Coates–Sujatha fine Selmer conjecture

Prove that for every elliptic curve E defined over a number field K, the Pontryagin dual of the fine Selmer group R(E/K_{\mathrm{cyc}}) over the cyclotomic \mathbb{Z}_p-extension K_{\mathrm{cyc}}/K is a finitely generated \mathbb{Z}_p-module.

Background

The fine Selmer group is a refinement of the usual Selmer group obtained by imposing trivial local conditions at the specified places. Coates and Sujatha related the structure of the dual fine Selmer group over the cyclotomic extension to the vanishing of Iwasawa’s \mu-invariant for elliptic curves.

The cited conjecture predicts that the dual fine Selmer group over the cyclotomic \mathbb{Z}_p-extension is finitely generated over \mathbb{Z}_p. The paper proves a neighbourhood-propagation statement conditional on an analogous finite-generation hypothesis, but does not establish the conjecture itself.

References

Motivated by this observation, they further conjectured that for all elliptic curves $E$ over $K$, $R(E/K_{\mathrm{cyc})\vee$ is a finitely generated $\mathbb{Z}_p$-module.

— Variation of Iwasawa Invariants for Ordinary Representations  (2608.20130 - Abhishek et al., 20 Aug 2026) in Section 3, subsection “Greenberg fine Selmer groups,” unnumbered remark following Theorem \ref{Gr fine selfukuda}

For any number field F, the dual fine Selmer group Y_{S}(A/F_{\cyc}) is a finitely generated R-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 A