ON THE EULER CHARACTERISTICS OF SIGNED SELMER GROUPS

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.