Coates–Sujatha Conjecture B

Prove 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 pseudonull over R[[Gal(\tilde{F}/F)]].

Background

Conjecture B is the higher-dimensional pseudonullity prediction originally formulated by Coates and Sujatha for elliptic curves and generalized here to the paper’s Galois representations. It predicts that the dual fine Selmer group over an admissible p-adic Lie extension containing the cyclotomic extension has support of codimension at least two.

The paper’s main results establish this prediction in a vertical setting under Greenberg finiteness conditions and, for the pseudonullity descent argument, suitable Cohen–Macaulay assumptions. The general conjecture is not proved unconditionally.

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 dual fine Selmer group Y_{S}(A/\tildeF) is a pseudonull R\lrbracket{\Gal(\tildeF/F)}-module.

— 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

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 that Y_{S}(A/F_{\cyc}) is a pseudonull R\lrbracket{\Gal(F_{\cyc}/F)}-module, the dual fine Selmer group Y_{S}(A/\tildeF) is a pseudonull R\lrbracket{\Gal(\tildeF/F)}-module.

— 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+