On Selmer Groups in the Supersingular Reduction Case

Abstract

Let $p$ be a fixed odd prime. Let $E$ be an elliptic curve defined over a number field $F$ with good supersingular reduction at all primes above $p$. We study both the classical and plus/minus Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of $F$. In particular, we give sufficient conditions for these Selmer groups to not contain a non-trivial sub-module of finite index. Furthermore, when $p$ splits completely in $F$, we calculate the Euler characteristics of the plus/minus Selmer groups over the compositum of all $\mathbb{Z}_p$-extensions of $F$ when they are defined.