On the structure of signed Selmer groups

Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, B\"uy\"ukboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary $\mathrm{Gal}(\overline{F}/F)$-representations. In particular, their construction applies to abelian varieties defined over $F$ with good supersingular reduction at primes of $F$ dividing $p$. Assuming that these Selmer groups are cotorsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we show that they have no proper sub-$\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler-Poincar\'e characteristic of these Selmer groups, we obtain an explicit formula on the size of the Bloch-Kato Selmer group attached to these representations. Furthermore, for two such representations that are isomorphic modulo $p$, we compare the Iwasawa-invariants of their signed Selmer groups.


INTRODUCTION
Let F be a number field and E be an elliptic curve defined over F . Let p be an odd prime and F ∞ the Z p -cyclotomic extension of F (see §1.1). On the algebraic side of the Iwasawa theory for E developed by Mazur [Maz72] is the p-Selmer group associated to E over F ∞ , denoted by Sel p (E/F ∞ ), which is naturally a discrete Z p [[Gal(F ∞ /F )]]module. The Selmer group contains arithmetic information of the curve, e.g. it fits in the exact sequence of groups where E(F ∞ ) is the group of F ∞ -rational points and X p (E/F ∞ ) the p-primary component of the Tate-Shafarevich group of E over F ∞ . One goal of Iwasawa theory is to understand the structure of Sel p (E/F ∞ ) as Z p [[Gal(F ∞ /F )]]-module.
When E has good ordinary reduction at primes of F dividing p, a conjecture of Mazur (proved by Kato [Kat04] when F = Q) states that the Pontryagin dual of Sel p (E/F ∞ ) is a torsion Z p [[Gal(F ∞ /F )]]-module. Furthemore, assuming Mazur's conjecture and that the E(F ) has no p-torsion, Greenberg [Gre99,Proposition 4.14] showed that Sel p (E/F ∞ ) has no proper sub-Z p [[Gal(F ∞ /F )]]-module of finite index.
When E has good supersingular reduction at some prime above p, the Selmer group over F ∞ is no longer a cotorsion Z p [[Gal(F ∞ /F )]]-module. In the case F = Q and a p = 0 (holds whenever p 5), Kobayashi [Kob03] has defined the so called plus and minus (or signed) Selmer groups, and proves that the Pontryagin duals of these Selmer groups are torsion Z p [[Gal(Q ∞ /Q)]]-modules. Kim [Kim13] has then extended the definition of these Selmer groups to number fields F where p is unramified and generalized Greenberg's result and showed that if the signed Selmer groups of E over F ∞ are cotorsion Z p [[Gal(F ∞ /F )]]-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 [KO18].
Kobayashi's construction of signed Selmer groups has been generalized to many situations ( [IP06,LLZ10,Kim14,BL17]). In [BL17], using p-adic Hodge theory machinery, Büyükboduk and Lei have defined signed Selmer groups for certain non-ordinary Galois representations of Gal(F /F ) (see §1.2 and 1.3 for hypotheses). In particular, their construction applies to abelian varieties defined over F with good supersingular reduction at primes of F dividing p. The definition of the signed Selmer groups depends on a choice of a basis for the Dieudonné module associated to the representation. For such a basis, we may attach to each of its subset I of some prescribed cardinality a signed Selmer group, which we denote in this introduction by Sel I (T /F ∞ ), where T is a Galois representation (a free Z p -module of finite rank with a continuous action of the absolute Galois group of F ) to which the construction of op. cit. applies. They conjectured these signed Selmer groups to be cotorsion Z p [[Gal(F ∞ /F )]]-modules. Let T * be the Tate dual of T . The Dieudonné module associated to T * is the dual of the Dieudonné module of T and we denote by I c the subbasis dual to I. We prove: Theorem (Theorem 2.1). Assume that the Pontryagin dual of both Sel I (T /F ∞ ) and For a good choice of basis of the Dieudonné module, one can relate the signed Selmer groups to Bloch-Kato's Selmer groups. For such a basis, assuming that the Bloch-Kato Selmer group of T over F is finite, our theorem above allows us to employ Greenberg's strategy in [Gre99,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 cardinal of the Bloch-Kato's Selmer group (see Corollary 2.9).
In the final part of the article, we study congruences of signed Selmer groups. If E and E ′ are elliptic curves defined over Q with good ordinary reduction at p and such that E[p] ≃ E ′ [p] as Galois modules, Greenberg and Vatsal [GV00] have studied the consequences of such a congruences in Iwasawa theory. In particular, assuming Mazur's conjecture, they proved that the µ-invariant of Sel p (E/Q ∞ ) vanishes if and only if that of Sel p (E/Q ∞ ) vanishes, and that when these µ-invariants do vanish, the λ-invariants of some non-primitive Selmer groups associated to E and E ′ over Q ∞ are equal. Kim [Kim09] 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 [BL17].
Theorem (Theorem 3.11). Let T and T ′ be Galois representations to which the construction of [BL17] applies. Assume that T /pT ≃ T ′ /pT ′ as Galois modules and that the Pontryagin dual of the signed Selmer groups associated to T , T * , T ′ and T ′, * are torsion Furthemore, when these µ-invariants do vanish, the λinvariants of the I -signed non-primitive Selmer groups associated to T and T ′ over F ∞ are equal.
The main ingredient is a result of Berger [Ber04] who showed that the congruence T /pT ≃ T ′ /pT ′ of Galois module induces a congruence modulo p on the Wach module associated to T and T ′ . This allows to keep track of the congruence through Büyükboduk and Lei's construction.
Acknowledgement. This article is part of the author's Ph.D. thesis. The author would like to thank his supervisor Antonio Lei for his support and patient explanation throughout this project, as well as Kazim Büyükboduk, Daniel Delbourgo and Jeff Hatley for their time and help. The author would also like to thank B.D. Kim.

