P-ADIC HEIGHTS and P-ADIC HODGE THEORY Denis Benois

P-ADIC HEIGHTS AND P-ADIC HODGE THEORY Denis Benois

To cite this version:

Denis Benois. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY. 2014. hal-01260908

HAL Id: hal-01260908 https://hal.archives-ouvertes.fr/hal-01260908 Preprint submitted on 22 Jan 2016

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY

DENIS BENOIS

ABSTRACT. Using the theory of (ϕ,Γ)-modules and the formalism of Selmer complexes we construct the p-adic height pairing for p-adic representations with coefﬁcients in an afﬁnoid algebra over Qp. For p-adic representations that are potentially semistable at p, we relate our con- truction to universal norms and compare it to the p-adic height pairing of Nekovár.ˇ

CONTENTS Introduction 2 0.1. Selmer complexes 2 0.2. Selmer complexes and (ϕ,Γ)-modules 4 0.3. p-adic height pairings 6 0.4. The organization of this paper 8 0.5. Remarks 8 1. Complexes and products 9 1.1. The complex T •(A•) 9 1.2. Products 16 2. Cohomology of (ϕ,ΓK)-modules 22 2.1. (ϕ,ΓK)-modules 22 2.2. Relation to the p-adic Hodge theory 26

2.3. The complex Cϕ,γK (D) 29 2.4. The complex K•(V) 31 2.5. Transpositions 36 2.6. The Bockstein map 38 2.7. Iwasawa cohomology 41 1 2.8. The group Hf (D) 44 3. p-adic height pairings I: Selmer complexes 46 3.1. Selmer complexes 46 3.2. p-adic height pairings 53 4. Splitting submodules 60

Date: December 18, 2014. 1991 Mathematics Subject Classiﬁcation. 2000 Mathematics Subject Classiﬁcation. 11R23, 11F80, 11S25, 11G40. 1 2 DENIS BENOIS

4.1. Splitting submodules 60 4.2. The canonical splitting 62 4.3. Filtration associated to a splitting submodule 64 5. p-adic height pairings II: universal norms 71 norm 5.1. The pairing hV,D 71 sel 5.2. Comparision with hV,D 76 6. p-adic height pairings III: splitting of local extensions 80 spl 6.1. The pairing hV,D 80 6.2. Comparision with Nekovár’sˇ height pairing 83 norm 6.3. Comparision with hV,D 85 7. p-adic height pairings IV: extended Selmer groups 87 7.1. Extended Selmer groups 87 norm 7.2. The pairing hV,D for extended Selmer groups 89 References 93

INTRODUCTION 0.1. Selmer complexes.

0.1.1. Let F be a number field. We denote by S f and S∞ the set of non- archimedean and archimedian places of F respectively. Fix a prime number p and denote by Sp the set of places q above p. Let S be a finite set of non-archimedian places of F containing Sp. To simplify notation, set Σp = S \ Sp. We denote by GF,S the Galois group of the maximal algebraic extension of F unramified outside S ∪ S∞. For each q ∈ S we denote by Fq the completion of F with respect to q and by GFq the absolute Galois group of Fq which we identify with the decomposition group at q. We will write Iq for the inertia subgroup of GFq and Frq for the relative Frobenius over Fq. If G is a topological group and M is a topological G-module, we denote by C•(G,M) the complex of continuous cochains of G with coefficients in M.

0.1.2. Let T be a continuous representation of GF,S with coefficients in a complete local noetherian ring A with a finite residue field of characteristic p. A local condition at q ∈ S is a morphism of complexes

• • gq : Uq (T) → C (GFq ,T). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 3

• • To each collection U (T) = (Uq (T),gq)q∈S of local conditions one can associate the following diagram • L • (1) C (GF,S,T) / C (GFq ,T) q∈S O (gq) L • Uq (T), q∈S where the upper row is the restriction map. The Selmer complex associated to the local conditions U•(T) is deﬁned as the mapping cone ! ! • • • M • M • S (T,U (T)) = cone C (GF,S,T) ⊕ Uq (T) → C (GFq ,T) [−1]. q∈S q∈S This notion was introduced by Nekovárˇ in [34], where the machinery of Selmer complexes was developed in full generality. 0.1.3. The most important example of local conditions is provided by Green- berg’s local conditions. If q ∈ S, we will denote by Tq the restriction of T on GFq . Fix, for each q ∈ Sp, a subrepresentation Mq of Tq and deﬁne • • Uq (T) = C (GFq ,Mq) q ∈ Sp.

For q ∈ Σp we consider the unramiﬁed local conditions Fr −1 • • Iq q Iq Uq (T) = Cur(Tq) = T −−−→ T , where the terms are placed in degrees 0 and 1. To simplify notation, we will write S•(T,M) for the Selmer complex associated to these conditions and RΓ(T,M) for the corresponding object of the derived category. Let T ∗(1) be the Tate dual of T equipped with Greenberg local conditions N = (Nq)q∈Sp such that M and N are orthogonal to each other under the canonical duality T × T ∗(1) → A. In this case, the general construction of cup products for cones gives a pairing L ∗ ∪ : RΓ(T,M) ⊗A RΓ(T (1),N) → A[−3] (see [34], Section 6.3). Nekovárˇ constructed the p-adic height pairing sel L ∗ h : RΓ(T,M) ⊗A RΓ(T (1),N) → A[−2] 1 as the composition of ∪ with the Bockstein map βT,M : RΓ(T,M) → RΓ(T,M)[1]: sel h (x,y) = βT,M(x) ∪ y.

1See [34], Section 11.1 or Section 3.2 below for the deﬁnition of the Bockstein map. 4 DENIS BENOIS

Passing to cohomology groups Hi(T,M) = RiΓ(T,M), we obtain a pairing sel 1 1 ∗ h1 : H (T,M) ⊗A H (T (1),N) → A. 0.1.4. The relationship of these constructions to the traditional treatements is the following. Let A = OE be the ring of integers of a local field E/Qp and let T be a Galois stable OE-lattice of a p-adic Galois representation V with coefficients in E. Assume that V is semistable at all q ∈ Sp. We say that V satisfies the Panchishkin condition at p if, for each q ∈ Sp, there + 2 exists a subrepresentation Vq ⊂ Vq such that all Hodge–Tate weights of + + + + + Vq/Vq are > 0. Set Tq = T ∩Vq , T = (Tq )q∈Sp and consider the associated Selmer complex RΓ(T,T +). The first cohomology group H1(T,T +) of RΓ(T,T +) is very close to the Selmer group defined by Greenberg [22], [24] and therefore to the Bloch–Kato Selmer group [17]. It can be shown sel (see [34], Chapter 11), that, under some mild conditions, the pairing h1 coincides with the p-adic height pairing constructed in [33] and [36] using universal norms (see also [39]). Note, that Nekovár’sˇ construction has many advantages over the classical definitions. In particular, it allows to study the variation of the p-adic heights in ordinary families of p-adic representations (see [34], Section 0.16 and Chapter 11 for further discussion). 0.2. Selmer complexes and (ϕ,Γ)-modules. 0.2.1. In this paper we consider Selmer complexes with local conditions coming from the theory of (ϕ,Γ)-modules and extend a part of Nekovár’sˇ machinery to this situation. Let now A be a Qp-affinoid algebra and let V be a p-adic representation of GF,S with coefficients in A. In [37], Pottharst studied Selmer complexes associated to the diagrams of the form (1) in this context. We will consider a slightly more general situation because, for • • the local conditions Uq (V) that we have in mind, the maps gq : Uq (V) → • C (GFq ,V) are not defined on the level of complexes but only in the derived category of A-modules. For each q ∈ Sp we denote by Γq the Galois group of the cyclotomic p-extension of Fq. As before, we denote by Vq the restriction of V on the decomposition group at q. The theory of (ϕ,Γ)-modules associates to Vq a † finitely generated projective module Drig,A(V) over the Robba ring RFq,A equipped with a semilinear Frobenius map ϕ and a continuous action of Γq which commute to each other (see [18], [13], [16], [29]). In [30], Kedlaya, Pottharst and Xiao extended the results of Liu [31] about the cohomology of (ϕ,Γ)-modules to the relative case. Their results play a key role in this paper.

2We call Hodge–Tate weights the jumps of the Hodge–Tate ﬁltration on the associated de Rham module. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 5

Namely, to each (ϕ,Γq)-module D over RFq,A one can associate the • ∗ Fontaine–Herr complex Cϕ,γq (D) of D. The cohomology H (D) of D is de- • † ﬁned as the cohomology of Cϕ,γq (D). If D = Drig,A(V), there exist isomor- ∗ † ∗ • † phisms H (Drig,A(V)) ' H (Fq,V), but the complexes Cϕ,γq (Drig(V)) and • C (GFq ,Vq) are not quasi-isomorphic. A simple argument allows us to con- • struct a complex K (Vq) together with quasi-isomorphisms • • • † • 3 ξq : C (GFq ,V) → K (Vq) and αq : Cϕ,γq (Drig,A(Vq)) → K (Vq) . For each † q ∈ Sp, we choose a (ϕ,Γq)-submodule Dq of Drig,A(Vq) that is a RFq,A- † module direct summand of Drig,A(Vq) and set D = (Dq)q∈Sp . Set     • M • M M • K (V) =  C (GFq ,V)  K (Vq) q∈Σp q∈Sp and ( • • Cϕ,γq (Dq), if q ∈ Sp, Uq (V,D) = • Cur(Vq), if q ∈ Σp. For each q ∈ Sp, we have morphisms

• resq • ξq • fq : C (GF,S,V) −−→ C (GFq ,V) −→ K (Vq), • • † αq • gq : Uq (V,D) −→ Cϕ,γq (Drig,A(Vq)) −→ K (Vq). • • If q ∈ Σp, we deﬁne the maps fq : C (GF,S,V) → C (GFq ,V) and gq : • • Cur(Vq) → C (GFq ,V) exactly as in the case of Greenberg local conditions. Consider the diagram ( f ) • q q∈S • C (GF,S,V) / K (V) O ⊕ gq q∈S L • Uq (V,D). q∈S We denote by S•(V,D) the Selmer complex associated to this diagram and by RΓ(V,D) the corresponding object in the derived category of A-modules.

3This complex was ﬁrst introduced in [5] 6 DENIS BENOIS

0.3. p-adic height pairings. 0.3.1. Let V ∗(1) be the Tate dual of V. We equip V ∗(1) with orthogonal local conditions D⊥ setting D⊥ = Hom D† (V )/D , , q ∈ S . q RFq,A rig,A q q RFq,A p The general machinery gives us a cup product pairing L ∗ ⊥ ∪V,D : RΓ(V,D) ⊗A RΓ(V (1),D ) → A[−3]. This allows us to construct the p-adic height pairing exactly in the same way as in the case of Greenberg local conditions. Deﬁnition. The p-adic height pairing associated to the data (V,D) is de- ﬁned as the morphism

sel L ∗ ⊥ δV,D hV,D : RΓ(V,D) ⊗A RΓ(V (1),D ) −−→ L ∗ ⊥ ∪V,D → RΓ(V,D)[1] ⊗A RΓ(V (1),D ) −−→ A[−2], where δV,D denotes the Bockstein map. sel The height pairing hV,D,M induces a pairing on cohomology groups sel 1 1 ∗ ⊥ hV,D,1 : H (V,D) × H (V (1),D ) → A. Applying the machinery of Selmer complexes, we obtain the following result (see Theorem 3.2.4 below). Theorem I. We have a commutative diagram

hsel L ∗ ⊥ V,D RΓ(V,D) ⊗A RΓ(V (1),D ) / A[−2]

s12 = hsel ∗ ⊥ ∗ ⊥ L V (1),D RΓ(V (1),D ) ⊗A RΓ(V,D) / A[−2], deg(a)deg(b) sel where s12(a ⊗b) = −(1) b ⊗a. In particular, the pairing hV,D,1 is skew symmetric.

0.3.2. Assume that A = E, where E is a ﬁnite extension of Qp. For each family D = (Dq)q∈Sp satisfying the conditions N1-2) of Section 5 we construct a pairing norm 1 1 ∗ ⊥ hV,D : H (V,D) × H (V (1),D ) → E, which can be seen as a direct generalization of the p-adic height pairing, constructed for representations satisfying the Panchishkin condition using universal norms. The following theorem generalizes Theorem 11.3.9 of [34] (see Theorem 5.2.2 below). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 7

Theorem II. Let V be a p-adic representation of GF,S with coefﬁcients in a

ﬁnite extension E of Qp. Assume that the family D = (Dq)q∈Sp satisﬁes the conditions N1-2). Then norm sel hV,D = hV,D,1.

0.3.3. We denote by DdR, Dst and Dst Fontaine’s classical functors. Let V be a p-adic representation with coefficients in E/Qp. Assume that the restriction of V on GFq is potentially semistable for all q ∈ Sp, and that V satisfies the following condition ϕ=1 ∗ ϕ=1 S) Dcris(V) = Dcris(V (1)) = 0, ∀q ∈ Sp. 0 For each q ∈ Sp we fix a splitting wq : DdR(Vq) → DdR(Vq)/F DdR(Vq) 0 of the canonical projection DdR(Vq) → DdR(Vq)/Fil DdR(Vq) and set w =

(wq)q∈Sp . In this situation, Nekovárˇ [33] constructed a p-adic height pairing

Hodge 1 1 ∗ hV,w : Hf (V) × Hf (V (1)) → E on the Bloch–Kato Selmer groups of V and V ∗(1), which is defined using the Bloch–Kato exponential map and depends on the choice of splittings w. Let q ∈ Sp, and let L be a finite extension of Fq such that Vq is semistable over L. The semistable module Dst/L(Vq) is a finite dimensional vector space over the maximal unramified subextension L0 of L, equipped with a Frobe- nius ϕ, a monodromy N, and an action of GL/Fq = Gal(L/Fq).

Deﬁnition. Let q ∈ Sp. We say that a (ϕ,N,GL/Fq )-submodule Dq of Dst/L(Vq) is a splitting submodule if 0 DdR/L(Vq) = Dq,L ⊕ Fil DdR/L(Vq), Dq,L = Dq ⊗L0 L as L-vector spaces.

It is easy to see, that each splitting submodule Dq deﬁnes a splitting of the Hodge ﬁltration of DdR(V), which we denote by wD,q. For each family

D = (Dq)q∈Sp of splitting submodules we construct a pairing

spl 1 1 ∗ hV,D : Hf (V) × Hf (V (1)) → E using the theory of (ϕ,Γ)-modules and prove that

hspl = hHodge V,D V,wD † (see Proposition 6.2.3). Let Dq denote the (ϕ,Γq)-submodule of Drig(Vq) associated to Dq by Berger [10] and let D = (Dq)q∈Sp . In the following theorem we compare this pairing with previous constructions (see Theo- rem 6.3.4 and Corollary 6.3.5). 8 DENIS BENOIS

Theorem III. Assume that (V,D) satisﬁes the conditions S) and N2). Then 1 1 1 ∗ ⊥ 1 ∗ i) H (V,D) = Hf (V) and H (V (1),D ) = Hf (V (1)); sel norm spl ii) The height pairings hV,D,1, hV,D and hV,D coincide. 0.3.4. If F = Q, we can slightly relax the conditions on (V,D) in Theo- rem III. We replace these conditions by the conditions F1-2) of Section 7.1, which reﬂect the conjectural behavior of V at p in the presence of trivial zeros [4], [6]. We show, that under these conditions we have a canonically splitting exact sequence

splV, f 0 0 1 o 1 0 / H (D ) / H (V,D) / Hf (V) / 0,

0 † where D = Drig(Vp)/D. By modifying previous constructions, one can de- ﬁne a pairing norm 1 1 ∗ hV,D : Hf (V) × Hf (V (1)) → E The following result is a simpliﬁed form of Theorem 7.2.2 of Section 7, which can be seen as a generalization of Theorem 11.4.6 of [34].

Theorem IV. Let V be a p-adic representation of GQ,S that is potentially semistable at p and satisfies the conditions F1-2). Then norm spl i) hV,D = hV,D; 1 1 ∗ ii) For all x ∈ Hf (V) and y ∈ Hf (V (1)) we have sel norm hV,D(splV, f (x),splV ∗(1), f (y)) = hV,D (x,y). 0.4. The organization of this paper. This paper is very technical by the nature, and in Sections 1-2 we assemble necessary preliminaries. In Sec- tion 1, we recall the formalism of cup products for cones following [34] (see also [35]), and prove some auxiliary cohomological results. In Section 2, • we review the theory of (ϕ,Γ)-modules and define the complex K (Vq). The reader familiar with (ϕ,Γ)-modules can skip these sections on first reading. In Section 3.1, we construct Selmer complexes RΓ(V,D) and study their sel first properties. The p-adic height pairing hV,D is defined in Section 3.2. Theorem I (Theorem 3.2.4 of Section 3.2) follows directly from the defi- sel nition of hV,D and the machinery of Section 1. Preliminary results about splitting submodules are assembled in Section 4. In Sections 5-7 we con- norm spl struct the pairings hV,D and hV,D and prove Theorems II, III and IV. 0.5. Remarks. 1) In the forthcoming joint paper with K. Büyükboduk [7], we prove the Rubin style formula for our height pairing and apply it to the study of extra-zeros of p-adic L-functions. 2) During the preparation of this paper, I learned from L. Xiao of an independent work in progress of Rufei Ren on this subject. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 9

1. COMPLEXESANDPRODUCTS 1.1. The complex T •(A•). 1.1.1. If R is a commutative ring, we write K (R) for the category of complexes of R-modules and Kft(R) for the subcategory of K (R) consisting • n n n • of complexes C = (C ,dC• ) such that H (C ) are ﬁnitely generated over R for all n ∈ Z. We write D(R) and Dft(R) for the corresponding derived categories and denote by [·] : K∗(R) → D∗(R), (∗ ∈ {/0,ft}) the obvious func- [a,b] tors. We will also consider the subcategories Kft (R), (a 6 b) consisting of objects of Kft(R) whose cohomologies are concentrated in degrees [a,b]. A perfect complex of R-modules is one of the form

0 → Pa → Pa+1 → ... → Pb → 0, where each Pi is a finitely generated projective R-module. If R is noetherian, [a,b] we denote by Dperf (R) the full subcategory of Dft(R) consisting of objects quasi-isomorphic to perfect complexes concentrated in degrees [a,b]. • n n If C = (C ,dC• )n∈Z is a complex of R-modules and m ∈ Z, we will • • n n+m n denote by C [m] the complex defined by C [m] = C and dC•[m](x) = m n n (−1) dC• (x). We will often write d or just simply d instead dC• . For each • • m, the truncation τ>mC of C is the complex 0 → coker(dm−1) → Cm+1 → Cm+2 → ··· . Therefore ( i • 0, if i < m, H (τ>mC ) = i • H (C ), if i > m . The tensor product A• ⊗ B• of two complexes A• and B• is defined by (A• ⊗ B•)n = MAi ⊗ Bn−i, i∈Z i i n−i d(ai ⊗ bn−i) = dxi ⊗ yn−i + (−1) ai ⊗ bn−i, ai ∈ A , bn−i ∈ B . • • • • We denote by s12 : A ⊗ B → B ⊗ A the transposition nm n m s12(an ⊗ bm) = (−1) bm ⊗ an, an ∈ A , bm ∈ B .

It is easy to check that s12 is a morphism of complexes. We will also con- ∗ • • • • sider the map s12 : A ⊗ B → B ⊗ A given by ∗ s12(an ⊗ bm) = bm ⊗ an, which is not a morphism of complexes in general. • • Recall that a homotopy h : f g between two morphisms f ,g : A → B is a family of maps h = (hn : An+1 → Bn) such that dh+hd = g− f . We will n sometimes write h instead h . A second order homotopy H : h k between 10 DENIS BENOIS

n n+2 n homotopies h,k : f g is a collection of maps H = (H : A → B ) such that Hd − dH = k − h. • • • • If fi : A1 → B1 (i = 1,2) and gi : A2 → B2 (i = 1,2) are morphisms of complexes and h : f1 f2 and k : g1 g2 are homotopies between them, then the formula n (2) (h ⊗ k)1(xn ⊗ ym) = h(xn) ⊗ g1(ym) + (−1) f2(xn) ⊗ k(ym), n m where xn ∈ A1, ym ∈ A2 , deﬁnes a homotopy

(h ⊗ k)1 : f1 ⊗ g1 f2 ⊗ g2.

1.1.2. For the content of this subsection we refer the reader to [41], §3.1. If f : A• → B• is a morphism of complexes, the cone of f is deﬁned to be the complex cone( f ) = A•[1] ⊕ B•, with differentials n n+1 n d (an+1,bn) = (−d (an+1), f (an+1) + d (bn)). We have a canonical distinguished triangle

f A• −→ B• → cone( f ) → A•[1]. We say that a diagram of complexes of the form

• f1 • (3) A1 / B1

α1 2: α2 h

• • A2 / B2 f2 is commutative up to homotopy, if there exists a homotopy

h : α2 ◦ f1 f2 ◦ α1. In this case, the formula n n c(α1,α2,h) (an+1,bn) = (α1(an+1),α2(bn) + h (an)) deﬁnes a morphism of complexes

(4) c(α1,α2,h) : cone( f1) → cone( f2). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 11

Assume that, in addition to (3), we have a diagram

• f1 • A1 / B1

0 0 α1 2: α2 h0

• • A2 / B2 f2 together with homotopies 0 k1 : α1 α1 0 k2 : α2 α2 and a second order homotopy 0 H : f2 ◦ k1 + h k2 ◦ f1 + h. Then the map

(5) (an+1,bn) 7→ (−k1(an+1),k2(bn) + H(an+1)) 0 0 0 deﬁnes a homotopy c(α1,α2,h) c(α1,α2,h ). 1.1.3. In the remainder of this section R is a commutative ring and all complexes are complexes of R-modules. Let A• = (An,dn) be a complex equipped with a morphism ϕ : A• → A•. By deﬁnition, the total complex ϕ−1 T •(A•) = Tot(A• −−→ A•). is given by T n(A•) = An−1 ⊕ An with differentials n n−1 n n n • d (an−1,an) = (d an−1 +(−1) (ϕ −1)an,d an), (an−1,an) ∈ T (A ). If A• and B• are two complexes equipped with morphisms ϕ : A• → A• and ψ : B• → B•, and if α : A• → B• is a morphism such that α ◦ ϕ = ψ ◦ α, then α induces a morphism T(α) : T •(A•) → T •(B•). We will often write α instead T(α) to simplify notation. Lemma 1.1.4. Let A• and B• be two complexes equipped with morphisms • • • • • • ϕ : A → A and ψ : B → B , and let αi : A → B (i = 1,2) be two morphisms such that

αi ◦ ϕ = ψ ◦ αi i = 1,2.

If h : α1 α2 is a homotopy between α1 and α2 such that n n+1 • n • h ◦ ϕ = ψ ◦ h, then the collection of maps hT = (hT : T (A ) → T (B )) n deﬁned by hT (an,an+1) = (h(an),h(an+1)) is a homotopy between T(α1) and T(α2). 12 DENIS BENOIS

Proof. The proof of this lemma is a direct computation and is omitted here. • • • In the remainder of this subsection we will consider triples (A1,A2,A3) of complexes of R-modules equipped with the following structures • • A1) Morphisms ϕi : Ai → Ai (i = 1,2,3). • • • A2) A morphism ∪A : A1 ⊗ A2 → A3 which satisﬁes

∪A ◦ (ϕ1 ⊗ ϕ2) = ϕ3 ◦ ∪A. • Proposition 1.1.5. Assume that a triple (Ai ,ϕi) (1 6 i 6 3) satisﬁes the conditions A1-2). Then the map T • • • • • • ∪A : T (A1) ⊗ T (A2) → T (A3) given by T m (xn−1,xn) ∪A (ym−1,ym) = (xn ∪A ym−1 + (−1) xn−1 ∪A ϕ2(ym),xn ∪A ym), is a morphism of complexes. Proof. This proposition is well known to the experts (compare, for example, to [35] Proposition 3.1). It follows from a direct computation which n • we recall for the convenience of the reader. Let (xn−1,xn) ∈ T (A1) and m • (ym−1,ym) ∈ T (A2). Then T d((xn−1,xn) ∪A (ym−1,ym) = m = d(xn ∪A ym−1 + (−1) xn−1 ∪A ϕ2(ym),xn ∪A ym) = (zn+m,zn+m+1), where

n m zn+m = dxn ∪A ym−1 + (−1) xn ∪A dym−1 + (−1) dxn−1 ∪A ϕ2(ym)+ m+n−1 n+m (−1) xn−1 ∪A d(ϕ2(ym)) + (−1) (ϕ3 − 1)(xn ∪A ym) and zn+m+1 = d(xn ∪A ym). On the other hand T ∪A ◦ d((xn−1,xn) ⊗ (ym−1,ym)) = T n = ∪A ◦ ((dxn−1 + (−1) (ϕ1 − 1)xn,dxn) ⊗ (ym−1,ym)) + n T m +(−1) ∪A ◦((xn−1,xn) ⊗ (dym−1 + (−1) (ϕ2 − 1)ym,dym)) = = (un+m,un+m+1), where

m n un+m = dxn ∪A ym−1 + (−1) (dxn−1 + (−1) (ϕ1 − 1)xn) ∪ ϕ2(ym)+ n m n+m−1 (−1) xn ∪ (dym−1 + (−1) (ϕ2 − 1)ym) + (−1) xn−1 ∪ ϕ2(dym), P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 13

n and un+m+1 = dxn ∪A ym + (−1) xn ∪A dym. Now the proposition follows from the formula n d(xn ∪A ym) = dxn ∪A ym + (−1) xn ∪A dym and the assumption A2) that reads ϕ1(xn) ∪A ϕ2(ym) = ϕ3(xn ∪A ym). • • Proposition 1.1.6. Let (Ai ,ϕi) and (Bi ,ψi) (1 6 i 6 3) be two triples of complexes that satisfy the conditions A1-2). Assume that they are equipped with morphisms • • αi : Ai → Bi , such that αi ◦ ϕi = ψi ◦ αi for all 1 6 i 6 3. Assume, in addition, that in the diagram

• • ∪A • A1 ⊗ A2 / A3

α1⊗α2 α3 h

∪ • • B Ô • B1 ⊗ B2 / B3 there exists a homotopy

h : α3 ◦ ∪A ∪B ◦ (α1 ⊗ α2). such that h ◦ (ϕ1 ⊗ ϕ2) = ψ3 ◦ h. Then the collection hT of maps k M n • m • k • hT : (T (A1) ⊗ T (A2)) → T (B3) m+n=k+1 deﬁned by

k hT ((xn−1,xn) ⊗ (ym−1 ⊗ ym)) = m = h(xn ⊗ ym−1) + (−1) h(xn−1 ⊗ ϕ2(ym)),h(xn ⊗ ym) .

