Elliptic Curves and Iwasawa’s µ = 0 Conjecture

Sujatha, R.

doi:10.1007/978-1-4419-6211-9_7

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…Using (6), it can be shown that G S (p, F cyc ) is free if and only if µ(F cyc /F ) = 0 (see [13, Chap. XI, §3] and [18] for more details). This provides an interesting connection between the classical Iwasawa µ-invariant for a number field F and the freeness of the pro-p Galois group G S (p, F cyc ).…”

Section: The µ-Invariant In Iwasawa Theorymentioning

confidence: 99%

On the 𝜇-invariant in Iwasawa theory

Ramdorai¹

2011

WIN—Women in Numbers

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Section: The µ-Invariant In Iwasawa Theorymentioning

confidence: 99%

On the 𝜇-invariant in Iwasawa theory

Ramdorai¹

2011

WIN—Women in Numbers

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“…Modelling after Iwasawa's ideas, Mazur developed an analogous approach towards studying the arithmetic of an abelian variety via examining the variations of its p-primary Selmer groups in a Z p -extension (see [34]). Recently, there have been great interest in the study of a certain subgroup of the p-primary Selmer group, called the fine Selmer group (for instances, see [6,18,28,30,31,32,45,48]). In the fundamental paper of Coates-Sujatha [6], they have examined this fine Selmer group in great depth and made several conjectures on its structure.…”

Section: Introductionmentioning

confidence: 99%

On the Control Theorem for Fine Selmer Groups and the Growth of Fine Tate-Shafarevich Groups in $\mathbb{Z}_p$-Extensions

Lim

2020

Doc. Math.

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Let A be an abelian variety defined over a number field F . We prove a control theorem for the fine Selmer group of the abelian variety A which essentially says that the kernel and cokernel of the natural restriction maps in an arbitrarily given Z p -extension F ∞ /F are finite and bounded. We emphasise that our result does not have any constraints on the reduction of A and the ramification of F ∞ /F . As a first consequence of the control theorem, we show that the fine Tate-Shafarevich group over an arbitrary Z p -extension has trivial Λcorank. We then derive an asymptotic growth formula for the ptorsion subgroup of the dual fine Selmer group in a Z p -extension. However, as the fine Mordell-Weil group need not be p-divisible in general, the fine Tate-Shafarevich group need not agree with the ptorsion of the dual fine Selmer group, and so the asymptotic growth formula for the dual fine Selmer groups do not carry over to the fine Tate-Shafarevich groups. Nevertheless, we do provide certain sufficient conditions, where one can obtain a precise asymptotic formula.

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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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Elliptic Curves and Iwasawa’s µ = 0 Conjecture

Cited by 6 publications

References 14 publications

On the 𝜇-invariant in Iwasawa theory

On the 𝜇-invariant in Iwasawa theory

On the Control Theorem for Fine Selmer Groups and the Growth of Fine Tate-Shafarevich Groups in $\mathbb{Z}_p$-Extensions

Fine Selmer Groups and Isogeny Invariance

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