© Gautier Ponsinet, 2018

On the algebraic side of the Iwasawa theory of some non-ordinary Galois representations

Thèse

Gautier Ponsinet

Doctorat en mathématiques

Philosophiæ doctor (Ph. D.)

Québec, Canada

On the algebraic side of the Iwasawa theory of somenon-ordinary Galois representations

Thèse

Gautier Ponsinet

Sous la direction de:

Antonio Lei, directeur de recherche

Résumé

Soit F un corps de nombres non-ramifié en un nombre premier impair p. Soit F∞la Zp-extension cyclotomique de F et Λ = Zp[[Gal(F∞/F )]] l’algèbre d’Iwasawa deGal(F∞/F ) ' Zp sur Zp. Généralisant les groupes de Selmer plus et moins de Kobayashi,Büyükboduk et Lei ont défini des groupes de Selmer signés sur F∞ pour certainesreprésentations galoisiennes. En particulier, leurs constructions s’appliquent aux casdes variétés abéliennes définies sur F ayant bonne réduction supersingulière en chaquepremier de F divisant p. Ces groupes de Selmer signés ont naturellement une structurede Λ-modules de type fini.

Nous commençons par prouver une équation fonctionnelle pour ces groupes de Selmersignés qui relie les groupes de Selmer signés d’une telle représentation aux groupes deSelmer signés du dual de Tate de la représentation.

Puis, nous étudions la structure de Λ-module des groupes de Selmer signés. Sousl’hypothèse qu’ils sont des Λ-modules de cotorsion, nous montrons qu’ils ne possèdentpas de sous-Λ-module propre d’indice fini. Nous déduisons de ce résultat quelquesapplications arithmétiques. Nous calculons le Λ-corang du groupe de Selmer de Bloch-Kato sur F∞ associé à la représentation, et, en étudiant la caractéristique d’Euler-Poincaré de ces groupes de Selmer signés, nous obtenons une formule explicite de lataille du groupe de Selmer de Bloch-Kato sur F . De plus, pour deux telles representationsisomorphes modulo p, nous comparons les invariants d’Iwasawa de leurs groupes deSelmer signés.

Finalement, en supposant que les groupes de Selmer signés associés à une variétéabélienne supersingulière sont des Λ-modules de cotorsion, nous montrons que le rangdes groupes de Mordell-Weil de la varitété abélienne est borné le long de l’extensioncyclotomique.

iii

Abstract

Let F be a number field unramified at an odd rational prime p. Let F∞ be theZp-cyclotomic extension of F and Λ = Zp[[Gal(F∞/F )]] be the Iwasawa algebra ofGal(F∞/F ) ' Zp over Zp. Generalizing Kobayashi’s plus and minus Selmer groups,Büyükboduk and Lei have defined signed Selmer groups over F∞ for some non-ordinaryGalois representations. In particular, their construction applies to abelian varietiesdefined over F with good supersingular reduction at primes of F dividing p. Thesesigned Selmer groups have a natural structure of finitely generated Λ-modules.

We first prove a functional equation for these signed Selmer groups, relating the signedSelmer groups of such a representation to the signed Selmer groups of Tate dual of therepresentation.

Second, we study the structure of Λ-module of the signed Selmer groups. Assumingthat they are cotorsion Λ-modules, we show that they have no proper sub-Λ-module offinite index. We deduce from this a number of arithmetic applications. We compute theΛ-corank of the Bloch-Kato Selmer group attached to the representation over F∞, and,on studying the Euler-Poincaré characteristic of these signed Selmer groups, we obtain anexplicit formula on the size of the Bloch-Kato Selmer group over F . Furthermore, for twosuch representations that are isomorphic modulo p, we compare the Iwasawa-invariantsof their signed Selmer groups.

Finally, under the hypothesis that the signed Selmer groups associated to a supersingularabelian variety are cotorsion Λ-modules, we show that the rank of Mordell-Weil groupsof the abelian variety is bounded along the cyclotomic extension.

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Contents

Résumé iii

Abstract iv

Contents v

Remerciements vi

Introduction 1

Notations and setup 6

1 Functional equation of signed Selmer groups 221.1 Orthogonality of local conditions . . . . . . . . . . . . . . . . . . . . 231.2 Control theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Λ-module structure and congruences of signed Selmer groups 322.1 Sub-Λ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2 An application: computation of the Euler-Poincaré characteristic . . . 392.3 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Mordell-Weil ranks of supersingular abelian varieties over cyclo-tomic extensions 49

Bibliography 56

v

Remerciements

Je tiens en tout premier lieu à remercier Antonio Lei de m’avoir fait découvrir toutesces mathématiques au cours de ces quatre années ainsi que pour ses encouragements etprécieux conseils.

Je remercie Kazim Büyükboduk, Daniel Delbourgo et Byoung Du Kim d’avoir acceptéd’être les rapporteurs de cette thèse ainsi que pour leurs soutiens et encouragements.

Je remercie également Jeffrey Hatley d’avoir répondu à mes questions et pour sa lectureattentive des versions préliminaires des articles qui composent cette thèse.

Enfin, j’ai eu la chance d’être entouré de personnes qui m’ont inspiré, encouragé,reconforté et sans qui je n’aurais certainement pas pu arriver jusqu’ici. Je pense à mesparents Marie-Christine et Norbert, mes frères Thomas et Lucas, mes grands-parentsJean, Christian et Claude, et puis Arnaud et Florence et toute cette chouette famille, àmes amis de toujours : Vinh-Lôc, Thibaut, Édouard, Aurélien, à Macarena, à Joaquinet Louis pour nos échanges musicaux respectifs, à Donato pour nos soirées au jazz club,à Gérard Freixas pour ses encouragements, à Hugo pour nos discussions mathématiques(et politiques!), et à toutes ces merveilleuses rencontres faites durant ces années de thèse: Marine, Christine, Coline, Ibra, Jan et tant d’autres!

vi

Introduction

Let E be an elliptic curve defined over a number field F . Let p be an odd prime. Let Fbe an algebraic closure of F . Let F∞ = ∪n>0Fn be the Zp-cyclotomic extension of Fin F with Gal(Fn/F ) ' Z/pnZ, and Λ = Zp[[Gal(F∞/F )]] ' Zp[[X]] be the Iwasawaalgebra of Gal(F∞/F ) over Zp.

The p-Selmer group of E over an algebraic extension K of F is defined to be the kernelin Galois cohomology:

Sel(E/K) = Ker

(H1(K,E[p∞])→

∏v

H1(Kv, E[p∞])

E(Kv)⊗Qp/Zp

).

It fits in the short exact sequence of groups

0→ E(K)⊗Z Qp/Zp → Sel(E/K)→X(E/K)[p∞]→ 0, (1)

where X(E/K) is the Tate-Shafarevich group of E over K. Set Sel(E/F∞) =

lim−→nSel(E/Fn), then Sel(E/F∞) is naturally equipped with a structure of finitely

generated Λ-module.

When E has good ordinary reduction at primes of F dividing p, a conjecture of Mazur [32]asserts that the Pontryagin dual of Sel(E/F∞) is a torsion Λ-module. In that case, wecan associated a characteristic ideal in Λ to the Selmer group and the Iwasawa mainconjecture states that this ideal is generated by the p-adic L-function associated toE by Mazur and Swinnerton-Dyer which p-adically interpolates values of the complexL-function associated to E. When F = Q and under some minor additional hypotheses,Mazur’s conjecture has been proved by Kato [21], and the Iwasawa main conjecture byKato and Skinner-Urban [43].

When E has good supersingular reduction at primes of F dividing p, already Mazur’sconjecture fails, and, on the analytic side, the p-adic L-functions of Višik [42] and Amice-Vélu [1] are no longer elements of Λ. When F = Q and ap = 0, Kobayashi [28] has defined

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two modified Selmer groups, referred as the plus and minus Selmer groups Sel±(E/F∞),and proved that they are cotorsion Λ-modules. On the analytic side, Pollack [39]has defined two p-adic L-functions in Λ which are related to the p-adic L-functions.Furthermore, under minor additional hypotheses, the Iwasawa main conjecture in thatcontext has been proved by Wan [47], that is, the two p-adic L-functions of Pollackgenerate the characteristic ideals of the plus and minus Selmer groups.

Kobayashi and Pollack constructions have been generalized to different situations [44,19, 30, 25, 7]. We focus on the construction of Büyükboduk and Lei [7]. Using p-adicHodge theory machinery, they defined signed Selmer groups for certain non-ordinaryGalois representations of GF = Gal(F/F ). In particular, the construction apply to theabelian varieties with good supersingular reduction at primes of F dividing p. Thisconstruction depends on the choice of a basis of the Dieudonné module attached to therepresentation which respect the filtration, in particular, for an elliptic curve definedover Q with ap = 0 and a good choice of basis, the signed Selmer groups of op. cit.coincide with Kobayashi plus and minus Selmer groups.

In the first chapter, we study functional equation for the signed Selmer groups. In boththe ordinary and the supersingular elliptic curve case, the p-adic L-functions (of Mazur-Swinnerton-Dyer and Pollack respectively) are known to satisfy a functional equation.Through the Iwasawa main conjecture, the characteristic ideal of the associated Selmergroup satisfy a functional equation as well. This “algebraic” functional equation can bechecked directly for the Selmer group, even without knowing that the Selmer groups arecotorsion. Greenberg [13] has proved such a functional equation for general ordinaryGalois representations and B.D. Kim [22] has proved such a functional equation for theplus and minus Selmer groups (before Wan’s proof of the main conjecture).

Let V be a p-adic representation of GF and T be a Zp-lattice GF -stable in V satisfyingthe hypothesis of [7]. We fix a basis for the Dieudonné module D(T ) of T . Choose Ia subset of that basis of cardinal dimQp(IndQ

F V )+ the dimension of the +1-eigenspaceunder the action of a complex conjugation on the induced representation IndQ

F V . LetV ∨(1) (respectively T∨(1)) be the Tate dual of V (respectively T ). We denote bySelI(V

∨(1)/T∨(1), F∞) the signed Selmer group associated to V ∨(1) and I, and bySelI(V

∨(1)/T∨(1), F∞)∧ its Pontryagin dual, which is a finitely generated Λ-module.By the structure theorem for finitely generated Λ-modules, if M is a finitely generated

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Λ-module, there exists a pseudo-isomorphism

M →r⊕i=1

Λ⊕n⊕j=1

Λ/(paj)⊕m⊕k=1

Λ/(f bkk ) (2)

where fk ∈ Λ ' Zp[[X]] are distinguished irreducible polynomials. There is a naturalpairing on the Dieudonné modules D(T )×D(T∨(1))→ Zp and we choose as a basis ofD(T∨(1)) the dual basis of the one we choose for D(T ) and let Ic be the orthogonalcomplement of I. Let SelIc(V/T, F∞) be the signed Selmer group associated to V and Ic.Let ι be the automorphism of Λ induced by the automorphism of Gal(F∞/F ), g 7→ g−1.Let SelIc(V/T, F∞)∧,ι be the Λ-module with same underlying set as SelIc(V/T, F∞)∧

but with Λ acting through ι. We prove the functional equation:

Theorem. Assume that F is abelian over Q with degree prime to p and that

dimQp(IndQF V )+ = dimQp(IndQ

F V )−.

Then SelI(V∨(1)/T∨(1), F∞)∧ and SelIc(V/T, F∞)∧,ι are pseudo-isomorphic to the same

Λ-module (up to isomorphism) by the structure theorem (2).

In particular, if one of SelI(V∨(1)/T∨(1), F∞)∧ or SelIc(V/T, F∞)∧,ι is a torsion Λ-

module, then they both are and are pseudo-isomorphic.

The proof makes a crucial use of Perrin-Riou’s explicit reciprocity law [36] (proved byColmez [9]).

In the second chapter, we study the Λ-module structure of SelI(V∨(1)/T∨(1), F∞).

Let E be an elliptic curve with ordinary reduction at prime above p. Assuming Mazur’sconjecture and that the E(F ) has no p-torsion, Greenberg [14, Proposition 4.14] showedthat Selp(E/F∞) has no proper sub-Λ-module of finite index. When E is an elliptic curvewith supersingular reduction at prime above p, B.D. Kim [24] has then extended thedefinition of the plus and minus Selmer groups to number fields F where p is unramifiedand generalized Greenberg’s result and showed that if the plus and minus Selmer groupsof E over F∞ are cotorsion Λ-modules, then they have no proper submodule of finiteindex (for one of the signed Selmer group, namely the plus one, he requires the additionalassumption that p splits completely in F and is totally ramified in F∞). This assumptionhas recently been removed by Kitajima and Otsuki, see [27]. We prove a similar resultfor the signed Selmer groups.

Theorem. Assuming that SelI(V∨(1)/T∨(1), F∞)∧ and SelIc(V/T, F∞)∧ are torsion

Λ-modules, then SelI(V∨(1)/T∨(1), F∞) has no proper sub-Λ-modules of finite index.

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As a byproduct of the proof, we compute the Λ-corank of the Bloch-Kato Selmer groupof V over F∞ (see Corollary 2.8). This theorem also allows us to employ Greenberg’sstrategy in [14, Theorem 4.1] to compute the Euler-Poincaré characteristic of the signedSelmer groups. We may relate the leading term of the characteristic series of theseSelmer groups to a product of Tamagawa numbers associated to the represenatationand the cardinality of the Bloch-Kato’s Selmer group over F (see Corollary 2.10).

As another consequence, we study congruences of signed Selmer groups. If E andE ′ are elliptic curves defined over Q with good ordinary reduction at p and suchthat E[p] ' E ′[p] as Galois modules, Greenberg and Vatsal [17] have studied theconsequences of such a congruences in Iwasawa theory. In particular, assuming Mazur’sconjecture, they proved that the µ-invariant of Selp(E/Q∞) vanishes if and only if thatof Selp(E

′/Q∞) vanishes, and that when these µ-invariants do vanish, the λ-invariants ofsome non-primitive Selmer groups associated to E and E ′ over Q∞ are equal. Kim [23]generalized this result to the plus and minus Selmer groups in the supersingular case.We prove a version of this result in the settings of [7].

Theorem. Let T and T ′ be Galois representations to which the construction of [7] applies.Assume that T/pT ' T ′/pT ′ as Galois modules and that the the signed Selmer groupsassociated to T , T∨(1), T ′ and (T ′)∨(1) are cotorsion Λ-modules. Then the µ-invariantof SelI(V

∨(1)/T∨(1), F∞) vanishes if and only if that of SelI((V′)∨(1)/(T ′)∨(1), F∞)

vanishes. Furthermore, when these µ-invariants do vanish, the λ-invariants of theI-signed non-primitive Selmer groups associated to T and T ′ over F∞ are equal.

The main ingredient of the proof is a result of Berger [3] who showed that the congruenceT/pT ' T ′/pT ′ of Galois module induces a congruence modulo p on the Wach moduleassociated to T and T ′. This allows to keep track of the congruence through Büyükbodukand Lei’s construction.

In the final chapter, we focus on abelian varieties with supersingular reduction at primesof F dividing p. Let X be an abelian variety defined over F . The Mordell-Weil groupsX(Fn) of Fn-rational points are finitely generated abelian groups [33]. When X has goodordinary reduction at primes above p, the Pontryagin dual of Selp(X/F∞) is conjecturedto be a torsion Λ-module. Under this conjecture, Mazur’s control theorem on the Selmergroups implies that the rank of the Mordell-Weil group X(Fn) is bounded independentlyof n. In the case where X is an elliptic curve with ap = 0 and F = Q, Kobayashi [28]proved a control theorem à la Mazur for the plus and minus Selmer groups. As aconsequence, one deduces that the rank of the Mordell-Weil group X(Fn) is bounded

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independently of n. See [28, Corollary 10.2]. Alternatively, it is possible to deduce thesame result using Kato’s Euler system in [21] and Rohrlich’s non-vanishing results onthe complex L-values of X in [40]. See the discussion after Theorems 1.19 and 1.20 in[15].