COLEMAN MAPS AND SIGNED SELMER GROUPS
In this section, we fix notations and recall results from [BL17] that we shall need.
1.1. Cyclotomic extension and Iwasawa algebra. Choose once and for all an odd prime p. Let F be a number field unramified at p. We fix F an algebraic closure of F and denote by G F = Gal(F /F ) the absolute Galois group of F . If v is a prime of F , we denote by F v the completion of F at v, F v its ring of integers and G F v the decomposition subgroup of v in G F . Let µ p n be the group of p n -th roots of unity for every n 1 and µ p ∞ = ∪ n 1 µ p n . We set F (µ p ∞ ) = ∪ n 1 F (µ p n ) the p ∞ -cyclotomic extension of F inside F . For every n 1, we choose a generator ǫ (n) of µ p n with the compatibilities (ǫ (n+1) ) p = ǫ (n) , so that lim ← −n ǫ (n) is a generator of lim ← −n µ p n ≃ Z p (1). The cyclotomic character χ : G F → Z * p is defined by the relations g(ǫ (n) ) = (ǫ (n) ) χ(g) and it induces an isomorphism χ : Gal(F (µ p ∞ )/F ) ≃ Z * p . In particular, the group Gal(F (µ p ∞ )/F ) decomposes as Γ × ∆ with Γ ≃ Z p and ∆ ≃ Z/(p − 1)Z. For every n 0, we denote by induced by γ → X + 1 where γ is a topological generator of Γ . For n 1, let ω n (X ) = (X + 1) p − 1, then this isomorphism induces Λ n ≃ Z p [∆][X ]/(ω n ).
For a Dirichlet character η on ∆ and a Λ-module M , let M η be the isotypic component of M , which is given by e η · M where e η = 1 |∆| δ∈∆ η −1 (δ)δ. Note that M η is naturally a Z p [[Γ ]]-module. We will say that a Λ-module M has rank r if M η has rank r over Z p [[Γ ]] for all characters η on ∆.
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).
Let be a motive defined over F with coefficients in Q [FPR94]. We denote by p its p-adic realization and we fix T a G F -stable Z p -lattice inside p . Let We will assume that, for every prime v of F dividing p, (H.-T.) the Hodge-Tate weights of p , as a G F v -representation, are in [0, 1], (Cryst.) the G F v -representation p is crystalline, (Tors.) the Galois cohomology groups H 0 (F v , T /pT ) and H 2 (F v , T /pT ) are trivial.
We denote by T * = Hom(T, Z p (1)) the Tate dual of T and we set We remark that the dual of , which we denote by * , satisfies the hypothesis (Cryst.) and (H.-T.), and T * , which is a G F -stable Z p -lattice inside its p-adic realization * p , satisfies (Tors.).
We also fix Σ a finite set of primes of F containing the primes dividing p, the archimedean primes and the primes of ramification of M * . Let F Σ be the maximal extension of F unramified outside Σ, so that M * is a Gal(F Σ /F )-module. We remark that F (µ p ∞ ) ⊆ F Σ since only primes above p and ∞ can be ramified in F (µ p ∞ ). If F ′ is an extension of F in F (µ p ∞ ), we will say by abuse that a prime of F ′ lies in Σ if it divides a prime of F which is in Σ.

