We study the growth of the -primary fine Selmer group, , of an elliptic curve over an intermediate sub-extension of a -adic Lie extension, . We estimate the -corank of the kernel and cokernel of the restriction map with a finite extension of contained in . 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) -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 -primaire d’une courbe elliptique sur une sous-extension intermédiaire d’une extension de Lie -adique, . Nous estimons le -corang du noyau et du conoyau de l’application de restriction où est une extension finie de contenue dans . 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 -adiques classiques (éventuellement non commutatives), nous prouvons la finitude du noyau et du conoyau et fournissons des estimations de croissance de leurs ordres.
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Keywords: control theorem, fine Selmer groups
@article{JTNB_2022__34_3_851_0, 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}, volume = {34}, number = {3}, year = {2022}, doi = {10.5802/jtnb.1231}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/} }
TY - JOUR AU - Debanjana Kundu AU - Meng Fai Lim TI - Control Theorems for Fine Selmer Groups JO - Journal de théorie des nombres de Bordeaux PY - 2022 SP - 851 EP - 880 VL - 34 IS - 3 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/ DO - 10.5802/jtnb.1231 LA - en ID - JTNB_2022__34_3_851_0 ER -
%0 Journal Article %A Debanjana Kundu %A Meng Fai Lim %T Control Theorems for Fine Selmer Groups %J Journal de théorie des nombres de Bordeaux %D 2022 %P 851-880 %V 34 %N 3 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1231/ %R 10.5802/jtnb.1231 %G en %F JTNB_2022__34_3_851_0
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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