On the Plus and the Minus Selmer Groups for Elliptic Curves at Supersingular Primes

Kitajima, Takahiro; Otsuki, Rei

doi:10.3836/tjm/1502179270

Cited by 26 publications

(55 citation statements)

References 14 publications

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“…for all v|p and v / ∈ S ss p,F . For all v ∈ S ss p,F , local Tate duality and [22,Theorem 3.34 Iw (F S , T ) is also Λ(Γ)-torsion. This is equivalent to H 2 (F S /F cyc , E p ∞ ) = 0 (see [27,Lemma 7.1] or [29,Proposition 1.3.2]).…”

Section: Euler Characteristicsmentioning

confidence: 99%

“…where E ± (F n,v ) ⊗ Q p /Z p are identified with their images inside H 1 (F n,v , E p ∞ ) via the Kummer map. By [22,Proposition 3.32], there is an isomorphism of Λ(Γ)modules…”

Section: Signed Selmer Groupsmentioning

confidence: 99%

“…E p ∞ (k ur cyc ) = 0.Proof. This follows from[22, Proposition 3.1], which says that E p ∞ k Let H k = Gal(k ur cyc /k cyc ) ∼ = Z p . Then, (a) H 1 (H k , E + (k ur cyc )) ⊕ H 1 (H k , E − (k ur cyc )) ∼ = H 1 (H k ,Ê(k ur )); (b) E ± (k ur cyc ) ⊗ Q p /Z p H k = E ± (k cyc ) ⊗ Q p /Z p .Proof.…”

mentioning

confidence: 95%

“…All these can be suitably extended to the Galois representation arising from elliptic modular forms, and these questions are investigated in this paper. In the last couple of decades, there are important subgroups of the Selmer groups, which we simply refer to as the 'signed Selmer groups' that have been defined by Kobayashi in [23] and later studied by various authors notably by Kim, Kitajima, Otsuki et al under the assumption that a p (E) = 0 (see [21,20,22,18,24,25,26]). These Selmer groups exhibit properties remarkably similar to the Selmer groups in the ordinary case and results on their Galois cohomology and Euler characteristic are known ( [20,Theorem 1.2]).…”

Section: Introductionmentioning

confidence: 99%

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On Selmer Groups in the Supersingular Reduction Case

Lei¹,

Sujatha²

2020

Tokyo J. Math.

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Let p be a fixed odd prime. Let E be an elliptic curve defined over a number field F with good supersingular reduction at all primes above p.We study both the classical and plus/minus Selmer groups over the cyclotomic Zp-extension of F . In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when p splits completely in F , we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all Zp-extensions of F when they are defined.

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Section: Euler Characteristicsmentioning

confidence: 99%

“…where E ± (F n,v ) ⊗ Q p /Z p are identified with their images inside H 1 (F n,v , E p ∞ ) via the Kummer map. By [22,Proposition 3.32], there is an isomorphism of Λ(Γ)modules…”

Section: Signed Selmer Groupsmentioning

confidence: 99%

mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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On Selmer Groups in the Supersingular Reduction Case

Lei¹,

Sujatha²

2020

Tokyo J. Math.

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“…Kim's result [Kim13, Theorem 1.1] on the analogue of Theorem 2.5 for the supersingular setting. For the further developments along this direction, see [KO18]. 3.4.…”

Section: 3mentioning

confidence: 99%

On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

Kim

Kurihara

2019

International Mathematics Research Notices

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In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic Zp-extension of Q of an elliptic curve over Q. Especially, we present a proof of the "weak main conjecture"à la Mazur and Tate for elliptic curves with good (supersingular) reduction at an odd prime p. We also prove the "strong main conjecture" suggested by the second named author under the validity of the ±-main conjecture and the vanishing of a certain error term. The key idea is the explicit comparison among "finite layer objects", "±-objects", and "fine objects" in Iwasawa theory. The case of good ordinary reduction is also treated.

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On the structure of signed Selmer groups

Ponsinet

2019

Math. Z.

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Let F be a number field unramified at an odd prime p and F∞ be the Zp-cyclotomic extension of F . Generalizing Kobayashi plus/minus Selmer groups for elliptic curves, Büyükboduk and Lei have defined modified Selmer groups, called signed Selmer groups, for certain non-ordinary Gal(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 Zp[[Gal(F∞/F )]]modules, we show that they have no proper sub-Zp[[Gal(F∞/F )]]-module of finite index. We deduce from this a number of arithmetic applications. On studying the Euler-Poincaré 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.

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On the Plus and the Minus Selmer Groups for Elliptic Curves at Supersingular Primes

Cited by 26 publications

References 14 publications

On Selmer Groups in the Supersingular Reduction Case

On Selmer Groups in the Supersingular Reduction Case

On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction

On the structure of signed Selmer groups

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