T T provides a homotopy hT : α3 ◦ ∪A ∪B ◦ (α1 ⊗ α2) :

∪T • • • • A • • T (A1) ⊗ T (A2) / T (A3)

α1⊗α2 α3 hT

∪T • • • • B ÑÙ • • T (B1) ⊗ T (B2) / T (B3). 14 DENIS BENOIS

n • Proof. Again, the proof is a routine computation. Let (xn−1,xn) ∈ T (A1) m • and (ym−1,ym) ∈ T (A2). We have n d((xn−1,xn)⊗(ym−1,ym)) = (dxn−1 +(−1) (ϕ1 −1)xn,dxn)⊗(ym−1,ym)+ n m + (−1) (xn−1,xn) ⊗ (dym−1 + (−1) (ϕ2 − 1)ym,dym), and therefore

hT ◦ d((xn−1,xn) ⊗ (ym−1,ym)) = (a,b), where m n a = h(dxn ⊗ ym−1) + (−1) h((dxn−1 + (−1) (ϕ1 − 1)xn) ⊗ ϕ2(ym))+ n m + (−1) (h(xn ⊗ (dym−1 + (−1) (ϕ2 − 1)ym))+ n+m−1 + (−1) h(xn−1 ⊗ ϕ2(dym)) = m = h ◦ d(xn ⊗ ym−1) + (−1) h ◦ d(xn−1 ⊗ ϕ2(ym))+ n+m + (−1) (ψ3 − 1) ◦ h(xn ⊗ ym) and n b = h(dxn ⊗ ym) + (−1) h(xn ⊗ dym) = h ◦ d(xn ⊗ ym). On the other hand

d ◦ hT ((xn−1,xn) ⊗ (ym−1,ym)) = m = d(h(xn ⊗ ym−1) + (−1) h(xn−1 ⊗ ϕ2(ym)),h(xn ⊗ ym)) = m = (d ◦ h(xn ⊗ ym−1) + (−1) d ◦ h(xn−1 ⊗ ϕ2(ym))+ n+m−1 + (−1) (ψ3 − 1)h(xn ⊗ ym),d ◦ h(xn ⊗ ym)). Thus

(hT d + dhT )((xn−1,xn) ⊗ (ym−1,ym)) = m = ((hd + dh)(xn ⊗ ym−1) + (−1) (hd + dh)(xn−1 ⊗ ϕ2(ym)), (hd + dh)(xn ⊗ ym)) =

= ((α1(xn) ∪B α2(ym−1) − α3(xn ∪A ym−1))+ m (−1) (α1(xn−1) ∪B ϕ2(α2(ym)) − α3(xn−1 ∪A ϕ2(ym)),

α1(xn) ∪B α2(ym) − α3(xn ∪A ym)) = T T = (∪B ◦ (α1 ⊗ α2) − α3 ◦ ∪A)((xn−1,xn) ⊗ (ym−1,ym)). and the proposition is proved. • Proposition 1.1.7. Let Ai (1 6 i 6 4) be four complexes equipped with • • morphisms ϕi : Ai → Ai and such that • • • • • • a) The triples (A1,A2,A3) and (A1,A2,A4) satisfy A1-2). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 15

• • • b) The complexes Ai (i = 1,2) are equipped with morphisms Ti : Ai → Ai which commute with morphisms ϕi

Ti ◦ ϕi = ϕi ◦ Ti, i = 1,2. • • c) There exists a morphism T34 : A3 → A4 such that T34 ◦ ϕ3 = ϕ4 ◦ T34. d) The diagram • • ∪A • A1 ⊗ A2 / A3

s12◦(T1⊗T2) T34

• • ∪A • A2 ⊗ A1 / A4. commutes. • • • • • • • • Let Ti : T (Ai ) → T (Ai ) (i = 1,2) and T34 : T (A3) → T (A4) be the morphisms (which we denote again by the same letter) deﬁned by

Ti(xn−1,xn) = (Ti(xn−1),Ti(xn)), T34(xn−1,xn) = (T34(xn−1),T34(xn)). Then in the diagram

∪T • • • • A • • T (A1) ⊗ T (A2) / T (A3)

s12◦(T1⊗T2) T34 hT

∪T • • • • A ÑÙ • • T (A2) ⊗ T (A1) / T (A4) T T the maps T34 ◦ ∪A and ∪A ◦ s12 ◦ (T1 ⊗ T2) are homotopic. n • m • Proof. Let (xn−1,xn) ∈ T (A1) and (ym−1,ym) ∈ T (A2). Then T (6) T34((xn−1,xn) ∪A (ym−1,ym)) = m = (T34(xn ∪A ym−1) + (−1) T34(xn−1 ∪A ϕ2(ym)),T34(xn ∪A ym)) and T ∪A ◦s12 ◦ (T1 ⊗ T2)((xn−1,xn) ⊗ (ym−1,ym)) = mn = (−1) T2(ym−1,ym) ∪A T1(xn−1,xn) = mn n (7) = (−1) (T2(ym) ∪A T1(xn−1) + (−1) T2(ym−1) ∪A ϕ1(T1(xn)), T2(ym) ∪A T1(xn)) = m = ((−1) T34(xn−1 ∪A ym) + T34(ϕ1(xn) ∪A ym−1),T34(xn ∪A ym)). 16 DENIS BENOIS

Deﬁne k M n • m • k • hT : (T (A1) ⊗ T (A2)) → T (A4), m+n=k+1 by k n−1 (8) hT ((xn−1,xn) ⊗ (ym−1 ⊗ ym)) = (−1) (T34(xn−1 ∪A ym−1),0). Then

(9) dhT ((xn−1,xn) ⊗ (ym−1 ⊗ ym)) = n−1 = (−1) d(T34(xn−1 ∪A ym−1),0) = n−1 n−1 = (−1) (T34(dxn−1 ∪A ym−1 + (−1) xn−1 ∪A dym−1),0) = n−1 = ((−1) T34(dxn−1 ∪A ym−1) + T34(xn−1 ∪A dym−1),0), and

(10) hT d((xn−1,xn) ⊗ (ym−1 ⊗ ym)) = n = hT ((dxn−1 + (−1) (ϕ1 − 1)xn,dxn) ⊗ (ym−1,ym)+ n m + (−1) (xn−1,xn) ⊗ (dym−1 + (−1) (ϕ2 − 1)ym,dym)) = n = ((−1) T34(dxn−1 ∪A ym−1) + T34(ϕ1(xn) ∪A ym−1)−

− T34(xn ∪A ym−1) − T34(xn−1 ∪A dym−1)− m m − (−1) T34(xn−1 ∪A ϕ2(ym)) + (−1) T34(xn−1 ∪A ym),0). From (6)-(10) it follows that T T ∪A ◦ s12 ◦ (T1 ⊗ T2) − T34 ◦ ∪A = dhT + hT d and the proposition is proved. 1.2. Products.

1.2.1. In this subsection we review the construction of products for cones following Nekovárˇ [34] and Nizioł[35]. We will work with the following data P1) Diagrams • fi • gi • Ai −→ Ci ←− Bi , i = 1,2,3, • • • where Ai , Bi and Ci are complexes of R-modules. P2) Morphisms • • • ∪A : A1 ⊗ A2 → A3, • • • ∪B : B1 ⊗ B2 → B3, • • • ∪C : C1 ⊗C2 → C3. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 17

P3) A pair of homotopies h = (h f ,hg)

h f : ∪C ◦ ( f1 ⊗ f2) f3 ◦ ∪A, hg : ∪C ◦ (g1 ⊗ g2) g3 ◦ ∪B. Deﬁne

• • • fi−gi • (11) Ei = cone Ai ⊕ Bi −−−→ Ci [−1].

Thus n n n n−1 Ei = Ai ⊕ Bi ⊕Ci with d(an,bn,cn−1) = (dan,dbn,− fi(an) + gi(bn) − dcn−1). Proposition 1.2.2. i) Given the data P1-3), for each r ∈ R the formula

0 0 0 (an,bn,cn−1) ∪r,h (am,bm,cm−1) = 0 0 0 0 (an ∪A am,bn ∪B bm,cn−1 ∪C (r f2(am) + (1 − r)g2(bm)) + n 0 0 0 (−1) ((1 − r) f1(an) + rg1(bn)) ∪C cm−1 − (h f (an ⊗ am) − hg(bn ⊗ bm))) deﬁnes a morphism in K (R) • • • ∪r,h : E1 ⊗ E2 → E3 .

ii) If r1,r2 ∈ R, then the map • • • k : E1 ⊗ E2 → E3 [−1], given by

0 0 0 n 0 k((an,bn,cn−1) ⊗ (am,bm,cm−1)) = (0,0,(−1) (r1 − r2)cn−1 ∪C cm−1) n 0 0 0 m for all (an,bn,cn−1) ∈ E1 and (am,bm,cm−1) ∈ E2 , deﬁnes a homotopy k : ∪r1,h ∪r2,h. 0 0 0 iii) If h = (h f ,hg) is another pair of homotopies as in P3), and if α : 0 0 h f h f and β : hg hg is a pair of second order homotopies, then the map

• • • s : E1 ⊗ E2 → E3 [−1], 0 0 0 0 0 s((an,bn,cn−1) ⊗ (am,bm,cm−1)) = (0,0,α(an ⊗ am) − β(bn,bm)) deﬁnes a homotopy s : ∪r,h ∪r,h0 .

Proof. See [35], Proposition 3.1. 18 DENIS BENOIS

1.2.3. Assume that, in addition to P1-3), we are given the following data T1) Morphisms of complexes

• • TA : Ai → Ai , • • TB : Bi → Bi , • • TC : Ci → Ci , for i = 1,2,3. T2) Morphisms of complexes

0 • • • ∪A : A2 ⊗ A1 → A3, 0 • • • ∪B : B2 ⊗ B1 → B3, 0 • • • ∪C : C2 ⊗C1 → C3.

0 0 0 T3) A pair of homotopies h = (h f ,hg)

0 0 0 h f : ∪C ◦ ( f2 ⊗ f1) f3 ◦ ∪A, 0 0 0 hg : ∪C ◦ (g2 ⊗ g1) g3 ◦ ∪B.

T4) Homotopies

Ui : fi ◦ TA TC ◦ fi, Vi : gi ◦ TB TC ◦ gi, for i = 1,2,3. T5) Homotopies

0 tA : ∪A ◦ s12 ◦ (TA ⊗ TA) TA ◦ ∪A, 0 tB : ∪B ◦ s12 ◦ (TB ⊗ TB) TB ◦ ∪B, 0 tC : ∪C ◦ s12 ◦ (TC ⊗ TC) TC ◦ ∪C. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 19

T6) A second order homotopy Hf trivializing the boundary of the cube

• • ∪A • A1 ⊗ A2 / A3 TA

f3 TA⊗TA

tA 08 f1⊗ f2 ∪0 ◦s #• • A 12 • A1 ⊗ A2 / A3 2: h f f1⊗ f2 • • ∪C • f3 C1 ⊗C2 / C3 5= TC

(U1⊗U2)1 TC⊗TC 0 U × h f ◦s12 Ô 3 tC 08 #• • • C1 ⊗C2 0 / C3, ∪C◦s12

i i i i−2 i.e. a system Hf = (Hf )i∈Z of maps Hf : (A1 ⊗ A2) → C3 such that

dHf − Hf d = −tC ◦ ( f1 ⊗ f2) − TC ◦ h f +U3 ◦ ∪A+ 0 0 + f3 ◦tA + h f ◦ (s12 ◦ (TA ⊗ TA)) − (∪C ◦ s12) ◦ (U1 ⊗U2)1.

In this formula, (U1 ⊗U2)1 denotes the homotopy deﬁned by (2). T7) A second order homotopy Hg trivializing the boundary of the cube

• • ∪B • B1 ⊗ B2 / B3 TB

g3 TB⊗TB

tB 08 g1⊗g2 ∪0 ◦s #• • B 12 • B1 ⊗ B2 / B3 2: hg g1⊗g2 g • • ∪C • 3 C1 ⊗C2 / C3 5= TC

(V1⊗V2)1 TC⊗TC 0 V × hg◦s12 Ô 3 tC 08 #• • • C1 ⊗C2 0 / C3, ∪C◦s12 20 DENIS BENOIS

i i i i−2 i.e. a system Hg = (Hg)i∈Z of maps Hg : (B1 ⊗ B2) → C3 such that

dHg − Hgd = −tC ◦ (g1 ⊗ g2) − TC ◦ hg +V3 ◦ ∪B+ 0 0 + g3 ◦tB + hg ◦ (s12 ◦ (TB ⊗ TB)) − (∪C ◦ s12) ◦ (V1 ⊗V2)1. Proposition 1.2.4. i) Given the data P1-3) and T1-7), the formula

Ti(an,bn,cn−1) = (TA(an),TB(bn),TC(cn−1) +Ui(an) −Vi(bn)) deﬁnes morphisms of complexes

• • Ti : Ei → Ei , i = 1,2,3 such that, for any r ∈ R, the diagram

∪ • • r,h • E1 ⊗ E2 / E3

s12◦(T1⊗T2) T3

0 ∪ 0 • • 1−r,h • E2 ⊗ E1 / E3 commutes up to homotopy.

Proof. See [34], Proposition 1.3.6.

1.2.5. Bockstein maps. Assume that, in addition to P1-3), we are given the following data B1) Morphisms of complexes

• • • • • • βZ,i : Zi → Zi [1], Zi = Ai ,Bi ,Ci , i = 1,2. B2) Homotopies

ui : fi[1] ◦ βA,i βC,i ◦ fi, vi : gi[1] ◦ βB,i βC,i ◦ gi for i = 1,2. B3) Homotopies

hZ : ∪Z[1] ◦ (id ⊗ βZ,2) ∪Z[1] ◦ (βZ,1 ⊗ id), for Z• = A•,B•,C•. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 21

B4) A second order homotopy trivializing the boundary of the following diagram

β ⊗id • • A,1 • • A1 ⊗ A2 / A1[1] ⊗ A2 ∪A[1] f1[1]⊗ f2 id⊗βA,2

hA .6 f1⊗ f2 •$ • ∪A[1] # • A1 ⊗ A2[1] / A3[1] u1⊗ f2 f1⊗ f2[1] βC, ⊗id f [1] • • 1 Ô • • 3 C1 ⊗C2 / C1[1] ⊗C2 3; ∪C[1]

f ⊗u h f [1] 1 2 )1 id⊗ βC,2 h f [1] ~Ö hC .6 •$ • # • C1 ⊗C2[1] / C3[1]. ∪C[1]

B5) A second order homotopy trivializing the boundary of the cube

β ⊗id • • B,1 • • B1 ⊗ B2 / B1[1] ⊗ B2 ∪B[1] g1[1]⊗g2 id⊗βB,2

hB .6 g1⊗g2 •$ • ∪B[1] # • B1 ⊗ B2[1] / B3[1] v1⊗ f2 g1⊗g2[1] βC, ⊗id g [1] • • 1 Ô • • 3 C1 ⊗C2 / C1[1] ⊗C2 3; ∪C[1]

g ⊗v hg[1] 1 2 )1 id⊗ βC,2 hg[1] ~Ö hC .6 •$ • # • C1 ⊗C2[1] / C3[1]. ∪C[1]

Proposition 1.2.6. i) Given the data P1-3) and B1-5), the formula

βE,i(an,bn,cn−1) = (βA,i(an),βB,i(bn),−βC,i(cn−1) − ui(an) + vi(bn)) 22 DENIS BENOIS deﬁnes a morphism of complexes • • βE,i : Ei → Ei [1] such that for any r ∈ R the diagram

β ⊗id • • E,1 • • E1 ⊗ E2 / E1 [1] ⊗ E2

id⊗βE,2 ∪r,h[1] ∪ [1] • • r,h • E1 ⊗ E2 [1] / E3 [1] commutes up to homotopy. ii) Given the data P1-3), T1-7) and B1-5), for each r ∈ R the diagram

β ⊗id ∪ [1] • • E,1 • • r,h • T3[1] • E1 ⊗ E2 / E1 [1] ⊗ E2 / E3 [1] / E3 [1]

s12 id 0 β ⊗id ∪ 0 [1] • • E,2 • • T2[1]⊗T1 • • 1−r,h • E2 ⊗ E1 / E2 [1] ⊗ E1 / E2 [1] ⊗ E1 / E3 [1] is commutative up to a homotopy.

Proof. See [34], Propositions 1.3.9 and 1.3.10.

2. COHOMOLOGYOF (ϕ,ΓK)-MODULES

2.1. (ϕ,ΓK)-modules.

2.1.1. Throughout this section, K denotes a finite extension of Qp. Let kK be the residue field of K, OK its ring of integers and K0 the maximal unram- ur ified subfield of K. We denote by K0 the maximal unramified extension of ur K0 and by σ the absolute Frobenius acting on K0 . Fix an algebraic closure K of K and set GK = Gal(K/K). Let Cp be the p-adic completion of K. We denote by vp : Cp → R ∪ {∞} the p-adic valuation on Cp normalized so v (x) 1 p that vp(p) = 1 and set |x|p = p . Write A(r,1) for the p-adic annulus

A(r,1) = {x ∈ Cp | r 6 |x|p < 1}.

n n Fix a system of primitive p -th roots of unity ε = (ζp )n>0 such that p n cyc S∞ n cyc ζpn+1 = ζp for all n > 0. Let K = n=0 K(ζp ), HK = Gal(K/K ), cyc ∗ ΓK = Gal(K /K) and let χK : ΓK → Zp denote the cyclotomic character. Recall the constructions of some of Fontaine’s rings of p-adic periods. Deﬁne E+ = lim O / pO = {x = (x ,x ,...,x ,...) | xp = x , ∀i ∈ N}. e ←− Cp Cp 0 1 n i i x7→xp P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 23

+ Let x = (x0,x1,...) ∈ Ee . For each n, choose a liftx ˆn ∈ OCp of xn. Then, pn (m) pn for all m > 0, the sequencex ˆm+n converges to x = limn→∞ xˆm+n ∈ OCp , which does not depend on the choice of lifts. The ring Ee+, equipped with (0) the valuation vE(x) = vp(x ), is a complete local ring of characteristic p with residue field k¯K. Moreover, it is integrally closed in its field of fractions Ee = Fr(Ee+). Let Ae = W(Ee) be the ring of Witt vectors with coefficients in Ee. Denote by [·] : Ee → W(Ee) the Teichmüller lift. Each u = (u0,u1,...) ∈ Ae can be written in the form ∞ p−n n u = ∑ [un ]p . n=0 Set π = [ε]−1, A+ = Z [[π]] and denote by A the p-adic completion Qp p Qp of A+ [1/π] in A. Qp e

Let Be = Ae [1/p], BQp = AQp [1/p] and let B denote the completion of the maximal unramiﬁed extension of BQp in Be. All these rings are endowed with natural actions of the Galois group GK and the Frobenius operator ϕ, H and we set BK = B K . Note that

χK(τ) γ(π) = (1 + π) − 1, γ ∈ ΓK, ϕ(π) = (1 + π)p − 1. For any r > 0 deﬁne †,r pr Be = x ∈ Be | lim vE(xk) + k = +∞ . k→+∞ p − 1

†,r †,r †,r †,r † [ †,r † [ †,r Set B = B ∩ Be , BK = BK ∩ B , B = B and BK = BK . r>0 r>0 cyc Let L denote the maximal unramiﬁed subextension of K /Qp and let cyc cyc eK = [K : L ]. It can be shown (see [13]) that there exists rK > 0 and †,rK †,r πK ∈ BK such that for all r > rK the ring BK has the following explicit description

( †,r k BK = f (πK) = ∑ akπK | ak ∈ L and f is holomorphic k∈Z o and bounded on A(p−1/eKr,1) .

Note that, if K/Qp is unramiﬁed, L = K0 and one can take πK = π. 24 DENIS BENOIS

Deﬁne ( †,r k Brig,K = f (πK) = ∑ akπK | ak ∈ L and f is holomorphic k∈Z o on A(p−1/eKr,1) .

†,r †,r The rings BK and Brig,K are stable under ΓK, and the Frobenius ϕ sends †,r †,pr †,r †,pr BK into BK and Brig,K into Brig,K. The ring [ †,r RK = Brig,K r>rK is isomorphic to the Robba ring over L. Note that it is stable under ΓK and ϕ. As usual, we set ∞ n n+1 π t = log(1 + π) = ∑ (−1) ∈ RQp . n=1 n

Note that ϕ(t) = pt and γ(t) = χK(γ)t, γ ∈ ΓK. (r) †,r (r) To simplify notation, for each r > rK we set RK = Brig,K. The ring RK is equipped with a canonical Fréchet topology (see [8]). Let A be an affinoid algebra over Qp. Define (r) (r) (r) RK,A = A⊗b Qp RK , RK,A = ∪ RK,A. r>rK (r) If the field K is clear from the context, we will often write RA instead (r) RK,A and RA instead RK,A. (r) Definition. i) A (ϕ,ΓK)-module over RA is a finitely generated projective (r) (r) RA -module D equipped with the following structures: a) A ϕ-semilinear map (r) (r) (pr) D → D ⊗ (r) RA RA such that the induced linear map ∗ (r) (pr) (r) (pr) ϕ : D ⊗ (r) RA → D ⊗ (r) RA RA ,ϕ RA (pr) is an isomorphism of RA -modules; (r) b) A semilinear continuous action of ΓK on D . (r) ii) D is a (ϕ,ΓK)-module over RA if D = D ⊗ (r) RA for some (ϕ,ΓK)- RA (r) (r) module D over RA , with r > rK. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 25

∗ If D is a (ϕ,ΓK)-module over RA, we write D = HomRA (D,A) for the dual (ϕ,Γ)-module. Let Mϕ,Γ denote the ⊗-category of (ϕ,Γ )-modules RA K over RA.

2.1.2. A p-adic representation of GK with coefficients in an affinoid Qp- algebra A is a finitely generated projective A-module equipped with a continuous A-linear action of GK. Note that, as A is a noetherian ring, a finitely generated A-module is projective if and only if it is flat. Let RepA(GK) denote the ⊗-category of p-adic representations with coefficients in A. The relationship between p-adic representations and (ϕ,ΓK)-modules first ap- peared in the pioneering paper of Fontaine [18]. The key result of this theory is the following theorem.

Theorem 2.1.3 (FONTAINE,CHERBONNIER–COLMEZ,KEDLAYA). Let A be an afﬁnoid algebra over Qp. i) There exists a fully faithul functor D† : Rep (G ) → Mϕ,Γ, rig,A A K RA which commutes with base change. More precisely, let X = Spm(A). For each x ∈ X , denote by mx the maximal ideal of A associated to x and set E = A/m . If V (resp. D) is an object of Rep (G ) (resp. of Mϕ,Γ), set x x A Qp RA Vx = V ⊗A Ex (resp. Dx = D ⊗A Ex). Then the diagram

D† rig,A ϕ,Γ Rep (G ) / M A Qp RA

⊗Ex ⊗Ex D† rig,Ex ϕ,Γ Rep (GQ ) / M Ex p REx † † commutes, i.e., Drig,A(V)x ' Drig(Vx). † ii) If E is a ﬁnite extension of Qp, then the essential image of Drig,E is the subcategory of (ϕ,ΓK)-modules of slope 0 in the sense of Kedlaya [28] Proof. This follows from the main results of [18], [13] and [28]. See also [16]. † 2.1.4. Remark. Note that in general the essential image of Drig,A does not coincide with the subcategory of étale modules. See [11] [30], [25] for further discussion. 26 DENIS BENOIS

2.2. Relation to the p-adic Hodge theory.

2.2.1. In [18], Fontaine proposed to classify the p-adic representations arising in the p-adic Hodge theory in terms of (ϕ,ΓK)-modules (Fontaine’s program). More precisely, the problem is to recover classical Fontaine’s † functors DdR(V), Dst(V) and Dcris(V) (see for example [20]) from Drig(V). The complete solution was obtained by Berger in [8], [10]. His theory also allowed him to prove that each de Rham representation is potentially semistable. In this subsection, we review some of results of Berger. See also [15] for introduction and relation to the theory of p-adic differential equations. Let E be a fixed finite extension of Qp. Definition. i) A filtered module over K with coefficients in E is a free

K ⊗Qp E-module M of finite rank equipped with a decreasing exhaustive i filtration (Fil M)i∈Z. We denote by MFK,E the ⊗-category of such modules. ii) A filtered (ϕ,N)-module over K with coefficients in E is a free

K0 ⊗Qp E-module M of ﬁnite rank equipped with the following structures: i a) An exhaustive decreasing ﬁltration (Fil MK)i∈Z on MK = M ⊗K0 K; b) A σ-semilinear bijective operator ϕ : M → M;

c) A K0 ⊗Qp E-linear operator N such that N ϕ = pϕN. iii) A filtered ϕ-module over K with coefficients in E is a filtered (ϕ,N)- module such that N = 0. ϕ,N We denote by MFK,E the ⊗-category of filtered (ϕ,N)-module over K ϕ with coefficients in E and by MFK,E the category of filtered ϕ-modules. iv) If L/K is a finite Galois extension and GL/K = Gal(L/K), then a filtered (ϕ,N,GL/K)-module is a filtered (ϕ,N)-module M over L equipped i with a semilinear action of GL/K such that the filtration (Fil ML)i∈Z is stable under the action of GL/K. ur 0 v) We say that M is a filtered (ϕ,N,GK)-module if M = K0 ⊗L0 M , where 0 ϕ,N,GK M is a filtered (ϕ,N,GL/K)-module for some L/K. We denote by MFK,E the ⊗-category of (ϕ,N,GK)-modules. Let Kcyc((t)) denote the ring of formal Laurent power series with coefficients in Kcyc equipped with the filtration FiliKcyc((t)) = tiKcyc[[t]] and the action of ΓK given by ! k k k γ ∑ akt = ∑ γ(ak)χK(γ) t , γ ∈ ΓK. k∈Z k∈Z cyc The ring RK,E can not be naturally embedded in E ⊗Qp K ((t)), but for (r) any r > rK there exists a ΓK-equivariant embedding in : RK,E → E ⊗Qp P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 27

cyc t/pn K ((t)) which sends π to ζpn e − 1. Let D be a (ϕ,ΓK)-module over (r) RK,E and let D = D ⊗ (r) RK,E for some r > rK. Then RK,E

Γ cyc (r) K DdR/K(D) = E ⊗Qp K ((t)) ⊗in D

is a free E ⊗Qp K-module of ﬁnite rank equipped with a decreasing ﬁltration

Γ i i cyc (r) K Fil DdR/K(D) = E ⊗Qp Fil K ((t)) ⊗in D , which does not depend on the choice of r and n. Let RK,E[logπ] denote the ring of power series in variable logπ with coefﬁcients in RK,E. Extend the actions of ϕ and ΓK to RK,E[logπ] setting

ϕ(π) ϕ(logπ) = plogπ + log , π p γ(π) γ(logπ) = logπ + log , γ ∈ ΓK. π

p ( Note that log(ϕ(π)/π ) and log(τ(π)/π) converge in RK,E.) Deﬁne a monodromy operator N : RK,E[logπ] → RK,E[logπ] by

1 −1 d N = − 1 − . p d logπ

For any (ϕ,ΓK)-module D deﬁne

ΓK Dst/K(D) = D ⊗RK,E RK,E[logπ,1/t] , t = log(1 + π), N=0 ΓK Dcris/K(D) = Dst(D) = (D[1/t]) .