We note that a different approach was developed by Perrin-Riou [34, §6] to study theMordell-Weil ranks of an elliptic curve with supersingular reduction and ap = 0. Sheshowed that when a certain algebraic p-adic L-function is non-zero and does not vanishat the trivial character, then the Mordell-Weil ranks of the elliptic curve over Fn arebounded independently of n. Kim [26] as well as Im and Kim [18] have generalizedthis method to abelian varieties which can have mixed reduction types at primes abovep. Furthermore, unlike the present text, they did not assume that p is unramified inF . Following Perrin-Riou’s construction, Im and Kim defined certain algebraic p-adicL-functions and showed that when this is non-zero, then the Mordell-Weil ranks of theabelian variety over Fn are bounded by certain explicit polynomials in n.

In our context, we prove:

Theorem. Assume that one of the signed Selmer groups of X over F∞ is a cotorsionΛ-module. Then the rank of the Mordell-Weil group X(Fn) is bounded as n varies.

As opposed to the ordinary case, we do not have a direct relation between the signedSelmer groups and the Mordell-Weil group as in the exact sequence (1). However, wemay nonetheless obtain information about the Selmer groups Selp(X/Fn) from the signedSelmer groups via the Poitou-Tate exact sequence and some calculations in multi-linearalgebra.

Finally, we recall that Kobayashi’s plus and minus Selmer groups have been generalized toelliptic curves with supersingular reduction at p and ap 6= 0 by Sprung [44]. Furthermore,as shown in [45], these Selmer groups can be used to study the growth of the Tate-Shafarevitch group of an elliptic cuve over Fn as n grows, generalizing the results ofKobayashi in [28, §10]. At present, we do not know how to generalize this to abelianvarieties since we do not have any explicit description of the action of the Frobeniusoperator on Dieudonné module attached to X. This prevents us from carrying out theexplicit matrix calculations in [45] in this setting.

5

Notations and setup

Unless stated otherwise, in this text p is an odd prime number. If K is a number field,we denote by OK the ring of integers of K. If v is prime of K, let Kv be the completionof K at v. When v is non-archimedean, let OKv be the ring of integers of Kv, mKv bethe maximal ideal of OKv and kKv = OKv/mKv be the residual field of OKv . We fix Kan algebraic closure of K and, for each prime v of K, an algebraic closure Kv of Kv

as well as an embedding K ↪→ Kv, and all algebraic extensions of K (respectively Kv)are considered contained in K (respectively Kv). Let GK = Gal(K/K) be the absoluteGalois group of K and GKv = Gal(Kv/Kv) ⊂ GK the absolute Galois group of Kv.

Cyclotomic extensions

For n > 1, let µpn be the group of pn-th root of unity in K and µp∞ = ∪n>1µpn . LetK(µpn) be the pn-cyclotomic extension of K and Kcyc = K(µp∞) = ∪n>1K(µpn) be thep∞-cyclotomic extension of K. We denote by Γcyc the Galois group of Kcyc over K. Letχ : GK → Z∗p be the cyclotomic character. For n > 1, the cyclotomic character inducesmorphisms χ : Gal(K(µpn)/K) ↪→ (Z/pnZ)∗, and χ : Γcyc = Gal(Kcyc/K) ↪→ Z∗p.Since p is odd, the group Z∗p decomposes as Z∗p ' Z/(p− 1)Z× Zp, and we denote byΓcyc = ∆×Γ the decomposition induces by χ, with χ : Γ ' Zp and χ : ∆ ↪→ Z/(p− 1)Z.For n > 1, let Γn be the unique subgroup of Γ of index pn and Γ0 = Γ. We setK∞ = K∆

cyc

the Zp-cyclotomic extension of K, and for n > 0, Kn = KΓn∞ (thus K0 = K).

Iwasawa algebra

If G is a profinite group, let Zp[[G]] be the Iwasawa algebra of G over Zp defined asthe projective limit of the group algebras Zp[G/U ] where U runs through the opensubgroups of G and relative to the natural projection maps Zp[G/U ′]→ Zp[G/U ], forU ′ ⊂ U .

Let Λcyc = Zp[[Γcyc]] be the Iwasawa algebra of Γcyc over Zp and Λ = Zp[[Γ]] be the

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Iwasawa algebra of Γ over Zp. The decomposition Γcyc = ∆×Γ induces an isomorphismΛcyc ' Zp[∆][[Γ]]. Furthermore, we fix γ a topological generator of Γ. The Iwasawaalgebra Zp[[Γ]] is isomorphic to the topological ring of formal power series Zp[[X]], theisomorphism being induced by γ 7→ X+1. For n > 1, let ωn(X) = (X+1)p

n−1 ∈ Zp[X].The isomorphism Zp[[Γ]] ' Zp[[X]] induces isomorphisms Zp[Γ/Γn] ' Zp[[X]]/(ωn).

Λcyc-modules

Let η be a Dirichlet character on ∆. We denote by eη the element of Λcyc defined by1|∆|∑

δ∈∆ η−1(δ)δ. For a Dirichlet character η on ∆ and a Λcyc-module M , let Mη be

the η-isotypic component of M which is defined as eη ·M . Note that Mη has a naturalstructure of Λ-module. We say that M has rank r over Λcyc if Mη has rank r over Λ forall Dirichlet character η on ∆.

Let M be a finitely generated Λ-module. There exists a pseudo-isomorphism of Λ-modules (i.e. a morphism of Λ-modules with finite kernel and cokernel) [41],

M →r⊕i=1

Λ⊕n⊕j=1

Λ/(paj)⊕m⊕k=1

Λ/(f bkk ) (3)

where fk ∈ Zp[X] are distinguished irreducible polynomials (identifying Zp[[X]] with Λ).Furthermore, the ideals (paj ), (f bkk ) and r are uniquely determined by M up to ordering.The rank of M over Λ is the integer r. The λ and µ-invariants of M are the integers

λ(M) =m∑k=1

bk deg fk and µ(M) =n∑j=1

aj.

The characteristic ideal of a torsion Λ-module M is the ideal charΛ(M) of Λ generatedby the element

char(M) = pµ(M)

m∏k=1

f bkk .

A Λ-module M is said to be cofinitely generated (respectively cotorsion, of corank r) ifthe Pontryagin dual of M , HomZp(M,Qp/Zp), is finitely generated (respectively torsionΛ-module, of Λ-rank r).

Proposition 1. Let M and M ′ be cofinitely generated Λ-modules. Assume that

1. for all irreducible distinguished polynomials f ∈ Λ and positive integers e,

corankΛ(M ⊗ Λ/(f e))Γ = corankΛ(M ′ ⊗ Λ/(f e))Γ,

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2. for all positive integers n and m,

#MΓn [pm]/#M ′,Γn [pm]

is bounded as n vary.

Then, the Pontryagin duals of M and M ′ are pseudo-isomorphic to the same Λ-module(up to isomorphism) by the structure theorem (3).

In particular, if M and M∧ are cotorsion (which is equivalent to if one of them is bythe previous statement), then their Pontryagin duals are pseudo-isomorphic.

This Proposition can be found in [13] but is spread across the article, therefore, we givea sketch of the proof.

Proof. Let M be a cofinitely generated Λ-module and M∧ its Pontryagin dual. Letf ∈ Λ be an irreducible distinguished polynomial and e > 1. The Zp-corank rf,e of

(M ⊗Zp Λ/(f e))Γ = HomΛ(M∧, (Qp/Zp)⊗Zp Λ/(f e))

is the Zp-rank of the Tate module of this last group which is HomΛ(M∧,Λ/(f e)). IfM∧ = Λ, then rf,e = deg(f e) = e · deg(f). If M∧ = Λ/(ga) with g an irreducibledistinguished polynomial and a > 1, then

rf,e =

{0 if (f) 6= (g),

min(e, a) · deg(f) otherwise.

Thus, if M∧ is finitely generated, then M∧/(M∧)Zp−torsion is pseudo-isomorphic to some

r⊕i=1

Λ⊕m⊕k=1

Λ/(f bkk )

and the knowledge of rf,e for all f and e determines r and the f bkk .

The second hypothesis implies that µ(M∧/pmM∧) = µ(M ′,∧/pmM ′,∧). We haveµ(Λ/pm) = m and µ((Λ/pb)/pm) = min(b,m). Thus, we deduce that M∧ and M ′,∧ havesame Λ-rank and µ-invariant.

Hence, the two hypotheses implies the first statement. Since pseudo-isomorphism isan equivalence relation on finitely generated torsion Λ-modules, the last statementfollows. �

8

Galois representations and Fontaine’s rings of periods

Let ` be a prime number and H a profinite group. A `-adic representation of H is thedata of a finite dimensional Q`-vector space V equipped with a continuous and Q`-linearaction of H. Similarly, a Z`-representation of H is a finitely generated Z`-module Tequipped with a continuous and Z`-linear action of H. For V a `-adic representation ofH, there exists a H-stable Z`-lattice T ⊂ V (i.e. T is a finite free Z`-submodule of Vand T ⊗Z` Q` = V ). When H is the Galois group of a field, we say that V and T areGalois representations.

Let K be a finite extension of Qp. To classify the p-adic representations of GK =

Gal(K/K), Fontaine [10, 11] has defined various subcategories of the category of allp-adic representations of GK . To do so, he has introduced topological Qp-algebrasendowed with a continuous action of GK and additional structures. For such an algebraB, a p-adic representation V of GK is said B-admissible if the BGK -module (V ⊗QpB)GK

is free of rank dimQp V . Then the aforementioned subcategories are the categories ofB-admissible representations.

We briefly recall the definition of some of these algebras. Let Zp(1) be the free Zp-module of rank 1 defined as the projective limit lim←−ζ 7→ζp µpn . The natural action ofthe Galois group GK on Zp(1) is given by the cyclotomic character. For n > 0, defineZp(n) = Zp(1)⊗n and Zp(−n) = HomZp(Zp(n),Zp) with the natural associated GK-actions. Let CK be the p-adic completion of K. The field CK is algebraically closedand GK acts continuously on it. For n ∈ Z, we set CK(n) = CK ⊗Zp Zp(n). The Galoisgroup GK acts on CK(n) by g(x⊗ y) = g(x)⊗ g(y) for g ∈ GK , x ∈ CK and y ∈ Zp(n).

The Hodge-Tate ring of K is the CK-algebra BHT = ⊕n∈ZCK(n) (with multiplicationin BHT defined via the natural maps CK(n) ⊗CK CK(m) ' CK(n + m)). We haveBGK

HT = K. A p-adic representation V of GK is Hodge-Tate if the K-vector spaceDHT(V ) = (V ⊗Qp BHT)GK has dimension dimQp V over K. The Hodge-Tate weightsof V are the n ∈ Z for which (CK(−n)⊗Qp V )GK is non-trivial and the multiplicity ofa weight n of V is dimK(CK(−n)⊗Qp V )GK . The K-vector space DHT(V ) is a gradedvector space, the direct summand of the grading being the non-trivial sub-vector spaces(CK(n)⊗Qp V )GK .

Let OCK be the ring of integers of CK . The ring

E+ = lim←−x7→xp

OCK/(p) = {(xn)n>0 ∈∞∏n>0

OCK/(p), (xn+1)p = x(n), for all n > 0}

9

is perfect of characteristic p. We denote by Ainf = W (E+) the ring of Witt vectors withcoefficients in E+. There is a surjective ring homomorphism

θ : Ainf → CK ,∑i�−∞

pi[xi] 7→∑i�−∞

pix(0)i ,

where [·] is the Teichmuller’s lift. The ring of p-adic periods B+dR is the (Ker θ)-adic

completion of Ainf [1/p]. If we fix ε ∈ E+ with ε(0) = 1 and ε(1) 6= 1 (i.e. a compatiblesystem of pn-th roots of unity), the series

log([ε]) =∞∑n=1

(−1)n+1 ([ε]− 1)n

n

converges in B+dR to an element t, the “2iπ” of Fontaine. The ring B+

dR is a discretevaluation ring and t is an uniformizer. We denote by BdR its field of fractions. Thefield BdR is equipped with a continuous action of GK and a filtration FiliBdR = tiB+

dR

stable for the Galois action, and BGKdR = K. A p-adic representation V of GK is de

Rham if it is BdR-admissible, i.e. if DdR(V ) = (V ⊗Qp BdR)GK is a K-vector spaceof dimension dimQp V . The K-vector space DdR(V ) is equipped with a decreasing,separated and exhaustive filtration induced by the one of BdR. If V is de Rham, then Vis Hodge-Tate and gr(DdR(V )) = ⊕i FiliDdR(V )/Fili+1 DdR(V ) = DHT(V ) as gradedK-vector spaces.

Let Acris be the p-adic completion of the W (E+)-subalgebra of W (E+)[1/p]

W (E+)[αn/n!]n>1,α∈Ker θ.

The ring Acris is stable under the continuous action of GK . Furthermore, the naturalFrobenius ϕ which acts on the ring of the Witt vectors, extends to Acris and commuteswith the Galois action. The element t lies in the ring B+

cris = Acris[1/p] and the ring ofcristalline periods for K is defined by Bcris = B+

cris[1/t]. If K0 is the maximal unramifiedextension of Qp inside K, we have BGK

cris = K0. Besides, there is a natural GK-equivariantinjective map j : K ⊗K0 Bcris → BdR, and K ⊗K0 Bcris is endowed with the filtrationinduced by the one of BdR. A p-adic representation V of GK is crystalline if theK0-vector space Dcris(V ) = (V ⊗Qp Bcris)

GK has dimension dimQp V . The Frobenius ϕnaturally acts on Dcris(V ) and the K-vector space K ⊗K0 Dcris(V ) is naturally equippedwith a decreasing, separated and exhaustive filtration the one of K ⊗K0 Bcris. If Vis crystalline, then it is de Rham and K ⊗K0 Dcris(V ) ' DdR(V ) as filtered K-vectorspaces.

10

Let V ∨(1) = HomQp(V,Qp(1)). If V is crystalline, then V ∨(1) is as well, and thereexists a natural pairing

[·, ·] : Dcris(V )×Dcris(V∨(1))→ Dcris(Qp(1)) ' K0

trace−−→ Qp. (4)

for which [ϕ(x), y] = [x, (pϕ)−1(y)] for x ∈ Dcris(V ) and y ∈ Dcris(V∨(1)) and, after

linearly extending the coefficients to K, gives

[·, ·] : DdR(V )×DdR(V ∨(1))→ DdR(Qp(1)) ' Ktrace−−→ Qp. (5)

for which FiliDdR(V ∨(1)) is the orthogonal complement of Fil1−iDdR(V ) (if V is onlyassume to be de Rham, we have the second pairing).

Let H i(K,V ) = H i(Gal(K/K), V ) be the i-th continuous Galois cohomology group ofGK with coefficients in V . Assume that V is a de Rham representation of GK . Blochand Kato [4] have defined two maps, the exponential

expK :DdR(V )

Fil0 DdR(V ) + Dcris(V )ϕ=1↪→ H1(K,V ),

which is the connecting homomorphism in the Galois cohomology of the fundamentalexact sequence

0→ V → V ⊗Qp Bϕ=1cris → V ⊗Qp BdR/B

+dR → 0,

and the dual exponential [20, II §1.2]

exp∗K : H1(K,V ∨(1))→ Fil0 DdR(V ∨(1)).

