aa r X i v : . [ m a t h . N T ] S e p On Perrin-Riou’s exponential map for ( ϕ, Γ)-modules
Andreas RiedelNovember 1, 2014
Abstract
Let K/ Q p be a finite Galois extension and D a ( ϕ, Γ)-module over the Robba-ring B † rig ,K . We give a generalization of the Bloch-Kato exponential map for D usingcontinuous Galois-cohomology groups H i ( G K , W ( D )) for the B -pair W ( D ) associatedto D . We construct a big exponential map Ω D,h ( h ∈ N ) for cyclotomic extensionsof K for D in the style of Perrin-Riou using the theory of Berger’s B -pairs, whichinterpolates the generalized Bloch-Kato exponential maps on the finite levels. Contents p -adic Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 ( ϕ, Γ K )-modules over B † rig ,K . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 e B † rig -modules and B -pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Cohomology of ( ϕ, Γ K )-modules . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 ( ϕ, N, Gal(
L/K ))-modules associated to ( ϕ, Γ K )-modules . . . . . . . . . . 13 ϕ, Γ K )-modules . . . . . . . . . . . . . . 183.2 Perrin-Riou exponential maps for ( ϕ, Γ K )-modules . . . . . . . . . . . . . . 31 We fix some notation. Let K be a finite extension of Q p and denote by F the biggestsubextension of K that is unramified over Q p . Let µ p n denote the roots of unity in a fixed p -adic Hodge theory, B -pairs, big exponential map, ( ϕ, Γ)-modules. lgebraic closure K of K and set K n = K ( µ p n ) and K ∞ = S n K n . As usual G K denotesthe absolute Galois group of K , and we set H K = Gal( K/K ∞ ) and Γ K = G K /H K . Perrin-Riou considers a distribution algebra H (Γ K ) that contains the usual Iwasawa algebraΛ(Γ K ).Recall that by the theory of Fontaine one may then associate to any p -adic represen-tation of V of G K finite dimensional F -vector spaces D cris ( V ) ⊂ D st ( V ) ⊂ D dR ( V ) , via the Q p -algebras B cris , B st , B dR , where the first two come equipped with an action of aFrobenius ϕ and a nilpotent monodromy operator N , and the third one is equipped witha filtration.Bloch and Kato constructed the exponential map exp : D dR ( V ) −→ H ( K, V ), whichis nothing but a transition morphism arising from a long exact sequence of continuousGalois cohomology. They showed that there exists a deep connection between this mapand the special values of the complex L -function attached to V .Perrin-Riou set out to adapt this construction to the theory of p -adic L -functions.Explicitly, for K/ Q p unramified and V crystalline (i.e. dim F D cris ( V ) = dim Q p V ) sheconstructed a map Ω V ( j ) ,h that fits into the following diagram H (Γ K ) ⊗ Q p D cris ( V ( j )) Ω V ( j ) ,h / / Ξ n,j (cid:15) (cid:15) H (Γ K ) ⊗ Λ H ( K, V ( j )) /V ( j ) G Q p,n pr n (cid:15) (cid:15) K n ⊗ D st ( V ( j )) ( h − Knn,V ( j ) / / H ( K n , V ( j )) (1)for h ≫ j ≫ n , where Ξ n,j and pr n are certain canonical projections and H denotes Iwasawa cohomology with respect to the tower ( K n ) n . The point here is that Ω V,h interpolates infinitely many Bloch-Kato exponential maps on the finite levels.In [28], Perrin-Riou extended her construction to semi-stable representations over un-ramified extensions. She gave a definition of a free H (Γ K )-module D ∞ ,g ( V ) and a mapΩ V,h : D ∞ ,g ( V ) −→ H (Γ Q p ) ⊗ Λ H ( K, V ) /V G K ∞ that has a similar interpolation property as (1) for j ≫ n ≫ H (Γ K ) ⊗ D cris ( V ) and H ( K, V ). His fundamental insight is a comparison isomorphismdepending on the construction of another ring B † rig ,K resp. B † log ,K .Berger considered in the crystalline case the element ∇ h − ◦ . . . ◦∇ , where ∇ i ∈ H (Γ K )is Perrin-Riou’s differential operator, and showed that one obtains a map ∇ h − ◦ . . . ◦ ∇ : ( ϕ − B † rig ,K ⊗ D cris ( V ( j ))) ψ =1 −→ ( ϕ − D † rig ( V ( j )) ψ =1 = H (Γ Q p ) ⊗ Λ H ( K, V ( j )) /V ( j ) G K ∞ that actually coincides with Perrin-Riou’s Ω V ( j ) ,h (see [4], Theorem II.13).2ince one has an embedding of the category of p -adic representations into the categoryof all ( ϕ, Γ)-modules over B † rig ,K via the functor D † rig ( ), one might be inclined to generalizethe framework of exponential maps to this setting. Similarly as in the ´etale case, one maydefine finite-dimensional vector spaces D cris ( D ), D st ( D ) and D dR ( D ), generalized Bloch-Kato exponential maps exp : D dR ( D ) → H ( K, D ) , and develop the notion of a ( ϕ, Γ)-module being crystalline, semi-stable or de Rham. Wedefine a H (Γ K )-module D ∞ ,g ( D ) and show that there exists a map for h ≫ D,h := ∇ h − ◦ . . . ◦ ∇ : D ∞ ,g ( D ) −→ ( ϕ − D ψ =1 . The main result of the third section is then the following interpolation property (seeTheorem 3.41 for the precise statement):
Theorem.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K , g ∈ D ∞ ,g ( D ) and G a“complete solution” (cf. Definition 3.32) for g in L and let h ≫
0. Then for k ≥ − h and n ≫ h K n ,D ( k ) ( ∇ h − ◦ . . . ◦ ∇ ( g ) ⊗ e k )= p − n ( K n ) ( − h + k − ( h + 1 − k )! 1[ L n : K n ] Cor L n /K n exp K n ,D ( k ) (Ξ n,k ( G )) . If one is interested in the construction of p -adic L -functions, one needs to construct acertain “inverse” of the map Ω h . This construction depends on the so-called reciprocitylaw for ( ϕ, Γ)-modules, which we will return to in a future paper, using the results of thisarticle.We remark that during this work learned of the results of K. Nakamura [23], whogave a description of a “big exponential map” for ( ϕ, Γ)-modules. We briefly outline howour constructions differ from [23]. Firstly, we show the existence of a fundamental exactsequence 0 −→ X ( e D ) −→ e D log [1 /t ] −→ X −→ X ( e D ) −→ G K -modules associated to any ( ϕ, Γ)-module D , generalizing the Bloch-Kato fundamental exact sequence (cf. p. 19 for the definition of X ). Taking continuousGalois-cohomology one obtains, in a completely analoguous fashion to the ´etale case, ageneralized Bloch-Kato exponential map as the transition map for cohomology, which isautomatically functorial by construction.Secondly, we introduce certain finitely generated H (Γ)-submodules D ∞ , ∗ ( D ) of thefree B (Γ)-module N dR ( D ) ψ =0 such that X arises in a natural way after projecting tosome finite level K n and looking at the Bloch-Kato exponential map on this level. Usingthese two different ingredients we are able to show the main theorem above.Some important facts about these modules are: • the D ∞ , ∗ ( D ) are invariant under Tate-twists (as opposed to (1 − ϕ ) N dR ( D ) ψ =1 ),and 3 the D ∞ , ∗ ( D ) remove the ambiguity in the statements [23], Theorem 3.10 (1) and [4],Theorem II.16 about the existence of an element y such that (1 − ϕ )( y ) = x .These points and further examples suggest that, in order to study reciprocity laws andthe connection of exponential maps with p -adic L -functions, one should look at thesemodules instead of (1 − ϕ ) N dR ( D ) ψ =1 . We also refer to the introduction of [27] in the´etale unramified case for some further motivation. Acknowledgements . This article is based on a part of my thesis, and I would like tothank my advisor Otmar Venjakob for his encouragement, patience and advice.
The general strategy of Fontaine is to study p -adic representations by certain admissibility conditions. Recall that if V is a finite dimensional Q p -vectorspace endowed with a con-tinuous action of a topological group G and if B is a topological Q p -algebra which alsocarries an action of G , then Fontaine considers the B G -modules D B ( V ) = ( B ⊗ Q p V ) G .It inherits actions from B and V . One says that V is B -admissible if B ⊗ Q p V ∼ = B d as G -modules.Let k be a perfect field of characteristic p . We denote by W ( k ) the ring of Witt-vectorsfor k and set F = Quot( W ( k )). Let K/F be a totally ramified extension of F . Fix analgebraic closure F of F and denote by C p = b F the p -adic completion of this closure. Let G K = Gal( K/K ) be the group of automorphisms of K which fix K . By continuity theseare also the K -linear automorphisms of C p . Let O C p be the ring of integers of C p and m C p its maximal ideal. We have O C p / m C p = k .We denote by µ p n the group of roots of unity of p n -order in C p and set K n = K ( µ p n ).Further we pose K ∞ = S n K n . We fix once and for all a compatible set of primitive p -throots of unity { ζ p n } n ≥ such that ζ = 1, ζ p = 1, ζ pp n +1 = ζ p n . One has the cyclotomiccharacter χ : G K → Z × p which is defined by the formula g ( ζ p n ) = ζ χ ( g ) p n for n ≥ g ∈ G K .We set H K = ker( χ ) and Γ K = G K /H K , which is the Galois group of K ∞ /K . Weknow that this can also be identified via the cyclotomic character with an open subgroupof Z × p .If K/ Q p is a finite extension denote by F = K the maximal unramified extension of Q p in K . Further denote by K ′ the biggest unramified subextension of K in K ∞ .By a p -adic representation we mean a finite dimensional Q p -vectorspace endowed witha continuous and linear action of G K . A Z p -representation is a free Z p -module of finiterank equipped with a linear and continuous action of G K . It is known that if V is a p -adicrepresentation then there exists a Z p -lattice T in V that is stable under the action of G K .If C • ( − ) denotes complex of R -modules for some commutative ring (for example, C • ( G K , M )) R we denote as usual R Γ( − ) the complex which we regard as an object inthe derived category of R -modules. 4 .2 Rings in p -adic Hodge theory We first recall certain rings constructed by Fontaine, see for instance [16]. Let e E = lim ←− x x p C p = { ( x (0) , x (1) , . . . ) | x ( i ) ∈ C p , ( x ( i +1) ) p = x ( i ) ∀ i } . Similarly, let e E + = lim ←− x x p O C p = { ( x (0) , x (1) , . . . ) | x ( i ) ∈ O C p , ( x ( i +1) ) p = x ( i ) ∀ i }∼ = { ( x n ) n ∈ N | x n ∈ O C p /p O C p , x pn +1 = x n ∀ n } . This is the set of elements of e E such that x (0) ∈ O C p . One can define multiplication andaddition on these sets. Also, one knows that e E is the fraction field of e E + .With the choices of the primitive p n -th roots of unity one defines the elements ε =(1 , ζ p , . . . ) ∈ e E + and π = ε − ∈ e E + . One has the usual commuting actions of a Frobenius ϕ and the Galois group G Q p on e E , which restrict to actions of e E + . For K/ Q p finite we set E + K = { ( x n ) ∈ e E + | x n ∈ O K n /p O K n ∀ n ≥ n ( K ) } , where n ( K ) is some constant depending on K which arises in the fields of norm theoryof Fontaine-Wintenberger (cf. [15]). We put E K = E + K [1 /π ]. One can show that that E F = κ (( π )) and one defines E as the seperable closure of E F in e E . Let E + = E ∩ e E + and m E = E ∩ m e E . One can show that E K = E H K and one knows that Gal( E / E K ) ∼ = H K .Let W be the Witt functor. We set e A + = W ( e E + ) , e A = W ( e E ) = W (Frac( e E + )) , e B + = e A + [1 /p ] . We write elements x ∈ e B + as x = P ∞ k ≫−∞ p k [ x k ] where x k ∈ e E + and [ x k ] is its Teichm¨ullerrepresentative. The commuting actions of ϕ and G Q p on e E + extend to an action of e B + (and e A , e B , . . . ).We have a ring homomorphism θ : e B + −→ C p , ∞ X k ≫−∞ p k [ x k ] ∞ X k ≫−∞ p k x (0) k . We set π = [ π ] = [ ε ] − π n = [ ε p − n ] − ω = π/π and q = ϕ ( ω ) = ϕ ( π ) /π . Then ker( θ )is a principal ideal generated by ω .The ring B +dR is defined by completing e B + with the ker( θ )-adic topology, i.e., B +dR =lim ←− n ≥ e B + / (ker( θ ) n ). This gives a complete discrete valuation ring with maximal idealker( θ ). One can show that log([ ε ]) converges in B +dR , and we denote this element by t . Itis a generator of the maximal ideal, hence we can form the field B dR = B +dR [1 /t ]. Thisfield is equipped with an action of G Q p and a canonical filtration defined by Fil i ( B dR ) = t i B +dR , i ∈ Z . 5e say that a p -adic representation V of G K is de Rham if it is B dR -admissible. Weput D dR ( V ) = ( B dR ⊗ Q p V ) G K , Fil i D dR ( V ) = (Fil i B dR ⊗ Q p V ) G K . From Fontaine’s theory it is known that D dR ( V ) is finite dimensional K -vectorspace whichwe endow with the above (exhaustive, seperated and decreasing) filtration.We say that a p -adic representation V is Hodge-Tate with Hodge-Tate weights h , . . . , h d if one has a decomposition C p ⊗ Q p V ∼ = L di =1 C p ( h i ). We say that V is positive if its Hodge-Tate weights are negative. It is known that every de Rham representationis Hodge-Tate and that the Hodge-Tate weights are those integers h such that there is ajump in the filtration at − h , i.e. Fil − h D dR ( V ) = Fil − h +1 D dR ( V ). With this conventionthe representation Q p (1) is of weight 1.Let A Q p = \ Z [[ π ]][1 /π ] = (X k ∈ Z a k π k (cid:12)(cid:12)(cid:12)(cid:12) a k ∈ Z p , lim k →−∞ v p ( a k ) = + ∞ ) ֒ → e A , and set B Q p = A Q p [1 /p ]. Then B Q p is a field, complete for the p -adic valuation with ringof integers A Q p and residue field E Q p . Let B be the p -adic completion of the maximalunramified extension of B Q p in e B . We define A = B ∩ e A , A + = A ∩ e A + . These ringsstill have the commuting action of ϕ and G Q p . We put A K = A H K and B K = A K [1 /p ].By Hensel’s Lemma there exists a unique lift π K ∈ A K such that the reduction mod p isequal to π K , viewed as an element in e A .Colmez has defined the ring B +max = { X n ≥ a n ω n p n | a n ∈ e B + , a n → n → ∞} which is ”very close” to B +cris . We set B max = B +max [1 /t ] . There is a canonical injection of B max into B dR and it is therefore equipped with a canonical filtration. There are actionsof ϕ and G Q p on B max , which extend the actions on e A + → e A + . Let e B +rig = ∞ \ n =0 ϕ n ( B +max )and e B rig = e B +rig [1 /t ]. We remark that one has e B rig = T ∞ n =0 ϕ n ( B cris ) and hence inparticular e B ϕ =1rig = B ϕ =1max = B ϕ =1cris . We say that a representation is crystalline if it is B max -admissible, which is the same as asking that it be e B +rig [1 /t ]-admissible. We put D cris ( V ) = ( B max ⊗ Q p V ) G K = ( e B +rig [1 /t ] ⊗ Q p V ) G K . This is a K -vectorspace of dimension d , equipped with a filtration induced by B dR and anaction of Frobenius induced by B max . If V is crystalline we have D dR ( V ) = K ⊗ K D cris ( V )which shows that a crystalline representation is also de Rham.6ollowing Berger the series log( π (0) ) + log( π/π (0) ), after a choice of log p , converges in B +dR , and we denote the limit by log[ π ]. This element is transcendent over Frac( B +max ),and we set B st = B max [log[ π ]] and e B +log = e B +rig [log[ π ]]. We say that a representation is semistable if it is B st -admissible, which is the same as asking it being e B +log [1 /t ]-admissible.Similarly, as in the crystalline case we put D st ( V ) = ( B st ⊗ Q p V ) G K = ( e B +log [1 /t ] ⊗ Q p V ) G K . Again this is a K -vectorspace of dimension d , equipped with a filtration and an actionof Frobenius induced by B st . As before we have in this case D dR ( V ) = K ⊗ K D st ( V ).Additionally one can define the monodromy operator N = − d/d log[ π ] on B st whichinduces a nilpotent endomorphism on D st ( V ) and satisfies the relation N ϕ = pϕN . Wealso make use of the finite dimensional K -vectorrspace D +st ( V ) = ( e B +log ⊗ Q p V ) G K .Recall that elements x ∈ e B may be written in the form x = P k ≫−∞ p k [ x k ] with x k ∈ e E . For r > e B † ,r = (cid:26) x ∈ e B (cid:12)(cid:12)(cid:12)(cid:12) lim k → + ∞ v E ( x k ) + prp − k = + ∞ (cid:27) . We note that x as above converges in B dR if and only if P k ≫−∞ p k x (0) k converges in C p .For n ≥ r n = ( p − p n − . Colmez and Cherbonnier showedthat for n big enough such that r n ≥ r there is an injection ι n = ϕ − n : e B † ,r −→ B +dR , X k ≫−∞ p k [ x k ] X k ≫−∞ p k [ x p − n k ] . We put e B † ,n = e B † ,r n . Let B † ,r = B ∩ e B † ,r , e B † = S r ≥ e B † ,r , B † = S r ≥ B † ,r . Let e A † ,r bethe elements of e B † ,r ∩ e A such that v E ( x ) + prp − k ≥ k ≥
0. Let A † ,r = e A † ,r ∩ A , A † = e A † ∩ A , e A † = e B † ∩ e A . Let B † ,rK = ( B † ,r ) H K , A † ,rK = ( A † ,r ) H K , e B † ,rK = ( e B † ,r ) H K , e A † ,rK = ( e A † ,r ) H K . Proposition 2.1. If L/K be a finite extension then B † L is a finite field extension of B † K of degree [ L ∞ : K ∞ ] = [ H K : H L ], and if L/K is Galois, then the same holds for B † L / B † K ,which then has galois group Gal( L ∞ /K ∞ ). Proof.
