Let K be a number field, let p be a prime, and let R(E/L) denote the fine Selmer group of an elliptic curve E/K over an algebraic extension L/K. Write K^cyc for the cyclotomic Z_p-extension of K, and let K_∞/K be an admissible p-adic Lie extension containing K^cyc, with G=operatornameGal(K_∞/K) and Lambda(G)=Z_p[[G]]. The Coates–Sujatha conjectures are: (A) the Pontryagin dual R(E/K^cyc)^vee is finitely generated as a Z_p-module; and (B) the Pontryagin dual R(E/K_∞)^vee is pseudonull as a Lambda(G)-module.

Coates–Sujatha Conjectures A and B

Let $K$ be a number field, let $p$ be a prime, and let $R(E/L)$ denote the fine Selmer group of an elliptic curve $E/K$ over an algebraic extension $L/K$ . Write $K^{\mathrm{cyc}}$ for the cyclotomic $\mathbb{Z}_p$ -extension of $K$ , and let $K_\infty/K$ be an admissible $p$ -adic Lie extension containing $K^{\mathrm{cyc}}$ , with $G=\operatorname{Gal}(K_\infty/K)$ and $\Lambda(G)=\mathbb{Z}_p[[G]]$ . The Coates–Sujatha conjectures are: (A) the Pontryagin dual $R(E/K^{\mathrm{cyc}})^\vee$ is finitely generated as a $\mathbb{Z}_p$ -module; and (B) the Pontryagin dual $R(E/K_\infty)^\vee$ is pseudonull as a $\Lambda(G)$ -module.

References

Primary source

arXiv

Additional references

Pseudonullity for fine Selmer groups over multiple Z_p^d-extensions — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

New special cases support the conjectures, but neither conjecture has been proved in full and both remain open.

Conjecture A predicts finite generation of the dual fine Selmer group over the cyclotomic $\mathbb{Z}_p$ -extension; Conjecture B predicts pseudonullity over suitable higher-dimensional admissible $p$ -adic Lie extensions. No proposer or original date is specified in the retrieved sources.

Known results

Kundu, Nuccio, and Ramdorai (2023): Conjecture A holds in rank-zero elliptic-curve cases for a positive-density set of primes, under finiteness hypotheses.
Kundu, Nuccio, and Ramdorai (2023): Conjecture B holds for certain CM elliptic curves when the cyclotomic fine Selmer group is finite.
Lim and related work (2023): results on fine Selmer groups over non-cyclotomic $\mathbb{Z}_p$ -extensions provide evidence for pseudonullity.
Isogeny-invariance work (2017): gives partial information on Conjecture A and its relation to the classical $\mu=0$ conjecture.

August 26, 2026 pseudonullity results

A new report claims pseudonullity for fine Selmer groups attached to cyclotomic characters, elliptic curves, cuspidal newforms, and Hida families. This extends the known evidence toward Conjectures A and B, but it does not claim a general proof and remains unverified.

Current status (as of August 2026): Special cases and new claimed results provide evidence for Conjectures A and B, while their general statements remain open and the latest claim is unverified.

Sources

No solutions have been posted yet.