B-minus conjecture for fine Selmer groups

Establish that, assuming Conjecture A, for every admissible p-adic Lie extension \tilde{F}/F of dimension greater than one containing F_cyc/F, the dual fine Selmer group Y_S(A/\tilde{F}) is finitely generated over R[[Gal(\tilde{F}/F_cyc)]].

Background

The B− conjecture is a weaker form of the pseudonullity conjecture proposed for fine Selmer groups over higher-dimensional admissible p-adic Lie extensions. It asks for finite generation over the Iwasawa algebra associated with the quotient Gal(\tilde{F}/F_cyc), conditional on the cyclotomic finiteness conjecture.

The paper provides evidence and proves a vertical propagation theorem under Greenberg finiteness conditions, but the conjecture remains unresolved without those additional hypotheses.

References

Let F be any number field, and let \tildeF/F be any admissible p-adic Lie extension of dimension > 1 containing F_{\cyc}/F. Assuming Conjecture \ref{myconj:A}, the module Y_{S}(A/\tildeF) is finitely generated over R\lrbracket{\Gal(\tildeF/F_{\cyc})}.

— Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions  (2608.25309 - Qi et al., 26 Aug 2026) in Section 1, subsection “Conjectures for fine Selmer groups,” Conjecture B−