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 to study analogous questions 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 results that are analogous to those 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.