On the pseudo-nullity of the dual fine Selmer groups

Lim, Meng Fai

doi:10.1142/s1793042115500888

Cited by 5 publications

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References 12 publications

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“…They formulate their conjecture for the fine Selmer group of an elliptic curve over certain (admissible) p-adic Lie extensions whose Galois group is a p-adic Lie group with dimension ≥ 2. We refer the reader to works of Ochi [39], Lim [31] and Shekhar [50] where there are other examples verifying the pseudo-nullity conjecture of Coates-Sujatha. The setups in their works and their approaches are completely different to ours.…”

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confidence: 99%

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Lei

Palvannan

2019

Forum of Mathematics, Sigma

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A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable p-adic L-functions) and algebraic objects (two "everywhere unramified" Iwasawa modules) involving codimension two cycles in a 2-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field K (where an odd prime p splits) of an elliptic curve E, defined over Q, with good supersingular reduction at p. On the analytic side, we consider eight pairs of 2-variable p-adic L-functions in this setup (four of the 2-variable p-adic L-functions have been constructed by Loeffler and a fifth 2-variable p-adic L-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the Z 2 p -extension of K. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.1 The first subscript d of ρ d,n indicates the dimension of the Galois representation, while the second subscript n denotes a number one less than the Krull dimension of the ring R. In the settings we are interested in, the number n would denote the number of variables in the corresponding p-adic L-functions.2 The Panchishkin condition is a type of "ordinariness" assumption, introduced by Greenberg, while formulating the Iwasawa main conjecture for Galois deformations. See Section 4 in [14] for the precise definition. 2

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confidence: 99%

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Lei

Palvannan

2019

Forum of Mathematics, Sigma

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“…This conjecture is very much open. Some examples verifying Conjecture B are given in [3,17,23,25,36,43]. As a corollary of Theorem 5.5, we show that Conjecture B can be characterized in terms of the growth of the fine Selmer groups in the intermediate cyclotomic extensions.…”

Section: Corollary 54 Let E Be An Elliptic Curve Defined Over F and Letmentioning

confidence: 58%

“…Recently, there has been an interest in the study the fine Selmer group (see [6,17,23,25,27,30,36,49,51]). This is a subgroup of the classical Selmer group obtained by imposing stronger vanishing conditions at primes above p (see Section 5 for its definition).…”

Section: Introductionmentioning

confidence: 99%

Control Theorems for Fine Selmer Groups

Kundu¹,

Lim²

2023

Journal De Théorie Des Nombres De Bordeaux

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“…; also see [12,23,31]). Subsequently, there have been much interest on the fine Selmer group of a modular form (for instance, see [14,16,17]) or even more general classes of Galois representations (see [21,26,27,32]). A common feature in these cited works is that they are mainly concerned with working over the cyclotomic Z p -extension.…”

Section: Introductionmentioning

confidence: 99%

Structure of fine Selmer groups over -extensions

LIM

2023

Math. Proc. Camb. Phil. Soc.

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This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_{p}$ -extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_{p}[[ \Gamma ]]$ , where $\Gamma$ is the Galois group of the $\mathbb{Z}_{p}$ -extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates–Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_{p}$ -extension, then it holds for almost all $\mathbb{Z}_{p}$ -extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary p-adic Lie extension.

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On the pseudo-nullity of the dual fine Selmer groups

Cited by 5 publications

References 12 publications

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

Control Theorems for Fine Selmer Groups

Structure of fine Selmer groups over -extensions

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