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On the equivariant Tamagawa number conjecture in tame CM-extensions, II

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  04 May 2011

Andreas Nickel
Andreas Nickel*
Affiliation:
Universität Regensburg, Fakultät für Mathematik, Universitätsstr. 31, 93053 Regensburg, Germany (email: [email protected])
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Abstract

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We use the notion of non-commutative Fitting invariants to give a reformulation of the equivariant Iwasawa main conjecture (EIMC) attached to an extension F/K of totally real fields with Galois group 𝒢, where K is a global number field and 𝒢 is a p-adic Lie group of dimension one for an odd prime p. We attach to each finite Galois CM-extension L/K with Galois group G a module SKu(L/K) over the center of the group ring ℤG which coincides with the Sinnott–Kurihara ideal if G is abelian. We state a conjecture on the integrality of SKu (L/K) which follows from the equivariant Tamagawa number conjecture (ETNC) in many cases, and is a theorem for abelian G. Assuming the vanishing of the Iwasawa μ-invariant, we compute Fitting invariants of certain Iwasawa modules via the EIMC, and we show that this implies the minus part of the ETNC at p for an infinite class of (non-abelian) Galois CM-extensions of number fields which are at most tamely ramified above p, provided that (an appropriate p-part of) the integrality conjecture holds.

Keywords

Tamagawa numberequivariant L-valuesIwasawa main conjecture

MSC classification

Secondary: 11R23: Iwasawa theory 11R42: Zeta functions and $L$-functions of number fields
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 4 , July 2011 , pp. 1179 - 1204
Copyright
Copyright © Foundation Compositio Mathematica 2011

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