Then Dst(D) is a free E ⊗Qp K0-module of finite rank equipped with natural actions of ϕ and N such that Nϕ = pϕN. Moreover, it is equipped with a canonical exhaustive decreasing filtration induced by the embeddings in. If L/K is a finite extension and D is a (ϕ,ΓK)-module, the tensor product

DL = RL,E ⊗RK,E D has a natural structure of a (ϕ,ΓL)-module, and we deﬁne ( ) = ( ). Dpst/K D −→limDst/L DL L/K 28 DENIS BENOIS

ur Then Dpst/K(D) is a K0 -vector space equipped with natural actions of ϕ and N and a discrete action of GK. Therefore, we have four functors : Mϕ,Γ → MF , DdR/K RK,E K,E : Mϕ,Γ → MFϕ,N, Dst/K RK,E K,E : Mϕ,Γ → MFϕ,N,GK , Dpst/K RK,E K,E : Mϕ,Γ → MFϕ . Dcris/K RK,E K,E If the ﬁeld K is ﬁxed and understood from context, we will omit it and simply write DdR, Dst, Dpst and Dcris.

Theorem 2.2.2 (BERGER). Let V be a p-adic representation of GK. Then

D∗/K(V) ' D∗/K(V), ∗ ∈ {dR,st,pst,cris}. Proof. See [8]. For any (ϕ,ΓK)-module over RK,E one has

rkE⊗K0 Dcris/K(D) 6 rkE⊗K0 Dst/K(D) 6 rkE⊗K0 DdR/K(D) 6 rkRK,E (D). Deﬁnition. One says that D is de Rham (resp. semistable, resp. potentially semistable, resp. crystalline) if rkE⊗K0 DdR/K(D) = rkRK,E (D) (resp. rkE⊗K0 Dst/K(D) = rkRK,E (D), resp. rkE⊗K0 Dpst/K(D) = rkRK,E (D)), resp. rkE⊗K0 Dcris/K(D) = rkRK,E (D)). Let Mϕ,Γ , Mϕ,Γ and Mϕ,Γ denote the categories of semistable, RE ,st RE ,pst RE ,cris potentially semistable and crystalline (ϕ,Γ)-modules respectively. If D is i de Rham, the jumps of the ﬁltration Fil DdR(D) will be called the Hodge– Tate weights of D.

Theorem 2.2.3 (BERGER). i) The functors : Mϕ,Γ → MFϕ,N, Dst RK,E ,st K,E : Mϕ,Γ → MFϕ,N,GK , Dpst RK,E ,pst K,E : Mϕ,Γ → MFϕ Dcris RK,E ,cris K,E are equivalences of ⊗-categories. ii) Let D be a (ϕ,ΓK)-module. Then D is potentially semistable if and only if D is de Rham. Proof. These results are proved in [10]. See Theorem A, Theorem III.2.4 and Theorem V.2.3 of op. cit.. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 29

2.3. The complex Cϕ,γK (D).

2.3.1. We keep previous notation and conventions. Set ∆K = Gal(K(ζp)/K). 0 0 Then ΓK = ∆K × ΓK, where ΓK is a pro-p-group isomorphic to Zp. Fix a 0 topological generator γK of ΓK. For each (ϕ,ΓK)-module D over RA = RK,A deﬁne

−1 • ∆K γK ∆K CγK (D) : D −−−→ D ,

0 00 where the ﬁrst term is placed in degree 0. If D and D are two (ϕ,ΓK)- modules, we will denote by

• 0 • 00 • 0 00 ∪γ : CγK (D ) ⊗CγK (D ) → CγK (D ⊗ D ) the bilinear map  x ⊗ γn (y ) if x ∈ Cn (D0), y ∈ Cm (D00),  n K m n γK m γK ∪γ (xn ⊗ ym) = and n + m = 0 or 1,  0 if n + m > 2.

Consider the total complex

• • ϕ−1 • Cϕ,γK (D) = Tot CγK (D) −−→ CγK (D) .

More explicitly,

d • ∆K 0 ∆K ∆K d1 ∆K Cϕ,γK (D) : 0 → D −→ D ⊕ D −→ D → 0, where d0(x) = ((ϕ − 1)x,(γK − 1)x) and d1(x,y) = (γK − 1)x − (ϕ − 1)y. • Note that Cϕ,γK (D) coincides with the complex of Fontaine–Herr (see [26], ∗ • 0 [27] [31]). We will write H (D) for the cohomology of Cϕ,γ (D). If D 00 and D are two (ϕ,ΓK)-modules, the cup product ∪γ induces, by Proposi- tion 1.1.5, a bilinear map

• 0 • 00 • 0 00 ∪ϕ,γ : Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ).

Explicitly

m ∪ϕ,γ ((xn−1,xn)⊗(ym−1,ym)) = (xn ∪γ ym−1 +(−1) xn−1 ∪γ ϕ(ym),xn ∪γ ym), 30 DENIS BENOIS

n 0 n−1 0 n 0 m 00 if (xn−1,xn) ∈Cϕ,γK (D ) =CγK (D )⊕CγK (D ) and (ym−1,ym) ∈Cϕ,γ (D ) = m−1 00 m 00 Cγ (D ) ⊕Cγ (D ). An easy computation gives the following formulas ( 0 0 0 00 0 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), x0 ⊗ y0 7→ x0 ⊗ y0,

( 0 0 1 00 1 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), x0 ⊗ (y0,y1) 7→ (x0 ⊗ y0,x0 ⊗ y1),

( 1 0 0 00 1 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), (x0,x1) ⊗ y0 7→ (x0 ⊗ ϕ(y0),x1 ⊗ γK(y0)),

( 1 0 1 00 2 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), (x0,x1) ⊗ (y0,y1) 7→ (x1 ⊗ γK(y0) − x0 ⊗ ϕ(y1)),

( 0 0 2 00 2 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), x0 ⊗ y1 7→ x0 ⊗ y1,

( 2 0 0 00 2 0 00 Cϕ,γK (D ) ⊗Cϕ,γK (D ) → Cϕ,γK (D ⊗ D ), x1 ⊗ y0 7→ x1 ⊗ γK(ϕ(y1)). Here the zero components are omitted.

2.3.2. For each (ϕ,ΓK)-module D we denote by h • i RΓ(K,D) = Cϕ,γK (D) the corresponding object of the derived category D(A). The cohomology of D is deﬁned by i i i • H (D) = R Γ(K,D) = H (Cϕ,γK (D)), i > 0. There exists a canonical isomorphism in D(A)

TRK : τ>2RΓ(K,RA(χK)) ' A[−2] ( see [27], [31], [30]). Therefore, for each D we have morphisms

L ∗ ∪ϕ,γ ∗ (12) RΓ(K,D) ⊗A RΓ(K,D (χK)) −−→ RΓ(K,D ⊗ D (χK)) duality −−−−→ RΓ(K,RA(χK)) → τ>2RΓ(K,RA(χK)) ' A[−2].

Theorem 2.3.3 (KEDLAYA–POTTHARST–XIAO). Let D be a (ϕ,ΓK)-module over RK,A, where A is an afﬁnoid algebra. [0,2] i) Finiteness. We have RΓ(K,D) ∈ Dperf (A). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 31

ii) Euler–Poincaré characteristic formula. We have

2 i i ∑(−1) rkAH (D) = −[K : Qp],rkRK,A (D). i=0 iii) Duality. The morphism (12) induces an isomorphism ∗ RΓ(K,D ) ' RHomA(RΓ(K,D),A). In particular, we have cohomological pairings i 2−i ∗ 2 ∪ : H (D) ⊗ H (D (χK)) → H (RA(χK)) ' A, i ∈ {0,1,2}. iv) Comparision with Galois cohomology. Let V is a p-adic representation of GK with coefﬁcients in A. There exist canonical (up to the choice of γK) and functorial isomorphisms i ∼ i † H (K,V) → H (Drig(V)) which are compatible with cup-products. In particular, we have a commutative diagram 2 TRK H (RA(χK)) / A

' = 2 invK H (K,A(χK)) / A, where invK is the canonical isomorphism of the local class ﬁeld theory. Proof. See Theorem 4.4.5 of [30] and Theorem 2.8 of [37].

2.3.4. Remark. The explicit construction of the isomorphism TRK is given in [27] and [3], Theorem 2.2.6. 2.4. The complex K•(V). 2.4.1. In this section, we give the derived version of isomorphisms Hi K V ∼ Hi † V C• V ( , ) → (Drig( )) of Theorem 2.3.3 iv). We write ϕ,γK ( ) instead C• † V K M ϕ,γK (Drig( )) to simplify notation. Let be a finite extension of Qp. If • is a topological GK-module, we denote by C (GK,M) the complex of continuous cochains with coefficients in M. Let V be a p-adic representation of GK with coefficients in an affinoid algebra A. Then • [0,2] C (GK,V) ∈ Kft (A) and for the associated object RΓ(K,V) of the derived category • [0,2] RΓ(K,V) = [C (GK,V)] ∈ Dperf (A) (see [37], Theorem 1.1). 32 DENIS BENOIS

The continuous Galois cohomology of GK with coefﬁcients in V is de- ﬁned by ∗ ∗ • H (K,V) = H (C (GK,V)). †,r In [8], Berger constructed, for each r > rK, a ring Berig which is the com- †,r †,r †,r pletion of B with respect to Frechet topology. Set Berig,A = Berig ⊗b Qp A and † †,r Berig,A = ∪ Berig,A. For each r > rK we have an exact sequence r>rK

†,r ϕ−1 †,rp 0 → Qp → Berig −−→ Berig → 0 (see [9], Lemma I.7). Since the completed tensor product by an orthonor- malizable Banach space is exact in the category of Frechet spaces (see, for example, [1], proof of Lemma 3.9), the sequence

†,r ϕ−1 †,rp 0 → A → Berig,A −−→ Berig,A → 0. is also exact. Passing to the direct limit we obtain an exact sequence

† ϕ−1 † (13) 0 → A → Berig,A −−→ Berig,A → 0.

† † • † Set Vrig = V ⊗A Berig,A and consider the complex C (GK,Vrig). Then (13) induces an exact sequence

• • † ϕ−1 • † 0 → C (GK,V) → C (GK,Vrig) −−→ C (GK,Vrig) → 0. Deﬁne

• • • † • † ϕ−1 • † K (V) = T (C (GK,Vrig)) = Tot C (GK,Vrig) −−→ C (GK,Vrig) . Consider the map C• V C• G V † αV : γK ( ) → ( K, rig) deﬁned by  α (x ) = x , x ∈ C0 (V),  V 0 0 0 γK κ(g) γK − 1 1 αV (x1)(g) = (x1), x1 ∈ CγK (V),  γK − 1 κ(g) where g ∈ GK and γ = g| 0 . It is easy to check that αV is a morphism of K ΓK complexes which commutes with ϕ. By fonctoriality, we obtain a morphism (which we denote again by αV ): • • αV : Cϕ,γK (V) → K (V). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 33

• • Proposition 2.4.2. The map αV : Cϕ,γK (V) → K (V) and the map • • ξV : C (GK,V) → K (V), n xn 7→ (0,xn), xn ∈ C (GK,V) are quasi-isomorphisms and we have a diagram

• ξV • C (GK,V) ' / K (V) O ' αV

• Cϕ,γK (V).

Proof. This is Proposition 9 of [6].

2.4.3. If M and N are two Galois modules, the cup-product • • • ∪c : C (M) ⊗C (M) → C (M ⊗ N) deﬁned by

(xn ∪c ym)(g1,g2,...,gn+m) =

= xn(g1,...,gn) ⊗ (g1g2 ···gn)ym(gn+1,...,gn+m), n m where xn ∈ C (GK,M) and ym ∈ C (GK,N), is a morphism of complexes. Let V and U be two Galois representations of GK. Applying Proposition 1.1.5 • † • † to the complexes C (GK,Vrig) and C (GK,Urig) we obtain a morphism • • • ∪K : K (V) ⊗ K (U) → K (V ⊗U). The following proposition will not be used in the remainder of this paper, but we state it here for completeness. Proposition 2.4.4. In the diagram

∪ • • ϕ,γK • Cϕ,γK (V) ⊗Cϕ,γK (U) / Cϕ,γ (V ⊗U)

αV ⊗αU αV⊗U hϕ,γ

∪K K•(V) ⊗ K•(U) }Õ / K•(V ⊗U) the maps αV⊗U ◦ ∪ϕ,γ and ∪K ◦ (αV ⊗ αU ) are homotopic. We need the following lemma. 34 DENIS BENOIS

For any x C1 V , y C1 U let c C1 0 † V Lemma 2.4.5. ∈ γK ( ) ∈ γK ( ), x,y ∈ (ΓK,Drig( ⊗ U)) denote the 1-cochain deﬁned by

n−1 n i+1 ! n i γK − γK (14) cx,y(γK) = ∑ γK(x) ⊗ (y), if n 6= 0,1, i=0 γK − 1 and cx,y(1) = cx,y(γK) = 0. Then 1 0 i) For each x ∈ Cϕ,γK (V) and y ∈ Cϕ,γK (U)

cx,(γK−1)y = αV (x) ∪c αU (y) − αV⊗U (x ∪γ y).

0 1 ii) If x ∈ Cϕ,γK (V) and y ∈ Cϕ,γK (U) then

c(γK−1)x,y = αV⊗U (x ∪γ y) − αV (x) ∪c αU (y). iii) One has 1 d (cx,y) = −αV (x) ∪c αU (y).

0 Proof of the lemma. i) Note that ΓK is the proﬁnite completion of the cyclic group hγKi, and an easy computation shows that the map cx,y, deﬁned on 0 hγKi by (14), extends by continuity to a unique cochain on ΓK which we denote again by cx,y. For any natural n 6= 0,1 one has

n−1 n i n i+1 cx,(γ−1)y(γK) = ∑ γK(x) ⊗ (γK − γK )(y) = i=0 n−1 n−1 i n i i+1 = ∑ γK(x) ⊗ γK(y) − ∑ γK(x) ⊗ γK (y) = i=0 i=0 n n γK − 1 n γK − 1 = (x) ⊗ γK(y) − (x ⊗ γK(y)) = γK − 1 γK − 1 n n = (gV (x) ∪c gU (y))(γK) − (gV⊗U (x ∪γ y))(γK).

By continuity, this implies that cx,(γK−1)y = αV (x)∪c αU (y)−αV⊗U (x∪γ y), and i) is proved. ii) An easy induction proves the formula

m i+1 i γK − 1 (15) ∑γK(γK − 1)(x) ⊗ (y) = i=0 γK − 1 m+1 m+1 m+1 γK − 1 γK − 1 = γK (x) ⊗ (y) − (x ⊗ y). γK − 1 γK − 1 P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 35

Therefore n−1 γn − γi+1 c ( n ) = ( i+1 − i )(x) ⊗ K K (y) = (γK−1)x,y γK ∑ γK γK i=0 γK − 1 n−1 n n−1 i+1 i+1 i γK − 1 i γK − 1 = ∑ (γK − γK)(x) ⊗ (y) − ∑ γK(γK − 1)(x) ⊗ (y) = i=0 γK − 1 i=0 γK − 1 n n n by (15) n γK − 1 γK − 1 n γK − 1 = (γK − 1)(x) ⊗ (y) + (x ⊗ y) − γK(x) ⊗ (y) = γK − 1 γK − 1 γK − 1 γn − 1 γn − 1 = K (x ⊗ y) − x ⊗ K (y) = γK − 1 γK − 1 n n = (αV⊗U (x ∪γ y))(γK) − (αV (x) ∪c αU (y))(γK), and ii) is proved. iii) One has 1 n m n m n+m n d cx,y(γK,γK ) = γKcx,y(γK ) − cx,y(γK ) + cx,y(γK) = m−1 n+m i+n+1 n+i γK − γK = ∑ γK (x) ⊗ (y) − i=0 γK − 1 n+m−1 n+m i+1 n−1 n i+1 i γK − γK i γK − γK − ∑ γK(x) ⊗ (y) + ∑ γK(x) ⊗ (y) = i=0 γK − 1 i=0 γK − 1 n−1 n+m i+1 n−1 n i+1 i γK − γK i γK − γK = − ∑ γK(x) ⊗ (y) + ∑ γK(x) ⊗ (y) = i=0 γK − 1 i=0 γK − 1 n−1 n n+m n m i γK − γK γK − 1 n γK − 1 = ∑ γK(x) ⊗ (y) = − (x) ⊗ γK (y) = i=0 γK − 1 γK − 1 γK − 1 n m = −(αV (x) ∪c αU (y))(γK,γK ). 1 By continuity, d cx,y = −αV (x) ∪c αU (y), and the lemma is proved. Proof of Proposition 2.4.4. Let h C• V C• U C• G V † U† γ : γK ( ) ⊗ γK ( ) → ( K, rig ⊗ rig)[−1] be the map deﬁned by ( −c if x ∈ C1 (V), y ∈ C1 (U), h (x,y) = x,y γK γK γ 0 elsewhere.

From Lemma 2.4.5 it follows that hγ deﬁnes a homotopy hγ : αV⊗U ◦ ∪γ ∪c ◦ (αV ⊗ αU ). By Proposition 1.1.6, hγ induces a homotopy hϕ,γ : αV⊗U ◦ ∪ϕ,γ ∪K ◦ (αV ⊗ αU ). The proposition is proved. 36 DENIS BENOIS

2.5. Transpositions.

• 2.5.1. Let M be a continuous GK-module. The complex C (GK,M) is equipped with a transposition

• • TV,c : C (GK,M) → C (GK,M) which is deﬁned by

n(n+1)/2 −1 −1 TV,c(xn)(g1,g2,...,gn) = (−1) g1g2 ···gn(xn(g1 ,...,gn ))

(see [34], Section 3.4.5.1). We will often write Tc instead TV,c. The map Tc satisﬁes the following properties 2 a) Tc is an involution, i.e. Tc = id. b) Tc is functorially homotopic to the identity map. ∗ • • c) Let s12 : C (GK,M ⊗N) → C (CK,N ⊗M) denote the map induced by the involution M ⊗ N → N ⊗ M given by x ⊗ y 7→ y ⊗ x (see Section 1.1.1). ∗ n m Set T12 = Tc ◦ s12. Then for all xn ∈ C (GK,M) and ym ∈ C (GK,N) one has nm T12(xn ∪ ym) = (−1) (Tcym) ∪ (Tcxn), i.e. the diagram

• • ∪c • (16) C (GK,M) ⊗C (GK,N) / C (GK,M ⊗ N)

s12 T12 • • ∪c • C (GK,N) ⊗C (GK,M) / C (GK,N ⊗ M) commutes ([34], Section 3.4.5.3).

2.5.2. There exists a homotopy

n (17) a = (a ) : id Tc which is functorial in M ([34], Section 3.4.5.5). We remark, that from the discussion in op. cit. it follows, that one can take a such that a0 = a1 = 0.

2.5.3. Let V be a p-adic representation of GK. We denote by TK(V), or • simply by TK, the transposition induced on the complex K (V) by Tc, thus

TK(V)(xn−1,xn) = (Tc(xn−1),Tc(xn)). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 37

From Proposition 1.1.7 it follows that in the diagram

∪K (18) K•(V) ⊗ K•(U) / K•(V ⊗U)

∗ s12◦(TK(V)⊗TK(U)) TK(V⊗U)◦s12 hT

∪K K•(U) ⊗ K•(V) ~Ö / K•(U ⊗V) ∗ the morphisms TK(V⊗U) ◦ s12 ◦ ∪K and ∪K ◦ s12 ◦ (TK(V) ⊗ TK(U)) are homotopic. Proposition 2.5.4. i) The diagram

• ξV • C (GK,V) / K (V)

Tc TK(V) • ξV • C (GK,V) / K (V). is commutative. The map aK(V) = (a,a) deﬁnes a homotopy aK(V) : idK(V) TK(V) such that aK(V) ◦ ξV = ξV ◦ a. ii) We have a commutative diagram

• αV • Cϕ,γK (V) / K (V)

id TK • αV • Cϕ,γK (V) / K (V). 0 1 If a : id Tc is a homotopy such that a = a = 0, then aK(V) ◦ αV = 0. Proof. i) The ﬁrst assertion follows from Lemma 1.1.6. x C1 V x C• G V † ii) If 1 ∈ γK ( ) then αV ( 1) ∈ ( K, rig) satisﬁes −1 Tc(αV (x1))(g) = −g(αV (x1)(g )) = −κ(g) ! κ(g) κ(g) γK − 1 γK − 1 = −γK (x1) = (x1) = (αV (x1))(g). γK − 1 γK − 1

Thus Tc ◦ αV = αV . By functoriality, TK ◦ αV = αV . Finally, the identity aK(V) ◦αV = 0 follows directly from the deﬁnition of ξV and the assumption 0 1 that a = a = 0. 38 DENIS BENOIS

2.6. The Bockstein map. 0 0 0 2.6.1. Consider the group algebra A[ΓK] of ΓK over A. Let ι : A[ΓK] → 0 −1 0 A[ΓK] denote the A-linear involution given by ι(γ) = γ , γ ∈ ΓK. We equip 0 A[ΓK] with the following structures: 0 a) The natural Galois action given by g(x) = gx¯ , where g ∈ GK, x ∈ A[ΓK] 0 andg ¯ is the image of g under canonical projection of GK → ΓK. 0 0 ι b) The A[ΓK]-module structure A[ΓK] given by the involution ι, namely 0 0 ι λ(x) = ι(λ)x for λ ∈ A[ΓK], x ∈ A[ΓK] . 0 Let JK denote the kernel of the augmentation map A[ΓK] → A. Then the element −1 2 2 Xe = log (χK(γ))(γ − 1)(mod JK) ∈ JK/JK 0 does not depend on the choice of γ ∈ ΓK and we have an isomorphism of A-modules 2 2 θK : A → JK/JK,

θK(a) = aXe. ι 0 ι 2 The action of GK on the quotient AeK = A[ΓK] /JK is given by

g(1) = 1 + log(χK(g))Xe, g ∈ GK.

We have an exact sequence of GK-modules

θK ι (19) 0 → A −→ AeK → A → 0.

