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

Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic polynomials in a certain precise manner. A recent work of Lei has made progress on this problem by proving that the fine Mordell-Weil group (in the sense of Wuthrich) does have this required property. The goal of this paper is study the analogous question of Greenberg over various ${\mathbb{Z}}_p$-extensions of an imaginary quadratic field $F$. In particular, when the elliptic curve has complex multiplication by the ring of integers of the imaginary quadratic field, 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, we study the situation of an elliptic curve over the anticyclotomic ${\mathbb{Z}}_p$-extension under the generalized Heegner hypothesis. Along the way, we establish an analogous result for the BDP-Selmer group. This latter result is then applied to obtain a relation between the BDP $p$-adic $L$-function and the Mordell-Weil rank growth in the anticyclotomic ${\mathbb{Z}}_p$-extension which may be of independent interest.