See [10], Proposition II.4.1.If A is a ring which is complete for the p -adic topology and X, Y are indeterminanteswe let A { X, Y } = lim ←− n A [ X, Y ] /p n A [ X, Y ] , that is, A { X, Y } is the p -adic completion of A [ X, Y ]. Every element of A { X, Y } canbe written as P i,j ≥ a ij X i Y j where a ij is a sequence in A tending to 0 in the p -adic7opology. We let r, s ∈ N [1 /p ] ∪ { + ∞} such that r ≤ s . By definition one has (in Fr( e B )) p/ [ π ] + ∞ = 1 / [ π ] and [ π ] + ∞ /p = 0. Let e A [ r ; s ] = e A + { p [ π ] r , [ π ] s p } = e A + { X, Y } / ([ π ] r X − p, pY − [ π ] s , XY − [ π ] s − r ) , e B [ r ; s ] = e A [ r ; s ] [1 /p ] . If I is any interval of R ∪ { + ∞} we let e B I = T [ r ; s ] ⊂ I e B [ r ; s ] . It is clear that if I ⊂ J aretwo closed intervals then e B J ⊂ e B I . One has a p -adic valuation V I on e B I defined by thecondition V I ( x ) = 0 if and only if x ∈ e A I \ p e A I and such that the image of V I is Z . Withthis valuation e B I becomes a p -adic Banach space.The action of G F on e A + extends to e A + [ p/ [ π ] r , [ π ] s /p ] and by continunity furtherextends to e A I and e B I . The Frobenius ϕ extends to a morphism ϕ : e A + [ p [ π ] r , [ π ] s p ] −→ e A + [ p [ π ] pr , [ π ] ps p ]and finally to a map ϕ : e A I → e A pI for every I .Berger defines e B † ,r rig = e B [ r, + ∞ [ , e B † rig = S r ≥ e B † ,r rig . e B † ,r rig is endowed with the Fr´echettopology defined by the family of valuations V I for closed subsets I ⊂ [ r, + ∞ [, and sub-sequently e B † rig is an LF-space. One can define e A † ,r rig as the ring of integers of e B † ,r rig withrespect to the valuation V [ r ; r ] . We put e A † rig = S r ≥ e A † ,r rig . One defines B † rig ,K to be theLF-space arising from the completion of the B † ,rK with respect to the Fr´echet topologyinduced by the V I . Further, let B † rig = B † rig ,F ⊗ B † F B † . Lemma 2.2. a) B † rig ,K = B † rig ,F ⊗ B † F B † K .b) B † rig = B † rig ,K ⊗ B † K B † .c) ( B † rig ) H K = B † rig ,K . Proof.
See [3], section 3.4.Berger has shown the existence of unique map log : e A + → e B † rig [ X ] such that log([ x ]) =log[ x ] , log( p ) = 0 and log( xy ) = log( x ) + log( y ). Hence one defines log π := log( π ) andsets e B † log = e B † rig [log π ], B † log = B † rig [log π ] and B † log ,K = B † rig ,K [log π ]. One defines amonodromy operator N on e B † log by extending N log π := − p/ ( p −
1) in the usual way. ( ϕ, Γ K ) -modules over B † rig ,K We describe how to extend certain results of [3] to (in general non-´etale) ( ϕ, Γ K )-modules,cf. also [6]. 8e make use of the following notation: Suppose R is a commutative ring equippedwith an endomorphism f : R → R , and M is a R -module. We may then consider the R -module R ⊗ f,R M , where R is considered as an R -module via r · s := f ( r ) s ( r, s ∈ R ).a) A ( ϕ, Γ K ) -module D over B † rig ,K is a free, finitely generated B † rig ,K -module with asemi-linear continuous map ϕ D (i.e. ϕ D ( λx ) = ϕ ( λ ) ϕ D ( x ) for λ ∈ B † rig ,K , x ∈ D )and a continuous action of Γ K which commutes with ϕ D , such that the map ϕ ∗ : B † rig ,K ⊗ ϕ, B † rig ,K D −→ D, a ⊗ x aϕ ( x )is an isomorphism of B † rig ,K -modules.b) ( ϕ, Γ K )-module D over B † rig ,K is ´etale (or of slope 0 ) if there exists p -adic repre-sentation V such that D = D † rig ,K ( V ).For example, for a p -adic representation V we set D † rig ,K ( V ) := ( B † rig ,K ⊗ Q p V ) H K .Furthermore, let us define D † log ,K ( V ) := ( B † log ⊗ Q p V ) H K and D +rig ,K := ( B +rig ⊗ Q p V ) H K .Then D † rig ,K ( V ) is a ( ϕ, Γ)-module over B † rig ,K .Let D be a ( ϕ, Γ K )-module over B † rig ,K . ϕ D will henceforth simply be denoted by ϕ .For the ring B † rig ,K we have a decomposition B † rig ,K = L p − i =0 (1 + π ) i ϕ ( B † rig ,K ) so that onemay define an operator ψ on B † rig ,K by sending P p − i =0 (1 + π ) i ϕ ( x i ) to x , that extendsa similarly defined operator ψ on B † K . More generally, if D is a ( ϕ, Γ K )-module over B † rig ,K we have thanks to condition a) in the definition of ( ϕ, Γ)-modules that there existsa unique operator ψ on D that is defined by the same formula and that and commuteswith the action of Γ K . Proposition 2.3.
If 0 → D ′ → D → D ′′ → ϕ, Γ K )-modulesover B † rig ,K then 0 → D ′ ψ =0 → D ψ =0 → D ′′ ψ =0 → K -modules. Proof.
For the proof of the right-exactness one just uses the fact that if x ∈ D ψ =0 then(uniquely) x = P p − i =1 (1 + π ) i ϕ ( x i ) with x i ∈ D . The compatibility with the action of Γ K is clear since it commutes with ψ .If L/K is a finite extension, we denote the restriction D | L by D | L := B † rig ,L ⊗ B † rig ,K D, with actions of ϕ and Γ L defined diagonally. Hence, D | L is a ( ϕ, Γ L )-module over B † rig ,L .The dual D ∗ of a ( ϕ, Γ K )-module D over B † rig ,K is defined by D ∗ := Hom B † rig ,K ( D, B † rig ,K ) , f ∈ D ∗ the actions of Γ K and ϕ are defined via γ ( f )( x ) := γ ( f ( γ − x )) , γ ∈ Γ K , x ∈ D, ϕ ( f )( x ) := X a i ϕ ( f ( x i )) , x = X a i ϕ ( x i ) ∈ D. If D , D are two ( ϕ, Γ K )-modules over B † rig ,K then the tensor product of D and D is defined by D ⊗ D := D ⊗ B † rig ,K D , where ϕ and Γ K act diagonally. Note that this does not imply that ψ acts diagonally.Let D be a ( ϕ, Γ K )-module over B † rig ,K of rank d . By [6], Theorem I.3.3 there existsan n ( D ) and a unique finite free B † ,r n ( D ) rig ,K -module D ( n ( D )) ⊂ D of rank d witha) B † rig ,K ⊗ B † ,rn ( D ) rig ,K D ( n ( D )) = D ,b) Let D ( n ) = B † ,r n rig ,K ⊗ B † ,rn ( D ) rig ,K D ( n ( D )) for each n ≥ n ( D ). Then ϕ ( D ( n ) ) ⊂ D ( n +1) andthe map B † ,r n +1 rig ,K ⊗ ϕ, B † ,rn rig ,K D ( n ) → D ( n +1) , a ⊗ x aϕ ( x ) , is an isomorphism. e B † rig -modules and B -pairs Let us collect some facts about ϕ -modules over e B † rig . Definition 2.4.
Let h ≥ a ∈ Z . The elementary ϕ -module M a,h is the ϕ -moduleover e B † rig with basis e , . . . , e h − and ϕ ( e ) = e , . . . , ϕ ( e h − ) = e h − , ϕ ( e h − ) = p a e . Proposition 2.5. If M is a ϕ -module over e B † rig then there exist integers a i , h i such that M ∼ = L i M a i ,h i . Proof.
See [19], Theorem 4.5.7.
Definition 2.6.
Let M be a ϕ -module over e B † rig . If M = M a,h is elementary one definesthe slope of M as µ ( M ) = a/h and one says that M is pure of this slope. In general if M ≡ L M a i ,h i one define µ ( M ) = P µ ( M a i ,h i ), so that µ is compatible with short exactsequences.Let D now be a ( ϕ, Γ)-module over B † rig ,K . One sets B e := ( e B † rig [1 /t ]) ϕ =1 . From [5],Proposition 2.2.6, we know thata) W e ( D ) := ( e B † rig [1 /t ] ⊗ B † rig ,K D ) ϕ =1 is a free B e -module of rank d which inherits anaction of G K ,b) W +dR ( D ) := B +dR ⊗ ι n , B † ,rn rig ,K D ( n ) does not depend on n ≫ B +dR -moduleof rank d which inherits an action of G K .10ith this in mind, Berger defined: Definition 2.7.
A tuple W = ( W e , W +dR ), where W e is a free B e -module of finite rankequipped with an semi-linear action of G K and W +dR is a B +dR -lattice in B dR ⊗ B e W e thatis stable under the action of G K , is called a B-pair .From [5], Proposition 2.2.6 it follows that the tuple W ( D ) = ( W e ( D ) , W +dR ( D )) ac-tually is a B -pair. Furthermore, Berger proved: Theorem 2.8.
The functor D W ( D ) gives rise to an equivalence of categories betweenthe category of ( ϕ, Γ K )-modules over B † rig ,K and the category of B-pairs.One knows (cf. [6], section 2.2.) how to construct a functor e D from the categoryof B -pairs to the category of ( ϕ, G K )-modules over e B † rig such that there exists a unique( ϕ, Γ K )-module D ( W ) over B † rig ,K with e B † rig ⊗ B † rig ,K D ( W ) = e D ( W ). Hence, one has,similarly as in the preceding theorem: Theorem 2.9.
