Characteristic elements for p-torsion iwasawa modules - Ardakov, K; Wadsley, S (2006)
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 GL2 main conjecture for elliptic curves without complex multiplication, preprint) of any finitely generated p-torsion module M over the Iwasawa algebra AG 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 ΩG of G with coefficients in double-struck F signp 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 ΩG is local. Finally, we compute the ranks of the K0 group of ΩG and of its classical ring of quotients Q(ΩG) whenever the latter is semisimple.