Let V be a p-adic representation of GK with coefﬁcients in A. Set VeK = ι V ⊗A AeK. Then the sequence (19) induces an exact sequence of p-adic representations 0 → V → VeK → V → 0. Therefore, we have an exact sequence of complexes • • • 0 → C (GK,V) → C (GK,VeK) → C (GK,V) → 0 which gives a distinguished triangle

(20) RΓ(K,V) → RΓ(K,VeK) → RΓ(K,V) → RΓ(K,V)[1]. ι 2 The map s : A → AeK that sends a to a (mod JK) induces a canonical non- equivariant section sV : V → VeK of the projection VeK → V. Deﬁne a mor- • • phism βV,c : C (GK,V) → C (GK,V)[1] by

1 • βV,c(xn) = (d ◦ sV − sV ◦ d)(xn), xn ∈ C (GK,V). Xe We will write βc instead βV,c if the representation V is clear from the context. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 39

Proposition 2.6.2. i) The distinguished triangle (20) can be represented by the following distinguished triangle of complexes

• • • βV,c • C (GK,V) → C (GK,VeK) → C (GK,V) −−→ C (GK,V)[1]. n ii) For any xn ∈ C (GK,V) one has

βV,c(xn) = −log χK ∪c xn. Proof. See [34], Lemma 11.2.3. • 2.6.3. We will prove analogs of this proposition for the complexes Cϕ,γK (D) • and K (V). Let D be a (ϕ,ΓK)-module with coefﬁcients in A. Set De = ι D ⊗A AeK. The splitting s induces a splitting of the exact sequence

sD (21) 0 / D / De o / D / 0 which we denote by sD. Deﬁne • • (22) βD : Cϕ,γK (D) → Cϕ,γK (D)[1], 1 n βD(x) = (d ◦ sD − sD ◦ d)(x), x ∈ Cϕ,γK (D). Xe n Proposition 2.6.4. i) The map βD induces the connecting maps H (D) → Hn+1(D) of the long cohomology sequence associated to the short exact sequence (21). n ii) For any x ∈ Cϕ,γK (D) one has

βD(x) = −(0,log χK(γK)) ∪ϕ,γ x, 1 where (0,log χK(γK)) ∈ Cϕ,γK (Qp(0)). Proof. The ﬁrst assertion follows directly from the deﬁnition of the con- n necting map. Now, let x = (xn−1,xn) ∈ Cϕ,γK (D). Then (dsD − sDd)(x) = n = d(xn−1 ⊗ 1,xn ⊗ 1) − sD((γK − 1)xn−1 + (−1) (ϕ − 1)xn,(γK − 1)xn) = n = (γK(xn−1) ⊗ γK − xn−1 ⊗ 1 + (−1) (ϕ − 1)xn ⊗ 1,γK(xn) ⊗ γK − xn ⊗ 1)− n − ((γK − 1)(xn−1) ⊗ 1 + (−1) (ϕ − 1)xn ⊗ 1,(γK − 1)(xn) ⊗ 1) =

= (γK(xn−1) ⊗ (γK − 1),γK(xn) ⊗ (γK − 1)). −1 2 From γK = 1+Xe log χK(γK) it follows that γK −1 ≡ −Xe log χK(γK)(mod JK) and we obtain 1 βD(x) = ((γK(xn−1),γK(xn)) ⊗ (γK − 1))) = Xe n+1 = −log χK(γK)(γK(xn−1),γK(xn)) ∈ Cϕ,γK (D). 40 DENIS BENOIS

On the other hand,

(0,log χK(γK)) ∪ϕ,γ (xn−1,xn) = log χK(γK)(γK(xn−1),γK(xn)) and ii) is proved. The exact sequence • † • † • † 0 → C (GK,Vrig) → C (GK,(VeK)rig) → C (GK,Vrig) → 0, induces an exact sequence • • • (23) 0 → K (V) → K (VeK) → K (V) → 0. • • Again, the splitting sV : V → VeK induces a splitting sK : K (V) → K (VeK) of (23) and we have a distinguished triangle of complexes

• • • βK(V) • K (V) → K (Ve) → K (V) −−−→ K (V)[1].

We will often write βK instead βK(V). Proposition 2.6.5. i) One has n βK(x) = −(0,log χK) ∪K x, x ∈ K (V). ii) The following diagrams commute

β D† (V) • βc • • rig • C (GK,V) / C (GK,V)[1] , Cϕ,γK (V) / Cϕ,γK (V)[1]

ξV ξV [1] αV αV [1] • βK • βK K (V) / K (V)[1] K•(V) / K•(V)[1].

n Proof. i) The proof is a routine computation. Let x = (xn−1,xn) ∈ K (V), n−1 † n † where xn−1 ∈ C (GK,Vrig), xn ∈ C (GK,Vrig). Since sK commutes with ϕ one has

(dsK − sKd)x = ((dsV − sV d)xn−1,(dsV − sV d)xn). On the other hand,

((dsV − sV d)xn−1)(g1,g2,...,gn) = g1xn−1(g2,...,gn) ⊗ (g¯1 − 1), whereg ¯1 denote the image of g1 ∈ GK in ΓK. As in the proof of Proposi- 2 tion 2.6.4, we can writeg ¯1 − 1 (mod JK) = Xe log χK(g1). Therefore

(d ◦ sV − sV ◦ d)xn−1(g1,g2,...,gn) = log χK(g1)g1xn−1(g2,...,gn) ⊗ Xe. and

(d ◦ sV − sV ◦ d)xn(g1,g2,...,gn,gn+1) =

= log χK(g1)g1xn−1(g2,...,gn,gn+1) ⊗ Xe. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 41

Since ι(g1 − 1) = −Xe log χK(g1), we have 1 βK(x)(g1,...,gn) = (d ◦ sK − sK ◦ d)x(g1,g2,...,gn) = Xe = −log χK(g1)(g1xn−1(g2,...,gn,gn),g1xn−1(g2,...,gn,gn+1)).

On the other hand, (0,log χK) ∪K (xn−1,xn) = (zn,zn+1), where

zi(g1,g2,...,gi) = log χK(g1)g1xi(g2,...,gi), i = n, n + 1, and i) is proved. ii) The second statement follows from the compatibility of the Bockstein morphisms βc, β † and βK with the maps αV and βV . This can be also Drig(V) proved using i) and Propositions 2.6.2 and 2.6.4. 2.7. Iwasawa cohomology.

cyc ∆ 2.7.1. We keep previous notation and conventions. Set K∞ = (K ) K , 0 where ∆K = Gal(K(ζp)/K). Then Gal(K∞/K) ' ΓK and we denote by Kn n the unique subextension of K∞ of degree [Kn : K] = p . Let E be a finite extension of Qp and let OE be its ring of integers. We denote by 0 0 ΛOE = OE[[ΓK]] the Iwasawa algebra of ΓK with coefficients in OE. The 0 choice of a generator γK of ΓK fixes an isomorphism ΛOE ' OE[[X]] such that γK 7→ X + 1. Let HE denote the algebra of formal power series f (X) ∈ E[[X]] that converge on the open unit disk A(0,1) = {x ∈ Cp | |x|p < 1} and let 0 HE(ΓK) = { f (γK − 1) | f (X) ∈ HE}. 0 We consider ΛOE as a subring of HE(ΓK). The involution ι : ΛOE → ΛOE extends to (Γ0 ). Let Λ ι (resp. (Γ0 )ι ) denote Λ (resp. (Γ0 )) HE K OE HE K OE HE K 0 equipped with the ΛOE -module (resp. HE(ΓK)-module) structure given by α ? λ = ι(α)λ. Let V be a p-adic representation of GK with coefficients in E. Fix a OE- lattice T of V stable under the action of GK and set IndK∞/K(T) = T ⊗OE Λ ι . We equip Ind (T) with the following structures: OE K∞/K a) The diagonal action of GK, namely g(x ⊗ λ) = g(x) ⊗ g¯λ, for all g ∈

GK and x ⊗ λ ∈ IndK∞/K(T);

b) The structure of ΛOE -module given by α(x ⊗ λ) = x ⊗ λι(α) for all

α ∈ ΛOE and x ⊗ λ ∈ IndK∞/K(T). • Let RΓIw(K,T) denote the class of the complex C (GK,IndK∞/K(T)) in the derived category D(ΛOE ) of ΛOE -modules. The augmentation map

ΛOE → OE induces an isomorphism L RΓIw(K,T) ⊗ OE ' RΓ(K,T). ΛOE 42 DENIS BENOIS

i i We write HIw(K,T) = R ΓIw(K,T) for the cohomology of RΓIw(K,T). From Shapiro’s lemma it follows that Hi (K,T) = Hi(K ,T) Iw ←−lim n cores (see, for example, [34], Sections 8.1-8.3). We review the Iwasawa cohomology of (ϕ,ΓK)-modules (see [14] and †,r †,pr †,pr [30]). The map ϕ : Brig,K → Brig,K equips Brig,K with the structure of a free †,r ϕ : Brig,K-module of rank p. Deﬁne

†,pr †,r 1 −1 ψ : Brig,K → Brig,K, ψ(x) = ϕ ◦ Tr †,pr †,r (x). p Brig,K/ϕ(Brig,K) †,r Since RK,Qp = ∪ Brig,K, the operator ψ extends by linearity to an operator r>rK ψ : RK,E → RK,E such that ψ ◦ ϕ = id.

Let D is a (ϕ,ΓK)-module over RK,E = RK ⊗Qp E. If e1,e2,...,ed is a base of D over RK,E, then ϕ(e1),ϕ(e2),...,ϕ(ed) is again a base of D, and we deﬁne ψ : D → D, d ! d ψ ∑aiϕ(ei) = ∑ψ(ai)ei. i=1 i=1

0 ∆K 0 The action of ΓK on D extends to a natural action of HE(ΓK) and we 0 consider the complex of HE(ΓK)-modules −1 • ∆K ψ ∆K CIw(D) : D −−−→ D , • where the terms are concentrated in degrees 1 and 2. Let RΓIw(D) = CIw(D) • 0 denote the class of CIw(D) in the derived category D(HE(ΓK)). We also • consider the complex Cϕ,γK (IndK∞/K(D)), where IndK∞/K(D) = 0 ι h • i D ⊗E HE(ΓK) , and set RΓ(K,IndK∞/K(D)) = Cϕ,γK (D) .

Theorem 2.7.2. Let D be a (ϕ,ΓK)-module over RK,E. Then • i) The complexes CIw(D) and Cϕ,γK (D) are quasi-isomorphic and therefore

RΓIw(D) ' RΓ(K,IndK∞/K(D)). i i ii) The cohomology groups HIw(D) = R ΓIw(D) are finitely-generated 0 1 HE(Γ )-modules. Moreover, rk 0 H (D) = [K : Qp]rk D and K HE (ΓK) Iw RK,E 1 2 HIw(D)tor and HIw(D) are finite-dimensional E-vector spaces. iii) We have an isomorphism • ∼ • C (IndK /K(D)) ⊗ 0 E → C (D) ϕ,γK ∞ HE (ΓK) ϕ,γK P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 43 which induces the Hochschild–Serre exact sequences i i i+1 Γ0 0 → H (D) 0 → H (D) → H (D) K → 0. Iw ΓK Iw 0 0 0 0 iv) Let ω = cone KE(ΓK) → KE(ΓK)/HE(ΓK) [−1], where KE(ΓK) 0 is the field of fractions of HE(Γ ). Then the functor D = Hom 0 (−,ω) K HE (ΓK) furnishes a duality ∗ ι DRΓIw(D) ' RΓIw(D (χK)) [2].

v) If V is a p-adic representation of GK, then there are canonical and functorial isomorphisms

L 0 0 ι RΓIw(K,T) ⊗ HE(ΓK) ' RΓ K,T ⊗O HE(ΓK) ' ΛOE E † ' RΓ(K,IndK∞/K(Drig(V))). Proof. See [38], Theorem 2.6. 2.7.3. We will need the following lemma.

Lemma 2.7.4. Let E be a finite extension of Qp and let D be a potentially semistable (ϕ,ΓK)-module over RK,E. Then =1 1 ∆K ϕ i) HIw(D)tor ' D . ii) Assume that ∗ ϕ=pi Dpst(D (χK)) = 0, ∀i ∈ Z. 2 Then HIw(D) = 0. Proof. i) Consider the exact sequence ϕ−1 0 → Dϕ=1 → Dψ=1 −−→ Dψ=0. =1 ∆K ψ 1 ψ=0 0 Since D ' HIw(D) and D is HE(ΓK)-torsion free ([30], Theo- =1 1 ∆K ϕ ϕ=1 rem 3.1.1), HIw(D)tor ⊂ D . On the other hand, D is a finitely dimensional E-vector space (see, for example [30], Lemma 4.3.5) and there- 0 fore is HE(ΓK)-torsion. This proves the first statement. 2 1 ∗ ii) By Theorem 2.7.2 iv), HIw(D) and HIw(D (χK))tor are dual to each ∗ ϕ=1 ∗ ϕ=1 other and it is enough to show that D (χK) = 0. Since dimE D (χK) < ∗ ϕ=1 ∗ (r) ∞, there exists r such that D (χK) ⊂ D (χK) , and for n 0 the map −n (r) cyc in = ϕ : RK,E → E ⊗Qp K [[t]] gives an injection

∗ ϕ=1 ∗ (r) cyc ∼ D (χK) → D (χK) ⊗in E ⊗Qp K [[t]] → ∼ 0 ∗ cyc → Fil (DdR(D (χK)) ⊗K ⊗K ((t))). 44 DENIS BENOIS

0 ∗ cyc Looking at the action of ΓK on Fil (DdR(D (χK)) ⊗K K ((t))) and using ∗ ϕ=1 the fact that D (χK) is finite-dimensional over E, it is easy to prove, ∗ ϕ=1 that there exists a finite extension L/K such that D (χK) , viewed as GL-module, is isomorphic to a finite direct sum of modules Qp(i), i ∈ Z. Therefore ∗ ϕ=1 ∗ ϕ=1 ΓL D (χK) ' D (χK) ⊗Qp Qp(−i) ⊗Qp Qp(i) as GL-modules. Since −i ∗ ϕ=1 ΓL ∗ ϕ=p ,ΓL D (χK) ⊗Qp Qp(−i) ⊂ D (χK) ⊗RK,E RL,E[1/t,`π ] = L ∗ ϕ=p−i = Dst(D (χK)) = 0, ∗ ϕ=1 we obtain that D (χK) = 0, and the lemma is proved. 1 2.8. The group Hf (D).

2.8.1. Let D be a potentially semistable (ϕ,ΓK)-module over RK,E, where E is a ﬁnite extension of Qp. As usual, we have the isomorphism H1(D) ' 1 ( ,D) ExtRK,E RK,E 1 which associates to each cocycle x = (a,b) ∈ Cϕ,γK (D) the extension 0 → D → Dx → RK,E → 0 such that Dx = D⊕RK,Ee with ϕ(e) = e+a and γK(e) = e+b. We say that [x] = class(x) ∈ H1(D) is crystalline if

rkE⊗K0 (Dcris(Dx)) = rkE⊗K0 (Dcris(D)) + 1 and deﬁne 1 1 Hf (D) = {[x] ∈ H (D) | cl(x) is crystalline}.

Proposition 2.8.2. Let D be a potentially semistable (ϕ,ΓK)-module over RK,E. Then 0 0 ϕ=1,N=0,GK 1 1 i) H (D) = Fil (Dpst(D)) and Hf (D) is a E-subspace of H (D) of dimension 1 0 0 dimE Hf (D) = dimE DdR(D) − dimE Fil DdR(D) + dimE H (D). ii) There exists an exact sequence

0 (pr,1−ϕ) 1 0 → H (D) → Dcris(D) −−−−−→ tD(K) ⊕ Dcris(D) → Hf (D) → 0, 0 where tD(K) = DdR(D)/Fil DdR(D). 1 ∗ 1 iii) Hf (D (χK)) is the orthogonal complement to Hf (D) under the dual- 1 1 ∗ ity H (D) × H (D (χK)) → E. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 45

iv) Let 0 → D1 → D → D2 → 0 be an exact sequence of potentially semistable (ϕ,ΓK)-modules. Assume that one of the following conditions holds a) D is crystalline; 0 1 1 b) Im((H (D2) → H (D1)) ⊂ Hf (D1). Then one has an exact sequence 0 0 0 1 1 1 0 → H (D1) → H (D) → H (D2) → Hf (D1) → Hf (D) → Hf (D2) → 0. Proof. This proposition is proved in [4], Proposition 1.4.4, and Corollaries 1 1.4.6 and 1.4.10. For another approach to Hf (D) and an alternative proof see [32], Section 2.

2.8.3. In this subsection we assume that K = Qp. We review the computation of the cohomology of some isoclinic (ϕ,ΓQp )-modules given in [4]. To simplify notation, we write χ and Γ0 instead χ and Γ0 respectively. p p Qp Qp

Proposition 2.8.4. Let D be a semistable (ϕ,ΓQp )-module of rank d over ϕ=1 0 RQp,E such that Dst(D) = Dst(D) and Fil Dst(D) = Dst(D). Then 0 i) D is crystalline and H (D) = Dcris(D). 0 1 2 ii) One has dimE H (D) = d, dimE H (D) = 2d and H (D) = 0. iii) The map 1 iD : Dcris(D) ⊕ Dcris(D) → H (D),

iD = cl(−x,log χp(γQp )y) is an isomorphism of E-vector spaces. Let iD, f and iD,c denote the restrictions of iD on the ﬁrst and the second summand respectively. Then 1 Im(iD, f ) = Hf (D) and we have a decomposition 1 1 1 H (D) = Hf (D) ⊕ Hc (D), 1 where Hc (D) = Im(iD,c). ∗ iv) Let D (χp) be the Tate dual of D. Then ∗ ϕ=p−1 ∗ Dcris(D (χp)) = Dcris(D (χp)) 0 ∗ 0 ∗ and Fil Dcris(D (χp)) = 0. In particular, H (D (χp)) = 0. Let ∗ [ , ]D : Dcris(D (χp)) × Dcris(D) → E denote the canonical duality. Deﬁne a morphism ∗ ∗ 1 ∗ ∗ iD (χp) : Dcris(D (χp)) ⊕ Dcris(D (χp)) → H (D (χp)) by ∗ iD (χp)(α,β) ∪ iD(x,y) = [β,x]D − [α,y]D 46 DENIS BENOIS

∗ ∗ ∗ and denote by Im(iD (χp), f ) and Im(iD (χp),c) the restrictions of iD (χp) on ∗ the ﬁrst and the second summand respectively. Then Im(iD (χp), f ) = 1 ∗ Hf (D (χp)) and again we have 1 ∗ 1 ∗ 1 ∗ H (D (χp)) = Hf (D (χp)) ⊕ Hc (D (χp)), 1 ∗ ∗ where Hc (D (χp)) = Im(iD (χp),c). Proof. See [4], Proposition 1.5.9 and Section 1.5.10. We also need the following result.

Proposition 2.8.5. Let D be a crystalline (ϕ,ΓQp )-module over RQp,E such ϕ=p−1 0 that Dcris(D) = Dcris(D) and Fil Dcris(D) = 0. Then 1 1 H (D) 0 = H (D). Iw Γp c Proof. See [6], Proposition 4.

3. p-ADIC HEIGHT PAIRINGS I:SELMERCOMPLEXES 3.1. Selmer complexes.

3.1.1. In this section we construct p-adic height pairings using Nekovár’sˇ formalism of Selmer complexes. Let F be a number field. We denote by S f (resp. S∞) the set of all non-archimedean (resp. archimedean) absolute values on F. Fix a prime number p and a finite subset S ⊂ S f containing the set Sp of all q ∈ S f such that q | p. We will write Σp for the complement of Sp in S. Let GF,S denote the Galois group of the maximal algebraic extension of F unramified outside S ∪ S∞. For each q ∈ S, we fix a decomposition group 0 at q which we identify with GFq , and set Γq = ΓFq . Fix a generator γq ∈ Γq.

3.1.2. Let V be a p-adic representation of GF,S with coefficients in a Qp- affinoid algebra A. We will write Vq for the restriction of V on the decompo- † sition group at q. For each q ∈ Sp, we fix a (ϕ,Γq)-submodule Dq of Drig(Vq) † that is a RFq,A-module direct summand of Drig,A(Vq). Set D = (Dq)q∈Sp and define ( • • Cϕ,γq (Dq), if q ∈ Sp, Uq (V,D) = • Cur(Vq), if q ∈ Σp, where • Iq Frq−1 Iq Cur(Vq) : Vq −−−→ Vq , q ∈ Σp, and the terms are concentrated in degrees 0 and 1. Note that, if q ∈ Sp, • [0,2] we have [Uq (V,D)] = RΓ(Fq,Dq) ∈ Dperf (A) by Theorem 2.3.3, i). On the P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 47

• other hand, if q ∈ Σp, then in general the complex Uq (V,D) is not quasi- isomorphic to a perfect complex of A-modules. First assume that q ∈ Σp. Then we have a canonical morphism • • (24) gq : Uq (V,D) → C (GFq ,V) deﬁned by 0 gq(x0) = x0, if x0 ∈ Uq (V,D), 1 gq(x1)(Frq) = x1, if x1 ∈ Uq (V,D) and the restriction map • • (25) fq = resq : C (GF,S,V) → C (GFq ,V). † Now assume that q ∈ Sp. The inclusion Dq ⊂ Drig(Vq) induces a morphism • • • Uq (V,D) = Cϕ,γ (Dq) → Cϕ,γ (Vq). We denote by • • (26) gq : Uq (V,D) → K (Vq), q | p the composition of this morphism with the quasi-isomorphism • • αVq : Cϕ,γ (Vq) ' K (Vq) constructed in Section 2.4 and by • • (27) fq : C (GF,S,V) → K (Vq), q | p • • the composition of the restriction map resq : C (GF,S,V) →C (GFq ,V) with • • the quasi-isomorphism ξVq : C (GFq ,V) → K (Vq) (see Proposition 2.4.2). Set     • M • M • K (V) =  C (GFq ,V) ⊕  K (Vq) q∈Σp q∈Sp • • and U (V,D) = ⊕ Uq (V,D). q∈S Recall that • [0,3] C (GF,S,V) ∈ Kft (A) and the associated object of the derived category • [0,3] RΓS(V) := C (GF,S,V) ∈ Dperf (A) [0,3] (see [34] and [37]). Therefore, we have a diagram in Kft (A)

• f • C (GF,S,V) / K (V) O g

U•(V,D), 48 DENIS BENOIS where f = ( fq)q∈S and g = (gq)q∈S, and the corresponding diagram in [0,3] Dft (A) L RΓS(V) / RΓ(Fq,V) q∈S O

L RΓ(Fq,V,D), q∈S

• where RΓ(Fq,V,D) = Uq (V,D) . The associated Selmer complex is deﬁned as

• h • • f −g • i S (V,D) = cone C (GF,S,V) ⊕U (V,D) −−→ K (V) [−1].

We set RΓ(V,D) := [S•(V,D)] and write H•(V,D) for the cohomology of • • [0,3] S (V,D). From the deﬁnition, it follows directly that S (V,D) ∈ Kft (A) • [0,1] and if, in addition, Uq (V,D) ∈ Dperf (A) for all q ∈ Σp, then RΓ(V,D) ∈ [0,3] Dperf (A).

3.1.3. The previous construction can be slightly generalized. Fix a ﬁnite subset Σ ⊂ Σp and, for each q ∈ Σ, a locally direct summand Mq of the

A-module Vq stable under the action of GFq . Let M = (Mq)q∈Σ. Deﬁne  C• (D ), if q ∈ S ,  ϕ,γq q p • • Uq (V,D,M) = Cur(Vq), if q ∈ Σp \ Σ,  • C (GFq ,Mq), if q ∈ Σ. We denote by S•(V,D,M) the associated Selmer complex and set • • RΓ(V,D,M) := [S (V,D,M)]. If Mq = 0 for all q ∈ Σ, we write S (V,D,Σ) and RΓ(V,D,Σ) for S•(V,D,M) and RΓ(V,D,M) respectively. Note that [0,3] RΓ(V,D,Σp) ∈ Dperf (A).

3.1.4. We construct cup products for our Selmer complexes RΓ(V,D,M). Consider the dual representation V ∗(1) of V. We equip V ∗(1) with the dual local conditions setting ⊥ † ∗ Dq = HomRA (Drig(Vq (1))/Dq,RA), ∀q ∈ Sp, ⊥ Mq = HomA(Vq/Mq,A), ∀q ∈ Σ, ⊥ ⊥ and denote by fq and gq the morphisms (24-27) associated to (V ∗(1),D⊥,M⊥). Consider the following data P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 49

• • • • • • 1) The complexes A1 = C (GF,S,V), B1 = U (V,D,M), and C1 = K (V) • • equipped with the morphisms f1 = ( fq)q∈S : A1 → C1 and g1 = ⊕ gq : q∈S • • B1 → C1; • • ∗ • • ∗ ⊥ ⊥ 2) The complexes A2 = C (GF,S,V (1)), B2 = U (V (1),D ,M ), and • • ∗ ⊥ • • C2 = K (V (1)) equipped with the morphisms f2 = ( fq )q∈S : A2 → C2 and ⊥ • • g2 = ⊕ gq : B2 → C2; q∈S • • • 3) The complexes A3 = τ>2C (GF,S,A(1)), B3 = 0 and • • • • C3 = τ>2K (A(1)) equipped with the map f3 : A3 → C3 given by

• (resq)q M • • τ>2C (GF,S,A(1)) −−−−→ τ>2C (GFq ,A(1)) → τ>2K (A(1)) q • • and the zero map g3 : B3 → C3. • • • 4) The cup product ∪A : A1 ⊗ A2 → A3 deﬁned as the composition

• • ∗ ∪c • ∗ ∪A : C (GF,S,V) ⊗C (GF,S,V (1)) −→ C (GF,S,V ⊗V (1)) → • ∗ • ∗ C (GF,S,A (1)) → τ>2C (GF,S,A (1)), • • • 5) The zero cup product ∪B : B1 ⊗ B2 → B3. • • • 6) The cup product ∪C : C1 ⊗C2 → C3 deﬁned as the composition

• • ∗ ∪K • ∗ • • K (V) ⊗ K (V (1)) −→ K (V ⊗V (1)) → K (A(1)) → τ>2K (A(1)). • • • • • • 7) The zero maps h f : A1 ⊗ A2 → C3[−1] and hg : B1 ⊗ B2 → C3[−1]. Theorem 3.1.5. i) There exists a canonical, up to homotopy, quasi-isomorphism • rS : E3 → A[−2]. ii) The data 1-7) above satisfy the conditions P1-3) of Section 1.2 and therefore deﬁne, for each a ∈ A and each quasi-isomorphism rS, the cup product

• • ∗ ⊥ ⊥ ∪a,rS : S (V,D,M) ⊗A S (V (1),D ,M ) → A[−3].

iii) The homotopy class of ∪a,rS does not depend on the choice of r ∈ A and, therefore, deﬁnes a pairing

L ∗ ⊥ ⊥ (28) ∪V,D,M : RΓ(V,D,M) ⊗A RΓ(V (1),D ,M ) → A[−3]. Proof. i) We repeat verbatim the argument of [34], Section 5.4.1. For each q ∈ S, let iq denote the composition of the canonical isomorphism A ' 2 H (Fq,A(1)) of the local class ﬁeld theory with the morphism 50 DENIS BENOIS

• • τ>2C (GFq ,A(1)) → K (Aq(1)). Then we have a commutative diagram (res ) • q q L • j • τ>2C (GF,S,A(1)) / τ>2K (Aq(1)) / E3 q∈S O O (iq)q iS

Σ L A[−2] / A[−2], q∈S where iS = j ◦ iq0 for some fixed q0 ∈ S and Σ denotes the summation over q ∈ S. By global class field theory, iS is a quasi-isomorphism and, because A[−2] is concentrated in degree 2, there exists a homotopy inverse rS of iS which is unique up to homotopy. ii) We only need to show that the condition P3) holds in our case. Note that ∪A = ∪c, ∪B = 0 and ∪C = ∪K. From the definition of ∪K it follows immediately that

(29) ∪K ◦ ( f1 ⊗ f2) = f3 ◦ ∪c. ⊥ If q ∈ Sp (resp. if q ∈ Σ), from the orthogonality of Dq and Dq (resp. from ⊥ ⊥ the orthogonality of Mq and Mq ) it follows that ∪K ◦ (gq ⊗ gq ) = 0. If q ∈ ⊥ Σp \Σ, we have ∪c ◦(gq ⊗gq ) = 0 by [34], Lemma 7.6.4. Since g3 ◦∪B = 0, this gives

(30) ∪C ◦ (g1 ⊗ g2) = g3 ◦ ∪B = 0.

