Non-commutative twisted Euler characteristic

TL;DR

This paper generalizes the concept of twisted Euler characteristics from modules over the Iwasawa algebra of initely generated torsion modules over initely generated torsion modules over the Iwasawa algebra of initely generated torsion modules over the Iwasawa algebra of initely generated torsion modules over the Iwasawa algebra of a general p-adic Lie group, relating it to Akashi series and applications in elliptic curve Iwasawa theory.

Contribution

It extends the twisted Euler characteristic concept to modules over the Iwasawa algebra of any p-adic Lie group, connecting it with Akashi series and elliptic curve Iwasawa theory.

Findings

Established finiteness of twisted Euler characteristics for modules over general p-adic Lie groups.

Linked the twisted Euler characteristic to the evaluation of Akashi series.

Indicated applications to the Iwasawa theory of elliptic curves.

Abstract

It is well known that given a finitely generated torsion module $M$ over the Iwasawa algebra $\mathbb Z_p[[\Gamma]]$, where $\Gamma \cong \mathbb Z_p$, there exists a continuous $p$-adic character $\rho$ of $\Gamma$ such that, for the twist $M(\rho)$ of $M$, the $\Gamma_n := \Gamma^{p^n}$ Euler characteristic, i.e. $\chi(\Gamma_n, M(\rho))$, is finite for every $n$. We prove a generalization of this result by considering modules over the Iwasawa algebra of a general $p$-adic Lie group $G$, instead of $\Gamma$. We relate this twisted Euler characteristic to the evaluation of the {\it Akashi series} at the twist and in turn use it to indicate some application to the Iwasawa theory of elliptic curves. This article is a natural generalization of the result established in [JOZ].