ON THE EULER CHARACTERISTICS OF SIGNED SELMER GROUPS

Abstract

Let$p$be an odd prime number and$E$an elliptic curve defined over a number field$F$with good reduction at every prime of$F$above$p$. We compute the Euler characteristics of the signed Selmer groups of$E$over the cyclotomic$\mathbb{Z}_{p}$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above$p$and mixed signs in the definition of the signed Selmer groups.