The equations (29) and (30) show that P3) holds with h f = hg = 0. We deﬁne ∪a,rS as the composition of the cup product constructed in Proposi- tion 1.2.2 with rS. The rest of the theorem follows from Proposition 1.2.2.

3.1.6. Remark. Consider the case Σ = /0. The cup product ∪V,D induces a map ∗ ⊥ (31) RΓ(V (1),D ) → RHomA(RΓ(V,D),A)[−3], which, in general, is not an isomorphism. For each q ∈ S deﬁne • • gq • Ueq (V,D) = cone Uq (V,D) −→ K (Vq) [−1]

h • i and RfΓ(Fq,V,D) = Ueq (V,D) . Then the cup product • • ∗ K (Vq) ⊗ K (Vq (1)) → A[−2] induces a pairing • • ∗ ⊥ Ueq (V,D) ⊗Uq (V (1),D ) → A[−2] which gives rise to a morphism ∗ ⊥ (32) RΓ(Fq,V (1),D ) → RHomA(RfΓ(Fq,V,D),A)[−2]. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 51

Repeating the arguments of [34] (see the proofs of Proposition 6.3.3 and Theorem 6.3.4 of op. cit.) it is easy to show that if RΓ(Fq,V,D) and ∗ ⊥ RΓ(Fq,V (1),D ) are perfect and (32) holds for all q ∈ S, then (31) is an isomorphism. First assume that q ∈ Sp. The complexes RΓ(Fq,D) and ⊥ RΓ(Fq,D ) are perfect by Theorem 2.3.3. Consider the tautological exact sequence

† 0 → Dq → Drig(Vq) → Deq → 0,

† where Deq = Drig(Vq)/Dq. Applying the functor RΓ(Fq,−) to this sequence, we obtain an exact sequence

† 0 → RΓ(Fq,Dq) → RΓ(Fq,Drig(Vq)) → RΓ(Fq,Deq) → 0,

and therefore in this case RfΓ(Fq,V,D) ' RΓ(Fq,Deq). From the deﬁnition ⊥ ⊥ ∗ of Dq we have Dq ' Deq(χ). Using Theorem 2.3.3, we obtain

⊥ ∗ RΓ(Fq,Dv ) ' RΓ(Fq,Deq(χ)) '

' RHomA(RΓ(Fq,Deq),A)[−2] ' RHomA(RfΓ(Fv,V,D),A)[−2],

and therefore (32) holds automatically for all q ∈ Sp. ∗ ⊥ If q ∈ Σp, the complexes RΓ(Fq,V,D) and RΓ(Fq,V (1),D ) are not perfect in general and, in addition, (32) does not always hold. However, (32) holds in many important cases, in particular if A is a ﬁnite extension of Qp. 52 DENIS BENOIS

• • • 3.1.7. Equip the complexes Ai , Bi and Ci with the transpositions given by

TA1 = TV,c, ! !

TB1 = ⊕ idCϕ,γ (Dq) ⊕ ⊕ idCur(Vq) ⊕ ⊕ TMq,c , q∈Sp q∈Σp\Σ q∈Σ ! !

TC1 = ⊕ TK(Vq) ⊕ ⊕ TVq,c , q∈Sp q∈Σp

∗ TA2 = TV (1),c, ! ! (33) = ⊕ id ⊥ ⊕ ⊕ id ∗ ⊕ ⊕ ⊥ , TB2 Cϕ,γ (Dq ) Cur(Vq (1)) TMq ,c q∈Sp q∈Σp\Σ q∈Σ ! ! = ⊕ ∗ ⊕ ⊕ ∗ , TC2 TK(Vq (1)) TVq (1),c q∈Sp q∈Σp

TA3 = TA(1),c,

TB3 = id, ! !

TC3 = ⊕ TK(A(1)q) ⊕ ⊕ TA(1)q,c . q∈Sp q∈Σp

Theorem 3.1.8. i) The data (33) satisfy the conditions T1-7) of Section 1.2. ii) We have a commutative diagram

∪ L ∗ ⊥ ⊥ V,D RΓ(V,D,M) ⊗A RΓ(V (1),D ,M ) / A[−3]

s12 = ∪ ∗ ⊥ ∗ ⊥ ⊥ L V (1),D RΓ(V (1),D ,M ) ⊗A RΓ(V,D,M) / A[−3].

0 0 0 Proof. i) We check the conditions T3-7) taking ∪A = ∪c, ∪B = 0 and ∪C = 0 0 ∪K. From (29) and (30) it follows that T3) holds if we take h f = hg = 0. To check the condition T4) we remark that, by Proposition 2.5.4,i) we have fi ◦ TA = TC ◦ fi and we can take Ui = 0. The existence of a homotopy Vi follows from Proposition 2.5.4 ii) and [34], Proposition 7.7.3. Note that again we can set Vi = 0. We prove the existence of homotopies tA, tB and tC satisfying T5). From the commutativity of the diagram (16), it follows that ∪c ◦s12 ◦(TA ⊗TA) = 0 TA ◦∪c and we can take tA = 0. Since ∪B = ∪B = 0, we can take tB = 0. We construct tC as a system of homotopies (tC,q)q∈S such that tC,q : ∪c ◦ s12 ◦

(TVq,c ⊗ TV(1)q,c) TA(1)q,c ◦ ∪c for q ∈ Σp and tC,q : ∪K ◦ s12 ◦ (TK(Vq) ⊗

TK(V(1)q)) TK(A(1)q) ◦∪K for q ∈ Sp. As before, from (16) it follows that P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 53 for q ∈ Σp one can take tC,q = 0. If q ∈ Sp, by Proposition 1.1.7 we can set n (34) tC,q((xn−1,xn) ⊗ (ym−1 ⊗ ym)) = (−1) (TA(1)q,c(xn−1 ∪c ym−1),0) n m ∗ for (xn−1,xn) ∈ K (Vq) and (ym−1,ym) ∈ K (V (1)q) (see (8)). This proves T5). From (34) it follows that tC ◦( f1 ⊗ f2) = 0 and it is easy to see that T6) and T7) hold if we take Hf = Hg = 0. ii) For each Galois module X, we denote by aX : id TX,c the homo- 0 1 topy (17). Recall that we can take aX such that aX = aX = 0. Consider the following homotopies • k = a : id • , on A , A1 V TA1 1 ! • kB1 = ⊕ 0Uq(V,D,M) ⊕ ⊕ aMq : id FB1 on B1, (35) q∈Sp∪Σp\Σ q∈Σ ! ! • k = ⊕ a ⊕ ⊕ a : id • , on C . C1 K(Vq) Vq TC1 1 q∈Sp q∈Σp • • • We will denote by kA2 , kB2 kC2 the homotopies on A2, B2 and C2 deﬁned by the analogous formulas. From Proposition 2.5.4, ii) it follows that ⊥ ⊥ f ◦ kA1 = kC1 ◦ f , f ◦ kA2 = kC2 ◦ f , ⊥ ⊥ g ◦ kB1 = kC1 ◦ g, g ◦ kB2 = kC2 ◦ g . sel sel • By (4), these data induce transpositions TV and TV ∗(1) on S (V,D,M) and S•(V ∗(1),D⊥,M⊥), and the formula (5) of Subsection 1.1.2 deﬁnes sel sel sel sel homotopies kV : id TV and kV ∗(1) : id TV ∗(1). By Proposition 1.2.4, the following diagram commutes up to homotopy: ∪ • • ∗ ⊥ ⊥ a,rS S (V,D,M) ⊗A S (V (1),D ,M ) / A[−3]

sel sel s12◦(TV ⊗TV∗(1)) = ∪ • ∗ ⊥ ⊥ • 1−a,rS S (V (1),D ,M ) ⊗A S (V,D,M) / A[−3]. sel sel Now the theorem follows from the fact that the map (kV ⊗ kV ∗(1))1, given sel sel by (2), furnishes a homotopy between id and TV ⊗ TV ∗(1). 3.2. p-adic height pairings. 3.2.1. We keep notation and conventions of the previous subsection. Let ∞ cyc F = ∪F(ζpn ) denote the cyclotomic p-extension of F. The Galois group n=1 cyc 0 ΓF = Gal(F /F) decomposes into the direct sum ΓF = ∆F × ΓF of the 0 group ∆F = Gal(F(ζp)/F) and a p-procyclic group ΓF . We denote by χ : 54 DENIS BENOIS

∗ ΓF → Zp the cyclotomic character and by χq the restriction of χ on Γq, q ∈ 0 S. As in Section 2.6, we equip the group algebra A[ΓF ] with the involution 0 0 −1 0 0 ι : A[ΓF ] → A[ΓF ] such that ι(γ) = γ , γ ∈ ΓF . Fix a generator γF of ΓF . ι 0 ι 2 0 Set AeF = A[ΓF ] /(JF ), where JF is the augmentation ideal of A[ΓF ]. We have an exact sequence

θF ι (36) 0 → A −→ AeF → A → 0, −1 where θF (a) = aXe, and Xe = log (χ(γF ))(γ − 1) does not depend on the 0 choice of γF ∈ ΓF . For each p-adic representation V with coefﬁcients in A, (36) induces an exact sequence

(37) 0 → V → VeF → V → 0, ι where VeF = AeF ⊗A V. As in Section 2.6, passing to continuous Galois cohomology, we obtain a distinguished triangle

• • • βV,c • C (GF,S,V) → C (GF,S,VeF ) → C (GF,S,V) −−→ C (GF,S,V)[1]. 0 0 For each q ∈ S, the inclusion Γq ,→ ΓF induces a commutative diagram

θq 0 / Vq / VeFq / Vq / 0

= = θF 0 / Vq / (VeF )q / Vq / 0, where the vertical middle arrow is an isomorphism by the ﬁve lemma. Tak- ing into account Proposition 2.6.2, we see that the exact sequence (37) induces a distinguished triangle

• • • βVq,c • C (GFq ,V) → C (GFq ,VeF ) → C (GFq ,V) −−−→ C (GFq ,V)[1]. where βVq,c(x) = −log χq ∪ x. † ι Let Dq be a (ϕ,Γq)-submodule of Drig(Vq) and let DeF,q = AeF ⊗A Dq. As in Section 2.6, we have an exact sequence

(38) 0 → Dq → DeF,q → Dq → 0 which sits in the diagram

θq 0 / Dq / Deq / Dq / 0

= = θF 0 / Dq / DeF,q / Dq / 0. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 55

Taking into account Proposition 2.6.4, we obtain that (38) induces the distingushed triangle

• • • βDq • Cϕ,γq (Dq) → Cϕ,γq (DeF,q) → Cϕ,γq (Dq) −−→ Cϕ,γq (Dq)[1],

where βDq (x) = −(0,log χq(γq))∪x. Finally, replacing in the exact sequence (23) Ve by VeF , and taking into account Proposition 2.6.5 we obtain the distinguished triangle

β • • • K(Vq) • K (Vq) → K ((VeF )q) → K (Vq) −−−→ K (Vq)[1],

where βK(Vq)(x) = −(0,log χq) ∪ x. • If q ∈ Σp, we construct the Bockstein map for Uq (V,D,M) following • • [34], Section 11.2.4. Namely, if q ∈ Σ, then Uq (V,D,M) = C (GFq ,Mq) and the exact sequence

(39) 0 → Mq → MeF,q → Mq → 0

• • gives rise to a map βMq,c : C (GFq ,Mq) → C (GFq ,Mq). If q ∈ Σp \ Σ, then Iq Iq ι Iq Iq (VeF ) = V ⊗ AeF and we denote by s : V → (VeF ) the section given by s(x) = x ⊗ 1. There exists a distingushed triangle

• • • βVq,ur • Cur(Vq) → Cur((VeF )q) → Cur(Vq) −−−→ Cur(Vq)[1],

0 1 where βVq,ur : Cur(Vq) → Cur(Vq) is given by

F 1 βVq,ur(x) = (ds − sd)(x) = −log χq(Frq)x. Xe 56 DENIS BENOIS

• • • Proposition 3.2.2. In addition to (33), equip the complexes Ai ,Bi and Ci (1 6 i 6 3) with the Bockstein maps given by

βA1 = βV,c, ! !

βB1 = ⊕ βDq ⊕ ⊕ βVq,ur , q∈Sp q∈Σp ! !

βC1 = ⊕ βK(Vq) ⊕ ⊕ βVq,c , q∈Sp q∈Σp

∗ βA2 = βV (1),c, ! !

β = ⊕ β ⊥ ⊕ ⊕ β ∗ , B2 Dq Vq (1),ur q∈Sp q∈Σp ! ! = ⊕ ∗ ⊕ ⊕ ∗ , βC2 βK(Vq (1)) βVq (1),c q∈Sp q∈Σp

βA3 = βA(1),c,

βB3 = 0, ! !

βC3 = ⊕ βK(A(1)q) ⊕ ⊕ βA(1)q,c . q∈Sp q∈Σp Then these data satisfy the conditions B1-5) of Section 1.2.

Proof. We check B2-5) for our Bockstein maps. For each q ∈ Σp, Nekovárˇ constructed homotopies

vV,q : gq ◦ βVq,ur βVq,c ◦ gq, ⊥ ⊥ v ∗ : g ◦ ∗ ∗ ◦ g . V (1),q q βVq (1),ur βVq (1),c q

From Proposition 2.6.5, ii) it follows that for all q ∈ Sp

gq ◦ βDq = βK(Vq) ◦ gq, ⊥ ⊥ g ◦ = ∗ ◦ g . q βDq βK(Vq (1)) q

Set vV,q = vV ∗(1),q = 0 for all q ∈ Sp. Then the condition B2) holds for ui = 0 and vi = (vi,q)q∈S. In B3), we can set hB = 0 because ∪B = 0. The existence of a homotopy hA between ∪A[1]◦(id⊗βA,2) and ∪A[1]◦(βA,1 ⊗id) is proved in [34], Section 11.2.6 and the same method allows to construct hC. Namely, we construct a system hC = (hC,q)q∈S of homotopies such that hC,q : ∪c[1] ◦ (id ⊗ ∗ ) ∪ [1] ◦ ( ⊗ id) for q ∈ and h : ∪ [1] ◦ (id ⊗ βVq (1),c c βVq,c Σp C,q K ∗ ) ∪ [1]◦( ⊗id) for q ∈ S . For q ∈ , the construction of βK(Vq (1)) K βK(Vq) p Σp P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 57 hC,q is the same as those of hA. Now, let q ∈ Sp. By Proposition 2.6.5, one has βK(Vq)(x) = −(0,log χq) ∪K x. Consider the following diagram, where zq = (0,log χq) (40) • • ∗ id • • ∗ K (Vq) ⊗A K (Vq (1)) / K (Vq) ⊗A K (Vq (1))

= =

• • ∗ id • • ∗ K (Vq) ⊗A A ⊗A K (Vq (1)) / A ⊗A K (Vq) ⊗A K (Vq (1))

id⊗(−zq)⊗id (−zq)⊗id⊗id

• • • ∗ s12⊗id • • • ∗ K (Vq) ⊗A K (A)[1] ⊗A K (Vq (1)) / K (A)[1] ⊗A K (Vq) ⊗A K (Vq (1))

∪K⊗id ∪K⊗id • • ∗ id • • ∗ K (Vq)[1] ⊗A K (Vq (1)) / K (Vq)[1] ⊗A K (Vq (1))

∪K ∪K

• ∗ id • ∗ K (Vq ⊗Vq (1))[1] / K (Vq ⊗Vq (1))[1].

The ﬁrst, second and fourth squares of this diagram are commutative. From Proposition 1.1.7 (see also (18)) it follows that the diagram

• • s12◦(TK⊗TK) • • K (Vq) ⊗ K (A)[1] / K (A)[1] ⊗ K (Vq)

∪K ∪K • TK • K (Vq)[1] / K (Vq)[1] is commutative up to some homotopy k1 : TK ◦∪K ∪K ◦s12 ◦(TK ⊗TK). 2 Since TK = id, we have a homotopy

TK ◦ k1 : ∪K TK ◦ ∪K ◦ s12 ◦ (TK ⊗ TK).

By [34], Section 3.4.5.5 (see also Section 2.5.2), for any topological GFq - module M there exists a functorial homotopy a : id Tc. By Proposi- • • tion 2.5.4, a induces a homotopy between id : K (Vq) → K (Vq) and TK : • • K (Vq) → K (Vq) which we denote by aK. Let (aK ⊗ aK)1 : id TK ⊗ • TK denote the homotopy between the maps id and TK ⊗ TK : K (Vq) ⊗ • • • K (Qp)[1] → K (Vq) ⊗ K (Qp)[1] given by (2). Then

d(aK ◦TK ◦∪K ◦s12 ◦(TK ⊗TK))+(aK ◦TK ◦∪K ◦s12 ◦(TK ⊗TK))d =

= (TK − id) ◦ ∪K ◦ s12 ◦ (TK ⊗ TK), 58 DENIS BENOIS and

d(∪K ◦ s12 ◦ (aK ⊗ aK)1) + (∪K ◦ s12 ◦ (aK ⊗ aK)1) =

= ∪K ◦ s12 ◦ (TK ⊗ TK − id). Therefore the formula

(41) k2 = aK ◦ TK ◦ ∪K ◦ s12 ◦ (TK ⊗ TK) + ∪K ◦ s12 ◦ (aK ⊗ aK)1 deﬁnes a homotopy

k2 : ∪K ◦ s12 TK ◦ ∪K ◦ s12 ◦ (TK ⊗ TK).

Then kC,q = FK ◦ k1 − k2 deﬁnes a homotopy

kC,q : ∪K ∪K ◦ s12 and we proved that the third square of the diagram (40) commutes up to a homotopy. We deﬁne the homotopy

h : ∪ [1] ◦ (id ⊗ ∗ ) ∪ [1] ◦ ( ⊗ id) C,q K βK(Vq (1)),c K βK(Vq) by

(42) hC,q = ∪K ◦ (kC,q ⊗ id) ◦ (id ⊗ (−zq) ⊗ id). This proves B3). Since u1 = u2 = h f = 0, the condition B4) reads

(43) dKf − Kf d = −hC ◦ ( f1 ⊗ f2) + f3[1] ◦ hA for some second order homotopy Kf . It is proved in [34], Section 11.2.6, that if q ∈ Σp, then

(44) hC,q ◦ ( f1 ⊗ f2) = resq ◦ hA.

Assume that q ∈ Sp. Recall (see [34], Section 11.2.6) that the homotopy hA is given by

(45) hA = ∪c ◦ (kA ⊗ id) ◦ (id ⊗ (−z) ⊗ id), where z = log χ and

(46) kA = −a ◦ (∪c ◦ s12 ◦ (Tc ⊗ Tc)) − (Tc ◦ ∪c ◦ s12) ◦ (a ⊗ a)1. n m ∗ From (8), it follows that for all x ∈ C (GF,S,V) and y ∈ C (GF,S,V (1)) we have

(47) (k1 ⊗ id) ◦ (id ⊗ (−zq) ⊗ id) ◦ ( f1 ⊗ f2)(x ⊗ y) =

= (k1 ⊗ id)((0,−log χq) ⊗ (0,xq) ⊗ (0,yq)) =

= k1((0,−log χq) ⊗ (0,xq)) ⊗ (0,yq) = 0, P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 59 where xq = resq(x), yq = resq(y). On the other hand, comparing (41) and (46) we see that

(48) (k2 ⊗ id) ◦ (id ⊗ (−zq) ⊗ id) ◦ ( f1 ⊗ f2)(x ⊗ y) =

= k2((0,−log χq) ⊗ (0,xq)) ⊗ (0,yq) =

= −(0,resq(kA(−z ⊗ x))) ⊗ (0,yq). From (47), (48), (42) and (46) we obtain that

(49) hC,q ◦ ( f1 ⊗ f2)(x ⊗ y) =

= (0,resq(kA(−z ⊗ x))) ∪K (0,yq) =

= (0,resq(kA(−z ⊗ x)) ∪c y) = (0,resq(hA(x ⊗ y))).

From (49) and (44) it follows that hC ◦ ( f1 ⊗ f2) = f3[1] ◦ hA and therefore we can set Kf = 0 in (43). Thus, B4) is proved. It remains to check B5). Since v1 = v2 = hg = 0, this condition reads

(50) dKg − Kgd = −hC ◦ (g1 ⊗ g2) + ∪C[1] ◦ (v1 ⊗ g2) − ∪C[1] ◦ (g1 ⊗ v2) for some second order homotopy Kg. Write Kg = (Kg,q)q∈S. For q ∈ Σp, Nekovárˇ proved that the q-component of the right hand side of (50) is equal to zero. For q ∈ Sp, we have v1,q = v2,q = 0 and hC,v ◦(g1 ⊗g2) = 0 because ⊥ of orthogonality of D and D , and again we can set Kg,q = 0. To sum up, the condition (50) holds for Kg = 0. The proposition is proved.

3.2.3. The exact sequences (37), (38) and (39) give rise to a distinguished triangle

δV,D,M RΓ(V,D,M) → RΓ(VeF ,DeF ,MeF ) → RΓ(V,D,M) −−−→ RΓ(V,D,M)[1] Deﬁnition. The p-adic height pairing associated to the data (V,D,M) is deﬁned as the morphism

sel L ∗ ⊥ ⊥ δV,D,M hV,D,M : RΓ(V,D,M) ⊗A RΓ(V (1),D ,M ) −−−→ L ∗ ⊥ ⊥ ∪V,D,M → RΓ(V,D,M)[1] ⊗A RΓ(V (1),D ,M ) −−−−→ A[−2], where ∪V,D,M is the pairing (28).

sel The height pairing hV,D,M induces a pairing

sel 1 1 ∗ ⊥ ⊥ (51) hV,D,M,1 : H (V,D,M) ⊗A H (V (1),D ,M ) → A. 60 DENIS BENOIS

Theorem 3.2.4. The diagram

hsel L ∗ ⊥ ⊥ V,D,M RΓ(V,D,M) ⊗A RΓ(V (1),D ,M ) / A[−2]

s12 = hsel ∗ ⊥ ⊥ ∗ ⊥ ⊥ L V (1),D ,M RΓ(V (1),D ,M ) ⊗A RΓ(V,D,M) / A[−2] sel is commutative. In particular, the pairing hV,D,1 is skew-symmetric. Proof. From Propositions 1.2.6 and 3.2.2 it follows, that the diagram

hsel • • ∗ ⊥ ⊥ V,D,M S (V,D,M) ⊗A S (V (1),D ,M ) / E3

sel sel s12◦(TV ⊗TV∗(1)) = sel h ∗ ⊥ ⊥ • ∗ ⊥ ⊥ • V (1),D ,M S (V (1),D ,M ) ⊗A S (V,D,M) / E3 is commutative up to homotopy. Now the theorem follows from the fact, sel sel that (TV ⊗TV ∗(1)) is homotopic to the identity map (see the proof of The- orem 3.1.8).

4. SPLITTINGSUBMODULES 4.1. Splitting submodules.

4.1.1. Let K be a finite extension of Qp, and let V be a potentially semistable representation of GK with coefficients in a finite extension E of Qp. For each finite extension L/K we set GL D∗/L(V) = (B∗ ⊗V) , where ∗ ∈ {cris,st,dR} and write D∗(V) = D∗/K(V) if L = K. We will use the same convention for the functors D∗/L. Fix a finite Galois extension L/K such that the restriction of V on GL is semistable. Then Dst/L(V) is a filtered (ϕ,N,GL/K)-module and DdR/L(V) =

Dst/L(V) ⊗L0 L. A (ϕ,N,GL/K)-submodule of Dst/L(V) is a L0-subspace D of Dst/L(V) stable under the action of ϕ, N and GL/K.

