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<inf>2</inf> main conjecture for elliptic curves without complex multiplication, preprint) of any finitely generated p-torsion module M over the Iwasawa algebra A<inf>G</inf> of G in terms of twisted μ-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 Ω<inf>G</inf> of G with coefficients in double-struck F sign<inf>p</inf> 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 Ω<inf>G</inf> is local. Finally, we compute the ranks of the K<inf>0</inf> group of Ω<inf>G</inf> and of its classical ring of quotients Q(Ω<inf>G</inf>) whenever the latter is semisimple.