Characteristic elements for $p$-torsion Iwasawa modules

Let $G$ be a compact $p$-adic analytic group with no elements of order $p$. We provide a formula for the characteristic element (J. Coates, et. al., The $GL_2$ main conjecture for elliptic curves without complex multiplication, preprint) of any finitely generated $p$-torsion module $M$ over the Iwasawa algebra $\Lambda _G$ of $G$ in terms of twisted $\mu$-invariants of $M$, which are defined using the Euler characteristics of $M$ and its twists. A version of the Artin formalism is proved for these characteristic elements. We characterize those groups having the property that every finitely generated pseudo-null $p$-torsion module has trivial characteristic element as the $p$-nilpotent groups. It is also shown that these are precisely the groups which have the property that every finitely generated $p$-torsion module has integral Euler characteristic. Under a slightly weaker condition on $G$ we decompose the completed group algebra $\Omega _G$ of $G$ with coefficients in $\mathbb {F}_p$ into blocks and show that each block is prime; this generalizes a result of Ardakov and Brown (Primeness, semiprimeness and localisation in Iwasawa Algebras, submitted). We obtain a generalization of a result of Osima (On primary decomposable group rings, Proc. Phy-Math. Soc. Japan (3) 24 (1942) 1–9), characterizing the groups $G$ which have the property that every block of $\Omega _G$ is local. Finally, we compute the ranks of the $K_0$ group of $\Omega _G$ and of its classical ring of quotients $Q(\Omega _G)$ whenever the latter is semisimple.