Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

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: 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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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
Email: S.J Wadsley@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-05-00415-7
Received by editor(s): February 27, 2005
Received by editor(s) in revised form: March 30, 2005
Published electronically: June 7, 2005

American Mathematical Society