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

On the algebraic side of the Iwasawa theory of some

non-ordinary Galois representations

Thèse

Gautier Ponsinet

Doctorat en mathématiques

Philosophiæ doctor (Ph. D.)

On the algebraic side of the Iwasawa theory of some

non-ordinary Galois representations

Thèse

Gautier Ponsinet

Sous la direction de:

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 de Gal(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 certaines représentations galoisiennes. En particulier, leurs constructions s’appliquent aux cas des variétés abéliennes définies sur F ayant bonne réduction supersingulière en chaque premier de F divisant p. Ces groupes de Selmer signés ont naturellement une structure de Λ-modules de type fini.

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

Puis, nous étudions la structure de Λ-module des groupes de Selmer signés. Sous l’hypothèse qu’ils sont des Λ-modules de cotorsion, nous montrons qu’ils ne possèdent pas de sous-Λ-module propre d’indice fini. Nous déduisons de ce résultat quelques applications 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 la taille du groupe de Selmer de Bloch-Kato sur F . De plus, pour deux telles representations isomorphes modulo p, nous comparons les invariants d’Iwasawa de leurs groupes de Selmer signés.

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

Abstract

Let F be a number field unramified at an odd rational prime p. Let F∞ be the Zp-cyclotomic extension of F and Λ = Zp[[Gal(F∞/F )]] be the Iwasawa algebra of Gal(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-ordinary Galois representations. In particular, their construction applies to abelian varieties defined over F with good supersingular reduction at primes of F dividing p. These signed Selmer groups have a natural structure of finitely generated Λ-modules.

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

Second, we study the structure of Λ-module of the signed Selmer groups. Assuming that they are cotorsion Λ-modules, we show that they have no proper sub-Λ-module of finite index. We deduce from this a number of arithmetic applications. We compute the Λ-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 an explicit formula on the size of the Bloch-Kato Selmer group over F . Furthermore, for two such representations that are isomorphic modulo p, we compare the Iwasawa-invariants of their signed Selmer groups.

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

Contents

Notations and setup 6

1 Functional equation of signed Selmer groups 22

1.1 Orthogonality of local conditions . . . 23

1.2 Control theorem . . . 25

2 Λ-module structure and congruences of signed Selmer groups 32

2.1 Sub-Λ-modules . . . 32

2.2 An application: computation of the Euler-Poincaré characteristic. . . 39

2.3 Congruences . . . 41

3 Mordell-Weil ranks of supersingular abelian varieties over

cyclo-tomic extensions 49

Remerciements

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

Je remercie Kazim Büyükboduk, Daniel Delbourgo et Byoung Du Kim d’avoir accepté d’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 lecture attentive 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 mes parents Marie-Christine et Norbert, mes frères Thomas et Lucas, mes grands-parents Jean, Christian et Claude, et puis Arnaud et Florence et toute cette chouette famille, à mes amis de toujours : Vinh-Lôc, Thibaut, Édouard, Aurélien, à Macarena, à Joaquin et 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

Introduction

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

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

Sel(E/K) = Ker H1(K, E[p∞]) →Y v H1_(K v, E[p∞]) E(Kv) ⊗ Qp/Zp ! .

It fits in the short exact sequence of groups

0 → E(K) ⊗ZQp/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, we can associated a characteristic ideal in Λ to the Selmer group and the Iwasawa main conjecture states that this ideal is generated by the p-adic L-function associated to E by Mazur and Swinnerton-Dyer which p-adically interpolates values of the complex L-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 by Kato and Skinner-Urban [43].

When E has good supersingular reduction at primes of F dividing p, already Mazur’s conjecture 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

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 that context has been proved by Wan [47], that is, the two p-adic L-functions of Pollack generate 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-adic Hodge theory machinery, they defined signed Selmer groups for certain non-ordinary Galois representations of GF = Gal(F /F ). In particular, the construction apply to the abelian varieties with good supersingular reduction at primes of F dividing p. This construction depends on the choice of a basis of the Dieudonné module attached to the representation which respect the filtration, in particular, for an elliptic curve defined over 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 both the 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 Selmer group satisfy a functional equation as well. This “algebraic” functional equation can be checked directly for the Selmer group, even without knowing that the Selmer groups are cotorsion. Greenberg [13] has proved such a functional equation for general ordinary Galois representations and B.D. Kim [22] has proved such a functional equation for the plus 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 satisfying the hypothesis of [7]. We fix a basis for the Dieudonné module D(T ) of T . Choose I a subset of that basis of cardinal dimQp(Ind

Q

F V )+ the dimension of the +1-eigenspace under the action of a complex conjugation on the induced representation IndQ_F V . Let V∨(1) (respectively T∨(1)) be the Tate dual of V (respectively T ). We denote by SelI(V∨(1)/T∨(1), F∞) the signed Selmer group associated to V∨(1) and I, and by SelI(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

Λ-module, there exists a pseudo-isomorphism M → r M i=1 Λ ⊕ n M j=1 Λ/(paj_{) ⊕} m M k=1 Λ/(fbk k ) (2)

where fk ∈ Λ ' Zp[[X]] are distinguished irreducible polynomials. There is a natural pairing on the Dieudonné modules D(T ) × D(T∨(1)) → Zp and we choose as a basis of D(T∨(1)) the dual basis of the one we choose for D(T ) and let Ic be the orthogonal complement 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(Ind Q F V ) + = dimQp(Ind Q 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 by Colmez [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’s conjecture and that the E(F ) has no p-torsion, Greenberg [14, Proposition 4.14] showed that Selp(E/F∞) has no proper sub-Λ-module of finite index. When E is an elliptic curve with supersingular reduction at prime above p, B.D. Kim [24] has then extended the definition of the plus and minus Selmer groups to number fields F where p is unramified and generalized Greenberg’s result and showed that if the plus and minus Selmer groups of E over F∞ are cotorsion Λ-modules, then they have no proper submodule of finite index (for one of the signed Selmer group, namely the plus one, he requires the additional assumption that p splits completely in F and is totally ramified in F∞). This assumption has recently been removed by Kitajima and Otsuki, see [27]. We prove a similar result for the signed Selmer groups.

