Selmer groups 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 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 $p$.

Made a slight change to the conjecture