The functor D e D := e B † rig ⊗ B † rig ,K D gives rise to an equivalence of cat-egories between the category of ( ϕ, Γ K )-modules over B † rig ,K and the category of ( ϕ, G K )-modules over B † rig .We shall also abbreviate e D log = e B † log ⊗ B † rig ,K D and W dR ( D ) := B dR ⊗ ι n B † ,rn rig ,K D ( n ) ,which is independent of the choice of n for n ≫ e B † ,r n rig ⊗ B e W e ( D ) → e B † ,r n rig ⊗ B † ,rn rig ,K D ( n ) , induced by a ⊗ x ax , is an isomorphism of G K -modules for every n ≥ n ( D ). One definesthe following map of G K -modules: β : W e ( D ) ֒ → B dR ⊗ B e W e ( D ) ∼ = B dR ⊗ ι n , B † ,rn rig ,K ( e B † ,r n rig ⊗ B e W e ( D )) ∼ = W dR ( D ) . (2)We use the same symbol for the map β : W e ( D ) → B dR / B +dR ⊗ B e W e ( D ). Set W + e ( D ) =( e B † rig ⊗ B † rig ,K D ) ϕ =1 . W e ( D ).Let now W be a B -pair and set X ( W ) = W e ∩ W +dR ⊂ W dR and X ( W ) = W dR / ( W e + W +dR ), which are nothing but the kernel and cokernel respectively of the natural map W e → W dR /W +dR . Hence, one has ([7], Theorem 3.1): Theorem 2.10. If W is a B -pair and e D = e D ( W ), there are natural identificationsa) X ( W ) ∼ = W + e ( D ) and X ( W ) ∼ = e D/ (1 − ϕ ),b) X ( W ) = 0 if and only if all slopes of e D are > X ( W ) = 0 if and only if all slopesof e D are ≤
0. 11e recall the following definition, introduced by Fontaine (see [14]):
Definition 2.11. An almost C p -representation is a p -adic Banach space X equippedwith a linear and continuous action of G K such that there exists a d ≥ p -adic representations V ⊂ X, V ⊂ C dp such that X/V ∼ = C dp /V .Berger has shown that X ( W ) and X ( W ) are almost C p -representations, cf. [7]. ( ϕ, Γ K ) -modules Liu (cf. [22]) has worked out reasonable definitions for cohomology of (in general non-´etale)( ϕ, Γ K )-modules over B K , B † K and B † rig ,K .Let D be a ( ϕ, Γ K )-module over one of these rings and let ∆ K be a torsion subgroupof Γ K . Γ K is an open subgroup of Z × p and ∆ K is a finite group of order dividing p − p = 2). Define the idempotent operator p ∆ K by p ∆ K = (1 / | ∆ K | ) P δ ∈ ∆ K δ , so that p ∆ K is the projection from D to D ′ := D ∆ K . If Γ ′ K := Γ K / ∆ K is procyclic with generator γ K define the exact sequence C • ϕ,γ K ( D ) : 0 / / D ′ d / / D ′ ⊕ D ′ d / / D ′ / / d ( x ) = (( ϕ − x, ( γ K − x ) , d ( x, y ) = ( γ K − x − ( ϕ − y. Define for i ∈ Z H i ( K, D ) := H i ( C • ϕ,γ K ( D )) , which is, up to canonical isomorphism, independent of the choice of γ K (cf. [22], section2), so that we shall now fix a choice of ∆ K and γ K .For applications in Iwasawa-theory one also considers the following complex: C • ψ,γ K ( D ) : 0 / / D ′ d / / D ′ ⊕ D ′ d / / D ′ / / d ( x ) = (( ψ − x, ( γ K − x ) , d ( x, y ) = ( γ K − x − ( ψ − y. If D and D are two ( ϕ, Γ K )-modules over B † rig ,K one may, following Herr ([17]), definethe following cup products (we always mean classes where appropriate): H ( K, D ) × H ( K, D ) −→ H ( K, D ⊗ D ) , ( x, y ) ( x ⊗ y ) ,H ( K, D ) × H ( K, D ) −→ H ( K, D ⊗ D ) , ( x, ( y, z )) ( x ⊗ y, x ⊗ z ) ,H ( K, D ) × H ( K, D ) −→ H ( K, D ⊗ D ) , ( x, y ) ( x ⊗ y ) ,H ( K, D ) × H ( K, D ) −→ H ( K, D ⊗ D ) , (( x, y ) , ( w, v )) y ⊗ γ K ( w ) − x ⊗ ϕ ( v ) . (5)We note that some authors swap the maps of the sequence C • ϕ,γ K ( D ) so that of courseone has to adjust the definition of the cup-product. We adhere to the conventions madein [17].Liu’s result is then ([22], Theorem 0.1 and Theorem 0.2):12 heorem 2.12. Let D be a ( ϕ, Γ K )-module over B † rig ,K .a) If D = D † rig ( V ) is ´etale one has canonical functorial isomorphisms H i ( K, D † rig ( V )) ∼ = H i ( G K , V ) for all i ∈ Z that are compatible with cup-products.b) H i ( K, D ) is a finite dimensional Q p -vectorspace and vanishes for i = 0 , , i = 0 , , H i ( K, D ) × H − i ( K, D ∗ (1)) −→ H ( K, D ⊗ D ∗ (1)) = H ( K, B † rig ,K (1))= H ( K, Q p (1)) ∼ = Q p where D ⊗ D ∗ (1) → B † rig ,K (1) is the map x ⊗ f f ( x ), is perfect.Recall that D | L = B † rig ,L ⊗ B † rig ,K D . Let m = [∆ K : ∆ L ] and n be such that γ p n K = γ L .Define τ L/K = P p n − i =0 γ iK and σ L/K = P g ∈ Γ K / Γ L g . We define the restriction maps Res : H i ( K, D ) → H i ( L, D | L ) via the map induced by the following map on complexes (where ∗ ′ means the invariants with respect to the “right” ∆):0 / / D ′ id (cid:15) (cid:15) d / / D ′ ⊕ D ′ id ⊕ ( m · τ L/K ) (cid:15) (cid:15) d / / D ′ / / id (cid:15) (cid:15) / / D | ′ L d / / D | ′ L ⊕ D | ′ L d / / D | ′ L / / H i ( K, D ) → H i ( L, D | L ) via the mapinduced by the following map on complexes:0 / / D | ′ Lσ L/K (cid:15) (cid:15) d / / D | ′ L ⊕ D | ′ Lσ L/K ⊕ id (cid:15) (cid:15) d / / D | ′ L / / id (cid:15) (cid:15) / / D ′ d / / D ′ ⊕ D ′ d / / D ′ / / Proposition 2.13.
The map Cor ◦ Res on H i ( K, D ) is nothing but multiplication by[ L : K ]. Proof.
It is clear that on H ( K, D ) = D ϕ =1 ,γ K =1 (thus γ K acts trivially) the map Cor ◦ Resis just the trace map and equal to multiplication by [ L : K ]. Since the H i ( K, D ) arecohomological δ -functors (see [21], Theorem 8.1) we get the claim. ( ϕ, N, Gal ( L/K )) -modules associated to ( ϕ, Γ K ) -modules We begin with a series of definitions (see [3], section 5, and [6]).13 efinition 2.14.
Let D be ( ϕ, Γ K )-module and n ≥ n ( D ). Set D +dif ,n ( D ) := K n [[ t ]] ⊗ ι n , B † ,rn rig ,K D ( n ) , D dif ,n ( D ) := K n (( t )) ⊗ ι n , B † ,rn rig ,K D ( n ) and, via the transition maps D +dif ,n ( D ) ֒ → D +dif ,n +1 , f ( t ) ⊗ x f ( t ) ⊗ ϕ ( x ) (and similarlyfor D dif ,n ( D ) ֒ → D dif ,n +1 ) D +dif ( D ) := lim −→ n D +dif ,n ( D ) , D dif ( D ) := lim −→ n D dif ,n ( D ) . Note that D +dif ( D ) (resp. D dif ( D )) is a free K ∞ [[ t ]] := S ∞ n =1 K n [[ t ]]- (resp. K ∞ (( t )) = K ∞ [[ t ]][1 /t ]-)module of rank d with a semi-linear action of Γ K . One defines a Γ K -equivariant injection ι n : D ( n ) ֒ → D +dif ,n ( D ) , x ⊗ x. Definition 2.15.
Let D be a ( ϕ, Γ K )-module. Set D K cris ( D ) := ( B † rig ,K [1 /t ] ⊗ B † rig ,K D ) Γ K , D K st ( D ) := ( B † log ,K [1 /t ] ⊗ B † rig ,K D ) Γ K , D K dR ( D ) := ( D dif ( D )) Γ K , and Fil i D K dR ( D ) := D K dR ( D ) ∩ t i D +dif ( D ) ⊂ D dif ( D ) , i ∈ Z . The filtration Fil i D K dR ( D ) is decreasing, separated and exhaustive. We also set D K, +dR ( D ) :=Fil ( D K dR ( D )) = D +dif ( D ) Γ K .One has canonical maps which we will denote by α ∗ for ∗ ∈ { cris , st , dR } , induced by a ⊗ d ad : B † rig ,K [1 /t ] ⊗ D K cris ( D ) → D [1 /t ] B † log ,K [1 /t ] ⊗ D K st ( D ) → B † log ,K [1 /t ] ⊗ D,K ∞ (( t )) ⊗ D K dR ( D ) → D dif ( D ) . Proposition 2.16.
All maps α ∗ above are injective. Hence, one always has inequalitiesdim K D K cris ( D ) ≤ dim K D K st ( D ) ≤ dim K D K dR ( D ) ≤ rank B † rig ,K D, and equalities dim D K ∗ ( D ) = rank B † rig ,K D for ∗ ∈ { cris , st , dR } if and only if the corre-sponding α is an isomorphism. Proof.
Standard proof. 14 efinition 2.17.
The
Hodge-Tate weights of a ( ϕ, Γ K )-module are those integers h such that Fil − h D K dR ( D ) = Fil − h +1 D K dR ( D ). We say that D is positive if h ≤ h , and that D is negative if h ≥ h . Proposition 2.18.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K . If D is positive then D K, +dR ( D ) = D K dR ( D ). More generally, let h ≥ − h D K dR ( D ) = D K dR ( D ).Then t h D K dR ( D ) = D K, +dR ( D ( − h )) (in D dif ( D )). Proof.
The first part is obvious from the definitions and can be shown the same way as inthe ´etale case. The second follows similarly from Lemma 2.19.One can define the
Tate-twist for a ( ϕ, Γ K )-module D: if k ∈ Z , then D ( k ) is the( ϕ, Γ K )-module with D as B † rig ,K -module, but with ϕ | D ( k ) = ϕ | D , γx = χ k ( γ ) γx, x ∈ D. Analoguously one define a
Tate-twist for a filtered ( ϕ, N )-module D over K . If k ∈ Z ,then D [ k ] is the filtered ( ϕ, N )-module with D as K -vectorspace and filtration Fil r ( D [ k ]) K =Fil r − k D K and N | D [ k ] = N | D , ϕ | D [ k ] = p k ϕ | D . Lemma 2.19.
One has D K st ( D ( k )) = D K st ( D )[ − k ]. Proof.
One has D ( k ) = D ⊗ Z p Z p ( k ), and if e k is a generator of Z p ( k ), the isomorphism( B † log ,K [1 /t ] ⊗ B † rig ,K D ) Γ K [ − k ] → ( B † log ,K [1 /t ] ⊗ B † rig ,K D ( k )) Γ K is given by d = X a n ⊗ d n X a n e − k ⊗ ( d n ⊗ e k ) = ( e − k ⊗ e k ) d. Definition 2.20.
A ( ϕ, Γ K )-module D is defined to be crystalline (resp. semi-stable ,resp. de Rham ) if dim K D K cris ( D ) = rank B † rig ,K D (resp. dim K D K st ( D ) = rank B † rig ,K D ,resp. dim K D K dR ( D ) = rank B † rig ,K D ).Similarly, we define D to be potentially crystalline (resp. potentially semi-stable ) if there exists a finite extension L/K such that D | L is cristalline (resp. semistable). Definition 2.21.
Let D be a de Rham ( ϕ, Γ K )-module of rank d . If n ≥ n ( D ), set N ( n )dR ( D ) := { x ∈ D ( n ) [1 /t ] | ι m ( x ) ∈ K m [[ t ]] ⊗ K D K dR ( D ) for any m ≥ n } and N dR ( D ) = lim −→ n N ( n )dR ( D ). 15 efinition 2.22. a) For a torsion free element γ K of Γ K Perrin-Riou’s differentialoperator ∇ is defined as ∇ = − log( γ )log p ( χ ( γ K )) = − p ( χ ( γ K )) X n ≥ (1 − γ K ) n n ∈ H (Γ K ) . b) The operator ∂ (on B † rig ,K [1 /t ]) is defined as ∂ := 1 /t · ∇ .We remark that ∇ is independent of the choice of γ , which may be checked with theseries properties of log. The module N dR ( D ) is denoted by D in [8], Theorem III.2.3.This theorem also implies: Theorem 2.23.
Let D be a de Rham ( ϕ, Γ K )-module of rank d . Then N dR ( D ) is a( ϕ, Γ K )-module of rank d with the following properties:a) N dR ( D )[1 /t ] = D [1 /t ],b) ∇ ( N dR ( D )) ⊂ t N dR ( D ).The following proposition is analoguous to [3], Theorem 3.6. Proposition 2.24.
Let D be a semistable ( ϕ, Γ K )-module. Then one has( e B † rig ⊗ B † rig ,K D ) G K = D Γ K , ( e B † rig [1 /t ] ⊗ B † rig ,K D ) G K = ( B † rig ,K [1 /t ] ⊗ B † rig ,K D ) Γ K , ( e B † log [1 /t ] ⊗ B † rig ,K D ) G K = ( B † log ,K [1 /t ] ⊗ B † rig ,K D ) Γ K . Proof.
We only treat the first case, as the proof of the others is similar.One has ( e B † rig ⊗ B † rig ,K D ) G K ⊂ ( e B † rig ⊗ B † rig ,K D ) H K = e B † rig ,K ⊗ B † rig ,K D since H K actstrivially on D (it is a free B † rig ,K -module). Let { e i } ≤ i ≤ d be a B † rig ,K -basis of D and { d i } ≤ i ≤ r be a K -basis for ( e B † rig ⊗ B † rig ,K D ) G K and M ∈ M r × d ( e B † rig ,K ) defined by therelation ( d i ) = M ( e i ). M has rang r (that is, the image of a basis of D under M form afree B † rig ,K -module of rank r ) and satisfies γ K ( M ) G = M (since the elements d i are fixedunder γ K ), where G ∈ GL d ( B † rig ,K ) is the matrix of γ K with respect to the basis { e i } .The operator R m of Colmez/Berger (cf. loc.cit., § γ K ( R m ( M )) G − R m ( M ) = 0for every m ∈ N . Further R m ( M ) m →∞ −→ M and N = ϕ m ( R m ( M )) ∈ B † rig ,K since R m is a section of ϕ − m ( B † ,p k r rig ,K ) ⊂ e B † ,r rig ,K . Hence, γ ( N ) ϕ m ( G ) = N and since the actionsof ϕ and Γ K commute on D one has ϕ ( G ) P = γ K ( P ) G , where P ∈ M d ( B † rig ,K ) is thematrix of ϕ with respect to the basis { e i } . If one sets Q = ϕ m − ( P ) . . . ϕ ( P ) P then ϕ m ( G ) Q = γ K ( Q ) G and hence γ K ( N Q ) G = N Q , so that
N Q determines r elements in D that are fixed under Γ K . But since for m big enough the matrix M has rank r and P has full rank, since it is an injection and B † rig ,K · ϕ ( D ) = D , one sees that these elementsgive a rank r -submodule of D . Hence, the K -vectorspace generated by these elements isalso of dimension r , whence the claim. 16efore stating the next result we recall the notion of a p -adic differential equation. If D is any ( ϕ, Γ K )-module over B † rig ,K it is known that the same definition as for ∇ givesrise to differential operator ∇ D : D → D that commutes with the action of ϕ and Γ K such that ∇ D ( λx ) = ∇ ( λ ) x + λ ∇ D ( x ) (see [6], Proposition III.1.1). With this one mayalso consider the operator ∂ D = 1 /t · ∇ D on D [1 /t ]. A p -adic differential equation isa ( ϕ, Γ K )-module D over B † rig ,K that is stable under the operator ∂ D .If there is no confusion we will drop the index D of the operators ∇ D and ∂ D . Theorem 2.25.