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

As a byproduct of the proof, we compute the Λ-corank of the Bloch-Kato Selmer group of V over F∞ (see Corollary 2.8). This theorem also allows us to employ Greenberg’s strategy in [14, Theorem 4.1] to compute the Euler-Poincaré characteristic of the signed Selmer groups. We may relate the leading term of the characteristic series of these Selmer groups to a product of Tamagawa numbers associated to the represenatation and 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 and E0 are elliptic curves defined over Q with good ordinary reduction at p and such that E[p] ' E0[p] as Galois modules, Greenberg and Vatsal [17] have studied the consequences of such a congruences in Iwasawa theory. In particular, assuming Mazur’s conjecture, they proved that the µ-invariant of Selp(E/Q∞) vanishes if and only if that of Selp(E0/Q∞) vanishes, and that when these µ-invariants do vanish, the λ-invariants of some non-primitive Selmer groups associated to E and E0 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 T0be Galois representations to which the construction of [7] applies. Assume that T /pT ' T0/pT0 as Galois modules and that the the signed Selmer groups associated to T , T∨(1), T0 and (T0)∨(1) are cotorsion Λ-modules. Then the µ-invariant of SelI(V∨(1)/T∨(1), F∞) vanishes if and only if that of SelI((V0)∨(1)/(T0)∨(1), F∞) vanishes. Furthermore, when these µ-invariants do vanish, the λ-invariants of the I-signed non-primitive Selmer groups associated to T and T0 over F∞ are equal.

The main ingredient of the proof is a result of Berger [3] who showed that the congruence T /pT ' T0/pT0 of Galois module induces a congruence modulo p on the Wach module associated to T and T0. This allows to keep track of the congruence through Büyükboduk and Lei’s construction.

In the final chapter, we focus on abelian varieties with supersingular reduction at primes of F dividing p. Let X be an abelian variety defined over F . The Mordell-Weil groups X(Fn) of Fn-rational points are finitely generated abelian groups [33]. When X has good ordinary reduction at primes above p, the Pontryagin dual of Selp(X/F∞) is conjectured to be a torsion Λ-module. Under this conjecture, Mazur’s control theorem on the Selmer groups implies that the rank of the Mordell-Weil group X(Fn) is bounded independently of 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 a consequence, one deduces that the rank of the Mordell-Weil group X(Fn) is bounded

independently of n. See [28, Corollary 10.2]. Alternatively, it is possible to deduce the same result using Kato’s Euler system in [21] and Rohrlich’s non-vanishing results on the 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 the Mordell-Weil ranks of an elliptic curve with supersingular reduction and ap = 0. She showed that when a certain algebraic p-adic L-function is non-zero and does not vanish at the trivial character, then the Mordell-Weil ranks of the elliptic curve over Fn are bounded independently of n. Kim [26] as well as Im and Kim [18] have generalized this method to abelian varieties which can have mixed reduction types at primes above p. Furthermore, unlike the present text, they did not assume that p is unramified in F . Following Perrin-Riou’s construction, Im and Kim defined certain algebraic p-adic L-functions and showed that when this is non-zero, then the Mordell-Weil ranks of the abelian 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 cotorsion Λ-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 signed Selmer groups and the Mordell-Weil group as in the exact sequence (1). However, we may nonetheless obtain information about the Selmer groups Selp(X/Fn) from the signed Selmer groups via the Poitou-Tate exact sequence and some calculations in multi-linear algebra.

Finally, we recall that Kobayashi’s plus and minus Selmer groups have been generalized to elliptic 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 of Kobayashi in [28, §10]. At present, we do not know how to generalize this to abelian varieties since we do not have any explicit description of the action of the Frobenius operator on Dieudonné module attached to X. This prevents us from carrying out the explicit matrix calculations in [45] in this setting.

Notations and setup

Unless stated otherwise, in this text p is an odd prime number. If K is a number field, we denote byOK the ring of integers of K. If v is prime of K, let Kv be the completion of K at v. When v is non-archimedean, let OKv be the ring of integers of Kv, mKv be

the maximal ideal of OKv and kKv =OKv/mKv be the residual field ofOKv. We fix K

an 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 absolute Galois 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>1µ_pn. Let

K(µpn) be the pn-cyclotomic extension of K and K_cyc= K(µ_p∞) = ∪_n>1K(µ_pn) be the

p∞-cyclotomic extension of K. We denote by Γcyc the Galois group of Kcyc over K. Let χ : GK → Z∗p be the cyclotomic character. For n> 1, the cyclotomic character induces morphisms χ : Gal(K(µpn)/K) ,→ (Z/pnZ)∗, and χ : Γ_cyc = Gal(K_cyc/K) ,→ Z∗_p.

Since p is odd, the group Z∗_p decomposes as Z∗_p ' Z/(p − 1)Z × Zp, and we denote by Γcyc = ∆ × 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 pnand Γ0 = Γ. We set K∞= Kcyc∆ the Zp-cyclotomic extension of K, and for n_{> 0, K}n= K∞Γn (thus K0 = K).

Iwasawa algebra

If G is a profinite group, let Zp[[G]] be the Iwasawa algebra of G over Zp defined as the projective limit of the group algebras Zp[G/U ] where U runs through the open subgroups of G and relative to the natural projection maps Zp[G/U0] → Zp[G/U ], for U0 ⊂ U .

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

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

Λ

-modules

Let η be a Dirichlet character on ∆. We denote by eη the element of Λcyc defined by 1

|∆| P

δ∈∆η

−1_{(δ)δ. For a Dirichlet character η on ∆ and a Λ}

cyc-module M , let Mη be the η-isotypic component of M which is defined as eη · M . Note that Mη has a natural structure of Λ-module. We say that M has rank r over Λcyc if Mη has rank r over Λ for all Dirichlet character η on ∆.

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

M → r M i=1 Λ ⊕ n M j=1 Λ/(paj_{) ⊕} m M k=1 Λ/(fbk k ) (3)

where fk ∈ Zp[X] are distinguished irreducible polynomials (identifying Zp[[X]] with Λ). Furthermore, the ideals (paj_{), (f}bk

k ) 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 X k=1 bkdeg fk and µ(M ) = n X j=1 aj.

The characteristic ideal of a torsion Λ-module M is the ideal charΛ(M ) of Λ generated by the element char(M ) = pµ(M ) m Y k=1 fbk k .

A Λ-module M is said to be cofinitely generated (respectively cotorsion, of corank r) if the Pontryagin dual of M , HomZp(M, Qp/Zp), is finitely generated (respectively torsion

Λ-module, of Λ-rank r).