Dieudonné modules.
If v is a prime of F dividing p, let D cris,v (T ) be the Dieudonné module associated to T considered as a G F v -representation [Ber04, Définition V.1.1]. Then D cris,v (T ) is a free F v -module of rank dim Q p p equipped with a filtration of We will assume that We call such a basis a good basis and fix one for the rest of the paper. Then, from our hypotheses, the matrix of the crystalline Frobenius ϕ with respect to this basis is of the form and I n is the identity matrix of size n. Let D cris,v (T * ) be the Dieudonné module associated to T * . There is a natural pairing

Decomposition of Perrin-Riou's big logarithm map.
Let v be a prime of F dividing p. For i 0, the projective limit of the Galois cohomology groups H As in [BL17], we may define for n 1, where Φ p n is the p n -th cyclotomic polynomial. By Proposition 2.5 in op. cit., the sequence (M v,n ) n 1 converges to some g v × g v logarithmic matrix over , which we denote by M v . This allows to decompose T,v into More details on the decomposition (4) are given in paragraph 3.2.

Signed Coleman maps. Let
These maps are called signed Coleman maps. We recall results about them that we shall need. (1) For any character η on ∆, the η-isotypic component of the image of the signed Coleman map Im Col are compatible under the natural projection maps thus, they define an element in Λ. This defines Perrin-Riou's pairing Then, we have the following relation.
Remark 1.4. In [LP17], there is an additional hypothesis that g + = g − and F is abelian over Q with degree prime to p. However, the proof of Lemma 3.2 in op. cit. applies in the setting considered in the present article in verbatim. Tate's local pairing passes to the limit relative to restriction and corestriction and defines a pairing The assumption H 2 (F v , T /pT ) = 0 (Tors.) implies by Tate's duality that In particular, by the inflation-restriction exact sequence, we have is finite of order a power of p, and for n 0, we have We set We also have signed Coleman maps for T * . For n 0, let (Ker Again by (Tors.), we have the exact sequence Proof. By Lemma 1.3 and bilinearity of Tate's pairing, the orthogonal complement of (Ker Col T,I v ) n under Tate's pairing contains (Ker Col T * ,I c v ) n . The reverse inclusion follows from the exactness of the sequence (7). As already remarked, by (Tors.), one has Let F ′ be one of F (µ p ∞ ), F ∞ or F n for some n 0, and w a prime not dividing p.
. Definition 1.6. Let F ′ be F (µ p ∞ ), F ∞ , or F n for some n 0. The I-Selmer group of M * over F ′ is defined by where the map is the composition of localization at each w ∈ Σ followed by the projection in the appropriate quotient. In the remainder of this article, we study these Selmer groups.
In the next section, we shall need twisted signed Selmer groups. Let us explain now what they are. For s ∈ Z, we set M * Therefore, for F ′ being F ∞ or F n for some n 0, we can define twisted I -Selmer groups Sel I (M * s /F ′ ) as above with local condition at p induced by H 1 ]-modules. Similarly, we can define signed Selmer groups for M using the signed Coleman maps Col T * ,I v , as well as twisted signed Selmer groups for M as above. We remark that if I is an element of p , then In particular, Conjecture 1.7 is expected to hold for the signed Selmer groups of M .

Bloch-Kato's Selmer groups.
Let n 0 and w be a prime of F n dividing p. .
Then, the Bloch-Kato's Selmer group of M * over F n is defined by and we set Sel BK (M * /F ∞ ) = lim − →n Sel BK (M * /F n ). Recall that the definition of the signed Coleman maps and thus of the signed Selmer groups depends on a choice of good basis of ⊕ v|p D cris,v (T ).

Lemma 1.10 ([BL15, Lemma 8.1]). There exists a good basis of ⊕ v|p D cris,v (T ) such that for any I
In particular, for such a basis, The basis of the lemma is a strongly admissible basis in the sense of [BL17, Definition 3.2].

SUBMODULES OF FINITE INDEX
We keep the notation of the previous section. Let I = (I v ) v|p ∈ p and set I c = (I c v ) v|p . The main goal of this section is to prove the following theorem. Remark 2.2. Under the additional hypothesis that F is abelian over Q with degree prime to p and that g + = g − , an algebraic functional equation relating Sel I (M * /F ∞ ) and Sel I c (M /F ∞ ) has been proved in [LP17]. In this situation, if one of these Z p [[Γ ]]modules is a cotorsion Z p [[Γ ]]-module, then they both are.
2.1. The proof of Theorem 2.1. We begin with a "control theorem" for these signed Selmer groups.