Let M be a p -adic differential equation equipped with a Frobenius. Thenthere exists a finite extension L/K such that the natural map B † log ,L ⊗ L ′ ( B † log ,L ⊗ B † rig ,K D ) ∂ =0 → B † log ,L ⊗ B † rig ,K D. is an isomorphism. Proof. [1].Recall that a ∇ -crystal over B † rig ,K is a free B † rig ,K -module equipped with an actionof a Frobenius and a connection (also denoted by ∇ ), compatible with ∇ on B † rig ,K , thatcommutes with the Frobenius. A ∇ -crystal over B † rig ,K is called unipotent if it admitsa filtration of sub-crystals such that each successive quotient has a basis consisting ofelements in the kernel of ∇ . More generally, a ∇ -crystal M is called quasi-unipotent ifthere exists a finite extension L/K such that B † rig ,L ⊗ B † rig ,K M (which is a ∇ -crystal over B † rig ,L in a natural way) is unipotent.We note the following result, which is known by the experts and may be proved as inthe ´etale case ([3], Proposition 5.6): Proposition 2.26.
Every de Rham ( ϕ, Γ K )-module is potentially semi-stable. Proof.
One defines the (faithful, exact, ...) functor D N dR ( D ) from the category ofde Rham ( ϕ, Γ K )-modules into the category of p -adic differential equations equipped witha Frobenius. Since by Andr´e’s theorem 2.25 one knows that any such equation is quasi-unipotent, it suffices to show that D is potentially semistable if and only if N dR ( D ) isquasi-unipotent.Now D is potentially semistable if and only if there exists a finite extension L/K suchthat dim L ( B † log ,L [1 /t ] ⊗ B † rig ,K D ) Γ L = rank B † rig ,K D =: d. This gives via [3], Proposition 5.5 a unipotent ∇ -subcrystal of D | L [1 /t ], which is nothingelse but N dR ( D | L ) ∼ = B † rig ,L ⊗ B † rig ,K N dR ( D ).Conversely if D | L ′ [1 /t ] contains a unipotent ∇ -subcrystal of rank d for some finiteextension L ′ /K then the again by loc.cit. there exist elements e , . . . , e d − which generatean L ′ -vectorspace of dimension d on which log( γ ) acts trivially. Hence, there exists afinite extension L/L ′ such that Γ L acts trivially on this basis, so that we obtain a basis of( B † log ,L [1 /t ] ⊗ B † rig ,K D ) Γ L of the right dimension, i.e. D is potentially semistable.17e briefly review the slope theory of ϕ -modules over B † rig ,K or B † K . Definition 2.27.
Let M is a ϕ -module over one of these rings. If M is of rank 1 and v a generator, then ϕ ( v ) = λv for some λ ∈ ( B † rig ,K ) × = ( B † K ) × (cf. [19]; see also [20],Hypothesis 1.4.1. resp. Example 1.4.2). We define the degree deg( M ) of M to be w ( λ ),where w is the p -adic valuation of B K . If M is of rank n then V n M has rank 1. Wedefine the slope µ ( M ) of M as µ ( M ) = deg( M ) / rk M .We remark that the definition of the degree (hence the slope) is independent of thechoice of the generator. Under the equivalence of Theorem 2.9 we have the followingcorrespondence of the slope theory: If D is a ( ϕ, Γ K )-module over B † rig ,K , one may considerthe ϕ -module e D over e B † rig . Then the two definitions of the slope for D coincide. Hence,we have the notion of a ( ϕ, Γ K )-module that is pure of some slope. The fundamentaltheorem is the following result by Kedlaya: Theorem 2.28. (Slope filtration theorem) Let M be a ϕ -module over B † rig ,K . Thenthere exists a unique filtration 0 = M ⊂ M ⊂ . . . ⊂ M l = M by saturated ϕ -submoduleswhose successive quotients are pure with µ ( M /M ) < . . . < µ ( M l − /M l ). If M is a( ϕ, Γ K )-module all M i are ( ϕ, Γ K )-submodules. Proof.
See [20]. ( ϕ, Γ K ) -modules In this section we define short exact sequences associated to ( ϕ, Γ K )-modules, generalizingthe “classical” Bloch-Kato sequence (see [9]) which one may use to study cohomologicalquestions relating to p -adic representations (i.e. the slope zero case). One interestingphenomenon that occurs in this more general setting is that, in order to get the generalversions of the exponential maps, it is necessary to distinguish between the slope ≤ > M be continuous G K -module and define the continuous imhomogeneous cochains in the usual way ( q ≥ C q cont ( G K , M ) := C q cont ( K, M ) := { x : G n −→ M | x continuous } with differential δ q : C q cont ( K, M ) → C q +1cont ( K, M ) defined by δ q ( x )( g , . . . , g q +1 ) = g x ( g , . . . , g q +1 ) + ( − q +1 x ( g , . . . , g q )+ q X i =1 ( − i x ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g q +1 ) .
18y convention C − i ( G K , M ) = 0 for i >
1. The continuous cochain complex is then definedvia C • cont ( K, M ) := (cid:20) C ( K, M ) δ −→ C ( K, M ) δ −→ . . . (cid:21) , and one defines continuous cohomology via H q cont ( K, M ) := H q ( C • cont ( K, M )) . Lemma 3.1.
If 0 → M ′ −→ M f −→ M ′′ −→ G K -modules suchthat f admits a continuous (but not necessarily G K -equivariant) splitting, then continuouscohomology induces a long exact sequence . . . → H i cont ( K, M ′ ) → H i cont ( K, M ) → H i cont ( K, M ′′ ) → H i +1cont ( K, M ′ ) → . . . Proof.
This is standard, see for example [30], § Proposition 3.2. If f : B −→ B be a linear continuous surjective map of p -adic Banachspaces, there exists a continuous splitting s : B −→ B of f , i.e. f ◦ s = id B . Proof.
See [12], Proposition I.1.5, (iii).We define the following set X , which will be used in the next few statements: X := { ( x, y, z ) ∈ e D log [1 /t ] ⊕ e D log [1 /t ] ⊕ W e ( D ) / W +dR ( D ) | N ( y ) = ( pϕ − x ) } . Lemma 3.3.
Let D be a ( ϕ, Γ)-module over B † rig ,K . We assume D is pure of slope µ ( D ) ≤
0. Then one has the following exact sequences of G K -modules (cf. (2) for thedefinition of β ): 0 −→ W + e ( D ) f −→ W e ( D ) g −→ W dR ( D ) / W +dR ( D ) −→ x β ( x )0 −→ W + e ( D ) f −→ e D [1 /t ] g −→ e D [1 /t ] ⊕ W dR ( D ) / W +dR ( D ) −→ x (( ϕ − x ) , β ( x ))0 −→ W + e ( D ) f −→ e D log [1 /t ] g −→ X −→ x ( N ( x ) , ( ϕ − x ) , β ( x ))Additionally, each g above admits a continuous (not necessarily G K -equivariant) splitting.19 roof. The exactness of the first sequence is tautological, see Theorem 2.10. For the secondrecall that for a ϕ -module M over e B † rig the map ϕ − M [1 /t ] → M [1 /t ] is surjective.This implies the exactness of the second sequence. For the exactness of the last sequencefirst observe that the map g is well-defined. Recall that N : e D log → e D log is extendedlinearly from the operator N on e B † log , so that N ( X i ≥ d i log i π ) = − X i ≥ i · d i log i − π (6)for P i ≥ d i log i π ∈ e D log . The exactness at e D log [1 /t ] is clear since from (6) one has( e D log [1 /t ]) N =0 = e D [1 /t ], so we only have to check the exactness at X . The surjectiv-ity of N : e B † log [1 /t ] → e B † log [1 /t ], which again follows from (6), implies that it is enough tocheck that if (0 , y, z ) ∈ X then there exists x ′ ∈ e D [1 /t ] such that g ( x ′ ) = (0 , y, z ), whichis nothing but exactness of the second sequence.The splitting property follows from Proposition 3.2 for the first sequence. For theremaining ones one has to observe that continuous surjections 1 − ϕ and N on e D [1 /t ] havecontinuous sections, which follows for example from the proof of Proposition 2.1.5 of [19]for the first map, and is obvious for the monodromy operator. Lemma 3.4.
Let D be a ( ϕ, Γ)-module over B † rig ,K . We assume D is pure of slope µ ( D ) >
0. Then one has the following exact sequences of G K -modules (cf. (2) for thedefinition of β ):0 −→ W e ( D ) f −→ W dR ( D ) / W +dR ( D ) g −→ W dR ( D ) / ( W e ( D ) + W +dR ( D )) −→ x x −→ e D [1 /t ] f −→ e D [1 /t ] ⊕ W dR ( D ) / W +dR ( D ) g −→ W dR ( D ) / ( W e ( D ) + W +dR ( D )) −→ f : x ((1 − ϕ )( x ) , x ) g : ( x, y ) y −→ e D log [1 /t ] f −→ X g −→ W dR ( D ) / ( W e ( D ) + W +dR ( D )) −→ f : x ( N ( x ) , ( ϕ − x ) , x ) g : ( x, y, z ) z Additionally, each g above admits a continuous (not necessarily G K -equivariant) splitting. Proof.
The exactness of the first sequence is again tautological by Theorem 2.10. The restof the proof follows analoguously to the previous proposition.Putting everything together, we also see:20 orollary 3.5.
Let D be a ( ϕ, Γ K )-module over B † rig ,K . Then one has the following exactsequence of G K -modules:0 −→ X ( e D ) i −→ e D log [1 /t ] f −→ X p −→ X ( e D ) −→ i : x xf : x ( N ( x ) , ( ϕ − x ) , x ) p : ( x, y, z ) z Following Nakamura, we now define for a B -pair W = ( W e , W +dR ) the following com-plex: C • ( G K , W ) := cone( C • ( G K , W e ) −→ C • ( G K , W dR /W +dR )) , which is induced by the canonical inclusion W e i −→ W dR . That is, we have C i ( G K , W ) = C i ( G K , W e ) ⊕ C i − ( G K , W dR /W +dR )with differentials δ iC : C i ( G K , W ) ∋ ( a, b ) ( δ iC i ( G K ,W e ) ( a ) , i ( a ) − δ i − C i ( G K ,W e ) ( b ))More generally, one may define the following complexes: C • ( G K , W ′ ) := cone( C • ( G K , e D [1 /t ]) (1 − ϕ,i ) / / C • ( G K , e D [1 /t ] ⊕ W dR /W +dR )) ,C • ( G K , W ′′ ) := cone( C • ( G K , e D log [1 /t ]) ( N, − ϕ,i ) / / C • ( G K , X )) , We recall:
Lemma 3.6.
Let 0 → A f → B g → C → G K -modules such that g admits a continuous, but not necessarely G K -equivariant, splitting.We write (by abuse of notation)cone( g ) := cone( C • ( G K , B ) g ∗ −→ C • ( G K , C ))cone( f ) := cone( C • ( G K , A ) f ∗ −→ C • ( G K , B )) . a) The natural map of complexes C • ( G K , A ) : (cid:15) (cid:15) C ( G K , A ) f (cid:15) (cid:15) / / C ( G K , A ) / / ( f, (cid:15) (cid:15) . . . cone( g ) : C ( G K , B ) / / C ( G K , B ) ⊕ C ( G K , C ) / / . . . is a quasi-isomorphism that is compatible with the long exact sequence, i.e. thefollowing diagram is commutative: . . . / / H i ( G K , A ) / / (cid:15) (cid:15) H i ( G K , B ) / / H i ( G K , C ) δ / / H i +1 ( G K , A ) / / (cid:15) (cid:15) . . .. . . / / H i (cone( g )) / / H i ( G K , B ) / / H i ( G K , C ) δ / / H i +1 (cone( g )) / / . . .
21) The natural map of complexes C • ( G K , C )[ −
1] : 0 = C − ( G K , C ) / / C ( G K , C ) / / . . . cone( f ) : O O C ( G K , A ) / / O O C ( G K , A ) ⊕ C ( G K , B ) (0 ,g ) O O / / . . . is a quasi-isomorphism that is compatible with the long exact sequence, i.e. thefollowing diagram is commutative: . . . / / H i ( G K , A ) / / H i ( G K , B ) / / H i ( G K , C ) δ / / H i +1 ( G K , A ) / / . . .. . . / / H i ( G K , A ) / / H i ( G K , B ) δ / / H i +1 (cone( f )) O O δ / / H i +1 ( G K , A ) / / . . . Proof.
This is left as an exercise, see for example [31], 1.5.8.
Lemma 3.7.
We have canonical quasi-isomorphisms C • ( G K , W ) ∼ = C • ( G K , W ′ ) ∼ = C • ( G K , W ′′ ) . Proof.
Let W = W ( D ). Observe that the inclusions W e ( D ) ⊂ e D [1 /t ] ⊂ e D log [1 /t ] and W dR ( D ) induce canonical maps on these complexes. If W = W ( D ) with D pure of someslope the statement then follows from Lemmas 3.3, 3.4 and 3.6.For general D we are by Kedlaya’s slope filtration theorem reduced to the case of anexact sequence 0 → D → D → D → D , D , hencethe claim follows by considering the long exact sequences associated to this.With this statement and the properties of the cone we obtain a long exact sequence ofcohomology groups: . . . → H i ( G K , W ) → H i ( G K , e D log [1 /t ]) → H i ( G K , X ) δ → H i +1 ( G K , W ) → . . . With these exact sequences in mind we suggest the following
Definition 3.8.