Proposition 1. Let M and M0 be cofinitely generated Λ-modules. Assume that

1. for all irreducible distinguished polynomials f ∈ Λ and positive integers e, corankΛ(M ⊗ Λ/(fe))Γ = corankΛ(M0⊗ Λ/(fe))Γ,

2. for all positive integers n and m,

#MΓn_[pm_]/#M0,Γn_[pm_]

is bounded as n vary.

Then, the Pontryagin duals of M and M0 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 by the previous statement), then their Pontryagin duals are pseudo-isomorphic.

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

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

(M ⊗ZpΛ/(f

e₎₎Γ_{= Hom}

Λ(M∧, (Qp/Zp) ⊗ZpΛ/(f

e₎₎

is the Zp-rank of the Tate module of this last group which is HomΛ(M∧, Λ/(fe)). If M∧ = Λ, then rf,e = deg(fe) = e · deg(f ). If M∧ = Λ/(ga) with g an irreducible distinguished 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 M i=1 Λ ⊕ m M k=1 Λ/(fbk k )

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

The second hypothesis implies that µ(M∧/pm_M∧_{) = µ(M}0,∧_/pm_M0,∧_{). We have} µ(Λ/pm_{) = m and µ((Λ/p}b_)/pm_{) = min(b, m). Thus, we deduce that M}∧ _{and M}0,∧ _have same Λ-rank and µ-invariant.

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

Galois representations and Fontaine’s rings of periods

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

Galois 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 all p-adic representations of GK. To do so, he has introduced topological Qp-algebras endowed with a continuous action of GK and additional structures. For such an algebra B, a p-adic representation V of GK is said B-admissible if the BGK_{-module (V ⊗Q}

pB)

GK

is free of rank dimQpV . Then the aforementioned subcategories are the categories of

B-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_←−

ζ7→ζpµpn. The natural action of

the Galois group GK on Zp(1) is given by the cyclotomic character. For n > 0, define Zp(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 closed and GK acts continuously on it. For n ∈ Z, we set CK(n) = CK⊗ZpZp(n). The Galois

group 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 multiplication in BHT defined via the natural maps CK(n) ⊗CK CK(m) ' CK(n + m)). We have

BGK

HT = K. A p-adic representation V of GK is Hodge-Tate if the K-vector space DHT(V ) = (V ⊗Qp BHT)

GK _{has dimension dim}

QpV over K. The Hodge-Tate weights

of V are the n ∈ Z for which (CK(−n) ⊗QpV )

GK _{is non-trivial and the multiplicity of}

a weight n of V is dimK(CK(−n) ⊗QpV )

GK_{. The K-vector space D}

HT(V ) is a graded vector space, the direct summand of the grading being the non-trivial sub-vector spaces (CK(n) ⊗QpV )

GK_.

Let OC_K be the ring of integers of CK. The ring ˜ E+= lim_←− x7→xp OCK/(p) = {(x n₎ n>0 ∈ ∞ Y n>0 OCK/(p), (x n+1₎p _{= x}(n) , for all n > 0}

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

θ : Ainf → CK, X i−∞ pi[xi] 7→ X 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 compatible system of pn_{-th roots of unity), the series}

log([ε]) = ∞ X 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 discrete valuation ring and t is an uniformizer. We denote by BdR its field of fractions. The field BdR is equipped with a continuous action of GK and a filtration FiliBdR = tiB+dR stable for the Galois action, and BGK

dR = 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 space}

of dimension dimQpV . 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 V is Hodge-Tate and gr(DdR(V )) = ⊕iFiliDdR(V )/ Fili+1DdR(V ) = DHT(V ) as graded K-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 natural Frobenius ϕ which acts on the ring of the Witt vectors, extends to Acris and commutes with the Galois action. The element t lies in the ring B+_cris = Acris[1/p] and the ring of cristalline periods for K is defined by Bcris = B+cris[1/t]. If K0 is the maximal unramified extension of Qp inside K, we have BGK

cris = K0. Besides, there is a natural GK-equivariant injective map j : K ⊗K0 Bcris → BdR, and K ⊗K0 Bcris is endowed with the filtration

induced by the one of BdR. A p-adic representation V of GK is crystalline if the K0-vector space Dcris(V ) = (V ⊗QpBcris)

GK _{has dimension dim}

QpV . The Frobenius ϕ

naturally acts on Dcris(V ) and the K-vector space K ⊗K0Dcris(V ) is naturally equipped

with a decreasing, separated and exhaustive filtration the one of K ⊗K0 Bcris. If V

is crystalline, then it is de Rham and K ⊗K0 Dcris(V ) ' DdR(V ) as filtered K-vector

Let V∨(1) = HomQp(V, Qp(1)). If V is crystalline, then V

∨_{(1) is as well, and there} exists a natural pairing

[·, ·] : Dcris(V ) × Dcris(V∨(1)) → Dcris(Qp(1)) ' K0 −−→ Qtrace p. (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)) ' K trace

−−→ Qp. (5) for which FiliDdR(V∨(1)) is the orthogonal complement of Fil1−iDdR(V ) (if V is only assume to be de Rham, we have the second pairing).

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

exp_K : DdR(V )

Fil0DdR(V ) + Dcris(V )ϕ=1

,→ H1(K, V ),

which is the connecting homomorphism in the Galois cohomology of the fundamental exact sequence 0 → V → V ⊗QpB ϕ=1 cris → V ⊗QpBdR/B + dR → 0, and the dual exponential [20, II §1.2]

exp∗_K : H1(K, V∨(1)) → Fil0DdR(V∨(1)). These maps satisfy the relation

hx, exp_K(y)i = [exp∗_K(x), y], for x ∈ H1(K, V ), y ∈ DdR(V∨(1)), where h·, ·i is Tate local pairing

h·, ·i : 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 representation V is unramified at a prime v of K if the inertia subgroup of v acts trivially on V . The representation V is pseudo-geometric if it is unramified outside a finite set of primes of K and de Rham at the primes of K dividing p.

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-cyclotomic extension K∞ of K and Λ• the associated Iwasawa algebra. We denote by

H_Iwi (K•, T ) = lim_←− K0

Hi(K0, T )

the projective limit relative to the corestriction maps of the Galois cohomology groups Hi_(K0_{, T ) and where K}0 _{runs through the finite extension of K contains in K}

•. The groups Hi

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). Let

T∨(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

H_Iw2 (K•, T ) ' H0(K•, T∨(1))∨,

in particular, H_Iw2 (K•, T ) is a Λ•-torsion module. The group H_Iw1 (K•, T ) has Λ-rank given by rankΛ•H 1 Iw(K•, T ) = ( 0 if ` 6= p, [K : Q`] rankZpT if ` = p.