Lemma 2.3. For all but finitely many s ∈ Z, the kernel and cokernel of the restriction map
Sel are finite of bounded orders as n varies.
Proof. The diagram where v is any prime of F dividing p, thus the central map is an isomorphism by the inflation-restriction exact sequence.
We now study the kernel of the rightmost vertical map. For a prime v of F dividing p, the diagram 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 (9) that the map is an injection. For a prime w of F n not dividing p and a prime w ′ of F ∞ above w, the diagram (10) ). If w is archimedean, it splits completely in F ∞ /F n so this group is trivial. If w is non-archimedean, it finitely decomposes in F ∞ /F n , so that Gal(F ∞,w ′ /F n,w ) ≃ Z p and is topologically generated by an element γ n .
One has the short exact sequence For all but finitely many s ∈ Z, M G F n,w s is finite for every n, hence M Thus, the snake lemma applied to the diagram (10) implies that the map has finite kernel of bounded orders as n varies. Finally, the result follows from the snake lemma applied to the diagram (8).

Proposition 2.4. Assume that
Then for all but finitely many s ∈ Z, the maps .
If w is archimedean, since p is odd, H 1 (F ∞,w , M * s ) is trivial. If v is a non-archimedean prime of F not dividing p, the surjection Thus, by definition of H 1 I v (F v , M * ) and the exact sequence (11), the map Proof. If w is archimedean, since p is odd, H 1 (F ∞,w , M * s ) is trivial. If w is a nonarchimedean prime not dividing p above a prime v of F , by [Gre89, Proposition 2], Finally, by definition, the Pontryagin dual of Proposition 2.4 and Lemma 2.6 enable to compute the corank of the Bloch-Kato Selmer group. Sel which induces, by the snake Lemma, the short exact sequence The Corollary follows from the hypothesis that Sel I (M * /F ∞ ) is cotorsion and (the proof of) Lemma 2.6. Proof. By Proposition 2.4, we have the short exact sequence where v runs through archimedean primes of F and, for a real prime v, dim Q p ( * p ) − is the dimension of the −1-eigenspace for a complex conjugation above v acting on p . From [Gre89, Eq. (34)], we have Proof of Theorem 2.1. For any s ∈ Z, since Γ ≃ Z p has p-cohomological dimension 1, the restriction map Thus, combined with Proposition 2.4 and Lemma 2.5, for all but finitely many s ∈ Z, we obtain the commutative diagram By Proposition 2.4 and possibly avoiding another finite set of s ∈ Z, we have the short exact sequence Taking Γ -invariants gives the long exact sequence For v a non-archimedean prime not dividing p, we recall the definition of the Tamagawa number of T at v [FPR94,I §4].
If N is Q p -vector space of finite dimension d (respectively a free Z p -module of rank d), we denote by N −1 its dual and we set det If N is now a finitely generated Z p -module, we define the determinant of N over Z p as is a resolution of N by free Z p -modules of finite ranks N −1 and N 0 . Let be the motive we have fixed before. Let Frob v be the Frobenius in Gal(F v,unr /F v ), we have an exact sequence of Q p -vector spaces ) −1 and the Tamagawa number of T at v, denoted by Tam v (T ), is defined as the unique power of p such that We can now deduce the following corollary on the leading term of the algebraic p-adic L-function, which is a generalization of Kim's result on Kobayashi Up to a unit, we have where the first relation is [Gre99, Lemma 4.2] and the second is Theorem 2.1.
It remains to relate the right hand side of the formula (12) to | Sel I (M * /F ∞ ) Γ |. It is done by studying the commutative diagram 0 where the surjection at the end of the top row is Proposition 2.4 and the one at the bottom row is due to Theorem 2.1. As we mentioned in the proof of Lemma 2.3, by (Tors.), the central map is an isomorphism by the inflation-restriction exact sequence. Hence, by the snake lemma, we have We now compute | Ker f v |. As we have already remarked, the archimedean part is trivial since p is odd, and if v divides p, then f v is injective (see the proof of Lemma 2.3). Finally, if v is a non-archimedean prime not dividing p, then Ker f v is the orthogonal complement under Tate's local pairing of the projection . Since Tam v (T ) = 1 at primes v where M * is unramified and all the ramified primes of M * are contained in Σ, we can extend the product in (14) over all nonarchimedean primes not dividing p. The corollary follows from (13) combined with (14).

CONGRUENCES Let
′ be another motive and T ′ a G F -stable Z p -lattice inside We shall simply add a superscript (·) ′ to the various object associated to T to denote the similar object associated to T ′ (e.g.
From now on, we assume that as G F -representations. The goal of this section is to compare the signed Selmer groups of M * and M ′, * under the hypothesis (Cong.). We begin by studying the implication of this congruence on the signed Coleman maps.

Wach modules.
We succinctly recall what we need for our purpose and refer the reader to [Ber03,Ber04] for details. Let v be a prime of F dividing p and let A + ] equipped with the semilinear action by the Frobenius ϕ which acts as the absolute Frobenius on F v and on π by ϕ(π) = (π + 1) p − 1, and with an action of Gal(F v (µ p ∞ )/F v ) given by There exists a Wach module , which we still denote by ϕ, commuting with the Galois action. We denote by ϕ * N v (T ) the A + F v -module generated by ϕ (N v (T )). The Dieudonné module associated to T is defined via the Wach module by where the filtration on N v (T ) inducing the one on D cris,v (T ) is One also recovers the first Iwasawa cohomology group from the Wach module by an where ψ is a left inverse for ϕ.
which is compatible with the filtration, the Galois action and the action of ϕ.

Congruences of signed Coleman maps.
We now follow the construction of the signed Coleman maps as given in [BL17,§2] keeping track of the congruences modulo p.
First, note that by (16) and Theorem 3. (15) and Theorem 3.1, the Dieudonné modules associated to T and T ′ are isomorphic modulo p. We fix good bases for D cris,v (T ) and D cris,v (T ′ ) compatible with the isomorphism given in Theorem 3.1 in the sense that they have the same image under (15). Proof. The first statement is Proposition 2.9 of op. cit.. We follow its proof to prove the second. Perrin-Riou's big logarithm T is given by

Lemma 3.2. For n 1, there exists a unique Λ-homomorphism
where M is the Mellin transform which maps elements of to overconvergent power series in π, whose set we denote B + rig,F v . The Mellin transform preserves integrality and the ideal (ω n ) corresponds to (ϕ n+1 (π)).
The first statement then follows from a study (Lemma 3.44 of op. cit.) of the map which shows that, for x ∈ N v (T ) ψ=1 , the element ϕ −n−1 • (1 − ϕ)(x) is congruent to an element of (A + F v ) ψ=0 ⊗ Z p D cris,v (T ) modulo ϕ n+1 (π)B + rig,F v ⊗ Z p D cris,v (T ). But by Theorem 3.1, the maps (18) for T and T ′ agree modulo p and we are done.
For i ∈ {1, . . . , g v }, we write  T with the projection on the i-th component of the fixed basis of D cris,v (T ). We set h n (respectively h ′ n ) the Λ nendomorphisms on ⊕ g v k=1 Λ n given by the left multiplication by the product C v,n · · · C v,1 (respectively C ′ v,n · · · C ′ v,1 ).
Proof. The first part is Proposition 2.10 of op. cit.. Again by Theorem 3.1 and (15), the matrices C v,n and C ′ v,n are congruent modulo p for all n. Thus, by the first part of the Lemma and Lemma 3.2, we have C v,n · · · C v,1 · Col (n) T ≡ C v,n · · · C v,1 · Col (n) T ′ mod (Ker h n , p). Since with C v ∈ GL g v (Z p ) and Φ p n (1 + X ) and p are coprime, the second part follows. Since M * is divisible, we deduce the second isomorphism.
Definition 3.6. Let Σ 0 ⊂ Σ be a subset that contains all the primes of ramification of M * but not the primes dividing p or the archimedean primes. We define the non-primitive I-Selmer groups of M * over F ∞ by Under the hypothesis of the above Theorem and the assumption that the µ-invariants vanish, we can relate the λ-invariant of Sel I (M * /F ∞ ) and that of Sel I (M ′, * /F ∞ ), which we denote by λ and λ ′ . The Theorem combined with (19) and the discussion that follows it implies Besides, we can compute the coranks of H 1 (F ∞,w , M * ) and H 1 (F ∞,w , M ′, * ) thanks to [GV00, Proposition 2.4]. Let v be the prime of F under w and ℓ be the rational prime (different from p) under v. We denote by ( * p ) F v,unr the maximal quotient of * p on which the group Gal(F v /F v,unr ) acts trivially and we set P v (X ) = det(1−Frob v X |( * p ) F v,unr ) ∈ Z p [X ]. Then the corank of H 1 (F ∞,w , M * ) is equal to the multiplicicy of ℓ −[F v :Q p ] as root of P v (X ) in F p [X ].