Let D be a ( ϕ, Γ K )-module over B † rig ,K . The transition mapexp K,D : H ( K, X ) → H ( K, W ( D ))from the exact sequence above is called generalized Bloch-Kato exponential map for D . Remark 3.9.
Let D be an ´etale ( ϕ, Γ K )-module, so that D = D † rig ,K ( V ) for some p -adicrepresentation V of Γ K . Then since the slope of D is equal to zero, the first exact sequencein Lemma 3.3 computes to0 −→ V → B e ⊗ Q p V −→ B dR / B +dR ⊗ Q p V −→ p -adicrepresentation V . 22ecall that if D is any ( ϕ, Γ K )-module over B † rig ,K the map ϕ − e D [1 /t ] → e D [1 /t ] issurjective. If x ∈ e D we write ( ϕ − − ( x ) for a choice of an element y ∈ e D [1 /t ] such that( ϕ − y ) = x . We want to consider the following maps: α : e D −→ W e ( D ) , x (cid:26) x, ϕ ( x ) = x, , otherwise .β : e D −→ W dR ( D ) / W +dR ( D ) , x ι n (( ϕ − − ( x )) , where the second map is well-defined due to the discussion in [7], Remark 3.4. α and β are continuous and fit into the following commutative diagram of G K -modules:0 / / e D ϕ =1 / / e D ϕ − / / α (cid:15) (cid:15) e D / / β (cid:15) (cid:15) e D/ ( ϕ − e D / / / / X ( e D ) / / W e ( D ) / / W dR ( D ) / W +dR ( D ) / / X ( e D ) / / , where we use the identifications for X and X from Theorem 2.10. Proposition 3.10.
One has a quasi-isomorphismcone( C • ( G K , e D ) ϕ − −→ C • ( G K , e D )) ∼ = C • ( G K , W ( D ))that is functorial in D . Proof.
We denote by A • the complex on the left hand side of the statement. One checksthat the commutativity of the preceeding diagram and the cohomological version of [31],Exercise 1.5.9 show that one has a commutative diagram . . . / / H n ( G K , e D ϕ =1 ) / / H n ( A • ) / / (cid:15) (cid:15) H n − ( G K , e D ( ϕ − e D ) / / H n +1 ( G K , e D ϕ =1 ) / / . . .. . . / / H n ( G K , X ( e D )) / / H n ( G K , W ( D )) / / H n − ( G K , X ( e D )) / / H n +1 ( G K , X ( e D )) / / . . . which gives the proof.Recall the following property of continuous cohomology: If f : M • → N • is map ofcomplexes of continuous G -modules for some profinite group G one has an identificationof complexes C • cont ( G, cone( M • f → N • )) = cone (cid:16) C • cont ( G, M • ) f ∗ → C • cont ( G, N • ) (cid:17) (7)(cf. the discussion in [24], 3.4.1.3, 3.4.1.4; it holds in this general setting).We recall that in the derived category of B † rig ,K -modules, the complex C • ϕ,γ is alsorepresented by R Γ( K, D ) = R Γ cont (Γ K , cone h D ϕ − −→ D i ) ∼ = cone h R Γ cont (Γ K , D ) ϕ − −→ R Γ cont (Γ K , D ) i , cf. [29], section 3.3, where the last identification is due to (7).The following is then a generalization of Proposition 2.24:23 roposition 3.11. One has an isomorphism R Γ( K, D ) ∼ = R Γ( K, e B † rig ,K ⊗ B † rig ,K D )that is functorial in D . Proof.
The proof is similar to [29], Proposition 3.8. It suffices to show that that thenatural map R Γ cont (Γ K , D ) −→ R Γ cont (Γ K , e B † rig ,K ⊗ B † rig ,K D )is an isomorphism, since applying cone h • ϕ − −→ • i induces the morphism in the statementagain due to (7). We apply the techniques of [2], Appendix I and use the notation there,as follows: Let e Λ := e B † ,r rig , G = G K , H = H K so that d = 0. Further, H ′ = H , Λ ( i ) m, H ′ = ϕ − m ( B † ,p m r rig ,K ) (since i = 0 is the only possible choice) and the maps τ ( i ) m, H ′ correspond tothe maps R m : e B † ,r rig ,K → ϕ − m ( B † ,p m r rig ,K ) (cf. [3], Proposition 2.32). As in [2], section 7.6, themaps R m induce maps (by the usual process of taking the direct limit over all sufficientlybig r ) R m : e B † rig ,K ⊗ B † rig ,K D → e B † rig ,K ⊗ B † rig ,K D for m ≥
0, and as in loc.cit. one obtainsa decomposition of Γ K -modules e B † rig ,K ⊗ B † rig ,K D ∼ = (1 − R m )( e B † rig ,K ⊗ B † rig ,K D ) ⊕ ( e B † rig ,K ⊗ B † rig ,K D ) R m =1 . By construction of the map R m it is clear that ( e B † rig ,K ⊗ B † rig ,K D ) R =1 = D . Furthermore,as in the proof of loc.cit., Proposition 7.7, one may infer that γ K − − R )( e B † rig ,K ⊗ B † rig ,K D ), so that R Γ cont (Γ K , (1 − R )( e B † rig ,K ⊗ B † rig ,K D )) = 0, whichgives the claim.Putting everything together, we see: Corollary 3.12.
One has an isomorphism R Γ( K, D ) ∼ = R Γ( G K , W ( D )) . that is functorial in D . Proof.
We observe that the natural map e B † rig ,K ∼ = R Γ cont ( H K , e B † rig ) (8)is a quasi-isomorphism. This, together with the preceeding isomorphisms implies R Γ( K, D ) ∼ = R Γ cont (Γ K , cone( D ϕ − −→ D ))= R Γ cont (Γ K , cone( e B † rig ,K ⊗ D ϕ − −→ e B † rig ,K ⊗ D ))= R Γ cont (Γ K , R Γ cont ( H K , cone( e D ϕ − −→ e D ))) ( ∗ ) = R Γ cont ( G K , cone( e D ϕ − −→ e D ))= R Γ( G K , W ( D )) . ∗ ) holds since the natural map H i ( G K /H K , e D H K ) → H i ( G K , e D ) is an isomor-phism, since again H n ( H K , e D ) = 0 for n > i = 1 this follows fromthe five term exact sequence in low degree, which extends in this case for continuouscohomology similarly as in e.g. [25], §
6, to higher degrees by induction.
Corollary 3.13. H i ( G K , W ( D )) = 0 for i = 0 , , H i ( G K , W ( D )) is a finite-dimensional Q p -vectorspace. Proof.
This follows from the preceeding Corollary and [21], Theorem 8.1.We wish to give a more explicit description of the isomorphisms on cohomology whichwe will need in the characterizing property of the big exponential map, where actuallyonly the map for the H ’s will be important for us. Hence, we may only sketch certainsteps for the higher cohomology groups (that is, H ).We briefly describe how one may interpret, in the slope ≤ H ( G K , W + e ( D )) as extensions of Q p by W + e ( D ). So let c ∈ H ( G K , W + e ( D )) andconsider the exact sequence of G K -modules0 −→ W + e ( D ) −→ E c −→ Q p −→ E c = Q p ⊕ W + e ( D ) as Q p -vectorspace and G K acts on E c via σ ( a, m ) = ( a, σm + ac σ ) . Since c is a 1-cocycle one has σ ( τ ( a, x )) = σ ( a, τ x + ac τ ) = ( a, στ x + aσc τ + c σ ) = στ ( a, x ) , so that one has a well-defined map Z ( K, D ) → Ext( Q p , W + e ( D )). E c is trivial if and onlyif there exists an element 1 ∈ E c such that g g , i.e.1 = (1 , x ) , g − , gx − x + c g ) = 0 , so that c g = (1 − g ) x is a coboundary, which implies that the above map factors through B ( K, D ). The fact that this map is an isomorphism can be checked as in the p -adicrepresentation case. Proposition 3.14.
Suppose we are in the situation of Lemma 3.3. Then the complex C • ϕ,γ K ( K, D ) (functorially) computes the cohomology of C • cont ( G K , X ( D )). Proof.
We may assume that Γ K is pro-cyclic with generator γ K . First we have H ( K, D ) = D Γ K ,ϕ =1 = e D G K ,ϕ =1 = X ( D ) G K = H ( G K , X ( D )) . thanks to Proposition 2.24.For H we apply the construction of Cherbonnier/Colmez ([11]). To wit, let ( x, y ) ∈ H ( K, D ) and pick b ∈ e D such that ( ϕ − b = x . Then h K,D (( x, y )) = log p ( χ ( γ )) · (cid:18) σ σ − γ K − y − ( σ − b (cid:19) e D but one easily checks that ( ϕ − h K,D (( x, y )) = 0 sothat we actually have a cocycle in H ( G K , X ( D )). Injectivity and surjectivity now followin the same way as in loc.cit. if one uses the description of extensions of Q p by X ( D )given above, so that we obtain the isomorphism in the H -case.For H one can show that since X ( D ) is an almost C p -representation that one has aHochschild-Serre spectral sequence H i (Γ K , H j ( H K , X ( D ))) ⇒ H i + j ( G K , X ( D )) associ-ated to the exact sequence 1 → H K → G K → Γ K →
1. Since the cohomology on the leftvanishes for j or i greater or equal to 2 one has with the fact that H ( G K , X ( D )) = 0 H ( G K , X ( D )) ∼ = H (Γ K , H ( H K , X ( D ))) . Now the exact sequence 0 → X ( D ) → e D ϕ − → e D → G K -modules gives rise to asequence . . . −→ e D H K ϕ − −→ e D H K −→ H ( H K , X ( D )) −→ , since H ( H K , e D ) = H ( H K , e B † rig ⊗ D ) ∼ = H ( H K , e B † rig ) d = 0. Hence, by Iwasawa theory H ( G K , X ( D )) ∼ = e D H K / ( ϕ − , γ K − . Looking at the quasi-isomorphisms in Corollary 3.12 one sees that using Lemma 3.6, sincewe are in the X ( D ) = 0-case, the map H ( K, D ) → H ( G K , X ( D )) is given by thecanonical inclusion of finite-dimensional Q p -vectorspaces H ( K, D ) = D/ ( ϕ − , γ K − ⊂ e D H K / ( ϕ − , γ K −
1) = H ( G K , X ( D )) , that are of the same dimension. This gives the description of the map for H . Lemma 3.15.
Let D be a ( ϕ, Γ K )-module over B † rig ,K and assume that Γ K is pro-cyclicwith generator γ K . Then one has an exact sequence0 −→ D ϕ =1 ( γ K − f −→ H ( K, D ) g −→ (cid:16) Dϕ − (cid:17) Γ K −→ y (0 , y )( x, y ) x Proof.
Recall that by definition H ( K, D ) = { ( x, y ) ∈ D ⊕ D | ( γ K − x = ( ϕ − y } / { (( ϕ − z, ( γ K − z ) | z ∈ D } , so that the first map is well-defined an injective. One checks that the map g is well-definedand if x ∈ D/ ( ϕ −
1) such that ( γ K − x ∈ ( ϕ − D then there exists an y ∈ D suchthat ( x, y ) ∈ H ( K, D ) and g ( x, y ) = x . Obviously g ◦ f = 0. Let g ( x, y ) = 0 so that x = ( ϕ − z for some z ∈ D , so that ( x, y ) ∼ (0 , y − ( γ K − z ) in H ( K, D ). Hence,( x, y ) is in the image of f .We remark that this sequence is nothing but the short exact sequence associated tothe inflation-restriction sequence if D is ´etale, i.e.,0 −→ H (Γ K , V H K ) −→ H ( G K , V ) −→ H ( H K , V Γ K ) −→ , see for example [13], section 5.2. 26 roposition 3.16. Suppose we are in the situation of Lemma 3.4. Then the complex C • ϕ,γ K ( K, D ) computes the cohomology of C • D := C • cont ( G K , X ( D ))[1] Proof.
We may assume that Γ K is procyclic with generator γ K . Since the slope of D is > X ( D ) = 0, so that D ϕ =1 = 0 since D ϕ =1 ⊂ e D ϕ =1 = 0, so that H ( K, D ) = 0.The same holds tautologically for H ( C • D ).For the case of the H ’s observe that since X ( D ) = 0 Lemma 3.15 implies thatthe canonical map H ( K, D ) → ( D/ ( ϕ − Γ K ) , ( x, y ) x, is an isomorphism. FromTheorem 2.10 we also know that X ( D ) = e D/ ( ϕ − H ( G K , X ( D )) = e Dϕ − ! G K = (cid:18) Dϕ − (cid:19) Γ K ∼ = H ( K, D ) . gives the identification.For H one has similarly as in the slope ≤ H i (Γ K , H j ( H K , X ( D ))) ⇒ H i + j ( G K , X ( D )). From the exact sequence in low degreeterms one then has0 → H (Γ K , H ( H K , e D/ ( ϕ − → H ( G K , e D/ ( ϕ − → H (Γ K , H ( H K , e D/ ( ϕ − . From the sequence 0 −→ e D ϕ − −→ e D → X ( D ) −→ H ( H K ,X ( D )) since H ( H K , e D ) = H ( H K , T D ) = H ( H K , e B † rig ) d = 0. Hence, we see H ( G K , X ( D )) = H (Γ K , H ( H K , e D/ ( ϕ − e D H K / ( ϕ − , γ K − . so that again by Corollary 3.12 and Lemma 3.6 the canonical inclusion of finite-dimensional Q p -vectorspaces H ( K, D ) = D/ ( ϕ − , γ K − ⊂ e D H K / ( ϕ − , γ K −
1) = H ( G K , X ( D )) , gives the description of the map for H .Finally we describe how one may piece together the isomorphisms H i ( K, D ) h −→ H i ( K, W ( D )) in the general case (where we only make the case H explicit, which isall we need for the application to Perrin-Riou’s exponential map): If ( x, y ) ∈ H ( K, D )write x = ( ϕ − b ′ ) + s ( b ′′ ), where s : e D/ ( ϕ − e D → e D is a continuous splitting of thenatural projection (which exists thanks to Proposition 3.2), b ′ ∈ e D and b ′′ ∈ e D/ ( ϕ − e D .Putting the two constructions together, we may consider the tuple h ( x, y )) := (cid:16) log p ( χ ( γ )) · (cid:16) σ σ − γ K − y − ( σ − b ′ (cid:17) , (0 , , ϕ − n (( ϕ − − ( s ( b ′′ ))) (cid:17) ∈ C ( G K , e D log ) ⊕ C ( G K , X ) , (9)and one sees that actually h i (( x, y )) ∈ H ( K, W ( D )), which gives the description of theisomorphism in the general case by the properties of the mapping cone.27e will briefly describe, similarly as in the slope ≤ H ( G K , W e ( D )) as extensions of B e by W e ( D ) (note however thatwe do not make any assumptions about the slopes of D ). So let c ∈ H ( G K , W e ( D )) andconsider the exact sequence of G K -modules0 −→ W e ( D ) −→ E c −→ B e −→ , where E c = B e ⊕ W e ( D ) as a B e -module with G K -action σ ( a, x ) = ( σa, σx + σa · c σ ).One has σ ( τ ( a, x )) = σ ( τ a, τ x + τ a · c τ ) = ( στ a, στ x + στ a · σc τ + στ a · c σ ) = στ ( a, x ) , so that one has a well-defined map Z ( K, W e ( D )) → Ext( Q p , W + e ( D )). E c is trivial ifand only if there exists an element 1 ∈ E c such that g g , i.e.1 = (1 , x ) , g − , gx − x + c g ) = 0 , so that c g = (1 − g ) x is a coboundary, which implies that the above map factors through B ( K, W e ( D )). The fact that this map is an isomorphism can be checked as before. Proposition 3.17.
