Euler Characteristics and their Congruences in the Positive Rank Setting

TL;DR

This paper investigates congruence relations between truncated Euler characteristics of dual Selmer groups of elliptic curves with isomorphic residual representations over p-adic Lie extensions, extending previous results from rank zero to higher rank cases.

Contribution

It extends congruence results for Euler characteristics from rank zero elliptic curves to those with higher rank in the context of p-adic Lie extensions.

Findings

Established congruences for truncated Euler characteristics in higher rank cases.

Extended previous rank zero results to higher rank elliptic curves.

Analyzed the behavior of Selmer groups over admissible p-adic Lie extensions.

Abstract

The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible $p$-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves.