Characteristic elements for 𝑝-torsion Iwasawa modules

Abstract

Let G G be a compact p p -adic analytic group with no elements of order p p . We provide a formula for the characteristic element (J. Coates, et. al., The G L 2 GL_2 main conjecture for elliptic curves without complex multiplication, preprint) of any finitely generated p p -torsion module M M over the Iwasawa algebra Λ G \Lambda _G of G G in terms of twisted μ \mu -invariants of M M , which are defined using the Euler characteristics of M 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 p -torsion module has trivial characteristic element as the p p -nilpotent groups. It is also shown that these are precisely the groups which have the property that every finitely generated p p -torsion module has integral Euler characteristic. Under a slightly weaker condition on G G we decompose the completed group algebra Ω G \Omega _G of G G with coefficients in F p \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 G which have the property that every block of Ω G \Omega _G is local. Finally, we compute the ranks of the K 0 K_0 group of Ω G \Omega _G and of its classical ring of quotients Q ( Ω G ) Q(\Omega _G) whenever the latter is semisimple.