Asian Journal of Mathematics

Volume 21 (2017)

Number 2

Notes on the fine Selmer groups

Pages: 337 – 362



Meng Fai Lim (School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China)


In this paper, we study the fine Selmer group attached to a Galois module defined over a commutative complete Noetherian ring with finite residue field of characteristic $p$. Namely, we are interested in its properties upon taking residual representation and within field extensions. In particular, we will show that the variation of the fine Selmer group in a cyclotomic $\mathbb{Z}_p$-extension is intimately related to the variation of the class groups in the cyclotomic tower.


fine Selmer groups, admissible $p$-adic Lie extensions, Auslander regular ring, pseudo-null

2010 Mathematics Subject Classification

Primary 11R23. Secondary 11F80, 11R34, 11S25, 16E65.

Received 25 May 2013

Published 5 July 2017