Non-commutative twisted Euler characteristic

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].

to appear in Münster Journal of Mathematics