Characteristic elements for $p$torsion Iwasawa modules
Authors:
Konstantin Ardakov and Simon Wadsley
Journal:
J. Algebraic Geom. 15 (2006), 339377
DOI:
https://doi.org/10.1090/S1056391105004157
Published electronically:
June 7, 2005
MathSciNet review:
2199061
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Abstract  References  Additional Information
Abstract: 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 pseudonull $p$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. PhyMath. 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.

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Additional Information
Konstantin Ardakov
Affiliation:
Christ’s College, University of Cambridge, Cambridge CB2 3BU, United Kingdom
Email:
K.Ardakov@dpmms.cam.ac.uk
Simon Wadsley
Affiliation:
DPMMS, University of Cambridge, Cambridge CB3 OWB, United Kingdom
MR Author ID:
770243
Email:
S.J Wadsley@dpmms.cam.ac.uk
Received by editor(s):
February 27, 2005
Received by editor(s) in revised form:
March 30, 2005
Published electronically:
June 7, 2005