The torsion Λ•-submodule of H_Iw1 (K•, T ) is isomorphic to H0(K•, T ). We recall some useful pairings. Let A∨(1) = (T∨(1) ⊗Zp Qp)/T

∨_{(1) = Hom}

Zp(T, µp∞).

Tate’s local pairing

Hi(K0, T ) × H2−i(K0, A∨(1)) → H2(K0, µp∞) ' Qp/Zp, (6)

is compatible with the limits and induces a pairing h·, ·i : Hi

Iw(K•, T ) × H 2−i

(K•, A∨(1)) → Qp/Zp. (7)

We recall Perrin-Riou’s pairing. Let h·, ·iK0 be Tate’s pairing

H1(K0, T ) × H1(K0, T∨(1)) → H2(K0, Zp(1)) ' Zp, (8) and let x = (xK0) and y = (y_K0) be elements in H_Iw1 (K•, T ) and H_Iw1 (K•, T∨(1))

respectively. Then the sequence whose K0-component is X

g∈Gal(K0_/K)

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

h·, ·iPR : H_Iw1 (K•, T ) × H_Iw1 (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 of GK and let Σ be a finite set of primes of K containing the primes of ramification of V , the archimedean primes and the primes dividing p. We denote by KΣ the maximal extension of K unramified outside Σ. Since only the archimedean primes and the primes dividing p are ramified in the cyclotomic extension, Kcyc is contained in KΣ. Let T a GK-stable Zp-lattice in V . As above, let K• be either the full p∞-cyclotomic extension Kcyc or the Zp-cyclotomic extension K∞ of K. We denote by

H_Iwi (KΣ/K•, T ) = lim_←− K0

H_Iwi (Gal(KΣ/K0), T )

the projective limit relative to corestriction maps where K0 runs through the finite extensions 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 by

ϕ(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

X ζ∈µp

f (ζ(1 + π) − 1).

Let T be a finite free crystalline Zp-representation of GK (we say that T is crystalline whenever V = T ⊗ Qp is). There exists a Wach module [3,2] N(T ) attached to T which is a free A+_K-module of rank rankZpT equipped with an action of Γcyc and a ϕ-linear

endomorphism of N(T )[_π1], 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 )).

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

h1_Iw,T : N(T )ψ=1 ∼−→ H1

Iw(Kcyc, T ). (11) Let N(V ) = N(T ) ⊗ZpQp. 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 )−→ D∼ cris(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 the one 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

X 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.

LetH (Γ) be the set of elements f(γ −1) with γ ∈ Γ and f(X) ∈ Qp[[X]] a formal power series which converges on the open p-adic unit disk. We set H = Qp[∆] ⊗QpH (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 isomorphismH ' (B+_rig,Q

p)

ψ=0_.

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

LT : H1 Iw(Kcyc, T ) N(T )ψ=1 ϕ∗N(T )ψ=0 (B+_rig,K)ψ=0_⊗ OK Dcris(T ) H ⊗ZpDcris(T ). (h1 Iw)−1 1−ϕ M−1⊗1 (13)

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 that χ(σ−1) = −1, i.e. the image of the complex conjugation inside Γcyc, and `0 = _{log χ(γ)}log γ . Then, for z ∈ H_Iw1 (Kcyc, T ) and z0 ∈ HIw1 (Kcyc, T∨(1)), we have

[LT(z),LT∨₍₁₎(z0)] = −σ₋₁· `₀· hz, z0i_PR. (14)

Selmer groups

Let K be a number field. Let V be a pseudo-geometric p-adic representation of GK and Σ a finite set of primes of K containing the primes of ramification of V , the archimedean primes 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 contained in KΣ. By abuse of notation, we shall say that a prime of L is contained in Σ if it divides a prime of K in Σ. For each prime v of L in Σ, let Fv be a subgroup of H1(Lv, A). The Selmer group of A over L associated to (Fv)v∈Σ is defined as

Sel_(Fv)(A/L) = Ker H

1 (KΣ/L, A) → Y v∈Σ H1(Lv, A)/Fv ! ,

where the maps H1(KΣ/L, A) → H1(Lv, A)/Fv are the localization maps composed with 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 generated

Λcyc-module.

Assume that L is a finite extension of K. Let F_v⊥ ⊂ H1_(L

v, T∨(1)) be the orthogonal complement of Fv under Tate local pairing

H1(Lv, A) × H1(Lv, T∨(1)) → Qp/Zp. We set H₍1_Fv)(KΣ/L, T∨(1)) = Ker H₍1_Fv)(KΣ/L, T∨(1)) → Y v∈Σ H1(Lv, T∨(1))/F_v⊥ ! .

If M is a Zp-module, we set M∧ to be the Pontryagin dual HomZp(M, Qp/Zp). The

Poitou-Tate exact sequence [37, Appendix A.3] induces the following exact sequences

0 H1 (Fv)(KΣ/L, T ∨_{(1)) H}1_(K Σ/L, T∨(1)) Q v∈ΣH1(Lv, T∨(1))/Fv⊥ Sel(Fv)(A/L) ∧ _H2_(K Σ/L, T∨(1)) Q v∈ΣH2(Lv, T∨(1)) H0(L, A)∧ 0, (15) and, 0 Sel(Fv)(A/L) H 1_(K Σ/L, A) Q v∈ΣH1(Lv, A)/Fv H1 (Fv)(KΣ/L, T ∨₍₁₎₎∧ _H2_(K Σ/L, A) Q v∈ΣH2(Lv, A) H0(L, T∨(1))∧ 0. (16) The exact sequence

0 → 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 Fv⊥ generates a sub-Qp-vector space of H1(Lv, V∨(1)) and we defineFv∨(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 of A∨(1) over L associated to (F_v∨(1))v∈Σ Sel(F∨ v(1))(A ∨ (1)/L) = Ker H1(KΣ/L, A∨(1)) → Y v∈Σ H1(Lv, A∨(1))/Fv∨(1) ! .

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

∨_{(1)/L) and that H}0_(L

v, A∨(1)) is finite for at least one v ∈ Σ. Then the cokernel of the map

f : H1(KΣ/L, A) → Y v∈Σ

H1(Lv, A)/Fv

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

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

Let v be a prime of L which does not divide p. If Lunr_v is the maximal unramified extension of Lv, we define

H_unr1 (Lv, V ) = Ker H1(Lv, V ) → H1(Lunrv , V )
.

