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On descent theory and main conjectures in non-commutative Iwasawa theory

Part of: Grothendieck groups and $K_0$ Algebraic number theory: global fields Whitehead groups and $K_1$ Arithmetic algebraic geometry

Published online by Cambridge University Press:  26 April 2010

David Burns  and
Otmar Venjakob
David Burns
Affiliation:
Department of Mathematics, King's College London, Strand, London WC2R 2LS, UK (david.burns@kcl.ac.uk)
Otmar Venjakob
Affiliation:
Mathematisches Institut der Universität Heidelberg, 69120 Heidelberg, Germany (venjakob@mathi.uni-heidelberg.de)
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Abstract

We develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

Keywords

Iwasawa theoryL-functionsmotivesK-theoryWhitehead groups

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R65: Class groups and Picard groups of orders 19A31: $K_0$ of group rings and orders 19B28: $K_1$ of group rings and orders

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 1 , January 2011 , pp. 59 - 118
Copyright
Copyright © Cambridge University Press 2010

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