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Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed aquestion whether the structure of the fine Selmer group over the cyclotomic$\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomicpolynomials in a certain precise manner. A recent work of Lei has made progresson this problem by proving that the fine Mordell-Weil group (in the sense ofWuthrich) does have this required property. The goal of this paper is study theanalogous question of Greenberg over various $\mathbb{Z}_p$-extensions of animaginary quadratic field $F$. In particular, when the elliptic curve hascomplex multiplication by the ring of integers of the imaginary quadraticfield, we obtain analogous results of Lei over the cyclotomic$\mathbb{Z}_p$-extension and anti-cyclotomic $\mathbb{Z}_p$-extension of $F$.In the event that the elliptic curve has good ordinary reduction at the prime$p$, we further obtain a result over the $\mathbb{Z}_p$-extension of $F$unramified outside precisely one of the prime of $F$ above $p$. Finally, westudy the situation of an elliptic curve over the anticyclotomic$\mathbb{Z}_p$-extension under the generalized Heegner hypothesis. Along theway, we establish an analogous result for the BDP-Selmer group which may be ofindependent interest. Motivated by our results, we pose questions analogous tothat of Greenberg in each of the above settings.

On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field