(
K,E ) γ ,τ ψ τ ( σ ) and c n,n ( σ ) = ( σ − ξ n + γ n, ψ ( σ ) + X τ ∈ Emb(
K,E ) γ n,τ ψ τ ( σ ) for any σ ∈ G K .Proof. Let ¯ c B be the image of c B in H B ( W ), and let ¯ c be the 1-cocycle¯ c = X j,i ∈{ ,n } c j,i e ∗ j,z ⊗ e i,z G K with values in B log ,E ⊗ K ⊗ Q p E D . Then the image of ¯ c B in H ( G K , B log ,E ⊗ K ⊗ Q p E D )is [¯ c ].Note that ¯ c has values in W e = ( B log ,E ⊗ K ⊗ Q p E D ) ϕ =1 ,N =0 . So, in particular c , and c n,n have values in B e,E . As N ¯ c = 0, we have N ( c n, ) = c , − c n,n , − N ( c , ) = N ( c n,n ) = c ,n . As [ c ,n ] = 0, the statement follows from Lemma 8.1.Write δ i = δ i,z (1 + Zǫ i ). Let ǫ ′ i be the additive character of G K with values in E such that ǫ ′ i ◦ rec K ( p ) = 0 and ǫ ′ i ◦ rec K | o × K = ǫ i | o × K . Then there are ǫ i,τ ( τ ∈ Emb(
K, E )) such that ǫ ′ i = P τ ∈ Emb(
K,E ) ǫ i,τ ψ τ . Lemma 8.4.
For h = 1 , n we have [ K : Q p ] γ h, = − v p ( π K ) ǫ h ( p ) and γ h,τ = ǫ h,τ .Proof. We keep to use notations in the proof of Theorem 7.2. By (7.3) and Lemma 8.3 we have( σ − λ h ) = − ( X − U σ X ) hh + X τ ∈ Emb(
K,E ) ǫ h,τ ψ τ ( σ )= − ( σ − ξ h − γ h, ψ ( σ ) + X τ ∈ Emb(
K,E ) ( ǫ h,τ − γ h,τ ) ψ τ ( σ ) . Note that there exists ω ∈ W( F p ) such that ϕ ( ω ) − ω = 1, where W( F p ) is the ring of Witt vectorswith coefficients in the algebraic closure of F p . Then ( σ − ω = ψ ( σ ). Hence X τ ∈ Emb(
K,E ) ( ǫ h,τ − γ h,τ ) ψ τ ( σ ) = ( σ − λ h + ξ h + γ h, ω ) . In other words, the cocycle P τ ∈ Emb(
K,E ) ( ǫ h,τ − γ h,τ ) ψ τ ( σ ) is de Rham. By Lemma 5.2 we have γ h,τ = ǫ h,τ and λ h + ξ h + γ h, ω ∈ E . Then( ϕ [ K : Q p ] − λ h = − ( ϕ − ξ h − γ h, ( ϕ [ K : Q p ] − ω = − [ K : Q p ] γ h, . (8.2)By our choice of the basis { e ,z , · · · , e n,z } , Y = ⊕ ni =2 Ze i,z is stable by ϕ . Put Y n = 0. Let x beas in the proof of Theorem 7.2. By Lemma 7.1 we have ϕ [ K : Q p ] e h,z = α h,z e h,z . Thus for h = 1 , n we have ϕ [ K : Q p ] ( x ) = (1 + Zϕ [ K : Q p ] ( λ h )) α h,z e h (mod Y h ) . On the other hand, ϕ [ K : Q p ] ( x ) = (1 + Zv p ( π K ) ǫ h ( p )) α h,z x = (1 + Zv p ( π K ) ǫ h ( p )) α h,z (1 + Zλ h ) e h (mod Y h ) . Hence we obtain ( ϕ [ K : Q p ] − λ h = v p ( π K ) ǫ h ( p ) . (8.3)22y (8.2) and (8.3) we have[ K : Q p ] γ h, = − ( ϕ [ K : Q p ] − λ h = − v p ( π K ) ǫ h ( p ) , as wanted.By Lemma 8.2 there exists some y ∈ B ϕ = p log ,E such that N ( y ) = ξ − ξ n . Let ¯ c ′ be the 1-cocycleof G K with values in B log ,E ⊗ K ⊗ Q p E D such that¯ c ′ = c ′ , e ∗ ,z ⊗ e ,z + c ′ n,n e ∗ n,z ⊗ e n,z + c ′ n, e ∗ n,z ⊗ e ,z with c ′ , = γ , ψ + X τ ∈ Emb(
K,E ) γ ,τ ψ τ , c ′ n,n = γ n, ψ + X τ ∈ Emb(
K,E ) γ n,τ ψ τ and c ′ n, ( σ ) = c n, ( σ ) − ( σ − y, σ ∈ G K . It is easy to check that ϕ (¯ c ′ ) = ¯ c ′ and N (¯ c ′ ) = 0. Hence ¯ c ′ is a 1-cocycle of G K with values in X log ( D ). Proposition 8.5.
The image of [¯ c ′ ] in H ( G K , X log ( D )) belongs to the kernel of H ( G K , X log ( D )) → H ( G K , X dR ( D )) . Proof.
Consider the following commutative diagram H ( G K , X log ( D )) / / (cid:15) (cid:15) H ( G K , X dR ( D )) (cid:15) (cid:15) H ( G K , X log ( D )) / / H ( G K , X dR ( D )) . The right vertical arrow in the above diagram is injective (see [13, Corollary 2.4]). So we only needto show that the image of [¯ c ′ ] in H ( G K , X dR ( D )) is zero. Note that[¯ c ′ ] = [¯ c ] − [ c ,n e ∗ ,z ⊗ e n,z ] = − [ c ,n e ∗ ,z ⊗ e n,z ]in H ( G K , X dR ( D )). As the image of [ c ,n ] in H ( G K , B log ,E ) is zero, so is its image in H ( G K , B dR ,E / Fil f B dR ,E ), where f is the smallest integer such that e ∗ ,z ⊗ e n,z ∈ Fil − f D K . Hence,the image of [¯ c ′ ] in H ( G K , X dR ( D )) is zero.Now, applying Lemma 5.3 to D with f = e ∗ n,z ⊗ e ,z , f = e ∗ ,z ⊗ e ,z and f = e ∗ n,z ⊗ e n,z , weget γ n, − γ , = X τ ∈ Emb(
K,E ) L τ ( γ n,τ − γ ,τ ) . Hence, by Lemma 8.4 we have v p ( π K )[ K : Q p ] ( ǫ n ( p ) − ǫ ( p )) + X τ ∈ Emb(
K,E ) L τ ( ǫ n,τ − ǫ ,τ ) = 0 . d δ h ( p ) δ h ( p ) = ǫ h ( p )d Z and d ~w ( ǫ h ) = ( ǫ h,τ d Z ) τ , we obtain1[ K : Q p ] (cid:18) d δ n ( p ) δ n ( p ) − d δ ( p ) δ ( p ) (cid:19) + ~ L F · (d ~w ( δ n ) − d ~w ( δ )) = 0 , as desired. This finishes the proof of Theorem 1.2. References [1] D. Benois,
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