Notes on the fine Selmer groups

Lim, Meng Fai

doi:10.4310/ajm.2017.v21.n2.a5

Cited by 26 publications

(16 citation statements)

References 12 publications

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“…We record another result (cf. [9,Theorem 5.4], see also [7,Theorem 10]) which is the main ingredient in proving our main results. In view of the discussion in this paper, we will state a slightly strengthened version of [9,Theorem 5.4].…”

Section: Question B ′mentioning

confidence: 82%

“…When T is the Tate module of an elliptic curve, this is precisely [2,Conjecture B]. In this context, the conjecture has also been studied and verified in certain numerical examples (see [1,7,9,11]). When T is the R(1)-dual of the Galois representation attached to a normalized eigenform ordinary at p, this is [7,Conjecture B].…”

Section: Introductionmentioning

confidence: 95%

“…Before continuing, we first compare our definition with that in [2,7,8,9]. Let W = Hom cts (T, µ p ∞ ), where µ p ∞ denotes the group of all p-power roots of unity.…”

Section: Introductionmentioning

confidence: 99%

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On the pseudo-nullity of the dual fine Selmer groups

Lim

2015

Int. J. Number Theory

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In this paper, we will study the pseudo-nullity of the fine Selmer group and its related question. Namely, we investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over a admissible $p$-adic Lie extension $F_{\infty}$ from the knowledge that pseudo-nullity of the Galois group of the maximal abelian unramified pro-$p$ extension of $F_{\infty}$ at which every prime of $F_{\infty}$ above $p$ splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a Galois module that is unramified outside $p$. We will illustrate our results with many examples.Comment: 10 pages; some minor change

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Section: Question B ′mentioning

confidence: 82%

Section: Introductionmentioning

confidence: 95%

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On the pseudo-nullity of the dual fine Selmer groups

Lim

2015

Int. J. Number Theory

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“…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]). This then gives the isomorphism Cokerλ ± cyc ∼ = ker loc ± .…”

Section: Euler Characteristicsmentioning

confidence: 99%

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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“…Conjecture A has been generalised in various other directions, for example by allowing more general coefficient rings [Lim13] or by considering Hida deformations and 'admissible' p-adic Lie extensions [JS11], [Jha12]. In the end, however, these generalisations turn out to be equivalent to Conjecture A(K, M ) for suitable M , as the following result shows.…”

Section: Conjecture a And Equivalent Formulationsmentioning

confidence: 99%

Fine Selmer Groups and Isogeny Invariance

Sujatha

Witte

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss Conjecture A, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated Zp-module. The relationship between this conjecture and Iwasawa's classical µ = 0 conjecture is clarified. We also present some partial results towards the question whether Conjecture A is invariant under isogenies.

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Notes on the fine Selmer groups

Cited by 26 publications

References 12 publications

On the pseudo-nullity of the dual fine Selmer groups

On the pseudo-nullity of the dual fine Selmer groups

On Selmer Groups in the Supersingular Reduction Case

Fine Selmer Groups and Isogeny Invariance

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