Let H_unr1 (Lv, A) be the image of Hunr1 (Lv, V ) under the map H1(Lv, V ) → H1(Lv, A) and H_unr1 (Lv, T ) be the inverse image of H_unr1 (Lv, V ) under the map H1(Lv, T ) → H1(Lv, V ). For v be a prime of L dividing p, let

H_f1(Lv, V ) = Ker H1(Lv, V ) → H1(Lv, V ⊗QpBcris)
.

Let H1

f(Lv, A) be the image of Hf1(Lv, V ) by the map H1(Lv, V ) → H1(Lv, A) and H_f1(Lv, T ) be the inverse image of Hf1(Lv, V ) by the map H1(Lv, T ) → H1(Lv, V ). The group H1

unr(Lv, A) (respectively Hf1(Lv, A)) is the orthogonal complement of Hunr1 (Lv, T∨(1)) (respectively H1

f(Lv, T∨(1))) for Tate local duality.

The Bloch-Kato Selmer group of A over L, which we denote by SelBK(A/L), is define by the choice Fv = ( H1 unr(Lv, A) if v - p, H1 f(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 = dimQpInd

Q

F V , g+ = dimQp(Ind

Q

F V )+ the dimension of the +1-eigenspace under the action of a complex conjugation on the induced representation IndQ_F V . We also set g− = g − g+. Similarly, for a prime v of F dividing p, we set gv = dimQpInd

Qp

Fv V . We have

P

v|pgv = 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_(F

v, T /pT ) and H2(Fv, T /pT ) are trivial, (Filtration) the equality

X v|p

dimQpFil

0

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 ) has a two-jump filtration

FiliDcris,v(T ) = (

Dcris,v(T ) for i6 −1, 0 for i_{> 1.}

We may choose a Zp-basis {u1, . . . , ugv} such that, for some d, {u1, . . . , ud} is a basis of

Fil0Dcris,v(T ). We call such a basis a Hodge-compatible basis of Dcris,v(T ). Since the Hodge-Tate weights of V are in the Fontaine-Laffaille range, the Dieudonné module is strongly divisible and we have

pϕDcris,v(T ) + ϕ(Fil0Dcris,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 1_pIdgv−d

!

, (17)

with C ∈ GLgv(Zp). For n > 1, let

Cn= Idd 0 0 Φpn(X + 1) Id_g

v−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 : HIw1 (Fv,cyc, T ) → ⊕gi=1v Λcyc, which satisfies LT : H_Iw1 (Fv,cyc, T ) ColT −−→ ⊕gv i=1Λcyc MT· −−→ H ⊗Z_p Dcris,v(T ).

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

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

ColT ,I : HIw1 (Fv,cyc, T ) → ⊕ |I| i=1Λcyc z 7→ ⊕i∈IColT ,i(z). We call these maps the signed Coleman maps.

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 Colη_T,I is contained in a free Λ-module of rank |I|. This containment is of finite index.

We define H_I1(Fv,cyc, A∨(1)) as the orthogonal complement of Ker ColT ,I for Tate pairing H1(Fv,cyc, A∨(1)) × HIw1 (Fv,cyc, T ) → Qp/Zp.

The hypothesis (Torsion) H2_(F

v, T /pT ) = 0 implies, by Tate local duality, that H0_(F

v, 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

H_I1(Fv,∞, A∨(1)) = HI1(Fv,cyc, A∨(1))∆⊂ H1(Fv,∞, A∨(1)), and

H_I1(Fv,n, A∨(1)) = HI1(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 where each Iv is a subset of {1, · · · , gv}. Let F0 be one of Fcyc, F∞ or Fn for some n> 0. The I-signed Selmer group of A∨(1) over F0, which we denote SelI(A∨(1)/F0), is defined by the choice of local conditions

Fv = ( H1 unr(F 0 v, A) if v - p, H1 Iv(F 0 v, A) if v | p.

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

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

H_I1

v(Fv, A

∨_{(1)) = H}1

f(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

X v|p

|Iv| = g−.

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

Conjecture 6 ([7, Remark 3.27]). For any I ∈I and any even Dirichlet character η on ∆, 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 Selmer groups for T∨(1), using the signed Coleman maps ColT∨_(1),I or by propagating the local

condition of T as in the paragraph titled “Selmer groups”. We shall see later that these two 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 F of dimension d. For an integer n_{> 1, we denote by X(F )[p}n_{] the groups of p}n_-torsion points of X in F . Let Tp(X) = lim_{←−x7→p·x}X(F )[pn] be the Tate module of X. Then Tp(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. Then Tp(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 and g+= [F : Q]d. Tate [46] has proved that (Hodge-Tate) is satisfied. Since X has good reduction at primes dividing p, it is cristalline and (Crystalline) and (Filtration) are satisfied. The supersingular reduction implies (Slopes), and since F is unramified and p 6= 2, we have X(F )[p] = 0 by [32, Lemma 5.11]. Similarly, if XD is the dual abelian

variety of X, then XD also has good supersingular reduction at primes of F dividing p and 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 we shall simply denote by SelI(X/Fcyc), coincide with Kobayashi’s plus and minus Selmer groups [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) ⊗ZQp/Zp ,→ H1(Fv, X[p∞]) and X(Fv) ⊗ZZp ,→ H1(Fv, Tp(X)). If v does not divide p then one has X(Fv) ⊗ZQp/Zp = 0. Furthermore, these subgroups coincide with Bloch and Kato H1

f(Fv, X[p∞]) and Hunr1 (Fv, X[p∞]). Hence, the Selmer group fits in a short exact sequence

0 → X(F ) ⊗ZQp/Zp → SelBK(X/F ) → X(X/F )[p∞] → 0, (19) where X(X/F ) is the Tate-Shafarevich group of X over F .

Chapter 1

Functional equation of signed Selmer

groups

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 the hypotheses (Hodge-Tate), (Crystalline), (Torsion), (Filtration) and (Slopes). We choose Hodge-compatible bases of the Dieudonné modules of T for each prime of F dividing 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 groups for 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 a finitely generated Λ-module, let Mι be the Λ-module with the same underlying set as M 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∞) and SelIc(A/F∞)ι are pseudo-isomorphic to the same Λ-module (up to isomorphism) by the

structure 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 are pseudo-isomorphic.

