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 Zp-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 Zp-extensions of F when they are defined.