Let D be a ( ϕ, Γ K )-module over B † rig ,K . Then the complex C • ϕ,γ K ( K,D [1 /t ]) computes the cohomology of C • cont ( G K , W e ( D )). Proof.
The proof is similar to the ones before; in fact, one may reduce to the case ofCorollary 3.5 by taking direct limits (see also [23], Theorem 4.5). We are interested in theexplicit description of the maps. From Proposition 2.24 again we have: H ( K, D [1 /t ]) = D [1 /t ] ϕ =1 , Γ K = e D [1 /t ] ϕ =1 ,G K = H ( G K , W e ( D )) . For H we apply the same construction as in Proposition 3.14. So let ( x, y ) ∈ H ( K, D [1 /t ])and pick b ∈ e D [1 /t ] such that ( ϕ − b = x . Then h K,D (( x, y )) = log p ( χ ( γ )) · (cid:18) σ σ − γ K − y − ( σ − b (cid:19) defines a 1-cocycle with values in e D [1 /t ] which lies actually in W e ( D ). Injectivity and sur-jectivity now follow in the same way as in loc.cit. if one uses the description of extensionsof B e by W e ( D ) given above, so that we obtain the isomorphism in the H -case.The case of the H ’s follows in the same way as in Proposition 3.14. Proposition 3.18.
One has an identification H ( K, W dR ( D )) = D K dR ( D ) Proof.
From [11], Proposition IV.1.1 (i) we know that K ∞ [[ t ]] is dense in ( B +dR ) H K , and theinclusion is compatible the action of Γ K . Also one has ( B +dR ) G K = K ∞ [[ t ]] Γ K = K . Since D is free as a B † rig ,K -module with trivial H K -action, we see that ( B +dR ⊗ D ) G K = (( B +dR ) H K ⊗ D ) Γ K = D +dif ( D ) Γ K . Since B dR = lim −→ n ≥ /t n · B +dR and K ∞ (( t )) = lim −→ n ≥ /t n · K ∞ [[ t ]]the claim follows, since taking invariants is compatible with direct limits.Alternatively, the claim also follows from [14], Theorem 2.14, B) i).28e shall make use of the following considerations. Let D be a semi-stable ( ϕ, Γ K )-module over B † rig ,K and consider the following complex C st ( K, D ) (concentrated in degrees0, 1, 2): D K st ( D ) → D K st ( D ) ⊕ D K st ( D ) ⊕ D K dR ( D ) / Fil D K dR ( D ) → D K st ( D ) x ( N ( x ) , ( ϕ − x ) , β ( x ))( x, y, z ) N ( x ) − ( pϕ − y ) . (10)Then an element in H ( C st ( K, D )) can be considered as an element in H ( K, X ) andhence be mapped via the exponential map to H ( K, W ( D )).We shall give two maps which will be important in the construction of the dual expo-nential map for de Rham ( ϕ, Γ K )-modules.First we remark that the canonical inclusion D → W dR ( D ) factors via D → D [1 /t ].This allows us to describe a map H ( K, D ) → H ( G K , W dR ( D )) explicitly via the compo-sition of the canonical map H ( K, D ) → H ( K, D [1 /t ]), the identification H ( K, D [1 /t ]) ∼ → H ( G K , W e ( D )) (cf. Proposition 3.17) and the canonical map H ( K, W e ( D )) −→ H ( K, W dR ( D )).Secondly, we show that the map D K dR ( D ) −→ H ( G K , W dR ( D )) , x [ g log( χ ( g )) x ] (11)which generalizes Kato’s formula of [18], § II.1, is an isomorphism, which may be provedas follows. First observe that H ( G K , B dR ⊗ D ) ∼ = H ( G K , B dR ⊗ K D K dR ( D )) = H ( G K , B dR ) ⊗ K D K dR ( D ) . From [16], Proposition 5.25, one knows that K = H ( G K , B dR ) → H ( G K , B dR ) , x x · log χ is an isomorphism. This gives the claim. Definition 3.19.
The generalized Bloch-Kato dual exponential map exp ∗ K,D ∗ (1) isthe composition of the above maps H ( K, D ) → H ( G K , W dR ( D )) with the inverse ofthe isomorphism D K dR ( D ) ∼ → H ( G K , W dR ( D )).Of course, in the ´etale case this is nothing but the dual exponential map considered byKato in [18]. But even in this more general case this map has the desired property withrespect to adjunction via pairings. First recall that one may define the K -bilinear perfectpairing [ , ] K,D by the natural map[ , ] K,D : D K dR ( D ) × D K dR ( D ∗ (1)) ev −→ D K dR ( B † rig ,K (1)) −→ K. For the next proposition we note that Nakamura uses a different definition of the dualexponential map (see [23], section 2.4), which we briefly recall (we refer to loc.cit for theproofs): one may define the cohomology groups H i ( K, D dif ( D )) by H i cont (Γ K , D dif ( D )),which is computed by the complex C • γ, ∆ ( D dif ( D )) : D dif ( D ) γ − −→ D dif ( D ) . K ∞ (( t )) ⊗ K D K dR ( D ) → D dif ( D ) is an isomorphism one has anidentification g D : D K dR ( D ) ∼ −→ H ( K, D dif ( D )) , x (log χ ( γ ))1 ⊗ x. The second definition of exp ∗ K,D is then given by the composition of the map H ( K, D ) → H ( K, D dif ( D )) , [( x, y )] ι n ( y ) (for n big enough) and the inverse of g D . Since H i ( H K , B dR ) = 0 for i > H ( G K , W dR ( D )) ∼ = H (Γ K , B H K dR ⊗ D ). Using the same argument as in Proposition 3.18 one sees that the natural map H ( K, D dif ( D )) → H ( G K , W dR ( D )) is an isomorphism. Further, the natural map H ( K, D ) → H ( G K , W dR ( D )) defined before is also given by [( x, y )] ι n ( y ). Hence,using all these identifications one obtains a commutative diagram H ( K, D ) / / H ( K, D dif ( D )) ∼ (cid:15) (cid:15) D K dR ( D ) ∼ o o ∼ (cid:15) (cid:15) H ( K, D ) / / H ( G K , W dR ( D )) H ( G K , W dR ( D )) , ∼ o o which shows that the two definitions of exp ∗ coincide. Proposition 3.20.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K and let x ∈ D K dR ( D )and y ∈ H ( K, D ∗ (1)). Then h exp K,D ( x ) , y i K,D = Tr K/ Q p [ x, exp ∗ K,D ( y )] K,D
Proof.
See [23], Proposition 2.16.
Proposition 3.21.
Let D be a semi-stable ( ϕ, Γ K )-module over B † rig ,K . Let y ∈ D ψ =1 and consider y as y ∈ ( B † log ,K [1 /t ] ⊗ F D K st ( D )) N =0 ,ψ =1 via the comparison isomorphism.Then for n ≫ ∗ V ∗ (1) ( h D,K n ( y )) = p − n ϕ − n ( y )(0) . Proof.
As before we have h D,K n ( y )( σ ) = σ − γ K n − y − ( σ − b, with ( γ K n − ϕ − b = ( ϕ − y for some b ∈ e D [1 /t ]. Further Let n be big enough sothat we may embed this cocycle into B dR ⊗ D , hence ϕ − n ( y ) ∈ K n (( t )) ⊗ D K st ( D ) andwe may consider ϕ − n ( b ) as an element in B dR ⊗ D . Since γ K n t = χ ( γ K n ) t the action of γ K n − t k K n ⊗ D K st ( D ) for every k = 0. Putting this together we see that h D,K n is equivalent in H ( K n , B dR ⊗ D ) to σ σ − γ K n − ϕ − n ( y ))(0) . acts via its image σ ∈ Γ nK (trivially) on K n . Furthermore, if n i ∈ Z is a sequence suchthat σ = lim i →∞ γ n i K n one checks by going to the limit that σ − γ K n − p χ ( γ K n )log p χ ( σ )acts trivially on K n . Hence, the above cycle is equivalent to σ p − n log( χ ( σ ))( ϕ − n ( y ))(0)The claim follows now from formula (11). ( ϕ, Γ K ) -modules We make the following definitions:
Definition 3.22.
Let M be a ( ϕ, N )-module over F . Define N dR ( M ) = ( B † log ,K ⊗ F M ) N =0 , where N = 1 ⊗ N + N ⊗ B † log ,K ⊗ F M .If D is a semi-stable ( ϕ, Γ K )-module over B † rig ,K then N dR ( D K st ( D )) = N dR ( D ) (seeDefinition 2.21). Definition 3.23.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K .a) Let D ∞ ,g ( D ) be the submodule of elements g ∈ N dR ( D ) ψ =0 such that there exists an r ∈ Z such that the equation (1 − p r ϕ ) G = ∂ r ( g ) has a solution in G ∈ N dR ( D ) ψ = p r .b) Let D ∞ ,f ( D ) be the submodule of elements g ∈ N dR ( D ) ψ =0 such that there existsa family ( G k ) k ∈ Z of elements G k ∈ N dR ( D ) with ∂ ( G k ) = G k +1 and an r ∈ Z suchthat (1 − p r ϕ ) G = ∂ r ( g )c) Let D ∞ ,e ( D ) be the submodule of elements g ∈ N dR ( D ) ψ =0 such that the equation(1 − p r ϕ ) G = ∂ r ( g ) has a solution in G ∈ N dR ( D ) ψ = p r for every r ∈ Z .We first note that if D −→ D ′ is a morphism of two de Rham ( ϕ, Γ K )-modules over B † rig ,K then this induces a map of Γ K -modules D ∞ , ∗ ( D ) → D ∞ , ∗ ( D ′ ). Also, one clearlyhas D ∞ ,e ( D ) ⊂ D ∞ ,f ( D ) ⊂ D ∞ ,g ( D ) ⊂ N dR ( D ) ψ =0 . By the above definition one may also define the modules D ∞ , ∗ ( ) by starting with a( ϕ, N )-module. Definition 3.24.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K . We say that D is of Perrin-Riou-type (or of PR-type) if D is semistable and K = K ′ . Lemma 3.25.
The map ∂ : B † log ,K → B † log ,K is surjective. Proof.
This amounts to an integration of power-series, cf. [3], Proposition 4.4.31 emma 3.26.
Suppose K = K ′ . Then the kernel of ∂ on B † log ,K is equal to K . Proof.
Let f ∈ B † rig ,K . Due to Proposition 2.1 and Lemma 2.2 there is a polynomial P in B † rig ,F such that P ( f ) = 0 and P ′ ( f ) = 0. Then ∂ ( f ) = − ( ∂P )( f ) /P ′ ( f ), so that ∂ ( f ) = 0if and only if f ∈ K .Now suppose f = P ri =1 f i log i π ∈ B † log ,K and ∂ ( f ) = 0. Since log π is a transcendentelement over any B † rig ,K this gives rise to relations ∂ ( f i )+( j +1) π +1 π f i +1 = 0 with f r +1 = 0.For i = r this implies f r = λ ∈ K , hence ∂ ( f r − ) = − λr π +1 π . Suppose there exists an f ∈ B † rig ,K with ∂ ( f ) = ππ . Then ∂ (log π − f ) = 0, so that log π = f + a with a ∈ K , acontradiction to the transcendency property of log π . Hence, π +1 π is not an element in theimage of ∂ on B † rig ,K , and we obtain λ = 0. By recurrence this shows that the kernel of ∂ on B † log ,K is contained in K .Let again D be a de Rham ( ϕ, Γ)-module over B † rig ,K . Lemma 3.27.
Let D be of PR-type. Then the map ∂ : B † log ,K ⊗ N dR ( D ) → B † log ,K ⊗ N dR ( D ) is surjective. Proof.
We have B † log ,K ⊗ B † rig ,K N dR ( D ) = B † log ,K ⊗ K D K st ( D ) , whence the claim follows from the Lemma above. Proposition 3.28.
Let D be of PR-type. The map ∂ : N dR ( D ) ψ =0 −→ N dR ( D [1]) ψ =0 (1)is an isomorphism of Γ K -modules. Proof.
With our preparations, namely, Lemma 3.25 and Lemma 3.26, this proof works thesame as in [27], Proposition 2.2.3.Obviously the operator ∂ induces a map of Γ K -modules ∂ : N dR ( D ) ψ =1 → N dR ( D [1]) ψ =1 (1)which however is in general neither injective nor surjective. This should be contrastedwith the ´etale case where D ψ =1 = D † rig ( V ) ψ =1 = H ( K, V ⊗ Q p H (Γ K )) and the fact that ∂ in this setting corresponds to the Tate-twist isomorphism.For a semistable ( ϕ, Γ K )-module consider the following complex: C K ( D ) : 0 → D K st ( D ) δ −→ D K st ( D ) × D K st ( D ) δ −→ D K st ( D ) → δ ( ν ) = ( N ν, (1 − ϕ ) ν ) ,δ ( λ, µ ) = N µ − (1 − pϕ ) λ. H ( C K ( D )) = D K st ( D ) ϕ =1 ,N =0 ,H ( C K ( D )) = { ( λ, µ ) ∈ D K st ( D ) × D K st ( D ) | N µ = (1 − pϕ ) λ } /δ ( D K st ( D )) ,H ( C K ( D )) = D K st ( D ) / ( N, − pϕ ) D K st ( D ) . One also checks that0 −→ D K st( D ) N =0 ( ϕ − D K st( D ) N =0 −→ H ( C K ( D )) −→ D K st( D ) N D K st( D )) ϕ = p − −→ µ (0 , µ )( λ, µ ) λ (12)furnishes an exact sequence for H ( C ( D )).We see that H ( C ( D ( k ))) = 0 for k ≫ k ≪ D st ( D ( k )) ϕ =1 and ( ϕ − D st ( D ( k )) vanish for those k . Similarly, H ( C ( D ( k ))) = 0 for k ≫ k ≪ D be a de Rham ( ϕ, Γ K )-module and fix a finite extension L/K such that D | L is semistable with L = L ′ . Lemma 3.29.