These maps satisfy the relation

〈x, expK(y)〉 = [exp∗K(x), y], for x ∈ H1(K,V ), y ∈ DdR(V ∨(1)),

where 〈·, ·〉 is Tate local pairing

〈·, ·〉 : H1(K,V )×H1(K,V ∨(1))→ H2(K,Qp(1)) ' Qp.

Now let K be a number field and V a p-adic representation of GK . The representationV is unramified at a prime v of K if the inertia subgroup of v acts trivially on V . Therepresentation V is pseudo-geometric if it is unramified outside a finite set of primes ofK and de Rham at the primes of K dividing p.

11

Iwasawa cohomology

Let K be a finite extension of Q`, for some prime `. Let T be a finite free Zp-representation of GK . Let K• be either the full p∞-cyclotomic Kcyc or the Zp-cyclotomicextension K∞ of K and Λ• the associated Iwasawa algebra. We denote by

H iIw(K•, T ) = lim←−

K′

H i(K ′, T )

the projective limit relative to the corestriction maps of the Galois cohomology groupsH i(K ′, T ) and where K ′ runs through the finite extension of K contains in K•. Thegroups H i

Iw(K•, T ) have a natural structure of Λ•-modules, which is well-known [37,Appendix A.2]. If N is a finitely generated Zp-module, let N∨ be HomZp(N,Zp). LetT∨(1) = HomZp(T,Zp(1)). The groups H i

Iw(K•, T ) are finitely generated Λ•-modules,trivial if i 6= {1, 2}. We have an isomorphism

H2Iw(K•, T ) ' H0(K•, T

∨(1))∨,

in particular, H2Iw(K•, T ) is a Λ•-torsion module. The group H1

Iw(K•, T ) has Λ-rankgiven by

rankΛ• H1Iw(K•, T ) =

{0 if ` 6= p,

[K : Q`] rankZp T if ` = p.

The torsion Λ•-submodule of H1Iw(K•, T ) is isomorphic to H0(K•, T ).

We recall some useful pairings. Let A∨(1) = (T∨(1)⊗Zp Qp)/T∨(1) = HomZp(T, µp∞).

Tate’s local pairing

H i(K ′, T )×H2−i(K ′, A∨(1))→ H2(K ′, µp∞) ' Qp/Zp, (6)

is compatible with the limits and induces a pairing

〈·, ·〉 : H iIw(K•, T )×H2−i(K•, A

∨(1))→ Qp/Zp. (7)

We recall Perrin-Riou’s pairing. Let 〈·, ·〉K′ be Tate’s pairing

H1(K ′, T )×H1(K ′, T∨(1))→ H2(K ′,Zp(1)) ' Zp, (8)

and let x = (xK′) and y = (yK′) be elements in H1Iw(K•, T ) and H1

Iw(K•, T∨(1))

respectively. Then the sequence whose K ′-component is∑g∈Gal(K′/K)

〈xK′ , g(yK′)〉K′ · g ∈ Zp[Gal(K ′/K)], (9)

12

is compatible under the corestriction maps and defines an element of Λ•. This defines apairing

〈·, ·〉PR : H1Iw(K•, T )×H1

Iw(K•, T∨(1))→ Λ•, (10)

called Perrin-Riou’s pairing.

Now let K be a number field. Let V be a pseudo-geometric p-adic representation ofGK and let Σ be a finite set of primes of K containing the primes of ramification ofV , the archimedean primes and the primes dividing p. We denote by KΣ the maximalextension of K unramified outside Σ. Since only the archimedean primes and the primesdividing p are ramified in the cyclotomic extension, Kcyc is contained in KΣ. Let T aGK-stable Zp-lattice in V . As above, let K• be either the full p∞-cyclotomic extensionKcyc or the Zp-cyclotomic extension K∞ of K. We denote by

H iIw(KΣ/K•, T ) = lim←−

K′

H iIw(Gal(KΣ/K

′), T )

the projective limit relative to corestriction maps where K ′ runs through the finiteextensions of K contains in K•.

Wach modules

Let K be a finite unramified extension of Qp. Let A+K be the ring OK [[π]] equipped

with a semilinear action of a Frobenius ϕ which acts as the absolute Frobenius on OK

and on π byϕ(π) = (π + 1)p − 1,

and with an action of Γcyc given by

g(π) = (π + 1)χ(g) − 1, for all g ∈ Γcyc.

Let ψ be a left-inverse of ϕ satisfying

ϕ ◦ ψ(f(π)) =1

p

∑ζ∈µp

f(ζ(1 + π)− 1).

Let T be a finite free crystalline Zp-representation of GK (we say that T is crystallinewhenever V = T ⊗Qp is). There exists a Wach module [3, 2] N(T ) attached to T whichis a free A+

K-module of rank rankZp T equipped with an action of Γcyc and a ϕ-linearendomorphism of N(T )[ 1

π], which we still denote by ϕ, commuting with the Galois

action. We denote by ϕ∗N(T ) the A+K-module generated by ϕ(N(T )).

13

Assuming that V has non-negative Hodge-Tate weights and no quotient isomorphic tothe trivial representation, then there is a canonical isomorphism of Λcyc-modules

h1Iw,T : N(T )ψ=1 ∼−→ H1

Iw(Kcyc, T ). (11)

Let N(V ) = N(T )⊗Zp Qp. The Wach module N(V ) is endowed with a filtration

FiliN(V ) = {x ∈ N(V ), ϕ(x) ∈ (ϕ(π)/π)iN(V )},

and there is a natural isomorphism of filtered ϕ-modules

N(V )/πN(V )∼−→ Dcris(V ). (12)

Thus, N(T )/πN(T ) with a similar filtration

FiliN(T ) = {x ∈ N(T ), ϕ(x) ∈ (ϕ(π)/π)iN(T )},

identifies with a lattice Dcris(T ) of Dcris(V ) equipped with the filtration induced by theone of N(T ) (or equivalently the one of Dcris(V )), called the Dieudonné module of T .Furthermore, if the Hodge-Tate weights of V are in the Fontaine-Laffaille range, i.e. in[a, (p− 1) + a] for some a ∈ Z, then Dcris(T ) is strongly divisible, that is∑

i∈Z

p−iϕ(FiliDcris(T )) = Dcris(T ).

Perrin-Riou’s regulator

We keep the notation of the previous paragraph. Let B+rig,K be the set of elements g(π)

with g(X) ∈ K[[X]] a formal power series which converges on the open p-adic unit disk.The operator ψ acts on B+

rig,K .

Let H (Γ) be the set of elements f(γ−1) with γ ∈ Γ and f(X) ∈ Qp[[X]] a formal powerseries which converges on the open p-adic unit disk. We set H = Qp[∆]⊗Qp H (Γ).

There exists a Λcyc-isomorphism, the Mellin transform,

M : Λcyc → (A+Qp

)ψ=0, f(λ− 1) 7→ f(λ− 1) · (π + 1),

which extends to an isomorphism H ' (B+rig,Qp

)ψ=0.

Assuming that V has non-negative Hodge-Tate weights and no quotient isomorphic tothe trivial representation, Perrin-Riou’s regulator map [36] is defined by

LT : H1Iw(Kcyc, T ) N(T )ψ=1 ϕ∗N(T )ψ=0

(B+rig,K)ψ=0 ⊗OK Dcris(T ) H ⊗Zp Dcris(T ).

(h1Iw)−1 1−ϕ

M−1⊗1

(13)

14

The map LT interpolates the dual exponential maps in the cyclotomic extension [31,Theorem B.5].

Perrin-Riou’s regulator satisfies an explicit reciprocity law (see Theorem B.6 of op.cit. for the formulation used here). Let σ−1 be the unique element of Γcyc such thatχ(σ−1) = −1, i.e. the image of the complex conjugation inside Γcyc, and `0 = log γ

logχ(γ).

Then, for z ∈ H1Iw(Kcyc, T ) and z′ ∈ H1

Iw(Kcyc, T∨(1)), we have

[LT (z),LT∨(1)(z′)] = −σ−1 · `0 · 〈z, z′〉PR. (14)

Selmer groups

Let K be a number field. Let V be a pseudo-geometric p-adic representation of GK andΣ a finite set of primes of K containing the primes of ramification of V , the archimedeanprimes and the primes dividing p. Let T be GK-stable Zp-lattice in V and set A = V/T .By definition A is a Gal(KΣ/K)-module. Let L be an algebraic extension of K containedin KΣ. By abuse of notation, we shall say that a prime of L is contained in Σ if it dividesa prime of K in Σ. For each prime v of L in Σ, let Fv be a subgroup of H1(Lv, A). TheSelmer group of A over L associated to (Fv)v∈Σ is defined as

Sel(Fv)(A/L) = Ker

(H1(KΣ/L,A)→

∏v∈Σ

H1(Lv, A)/Fv

),

where the maps H1(KΣ/L,A) → H1(Lv, A)/Fv are the localization maps composedwith quotient maps.

Since only archimedean primes or primes dividing p are ramified in the cyclotomic exten-sion, we have that Kcyc is contained in KΣ. Furthermore, a Selmer group Sel(Fv)(A/L)

has the structure of a discrete Λcyc-module, and a general result of Greenberg [13] en-sures that the Pontryagin dual of a Selmer group Sel(Fv)(A/Kcyc) is a finitely generatedΛcyc-module.

Assume that L is a finite extension of K. Let F⊥v ⊂ H1(Lv, T

∨(1)) be the orthogonalcomplement of Fv under Tate local pairing

H1(Lv, A)×H1(Lv, T∨(1))→ Qp/Zp.

We set

H1(Fv)(KΣ/L, T

∨(1)) = Ker

(H1

(Fv)(KΣ/L, T∨(1))→

∏v∈Σ

H1(Lv, T∨(1))/F⊥

v

).

15

If M is a Zp-module, we set M∧ to be the Pontryagin dual HomZp(M,Qp/Zp). ThePoitou-Tate exact sequence [37, Appendix A.3] induces the following exact sequences

0 H1(Fv)(KΣ/L, T

∨(1)) H1(KΣ/L, T∨(1))

∏v∈Σ H

1(Lv, T∨(1))/F⊥

v

Sel(Fv)(A/L)∧ H2(KΣ/L, T∨(1))

∏v∈Σ H

2(Lv, T∨(1)) H0(L,A)∧ 0,

(15)and,

0 Sel(Fv)(A/L) H1(KΣ/L,A)∏

v∈Σ H1(Lv, A)/Fv

H1(Fv)(KΣ/L, T

∨(1))∧ H2(KΣ/L,A)∏

v∈Σ H2(Lv, A) H0(L, T∨(1))∧ 0.

(16)

The exact sequence0→ T∨(1)→ V ∨(1)→ A∨(1)→ 0

induces maps in cohomology

H1(Lv, T∨(1))→ H1(Lv, V

∨(1))→ H1(Lv, A∨(1)).

Under the first map H1(Lv, T∨(1)) → H1(Lv, V

∨(1)), the image of F⊥v generates a

sub-Qp-vector space of H1(Lv, V∨(1)) and we define F∨

v (1) as its image in H1(Lv, A∨(1))

under the last map H1(Lv, V∨(1)) → H1(Lv, A

∨(1)). We define the Selmer group ofA∨(1) over L associated to (F∨

v (1))v∈Σ

Sel(F∨v (1))(A∨(1)/L) = Ker

(H1(KΣ/L,A

∨(1))→∏v∈Σ

H1(Lv, A∨(1))/F∨

v (1)

).

Proposition 2 ([14, Proposition 4.13]). Assume that Fv is a divisible group for eachv ∈ Σ. Assume also that m = corankZp Sel(F∨v (1))(A

∨(1)/L) and that H0(Lv, A∨(1)) is

finite for at least one v ∈ Σ. Then the cokernel of the map

f : H1(KΣ/L,A)→∏v∈Σ

H1(Lv, A)/Fv

has Zp-corank less than or equal to m. If m = 0, then the cokernel of f is isomorphicto the Pontryagin dual of H0(L,A∨(1)).

Let L be a finite extension of K. We recall the definition of the Bloch-Kato Selmergroup of A over L.

16

Let v be a prime of L which does not divide p. If Lunrv is the maximal unramified

extension of Lv, we define

H1unr(Lv, V ) = Ker

(H1(Lv, V )→ H1(Lunr

v , V )).

Let H1unr(Lv, A) be the image of H1

unr(Lv, V ) under the map H1(Lv, V )→ H1(Lv, A) andH1

unr(Lv, T ) be the inverse image of H1unr(Lv, V ) under the map H1(Lv, T )→ H1(Lv, V ).

For v be a prime of L dividing p, let

H1f (Lv, V ) = Ker

(H1(Lv, V )→ H1(Lv, V ⊗Qp Bcris)

).

Let H1f (Lv, A) be the image of H1

f (Lv, V ) by the map H1(Lv, V ) → H1(Lv, A) andH1f (Lv, T ) be the inverse image of H1

f (Lv, V ) by the map H1(Lv, T )→ H1(Lv, V ).

The groupH1unr(Lv, A) (respectivelyH1

f (Lv, A)) is the orthogonal complement ofH1unr(Lv, T

∨(1))

(respectively H1f (Lv, T

∨(1))) for Tate local duality.

The Bloch-Kato Selmer group of A over L, which we denote by SelBK(A/L), is defineby the choice

Fv =

{H1

unr(Lv, A) if v - p,H1f (Lv, A) if v | p.

Büyükboduk and Lei’s signed Selmer groups

We recall results of Büyükboduk and Lei [7].

Let F be a number field unramified at p. Let V be a pseudo-geomtric p-adic represen-tation of GF . We set g = dimQp IndQ

F V , g+ = dimQp(IndQF V )+ the dimension of the

+1-eigenspace under the action of a complex conjugation on the induced representationIndQ

F V . We also set g− = g − g+. Similarly, for a prime v of F dividing p, we setgv = dimQp Ind

Qp

FvV . We have

∑v|p gv = g.

Let T be a GF -stable Zp-lattice of V such that, at each prime of F dividing p:

(Hodge-Tate) the Hodge-Tate weights of V are in {0, 1},

(Crystalline) the representation V is crystalline,

(Torsion) the groups H0(Fv, T/pT ) and H2(Fv, T/pT ) are trivial,

(Filtration) the equality ∑v|p

dimQp Fil0 Dcris,v(T )⊗Qp = g−,

17

holds, where Dcris,v(T ) is the Dieudonné module of T as a GFv -representation,

(Slopes) the slopes of ϕ on Dcris,v(V ) are in ]− 1, 0[.

Let v be a prime dividing p. By (Hodge-Tate), the Dieudonné module Dcris,v(T ) hasa two-jump filtration

FiliDcris,v(T ) =

{Dcris,v(T ) for i 6 −1,

0 for i > 1.

We may choose a Zp-basis {u1, . . . , ugv} such that, for some d, {u1, . . . , ud} is a basis ofFil0 Dcris,v(T ). We call such a basis a Hodge-compatible basis of Dcris,v(T ). Since theHodge-Tate weights of V are in the Fontaine-Laffaille range, the Dieudonné module isstrongly divisible and we have

pϕDcris,v(T ) + ϕ(Fil0 Dcris,v(T )) = Dcris,v(T ).

Thus, the matrix of the Frobenius in the basis {u1, . . . , ugv} is of the form

Cϕ = C

(Idd 0

0 1p

Idgv−d

), (17)

with C ∈ GLgv(Zp). For n > 1, let

Cn =

(Idd 0

0 Φpn(X + 1) Idgv−d

)C−1, and , Mn = (Cϕ)n+1Cn · · ·C1, (18)

where Φpn(X) is the pn-th cyclotomic polynomial.

