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

TL;DR

This paper investigates the structure of fine Mordell-Weil groups over various -extensions of imaginary quadratic fields, extending Greenberg's questions and Lei's results from  to more general settings, including CM and anticyclotomic cases.

Contribution

It generalizes Lei's results on the fine Mordell-Weil group to -extensions of imaginary quadratic fields, including CM and anticyclotomic cases, and relates these to p-adic L-functions.

Findings

Results for CM elliptic curves over cyclotomic and anti-cyclotomic -extensions.

Established properties of the fine Mordell-Weil group in various -extensions.

Connected the BDP-Selmer group with Mordell-Weil rank growth in anticyclotomic -extensions.

Abstract

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.