Fine Selmer Groups, Heegner points and Anticyclotomic $\mathbb{Z}_p$-extensions

TL;DR

This paper explores conjectures about the fine Selmer group and Heegner points over anticyclotomic $Z_p$-extensions of imaginary quadratic fields, establishing their equivalence and deriving results under supersingular reduction assumptions.

Contribution

It formulates and proves the equivalence of conjectures concerning the structure of fine Selmer groups and Heegner points, extending previous work to supersingular cases.

Findings

Conjectures about the fine Selmer group and Heegner points are equivalent.

Under supersingular reduction, these conjectures align with earlier conjectures.

Results on the structure of the Selmer group over the anticyclotomic extension are obtained.

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, $p$ a prime and $K_{\infty}/K$ the anticyclotomic $\mathbb{Z}_p$-extension of a quadratic imaginary field $K$ satisfying the Heegner hypothesis. In this paper we make a conjecture about the fine Selmer group over $K_{\infty}$. We also make a conjecture about the structure of the module of Heegner points in $E(K_{\mathfrak{p}_{\infty}})/p$ where $K_{\mathfrak{p}_{\infty}}$ is the union of the completions of the fields $K_n$ at a prime of $K_{\infty}$ above $p$. We prove that these conjectures are equivalent. When $E$ has supersingular reduction at $p$ we also show that these conjectures are equivalent to the conjecture in our earlier work. Assuming these conjectures when $E$ has supersingular reduction at $p$, we prove various results about the structure of the Selmer group over $K_{\infty}$.