These matrices allow one to decompose Perrin-Riou’s regulator.

Theorem 3 ([7, Theorem 1.1]). The sequences (Mn)n>1 converges to some gv × gv

logarithmic matrix MT over H . There exists a Λcyc-homomorphism

ColT : H1Iw(Fv,cyc, T )→ ⊕gvi=1Λcyc,

which satisfies

LT : H1Iw(Fv,cyc, T )

ColT−−→ ⊕gvi=1ΛcycMT ·−−→H ⊗Zp Dcris,v(T ).

For i ∈ {1, . . . , gv}, let ColT,i be the composition of ColT with the projection ⊕gvi=1Λcyc

on the i-th components. For I a subset of {1, . . . , gv}, we set

ColT,I : H1Iw(Fv,cyc, T )→ ⊕|I|i=1Λcyc

z 7→ ⊕i∈I ColT,i(z).

We call these maps the signed Coleman maps.

18

Proposition 4 ([7, Proposition 2.20, Lemma 3.22]). Let I be a subset of {1, . . . , gv}and η a character on ∆.

1. The Λcyc-module Ker ColT,I is free of rank gv − |I|.

2. The η-isotypic component of the image of the signed Coleman map Im ColηT,I iscontained in a free Λ-module of rank |I|. This containment is of finite index.

We define H1I (Fv,cyc, A

∨(1)) as the orthogonal complement of Ker ColT,I for Tate pairing

H1(Fv,cyc, A∨(1))×H1

Iw(Fv,cyc, T )→ Qp/Zp.

The hypothesis (Torsion) H2(Fv, T/pT ) = 0 implies, by Tate local duality, thatH0(Fv, A

∨(1)) = 0. Hence, since Γ is pro-p group, the group H0(Fv,∞, A∨(1)) is trivial.

By the inflation-restriction exact sequence, we have

H1(Fv,∞, A∨(1)) ' H1(Fv,cyc, A

∨(1))∆,

since the order of the group ∆ is p− 1 and H0(Fv,cyc, A∨(1)) is finite of order a power

of p, and, for n > 0, we have

H1(Fv,n, A∨(1)) ' H1(Fv,∞, A

∨(1))Γn .

Through these isomorphisms, we set

H1I (Fv,∞, A

∨(1)) = H1I (Fv,cyc, A

∨(1))∆ ⊂ H1(Fv,∞, A∨(1)),

andH1I (Fv,n, A

∨(1)) = H1I (Fv,∞, A

∨(1))Γn ⊂ H1(Fv,n, A∨(1)).

Let I = (Iv)v|p denote a tuple of sets indexed by the primes of v dividing F and whereeach Iv is a subset of {1, · · · , gv}. Let F ′ be one of Fcyc, F∞ or Fn for some n > 0. TheI-signed Selmer group of A∨(1) over F ′, which we denote SelI(A

∨(1)/F ′), is defined bythe choice of local conditions

Fv =

{H1

unr(F′v, A) if v - p,

H1Iv

(F ′v, A) if v | p.

The definition depends on the choice of a Hodge-compatible bases of the Dieudonnémodules Dcris,v(T ). For some choice, the signed Selmer groups are related with theBloch-Kato Selmer group.

19

Proposition 5 ([6, Lemma 8.1]). There exists a choice of Hodge-compatible bases ofthe Dieudonné modules Dcris,v(T ) (called a strongly admissible basis in [7]) such that,for any I,

H1Iv(Fv, A

∨(1)) = H1f (Fv, A

∨(1)).

In particular, this implies

SelI(A∨(1)/F ) ' SelBK(A∨(1)/F ).

We define I to be the set of tuples I = (Iv)v|p where each Iv is a subset of {1, · · · , gv}and such that ∑

v|p

|Iv| = g−.

A Dirichlet character η on ∆ is said to be even (respectively odd) if the image of acomplex conjugation by η is +1 (respectively −1).

Conjecture 6 ([7, Remark 3.27]). For any I ∈ I and any even Dirichlet character ηon ∆, the Pontryagin dual of SelI(A

∨(1)/Fcyc)η is a torsion Λ-module.

Remark 7. The hypotheses (Hodge-Tate), (Crystalline), (Torsion), (Filtration)and (Slopes) holds for T∨(1). Thus, a priori, we have two ways to define signed Selmergroups for T∨(1), using the signed Coleman maps ColT∨(1),I or by propagating the localcondition of T as in the paragraph titled “Selmer groups”. We shall see later that thesetwo ways produce the same Selmer groups (see Lemma 2.4 and its proof).

Supersingular abelian varieties

Let F be a number field unramified at p and X be an abelian variety defined over Fof dimension d. For an integer n > 1, we denote by X(F )[pn] the groups of pn-torsionpoints of X in F . Let Tp(X) = lim←−x 7→p·xX(F )[pn] be the Tate module of X. ThenTp(X) is a free Zp-module of rank 2d equipped with a continuous action of GF .

Assume that X has good supersingular reduction at each prime of F dividing p. ThenTp(X) satisfies all the hypotheses: (Hodge-Tate), (Crystalline), (Torsion), (Filtra-tion) and (Slopes). Indeed, the Hodge structure of X implies that g = 2[F : Q]d andg+ = [F : Q]d. Tate [46] has proved that (Hodge-Tate) is satisfied. Since X has goodreduction at primes dividing p, it is cristalline and (Crystalline) and (Filtration) aresatisfied. The supersingular reduction implies (Slopes), and since F is unramified andp 6= 2, we have X(F )[p] = 0 by [32, Lemma 5.11]. Similarly, if XD is the dual abelian

20

variety of X, then XD also has good supersingular reduction at primes of F dividing pand XD(F )[p] = 0. Since Tp(XD) = (TpX)∨(1), (Torsion) is satisfied.

When F = Q and X is an elliptic curve with ap = 0 (which is true whenever p > 5),Büyükboduk and Lei have showed that for some basis, the signed Selmer groups, that weshall simply denote by SelI(X/Fcyc), coincide with Kobayashi’s plus and minus Selmergroups [28].

Let v be a prime of F , the exact sequence of GFv -module

0→ X(Fv)[pn]→ X(Fv)

pn·−→ X(Fv)→ 0,

induces the Kummer maps

X(Fv)⊗Z Qp/Zp ↪→ H1(Fv, X[p∞]) and X(Fv)⊗Z Zp ↪→ H1(Fv, Tp(X)).

If v does not divide p then one has X(Fv)⊗Z Qp/Zp = 0. Furthermore, these subgroupscoincide with Bloch and Kato H1

f (Fv, X[p∞]) and H1unr(Fv, X[p∞]). Hence, the Selmer

group fits in a short exact sequence

0→ X(F )⊗Z Qp/Zp → SelBK(X/F )→X(X/F )[p∞]→ 0, (19)

where X(X/F ) is the Tate-Shafarevich group of X over F .

21

Chapter 1

Functional equation of signed Selmergroups

The results of this chapter were obtained in collaboration with Antonio Lei [29].

Let F be a number field unramified at p and T a Zp-representation satisfying thehypotheses (Hodge-Tate), (Crystalline), (Torsion), (Filtration) and (Slopes).We choose Hodge-compatible bases of the Dieudonné modules of T for each prime of Fdividing p.

The Tate dual T∨(1) satisfies the hypotheses (Hodge-Tate), (Crystalline), (Tor-sion), (Filtration) and (Slopes) as well. In order to define the signed Selmer groupsfor T∨(1), we use the dual bases for the pairing (4).

Let ι be the automorphism of Λ induced by the automorphism of Γ, γ 7→ γ−1. If M is afinitely generated Λ-module, let M ι be the Λ-module with the same underlying set asM but with Λ acting through ι.

The goal of this chapter is to prove a functional equation for the signed Selmer groups.

Theorem 1.1. Assume that F is abelian over Q with degree prime to p. Furthermore,assume that T satisfies g+ = g−. Then the Pontryagin duals of SelI(A

∨(1)/F∞) andSelIc(A/F∞)ι are pseudo-isomorphic to the same Λ-module (up to isomorphism) by thestructure theorem (3).

In particular, if SelI(A∨(1)/F∞) and SelIc(A/F∞)ι are cotorsion (which is equivalent

to saying one of them is, by the previous statement), then their Pontryagin duals arepseudo-isomorphic.

22

Remark 1.2. The additional hypothesis g+ = g− is satisfied by abelian varieties.

1.1 Orthogonality of local conditions

Let v be a prime of F dividing p. We have chosen a Hodge-compatible basis {u1, . . . , ugv}of Dcris,v(T ). Let {u′1, . . . , u′gv} be the dual basis of Dcris,v(T

∨(1)) for the pairing (4)

[·, ·] : Dcris,v(T )×Dcris,v(T∨(1))→ Zp.

We denote by C ′ϕ the matrix of the Frobenius on Dcris,v(T∨(1)) with respect to the basis

{u′1, . . . , u′gv}. From duality we have the relation

C ′ϕ =1

p(C−1

ϕ )t = (C−1)t

(1pId 0

0 Igv

),

where (·)t is the transpose. We have the matrices of (18),

MT∨(1) = lim−→n

(C ′ϕ)n+1C ′n · · ·C ′1, (1.1)

where

C ′n =

(Φpn(X + 1)Id 0

0 Igv

)Ct.

By Theorem 3, Perrin-Riou’s regulator decomposes as

LT∨(1) = (u′1 . . . u′gv) ·MT∨(1) · ColT∨(1),

where ColT∨(1) is the column vector of Coleman maps with respect to the basis{u′1, . . . , u′gv}.

We linearly extend the pairing (4) to

[·, ·] : H ⊗Zp Dcris,v(T )×H ⊗Zp Dcris,v(T∨(1))→H .

Lemma 1.3. Let z ∈ H1Iw(Fv,cyc, T ) and z′ ∈ HIw(Fv,cyc, T

∨(1)). Then

[LT (z),LT∨(1)(z′)] =

log(1 +X)

X· ColT (z)t · ColT∨(1)(z

′).

Proof. Since {u1, . . . , ugv} is dual to {u′1, . . . , u′gv}, the decompositions of Perrin-Riou’sregulators LT and LT∨(1) give

[LT (z),LT∨(1)(z′)] = ColT (z)t ·M t

T ·MT∨(1) · ColT∨(1)(z′).

23

Using (18) and (1.1), we may compute

MTt ·MT∨(1) =

(lim−→ (Cϕ)n+1Cn · · ·C1

)t · (lim−→(C ′ϕ)n+1

C ′n · · ·C ′1)

= lim−→(Ct

1 · · ·Ctn

(Cϕ

t)n+1 ·

(C ′ϕ)n+1

C ′n · · ·C ′1)

= lim−→1

pn+1

n∏k=1

Φpk(1 +X) Idgv

Since

lim−→n→+∞

1

pn

n∏k=1

Φpk(1 +X) =log(1 +X)

X,

we haveMT

t ·MT∨(1) =log(1 +X)

pXIdgv .

�

We use the explicit reciprocity law (14) to prove.

Proposition 1.4. Let I ⊂ {1, · · · , gv} and Ic be its complement. Then Ker ColT∨(1),Ic

is the orthogonal complement of Ker ColT,I with respect to Perrin-Riou’s pairing (10).

Proof. Let z ∈ H1Iw(Fv,cyc, T ) and z′ ∈ HIw(Fv,cyc, T

∨(1)). By the explicit reciprocitylaw (14) and Lemma 1.3, we have

〈z, z′〉PR = 0 ⇔ [LT (z),LT∨(1)(z′)] = 0

⇔ ColT (z)t · ColT∨(1)(z′) = 0.

Thus, if z ∈ Ker ColT,I ,

〈z, z′〉PR = 0⇔∑i 6=I

ColT,i(z) · ColT∨(1),i(z′) = 0. (1.2)

So, Ker ColT∨(1),Ic is included in the orthogonal complement of Ker ColT,I . Proposition 4implies that for all i ∈ {1, . . . , gv}, there exists zi such that

ColT,j(zi)

{= 0 if j ∈ {1, . . . , gv} \ {i},6= 0 if j = i.

In particular, if i /∈ I, then such zi ∈ Ker ColT,I . If z′ is in the orthogonal complement ofKer ColT,I , then 〈zi, z′〉PR = 0. Therefore, equation (1.2) tells us that ColT∨(1),i(z

′) = 0.Since this is true for all i ∈ Ic, we have z′ ∈ Ker ColT∨(1),Ic as required. �

24

1.2 Control theorem

We first assume F = Q. Let f be an irreducible distinguished polynomial of Λ. Let eand m be positive integers. We set

A∨(1)fe = A∨(1)⊗Zp Λ/(f e), and A(fe)ι = A⊗Zp Λ/(f e)ι.

The pairing A[pm]×A∨(1)[pm]→ µpm induces a perfect Zp-linear GQ-equivariant pairingA∨(1)fe [p

m]× A(fe)ι [pm]→ µpm .

Since Γ is a pro-p-group, (Torsion) implies that H0(Qp,∞, A) and H0(Qp,∞, A∨(1)) are

trivial. Furthermore, since the Galois group Gal(Qp/Qp,∞) acts trivially on Λ/(f e), thehypothesis (Torsion) implies that H0(Qp,∞, A

∨(1)fe) and H0(Qp,∞, A(fe)ι) are trivial.Thus, the exact sequence of GQp-modules 0→ A∨(1)fe [p

m]→ A∨(1)fepm−→ A∨(1)fe → 0

inducesH1(Qp,n, A

∨(1)fe [pm]) ' H1(Qp,n, A

∨(1)fe)[pm].

For I ⊂ {1, . . . , g}, we define

H1I (Qp,∞, A

∨(1)fe) = H1I (Qp,∞, A

∨(1))⊗ Λ/(f e) ⊂ H1(Qp,∞, A∨(1)fe)

H1I (Qp,n, A

∨(1)fe) = H1I (Qp,∞, A

∨(1)fe)Γn ⊂ H1(Qp,n, A

∨(1)fe)

H1I (Qp,n, A

∨(1)fe [pm]) = H1

I (Qp,n, A∨(1)fe)[p

m] ⊂ H1(Qp,n, A∨(1)fe [p

m]),

and similarly for A(fe)ι .

We shall simply write Am for A∨(1)fe [pm] or A∨(1)[pm] and A∗m for A(fe)ι [p

m] or A[pm].

For n,m > 0, we define

SI(Am/Qn) = Ker

(H1(QΣ/Qn, Am)→

∏w∈Σ

H1(Qn,w, Am)

H1unr(Qn,w, Am)

× H1(Qp,n, Am)

H1I (Qp,n, Am)

)

Lemma 1.5. Let χglob.,Qn(Am) be the global Euler characteristic of Am over Qn. LetI ⊂ {1, . . . , g} and write Ic for its complement. Then,

#SI(Am/Qn) =#SIc(A

∗m/Qn)

χglob.,Qn(Am) · [H1(Qp,n, Am) : H1I (Qp,n, Am)]

.

Proof. We set

P iΣ :=

∏v∈Σ H

i(Qn,v, Am) and P ∗,iΣ :=∏

v∈Σ Hi(Qn,v, A

∗m),

Lp := H1I (Qp,n, Am) and L∗p := H1

Ic(Qp,n, A∗m),

Lv := H1unr(Qn,v, Am) and L∗v := H1

unr(Qn,v, A∗m) for v - p,

L :=∏

v∈Σ Lv and L∗ :=∏

v∈Σ L∗v.

25

Letλi : H i(QΣ/Qn, Am)→ P i

Σ

be the restriction map and write

Gi := Imλi, Ki := Kerλi. .

