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

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 $K_{\infty }$.

arXiv commentfixed mistake. This is the final version