Let k ∈ N . Then one has an exact sequence of Γ K -modules0 → M − k ≤ i< H ( C ( D | L ( − i )))( i ) ∩ N dR ( D ( k )) ψ =1 ( − k )) → N dR ( D ( k )) ψ =1 ( − k ) ∂ k −→ N dR ( D ) ψ =1 e R D → M − k ≤ i< H ( C ( D | L ( − i )))( i ) Proof.
The proof may be done in an analogous way as in [27], Lemma 2.2.5. We give adescription of the map e R D following the definition of a map R D (cf. equation (15)) sincethe constructions which give rise to it will be important later on. We just briefly mentionthat this map depends on the inclusion N dR ( D ) ⊂ N dR ( D | L ) which is induced by theinclusion D ⊂ D | L .From the lemma we see that, by considering the possible eigenvalues for ϕ , D ∞ ,e ( D ) = ∂ h (1 − p − h ϕ ) N dR ( D ( h )) ψ =1 , (13) D ∞ ,g ( D ) = ∂ − h (1 − p h ϕ ) N dR ( D ( − h )) ψ =1 (14)for h ≫ H i ( C ( D )), i = 0 ,
1, vanish in this case. More precisely, for ´etale( ϕ, Γ K )-module one has the following: Lemma 3.30.
Let D = D † rig ( V ) for a p -adic representation V that is de Rham. Let h ≥ − h D K dR ( D ) = D K dR ( D ). Then D ∞ ,g ( D ) = ∂ − ( h +1) (1 − p h +1 ϕ ) N dR ( D ( − ( h +1))) ψ =1 . 33 roof. We may reduce to the case that D is semi-stable with K = K ′ and further bytwisting that h = 1. We have to check that ∂ : N dR ( D ( − ψ =1 → N dR ( D ( − ψ =1 (1) isan isomorphism, i.e., we have to check the vanishing of H ( C ( D ( − H ( C ( D ( − ϕ, N )-module that is positivethe eigenvalues of the Frobenius are positive. Similarly, thanks to the exact sequence (12),we see that the H -part vanishes. Remark 3.31.
We suspect that in the cases where V is as above and does not containthe subrepresentation Q p ( h ) one actually has D ∞ ,g ( D ) = ∂ − h (1 − p h ϕ ) N dR ( D ( − h )) ψ =1 .This would fit in with the characterizing description of the big exponential map in the´etale case; cf. also the discussion in [26], section 5.1.We recall the application R D . For our purposes (since we may restrict/corestrict) itwill be enough for this part to assume that D of PR-type over B † rig ,K . Definition 3.32.
Let g ∈ D ∞ ,g ( D ) and r be big enough such that D ∞ ,g ( D ) admits thedescription in (14). A family of elements ( G k ) k ∈ Z in B † log ,K ⊗ B † rig ,K N dR ( D ) is called a complete solution for (1 − ϕ ) G = g if ∂ ( G k ) = G k +1 (cf. 3.27) and ∂ r ( g ) = (1 − p r ϕ ) G r for r big enough.If G = ( G k ) is a complete solution of g ∈ D ∞ ,g ( D ) we also write ∂ − k ( G ) = G k byabuse of notation. Let s ≫ − p s ϕ ) G s = ∂ s ( g ). Then one sees inductivelythanks to Lemma 3.29 that N ( G k ) = X j ≥− k λ j t j + k ( j + k )! =: L k , λ j ∈ D K st ( D )( ψ ⊗ − p − k ⊗ ϕ )( G k ) = p − k X j ≥− k µ j t j + k ( j + k )! =: ( ψ ⊗ M k ) , µ j ∈ D K st ( D ) , where for almost all j one has λ j = µ j = 0. On B † log ,K ⊗ K D K st ( D ), as one checks easily,we have the identity of operators( pN ⊗ ⊗ N )( ψ ⊗ − p − k ⊗ ϕ ) = ( ψ ⊗ − p − k +1 ⊗ ϕ )( N ⊗ ⊗ N ) = ( ψ ⊗ − p − k +1 ⊗ ϕ ) N, hence N (( ψ ⊗ M k )) = ( ψ ⊗ − p − k +1 ⊗ ϕ )( L k ) , since N ⊗ P t i · D K st ( D ), hence the relation (by applying ( ψ − ⊗ ϕ ⊗
1, which we may since ψ acts invertibly on P t i · D K st ( D )) N ( M k ) = (1 − p − k +1 ϕ )( L k ) . On the coefficients this implies the relation
N µ j = (1 − p − j +1 ϕ ) λ j . A = P j ≥− k ν j / ( j + k )! · t j + k and if one changes G k to G ′ k = G k + A so that still ∂ k ( G ′ k ) = ∂ k ( G k ), then λ j is changed to λ j + N ( ν j ) and µ j is changed to µ j + (1 − p j ϕ ) ν j .Hence, L k is changed to L k + N ( A ) and M k is changed to M k +(1 − ϕ )( A ), so that the classof ( λ i , µ i ) is well-defined in H ( C ( D | L ( − i ))). The tupel ( λ j , µ j ) may be considered as anelement of H ( C ( D | L ( − i )))( i ), and we denote the collection of these elements element by R D ( g ), i.e. one has a Γ K -equivariant map R D : D ∞ ,g ( D ) −→ M i ∈ Z H ( C ( D | L ( − i )))( i ) . (15)We note that the map e R D in Lemma 3.29 is the composition of (1 − ϕ ) with R D and thenatural projection to the sum L − k ≤ i< H ( C ( D | L ( − i )))( i ).Define for all k ∈ Z N ( G k ) = L k =: ∂ − k ( L )( ψ ⊗ − ⊗ ϕ )( G k ) = ψ ⊗ M k ) =: ψ ⊗ ∂ − k ( M )) . These definitions imply that (calculating again in B † log ,K ⊗ K D K st ( D )) ψ ((1 − ϕ )( G k ) − M k ) = ( ψ ⊗ − ϕ )( G k ) − M k )) = 0 , hence, since ∂ acts invertibly on ( B † log ,K ⊗ K D K st ( D )) ψ =0 , ∂ k ( g ) = (1 − p k ϕ ) G k − M k . Of course, M k = L k = 0 for k big enough. We will also refer to the system H =( L [1] k , M k , G k ) as a complete solution for g ∈ D ∞ ,g ( D ), where by L [1] k we mean that theaction of ϕ is multiplied by p . This extra factor is introduced so that the interpolationproperty holds.Following Perrin-Riou, we set U ( D ) := M i ∈ Z t i · D st ( D )and D ∞ ,g ( D ) := U ( D ) / (1 − pϕ, N ) U ( D ) . Proposition 3.33.
One has the following exact sequences of Γ K -modules:0 −→ D ∞ ,e ( D ) −→ D ∞ ,g ( D ) R D −→ M i ∈ Z H ( C L ( D | L ( − i )))( i )0 −→ D ∞ ,f ( D ) −→ D ∞ ,g ( D ) R D −→ ( U ( D | L ) /N U ( D | L )) ϕ = p − −→ D ∞ ,e ( D ) −→ D ∞ ,f ( D ) R D −→ ( U ( D | L )) N =0 / (1 − ϕ )( U ( D )) N =0 . roof. See [27], Proposition 2.3.4.We remark that in the case where K/ Q p is unramified one can show all the right-mostmaps in the preceeding Proposition are actually surjective. This can be deduced as in[27], Proposition 4.1.1. Additionally, using the preceding Proposition, one can show that D ∞ ,f ( ) need not be exact. Definition 3.34. a) For a torsion free element γ of Γ K and i ∈ Z Perrin-Riou’s differ-ential operator ∇ i = l i is defined as ∇ i = log( γ )log p ( χ ( γ )) − i = ∇ − i b) The operator ∇ / ( γ n −
1) for n such that Γ n is cyclic is defined as ∇ γ n − γ n )log p ( χ ( γ ))( γ n −
1) := 1log p ( χ ( γ n )) ∞ X i =1 (1 − γ n ) i − i . First, we remark that the second operator is not a quotient of two operators, althoughit behaves as one would like. To clarify we observe that the first definition is independentof the choice of γ since log( γ m ) / log p ( χ ( γ m )) = m/m · log( γ ) / log p ( χ ( γ )). Hence, if ∇ ( y )for some y ∈ D (for instance, y ∈ D ψ =0 ) is such that γ n − γ n − − ∇ ( y ) = ∇ γ n − ( y ). From this it also follows that ( γ n − ∇ γ n − = ∇ . Secondlywe observe that ∇ i = log( χ ( γ ) − i · γ )log p ( χ ( γ )) = Tw − i (cid:18) log( γ )log p ( χ ( γ )) (cid:19) where Tw i is the operator on B (Γ K ) which sends γ to χ ( γ ) k γ . Definition 3.35. If h ≥ h := ∇ h − · . . . · ∇ ∈ H (Γ K ). Lemma 3.36.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K and let h ∈ N such thatFil − h D K dR ( D ) = D K dR ( D ). Then Ω h ( N dR ( D )) ⊂ D . Proof.
Since Ω h = ∇ h − ◦ ∇ h − ◦ . . . ◦ ∇ = t h ∂ h it suffices to show that t h N dR ( D ) ⊂ D .First assume that D is semi-stable. We know from Proposition 2.18 that if D is positive,then D K st ( D ) = ( B † log ,K [1 /t ] ⊗ D ) Γ K ⊂ B † log ,K ⊗ B † rig ,K D , so that N dR ( D ) = ( B † log ,K ⊗ B † rig ,K D K st ( D )) N =0 ⊂ D . For general D if h ≥ D ( − h ) is positive, sothat t h N dR ( D ) ⊂ D . Now if D is de Rham and L/K a finite extension such that D | L issemi-stable, then we have that t h N dR ( D ) ⊂ t h N dR ( D | L ) ⊂ D | L and t h N dR ( D ) ⊂ D [1 /t ],so that t h N dR ( D ) ⊂ D as required. Definition 3.37.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K and h ≥ − h D K dR ( D ) = D K dR ( D ). We define Perrin-Riou’s big exponential map byΩ D,h : D ∞ ,g ( D ) −→ D ψ =0 g h − ◦ . . . ◦ ∇ ( g )36 emma 3.38. One has the following commutative diagram: D ∞ ,g ( D ) ∂ − k / / Ω h (cid:15) (cid:15) D ∞ ,g ( D ( k )) Ω h + k (cid:15) (cid:15) D ψ =0 t k / / D ( k ) ψ =0 Proof.
This is clear from the fact that Ω h = t h ∂ h . Lemma 3.39.
Let D be as before and assume that K is such that Γ K is torsion free.Then one has a canonical map h K,D : ( ϕ − D ψ =1 → H ( K, D ) / ( D ϕ =1 / ( γ K − ϕ − D ψ =1˜ h K,D (cid:15) (cid:15) D ψ =1 ϕ − o o o o h K,D (cid:15) (cid:15) H ( K, D ) / ( D ϕ =1 / ( γ K − H ( K, D ) o o o o is commutative. Proof.
Obviously D ψ =1 /D ϕ =1 ∼ = ( ϕ − D ψ =1 . It is clear that the map h K n ,D factorizesover D ψ =1Γ K . The claim follows. Remark 3.40. If D is of PR-type and let h be such that (14) is satisfied. If g ∈ D ∞ ,g ( V )and k ≥ − h we actually have Ω h ( g ) ⊗ e k ∈ (1 − ϕ ) D ( k ) ψ =1 . Proof.
Let ∂ − k ( g ) = (1 − ϕ ) ∂ − k ( G ) − ∂ − k ( M ). Then ∂ − k ( M ) = h + k − X j ≥ µ j − k t j j ! ∈ H ⊗ D st ( V ( k )) . Since ∇ h + k − ◦ . . . ◦ ∇ = t h + k ∂ h + k the ∂ − k ( M )-part of ∂ − k ( g ) is killed by Ω h .Hence, we see that if h is such that (14) is satisfied and h − r > B † log ,K ⊗ F D K st ( D ( − r )) N =0 ,ψ =1 Ω h − r / / − p r ϕ (cid:15) (cid:15) D ( − r ) ψ =11 − p r ϕ (cid:15) (cid:15) (1 − p r ϕ )( B † log ,K ⊗ F D K st ( D ( − r )) N =0 ,ψ =1 Ω h − r / / ∂ − r (cid:15) (cid:15) (1 − p r ϕ ) D ( r ) ψ =1Tw r (cid:15) (cid:15) D ∞ ,g ( D ) Ω h / / (1 − ϕ ) D ψ =1 commutes. 37et D be of PR-type, g ∈ D ∞ ,g ( D ) and G = ( L k , M k , G k ) be a complete solution for g . Then for each k and n ≫ n,k ( G ) := p n ( k − ϕ − n ∂ − k ( H )(0) := p n ( k − ( p − n ϕ − n ∂ − k ( L )(0) , ϕ − n ∂ − k ( M )(0) , ϕ − n ∂ − k ( G )(0))may be viewed as an element in H ( C st ( K, D ( k ))) (see (10)). Theorem 3.41.
Let D be a de Rham ( ϕ, Γ K )-module over B † rig ,K , g ∈ D ∞ ,g ( D ) and G acomplete solution for g in L . Let h be such that (14) is satisfied. Then for k ≥ − h and n ≫ h K n ,D ( k ) ( ∇ h − ◦ . . . ◦ ∇ ( g ) ⊗ e k )= p − n ( K n ) ( − h + k − ( h + 1 − k )! 1[ L n : K n ] Cor L n /K n exp K n ,D ( k ) (Ξ n,k ( G )) , where we consider the elements on both sides in H ( K n , D ) / ( D ϕ =1 / ( γ K n − Proof.