We have similarly λ∗,i, G∗,i, K∗,i for A∗. For all v ∈ Σ the orthogonal complement of Lvunder the local Tate pairing is L∗v (see [13, Remark at the end of §3] when v - p andProposition 1.4 when v = p). Thus we have

#SI(Am/Qn) = #K1 ·#(G1 ∩ L) = #K1 ·#G1 ·#L ·#(G1 · L)−1.

Since #K1 ·#G1 = #H1(QΣ/Qn, Am), and by duality #(G1 · L) = #P 1Σ/#(G∗,1 ∩ L∗),

we get#SI(Am/Qn) = #H1(QΣ/Qn, Am) ·#L ·#(G∗,1 ∩ L∗)/#P 1

Σ.

By (Torsion), we have

#H1(QΣ/Qn, Am) = χglob.,Qn(Am)−1 ·#H0(QΣ/Qn, Am) ·#H2(QΣ/Qn, Am)

= χglob.,Qn(Am)−1 ·#H2(QΣ/Qn, Am).

By global duality #K∗,1 = #K2, we obtain

#(G∗,1 ∩ L∗) = #SIc(A∗m/Qn)/#K∗,1 = #SIc(A

∗m/Qn)/#K2.

Thus, we have

#SI(Am/Qn) = χglob.,Qn(Am)−1·(#H2(QΣ/Qn, Am)/#K2)·(#L/#P 1Σ)·#SIc(A∗m/Qn).

(1.3)Now, the local Euler characteristic formula tells us that

#H1(Qn,v, Am) = #H0(Qn,v, Am) ·#H2(Qn,v, Am).

Also for v not dividing p, #H0(Qn,v, Am) = #H1unr(Qn,v, Am), so we deduce that

#L/#P 1Σ =

∏v∈Σ,v-p #(H1

unr(Qn,v, Am)/H1(Qn,v, Am))×#(H1I (Qp,n, Am)/H1(Qp,n, Am))

=∏

v∈Σ,v-p #(H0(Qn,v, Am)/H1(Qn,v, Am))×#(H1I (Qp,n, Am)/H1(Qp,n, Am))

=∏

v∈Σ,v-p #(H2(Qn,v, Am))−1 ×#(H1I (Qp,n, Am)/H1(Qp,n, Am)).

(1.4)On the other hand, global duality and (Torsion) tell us that # cokerλ2 = #H0(QΣ/Qn, A

∗m) =

1, so#H2(QΣ/Qn, Am)/#K2 = #G2 = #P 2

Σ/ cokerλ2 = #P 2Σ. (1.5)

26

Furthermore, local Tate duality implies that

#H2(Qn,v, Am) = #H0(Qn,v, A∗m) = 1.

Hence, the result follows on combining the equalities (1.3), (1.4) and (1.5) above. �

We give a generalization of [22, Lemma 3.3].

Lemma 1.6. Let I ⊂ {1, . . . , g} with #I = g+ = g−. Then #SI(Am/Qn)/#SIc(A∗m/Qn)

is bounded as n and m vary.

Proof. We prove the Lemma for Am = A∨(1)fe [pm]. We shall first of all compute the

quantityχglob.,Fn(Am)−1 · [H1(Qp,n, Am) : H1

I (Qp,n, Am)]−1

using Lemma 1.5.

SinceQn is totally real, one has χglob.,Qn(Am) = #(A−m)−[Qn:Q], where A−m is the subgroupof Am on which complex conjugation acts by −1. As complex conjugation acts triviallyon Λ/(f e), we get χglob.,Qn(Am) = p−m·[Qn:Q]·e·deg(f)·g− . Furthermore, we know that#H1(Qp,n, Am) = p−m·[Qp,n:Qp]·e·deg(f)·g.

It remains to compute #H1I (Qp,n, Am). By definition,

H1I (Qp,n, Am) =

(H1I (Qp,∞, A)⊗ Λ/(f e)

)Γn[pm].

Recall that#H1

I (Qp,∞, A∨(1)) = #H1

I (Qp(µp∞), A∨(1))∆,

and that we write (·)∧ for the Pontryagin dual. By definition of H1I (Qp(µp∞), A∨(1)),

we haveH1I (Qp(µp∞), A∨(1))∆ ' (Im Col∆T,I)

∧.

Proposition 4 gives an inclusion of Im Col∆T,I into a free Λ-module of rank g+, say NI,∆,with finite index:

0→ Im Col∆T,I → NI,∆ → KI,∆ → 0,

which gives the long exact sequence

0→ (K∧I,∆ ⊗ Λ/(f e))Γn → (N∧I,∆ ⊗ Λ/(f e))Γn

→ ((Im Col∆T,I)∧ ⊗ Λ/(f e))Γn → H1(Γn, K

∧I,∆ ⊗ Λ/(f e)).

27

Since H1(Qp,∞/Qp,n, K∧I,∆ ⊗ Λ/(f e)) is finite it implies that

#((Im Col∆T,I)∧[pm]⊗ Λ/(f e))Γn 6 C ·#(N∧I,∆ ⊗ Λ/(f e))Γn [pm]

6 C · pm·e·deg(f)·g+·pn

where C <∞ is independant of n and m. Hence the lemma follows. �

Note that our result is actually weaker than [22, Lemma 3.3], which in fact givesan equality. However, our result is sufficient to prove the following generalization ofLemma 3.4 in op. cit.

Lemma 1.7. Let A• ∈ {A∨(1)fe , A∨(1)}. For all positive integers n and m, the natural

mapSI(Am/Qn)→ SI(A•/Qn)[pm]

is injective and its cokernel is finite and bounded as n and m vary.

Proof. Consider the diagram

0 SI(Am/Qn) H1(QΣ/Qn, Am)∏

v∈Σ,v-pH1(Qn,v ,Am)

H1unr(Qn,v ,Am)

× H1(Qp,n,Am)

H1I (Qp,n,Am)

.

0 SI(A•/Qn)[pm] H1(QΣ/Qn, A•)[pm]

∏v∈Σ,v-p

H1(Qn,v ,A•)H1

unr(Qn,v ,A•)× H1(Qp,n,A•)

H1I (Qp,n,A•)

.

∏fv

We already know that the center vertical map is an isomorphism. Since H1I (Qp,n, Am) =

H1I (Qp,n, A•)[p

m], the map fp is injective. The local condition is the unramified conditionfor v - p, thus one has

H1(Qn,v, A•)/H1unr(Qn,v, A•) ⊂ H1(Qunr

n,v , A•)

H1(Qn,v, Am)/H1unr(Qn,v, Am) ⊂ H1(Qunr

n,v , Am)

with Qunrn,v the maximal unramified extension of Qn,v.

The short exact sequence 0→ Am → A•pm−→ A• → 0 implies

AGQunr

n,v• /pmA

GQunrn,v

• = Ker(H1(Qunrn,v , Am)→ H1(Qunr

n,v , A•))

and we have#A

GQunrn,v

• /pmAGQunr

n,v• 6 #A

GQunrn,v

• /(AGQunr

n,v• )div <∞.

Since no prime splits completely in Q∞/Q, we conclude that Ker∏fv is bounded as n

and m vary. Applying the Snake Lemma, the lemma follows. �

28

Remark 1.8. The results above are stated for F = Q and the isotypic component ofthe trivial character of ∆. Let η be a character on ∆ and denote by η its complexconjugate and by A∨(1)η the Zp-module A∨(1) with action of the Galois group twistedby η, then

H1(Qp(µp∞), A∨(1))η = H1(Qp(µp∞), A∨(1)η)∆ = H1(Qp,∞, A

∨(1)η).

The results above (for the isotypic component of the trivial character) in fact holdfor every isotypic component because we may replace A∨(1) and A by A∨(1)η and Aηrespectively in the proofs above.

Similarly, suppose that F/Q is a general abelian extension in which p is unramifiedwith [F : Q] coprime to p. We may prove the above results for an isotypic componentcorresponding to a character of Gal(F/Q).

We can now prove our main result.

Proof of Theorem 1.1. It is enough to consider the isotypic component for the trivialcharacter, as explained in Remark 1.8. Since Λ/(f e) is a free Zp-module on which GQ∞

acts trivially, we have

SelI(A∨(1)/Q∞)⊗ Λ/(f e) = Ker

H1(QΣ/Q∞, A

∨(1))⊗ Λ/(f e)

∏w∈Σ

H1(Q∞,w,A∨(1))⊗Λ(fe)

H1loc(Q∞,w,A∨(1))⊗Λ/(fe)

= Ker

H1(QΣ/Q∞, A

∨(1)⊗ Λ/(f e))

∏w∈Σ

H1(Q∞,w,A∨(1)⊗Λ(fe))

H1loc(Q∞,w,A∨(1))⊗Λ/(fe)

,

where H1loc(Q∞,w, A

∨(1)) is the appropriate local condition,

H1loc(Q∞,w, A

∨(1)) =

{H1

unr(Q∞,w, A∨(1)) if w - p,

H1I (Q∞,w, A

∨(1)) if w | p.

For the prime above p, we have by definition an injection

H1(Qp, A∨(1)fe)

H1I (Qp, A∨(1)fe)

↪→ H1(Qp,∞, A∨(1)fe)

H1I (Qp,∞, A∨(1))⊗ Λ/(f e)

.

29

For a prime w of Q∞ above a prime v 6= p of Q, the group H1unr(Q∞,w, A

∨(1)) is trivial[37, §A.2.4]. From the inflation-restriction exact sequence, the kernel of the restrictionmap

H1(Qv, A∨(1)fe)→ H1(Q∞,w, A

∨(1)fe)

is H1(Q∞,w/Qv, (A∨(1)fe)

GQ∞,w ). If v is archimedean, then this group is trivial. If v isnon-archimedean, it finitely decomposes in Q∞/Qv, so that Gal(Q∞,w/Qv) ' Zp andit is topologically generated by an element γn. Thus H1(Q∞,w/Qv, (A

∨(1)fe)GQ∞,w ) is

isomorphic to(A∨(1)fe)

GQ∞,w/(γn − 1)(A∨(1)fe)GQ∞,w .

One has the short exact sequence

0 (A∨(1)fe)GQv (A∨(1)fe)

GQ∞,w

(A∨(1)fe)GQ∞,w (A∨(1)fe)

GQ∞,w/(γ − 1)(A∨(1)fe)GQ∞,w 0.

(γn−1)

The group (A∨(1)fe)GQv is finite, hence (A∨(1)fe)

GQ∞,w/(γ − 1)(A∨(1)fe)GQ∞,w is finite.

So, ((A∨(1)fe)GQ∞,w )div the maximal divisible subgroup of (A∨(1)fe)

GQ∞,w is containedin (γn − 1)(A∨(1)fe)

GQ∞,w and the order of (A∨(1)fe)GQ∞,w/(γn − 1)(A∨(1)fe)

GQ∞,w isbounded by the order of (A∨(1)fe)

GQ∞,w/((A∨(1)fe)GQ∞,w )div.

Since H1(Q, A∨(1)fe) ' H1(Q∞, A∨(1)fe)

Γ, applying the Snake lemma to the diagram

0 0

SI(A∨(1)fe/Q) (SelI(A

∨(1)/Q∞)⊗ Λ/(f e))Γ

H1(QΣ/Q, A∨(1)fe) H1(QΣ/Q∞, A

∨(1)fe)Γ

∏v∈Σ

H1(Qv ,A∨(1)fe )

H1loc(Qv ,A∨(1)fe )

(∏v∈Σ

H1(Q∞,v ,A∨(1)fe )

H1loc(Q∞,v ,A∨(1))⊗Λ/(fe)

)Γ∏fv

we get an inclusion

SI(A∨(1)fe/Q) ↪→ (SelI(A

∨(1)/Q∞)⊗ Λ/(f e))Γ

30

with finite cokernel, and similarly one has

SelI(A∨(1)/Qn) ↪→ SelI(A

∨(1)/Q∞)Γn

with finite cokernel. Now by Lemmas 1.6 and 1.7, we see that SelI(A∨(1)/Q∞) and

SelIc(A/Q∞)ι satisfy Proposition 1 and the theorem follows. �

31

Chapter 2

Λ-module structure and congruences ofsigned Selmer groups

Let F be a number field unramified at p and T a Zp-representation satisfiying thehypotheses (Hodge-Tate), (Crystalline), (Torsion), (Filtration) and (Slopes).We choose Hodge-compatible bases of the Dieudonné modules for each prime of Fdividing p. As in the previous chapter, we use the dual basis for a Hodge-compatiblebasis of the Dieudonné modules of T∨(1).

2.1 Sub-Λ-modules

In this section, we prove.

Theorem 2.1. Let I = (Iv)v|p ∈ I and let Ic = (Icv)v|p be its complement. Assume thatSelI(A

∨(1)/F∞) and SelIc(A/F∞) are cotorsion Λ-modules. Then SelI(A∨(1)/F∞) has

no proper sub-Λ-modules of finite index.

Remark 2.2. Under the additional hypothesis that F is abelian over Q with degreeprime to p and that g+ = g−, the algebraic functional equation 1.1 relating SelIc(A/F∞)

and SelI(A∨(1)/F∞) implies that if one of these Λ-modules is a cotorsion, then they

both are.

We shall need twisted signed Selmer groups. For s ∈ Z, we set As = A ⊗ χs|Γ whereχ|Γ : Γ ' Zp. As a Gal(F/F∞)-module, As = A, thus H1(F∞, As) = H1(F∞, A) ⊗ χs

and for a prime v of F , H1(Fv,∞, As) = H1(Fv,∞, A) ⊗ χs and H0(Fv,∞, As) = 0. At

32

primes dividing p, we set

H1Iv(Fv,∞, A

∨(1)s) = H1Iv(Fv,∞, A

∨(1))⊗ χs.

Therefore, for F ′ being F∞ or Fn for some n > 0, we can define twisted I-Selmer groupsSelI(A

∨(1)s/F′) as in with local condition at p induced by H1

Iv(Fv,∞, A

∨(1)s). Weremark that SelI(A

∨(1)s/F∞) ' SelI(A∨(1)/F∞)⊗ χs as Λ-modules.

For F ′ being F∞ or Fn for some n > 0, we set

PΣ,I(A∨(1)s/F

′) =∏

w∈Σ,w-p

H1(F ′w, A∨(1)s)

H1unr(F

′w, A

∨(1)s)×∏v|p

H1(F ′v, A∨(1)s)

H1Iv

(F ′v, A∨(1)s)

.

We prove a “control theorem” for the signed Selmer groups.

Proposition 2.3. For all but finitely many s ∈ Z, the kernel and cokernel of therestriction map

SelIc(As/Fn)→ SelIc(As/F∞)Γn

are finite of bounded order as n varies.

Proof. The diagram

0 SelIc(As/Fn) H1(FΣ/Fn, As) PΣ,Ic(As/Fn)

0 SelIc(As/F∞)Γn H1(FΣ/F∞, As)Γn PΣ,Ic(As/F∞)Γn

(2.1)

is commutative.

By (Torsion), H0(Fv,∞, As) = 0 where v is any prime of F dividing p, thus the centralmap is an isomorphism by the inflation-restriction exact sequence, and the fact that Γn

has p-cohomological dimension one.

