Control Theorems for Fine Selmer Groups

Debanjana Kundu; Meng Fai Lim

doi:10.5802/jtnb.1231

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.

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 3, pp. 851-880.

Abstract
Résumé

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 $r_{ℒ / F^{'}} : R (E / F^{'}) \to R {(E / ℒ)}^{Gal (ℒ / F^{'})}$ 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 $r_{ℒ / F^{'}} : R (E / F^{'}) \to R {(E / ℒ)}^{Gal (ℒ / F^{'})},$ 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: 2021-06-14
Revised: 2021-10-16
Accepted: 2021-10-29
Published online: 2023-01-27

DOI: 10.5802/jtnb.1231

Classification: 11R23
Keywords: control theorem, fine Selmer groups

Author's affiliations:

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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     author = {Debanjana Kundu and Meng Fai Lim},
     title = {Control {Theorems} for {Fine} {Selmer} {Groups}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {851--880},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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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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