Characteristic elements for $p$-torsion Iwasawa modules

Authors:
Konstantin Ardakov and Simon Wadsley

Journal:
J. Algebraic Geom. **15** (2006), 339-377

DOI:
https://doi.org/10.1090/S1056-3911-05-00415-7

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 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.

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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