We now study the kernel of the rightmost vertical map. For a prime v of F dividing p,the diagram

0 H1Icv

(Fv,n, As) H1(Fv,n, As)H1(Fv,n,As)

H1Icv

(Fv,n,As)0

0 H1Icv

(Fv,∞, As)Γn H1(Fv,∞, As)

Γn

(H1(Fv,∞,As)H1Icv

(Fv,∞,As)

)Γn

(2.2)

33

is commutative. The central vertical map is an isomorphism by the inflation-restrictionexact sequence and the left-most vertical one is an isomorphism by definition, thus itfollows from the snake lemma applied to the diagram (2.2) that the map

H1(Fv,n, As)

H1Icv

(Fv,n, As)→

(H1(Fv,∞, As)

H1Icv

(Fv,∞, As)

)Γn

is an injection.

For a prime w of Fn not dividing p and a prime w′ of F∞ above w, the diagram

0 H1unr(Fn,w, As) H1(Fn,w, As)

H1(Fn,w,As)

H1unr(Fn,w,As)

0

0 H1unr(F∞,w′ , As)

Γn H1(F∞,w′ , As)Γn

(H1(F∞,w′ ,As)

H1unr(F∞,w′ ,As)

)Γn

(2.3)is commutative. If w is archimedean, since p is odd, then H1(F∞,w′ , As) is trivial, andif w is non-archimedean, then H1

unr(F∞,w′ , As) is trivial [37, §A.2.4].

We now look at the kernel of the central vertical map in (2.3). From the inflation-

restriction exact sequence, it is H1(F∞,w′/Fn,w, AGF∞,w′s ). If w is archimedean, then w

splits completely in F∞/Fn so this group is trivial. If w is non-archimedean, it finitelydecomposes in F∞/Fn, so that Gal(F∞,w′/Fn,w) ' Zp and is topologically generated by

an element γn. Thus H1(F∞,w′/Fn,w, AGF∞,w′s ) is isomorphic to

AGF∞,w′s /(γn − 1)A

GF∞,w′s .

One has the short exact sequence

0 AGFn,ws A

GF∞,w′s A

GF∞,w′s A

GF∞,w′s /(γ − 1)A

GF∞,w′s 0.

(γn−1)

For all but finitely many s ∈ Z, AGFn,ws is finite for every n, hence A

GF∞,w′s /(γ−1)A

GF∞,w′s

is finite. So, (AGF∞,w′s )div, the maximal divisible subgroup of A

GF∞,w′s , is contained in

(γn − 1)AGF∞,w′s and the order of A

GF∞,w′s /(γn − 1)A

GF∞,w′s is bounded by the one of

AGF∞,w′s /(A

GF∞,w′s )div for all but finitely many s ∈ Z.

Thus, the snake lemma applied to the diagram (2.3) implies that the map

H1(Fn,w, As)

H1unr(Fn,w, As)

→(H1(F∞,w′ , As)

H1unr(F∞,w′ , As)

)Γn

34

has finite kernel of bounded orders as n varies.

Finally, the result follows from the snake lemma applied to the diagram (2.1). �

Lemma 2.4. For any n > 0, the group H1Iv

(Fv,n, A∨(1)) is divisible.

Proof. For n > 0, let (Ker ColT∨(1),Icv)n be the image of Ker ColT∨(1),Icv under the naturalmap H1

Iw(Fv, T∨(1))→ H1(Fv,n, T

∨(1)). By (Torsion), we have the exact sequence

0→ H1(Fv,n, T∨(1))

in−→ H1(Fv,n, V∨(1))

πn−→ H1(Fv,n, A∨(1))→ 0. (2.4)

The image of (Ker ColT∨(1),Icv)n under in generates a Qp-vector space in H1(Fv,n, V∨(1)),

and we denote by (Ker ColT∨(1),Icv)n the image of this Qp-vector space in H1(Fv,n, A∨(1))

under πn.

By Proposition 1.4 and the bilinearity of Tate’s pairing, the orthogonal complementof (Ker ColT,Iv)n under Tate’s pairing contains (Ker ColT∨(1),Icv)n. The reverse inclusionfollows from the exactness of the sequence (2.4). As already remarked, by (Torsion),one has H1(Fv,n, A

∨(1)) = H1(Fv(µp∞), A∨(1))Γn×∆ and by duality HIw(Fv, T )Γn×∆ =

H1(Fv,n, T ). It follows from the definition of H1Iv

(Fv,n, A∨(1)) that (Ker ColT∨(1),Icv)n =

H1Iv

(Fv,n, A∨(1)). Hence, the group H1

Iv(Fv,n, A

∨(1)) is divisible. �

Proposition 2.5. Assume that SelIc(A/F∞) is a cotorsion Λ-module. Then for all butfinitely many s ∈ Z, the maps

H1(FΣ/F,A∨(1)s)→PΣ,I(A

∨(1)s/F )

andH1(FΣ/F∞, A

∨(1)s)→PΣ,I(A∨(1)s/F∞)

are surjective.

Proof. If SelIc(A/F∞) is a cotorsion Λ-module, then, for all but finitely many s ∈ Z,(SelIc(A/F∞)⊗χs)Γn = (SelIc(As/F∞))Γn is finite for every n. Thus, by Proposition 2.3and possibly avoiding another finite number of s ∈ Z, SelIc(As/Fn) is finite for everyn. For such an s and any n, the finiteness of SelIc(As/Fn) and Lemma 2.4 allow us toapply Proposition 2 which says that the cokernel of

H1(FΣ/Fn, A∨(1)−s)→PΣ,I(A

∨(1)−s/Fn)

is the Pontryagin dual of H0(Fn, As). By (Torsion), H0(F,A) = 0, thus H0(F∞, A) = 0

as Zp is a pro-p-group. FurthermoreAs ' A as Gal(F/F∞)-modules, henceH0(F∞, As) =

35

0 and finally H0(Fn, As) is trivial for any n. We obtain the surjection at level F , and,the surjection being valid for any n implies the surjection at level F∞. This concludesthe proof of the Proposition. �

Lemma 2.6. The restriction map

PΣ,I(A∨(1)s/F )→PΣ,I(A

∨(1)s/F∞)Γ

is surjective.

Proof. We have

PΣ,I(A∨(1)s/F∞) =

∏w∈Σ,w-p

H1(F∞,w, A∨(1)s)

H1unr(F∞,w, A

∨(1)s)×∏v|p

H1(Fv,∞, A∨(1)s)

H1Iv

(Fv,∞, A∨(1)s).

If w is archimedean, since p is odd, H1(F∞,w, A∨(1)s) is trivial. If v is a non-archimedean

prime of F not dividing p, the surjection

H1(Fv, A∨(1)s)

H1unr(Fv, A

∨(1)s)→

∏w|v

H1(F∞,w, A∨(1)s)

H1unr(F∞,w, A

∨(1)s)

Γ

follows from the fact that H1unr(F∞,w, A

∨(1)s) is trivial and Γ has p-cohomologicaldimension 1. Finally, if v is a prime of F dividing p, then the Pontryagin dual ofH1Iv

(Fv,∞, A∨(1)) is contained in a free Λ-module by Proposition 4, thusH1

Iv(Fv,∞, A

∨(1))Γ =

0. Hence, we have an exact sequence

0→ H1Icv

(Fv,∞, A∨(1))Γ → H1(Fv,∞, A

∨(1))Γ →

(H1(Fv,∞, A

∨(1))

H1Icv

(Fv,∞, A∨(1))

)Γ

→ 0. (2.5)

By (Torsion), we know that H1(Fv, A∨(1)) ' H1(Fv,∞, A

∨(1))Γ. Thus, by definitionof H1

Iv(Fv, A

∨(1)) and the exact sequence (2.5), the map

H1(Fv, A∨(1))

H1Iv

(Fv, A∨(1))→(H1(Fv,∞, A

∨(1))

H1Iv

(Fv,∞, A∨(1))

)Γ

is surjective. �

Lemma 2.7. The Λ-corank of PΣ,I(A∨(1)s/F∞) is g+.

Proof. If w is archimedean, since p is odd, H1(F∞,w, A∨(1)s) is trivial. If w is a

non-archimedean prime not dividing p, above a prime v of F , then by [13, Proposi-tion 2], H1(F∞,w, A

∨(1)s) is cotorsion. Finally, by definition, the Pontryagin dual of

36

H1(Fv,∞,A∨(1)s)

H1Iv

(Fv,∞,A∨(1)s)is isomorphic to Ker ColT∨(1),Iv which is of rank gv−|Icv| by Proposition 4.

Therefore, from our choice of I, the corank of PΣ,I(A∨(1)s/F∞) is∑

v|p

gv − |Iv| = g − g− = g+.

�

Before going any further, we compute the corank of the Bloch-Kato Selmer group.

Corollary 2.8. Assume that SelI(A∨(1)/F∞) and SelIc(A/F∞) are cotorsion Λ-modules.

Then the Λ-corank of SelBK(A∨(1)/F∞) is g+.

Proof. By our hypothesis (Slopes), the representation V does not contain a sub-GF -representation with strictly positive Hodge-Tate weights at each prime of F dividingp. Hence, by [38], we have

H1f (Fv,∞, A

∨(1)) = H1(Fv,∞, A∨(1)). (2.6)

We recall that H1(Fv,∞, A∨(1)) ⊗ χs = H1(Fv,∞, A

∨(1)s) and H1(F∞, A∨(1)) ⊗ χs =

H1(F∞, A∨(1)s). Thus, equation (2.6) combined with Proposition 2.5 gives the commu-

tative diagram

0 SelI(A∨(1)/F∞) H1(FΣ/F∞, A

∨(1)) PΣ,I(A∨(1)/F∞) 0

0 SelBK(A∨(1)/F∞) H1(FΣ/F∞, A∨(1))

∏w∈Σ,w-p

H1(F∞,w,A∨(1))

H1unr(F∞,w,A

∨(1)),

=

which induces, by the snake Lemma, the short exact sequence

0→ SelI(A∨(1)/F∞)→ SelBK(A∨(1)/F∞)→

∏v|p

H1(Fv,∞, A∨(1))

H1Iv

(Fv,∞, A∨(1))→ 0.

The Corollary follows from the hypothesis that SelI(A∨(1)/F∞) is cotorsion and (the

proof of) Lemma 2.7. �

Proposition 2.9. Assume that SelIc(A/F∞) and SelI(A∨(1)/F∞) are cotorsion Λ-

modules. Then, for all but finitely many s ∈ Z, H1(FΣ/F∞, A∨(1)s) has no proper

sub-Λ-modules of finite index.

37

Proof. By Proposition 2.5, we have the short exact sequence

0→ SelI(A∨(1)s/F∞)→ H1(FΣ/F∞, A

∨(1)s)→PΣ,I(A∨(1)s/F∞)→ 0,

for all but finitely many s ∈ Z. We assume that SelI(A∨(1)/F∞) is a cotorsion Λ-

module, hence for all but finitely many s ∈ Z, SelI(A∨(1)s/F∞) is a cotorsion Λ-module.

The above short exact sequence then forces the Λ-coranks of H1(FΣ/F∞, A∨(1)s) and

PΣ,I(A∨(1)s/F∞) to be equal. Thus, by Lemma 2.7, we have

corankΛH1(FΣ/F∞, A

∨(1)s) = g+.

On the other hand, from the global Euler-Poincaré characteristic formula [13, Proposition3], we have

corankΛH1(FΣ/F∞, A

∨(1)s) = corankΛH2(FΣ/F∞, A

∨(1)s) + δ(F, V ∨(1)),

withδ(F, V ∨(1)) =

∑v complex

dimQp V∨(1) +

∑v real

dimQp(V∨(1))−

where v runs through archimedean primes of F and, for a real prime v, and dimQp(V∨(1))−

is the dimension of the −1-eigenspace for a complex conjugation above v acting onV ∨(1). From [13, Eq. (34)], we have

δ(F, V ∨(1)) = dimQp(IndQF V

∨(1))− = g+.

Thus,H2(FΣ/F∞, A∨(1)s) is a cotorsion Λ-module. But by [13, Proposition 4],H2(FΣ/F∞, A

∨(1)s)

is a cofree Λ-module, hence H2(FΣ/F∞, A∨(1)s) = 0 and the Proposition follows from

[13, Proposition 5]. �

Proof of Theorem 2.1. For any s ∈ Z, since Γ ' Zp has p-cohomological dimension 1,the restriction map

H1(FΣ/F,A∨(1)s)→ H1(FΣ/F∞, A

∨(1)s)Γ

is surjective. Thus, combining Proposition 2.5 and Lemma 2.6, for all but finitely manys ∈ Z, we obtain the commutative diagram with exact rows

H1(FΣ/F,A∨(1)s) H1(FΣ/F∞, A

∨(1)s)Γ 0

PΣ,I(A∨(1)s/F ) PΣ,I(A

∨(1)s/F∞)Γ 0

0

38

which implies that

H1(FΣ/F∞, A∨(1)s)

Γ →PΣ,I(A∨(1)s/F∞)Γ

is surjective.

By Proposition 2.5 and possibly avoiding another finite set of s ∈ Z, we have the shortexact sequence

0→ SelI(A∨(1)s/F∞)→ H1(FΣ/F∞, A

∨(1)s)→PΣ,I(A∨(1)s/F∞)→ 0.

Taking Γ-invariants gives the long exact sequence

H1(FΣ/F∞, A∨(1)s)

Γ →PΣ,I(A∨(1)s/F∞)Γ → SelI(A

∨(1)s/F∞)Γ → H1(FΣ/F∞, A∨(1)s)Γ.

Since the first map is surjective, SelI(A∨(1)s/F∞)Γ → H1(FΣ/F∞, A

∨(1)s)Γ is injective.Furthermore, H1(FΣ/F∞, A

∨(1)s)Γ is trivial by Proposition 2.9. Thus,

SelI(A∨(1)s/F∞)Γ = 0,

which implies the result. �

2.2 An application: computation of the

Euler-Poincaré characteristic

We now choose a strongly admissible basis as in Proposition 5 and any I ∈ I . We finishthis paragraph by a computation of the Euler-Poincaré of the signed Selmer groups.

For v a non-archimedean prime not dividing p, we recall the definition of the Tamagawanumber of T at v [12, I §4].

If N is Qp-vector space of finite dimension d (respectively a free Zp-module of rank d),we denote by N−1 its dual and we set detQp N = ∧dQp

N (respectively detZp N = ∧dZpN).If N is now a finitely generated Zp-module, we define the determinant of N over Zp as

detZp N = (detZp N−1)−1 ⊗ detZp N0,

where0→ N−1 → N0 → N → 0

is a resolution of N by free Zp-modules of finite ranks N−1 and N0.

39

Let Frobv be the Frobenius in Gal(F unrv /Fv). We have an exact sequence of Qp-vector

spaces

0→ H0(Fv, V )→ H0(F unrv , V )

1−Frobv−−−−−→ H0(F unrv , V )→ H1

unr(Fv, V )→ 0

which induces an isomorphism of Qp-vector spaces

ιV,v : detQp H0(Fv, V )⊗ (detQp H

1unr(Fv, V ))−1 → Qp.

ThendetZp H

0(Fv, T )⊗ (detZp H1unr(Fv, T ))−1

is a Zp-lattice inside detQp H0(Fv, V )⊗(detQp H

1unr(Fv, V ))−1 and the Tamagawa number

of T at v, denoted by Tamv(T ), is defined as the unique power of p such that

ιV,v(detZp H0(Fv, T )⊗ (detZp H

1unr(Fv, T ))−1) = Zp · Tamv(T ).

We can now deduce the following corollary on the leading term of the algebraic p-adicL-function, which is a generalization of Kim’s result on Kobayashi’s plus/minus Selmergroups [24, Theorem 1.2].

Corollary 2.10. Assume that SelBK(A∨(1)/F ) is finite, and that SelIc(A/F∞) is acotorsion Λ-module. Denote by (fI) ⊂ Λ the characteristic ideal of SelI(A

∨(1)/F∞).Then, up to a unit,

fI(0) = | SelBK(A∨(1)/F )| ·∏v-p

Tamv(T ).

