Euler Characteristics and their Congruences for Multi-signed Selmer Groups

Abstract

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

arXiv comment23 pages

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Cited by in corpus (3)

• Statistics for Iwasawa invariants of elliptic curves
• Arithmetic statistics and diophantine stability for elliptic curves
• Iwasawa Invariants for elliptic curves over ${\mathbb{Z}}_p$-extensions and Kida's Formula