Coates–Sujatha pseudo-nullity conjecture for fine Selmer groups

Let $E$ be an elliptic curve defined over a number field $F$. Suppose that $F_\infty$ is a $p$-adic Lie extension of $F$ for which $\operatorname{Gal}(F_\infty/F)$ has dimension $\geq 2$ and which contains the cyclotomic $\mathbb{Z}_{p}$-extension $F^\mathrm{cyc}$. The fine Selmer group is denoted by $R(E/F_\infty)$, and for a module $M$ its Pontryagin dual is denoted by $M^\vee$. A finitely generated $\mathbb{Z}_{p}[[\operatorname{Gal}(F_\infty/F)]]$-module $M$ is pseudo-null if

$$\operatorname{Ext}^i_{\mathbb{Z}_{p}[[\operatorname{Gal}(F_\infty/F)]]}\bigl(M,\mathbb{Z}_{p}[[\operatorname{Gal}(F_\infty/F)]]\bigr)=0$$

for $i=0,1$.

Coates–Sujatha's pseudo-nullity conjecture. The module

$$R(E/F_\infty)^\vee$$

is pseudo-null over $\mathbb{Z}_{p}[[\operatorname{Gal}(F_\infty/F)]]$.

This conjecture remains wide open; the paper discusses theoretical support and numerical examples, while proving pseudo-nullity for the fine Mordell–Weil group in a specific $\mathbb{Z}_{p}^{2}$-extension setting.