Selmer groups and anticyclotomic $\mathbb{Z}_p$-extensions

TL;DR

This paper provides new proofs for theorems related to the structure of Selmer groups over anticyclotomic $Z_p$-extensions of imaginary quadratic fields, focusing on elliptic curves with both ordinary and supersingular reduction at $p$, and introduces a conjecture for the supersingular case.

Contribution

It offers new proofs of existing theorems on Selmer groups and introduces a conjecture on Heegner points for supersingular elliptic curves.

Findings

Determined the $Z_p$-corank of Selmer groups for ordinary reduction.

Proposed a conjecture on Heegner points mod $p$ for supersingular reduction.

Provided a new proof for the supersingular case assuming the conjecture.

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 give a new proof to a theorem of Bertolini which determines the value of the $\Lambda$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has ordinary reduction at $p$. In the case where $E$ has supersingular reduction at $p$ we make a conjecture about the structure of the module of Heegner points mod $p$. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the $\Lambda$-corank of $\text{Sel}(E/K_{\infty})$ in the case where $E$ has supersingular reduction at $p$.