Deﬁnition. We say that a (ϕ,N,GL/K)-submodule D of Dst/L(V) is a splitting submodule if 0 DdR/L(V) = DL ⊕ Fil DdR/L(V), DL = D ⊗L0 L as L-vector spaces. In Subsections 4.1-4.2 we will always assume that V satisﬁes the following condition:

ϕ=1 ∗ ϕ=1 S) Dcris(V) = Dcris(V (1)) = 0. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 61

If D is a splitting submodule, we denote by D the (ϕ,ΓK)-submodule of † Drig(V) associated to D by Theorem 2.2.3. The natural embedding D → † 1 1 † ∼ 1 Drig(V) induces a map H (D) → H (Drig(V)) → H (K,V). Passing to duals, we obtain a map H1(K,V ∗(1)) → H1(D∗(χ)). Proposition 4.1.2. Assume that V satisﬁes the condition S). Let D be a splitting submodule. Then 1 ∗ 1 ∗ i) Hf (K,V (1)) → Hf (D (χ)) is the zero map. 1 1 1 1 1 ii) Im(H (D) → H (K,V)) = Hf (K,V) and the map Hf (D) → Hf (K,V) is an isomorphism. 0 ϕ=1,N=0,GL/K iii) If, in addition, Fil (Dst/L(V)/D) = 0, then 1 1 H (D) = Hf (K,V). Proof. i) By Proposition 2.8.2 we have a commutative diagram

1 ∗ 1 ∗ (52) Hcris(V (1)) / Hcris(D (χ)) = = 1 ∗ 1 ∗ Hf (K,V (1)) / Hf (D (χ)), where we set 1 ∗ (1−ϕ,pr) Hcris(V (1)) = coker Dcris(V) −−−−−→ Dcris(V) ⊕tV (K) and 1 ∗ ∗ (1−ϕ,pr) ∗ Hcris(D (χ)) = coker Dcris(D (χ)) −−−−−→ Dcris(D (χ))) ⊕tD∗(χ)(K) to simplify notation. ∗ ϕ=1 ∗ ∗ Since Dcris(V (1)) = 0, the map 1−ϕ : Dcris(V (1)) → Dcris(V (1)) 1 ∗ is an isomorphism and Hcris(V (1)) = tV ∗(1)(K). On the other hand, all ∗ Hodge–Tate weights of D (χ) are > 0 and tD∗(χ)(K) = 0. Hence 1 ∗ ∗ Hcris(D (χ)) = coker(1 − ϕ | Dcris(D (χ))) and the upper map in (52) is zero because it is induced by the canonical projection of tV ∗(1)(K) on tD∗(χ)(K). This proves i). Now we prove ii). Using i) together with the orthogonality property of 1 Hf we obtain that the map

1 1 1 1 HomE(H (K,V)/Hf (K,V),E) → HomE(H (D)/Hf (D),E), 62 DENIS BENOIS induced by H1(D) → H1(K,V), is zero. This implies that the image of 1 1 1 H (D) is H (K,V) is contained in Hf (K,V). Finally one has a diagram

1 1 Hcris(D) / Hcris(V)

' ' 1 1 Hf (D) / Hf (K,V). From S) it follows that the top arrow can be identiﬁed with the natural map tD(K) → tV (K) which is an isomorphism by the deﬁnition of a splitting submodule. iii) Taking into account ii), we only need to prove that the natural map H1(D) → H1(K,V) is injective. This follows from the exact sequence † 0 0 † 0 → D → Drig(V) → D → 0, D = Drig(V)/D

0 0 0 ϕ=1,N=0,GL/K and the fact that H (D ) = Fil (Dst/L(V)/D) = 0 (see Proposi- tion 2.8.2, i)). 4.2. The canonical splitting.

4.2.1. Let ∗ y : 0 → V (1) → Yy → E → 0 be an extension of E by V ∗(1). Passing to (ϕ,ΓK)-modules, we obtain an extension † ∗ † 0 → Drig(V (1)) → Drig(Yy) → RK,E → 0. By duality, we have exact sequences ∗ 0 → E(1) → Yy (1) → V → 0 and † ∗ † 0 → RK,E(χ) → Drig(Yy (1)) → Drig(V) → 0. ∼ We denote by [y] the class of y in Ext1 (E,V ∗(1)) → H1(K,V ∗(1)). As- E[GK] 1 ∗ sume that y is crystalline, i.e. that [y] ∈ Hf (K,V (1)). Let D be a splitting submodule of Dst/L(V). Consider the commutative diagram

† ∗ † y : 0 / Drig(V (1)) / Drig(Yy) / RK,E / 0 pr =

∗ ∗ pr(y) : 0 / D (χ) / Dy(χ) / RK,E / 0 P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 63

† ∗ where Dy is the inverse image of D in Drig(Yy (1)). The class of pr(y) in H1(D∗(χ)) is the image of [y] under the map 1 † ∗ 1 ∗ Ext (RK,E,Drig(V (1))) → Ext (RK,E,D (χ)) which coincides with the map 1 ∗ 1 † ∗ 1 ∗ H (K,V (1)) = H (Drig(V (1))) → H (D (χ)) 1 1 after the identiﬁcation of Ext (RK,E,−) with the cohomology group H (−). 1 ∗ Since we are assuming that [y] ∈ Hf (K,V (1)), by Proposition 4.1.2 i), we obtain that [pr(y)] = 0. Thus the sequence pr(y) splits.

4.2.2. We will construct a canonical splitting of pr(y) using the idea of ∗ Nekovárˇ [33]. Since dimE Dcris(Yy) = dimE Dcris(V (1)) + 1, the sequence ∗ 0 → Dcris(V (1)) → Dcris(Yy) → Dcris(E) → 0 ∗ ϕ=1 is exact by the dimension argument. From Dcris(V (1)) = 0 and the snake lemma it follows that ϕ=1 Dcris(Yy) = Dcris(E) and we obtain a canonical ϕ-equivariant morphism of K0-vector spaces Dcris(E) → Dcris(Yy). By linearity, this map extends to a (ϕ,N,GL/K)-equivariant morphism of L0-vector spaces Dst/L(E) → Dst/L(Yy). Therefore we have a canonical splitting ∼ ∗ Dst/L(Yy) → Dst/L(V (1)) ⊕ Dst/L(E) of the sequence ∗ 0 → Dst/L(V (1)) → Dst/L(Yy) → Dst/L(E) → 0 in the category of (ϕ,N,GL/K)-modules. This splitting induces a (ϕ,N,GL/K)- equivariant isomorphism ∗ ∼ ∗ (53) Dst/L(Dy(χ)) → Dst/L(D (χ)) ⊕ Dst/L(E). Moreover, since all Hodge–Tate weights of D∗(χ) are positive, we have i ∗ ∼ i ∗ i Fil DdR/L(Dy(χ)) → Fil DdR/L(D (χ)) ⊕ Fil DdR/L(E) and therefore the isomorphism ∗ ∼ ∗ DdR/L(Dy(χ)) → DdR/L(D (χ)) ⊕ DdR/L(E) is compatible with ﬁltrations. Thus, we obtain that (53) is an isomorphism in the category of ﬁltered (ϕ,N,GL/K)-modules. This gives a canonical splitting

∗ ∗ o pr(y) : 0 / D (χ) / Dy(χ) / RK,E / 0 64 DENIS BENOIS of the extension pr(y). Passing to duals, we obtain a splitting

sD,y o (54) 0 / RK,E(χ) / Dy / D / 0.

Taking cohomology, we get a splitting

sy 1 1 o 1 (55) 0 / Hf (K,E(1)) / Hf (Dy) / Hf (D) / 0.

Our constructions can be summarized in the diagram

sy 1 1 o 1 0 / Hf (K,E(1)) / Hf (Dy) / Hf (D) / 0

= ' ' 1 1 ∗ 1 0 / Hf (K,E(1)) / Hf (K,Yy (1)) / Hf (K,V) / 0.

Here the vertical maps are isomorphisms by Proposition 4.1.2 and the ﬁve lemma.

4.2.3. Remark. Assume that H0(D∗(χ)) = 0. Then each crystalline extension of D by RK(χ) splits uniquely. This follows from Proposition 2.8.2 i) 1 ∗ which implies that Hf (D (χ)) = 0 and from the fact that various splittings are parametrized by H0(D∗(χ)).

4.3. Filtration associated to a splitting submodule.

4.3.1. In this subsection we assume that K = Qp. We review the construction of the canonical ﬁltration on Dst/L(V) associated to a splitting submodule D. Note that this construction is the direct generalization of the ﬁltration constructed by Greenberg [23] in the ordinary case. See [4] for more detail.

Let V be a potentially semistable representation of GQp with coefficients in a finite extension E of Qp. As before, we fixe a finite Galois extension L/Qp such that V is semistable over L and denote by Dst/L(V) the semistable module of the restriction of V on GL. Let D ⊂ Dst/L(V) be a 0 0 0 0 splitting submodule. Set D = Dst/L(V)/D. Then Fil D = D and we de- 0 ϕ=1,N=0,GL/Qp fine M1 = (D ) . For the dual filtered (ϕ,N,GL/Qp )-module ∗ 0 ∗ ∗ D = HomL0⊗E(D,Dst/L(E(1))) we have Fil D = D and we define M0 = ∗ ∗ ϕ=1,N=0,GL/Qp (D ) . We have canonical projections prD0 : Dst/L(V) → P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 65

D0 and pr : D → M . Define a five-step filtration M0 0

{0} = F−2Dst/L(V) ⊂ F−1Dst/L(V) ⊂ F0Dst/L(V) ⊂

F1Dst/L(V) ⊂ F2Dst/L(V) = Dst/L(V) by  ker(pr ) if i = −1,  M0 FiDst/L(V) = D if i = 0,  −1 prD0 (M1) if i = 1. Set W = F1Dst/L(V)/F−1Dst/L(V). These data can be represented by the diagram prD0 0 / D / DL (V) / D0 / 0 st O pr M0 ? 0 / M0 / W / M1 / 0. By Theorem 2.2.3, the ﬁltration F D (V)2 induces a ﬁltration i st/L i=−2 2 † † FiD (V) on the (ϕ,ΓQ )-module D (V) such that rig i=−2 p rig † Dst/L(FiDrig(V)) = FiDst/L(V). † † † Note that D = F0Drig(V). We also set M0 = F0Drig(V)/F−1Drig(V), † † † † M1 = F1Drig(V)/F0Drig(V) and W = F1Drig(V)/F−1Drig(V). We have a tautological exact sequence α β (56) 0 → M0 −→ W −→ M1 → 0.

By construction, M0 and M1 are crystalline (ϕ,ΓQp )-modules such that

Dcris/Qp (M0) = M0, Dcris/Qp (M1) = M1. Since ϕ=p−1 0 M0 = M0, Fil M0 = 0, ϕ=1 0 M1 = M1, Fil M1 = M1, the structure of modules M0 and M1 is given by Proposition 2.8.4. In particular, we have canonical decompositions (pr ,pr ) (pr ,pr ) 1 f c 1 1 1 f c 1 1 H (M0) ' Hf (M0)⊕Hc (M0), H (M1) ' Hf (M1)⊕Hc (M1). 0 1 The exact sequence (56) induces the cobondary map δ0 : H (M1) → H (M0). Passing to cohomology in the dual exact sequence ∗ ∗ ∗ (57) 0 → M1(χ) → W (χ) → M0(χ) → 0, 66 DENIS BENOIS

∗ 0 ∗ 1 ∗ we obtain the coboundary map δ0 : H (M0(χ)) → H (M1(χ)). 4.3.2. In the remainder of this subsection we assume that (V,D) satisﬁes the following conditions:

F1) For all i ∈ Z † † ϕ=pi † ϕ=pi Dpst(Drig(V)/F1Drig(V)) = Dpst(F−1Drig(V)) = 0. F2) The composed maps

0 δ0 1 prc 1 H (M1) −→ H (M0) −→ Hc (M0), pr 0 δ0 1 f 1 H (M1) −→ H (M0) −−→ Hf (M0), where the second arrows denote canonical projections, are isomorphisms.

† ∗ 4.3.3. Remarks. 1) It is easy to see, that the ﬁltration on Drig(V (1)) asso- ⊥ ciated to the dual splitting submodule D = Hom(Dst/L(V)/D,Dst/L(E(1))), is dual to the ﬁltration FiDst/L(V). In particular, † ∗ ∗ † † F−1Drig(V (1)) (χ) ' Drig(V)/F1Drig(V), † ∗ † ∗ † ∗ Drig(V (1))/F1Drig(V (1)) ' (F−1Drig(V)) (χ), and the sequence (56) for (V ∗(1),D⊥) coincides with (57). In particular, the conditions F1-2) hold for (V,D) if and only if they hold for (V ∗(1),D⊥). 2) The condition F1) implies that 0 † † 0 † H (Drig(V)/F1Drig(V)) = H (F−1Drig(V)) = 0.

3) If V is semistable over Qp, and the linear map ϕ : Dst(V) → Dst(V) is semisimple at 1 and p−1, it is easy to see that  (1 − p−1ϕ−1)D + N(Dϕ=1) if i = −1,  FiDst(V) = D if i = 0,  ϕ=1 −1 ϕ=p−1 D + Dst(V) ∩ N (D ) if i = 1.

In this form, the ﬁltration FiDst(V) was constructed in [4].

† We summarize some properties of the ﬁltration FiDrig(V).

Proposition 4.3.4. Let D be a regular submodule of Dst(V) such that (V,D) satisﬁes the conditions F1-2). Then i) rk(M0) = rk(M1). ii) H0(W) = 0 P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 67

iii) The representation V satisﬁes S), namely

ϕ=1 ∗ ϕ=1 Dcris(V) = Dcris(V (1)) = 0.

1 † 1 † 1 † iv) One has Hf (F−1Drig(V)) = H (F−1Drig(V)) and Hf (F1Drig(V)) = 1 Hf (Qp,V). v) We have exact sequences

0 1 1 (58) 0 → H (M1) → H (M0) → Hf (W) → 0 and

0 1 1 (59) 0 → H (M1) → H (D) → Hf (Qp,V) → 0.

0 Proof. i) From F2) and the fact that dimE H (M1) = rk(M1) and 1 dimE Hc (M0) = rk(M0) (see Proposition 2.8.4) we obtain that rk(M0) = rk(M1). 0 ii) By Proposition 2.8.4, iv), H (M0) = 0, and we have an exact sequence

0 0 δ0 1 0 → H (W) → H (M1) −→ H (M0).

0 By F2), the map δ0 is injective and therefore H (W) = 0. Applying the same argument to the dual exact sequence, we obtain that H0(W∗(χ)) = 0. iii) First prove that Dcris(W) = Dcris(M0). The exact sequence (56) gives an exact sequence

α β 0 → Dcris(M0) −→ Dcris(W) −→ Dcris(M1) and we have immediately the inclusion Dcris(M0) ⊂ Dcris(W). Thus, it is enough to check that dimE Dcris(W) = dimE Dcris(M0). Assume that dimE Dcris(W) > dimE Dcris(M0). Then there exists x ∈ Dcris(W) such that ΓQp m = β(x) 6= 0. Since ϕ acts trivially on Dcris(M1) = M1 , RQp,Em is a (ϕ,ΓQp )-submodule of M1, and there exists a submodule X ⊂ W which sits in the following commutative diagram with exact rows

0 / M0 / X / RQp,Em / 0 = 0 / M0 / W / M1 / 0. 68 DENIS BENOIS

ΓQ n Since Dcris(W) = (W[1/t]) p , there exists n > 0 such that t x ∈ X, and therefore x ∈ Dcris(X). This implies that X is crystalline, and by Proposi- tion 2.8.2 iv) we have a commutative diagram Em H1( ) _ / f M0 _

0 δ0 1 H (M1) / H (M0). 1 Thus, Im(δ0) ∩ Hf (M0) 6= {0} and the condition F2) is violated. This proves that Dcris(W) = Dcris(M0). Now we can ﬁnish the proof of iii). Taking invariants, we have ϕ=1 ϕ=1 Dcris(W) = Dcris(M0) = 0. By F1) † ϕ=1 † † ϕ=1 Dcris(F−1Drig(V)) = Dcris(Drig(V)/F1Drig(V)) = 0, ϕ=1 and, applying the functor Dcris(−) to the exact sequences † † † † 0 → F1Drig(V) → Drig(V) → Drig(V)/F1Drig(V), † † 0 → F−1Drig(V) → F1Drig(V) → W → 0, ϕ=1 ϕ=1 we obtain that Dcris(V) ⊂ Dcris(W) = 0. The same argument shows ∗ ϕ=1 that Dcris(V (1)) = 0. iv) By F1) together with Proposition 2.8.2 and the Euler–Poincaré characteristic formula, we have 1 † 1 † dimE H (F−1Drig(V)) − dimE Hf (F−1Drig(V)) = 0 † ∗ = dimE H ((F−1Drig(V)) (χ)) = 0, 1 † 1 † and therefore Hf (F−1Drig(V)) = H (F−1Drig(V)). Since 0 † † H (Drig(V)/F1Drig(V)) = 0, the exact sequence † † † † 0 → F1Drig(V) → Drig(V) → Drig(V)/F1Drig(V) → 0 induces, by Proposition 2.8.2 iv), an exact sequence 1 † 1 † 1 † † 0 → Hf (F1Drig(V)) → Hf (Drig(V)). → Hf (Drig(V)/F1Drig(V)) → 0 On the other hand, since † † 0 † † DdR(Drig(V)/F1Drig(V)) = Fil DdR(Drig(V)/F1Drig(V)), by Proposition 2.8.2, i) we have 1 † † 0 † † dimE Hf (Drig(V)/F1Drig(V)) = dimE H (Drig(V)/F1Drig(V)) = 0, 1 † 1 † 1 and therefore Hf (F1Drig(V)) = Hf (Drig(V)) = Hf (Qp,V). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 69

v) To prove the exacteness of (58), we only need to show that the image 1 1 of the map α : H (M0) → H (W), induced by the exact sequence (56), 1 1 coincides with Hf (W). By F2), Im(δ0) ∩ Hf (M0) = {0}, and therefore the 1 1 map Hf (M0) → Hf (W) is injective. Set e = rk(M0) = rk(M1). Since 1 0 1 dimE Hf (W) = dimE tW(Qp) − H (W) = e = dimE Hf (M0), 1 1 we obtain that Hf (M0) = Hf (W). On the other hand, the exact sequence

0 δ0 1 α 1 0 → H (M1) −→ H (M0) −→ H (W) 1 0 1 shows that dimE Im(α) = dimE H (M0)−dimE H (M1) = e = dimE Hf (M0). 1 1 Therefore Im(α) = Hf (M0) = Hf (W), and the exacteness of (58) is proved. 0 1 † 1 † Since H (W) = 0 and Hf (F−1Drig(V)) = H (F−1Drig(V)), by Proposi- tion 2.8.2 iv) we have an exact sequence 1 † 1 † 1 0 → H (F−1Drig(V)) → Hf (F1Drig(V)) → Hf (W) → 0, 1 † 1 which shows that Hf (F1Drig(V)) is the inverse image of Hf (W) under the 1 † 1 map H (F1Drig(V)) → H (W). Therefore we have the following commutative diagram with exact rows 0 0

0 = 0 H (M1) / H (M1)

1 † 1 1 0 / H (F−1Drig(V)) / H (D) / H (M0) / 0

= 1 † 1 † 1 0 / H (F−1Drig(V)) / Hf (F1Drig(V)) / Hf (W) / 0

0 0. Since the right column of this diagram is exact, the ﬁve lemma gives the exacteness of the middle column. Now the exacteness of (59) follows from 1 † 1 the fact that Hf (F1Drig(V)) = Hf (Qp,V) by iv). 4.3.5. The tautological exact sequence † 0 0 → D → Drig(V) → D → 0 70 DENIS BENOIS induces the coboundary map 0 0 1 ∂0 : H (D ) → H (D), 0 which is injective because H (Qp,V) = 0 by Proposition 4.3.4 ii).

Proposition 4.3.6. Let V be a p-adic representation of GQp which satisﬁes the conditions F1-2). Then 1 1 0 0 H (D) = HIw(D)Γ0 ⊕ ∂0 H (D ) . Qp ∗ ϕ=pi † Proof. Since Dpst F−1Drig(V) (χ) = 0 for all i ∈ Z, by Lemma 2.7.4 2 † we have HIw(F−1Drig(V)) = 0. Then the tautological exact sequence † 0 → F−1Drig(V) → D → M0 → 0 induces an exact sequence 1 † 1 1 0 → HIw(F−1Drig(V)) → HIw(D) → HIw(M0) → 0. 0 1 ΓQp 0 Since HIw(M0) = H (M0) = 0 by Proposition 4.3.4, the snake lemma gives an exact sequence 1 † 1 1 (60) 0 → HIw(F−1D (V))Γ0 → HIw(D)Γ0 → HIw(M0)Γ0 → 0. rig Qp Qp Qp The Hochschild–Serre exact sequence 0 1 † 1 † 2 † ΓQp 0 → HIw(F−1D (V))Γ0 → H (F−1D (V)) → HIw(F−1D (V)) → 0 rig Qp rig rig together with the fact that

0 2 † ΓQp 2 † dimE HIw(F−1D (V)) = dimE HIw(F−1D (V))Γ0 = rig rig Qp 0 † ∗ = dimE H (F−1Drig(V)) (χ) = 0 1 † 1 † implies that HIw(F−1D (V))Γ0 = H (F−1D (V)). On the other hand, rig Qp rig 1 1 HIw(M0)Γ0 = Hc (M0) by Proposition 2.8.5. Therefore, the sequence (60) Qp reads 1 † 1 1 0 → H (F−1D (V)) → HIw(D)Γ0 → Hc (M0) → 0 rig Qp and we have a commutative diagram

1 † 1 1 (61) 0 / H (F−1Drig(V)) / HIw(D)Γ0 / Hc (M0) / 0 Qp _ _ = 1 † 1 1 0 / H (F−1Drig(V)) / H (D) / H (M0) / 0. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 71

0 † † Since H (Drig(V)/F1Drig(V)) = 0, the exact sequence 0 † † 0 → M1 → D → Drig(V)/F1Drig(V) → 0 0 0 0 gives H (M1) = H (D ) and we have a commutative diagram

∂0 (62) H0(D0) / H1(D) O = 0 δ0 1 H (M1) / H (M0). 1 0 Finally, from F2) it follows that Hc (M0) ∩ δ0 H (M1) = {0}, and the dimension argument shows that 1 1 0 (63) H (M0) = Hc (M0) ⊕ δ0 H (M1) . Now, the proposition follows from (63) and the diagrams (61) and (62).

5. p-ADIC HEIGHT PAIRINGS II: UNIVERSALNORMS norm 5.1. The pairing hV,D . norm 5.1.1. In this section, we construct the pairing hV,D , which is a direct generalization of the pairing constructed in [33], Section 6 and [36]. Let V is a p-adic representation of GF,S with coefficients in a finite extension E of † Qp. We fix a system D = (Dq)q∈Sp of submodules Dq ⊂ Drig(Vq) and denote ⊥ ⊥ by D = (Dq )q∈Sp the orthogonal complement of D. We have tautological exact sequences † 0 0 → Dq → Drig(Vq) → Dq → 0, q ∈ Sp, 0 † where Dq = Drig(Vq)/Dq. Passing to duals, we have exact sequences 0 ∗ † ∗ ∗ 0 → (Dq) (χq) → Drig(Vq (1)) → Dq(χq) → 0, 0 ∗ ⊥ where (Dq) (χq) = Dq . If the contrary is not explicitly stated, in this section we will assume that the following conditions hold

0 0 ∗ N1) H (Fq,V) = H (Fq,V (1)) = 0 for all q ∈ Sp; 0 0 0 ∗ N2) H (Dq) = H (Dq(χq)) = 0 for all q ∈ Sp.

1 1 From N2), it follows immediately that H (Dq) injects into H (Fq,V). By our deﬁnition of Selmer complexes we have     1 1 1 1 M H (Fq,V) M M H (Fq,V) H (V,D) ' kerHS (V) → 1   1 , H (Fq,V) H (Dq) q∈Σp f v∈Sp 72 DENIS BENOIS

∗ ⊥ and the same formula holds for V (1) if we replace Dq by Dq . Using this + 1 isomorphism, we identify each cohomology class [(x,(xq ),(λq)] ∈ H (V,D) 1 with the corresponding class [x] ∈ HS (V).

1 ∗ ⊥ 5.1.2. Let [y] ∈ H (V (1),D ) and let Yy be the associated extention ∗ 0 → V (1) → Yy → E → 0. Passing to duals, we have an exact sequence ∗ 0 → E(1) → Yy (1) → V → 0.