The proof is divided into several parts. The first general assumption is that D isof PR-type.Let D be pure of slope ≤
0. Then the exponential map has the description given inProposition 3.14. We may assume n big enough so that Γ nK is torsion free. Recall therelation Ω D ( k ) ,h + k ( ∂ − k ( G )) = Ω D,h ( G ) ⊗ e k Hence, for the k ≥ − h we have h K n ,D ( k ) ( ∇ h − ◦ . . . ◦ ∇ ( G ) ⊗ e k ) = h K n ,D ( k ) ( ∇ h + k − ◦ . . . ◦ ∇ ( ∂ − k ( G ))) . Let y h = ∇ h + k − ◦ . . . ∇ ( ∂ − k ( G )) and w n,h = ∇ h + k − ◦ . . . ∇ γ n − ( ∂ − k ( G )). Then in thiscase h K n ,D ( k ) ( y h )( σ ) = σ − γ n − y h − ( σ − b n,h ∈ H ( K n , D ( k )) , where b n,h ∈ e D is such that ( γ n − ϕ − b n,h = ( ϕ − y h . Recall that ∂ − k ( g ) =(1 − ϕ ) ∂ − k ( G ) − ∂ − k ( M ) and Ω D ( k ) ,h + k ( ∂ − k ( g )) = (1 − ϕ )Ω D ( k ) ,h + k ( ∂ − k ( G )), hence ∇ h + k − ◦ . . . ◦ ∇ γ n − ∂ − k ( g )) = ∇ h + k − ◦ . . . ◦ ∇ γ n − − ϕ ) G − k ) −∇ h + k − ◦ . . . ◦ ∇ γ n − M − k ) . With this we may choose b n,h = ( ϕ − − (cid:18) Ω D ( k ) ,h + k γ n − − ϕ ) G − k ) − Ω D ( k ) ,h + k γ n − M − k ) (cid:19) ∈ e D. Now for n ≫ g ∈ B † ,n log ,K ⊗ D K st ( D ), hence the cocycle h K n ,V ( k ) ( y h )( σ ) =( σ −
1) ( w n,h − b n,h ) is cohomologuous to h K n ,V ( k ) ( y h )( σ ) = ( σ − (cid:0) ϕ − n ( w n,h ) − ϕ − n ( b n,h ) (cid:1) ϕ − w n,h − b n,h ) ∈ D K st ( D ( k )) so that G K acts trivially (and ϕ acts as usualinvertibly on D K st ( D ( k ))). We use the exact sequences from the generalized Bloch-Katomap from Proposition 3.14. By the general properties of the connecting homomorphismfor continuous cohomology we have the following: if ( x, y, z ) ∈ H ( C st ( K, D ( k ))) and˜ x ∈ e D log [1 /t ] is such that g (˜ x ) = ( x, y, z ) then exp K n ,D ( k ) (( x, y, z ))( σ ) = ( σ − x . Firstone has ϕ − n ( y ) − ϕ − n ( y )(0) ∈ tK [[ t ]] ⊗ K D K st ( D ) , hence ∇ γ n − ϕ − n ( y ) = p − n ϕ − n ( y )(0) + tz . The same recursion as in [3], Theorem II.3 shows that ϕ − n ( w n,h ) − ( − h − ( h − p − n ϕ − n ( y )(0) ∈ B +dR ⊗ D. Next we have N ( ϕ − n ( w n,h ) − ϕ − n ( b n,h )) = p − n ϕ − n ( ∇ h + k − ◦ . . . ∇ γ n − N ∂ − k ( G ))) . Again we see by recursion with our choice of h that since N ∂ − k ( G ) = L − k and L − k = h − X i =0 λ i · t i /i ! , that we obtain an equality p − n ϕ − n ( ∇ h + k − ◦ . . . ∇ γ n − L − k )) = ( − h − ( h − p − n ϕ − n ( L − k )(0) . Finally one has( ϕ − ϕ − n ( w n,h ) − ϕ − n ( b n,h )) = ϕ − n ( ∇ h + k − ◦ . . . ∇ γ n − M − k )) . Similarly, as before we have M − k = h − X i =0 µ i · t i /i ! , so that the recursion shows ϕ − n ( ∇ h + k − ◦ . . . ∇ γ n − L − k )) = ( − h − ( h − p − n ϕ − n ( M − k )(0) . Altogether this shows that( − h − ( h − p − n exp K n ,D ( k ) (Ξ n,k ( G ))( σ ) = ( σ − ϕ − n ( w n,h ) − ϕ − n ( b n,h )) , which is the claim in this case. 39ext assume D is pure of slope >
0. Then the exponential map has the descriptiongiven in Proposition 3.16. First we note that h K n ,D ( k ) (Ω D,h ( g ) ⊗ e k ) = ( x, y ) with y = Ω D ( k ) ,h + k ( G − k ) , x = ∇ h + k − ◦ . . . ◦ ∇ γ K − ϕ − G − k )) . The exponential map sends Ξ n,k ( G ) to ϕ − n ( G − k )(0) ∈ X ( e D ) G K . The identification e D/ ( ϕ − ∼ → X ( e D ) is given by the following construction (see [5], Remark 3.4): If x ∈ e D/ ( ϕ −
1) and y ∈ e D [1 /t ] is chosen so that ( ϕ − y = x then for n ≫ x is ϕ − n ( y ). With this we see that under these identifications the class of h K n ,D ( k ) (Ω D,h ( g ) ⊗ e k ) is send to ϕ − n ( ∇ h + k − ◦ . . . ◦ ∇ γ K − G − k )) ≡ ( − h − ( h − p − n ϕ − n ( G − k )(0) mod B +dR ⊗ D where we use the same recursion as before, hence the claim in this case.In the general case of semistable a D of PR-type one may use the exact 0 → D ≤ → D → D > →
0, where D ≤ is the biggest submodule of D with slopes ≤
0, and D > = D/D ≤ , which is a ( ϕ, Γ K )-module with slopes >
0. By using the description of theisomorphism (9) and the explicit description of the transition morphism for the cone oneis reduced, since all maps are compatible with exact sequences, to the case of a modulewith all slopes ≤ >
0. But in these cases we have just verified that thestatement holds.Now assume D is de Rham and let L/K be a finite extension such that D is of PR-typeover L . Then for y ∈ D ∞ ,g ( D ) one has, if we consider y ∈ D ∞ ,g ( D | L )Res L n /K n ( h K n ,D ( k ) (Ω D,h ( y ))) = h L n ,D | L ( k ) (Ω D,h ( y )) , so that the claim follows from Proposition 2.13.For the record we state the next proposition in case D is semi-stable. As before, let h ≥ D , and dually let h ∗ ≥ D ∗ (1) Proposition 3.42. a) If k ≥ − h and n ≥ h K n ,D ( k ) ( ∇ h − ◦ . . . ◦∇ ( g ) ⊗ e k ) = p − n ( K n ) ( − h + k − ( h + 1 − k )! exp K n ,D ( k ) (Ξ n,k ( G ))b) If k ≤ − h ∗ and n ≥ ∗ K n ,D ∗ (1) ( h K n ,D ( k ) ( ∇ h − ◦ . . . ◦∇ ( g ) ⊗ e k )) = p − n ( K n ) − h − k )! ϕ − n ( ∂ − k g ⊗ t − j e j )(0) Proof.
The first part is just the preceding theorem. For the second observe that due toProposition 3.21 one hasexp ∗ K n ,D ∗ (1) ( h K n ,D ( k ) ( ∇ h − ◦ . . . ◦ ∇ ( g ) ⊗ e k )) = p − n ( K n ) ϕ − n ( ∇ h − ◦ . . . ◦ ∇ ( g ) ⊗ e k )(0) .
40 computation with the Taylor series shows that p − n ( K n ) ϕ − n ( ∇ h − ◦ . . . ◦ ∇ ( g ) ⊗ e k )(0) = p − n ( K n ) − h − j )! ϕ − n ( ∂ − k g ⊗ t − k e k )(0) , hence the claim. 41 References [1] Yves Andr´e. Hasse-Arf filtrations and p -adic monodromy. (Filtrations de type Hasse-Arf et monodromie p -adique.). Invent. Math. , 148(2):285–317, 2002.[2] Fabrizio Andreatta and Adrian Iovita. Global applications of relative ( ϕ, Γ)-modules.I. Berger, Laurent (ed.) et al., Repr´esentation p -adiques de groupes p -adiquesI. Repr´esentations galoisiennes et ( ϕ, Γ)-modules. Paris: Soci´et´e Math´ematique deFrance. Ast´erisque 319, 339-419; erratum Ast´erisque 330, 543-554 (2010), 2008.[3] Laurent Berger. p -adic representations and differential equations. (Repr´esentations p -adiques et ´equations diff´erentielles.). Invent. Math. , 148(2):219–284, 2002.[4] Laurent Berger. Bloch and Kato’s exponential map: three explicit formulas.
Doc.Math., J. DMV Extra , pages 99–129, 2003.[5] Laurent Berger. Construction of ( φ, Γ)-modules: p -adic representations and B -pairs.(Construction de ( φ, Γ)-modules: repr´esentations p -adiques et B -paires.). AlgebraNumber Theory , 2(1):91–120, 2008.[6] Laurent Berger. p -adic differential equations and filtered ( ϕ, N )-modules. (´Equationsdiff´erentielles p -adiques et ( ϕ, N )-modules filtr´es.). In Ast´erisque 319 . Paris: Soci´et´eMath´ematique de France, 2008.[7] Laurent Berger. Almost C p -representations and ( ϕ, Γ)-modules. (Presque C p -repr´esentations et ( ϕ, Γ )-modules.). J. Inst. Math. Jussieu , 8(4):653–668, 2009.[8] Laurent Berger. On some modular representations of the Borel subgroup of GL ( Q p ). Compos. Math. , 146(1):58–80, 2010.[9] Pierre (ed.) Cartier, Luc (ed.) Illusie, Nicholas M. (ed.) Katz, G´erard (ed.) Laumon,Yuri I. (ed.) Manin, and Ken A. (ed.) Ribet.
The Grothendieck Festschrift. A collec-tion of articles written in honor of the 60th birthday of Alexander Grothendieck. Vol-ume I. Reprint of the 1990 edition.
Modern Birkh¨auser Classics. Basel: Birkh¨auser.xx, 498 p., 2007.[10] Fr´ed´eric Cherbonnier and Pierre Colmez. Overconvergent p -adic representations.(Repr´esentations p -adiques surconvergentes.). Invent. Math. , 133(3):581–611, 1998.[11] Fr´ed´eric Cherbonnier and Pierre Colmez. Th´eorie d’Iwasawa des repr´esentations p -adiques d’un corps local. (Iwasawa theory of p -adic representations of a local field). J. Am. Math. Soc. , 12(1):241–268, 1999.[12] Pierre Colmez. Iwasawa theory of de Rham representations of a local field. (Th´eoried’Iwasawa des repr´esentations de de Rham d’un corps local.).
Ann. Math. (2) ,148(2):485–571, 1998.[13] Pierre Colmez. Fontaine’s rings and p -adic L-functions. , 2004. Lecture notes.4214] Jean-Marc Fontaine. Almost C p -representation. (Presque C p -repr´esentations.). Doc.Math., J. DMV Extra , pages 285–385, 2003.[15] Jean-Marc Fontaine and Jean-Pierre Wintenberger. Le ”corps des normes” de cer-taines extensions alg´ebriques de corps locaux.
C. R. Acad. Sci., Paris, S´er. A ,288:367–370, 1979.[16] Jean-Mark Fontaine and Yi Ouyang. Theory of p -adic Galois representations. http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf . Book in preparation.[17] Laurent Herr. A new approach to Tate’s local duality. (Une approche nouvelle de ladualit´e locale de Tate.). Math. Ann. , 320(2):307–337, 2001.[18] Kazuya Kato. Lectures on the approach to Iwasawa theory for Hasse-Weil L -functionsvia B dR . Colliot-Th´el`ene, Jean-Louis et al., Arithmetic algebraic geometry. Lecturesgiven at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.),held in Trento, Italy, June 24 - July 2, 1991. Berlin: Springer-Verlag. Lect. NotesMath. 1553, 50-163 (1993)., 1993.[19] Kiran S. Kedlaya. Slope filtrations revisited. Doc. Math., J. DMV , 10:447–525, 2005.[20] Kiran S. Kedlaya. Slope filtrations for relative Frobenius. Berger, Laurent (ed.) etal., Repr´esentation p -adiques de groupes p -adiques I. Repr´esentations galoisienneset ( ϕ, Γ)-modules. Paris: Soci´et´e Math´ematique de France. Ast´erisque 319, 259-301,2008.[21] Kiran S. Kedlaya. Some new directions in p-adic Hodge theory.
J. Th´eor. NombresBordx. , 21(2):285–300, 2009.[22] Ruochuan Liu. Cohomology and duality for ( ϕ, Γ)-modules over the Robba ring.
Int.Math. Res. Not. , 2008:32 p, 2008.[23] Kentaro Nakamura. Iwasawa theory of de Rham ( φ ,Γ)-modules over the Robba rings. ArXiv e-prints , 1201.6475, January 2012.[24] Jan Nekov´aˇr.
Selmer complexes.
Ast´erisque 310. Paris: Soci´et´e Math´ematique deFrance. viii, 559 p., 2006.[25] J¨urgen Neukirch, Alexander Schmidt, and Kay Wingberg.
Cohomology of num-ber fields. 2nd ed.
Grundlehren der Mathematischen Wissenschaften 323. Berlin:Springer. xv, 825 p., 2008.[26] Bernadette Perrin-Riou. Iwasawa theory and explicit reciprocity law. A remake ofan article of P. Colmez. (Th´eorie d’Iwasawa et loi explicite de r´eciprocit´e. Un remaked’un article de P. Colmez.).
Doc. Math., J. DMV , 4:219–273, 1999.[27] Bernadette Perrin-Riou. Th´eorie d’Iwasawa des repr´esentations p -adiques semi-stables. (Iwasawa theory of semi-stable p -adic representations). M´em. Soc. Math.Fr., Nouv. S´er. , 84:vi, 111 p., 2001. 4328] Bernadette Perrin-Riou. Some remarks on Iwasawa theory for elliptic curves.(Quelques remarques sur la th´eorie d’Iwasawa des courbes elliptiques.). Natick, MA:A K Peters, 2002.[29] Jay Pottharst. Analytic families of finite-slope Selmer groups. http://math.bu.edu/people/potthars/writings/affssg-old.pdf . Preprint.[30] John Tate. Relations between K and Galois cohomology. Invent. Math. , 36:257–274,1976.[31] Charles A. Weibel.
An introduction to homological algebra.
Cambridge Studies inAdvanced Mathematics. 38. Cambridge: Cambridge University Press. xiv, 450 p.,1994.
Andreas RiedelUniversit¨at Heidelberg, Mathematisches InstitutIm Neuenheimer Feld, 22869115 Heidelberg, GermanyEmail: [email protected]