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 {u01, . . . , u 0

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_ϕ0 the matrix of the Frobenius on Dcris,v(T∨(1)) with respect to the basis {u0

1, . . . , u 0

gv}. From duality we have the relation

C_ϕ0 = 1 p(C −1 ϕ ) t_{= (C}−1 )t 1 pId 0 0 Igv ! ,

where (·)t _{is the transpose. We have the matrices of (18),} MT∨₍₁₎ = lim −→ n (C_ϕ0)n+1C_n0 · · · C₁0, (1.1) where C_n0 = Φpn(X + 1)Id 0 0 Igv ! Ct. By Theorem 3, Perrin-Riou’s regulator decomposes as

LT∨₍₁₎ = (u0

1. . . u 0

gv) · MT∨(1)· ColT∨(1),

where ColT∨₍₁₎ is the column vector of Coleman maps with respect to the basis

{u0

1, . . . , u0gv}.

We linearly extend the pairing (4) to

[·, ·] :H ⊗ZpDcris,v(T ) ×H ⊗ZpDcris,v(T

∨

(1)) →H . Lemma 1.3. Let z ∈ H1

Iw(Fv,cyc, T ) and z0 ∈ HIw(Fv,cyc, T∨(1)). Then [LT(z),LT∨₍₁₎(z0)] =

log(1 + X)

X · ColT(z) t_{· Col}

T∨₍₁₎(z0).

Proof. Since {u1, . . . , ugv} is dual to {u

0

1, . . . , u0gv}, the decompositions of Perrin-Riou’s

regulators LT and LT∨₍₁₎ give

[LT(z),LT∨₍₁₎(z0)] = Col_T(z)t· Mt

Using (18) and (1.1), we may compute MTt· MT∨₍₁₎ = lim −→(Cϕ)n+1Cn· · · C1
t ·lim_−→ C_ϕ0
n+1C_n0 · · · C0 1  = lim_−→  C₁t· · · Ct n Cϕ t
n+1_{· C}0 ϕ
n+1 C_n0 · · · C0 1  = lim_−→pn+11 n Q k=1 Φ_pk(1 + X) Id_gv Since lim −→ n→+∞ 1 pn n Y k=1 Φ_pk(1 + X) = log(1 + X) X , we have MTt· MT∨₍₁₎ = log(1 + X) pX Idgv. 

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 ∈ H_Iw1 (Fv,cyc, T ) and z0 ∈ HIw(Fv,cyc, T∨(1)). By the explicit reciprocity law (14) and Lemma1.3, we have

hz, z0_i

PR = 0 ⇔ [LT(z),LT∨₍₁₎(z0)] = 0

⇔ ColT(z)t· ColT∨₍₁₎(z0) = 0.

Thus, if z ∈ Ker ColT ,I,

hz, z0iPR= 0 ⇔ X

i6=I

ColT ,i(z) · ColT∨_(1),i(z0) = 0. (1.2)

So, Ker ColT∨_(1),Ic is included in the orthogonal complement of Ker Col_{T ,I}. Proposition4

implies 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 z0 is in the orthogonal complement of Ker ColT ,I, then hzi, z0iPR= 0. Therefore, equation (1.2) tells us that ColT∨_(1),i(z0) = 0.

Since this is true for all i ∈ Ic_{, we have z}0 _{∈ Ker Col}

1.2

Control theorem

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

A∨(1)fe = A∨(1) ⊗_Z pΛ/(f e ), and A(fe₎ι = A ⊗_Z pΛ/(f e )ι. The pairing A[pm_{] × A}∨_(1)[pm_{] → µ}

pm induces a perfect Z_p-linear G_Q-equivariant pairing

A∨(1)fe[pm] × A_(fe₎ι[pm] → µ_pm.

Since Γ is a pro-p-group, (Torsion) implies that H0_(Q

p,∞, A) and H0(Qp,∞, A∨(1)) are trivial. Furthermore, since the Galois group Gal(Qp/Qp,∞) acts trivially on Λ/(fe), the hypothesis (Torsion) implies that H0_(Q

p,∞, A∨(1)fe) and H0(Q_p,∞, A_(fe₎ι) are trivial.

Thus, the exact sequence of GQp-modules 0 → A

∨₍₁₎ fe[pm] → A∨(1)_fe pm −→ A∨₍₁₎ fe → 0 induces H1(Qp,n, A∨(1)fe[pm]) ' H1(Q_p,n, A∨(1)_fe)[pm]. For I ⊂ {1, . . . , g}, we define H1 I(Qp,∞, A∨(1)fe) = H_I1(Q_p,∞, A∨(1)) ⊗ Λ/(fe) ⊂ H1(Q_p,∞, A∨(1)_fe) H1 I(Qp,n, A∨(1)fe) = H_I1(Q_p,∞, A∨(1)_fe)Γn ⊂ H1(Q_p,n, A∨(1)_fe) H1 I(Qp,n, A∨(1)fe[pm]) = H_I1(Q_p,n, A∨(1)_fe)[pm] ⊂ H1(Q_p,n, A∨(1)_fe[pm]),

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[pm] or A[pm].

For n, m _{> 0, we define} SI(Am/Qn) = Ker H1(QΣ/Qn, Am) → Y w∈Σ H1(Qn,w, Am) H1 unr(Qn,w, Am) × H 1_(Q p,n, Am) H1 I(Qp,n, Am) !

Lemma 1.5. Let χglob.,Qn(Am) be the global Euler characteristic of Am over Qn. Let

I ⊂ {1, . . . , g} and write Ic _{for its complement. Then,} #SI(Am/Qn) = #SIc(A∗_m/Q_n) χglob.,Qn(Am) · [H 1_(Q p,n, Am) : HI1(Qp,n, Am)] . Proof. We set Pi Σ := Q v∈ΣH i_(Q n,v, Am) and P ∗,i Σ := Q v∈ΣH i_(Q n,v, A∗m), Lp := HI1(Qp,n, Am) and Lp∗ := HI1c(Qp,n, A∗m),

Lv := Hunr1 (Qn,v, Am) and Lv∗ := Hunr1 (Qn,v, A∗m) for v - p, L := Q v∈ΣLv and L∗ := Q v∈ΣL ∗ v.