For each q ∈ Sp, this sequence induces an exact sequence of (ϕ,Γq)-modules † ∗ † 0 → RFq,E(χq) → Drig(Yy (1)q) → Drig(Vq) → 0. Consider the commutative diagram

π δ 1 1 1 D,q 1 D,q 2 0 / H (Fq,E(1)) / H (Dq,y) / H (Dq) / H (Fq,E(1)) _ _ = gq,y gq = π δ 1 1 1 ∗ q 1 V,q 2 0 / H (Fq,E(1)) / H (Fq,Yy (1)) / H (Fq,V) / H (Fq,E(1)) O O O O resq resq resq δ 1 1 1 ∗ π 1 V 2 0 / HS (E(1)) / HS (Yy (1)) / HS (V) / HS (E(1)),

† where Dq,y denotes the inverse image of Dq in Drig(Vq). In the following lemma we do not assume that the condition N2) holds. Lemma 5.1.3. Assume that V is a p-adic representation satisfying the con- + 1 dition N1). Let [x] = [(x,(xq ),(λq))] ∈ H (V,D) and let xq = resq(x). Then 1 i) If q - p, then Hf (Fq,E(1)) = 0 and 1 ∗ 1 Hf (Fq,Yy (1)) ' Hf (Fq,V). 1 1 + ii) For each q ∈ Sp one has δV,q([xq]) = δD,q([xq ]) = 0. 1 iii) δV ([x]) = 0. iv) The sequence 1 1 ∗ 1 0 → H (E(1),R(χ)) → H (Yy (1),Dy) → H (V,D) → 0, where R(χ) = (RFq,E(χq))q∈Sp , is exact. 0 ur Proof. i) If q - p, then E(1) is unramiﬁed at q, H (Fq /Fq,E(1)) = 0 and 1 1 ur Hf (Fq,E(1)) = H (Fq /Fq,E(1)) = E(1)/(Frq − 1)E(1) = 0. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 73

Since [y] is unramiﬁed at q, the sequence

∗ Iq Iq 0 → E(1) → Yy (1) → V → 0 is exact. Passing to the associated long exact cohomology sequence of ur Gal(Fq /Fq) and taking into account that 1 ur 2 ur H (Fq /Fq,E(1)) = H (Fq /Fq,E(1)) = 0

1 ur ∗ Iq ∼ 1 ur Iq we obtain that H (Fq /Fq,Yy (1) ) → H (Fq /Fq,V ). This proves i). + ii) For each q ∈ Sp we have gq([xq ]) = [xq]. From the orthogonality of Dq ⊥ and Dq it follows that 1 + + + δD(xq ) = −xq ∪ yq = 0. 1 1 + Therefore, δV,q([xq]) = δD,q([xq ]) = 0 for each q ∈ Sp. 1 iii) Let q ∈ Σp. Since [xq] ∈ Hf (Fq,V), from i) it follows that again δV,q([xq]) = 0. As the localization map 2 M 2 HS (E(1)) → H (Fq,E(1)) v∈S 1 is injective, we obtain that δV (x) = 0. 1 1 iv) First prove the surjectivity of π : H (Yy(1),Dy) → H (V,D). For each q ∈ Σp we denote by 1 ∗ 1 sy,q : Hf (Fq,Yy (1)) ' Hf (Fq,V) 1 1 the inverse of the isomorphism i). Let [x] ∈ H (V,D). By ii), δV ([x]) = 0, 1 ∗ and there exists [a] ∈ HS (Yy (1)) such that π([a]) = [x]. Let q ∈ Σp. Since + 1 + 1 [xq ] ∈ Hf (Fq,V), there exists [bq ] ∈ H (Fq,E(1)) such that + + + [aq ] = sy,q([xq ]) + [bq ]. 1 L 1 The localization map HS (E(1)) → H (Fq,E(1)) is surjective, and there q∈Σp 1 + exists [b] ∈ HS (E(1)) such that [bq] = resq([bq]) = [bq ] for each q ∈ Σp. 1 ∗ Then [cx] = [a] − [b] ∈ H (Yy (1),Dy) and satisﬁes π([cx]) = [x]. Thus, the map π is surjective. Finally, from i) we have   1 1 M 1 H (E(1),R(χ)) = kerHS (E(1)) → H (Fq,E(1)), q∈Σp and it is easy to see that H1(E(1),R(χ)) coincides with the kernel of π. The lemma is proved. 74 DENIS BENOIS

∗ 5.1.4. Let logp : Qp → Qp denote the p-adic logarithm normalized by ∗ logp(p) = 0. For each ﬁnite place q we deﬁne an homomorphism `q : Fq → Qp by ( logp(NFq/Qp (x)), if q | p, `q(x) = logp |x|q, if q - p, where NFq/Qp denotes the norm map. By linearity, `q can be extended to ∗ ∗ ∼ 1 a map `q : Fq ⊗b Zp E → E, and the isomorphism Fq ⊗b Zp E → H (Fq,E(1)) 1 allows to consider `q as a map H (Fq,E(1)) → E which we denote again by `q. From the product formula

|NF/Q(x)|∞ ∏ |x|q = 1 q∈S f

and the fact that NF/Q(x) = ∏NFq/Qp (x) it follows that q|p

∗ (64) ∑ `q(x) = 1, ∀x ∈ F . q∈S f

0 We set ΛFq = OE[[Γv]] and ΛFq,E = ΛFq [1/p].

Lemma 5.1.5. Let V be a p-adic representation of GF,S that satisﬁes N1- 1 ∗ ⊥ 2) and let [y] ∈ H (V (1),D ). For each q ∈ Sp, the following diagram is commutative with exact rows and columns

(65) 0 0

`q ( 0) ⊗ H1 (F ,E( )) H1(F ,E( )) E H Γq ΛFq,E Iw q 1 / q 1 /

pr 1 q,y 1 HIw(Dq,y) / H (Dq,y)

Iw πD,q πD,q pr 1 q 1 HIw(Dq) / H (Dq) / 0

2 = 2 H (Fq,E(1)) / H (Fq,E(1)). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 75

Proof. The exacteness of the left column is clear. The exactness of the right column follows from the fact that the diagram inv 2 v,n H (Fq,n,E(1)) / E

cores id inv 2 v,n−1 H (Fq,n−1,E(1)) / E is commutative, and therefore 2 2 HIw(Fq,E(1)) ' H (Fq,E(1)) ' E. The diagram (65) is clearly commutative. Now, we prove that the projection 1 1 map HIw(Dq) → H (Dq) is surjective. We have an exact sequence 1 1 2 Γ0 0 → H (D ) 0 → H (D ) → H (D ) q → 0, Iw q Γq q Iw q 0 2 Γq and therefore it is enough to show that HIw(Dq) = 0. Consider the exact sequence

2 Γ0 2 γv−1 2 2 0 → H (D ) q → H (D ) −−−→ H (D ) → H (D ) 0 → 0. Iw q Iw q Iw q Iw q Γq 2 Since HIw(Dq) is a ﬁnite-dimensional E-vector space, we have 2 Γ0 2 2 0 ∗ dim H (D ) q = dim H (D ) 0 = dim H (D ) = dim H (D (χ)) = 0. E Iw q E Iw q Γq E q E q 1 1 Thus, the map HIw(Dq) → H (Dq) is surjective. To prove the exactness of the ﬁrst row, we remark that the sequence

1 1 `q HIw(Fq,E(1)) → H (Fq,E(1)) −→ E is known to be exact (see, for example, [34], Section 11.3.5), and that the ( 0) ⊗ H1 (F ,E( )) → H1(F ,E( )) image of the projection H Γq ΛFq,E Iw q 1 q 1 coin- 1 1 cides with the image of the projection HIw(Fq,E(1)) → H (Fq,E(1)).

5.1.6. By Lemma 5.1.5, for each q ∈ Sp we have the following commutative diagram with exact rows, where the map prq is surjective (66) πIw 1 D,q 1 2 HIw(Dq,y) / HIw(Dq) / H (Fq,E(1))

prq,y prq = 1 π δ 1 1 D,q 1 D,v 2 0 / H (Fq,E(1)) / H (Dq,y) / H (Dq) / H (Fq,E(1)) _ _ = gq,y gq π 1 1 ∗ q 1 0 / H (Fq,E(1)) / H (Fq,Yy (1)) / H (Fq,V). 76 DENIS BENOIS

1 + Let [x] ∈ H (V,D). By Lemma 5.1.3 ii), for each q ∈ Sp we have δD,q([xq ]) = Iw 1 Iw Iw 0, and therefore there exists [xq,y] ∈ HIw(Dq,y) such that prq ◦πD,q [xq,y] = + 1 ∗ [xq ]. By Lemma 5.1.3 iv), there exists a lift [cx] ∈ H (Yy (1),Dy) of [x]. For each v ∈ Sp we set

Iw (67) [uq] = gq,y ◦ prq,y([xq,y]) − [cxq],

1 where xbq = resq(xb). Then πq([uq]) = 0, and therefore [uq] ∈ H (Fq,E(1)).

Deﬁnition. Let V be a p-adic representation of GF,S equipped with a family

D = (Dq)q∈Sp of (ϕ,Γq)-modules satisfying the conditions N1-2). The p- norm adic height pairing hV,D associated to these data is deﬁned to be the map

norm 1 1 ∗ ⊥ hV,D : H (V,D) × H (V (1),D ) → E, norm hV,D ([x],[y]) = ∑ `q ([uq]). q∈Sp

1 ∗ 5.1.7. Remarks. 1) If [fx] ∈ H (Yy (1),Dy) is another lift of [x], then from (64) and the fact that [cxq] = [fxq] = sy,q([xq]) for all q ∈ Sp, it follows that the norm deﬁnition of hV,D ([x],[y]) does not depend on the choice of the lift [cxq]. 1 ∗ 1 ∗ 2) It is not indispensable to take [cx] in H (Yy (1),Dy). If [cx] ∈ HS (Yy (1)) is such that π([cx]) = [x], we can again deﬁne [uq] by (67). For q ∈ Σp we set

+ [uq] = gq,y ◦ sy,q([xq ]) − [cxq],

1 ∗ ∼ 1 ∗ where sy,q : Hf (Fq,V (1)) → Hf (Fq,Yy (1)) denotes the isomorphism from 1 Lemma 6.1.2 i). Note that again [uq] ∈ H (Fv,E(1)). Then norm hV,D ([x],[y]) = ∑`q ([uq]). q∈S norm 3) The map hV,D is bilinear. This can be shown directly, but follows from Theorem 5.2.2 below.

sel 5.2. Comparision with hV,D.

norm 5.2.1. In this subsection we compare hV,D with the p-adic height pairing constructed in Subsection 3.2. We take Σ = /0and denote by

sel 1 1 ∗ ⊥ hV,D,1 : H (V,D) × H (V (1),D ) → E the associated height pairing (51). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 77

Theorem 5.2.2. Let V be a p-adic representation of GF,S with coefficients in a finite extension E of Qp. Assume that the family D = (Dq)q∈Sp satisfies norm the conditions N1-2). Then hV,D is a bilinear map and norm sel hV,D = hV,D,1. Proof. The proof repeats the arguments of [34], Sections 11.3.9-11.3.12, where this statement is proved in the case of p-adic height pairings arising from Greenberg’s local conditions. We remark that in this case our defini- norm norm tion of hV,D differs from Nekovár’sˇ hπ by a sign. Let [x] ∈ H1(V,D) and [y] ∈ H1(V ∗(1),D⊥). We use the notation of Sec- tion 3.1 and denote by fq and gq the morphisms defined by (24–27). As ⊥ before, to simplify notation we set xq = fq(x) and yq = fq (y). We rep- sel + 1 sel resent [x] and [y] by cocycles x = (x,(xq ),(λq)) ∈ S (V,D) and y = + 1 ∗ ⊥ (y,(yq ),(µq)) ∈ S (V (1),D ), where 1 + 1 0 x ∈ C (GF,S,V), xq ∈ Uq (V,D), λq ∈ K (Vq), 1 ∗ + 1 ∗ ⊥ 0 ∗ y ∈ C (GF,S,V (1)), yq ∈ Uq (V (1),D ), µq ∈ K (Vq (1)) and dx = 0, dy = 0, + + dxq = 0 dyq = 0, + ⊥ + ⊥ gq(xq ) = fq(x) + dλq, gq (yq ) = fq (x) + dµq, ∀q ∈ S. By Propositions 2.6.2, 2.6.4 and 2.6.5 we have sel + 2 (68) βV,D(x ) = (−z ∪ x,(−wq ∪ xq ),(zq ∪ λq)) ∈ S (V,D), where 1 z = log χ ∈ C (GF,S,E(0)), ( 0, if q ∈ Σp, wq = 1 (0,log χq(γq)) ∈ Cϕ,γq (E(0)), if q ∈ Sp, ( 1 log χq ∈ C (GFq ,E(0)), if q ∈ Σp, zq = 1 (0,log χq) ∈ K (E(0)q), if q ∈ Sp.

1 ∗ 1 Lemma 5.2.3. Let [cx] ∈ H (Yy (1),Dy) be a lift of an element [x] ∈ H (V,D). + 1 + 1 ∗ Represent [cx] and [xbq ] ∈ H (Dq,y) for q ∈ Sp (respectively [xbq ] ∈ Hf (Fq,Yy (1)) for q ∈ Σ ) by cocycles x ∈ C1(G ,Y ∗(1)) and x+ ∈ C1 (D ) (respec- p b F,S y bq ϕ,γq q,y + 1 ∗ + tively by xbq ∈ Cur(Yy (1)q)). Since gq(xbq ) = fq(xb) + dbλq for some bλq ∈ 78 DENIS BENOIS

0 ∗ sel + 1 ∗ K (Yy (1)q), we obtain a cocycle xb = (xb,(xbq ),(bλq)) ∈ S (Yy (1),Dy). Then

sel 2 ∗ β ∗ (x ) = (a,(b ),(c )) ∈ S (Y (1),D ), Yy (1),Dy b b bq bq y y where ( 0, if q ∈ Σp, bq = 2 wq ∪ uq ∈ Cϕ,γq (E(1)q), if q ∈ Sp and uq is a representative of the cohomology class (67).

Proof. The proof is analogous to the proof of Lemma 11.3.10 of [34]. Since • ∗ the complex Cur(Yy (1)v) is concentrated in degrees 0 and 1, it is clear that 0 bq = 0 for q ∈ Σp. For each q ∈ Sp, we consider the algebra HE(ΓF ) = 0 ι { f (γF − 1) | f (X) ∈ HE} and deﬁne IndF∞/F (Dq) = Dq⊗b EHE(ΓF ) . The 0 0 natural inclusion HE(Γq) ⊂ HE(ΓF ) allows us to consider IndF∞/F (Dq) as a †,r 0 (ϕ,Γq)-module over the ring RF , = limB ⊗HE(Γ ). Let J denote q HE −→ rig,Fq b q H r 0 the augmentation ideal of HE(ΓF ). The obvious commutative diagram

mod Γ0 0 q 0 0 HE(ΓF ) / E[ΓF /Γq]

2 mod JH AeF / E, where the right vertical morphism is the augmentation map, induces a commutative diagram

1 1 0 0 (69) H (IndF∞/F (Dq)) / H Dq ⊗E E[ΓF /Γq]

1 1 H (DeF,q) / H (Dq).

0 0 0 1 0 0 Since Γq acts trivially of E[ΓF /Γq], the group H Dq ⊗E E[ΓF /Γq] is iso- 0 0 1 morphic to the direct sum of (ΓF : Γq) copies of H (Dq) and the right ver- 0 0 tical map of (69) is surjective. Since HE(ΓF ) is a free HE(Γq)-module of 0 0 1 rank [ΓF : Γq], the group H (IndF∞/F (Dq)) is isomorphic to the direct sum of 0 0 1 1 1 [ΓF : Γq]-copies of HIw(Dq). Since the projection prq : HIw(Dq) → H (Dq) is surjective by Proposition 5.1.5, the upper horizontal map of (69) is also P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 79

1 1 surjective. Therefore the projection H (IndF∞/F (Dq)) → H (Dq) is surjective and we have the following analog of the diagram (66) (70) πIw,F 1 D,q 1 H (IndF∞/F (Dq,y)) / H (IndF∞/F (Dq)) / 0

F F prq,y prq π 1 1 D,q 1 0 / H (Fq,E(1)) / H (Dq,y) / H (Dq) / 0 _ _ = gq,y gq π 1 1 ∗ q 1 0 / H (Fq,E(1)) / H (Fq,Yy (1)) / H (Fq,V) / 0.

1 From the above discussion it follows that Cϕ,γq (IndF∞/F (Dq,y)) is isomor- 0 0 1 phic to the direct sum of [ΓF : Γq] copies of Cϕ,γq (IndFq,∞/Fq (Dq,y)). There- fore there exists a cocycle x ∈ C1 (Ind (D )) such that eq,y ϕ,γq F∞/F q,y F Iw prq,y(xeq,y) = prq,y(xq,y). Since the map • • Cϕ,γq (IndF∞/F (Dq,y)) → Cϕ,γq (Dq,y) • ι factors through Cϕ,γq (DeF,q,y), where DeF,q,y = Dq,y ⊗ AeF , from the distinguished triangle

• • • βDq,y Cϕ,γq (Dq,y) → Cϕ,γq (DeF,q,y) → Cϕ,γq (Dq,y) −−−→ Cϕ,γq (Dq,y)[1] ( F (x )) = . it follows that βDq,y prq,y eq,y 0 Thus, b = (x+) = (x+ − (xIw ) + F (x )) = bq βDq,y bq βDq,y bq prq,y q,y prq,y eq,y = − (u ) + ( F (x )) = − (u ) = w ∪ u . βDq,y q βDq,y prq,y eq,y βDq,y q q q The lemma is proved.

For each q ∈ Sp, we have the canonical isomorphism of local class ﬁeld theory 2 ∼ invq : H (Fq,E(1)) → E. ∗ 1 Let κq : Fq ⊗bE → H (Fq,E(1)) denote the Kummer map. Then

(71) invq(log χq ∪ κq(x)) = logp(NFq/Qp (x)) = `q(x) ([40], Chapitre 14, see also [2], Corollaire 1.1.3). sel 2 Lemma 5.2.4. Assume that βV,D([x ]) ∈ H (V,D) is represented by a 2- cocycle e = (a,(bq),(cq)) of the form e = π(eb), where 2 ∗ eb= (ab,(bq),(cbq)) ∈ S (Yy (1),Dy) 80 DENIS BENOIS

2 ∗ 2 is also a 2-cocycle and π : S (Yy (1),Dy) → S (V,D) denotes the canonical projection. Then

sel sel ⊥ [βV,D(x ) ∪ y ] = ∑ invq(gq(bq) ∪ fq (αy) + gq(bq) ∪ µq), q∈Sp

0 0 where αy ∈ C (GF,S,Yy) is an element that maps to 1 ∈ C (GF,S,E) = E and satisﬁes dαy = y. If, in addition,

2 bq ∈ Cϕ,γq (E(1)q), ∀q ∈ Sp, then sel sel [βV,D(x ) ∪ y ] = ∑ invq(gq(bq)). q∈Sp Proof. The proof of this lemma is purely formal and follows verbatim the proof of [34], Lemma 11.3.11. Now we can proof Theorem 5.2.2. Combining Lemma 5.2.3 and Lemma 5.2.4 we have

sel sel sel hV,D,1([x],[y]) = [βV,D(x ) ∪ y ] = ∑ invq(gq(bq)) = q∈Sp norm = ∑ invq(gq(wq ∪ uq)) = ∑ `q(uq) = hV,D ([x],[y]). q∈Sp q∈Sp

6. p-ADIC HEIGHT PAIRINGS III: SPLITTINGOFLOCAL EXTENSIONS spl 6.1. The pairing hV,D.

6.1.1. Let F be a finite extension of Q. We keep notation of Sections 3-5. In particular, we fix a finite set S of places of F such that Sp ⊂ S and denote by GF,S the Galois group of the maximal algebraic extension of F which is unramified outside S ∪ S∞. For each topological GF,S-module M, we write ∗ HS (M) for the continuous cohomology of GF,S with coefficients in M. Let V be a p-adic representation of GF,S with coefficients in a finite extension E/Qp which is potentially semistable at all q | p. Following Bloch 1 1 and Kato, for each q ∈ S we define the subgroup Hf (Fq,V) of H (Fq,V) by

( 1 1 1 ker(H (Fq,V) → H (Fq,V ⊗Qp Bcris)) if q | p, Hf (Fq,V) = 1 1 ur ker(H (Fq,V) → H (Fq ,V)) if q - p. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 81

The Bloch–Kato Selmer group of V is deﬁned as 1 ! 1 1 MH (Fq,V) Hf (V) = ker HS (V) → 1 . q∈S Hf (Fq,V)

In this section, we assume that, for all q ∈ Sp, the representation Vq satisﬁes the condition S) of Section 4, namely that

ϕ=1 ∗ ϕ=1 S) Dcris(Vq) = Dcris(Vq (1)) = 0 for all q ∈ Sp.

For each q | p, we ﬁx a splitting (ϕ,N,GFq )-submodule Dq of Dpst(Vq) (see Section 4.1). We will associate to these data a pairing spl 1 1 ∗ hV,D : Hf (V) × Hf (V (1)) → E and compare it with the height pairing constructed in Section 4 of [33] using the exponential map and splitting of the Hodge ﬁltration. 1 ∗ 1 ∗ Let [y] ∈ Hf (V (1)). Fix a representative y ∈ C (GF,S,V (1)) of y and consider the corresponding extension of Galois representations ∗ (72) 0 → V (1) → Yy → E → 0. Passing to duals, we obtain an extension ∗ 0 → E(1) → Yy (1) → V → 0. 0 From S) it follows that HS (V) = 0, and the associated long exact sequence of global Galois cohomology reads 1 1 1 ∗ 1 δV 2 0 → HS (E(1)) → HS (Yy (1)) → HS (V) −→ HS (E(1)) → .... Also, for each place q ∈ S we have the long exact sequence of local Galois cohomology 0 1 1 ∗ H (Fq,V) → H (Fq,E(1)) → H (Fq,Yy (1)) → δ 1 1 V,q 2 → H (Fq,V) −−→ H (Fq,E(1)) → .... The following results, which can be seen as an analog of Lemma 5.1.3, are well known but we recall them for the reader’s convenience.

Lemma 6.1.2. Let V be a p-adic representation of GF,S that is potentially semistable at all q ∈ Sp and satisﬁes the condition S). Assume that [y] ∈ 1 ∗ Hf (V (1)). Then 1 1 i) δV ([x]) = 0 for all x ∈ Hf (V); ii) There exists an exact sequence 1 1 ∗ 1 0 → Hf (E(1)) → Hf (Yy (1)) → Hf (V) → 0. 82 DENIS BENOIS

1 1 Proof. i) For any x ∈ C (GF,S,V), let xq = resq(x) ∈ C (GFq ,V) denote the 1 1 localization of x at q. If [x] ∈ Hf (V), then for each q one has δV,q([xq]) = 1 1 ∗ −[xq]∪[yq] = 0 because Hf (Fq,V) and Hf (Fq,V (1)) are orthogonal to each 2 L 2 other under the cup product. Since the map HS (E(1)) → H (Fq,E(1)) is q∈S injective and the localization commutes with cup products, this shows that 1 δV ([x]) = 0. ii) This is a particular case of [21], Proposition II, 2.2.3.

1 1 ∗ 6.1.3. Let [x] ∈ Hf (V) and [y] ∈ Hf (V (1)). In Section 4.2, for each q ∈ Sp we constructed the canonical splitting (55) which sits in the diagram

sy,q 1 1 o 1 0 / Hf (Fq,E(1)) / Hf (Dq,y) / Hf (Dq) / 0

= gq,y ' gq ' 1 1 ∗ 1 0 / Hf (Fq,E(1)) / Hf (Fq,Yy (1)) / Hf (Fq,V) / 0.

1 1 ∗ By Lemma 6.1.2 ii), we can lift [x] ∈ Hf (V) to an element [cx] ∈ Hf (Yy (1)). 1 ∗ + Let [cxq] = resq [cx] ∈ Hf (Fq,Yy (1)). If q ∈ Sp, we denote by [xeq ] the 1 + unique element of Hf (Dq) such that gq([xeq ]) = [xq]. Deﬁnition. The p-adic height pairing associated to splitting submodules D = (Dq)q∈S is deﬁned to be the map

spl 1 1 ∗ hV,D : Hf (V) × Hf (V (1)) → E given by spl + hV,D([x],[y]) = ∑ `q gq,y ◦ sy,q([xeq ]) − [cxq] . q∈Sp

1 ∼ 1 ∗ 6.1.4. Remarks. 1) For each q ∈ Σp, denote by sy,q : Hf (Fq,V) → Hf (Fq,Yy (1)) 1 the isomorphism constructed in Lemma 5.1.3, i) and by gq : Hf (Fq,V) ,→ 1 1 ∗ 1 ∗ H (Fq,V) and gq,y : Hf (Fq,Yy (1)) ,→ H (Fq,Yy (1)) the canonical embed- + 1 + dings. Let [xeq ] ∈ Hf (Fq,V) be the unique element such that gq([xq ]) = [xq]. spl From the product formula (64) it follows, that hV,D can be deﬁned by

spl + hV,D([x],[y]) = ∑`q gq,y ◦ sy,q([xeq ]) − [cxq] , q∈S

1 where [cx] ∈ HS (V) is an arbitrary lift of [x]. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 83

spl 2) The pairing hV,D is a bilinear skew-symmetric map. This can be shown spl directly, but follows from the interpretation of hV,D in terms of Nekovár’sˇ height pairing (see Proposition 6.2.3 below). 6.2. Comparision with Nekovár’sˇ height pairing. spl 6.2.1. We relate the pairing hV,D to the p-adic height pairing constructed by Nekovárˇ in [33], Section 4. First recall Nekovár’sˇ construction. If [y] ∈ 1 ∗ Hf (V (1)), the extension (72) is crystalline at all q ∈ Sp, and therefore the sequence ∗ 0 → Dcris(Vq (1)) → Dcris(Yy,q) → Dcris(E(0)q) → 0 ∗ ϕ=1 is exact. Since Dcris(Vq (1)) = 0, we have an isomorphism of vector spaces ∼ ϕ=1 Dcris(E(0)q) → Dcris(Yy,q) , which can be extended by linearity to a map DdR(E(0)q) → DdR(Yy,q). Pass- ∗ ing to duals, we obtain a Fq-linear map DdR(Yy,q(1)) → DdR(E(1)q) which deﬁnes a splitting sdR,q of the exact sequence

sdR,q ∗ o 0 / DdR(E(1)q) / DdR(Yy,q(1)) / DdR(Vq) / 0. 0 Fix a splitting wq : DdR(Vq)/Fil DdR(Vq) → DdR(Vq) of the canonical projection 0 (73) prdR,Vq : DdR(Vq) → DdR(Vq)/Fil DdR(Vq). We have a commutative diagram w sy,q 1 1 ∗ o 1 0 / H (Fq,E(1)) / Hf (Fq,Yy (1)) / Hf (Fq,V) / 0 O O exp ∗ Yy,q(1) ' expVq ' ∗ DdR(Yy,q(1)) DdR(Vq) / 0 ∗ 0 Fil DdR(Yy,q(1)) Fil DdR(Vq) O pr ∗ dR,Vy,q(1) wq s ∗ dR,q DdR(Yy,q(1)) o DdR(Vq). w 1 1 ∗ Then the map sy,q : Hf (Fq,V) → Hf (Fq,Yy (1)) deﬁned by w −1 s = exp ∗ ◦pr ∗ ◦ s ◦ w ◦ exp y,q Yy,q(1) dR,Yy,q(1) dR,q q Vq gives a splitting of the top row of the diagram, which depends only on the choice of wq and [y]. 84 DENIS BENOIS

Deﬁnition (NEKOVÁRˇ ). The p-adic height pairing associated to a family w = (wq)q∈Sp of splitting wq of the projections (73) is deﬁned to be the map Hodge 1 1 ∗ hV,w : Hf (V) × Hf (V (1)) → E given by Hodge w hV,w ([x],[y]) = ∑`q sy,q([xq]) − [cxq] , q|p 1 ∗ 1 where [cx] ∈ Hf (Yy (1)) is a lift of [x] ∈ Hf (V) and [cxq] denotes its localization at q. Hodge In [33], it is proved that hV,w is a E-bilinear map.

6.2.2. Now, let Dq be a splitting submodule of Dst/L(Vq). We have 0 (74) DdR/L(Vq) = Dq,L ⊕ Fil DdR/L(Vq), Dq,L = Dq ⊗L0 L.

GFq Set Dq,Fq = (Dq,L) . Since the decomposition (74) is compatible with the Galois action, taking galois invariants we have 0 DdR(Vq) = Dq,Fq ⊕ Fil DdR(Vq). This decomposition deﬁnes a splitting of the projection (73) which we will denote by wD,q.