Proof. First, we remark that the hypothesis implies that SelI(A∨(1)/F∞) is a cotorsion

Λ-module since, by Proposition 5, SelBK(A∨(1)/F ) = SelI(A∨(1)/F ), and we have a

“control theorem” (Proposition 2.3). Thus SelI(A∨(1)/F∞)Γ is finite, which implies that

SelI(A∨(1)/F∞) is cotorsion.

Up to a unit, we have

fI(0) = | SelI(A∨(1)/F∞)Γ|/| SelI(A

∨(1)/F∞)Γ| (2.7)

= | SelI(A∨(1)/F∞)Γ|,

where the first relation is [14, Lemma 4.2] and the second is Theorem 2.1.

It remains to relate the right hand side of the formula of the corollary to | SelI(A/F∞)Γ|.It is done by studying the commutative diagram

40

0 SelBK(A∨(1)/F ) H1(FΣ/F,A∨(1)) PΣ,f (A

∨(1)/F ) 0

0 SelI(A∨(1)/F∞)Γ H1(FΣ/F∞, A

∨(1))Γ PΣ,I(A∨(1)/F∞)Γ 0,

(fv)v

where the surjection at the end of the top row is Proposition 2.5 and the one at thebottom row is due to Theorem 2.1. As we mentioned in the proof of Proposition 2.3,by (Torsion), the central map is an isomorphism by the inflation-restriction exactsequence. Hence, by the snake Lemma, we have

| SelI(A∨(1)/F∞)Γ| = | SelBK(A∨(1)/F )|.

∏v∈Σ

|Ker fv|. (2.8)

We now compute |Ker fv|. As we have already remarked, the archimedean part is trivialsince p is odd, and if v divides p, then fv is injective (see the proof of Proposition 2.3).Finally, if v is a non-archimedean prime not dividing p, then Ker fv is the orthogonalcomplement under Tate’s local pairing of the projection

Im(HIw(Fv,∞, T )f∗v−→ H1

unr(Fv, T )).

By [35, Lemme 2.2.5], we have | coker f ∗v | = Tamv(T ). Thus, |Ker fv| = Tamv(T ). SinceTamv(T ) = 1 at primes v where A is unramified and all the ramified primes of A arecontained in Σ, we can extend the product in (2.8) over all nonarchimedean primes notdividing p. The corollary follows from (2.7) combined with (2.8). �

2.3 Congruences

Let V ′ be another pseudo-geometric p-adic representation of GF and T ′ a GF -stableZp-lattice inside V ′ satisfying all the hypotheses (Hodge-Tate), (Crystalline), (Tor-sion), (Filtration) and (Slopes). We shall simply add a superscript (·)′ to thevarious object associated to T to denote the similar object associated to T ′ (e.g.A′ = T ′ ⊗Qp/Zp).

From now on, we assume that

T/pT ' T ′/pT ′, (Congruence)

as GF -representations. The goal of this paragraph is to compare the signed Selmergroups of A and A′ under the hypothesis (Congruence).

41

Let v be a prime of F dividing p. We begin by studying the implication of this congruenceon the signed Coleman maps. The crucial step is to compare the Wach module N(T )

and N(T ′) associated to T and T ′ modulo p. Since the Hodge-Tate weights of T and T ′

are in [0, 1], the following theorem is a special case of [3, Théorème IV.1.1].

Theorem 2.11. The isomorphism (Congruence) induces an isomorphism of A+Fv-

modulesN(T )/pN(T ) ' N(T ′)/pN(T ′),

which is compatible with the filtration, the Galois action and the action of ϕ.

We now follow the construction of the signed Coleman maps as given in [7, §2] keepingtrack of the congruences modulo p.

First, note that by (11) and Theorem 2.11, we have a Λcyc-isomorphism

HIw(Fv,cyc, T )/p ' HIw(Fv,cyc, T′)/p, for all v | p. (2.9)

Also, combining the definition of the Dieudonné modules (see (12)) and Theorem 2.11,the Dieudonné modules associated to T and T ′ are isomorphic modulo p. We fix Hodge-compatible bases for Dcris(T ) and Dcris(T

′) compatible with the isomorphism given inTheorem 2.11.

Lemma 2.12. For n > 1, there exists a unique Λ-homomorphism

L (n)T : HIw(Fv,cyc, T )→ Λn ⊗Zp Dcris(T )

such thatϕ−n−1 ◦LT ≡ L (n)

T mod ωn.

Furthermore, the homomorphisms L (n)T and L (n)

T ′ are congruent modulo p, i.e. thediagram

HIw(Fv,cyc, T )/p Λn ⊗Zp Dcris(T )/p

HIw(Fv,cyc, T′)/p Λn ⊗Zp Dcris(T

′)/p

L(n)T mod p

' 'L

(n)

T ′ mod p

is commutative.

42

Proof. The first statement is Proposition 2.9 of op. cit.. We follow its proof to provethe second. Since Perrin-Riou’s regulator LT is given by

(M⊗ 1)−1 ◦ (1− ϕ) ◦ (hT )−1,

the first statement then follows from a study (Lemma 3.44 of op. cit.) of the map

ϕ−n−1 ◦ (1− ϕ) : N(T )ψ=1 → (ϕ∗N(T ))ψ=0 ↪→ B+rig,Fv

⊗Zp Dcris,v(T ) (2.10)

which shows that, for x ∈ N(T )ψ=1, the element ϕ−n−1 ◦ (1 − ϕ)(x) is congruent toan element of (A+

Fv)ψ=0 ⊗Zp Dcris,v(T ) modulo ϕn+1(π)B+

rig,Fv⊗Zp Dcris,v(T ). But by

Theorem 2.11, the maps (2.10) for T and T ′ agree modulo p and we are done. �

For i ∈ {1, . . . , gv}, we write L (n)T,i for the composition of L (n)

T with the projection onthe i-th component of the fixed basis of Dcris(T ). We write hn (respectively h′n) for theΛn-endomorphisms on ⊕gvk=1Λn given by the left multiplication by the product Cn · · ·C1

(respectively C ′n · · ·C ′1).

Lemma 2.13. For n > 1, there exists a unique Λ-homomorphism

Col(n)T : HIw(Fv,cyc, T )→

gv⊕k=1

Λn

such that L (n)T,1...

L (n)T,gv

≡ Cn · · ·C1 · Col(n)T mod Kerhn.

Furthermore, we haveCol

(n)T ≡ Col

(n)T ′ mod p.

Proof. The first part is Proposition 2.10 of op. cit.. Again by Theorem 2.11 and thedefinition of the Dieudonné modules, the matrices Cn and C ′n are congruent modulo pfor all n. Thus, by the first part of the Lemma and Lemma 2.12, we have

Cn · · ·C1 · Col(n)T ≡ Cn · · ·C1 · Col

(n)T ′ mod (Kerhn, p).

Since

Cn =

(Idv 0

0 Φpn(1 +X)Igv−d

)C−1,

with C ∈ GLgv(Zp) and Φpn(1 +X) and p are coprime, the second part follows. �

43

By [7, Lemma 2.11 and Theorem 2.13], the maps (Col(n)T )n>1 are compatible with the

natural projection ⊕gvk=1Λn+1/Kerhn+1 → ⊕gvk=1Λn/Kerhn and thus define a map tolim←−n⊕

gvk=1Λn/Kerhn which naturally identifies with ⊕gvk=1Λ. By definition, this map is

ColT . Thus, by Lemma 2.13 we have :

Proposition 2.14. The Coleman maps associated to T and T ′ are congruent modulo p.More precisely, if z ∈ HIw(Fv,cyc, T ) and z′ ∈ HIw(Fv,cyc, T

′) have the same image underthe isomorphism given in (2.9), then

ColT (z) ≡ ColT ′(z′) mod p⊕gvk=1 Λ.

We now define and compare the non-primitive signed Selmer groups under the hypothesis(Congruence) and deduce the main result of this chapter.

Lemma 2.15. The exact sequence

0→ A∨(1)[p]→ A∨(1)p−→ A∨(1)→ 0

of GF -modules induces isomorphisms

H1(FΣ/F∞, A∨(1)[p]) ' H1(FΣ/F∞, A

∨(1))[p],

andH1(Fv,∞, A

∨(1)[p]) ' H1(Fv,∞, A∨(1))[p],

for any prime v dividing p, and, for any non-archimedean prime v not dividing p or aprime of ramification of A∨(1),

H1(F unrv , A∨(1)[p]) ' H1(F unr

v , A∨(1))[p].

Proof. We have the exact sequence

0→ H0(F∞, A∨(1))/p→ H1(FΣ/F∞, A

∨(1)[p])→ H1(FΣ/F∞, A∨(1))[p]→ 0.

SinceH0(F∞, A∨(1)) is trivial by our hypothesis (Torsion), we get the first isomorphism.

The same proof applies for the second isomorphism at v dividing p. Let v as in the thirdstatement, then v is not a prime of ramification for A∨(1), thus H0(F unr

v , A∨(1)) = A∨(1).Hence, we have the exact sequence

0→ A∨(1)/p→ H1(F unrv , A∨(1)[p])→ H1(F unr

v , A∨(1))[p]→ 0.

Since A∨(1) is divisible, we deduce the second isomorphism. �

44

Definition 2.16. Let Σ0 ⊂ Σ be a subset that contains all the primes of ramificationof A∨(1) but not the primes dividing p or the archimedean primes. We define thenon-primitive I-Selmer groups of A∨(1) over F∞ by

SelΣ0I (A∨(1)/F∞) = Ker(H1(FΣ/F∞, A

∨(1))→PΣ\Σ0,I(A∨(1)/F∞)),

with

PΣ\Σ0,I(A∨(1)/F∞) =

∏w∈Σ\Σ0,w-p

H1(F unrw , A∨(1))×

∏v|p

H1(Fw,∞, A∨(1))

H1Iv

(Fw,∞, A∨(1)).

If v is a prime dividing p, by Lemma 2.15 we haveH1(Fv,∞, A∨(1)[p]) ' H1(Fv,∞, A

∨(1))[p]

and we set

H1Iv(Fv,∞, A

∨(1)[p]) = H1Iv(Fv,∞, A

∨(1))[p] ⊂ H1(Fv,∞, A∨(1)[p]).

We set

PΣ\Σ0,I(A∨(1)[p]/F∞) =

∏w∈Σ\Σ0,w-p

H1(F unrw , A∨(1)[p])×

∏w|p

H1(Fw,∞, A∨(1)[p])

H1Iv

(Fw,∞, A∨(1)[p]).

We define the non-primitive I-Selmer groups of A∨(1)[p] over F∞ by

SelΣ0I (A∨(1)[p]/F∞) = Ker(H1(FΣ/F∞, A

∨(1)[p])→PΣ\Σ0,I(A∨(1)[p]/F∞)).

Remark 2.17. By Tate’s pairing (7), the Pontryagin dual of H1Iv

(Fv,∞, A∨(1)[p]) is

Im ColT,Iv /p.

From now on, we write by abuse of notation the µ and λ-invariants of the various Selmergroups to refer to the µ and λ-invariants of their Pontryagin duals.

Proposition 2.18. For any Σ0 ⊂ Σ as in Definition 2.16, we have an isomorphism ofΛ-modules

SelΣ0I (A∨(1)[p]/F∞) ' SelΣ0

I (A∨(1)/F∞)[p].

Proof. By Lemma 2.15, we have

H1(FΣ/F∞, A∨(1)[p]) ' H1(FΣ/F∞, A

∨(1))[p].

Therefore, in order to prove the Proposition, it is enough to compare the local conditionsdefining the two Selmer groups. At v ∈ Σ \ Σ0 and v not dividing p, the second partof Lemma 2.15 shows that the local conditions are equivalent. Since p is odd, thearchimedean part is trivial. At v dividing p, by definition the local conditions are thesame. �

45

We now relate the non-primitive signed Selmer groups to the signed Selmer groups.Assume that SelI(A

∨(1)/F∞) is a cotorsion Λ-module. Then by Proposition 2.5 and thedefinition of the non-primitive signed Selmer groups, we have

SelΣ0I (A∨(1)/F∞)/ SelI(A

∨(1)/F∞) '∏w∈Σ0

H1(F∞,w, A∨(1))

H1unr(F∞,w, A

∨(1)). (2.11)

We have already noted that H1unr(F∞,w, A

∨(1)) is trivial (by [37, §A.2.4]) and thatH1(F∞,w, A

∨(1)) is a cotorsion Λ-module for all w ∈ Σ0 (by [13, Proposition 2]).Furthermore, the µ-invariant of the Pontryagin dual of H1(F∞,w, A

∨(1)) is zero [17,Proposition 2.4]. Thus, if SelI(A

∨(1)/F∞) is a cotorsion Λ-module, then, by (2.11),SelΣ0

I (A∨(1)/F∞) is a cotorsion Λ-module as well, their µ-invariants are equal and

corankZp SelΣ0I (A∨(1)/F∞) = corankZp SelI(A

∨(1)/F∞)+∑w∈Σ0

corankZp H1(F∞,w, A

∨(1)).

We have an analogue of Theorem 2.1 for non-primitive signed Selmer groups.

Proposition 2.19. Assume that SelI(A∨(1)/F∞) and SelIc(A/F∞) are cotorsion Λ-

modules. Then SelΣ0I (A∨(1)/F∞) has no proper sub-Λ-modules of finite index.

Proof. From the definition of the non-primitive signed Selmer groups and Proposition 2.5,we have an analogue of Proposition 2.5 for the non-primitive signed Selmer groups.That is, we have the exact sequence

0→ SelΣ0I (A∨(1)/F∞)→ H1(FΣ/F∞, A

∨(1))→PΣ\Σ0,I(A∨(1)/F∞)→ 0,

and,

0→ SelΣ0I (A∨(1)/F )→ H1(FΣ/F,A

∨(1))→PΣ\Σ0,I(A∨(1)/F )→ 0.

The proof of the Proposition then follows precisely the one of Theorem 2.1, hence, weskip it. �

Corollary 2.20. Assume that SelI(A∨(1)/F∞) and SelIc(A/F∞) are cotorsion Λ-

modules. Furthermore, assume that the µ-invariant of SelI(A∨(1)/F∞) is zero. Then

the λ-invariant of SelΣ0I (A∨(1)/F∞) is equal to dimFp SelΣ0

I (A∨(1)[p]/F∞).

Proof. By the discussion preceding Proposition 2.19 and the hypotheses, SelΣ0I (A∨(1)/F∞)

is a cotorsion Λ-module and its µ-invariant is zero. Thus, the Pontryagin dual of

46

SelΣ0I (A∨(1)/F∞) is a finitely generated Zp-module, and, by Proposition 2.19, its Zp-

torsion submodule is trivial. Thus, SelΣ0I (A∨(1)/F∞) is Zp-divisible with Zp-corank its

λ-invariant (i.e. SelΣ0I (A∨(1)/F∞) ' (Qp/Zp)

λ). Therefore, by Proposition 2.18, wehave

λ = dimFp SelΣ0I (A∨(1)/F∞)[p] = dimFp SelΣ0

I (A∨(1)[p]/F∞).

�

We are now able to prove the main result of this section.

