Notes on the fine Selmer groups

Abstract

In this paper, we study the fine Selmer groups 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 $\Zp$-extension is intimately related to the variation of the class groups in the cyclotomic tower. We also discuss some examples of pseudo-nullity of fine Selmer groups.

31 pages, numerous corrections and changes

Cited by in corpus (15)

• Fine Selmer groups of congruent Galois representations
• Fine Selmer groups of congruent $p$-adic Galois representations
• On Selmer groups in the supersingular reduction case
• On the growth of even $K$-groups of rings of integers in $p$-adic Lie extensions
• A note on asymptotic class number upper bounds in $p$-adic Lie extensions
• On the codescent of étale wild kernels in $p$-adic Lie extensions
• Fine Selmer groups and ideal class groups
• Some remarks on Kida's formula when $\mu \ne 0$
• Structure of fine Selmer groups over ${\mathbb{Z}}_p$-extensions
• Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p
• Comparing direct limit and inverse limit of even $K$-groups in multiple ${\mathbb{Z}}_p$-extensions
• On pseudo-nullity of fine Mordell-Weil group
• On the weak Leopoldt conjecture and coranks of Selmer groups of supersingular abelian varieties in $p$-adic Lie extensions
• Fine Selmer groups of modular forms
• On order of vanishing of characteristic elements