Proposition 6.2.3. Let V be a p-adic representation of GF,S such that for each q ∈ Sp the restriction of V on the decomposition group at q is potentially semistable and satisﬁes the condition S). Let (Dq)q∈Sp be a family of splitting submodules and let wD = (wD,q)q∈Sp be the associated system of splittings. Then hspl = hHodge. V,D V,wD

We need the following auxiliary result. As before, we denote by Dq the (ϕ,Γq)-module associated to Dq. Lemma 6.2.4. The following diagram

sDq,y DdR(Dq) / DdR(Dq,y)

s dR,q ∗ DdR(Vq) / DdR(Yy,q(1)), where the vertical maps are induced by the canonical inclusions of corresponding (ϕ,Γq)-modules and sDq,y is the map induced by the splitting (54), is commutative.

Proof of the lemma. The proof is an easy exercice and is omitted here. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 85

Proof of Proposition 6.2.3. From the functoriality of the exponential map and Proposition 4.1.2 it follows that the diagram exp Dq 1 (75) DdR(Dq) / Hf (Dq)

= = exp 0 Vq 1 DdR(Vq)/Fil DdR(Vq) / Hf (Fq,V)

∗ is commutative. The same holds if we replace Vq and Dq by Yy,q(1) and Dq,y respectively. Consider the diagram

sDq,y (76) DdR(Dq) / DdR(Dq,y) O = ∗ D (V ) DdR(Y (1)) dR q y,q , 0 0 ∗ Fil DdR(Vq) Fil DdR(Yy,q(1)) O pr ∗ wD,q dR,Yy,q(1) s dR,q ∗ DdR(Vq) / DdR(Yy,q(1)).

From the deﬁnition of wD,q, it follows that the composition of vertical maps † in the left (resp. right) colomn is induced by the inclusion Dq ⊂ Drig(Vq) † ∗ (resp. by Dq,y ⊂ Drig(Yy,q(1))) and therefore the diagram (76) is commutative by Lemma 6.2.4. From the commutativity of (75) and (76) and the w w deﬁnition of sy,q and sy,q, it follows now that sy,q = sy,q for all q ∈ Sp, and the proposition is proved. norm 6.3. Comparision with hV,D .

spl norm 6.3.1. In this section, we compare the pairing hV,D with the pairing hV,D constructed in Section 5. Let V be a p-adic representation of GF,S that is potentially semistable at all q ∈ Sp. Fix a system (Dq)q∈Sp of splitting submodules and denote by (Dq)q∈Sp the system of (ϕ,Γq)-submodules of † Drig(Vq) associated to (Dq)q∈Sp by Theorem 2.2.3. We will assume, that (V,D) satisﬁes the condition S) of Section 6.1 and the condition N2) of Section 5.1. Note, that S) implies N1). We also remark, that from Proposi- tion 2.8.2 i) and the fact that the Hodge–Tate weights of Dst/L(Vq)/Dq and ∗ ⊥ Dst/L(Vq (1))/Dq are positive, it follows that, under our assumptions, N2) is equivalent to the following condition 86 DENIS BENOIS

N2’) For each q ∈ Sp,

ϕ=1,N=0,GL/Fq ∗ ⊥ ϕ=1,N=0,GL/Fq (Dst/L(Vq)/Dq) = (Dst/L(Vq (1))/Dq ) = 0, ∗ where L is a ﬁnite extension of Fq such that Vq (respectively Vq (1)) is semistable over L.

6.3.2. The following statement is known (see [37] and [6]), but we prove it here for completeness. Proposition 6.3.3. Assume that V is a p-adic representation satisfying the conditions S) and N2). Then 1 1 1 1 ∗ 1 ⊥ i) Hf (Fq,V) = Hf (Dq) = H (Dq) and Hf (Fq,V (1)) = Hf (Dq ) = 1 ⊥ = H (Dq ) for all q ∈ Sp. 1 1 1 ∗ 1 ∗ ⊥ ii) Hf (V) ' H (V,D) and Hf (V (1)) ' H (V (1),D ). Proof. i) The ﬁrst statement follows from N2) and Proposition 4.1.2 iii). ii) Note that by i) ( 1 1 Hf (Fq,V), if q ∈ Σp, R Γ(Fq,V,D) = 1 H (Dq), if q ∈ Sp. By deﬁnition, the group H1(V,D) is the kernel of the morphism     1 M M 1 M M 1 M 1 HS (V)  Hf (Fq,V)  H (Dq) → H (Fq,V) q∈Σp q∈Sp q∈S given by

([x],[yq]q∈S) 7→ ([xq] − gq(yq))q∈S, [xq] = resq([x]), 1 1 where gq denotes the canonical inclusion Hf (Fq,V) → H (Fq,V) if q ∈ Σp 1 1 and the map H (Dq) → H (Fq,V) if q ∈ Sp. In the both cases, gq is in- 1 1 jective and, in addition, for each q ∈ Sp we have H (Dq) = Hf (Fq,V) by 1 1 i). This implies that H (V,D) = Hf (V). The same argument shows that 1 ∗ ⊥ 1 ∗ H (V (1),D ) = Hf (V (1)).

Theorem 6.3.4. Let V be a p-adic representation such that Vq is potentially semistable for each q ∈ Sp, and let (Dq)q∈Sp be a family of splitting submodules. Assume that (V,D) satisfies the conditions S) and N2). Then norm spl hV,D = hV,D. + Proof. First note, that in our case the element [xeq ], defined in Section 6.1.3, + norm spl coincides with [xq ]. Comparing the definitions of hV,D and hV,D we see that P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 87

Iw + it is enough to show that `q prq,y([xq,y]) − sq,y([xq ]) = 0 for all q ∈ Sp. The splitting sy,q of the exact sequence 1 1 1 0 → Hf (Fq,E(1)) → Hf (Dq,y) → H (Dq) → 0 (see (55)) gives an isomorphism 1 1 1 1 1 H (D ) 0 ' H (D ) 0 ⊕H ( (χ)) 0 ' H (D )⊕H (F ,E(1)) 0 . Iw q,y Γq Iw q Γq Iw RFq,E Γq q Iw q Γq Iw + Since πD,q prq,y([xq,y]) − sq,y([xq ]) = 0, from this decomposition it follows that + 1 pr ([w ]) − s ([x ]) ∈ H (F ,E(1)) 0 = ker(` ), q,y q q,y q Iw q Γq q and the theorem is proved. Corollary 6.3.5. If (V,D) satisﬁes the conditions S) and N2), then the sel norm spl height pairings hV,D,1, hV,D and hV,D coincide. Proof. This follows from Theorems 5.2.2 and 6.3.4.

7. p-ADIC HEIGHT PAIRINGS IV: EXTENDED SELMERGROUPS 7.1. Extended Selmer groups.

7.1.1. Let F = Q. Let V be a p-adic representation of GQ,S that is potentially semistable at p. We ﬁx a splitting submodule D of V. In Sec- † tion 4.3, we associated to D a canonical ﬁltration FiDrig(V) . Re- −26i62 † call that F0Drig(V) = D. We maintain the notation of Section 4.3 and set † † † † M0 = D/F−1Drig(V), M1 = F1Drig(V)/D and W = F1Drig(V)/F−1Drig(V). The exact sequence

0 → M0 → W → M1 → 0 0 1 induces the coboundary map δ0 : H (M1) → H (M0). In this section, we norm generalize the construction of the height pairing hV,D to the case when V satisﬁes the conditions F1-2) of Subsection 4.3, namely F1) For all i ∈ Z † † ϕ=pi † ϕ=pi Dpst(Drig(V)/F1Drig(V)) = Dpst(F−1Drig(V)) = 0. F2) The composed maps

0 δ0 1 prc 1 δ0,c : H (M1) −→ H (M0) −→ Hc (M0), pr 0 δ0 1 f 1 δ0, f : H (M1) −→ H (M0) −−→ Hf (M0), where the second arrows denote the canonical projections, are isomorphisms. Note, that if V satisﬁes the conditions N1-2) of Section 5, we have M0 = M1 = 0. 88 DENIS BENOIS

7.1.2. Deﬁne a bilinear map 1 1 ∗ h , iD, f : Hf (M0) × Hf (M1(χ)) → E as the composition

(δ −1,id) 1 1 ∗ 0, f 0 1 ∗ ∪ Hf (M0) × Hf (M1(χ)) −−−−→ H (M1) × Hf (M1(χ)) −→ ` 1 Qp H (RQp,E(χ)) −−→ E. 1 1 ∗ Lemma 7.1.3. For all x ∈ Hf (M0) and y ∈ Hf (M1(χ)) we have −1 −1 hx,yiD, f = −[i ∗ (y),δ f (x)]M1 , M1(χ), f 0, ∗ where [ , ]M0 : Dcris(M1(χ)) × Dcris(M1) → E denotes the canonical du- ∗ 1 ∗ ality and i ∗ : D (M (χ)) → H (M (χ)) is the isomorphism con- M1(χ), f cris 1 f 1 structed in Proposition 2.8.4. 1 Proof. Recall that for each z ∈ H (RQp,E(χ)) we have invp(wp ∪z) = `p(z), where wp = (0,log χ(γQp )). Therefore, using Proposition 2.8.4, we obtain −1 −1 hx,yiD, f = `Qp (δ0, f (x) ∪ y) = invp(wp ∪ δ0, f (x) ∪ y) = −1 = −invp(iM1,c(δ0, f (x)) ∪ y)) = −1 −1 ∗ = −invp(iM1,c(δ f (x)) ∪ iM (χ), f ◦ i ∗ (y)) = 0, 1 M1(χ), f −1 −1 = −[i ∗ (y),δ f (x)]M1 . M1(χ), f 0,

7.1.4. Let ρD, f andρD,c denote the composed maps pr 1 1 f 1 ρD, f : H (D) → H (M0) −−→ Hf (M0), 1 1 prc 1 ρD,c : H (D) → H (M0) −→ Hc (M0). 0 0 0 Note that H (M1) = H (D ).

Proposition 7.1.5. Let V be a p-adic representation of GQ,S which is potentially semistable at p. Assume that the restriction of V on the decomposition group at p satisﬁes the conditions F1-2) of Section 4.3. Then i) There exists an exact sequence 0 0 1 1 (77) 0 → H (D ) → H (V,D) → Hf (V) → 0. ii) The map 1 0 0 splV,D : H (V,D) → H (D ), + −1 + (x,(xq ),(λq)) 7→ δ0,c ◦ ρD,c [xp ] P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 89 deﬁnes a canonical splitting of (77).

Proof. The first statement is proved in [6], Proposition 11. It follows directly from the definition of Selmer complexes and the exact sequence (59). The second statement follows immediately from the definition of splV,D.

From Proposition 7.1.5 it follows that we have a canonical decomposition

1 1 0 0 H (V,D) ' Hf (V) ⊕ H (D )

1 1 and we denote by jV,D : Hf (V) → H (V,D) the resulting injection.

norm 7.2. The pairing hV,D for extended Selmer groups.

1 ∗ 7.2.1. We keep previous notation and conventions. Let [y] ∈ Hf (V (1)) and let Yy denote the associated extention (72). As before, we denote by † ∗ Dy the inverse image of D in Drig(Yy (1)p). By Proposition 4.3.4 iii), the representation Vp satisﬁes the condition S), and therefore the exact sequence (54) have a canonical splitting sD,y. Thus, we have the following diagram which can be seen as an analog of the diagram (65) in our situation

0 0

`Qp (Γ0 ) ⊗ H1 (Q ,E(1)) / H1(Q ,E(1)) / E H Qp ΛE,Qp Iw p p

prD,y ∂ 1 1 0 ? _ 0 0 HIw(Dy) / H (Dy) o H (D ) Iw = πD πD pr ∂ 1 D 1 0 ? _ 0 0 HIw(D) / H (D) o H (D )

0 0. 90 DENIS BENOIS

In the diagram (66), the maps gv and gv,y are no more injective and we consider the following diagram with exact rows and columns 0 0

= H0(D0) / H0(D0)

∂0 ∂0 1 1 πD 1 0 / H (Qp,E(1)) / H (Dy) / H (D) / 0

= gp,y gp π 1 1 ∗ p 1 0 / H (Qp,E(1)) / H (Qp,Yy (1)) / H (Qp,V). 1 1 By Proposition 4.3.4 v), Im(gp) = Hf (Qp,V). Let [x] ∈ Hf (V) and let h + i + 1 (x,(xeq ),(eλq)) = jV,D([x]). Then [xep ] is the unique element of Hf (D) such + that gp([xep ]) = [xp]. Let 1 ∗ ! H (Qq,Y (1)) [x] ∈ ker H1(Y ∗(1)) → y c S y 1 ∗ Hf (Qq,Yy (1)) be an arbitrary lift of [x]. (Note, that by Lemma 6.1.2, we can even take 1 ∗ + 1 [cx] ∈ Hf (Yy (1)).) By the five lemma there exists a unique [xcp ] ∈ H (Dy) + + + such that gp,y([xcp ]) = fp([cx]) and πD([xcp ]) = [xep ]. On the other hand, from Iw 1 Proposition 4.3.6 it follows that there exist xp,y ∈ HIw(Dy) and [tp] ∈ H0(D0) such that + Iw Iw (78) [xep ] + ∂0([tp]) = prD ◦ πD xp,y . Set Iw + (79) [up] = prD,y xp,y − ∂0(tp) − [xcp ]. 1 Then [up] ∈ H (Qp,E(1)). Definition. Let V be a p-adic representation satisfying the conditions F1- 2). We define the height pairing norm 1 1 ∗ hV,D : Hf (V) × Hf (V (1)) → E by norm hV,D ([x],[y]) = `Qp ([up]). norm It is easy to see that hV,D ([x],[y]) does not depend on the choice of the lift Iw xp,y . The following result generalizes Theorem 11.4.6 of [34]. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 91

Theorem 7.2.2. Let V be a p-adic representation of GQ,S that is potentially semistable at p and satisﬁes the conditions F1-2). Then norm spl i) hV,D = hV,D; sel + 1 sel + ii) For all [x ] = [(x,(xq ),(λq))] ∈ H (V,D) and [y ] = [(y,(yq ),(µq))] ∈ H1(V ∗(1),D⊥) we have sel sel sel norm D + + E hV,D([x ],[y ]) = hV,D ([x],[y]) + ρD, f ([xp ]),ρD⊥, f (yp ) . D, f

norm 1 ∗ Proof. i) Recall that in the definition of hV,D we can take [cx] ∈ Hf (Yy (1)). norm spl Comparing the definitions of hV,D and hV,D, we see that it is enough to prove that [u ] − s ([x+]) − [x ] ∈ (` ), p y,p ep bp ker Qp where [up] is defined by (79) and sy,p denotes the splitting (55). Since the 1 restriction of gp,y on H (Qp,E(1)) is the identity map, we have Iw [up] = gp,y([up]) = gp,y([xp,y]) − [xbp], and it is enough to check that g ([xIw ]) − g ◦ s ([x+]) ∈ (` ). (80) p,y p,y p,y y,p ep ker Qp Iw First remark that the canonical splitting (54) induces splittings sp,y and sp,y in the diagram

Iw sp,y 1 1 o 1 0 / HIw(RQp,E(χ)) / HIw(Dy) / HIw(D) / 0

prD,y prD sp,y 1 1 o 1 0 / H (Qp,E(1)) / H (Dy) / H (D) / 0. Iw Write [xp,y] in the form Iw Iw Iw Iw Iw 1 Iw 1 [xp,y] = sp,y(a ) + b , a ∈ HIw(D), b ∈ HIw(RQp,E(χ)). Iw By the deﬁnition of [xp,y], we have Iw prD,y([xp,y]) = sy,p(a) + b, 1 where b ∈ ker(`Qp ) = H (Qp,E(1))Γ0 and Qp + 1 a = ∂([tp]) + sp,y([xep ]) ∈ H (D). Since gp,y(sy,p(∂0([tp])) = 0, we have Iw + gp,y(prD,y([xp,y])) = b + gp,y(sy,p(a)) = b + gp,y(sy,p([xep ])), and (80) is checked. 92 DENIS BENOIS

ii) The proof is the same as that of Theorem 11.4.6 of [34] with obvious sel + modiﬁcations. First note that jV,D([x ]) = (x,(xeq ),(eλq)), where

+ + −1 + xep = xp − ∂0 ◦ δ0,c ◦ ρD,c [xp ] . Set −1 + (81) [tp] = −δ0, f ◦ ρD, f ([xp ]). Then + + ρD, f ([xep ]) + ρD, f (∂0([tp])) = ρD, f ([xep ]) + δ0, f ([tp]) = 0. + 1 1 Thus, the image of [xep ] + ∂0([tp]) under the projection H (D) → H (M0) 1 + lies in Hc (M0). Now, from the diagram (61) we obtain that [xep ]+∂0([tp]) ∈ 1 Iw 1 HIw(D)Γ0 , and there exists xp,y ∈ HIw(Dy) which satisﬁes (78) with [tp] Qp given by (81). It is not difﬁcult to check that the statement of Lemma 5.2.3 holds if we replace bv by

bp = wp ∪ (up + ∂0(tp)). † ∗ Since gp,y(∂0([tp])) = 0, there exists etp ∈ Drig(Yy (1)p) such that etp 7→ tp † ∗ 0 under the projection Drig(Yy (1)p) → Dy and

gp,y(∂0(t0)) = d0(etp) = ((ϕ − 1)(etp),(γQp − 1)(etp)). Therefore

gp,y(bp) = wp ∪ up + wp ∪ gp,y(d0etp), gp(bp) ∪ µp = wp ∪ gp(detp) ∪ µp. ⊥ Since the projection of yb on E is 1, we have wp ∪ up ∪ fp (yb) = wp ∪ up. Thus,

⊥ (82) invp(gp,y(bp) ∪ fp (yb) + gp(bp) ∪ µp) = = ` (u ) + (w ∪ g (dt ) ∪ f ⊥(y) + w ∪ g ( (dt )) ∪ ) = Qp p invp p p,y ep p b p p πp ep µp = ` (u ) − (w ∪ g (t ) ∪ d f ⊥(y) + w ∪ g (t ) ∪ d ) = Qp p invp p p,y ep p b p p ep µp ⊥ = `Qp (up) − invp(wp ∪etp ∪ ( fp (y) + dµp)) = + + = `Qp (up) − invp(wp ∪etp ∪ gp(yp )) = `Qp (up) − invp(wp ∪tp ∪ gp(yp )) = + `Qp (up) − `Qp (tp ∪ gp(yp )). P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 93

+ + Now we remark that `Qp (tp ∪ gp(yp )) = `Qp tp ∪ ρD⊥, f (yp ) and, taking into account (81), we have + −1 + + (83) `Qp (tp ∪ gp(yp )) = −`Qp δ0, f ◦ ρD, f ([xp ]) ∪ ρD, f ([yp ]) = D + + E = − ρD, f ([xp ]),ρD⊥, f ([yp ]) . D, f The ﬁrst formula of Lemma 5.2.4 still holds in our case. From (82), (83) norm and the deﬁnition of hV,D it follows that

sel ⊥ hV,D([x f ],[y f ]) = invp(gp,y(bp) ∪ fp (yb) + gp(bp) ∪ µp) = D + + E `Qp (up) + ρD, f ([xp ]),ρD⊥, f ([yp ]) = D, f norm D + + E = hV,D ([x],[y]) + ρD, f ([xp ]),ρD⊥, f ([yp ]) D, f and the theorem is proved.

REFERENCES [1] J. Bellaïche, Critical p-adic L-functions, Invent. Math. 189 (2012), 1-60. [2] D. Benois, Périodes p-adiques et lois de réciprocité explicites, J. reine angew. Math. 493 (1997), 115-151 [3] D. Benois, On Iwasawa theory of p-adic representations, Duke Math. J. 104 (2000), 211-267. [4] D. Benois, A generalization of Greenberg’s L -invariant, Amer. J. Math. 133 (2011), 1573-1632. [5] D. Benois, On extra zeros of p-adic L-functions: the crystalline case, To appear in: Iwasawa theory 2012. State of the Art and Recent Ad- vances (T. Bouganis and O. Venjakob, eds.), Contributions in Mathematical and Computational Sciences, vol. 7, Springer, 2015. Preprint available on http://arxiv.org/abs/1305.0431. [6] D. Benois, Selmer Complexes and p-adic Hodge theory, to appear in “Proceed- ings Arithmetic Geometry Program, Hausdorff Institute for Mathematics, Bonn 2012/13” London Mathematical Society Lecture Note Series. Preprint available on http://arxiv.org/pdf/1404.7386.pdf, 47 pages. [7] D. Benois, K. Büyükboduk, On the exceptional zeros of p-adic L-functions of modular forms, (in progress) [8] L. Berger, Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219-284. [9] L. Berger, Bloch and KatoâA˘Zs´ exponential map: three explicit formulas, Doc. Math., Extra Volume: Kazuya Kato’s Fiftieth Birthday (2003), pp. 99-129. [10] L. Berger, Equations différentielles p-adiques et (ϕ,N)-modules ﬁltrés, In: Représentations p-adiques de groupes p-adiques I (L. Berger, P. Colmez, Ch. Breuil eds.), Astérisque 319 (2008), 13-38 [11] L. Berger, P. Colmez, Familles de représentations de de Rham et monodromie p-adique, In: Représentations p-adiques de groupes p-adiques I (L. Berger, P. Colmez, Ch. Breuil eds.), Astérisque 319 (2008), 303-337. 94 DENIS BENOIS

[12] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, In: The Grothendieck Festschrift, vol. 1 (P. Cartier, L. Illusie, N. M. Katz, G. Lau- mon, Yu. Manin, K. Ribet, eds.), Progress in Math. vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400. [13] F. Cherbonnier, P. Colmez, Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581-611. [14] F. Cherbonnier, P. Colmez, Théorie d’Iwasawa des représentations p-adiques d’un corps local, J. Amer. Math. Soc. 12 (1999), 241-268. [15] P. Colmez, Les conjectures de monodromie p-adiques, in: SÂt’eminaire Bour- baki. Vol 2001/2002, AstÂt’erisque 290 (2003), 53-101. [16] P. Colmez, Représentations triangulines de dimension 2, In: Représentations p-adiques de groupes p-adiques I (L. Berger, P. Colmez, Ch. Breuil eds.), Astérisque 319 (2008), 213-258. [17] M. Flach, A generalization of the Cassels–Tate pairing, J. Reine Angew. Math. 412 (1990), 113-127. [18] J.-M. Fontaine, Représentations p-adiques des corps locaux, In: The Grothendieck Festschrift, vol. 2 (P. Cartier, L. Illusie, N. M. Katz, G. Lau- mon, Yu. Manin, K. Ribet, eds.), Progress in Math. vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 249-309. [19] J.-M. Fontaine, Le corps des périodes p-adiques, In: Périodes p-adiques (J.-M. Fontaine, ed.), Astérisque 223 (1994), 59-102. [20] J.-M. Fontaine, Représentations p-adiques semi-stables, In: Périodes p-adiques (J.-M. Fontaine, ed.), Astérisque, 223 (1994), 59-102. [21] J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato; cohomologie galoisienne et valeurs de fonctions L, In: Motives (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. Symp. in Pure Math., vol. 55, part 1, 1994, pp. 599-706. [22] R. Greenberg, Iwasawa theory for p-adic representations, In: Algebraic Num- ber Theory – in honor of K. Iwasawa (J. Coates, R. Greenberg, B. Mazur and I. Satake, eds.), Adv. Studies in Pure Math. vol. 17 (1989), pp. 97-137. [23] R. Greenberg, Trivial zeros of p-adic L-functions, In: p-Adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (B. Mazur and G. Stevens, eds.), Contemp. Math., vol. 165, 1994, pp. 149-174. [24] R. Greenberg, Iwasawa Theory and p-adic Deformations of Motives, In: Mo- tives (U. Jannsen, S. Kleiman, J.-P. Serre, eds.), Proc. of Symp. in Pure Math., vol. 55, part 2, 1994, pp. 193-223. [25] E. Hellmann, Families of trianguline representations and ïˇn˛Aniteslope spaces, preprint available on http://arXiv:1202.4408v1. [26] L. Herr, Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563-600. [27] L. Herr, Une approche nouvelle de la dualité locale de Tate, Math. Annalen 320 (2001), 307-337. [28] K. Kedlaya, A p-adic monodromy theorem, Ann. of Math. 160 (2004), 94-184. [29] K. Kedlaya, R. Liu, On families of (ϕ,Γ)-modules, Algebra and Number The- ory 4 (2010), 943âA¸S967.˘ [30] K. Kedlaya, J. Pottharst, L. Xiao, Cohomology of arithmetic families of (ϕ,Γ)- modules, Journal of the Amer. Math. Soc. 27 (2014), 1043-1115. P-ADIC HEIGHTS AND P-ADIC HODGE THEORY 95

[31] R. Liu, Cohomology and Duality for (ϕ,Γ)-modules over the Robba ring, Int. Math. Research Notices (2007), no. 3, 32 pages. [32] K. Nakamura, Iwasawa theory of de Rham (ϕ,Γ)-modules over the Robba ring, J. of the Inst. of Math. of Jussieu, vol. 13, issue 1, pp 65-118. [33] J. Nekovár,ˇ On p-adic height pairing, In: Séminaire de Théorie des Nombres, Paris, 1990/91 (S. David, ed.), Progress in Mathematics, vol. 108, Birkhäuser Boston, 1993, pp. 127âA¸S202.˘ [34] J. Nekovár,ˇ Selmer complexes, Astérisque 310 (2006), 559 pages. [35] W. Nizioł, Cohomology of crystalline representations, Duke Math. J. 71 (1993), pp.747-791. [36] B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p-adiques, Invent. Math. 109 (1992), 137-185. [37] J. Pottharst, Analytic families of ﬁnite slope Selmer groups, Algebra and Num- ber Theory 7 (2013), 1571-1611. [38] J. Pottharst, Cyclotomic Iwasawa theory of motives, Preprint available on http://math.bu.edu/people/potthars/. [39] P. Schneider, p-adic height pairings. I, Invent. Math. 69 (1982), 401âA¸S409.˘ [40] J.-P. Serre, Corps locaux, Hermann, Paris, 1968. [41] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239, Soc. Math. France, Paris , 1996.

INSTITUTDE MATHÉMATIQUES,UNIVERSITÉDE BORDEAUX, 351, COURSDELA LIBÉRATION 33405 TALENCE,FRANCE E-mail address: [email protected]