On the $\Lambda$-cotorsion subgroup of the Selmer group

Matar, Ahmed

doi:10.4310/ajm.2020.v24.n3.a3

Cited by 8 publications

(6 citation statements)

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“…In the case where T f is the Tate module of an elliptic curve E{Q with good supersingular reduction at p, a similar result has been proved in [16]. One of the key ingredients of the proof given in [16] is a link between the the fine Selmer group of E over Q cyc and the maximal Λ-torsion submodule of the Pontryagin dual of the p-primary Selmer group of E over Q cyc , which was proved by Wingberg [31] (see also [20], where an alternative proof is given). In the present paper, we prove Theorem 4.4 by first establishing analogues of Wingberg's result in the context of modular forms (see Theorems 3.1 and 3.4).…”

Section: Introductionmentioning

confidence: 60%

“…In the present paper, the latter definition is used. Our proof is very different from the ones employed in both [31] and [20]. We make use of Nekovář's spectral sequence, which seems to give a somewhat simpler and more general proof than the previous proofs available in the literature.…”

Section: Introductionmentioning

confidence: 95%

“…In the present paper, we prove Theorem 4.4 by first establishing analogues of Wingberg's result in the context of modular forms (see Theorems 3.1 and 3.4). In [31], Wingberg worked with Selmer groups defined using flat cohomology, whereas Matar worked with Selmer groups defined using Galois cohomology in [20] instead. In the present paper, the latter definition is used.…”

Section: Introductionmentioning

confidence: 99%

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On fine Selmer groups and signed Selmer groups of elliptic modular forms

Lei,

Lim

2021

Preprint

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Let f be an elliptic modular form and p an odd prime that is coprime to the level of f . We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic Zp extension of Q and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalization of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

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Section: Introductionmentioning

confidence: 60%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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On fine Selmer groups and signed Selmer groups of elliptic modular forms

Lei,

Lim

2021

Preprint

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“…In the case where is the Tate module of an elliptic curve with good supersingular reduction at p , a similar result has been proved in [18]. One of the key ingredients of the proof given in [18] is a link between the fine Selmer group of E over and the maximal -torsion submodule of the Pontryagin dual of the p -primary Selmer group of E over that was proved by Wingberg [28] (see also Matar [19], where an alternative proof is given). In the present paper, we prove Theorem 4.4 by first establishing analogues of Wingberg’s result in the context of modular forms (see Theorems 3.1 and 3.4).…”

Section: Introductionmentioning

confidence: 72%

On Fine Selmer Groups and Signed Selmer Groups of Elliptic Modular Forms

Lei

Lim²

2022

Bull. Aust. Math. Soc.

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Let f be an elliptic modular form and p an odd prime that is coprime to the level of f. We study the link between divisors of the characteristic ideal of the p-primary fine Selmer group of f over the cyclotomic $\mathbb {Z}_p$ extension of $\mathbb {Q}$ and the greatest common divisor of signed Selmer groups attached to f defined using the theory of Wach modules. One of the key ingredients of our proof is a generalisation of a result of Wingberg on the structure of fine Selmer groups of abelian varieties with supersingular reduction at p to the context of modular forms.

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“…Remark 4.6. In [Mat19], Ahmed Matar gives a Galois theoretic proof of Wingberg's result (Theorem 4.3) with a slightly different hypothesis. See Theorem 1.1 of [Mat19].…”

Section: Conjecture a And µ-Invariant Of Torsion Part Of Dual Selmer ...mentioning

confidence: 99%

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

Hamidi¹,

Ray²

2020

Preprint

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Let p be an odd prime and let E be an elliptic curve defined over a number field F with good reduction at primes above p. In this survey article, we give an overview of some of the important results proven for the fine Selmer group and the signed Selmer groups over cyclotomic towers as well as the signed Selmer groups over Z 2 p -extensions of an imaginary quadratic field where p splits completely. We only discuss the algebraic aspects of these objects through Iwasawa theory. We also attempt to give some of the recent results implying the vanishing of the µ-invariant under the hypothesis of Conjecture A. Moreover, we draw an analogy between the classical Selmer group in the ordinary reduction case and that of the signed Selmer groups of Kobayashi in the supersingular reduction case. We highlight properties of signed Selmer groups (when E has good supersingular reduction) which are completely analogous to the classical Selmer group (when E has good ordinary reduction). In this survey paper, we do not present any proofs, however we have tried to give references of the discussed results for the interested reader.

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On the $\Lambda$-cotorsion subgroup of the Selmer group

Cited by 8 publications

References 0 publications

On fine Selmer groups and signed Selmer groups of elliptic modular forms

On fine Selmer groups and signed Selmer groups of elliptic modular forms

On Fine Selmer Groups and Signed Selmer Groups of Elliptic Modular Forms

Conjecture A and $μ$-invariant for Selmer groups of supersingular elliptic curves

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