Let

λi : Hi(QΣ/Qn, Am) → PΣi be the restriction map and write

Gi _{:= Im λ}i_{, K}i _{:= Ker λ}i_{. .}

We have similarly λ∗,i, G∗,i, K∗,i for A∗. For all v ∈ Σ the orthogonal complement of Lv under the local Tate pairing is L∗_v (see [13_{, Remark at the end of §3] when v - p and}

Proposition 1.4 when v = p). Thus we have

#SI(Am/Qn) = #K1· #(G1∩ L) = #K1· #G1· #L · #(G1· L)−1. Since #K1_{· #G}1 _{= #H}1_(Q Σ/Qn, Am), and by duality #(G1· L) = #PΣ1/#(G ∗,1_{∩ L}∗_), we get #SI(Am/Qn) = #H1(QΣ/Qn, Am) · #L · #(G∗,1∩ L∗)/#PΣ1. By (Torsion), we have

#H1(QΣ/Qn, Am) = χglob.,Qn(Am)

−1_{· #H}0_{(QΣ/Qn, Am) · #H}2_{(QΣ/Qn, Am)} = χglob.,Qn(Am)

−1_{· #H}2_(Q

Σ/Qn, Am). By global duality #K∗,1 = #K2_{, we obtain}

#(G∗,1∩ L∗) = #SIc(A∗_m/Q_n)/#K∗,1 = #S_Ic(A∗_m/Q_n)/#K2. Thus, we have #SI(Am/Qn) = χglob.,Qn(Am) −1_·(#H2_(Q Σ/Qn, Am)/#K2)·(#L/#PΣ1)·#SIc(A∗_m/Q_n). (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) = #Hunr1 (Qn,v, Am), so we deduce that #L/#P1 Σ = Q v∈Σ,v-p#(H 1 unr(Qn,v, Am)/H1(Qn,v, Am)) × #(HI1(Qp,n, Am)/H1(Qp,n, Am)) =Q v∈Σ,v-p#(H 0_(Q n,v, Am)/H1(Qn,v, Am)) × #(HI1(Qp,n, Am)/H1(Qp,n, Am)) =Q v∈Σ,v-p#(H 2_(Q n,v, Am))−1× #(HI1(Qp,n, Am)/H1(Qp,n, Am)). (1.4) On the other hand, global duality and (Torsion) tell us that # coker λ2 = #H0_(Q

Σ/Qn, A∗m) = 1, so

#H2(QΣ/Qn, Am)/#K2 = #G2 = #PΣ2/ coker λ

2 _{= #P}2

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/Q_n)

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

quantity

χglob.,Fn(Am)

−1_{· [H}1

(Qp,n, Am) : HI1(Qp,n, Am)]−1 using Lemma 1.5.

Since Qnis totally real, one has χglob.,Qn(Am) = #(A

−

m)−[Qn:Q], where A−mis the subgroup of Am on which complex conjugation acts by −1. As complex conjugation acts trivially on Λ/(fe), we get χglob.,Qn(Am) = p

−m·[Qn:Q]·e·deg(f )·g−_{. Furthermore, we know that}

#H1_(Q

p,n, Am) = p−m·[Qp,n:Qp]·e·deg(f )·g.

It remains to compute #H_I1(Qp,n, Am). By definition, H_I1(Qp,n, Am) = HI1(Qp,∞, A) ⊗ Λ/(fe)

Γn

[pm]. Recall that

#H_I1(Qp,∞, A∨(1)) = #HI1(Qp(µp∞), A∨(1))∆,

and that we write (·)∧ for the Pontryagin dual. By definition of H_I1(Qp(µp∞), A∨(1)),

we have

H_I1(Qp(µp∞), A∨(1))∆' (Im Col∆_{T ,I})∧.

Proposition 4gives 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/(fe₎₎Γn → (N∧ I,∆⊗ Λ/(fe))Γn → ((Im Col∆ T ,I) ∧ _{⊗ Λ/(f}e₎₎Γn _{→ H}1_(Γ n, KI,∆∧ ⊗ Λ/(fe)).

Since H1(Qp,∞/Qp,n, KI,∆∧ ⊗ Λ/(fe)) is finite it implies that #((Im Col∆_{T ,I})∧[pm_{] ⊗ Λ/(f}e₎₎Γn _{6 C · #(N}∧

I,∆⊗ Λ/(fe))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 gives an equality. However, our result is sufficient to prove the following generalization of Lemma 3.4 in op. cit.

Lemma 1.7. Let A• ∈ {A∨₍₁₎

fe, A∨(1)}. For all positive integers n and m, the natural

map

SI(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) Q_v∈Σ,v-p H 1_(Q n,v,Am) H1 unr(Qn,v,Am) × H1_(Q p,n,Am) H1 I(Qp,n,Am). 0 SI(A•/Qn)[pm] H1(QΣ/Qn, A•)[pm] Q v∈Σ,v-p H1_(Q n,v,A•) H1 unr(Qn,v,A•) × H1_(Q p,n,A•) H1 I(Qp,n,A•) . Q fv

We already know that the center vertical map is an isomorphism. Since H1

I(Qp,n, Am) = H1

I(Qp,n, A•)[pm], the map fpis injective. The local condition is the unramified condition for v - p, thus one has

H1_(Q

n,v, A•)/Hunr1 (Qn,v, A•) ⊂ H1(Qunrn,v, A•) H1_(Q

n,v, Am)/Hunr1 (Qn,v, Am) ⊂ H1(Qunrn,v, Am) with Qunr

n,v the maximal unramified extension of Qn,v.

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

• /pmA GQunrn,v

• = Ker(H1(Qunr_n,v, Am) → H1(Qunrn,v, A•)) and we have #AGQunrn,v • /pmA GQunrn,v • 6 #A GQunrn,v • /(A GQunrn,v • )div < ∞.

Since no prime splits completely in Q∞/Q, we conclude that KerQ fv is bounded as n and m vary. Applying the Snake Lemma, the lemma follows. _

Remark 1.8. The results above are stated for F = Q and the isotypic component of the trivial character of ∆. Let η be a character on ∆ and denote by η its complex conjugate and by A∨(1)η the Zp-module A∨(1) with action of the Galois group twisted by 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 hold for every isotypic component because we may replace A∨(1) and A by A∨(1)η and Aη respectively in the proofs above.

