Codimension two cycles in Iwasawa theory and tensor product of Hida families

Lei, Antonio; Palvannan, Bharathwaj

doi:10.1007/s00208-020-02072-8

Cited by 2 publications

(2 citation statements)

References 25 publications

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“…Even in the context of a cyclotomic Z p -extension, it is still an open question whether R(A/F ∞ ) ∨ is finitely generated over R (see [17, conjecture 1]). There are some recent studies on this structure by Lei-Palvannan [25] in this direction. However, to the best knowledge of the author, the subject of the fine Selmer group of a Hida deformation over a general Z p -extension does not seem to be covered in the literature.…”

Section: Hida Deformationsmentioning

confidence: 98%

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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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Section: Hida Deformationsmentioning

confidence: 98%

“…Unfortunately, we do not have an answer to this. In fact, to the best knowledge of the author, even in the context of a cyclotomic Z p -extension F cyc , the structure of R(A/F cyc ) does not seem well-understood (but see a very recent work of Lei-Palvannan [25] for some discussion in this direction).…”

Section: Introductionmentioning

confidence: 99%

Structure of fine Selmer groups over -extensions

LIM

2023

Math. Proc. Camb. Phil. Soc.

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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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Codimension two cycles in Iwasawa theory and tensor product of Hida families

Cited by 2 publications

References 25 publications

Structure of fine Selmer groups over -extensions

Structure of fine Selmer groups over -extensions

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

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