Euler characteristics and their congruences for multisigned Selmer groups

The notion of the truncated Euler characteristic for Iwasawa modules is a generalization usual to case when cohomology groups are not finite. Let $p$ be an odd prime, $E_1$ and $E_2$ elliptic curves over number field $F$ with semistable reduction at all primes $v|p$ such that $\operatorname{Gal}(\bar{F}/F)$-modules $E_1[p]$ $E_2[p]$ irreducible isomorphic. We compare invariants certain imprimitive multisigned Selmer $E_2$. Leveraging these results, congruence relations characteristics associated $\mathbb{Z}_p^m$-extensions studied. Our results extend earlier $\mathbb{Q}$ good ordinary $p$.