Similarly, suppose that F/Q is a general abelian extension in which p is unramified with [F : Q] coprime to p. We may prove the above results for an isotypic component corresponding 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 trivial character, as explained in Remark 1.8. Since Λ/(fe) is a free Zp-module on which GQ∞

acts trivially, we have

SelI(A∨(1)/Q∞) ⊗ Λ/(fe) = Ker        H1_(Q Σ/Q∞, A∨(1)) ⊗ Λ/(fe) Q w∈Σ H1(Q∞,w,A∨(1))⊗Λ(fe) H1 loc(Q∞,w,A∨(1))⊗Λ/(fe)        = Ker        H1(QΣ/Q∞, A∨(1) ⊗ Λ/(fe)) Q w∈Σ H1_(Q ∞,w,A∨(1)⊗Λ(fe)) H1 loc(Q∞,w,A ∨_(1))⊗Λ/(fe₎        , where H1 loc(Q∞,w, A

∨_{(1)) is the appropriate local condition,}

H_loc1 (Q∞,w, A∨(1)) = (

H_unr1 (Q∞,w, A∨(1)) _{if w - p,} H_I1(Q∞,w, A∨(1)) if w | p.

For the prime above p, we have by definition an injection H1(Qp, A∨(1)fe) H1 I(Qp, A∨(1)fe) ,→ H 1_(Q p,∞, A∨(1)fe) H1 I(Qp,∞, A∨(1)) ⊗ Λ/(fe) .

For a prime w of Q∞ above a prime v 6= p of Q, the group H_unr1 (Q∞,w, A∨(1)) is trivial [37, §A.2.4]. From the inflation-restriction exact sequence, the kernel of the restriction map

H1(Qv, A∨(1)fe) → H1(Q_∞,w, A∨(1)_fe)

is H1_(Q∞,w/Q

v, (A∨(1)fe)GQ∞,w). If v is archimedean, then this group is trivial. If v is

non-archimedean, it finitely decomposes in Q∞/Qv, so that Gal(Q∞,w/Qv) ' Zp and it 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 contained

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

bounded 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) (Sel_I(A∨(1)/Q∞) ⊗ Λ/(fe))Γ H1_(Q Σ/Q, A∨(1)fe) H1(Q_Σ/Q_∞, A∨(1)_fe)Γ Q v∈Σ H1_(Q v,A∨(1)f e) H1 loc(Qv,A ∨₍₁₎ f e)  Q v∈Σ H1_(Q ∞,v,A∨(1)f e) H1 loc(Q∞,v,A ∨_(1))⊗Λ/(fe₎ Γ Q fv we get an inclusion SI(A∨(1)fe/Q) ,→ (Sel_I(A∨(1)/Q_∞) ⊗ Λ/(fe))Γ

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. 

Chapter 2

Λ-module structure and congruences of

signed Selmer groups

Let F be a number field unramified at p and T a Zp-representation satisfiying the hypotheses (Hodge-Tate), (Crystalline), (Torsion), (Filtration) and (Slopes). We choose Hodge-compatible bases of the Dieudonné modules for each prime of F dividing p. As in the previous chapter, we use the dual basis for a Hodge-compatible basis 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= (Ivc)v|p be its complement. Assume that SelI(A∨(1)/F∞) and SelIc(A/F∞) are cotorsion Λ-modules. Then Sel_I(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 degree prime to p and that g+= g−, the algebraic functional equation 1.1relating 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 where χ|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_(F

primes dividing p, we set

H_I1_v(Fv,∞, A∨(1)s) = H_I1_v(Fv,∞, A∨(1)) ⊗ χs.

Therefore, for F0 being F∞ or Fn for some n> 0, we can define twisted I-Selmer groups SelI(A∨(1)s/F0) as in with local condition at p induced by HI1v(Fv,∞, A

∨₍₁₎

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

For F0 being F∞ or Fn for some n> 0, we set PΣ,I(A∨(1)s/F0) = Y w∈Σ,w-p H1_(F0 w, A ∨₍₁₎ s) H1 unr(Fw0, A∨(1)s) ×Y v|p H1_(F0 v, A ∨₍₁₎ s) H1 Iv(F 0 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 the restriction map

SelIc(A_s/F_n) → Sel_Ic(A_s/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_(F

v,∞, As) = 0 where v is any prime of F dividing p, thus the central map 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 H_I1c v(Fv,n, As) H 1_{(Fv,n, As)} H1_(F v,n,As) H1 Ic_v(Fv,n,As) 0 0 H_I1c v(Fv,∞, As) Γn _H1_(F v,∞, As)Γn  H1_(F v,∞,As) H1 Ic_v(Fv,∞,As) Γn (2.2)

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

H1_(F v,n, As) H1 Ic v(Fv,n, As) → H 1_(F v,∞, As) H1 Ic v(Fv,∞, As) !Γn is an injection.

For a prime w of Fn not dividing p and a prime w0 of F∞ above w, the diagram 0 H_unr1 (Fn,w, As) H1(Fn,w, As) H1_(F n,w,As) H1 unr(Fn,w,As) 0 0 H1 unr(F∞,w0, As)Γn H1(F∞,w0, A_s)Γn  _H1_(F ∞,w0,As) H1 unr(F∞,w0,As) Γn (2.3) is commutative. If w is archimedean, since p is odd, then H1_{(F∞,w0, A}

s) is trivial, and if w is non-archimedean, then H1

unr(F∞,w0, 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∞,w0/F

n,w, A G_F∞,w0

s ). If w is archimedean, then w splits completely in F∞/Fn so this group is trivial. If w is non-archimedean, it finitely decomposes in F∞/Fn, so that Gal(F∞,w0/F_n,w) ' Z_p and is topologically generated by

an element γn. Thus H1(F∞,w0/F_n,w, A G_F∞,w0 s ) is isomorphic to AGsF∞,w0/(γn− 1)A G_F∞,w0 s . One has the short exact sequence

0 AGsFn,w A G_F∞,w0 s A G_F∞,w0 s A G_F∞,w0 s /(γ − 1)A G_F∞,w0 s 0. (γn−1)

For all but finitely many s ∈ Z, AGsFn,w is finite for every n, hence A G_F∞,w0

s /(γ −1)A G_F∞,w0 s is finite. So, (AGsF∞,w0)div, the maximal divisible subgroup of A

G_F∞,w0

s , is contained in (γn − 1)A

G_F∞,w0

s and the order of A G_F∞,w0

s /(γn− 1)A G_F∞,w0

s is bounded by the one of AGsF∞,w0/(A

G_F∞,w0

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) H1 unr(Fn,w, As) → H 1_(F∞,w 0, A_s) H1 unr(F∞,w0, As) Γn