Control Theorems for Fine Selmer Groups
Debanjana Kundu1; Meng Fai Lim2
1 Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
2 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal University, Wuhan, 430079, P.R.China.
Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 851-880.

We study the growth of the p-primary fine Selmer group, R(E/F ′ ), of an elliptic curve over an intermediate sub-extension F ′ of a p-adic Lie extension, ℒ/F. We estimate the ℤ p -corank of the kernel and cokernel of the restriction map r ℒ/F ′ :R(E/F ′ )→R(E/ℒ) Gal(ℒ/F ′ ) with F ′ a finite extension of F contained in ℒ. We show that the growth of the fine Selmer groups in these intermediate sub-extension is related to the structure of the fine Selmer group over the infinite level. On specializing to certain (possibly non-commutative) p-adic Lie extensions, we prove finiteness of the kernel and cokernel and provide growth estimates on their orders.

Nous étudions la croissance du groupe de Selmer fin p-primaire R(E/F ′ ) d’une courbe elliptique sur une sous-extension intermédiaire F ′ d’une extension de Lie p-adique, ℒ/F. Nous estimons le ℤ p -corang du noyau et du conoyau de l’application de restriction r ℒ/F ′ :R(E/F ′ )→R(E/ℒ) Gal(ℒ/F ′ ) , où F ′ est une extension finie de F contenue dans ℒ. Nous montrons également que la croissance des groupes de Selmer fins dans ces sous-extensions intermédiaires est liée à la structure du groupe de Selmer fin au niveau infini. En spécialisant au cas des extensions de Lie p-adiques classiques (éventuellement non commutatives), nous prouvons la finitude du noyau et du conoyau et fournissons des estimations de croissance de leurs ordres.

Received:
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Accepted:
Published online:
DOI: 10.5802/jtnb.1231
Classification: 11R23
Keywords: control theorem, fine Selmer groups
Debanjana Kundu 1; Meng Fai Lim 2

1 Mathematics Department, 1984, Mathematics Road, University of British Columbia, Vancouver, Canada, V6T1Z2
2 School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences Central China Normal University, Wuhan, 430079, P.R.China.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Debanjana Kundu; Meng Fai Lim. Control Theorems for Fine Selmer Groups. Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 851-880. doi : 10.5802/jtnb.1231. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/

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