Theorem 2.21. Assume (Congruence) and choose Hodge-compatible bases for Tand T ′ compatible modulo p by Theorem 2.11. Let I ∈ I and Σ0 be a finite set ofprimes that contains all the primes of ramification of A∨(1) and A∨(1)′ but not theprimes dividing p or the archimedean primes. Further, assume that SelI(A

∨(1)/F∞),SelIc(A/F∞), SelI(A

∨(1)′/F∞) and SelIc(A′/F∞) are cotorsion Λ-modules. Then the

µ-invariant of SelI(A∨(1)/F∞) vanishes if and only if that of SelI(A

∨(1)′/F∞) vanishes.Furthermore, when these µ-invariants do vanish, the λ-invariants of SelΣ0

I (A∨(1)/F∞)

and SelΣ0I (A∨(1)′/F∞) are equal.

Proof. The hypothesis (Congruence) implies that

A∨(1)[p] ' A∨(1)′[p],

as GF -representations. Hence, we have

H1(FΣ/F∞, A∨(1)[p]) ' H1(FΣ/F∞, A

∨(1)′[p]),

H1(F unrv , A∨(1)[p]) ' H1(F unr

v , A∨(1)′[p]),

for v not dividing p. By Remark 2.17 and Proposition 2.14, for each v dividing p, wehave

H1Iv(Fv,∞, A

∨(1)[p]) ' H1Iv(Fv,∞, A

∨(1)′[p]).

It follows thatSelΣ0

I (A∨(1)[p]/F∞) ' SelΣ0I (A∨(1)′[p]/F∞).

Combined with Proposition 2.18, we have

SelΣ0I (A∨(1)/F∞)[p] ' SelΣ0

I (A∨(1)[p]/F∞) ' SelΣ0I (A∨(1)′[p]/F∞) ' SelΣ0

I (A∨(1)′/F∞)[p].

Therefore, the first assertion follows from the discussion preceding Proposition 2.19.The second claim follows from Corollary 2.20. �

47

Under the hypothesis of the above Theorem and the assumption that the µ-invariantsvanish, we can relate the λ-invariant of SelI(A

∨(1)/F∞) and that of SelI(A∨(1)′/F∞),

which we denote by λ and λ′. The Theorem combined with (2.11) and the discussionthat follows it implies

λ−∑w∈Σ0

corankZp H1(F∞,w, A

∨(1)) = λ′ −∑w∈Σ0

corankZp H1(F∞,w, A

∨(1)′).

Besides, we can compute the coranks of H1(F∞,w, A∨(1)) and H1(F∞,w, A

∨(1)′) thanksto [17, Proposition 2.4].

48

Chapter 3

Mordell-Weil ranks of supersingularabelian varieties over cyclotomicextensions

The results of this chapter were obtained in collaboration with Antonio Lei.

Let F be a number field unramified at p and X an abelian variety of dimension g

defined over F with good supersingular reduction at every prime of F dividing p. LetT = Tp(X) be the Tate module of X. Let XD be the dual abelian variety of X.

The goal of this chapter is to prove.

Theorem 3.1. Assume that for one I ∈ I , the Pontryagin dual of SelI(XD/F∞) is a

torsion Λ-module. Then the rank of SelBK(XD/Fn)∧ over Zp is bounded independentlyof n. Consequently, the Mordell-Weil rank of XD(Fn) is bounded as n varies.

Let F ′ be an algebraic extension of F contained in FΣ. The fine Selmer group of X overF ′ is defined by

Sel0(X/F ′) = Ker

(H1(FΣ/F

′, X[p∞])→∏v∈Σ

H1(F ′v, X[p∞])

).

One has a “control theorem” for the fine Selmer groups in the cyclotomic extension,which we prove following closely Greenberg [14, §3] and [16].

Lemma 3.2. The kernel and cokernel of the restriction map

Sel0(X/Fn)→ Sel0(X/F∞)Γn

49

are finite and have bounded order as n varies.

Proof. The diagram

0 Sel0(X/Fn) H1(FΣ/Fn, X[p∞])∏

vn∈ΣH1(Fn,vn , X[p∞])

0 Sel0(X/F∞)Γn H1(FΣ/F∞, X[p∞])Γn∏

w∈Σ H1(F∞,w, X[p∞])Γn

(3.1)is commutative.

One has the inflation-restriction exact sequence

0→ H1(Γn, X(F∞)[p∞])→ H1(FΣ/Fn, X[p∞])→ H1(FΣ/F∞, X[p∞])Γn → H2(Γn, X(F∞)[p∞]).

By (Torsion), the groupsH1(Γn, X(F∞)[p∞]) andH2(Γn, X(F∞)[p∞]) are trivial. Thus,the central vertical map of the diagram is an isomorphism.

We now study the rightmost vertical map prime by prime. Let v be any prime of F andvn be any prime of Fn above v. Let rn be the restriction map

H1(Fn,vn , X[p∞])→ H1(F∞,w, X[p∞])

where w is any prime of F∞ dividing vn. If v is archimedean, then v splits completely inF∞/F . Thus, Ker(rn) = 0. If v is a non-archimedean prime, by the inflation-restrictionexact sequence

Ker(rn) ' H1(Γvn , X(F∞,w)[p∞]).

By (Torsion), if v divides p, this last group is trival. We assume that v does not dividep. Then v is unramified and finitely decomposed in F∞. Thus, F∞,w/Fv is an unramifiedZp-extension. Let γvn be a topological generator of Γvn , where Γvn = Gal(F∞,w/Fv) ' Zp.Then

Ker(rn) ' H1(Γvn , X(F∞,w)[p∞]) ' X(F∞,w)[p∞]/(γvn − 1)X(F∞,w)[p∞].

As a group X(F∞,w)[p∞] ' (Qp/Zp)t × (a finite group), for some 0 6 t 6 2g. Since

X(Fn,vn) is finitely generated, the kernel of (γvn − 1) acting on X(F∞,w)[p∞] is finite.Thus, the restriction of (γvn − 1) on the maximal divisible subgroup, which we write as(X(F∞,w)[p∞])div, is surjective and we have

(X(F∞,w)[p∞])div ⊂ (γvn − 1)X(F∞,w)[p∞].

50

Therefore, the cardinality of Ker(rn) is bounded by [X(F∞,w)[p∞] : (X(F∞,w)[p∞])div]

which is independent of n. Furthermore, if X has good reduction at v, then the inertiasubgroup of GFv acts trivially and X(F∞,w)[p∞] is divisible. Hence, Ker(rn) is trivial.

Now the set of non-archimedean primes of F where X has bad reduction is finite andfor each of these primes, the order of Ker(rn) is bounded as n varies and the numberof primes vn of Fn dividing v is also bounded, hence, the order of the kernel of theright-most vertical map in the diagram is bounded as n varies.

The result follows by applying the snake lemma on the diagram (3.1). �

Taking the projective limit over n of the Poitou-Tate exact sequence (15) and (16), wehave the exact sequences of Λ-modules

0→ Sel0(X/F∞)∧ → H2Iw(FΣ/F∞, T )→

∏w∈Σ

H2Iw(F∞,w, T ), (3.2)

H1Iw(FΣ/F∞, T )→

∏w∈Σ

H1Iw(Fv, T )

Ker ColIv→ SelI(X/F∞)∧ → Sel0(X/F∞)∧ → 0, (3.3)

and, for any n > 0,

H1(FΣ/Fn, T )→∏w∈Σ

H1(Fn,w, T )

X(Fn,v)⊗Zp→ Selp(X/Fn)∧ → Sel0(X/Fn)∧ → 0. (3.4)

Lemma 3.3. Assume that SelI(X/F∞)∧ is a torsion Λ-module. Then

1. Sel0(X/F∞) is a torsion Λ-module,

2. H2Iw,Σ(F∞, T ) is a torsion Λ-module,

3. H1Iw,Σ(F∞, T ) is a Λ-module of rank g[F : Q].

Proof. By the exact sequence (3.3), the hypothesis of the Lemma implies (1).

By the structure of Λ-module of the 2nd Iwasawa cohomology group, for every w ∈ Σ,H2

Iw(Fv,∞, T ) is a torsion Λ-module. Hence, using the exact sequence (3.2), the firststatement of the lemma implies the second.

Finally, thanks to [37, Proposition 1.3.2 (i)⇒ (ii)], we conclude that (2) implies (3). �

We shall need a precise relation between the Iwasawa cohomology groups and the Galoiscohomology groups at level Fn.

51

Lemma 3.4. Assume that v divides p. The natural projection map

πn : H1Iw(Fv,∞, T )→ H1(Fn,vn , T )

is surjective.

Proof. It is enough to check that the corestriction map

H1(Fn+1,vn+1 , T )→ H1(Fn,vn , T )

is surjective for all n > 0. By Tate’s local duality, it is equivalent to check that therestriction map

H1(Fn,vn , XD[p∞])→ H1(Fn+1,vn+1 , X

D[p∞])

is injective for any n > 0. By the inflation-restriction exact sequence, the kernel ofthe restriction map is H1(Gal(Fn+1,vn+1/Fn,vn), (XD[p∞])

GFn+1,vn+1 ). By (Torsion), thelast group is trivial, hence the result. �

Recall that, for any n > 0, one has the Kummer map

X(Fn,vn)⊗Zp → H1(Fn,vn , T )

which is injective. We denote by πn the composition of πn with the natural surjectionH1(Fn,vn , T )→ H1(Fn,vn , T )/X(Fn,vn)⊗Zp.

Corollary 3.5. Assume that v divides p. The induced map

lim←−n

∧πn :

g[Fv :Q]∧H1

Iw(Fv,∞, T )→ lim←−n

g[Fv :Q]∧(H1(Fn,vn , T )/X(Fn,vn)⊗Zp)

is injective.

Proof. Consider the induced map on the exterior algebras over Zp[[Γ]]

∧πn :∧

H1Iw(Fv,∞, T )→

∧(H1(Fn,vn , T )/X(Fn,vn)⊗Zp).

By functoriality, we have Ker∧πn+1 ⊂ Ker∧πn and Ker lim←−∧πn = ∩n Ker∧πn. Sincethe projection map is surjective by Lemma 3.4, the induced map on the exterior algebras∧πn is surjective and its kernel is the sub-module of

∧H1

Iw(Fv,∞, T ) generated by thekernel of πn [5, A.III., §7.2, Proposition 3]. But ∩n Ker πn is zero, since there are no“universal norms” for supersingular abelian varieties over the cyclotomic Zp-extensionwhich is well known (see [8, Theorem 5.2]). �

52

Lemma 3.6. Let r, n > 1 be integers and x1, . . . , xr ∈ Qp[Γ/Γn]⊕r, then

dimQp

Qp[Γ/Γn]⊕r

(x1, . . . , xr)6 r × dimQp

Qp[Γ/Γn]

(∧xi).

Proof. Given a Qp[Γ/Γn]-module M , we have the equality

dimQpM =n∑k=0

dimQpMk,

where Mk = eθ ·M and eθ is the idempotent attached to any Dirichlet character θ thatfactors through Γ/Γk, but not Γ/Γk−1. Note that multiplying by eθ commutes withtaking wedge products. Therefore, it is enough to consider

eθQp[Γ/Γn]⊕r

(eθ · x1, . . . , eθ · xr)=

Qp(ζ)⊕r

(y1, . . . , yr)Qp(ζ)

,

where ζ is a p-power root of unity, yi is the image of xi under θ and (y1, . . . , yr)Qp(ζ)

denotes the Qp(ζ)-vector space generated by y1, . . . , yr. In particular, the quotienton the right-hand side is non-trivial if and only if the vectors y1, . . . , yr are linearlydependent, which is exactly the same as saying that ∧yi = 0. When this happens, thedimension on the right-hand side is bounded above by

r × dimQp Qp(ζ) = r × dimQp Qp(ζ)/(∧yi).

Otherwise, the quotient is trivial and so is Qp(ζ)/(∧yi). �

Proof of Theorem 3.1. The second statement of the theorem is a direct consequence ofthe first one thanks to the short exact sequence (19). We prove that rankZp Selp(X

D/Fn)∧

is bounded independently of n. Since Sel0(XD/F∞)∧ is a torsion Λ-module by Lemma 3.3,the “control theorem” for the fine Selmer groups (Lemma 3.2) implies that

rankZp Sel0(XD/Fn)∧

is bounded independently of n. Therefore, our theorem would follow from the exactsequence (3.4) if we show that

rankZp coker

H1(FΣ/Fn, T )→∏v|p

H1(Fn,vn , T )

X(Fn,vn)⊗Zp

is bounded independently of n.

53

To do so, we study the commutative diagram

H1Iw(FΣ/F∞, T )

∏w∈Σ H

1Iw(Fv,∞, T )

H1(FΣ/Fn, T )∏

vn∈ΣH1(Fn,vn ,T )

X(Fn,vn )⊗Zp.

(3.5)

We first look at the top localization map. In the exact sequence (3.3), bothH1Iw(FΣ/F∞, T )

and∏

w∈ΣH1

Iw(Fv,∞,T )

Ker ColIvare Λ-modules of rank g[F : Q] by Lemma 3.3 (3) and Proposi-

tion 4. By hypothesis, SelI(XD/F∞)∧ is a torsion Λ-module. In particular, the cokernel

of the first arrow of the sequence (3.3) is a torsion Λ-module. Thus, if we choose{c1, . . . , cg[F :Q]} to be a family of linearly independent elements of H1

Iw(FΣ/F∞, T ), itsimage in

∏w∈Σ

H1Iw(Fv,∞,T )

Ker ColIvand, therefore in

∏w∈Σ H

1Iw(Fv,∞, T ), is linearly independent.

Hence, we have the exterior product of diagram (3.5):

∧g[F :Q] H1Iw(FΣ/F∞, T )

∧g[F :Q]∏w∈Σ H

1Iw(Fv,∞, T )

∧g[F :Q] H1(FΣ/Fn, T )∧g[F :Q]∏

vn∈ΣH1(Fn,vn ,T )

X(Fn,vn )⊗Zp,

(3.6)

and a non-zero element ∧ci of∧g[F :Q] H1

Iw(FΣ/F∞, T ) which maps to a non-zero elementin∧g[F :Q]∏

w∈Σ H1Iw(Fv,∞, T ). By Corollary 3.5, this element maps to a non-trivial

element in∧g[F :Q]∏

vn∈ΣH1(Fn,vn ,T )

X(Fn,vn )⊗Zpfor n� 0.

Via Bloch-Kato’s dual exponential map,∏

vn∈ΣH1(Fn,vn ,T⊗Qp)

X(Fn,vn )⊗Qpis free of rank g[F : Q]

over Qp[Γ/Γn], we have

g[F :Q]∧ ∏vn∈Σ

H1(Fn,vn , T ⊗Qp)

X(Fn,vn)⊗Qp

∼= Qp[Γ/Γn].

Let us denote by cn the image of ∧ci in Qp[Γ/Γn] under this isomorphism. As in theproof of Lemma 3.6,

dimQp

Qp[Γ/Γn]

cn=

n∑k=0

dimQp

(Qp[Γ/Γn]

cn

)k

=n∑k=0

dimQp

Qp(ζk)

eθk cn,

where θk is a Dirichlet character that factors through Γ/Γk, but not Γ/Γk−1 and ζk issome p-power root of unity. Since cn are norm compatible, eθk cn is independent of n

54

whenever n > k. As ∧ci 6= 0, Weierstrass’ preparation theorem implies that eθk ∧ ci 6= 0

when k is sufficiently large. In particular eθk cn 6= 0 whenever n > k � 0. Therefore,

dimQp

Qp[Γ/Γn]

cn= O(1),

which does not depend on n. Lemma 3.6 tells us that

rankZp coker

H1(Fn,Σ, T )→∏v|p

H1(Fn,vn , T )

X(Fn,vn)⊗Zp

6 g[F : Q]× dimQp

Qp[Γ/Γn]

cn

and so we are done. �

55

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