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Noncommutative Iwasawa Main Conjectures over Totally Real Fields

Münster, April 2011

  • Conference proceedings
  • © 2013

Overview

  • Includes a self-contained and simplified proof of Kakde's main algebraic result, as well as introductory articles on related topics
  • Extremely useful for many years to come
  • Will almost certainly lead to major research progress in both Iwasawa theory and the theory of automorphic forms?
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Proceedings in Mathematics & Statistics (PROMS, volume 29)

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About this book

The algebraic techniques developed by Kakde will almost certainly lead eventually to major progress in the study of congruences between automorphic forms and the main conjectures of non-commutative Iwasawa theory for many motives. Non-commutative Iwasawa theory has emerged dramatically over the last decade, culminating in the recent proof of the non-commutative main conjecture for the Tate motive over a totally real p-adic Lie extension of a number field, independently by Ritter and Weiss on the one hand, and Kakde on the other. The initial ideas for giving a precise formulation of the non-commutative main conjecture were discovered by Venjakob, and were then systematically developed  in the subsequent papers by Coates-Fukaya-Kato-Sujatha-Venjakob and Fukaya-Kato. There was also parallel related work in this direction by Burns and Flach on the equivariant Tamagawa number conjecture. Subsequently, Kato discovered an important idea for studying the K_1 groups of non-abelian Iwasawa algebras in terms of the K_1 groups of the abelian quotients of these Iwasawa algebras. Kakde's proof is a beautiful development of these ideas of Kato, combined with an idea of Burns, and essentially reduces the study of the non-abelian main conjectures to abelian ones. The approach of Ritter and Weiss is more classical, and partly inspired by techniques of Frohlich and Taylor. Since many of the ideas in this book should eventually be applicable to other motives, one of its major aims is to provide a self-contained exposition of some of the main general themes underlying these developments. The present volume will be a valuable resource for researchers working in both Iwasawa theory and the theory of automorphic forms.

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Keywords

Table of contents (7 papers)

Reviews

“Its aim was to present progress on noncommutative main conjectures for Iwasawa theory over totally real fields. … The number theory community has to thank the editors and the authors for their efforts which make these intriguing recent developments more widely accessible.” (C. Baxa, Monatshefte für Mathematik, Vo. 180, 2016)

Editors and Affiliations

  • Department of Pure Mathematics, and Mathematical Statistics (DPMMS), University Cambridge, Cambridge, United Kingdom

    John Coates

  • Institute of Mathematics, Westphalian University of Münster, Münster, Germany

    Peter Schneider

  • Department of Mathematics, University of British Columbia, Vancouver, Canada

    R. Sujatha

  • Institute of Mathematics, University of Heidelberg, Heidelberg, Germany

    Otmar Venjakob

Bibliographic Information

  • Book Title: Noncommutative Iwasawa Main Conjectures over Totally Real Fields

  • Book Subtitle: Münster, April 2011

  • Editors: John Coates, Peter Schneider, R. Sujatha, Otmar Venjakob

  • Series Title: Springer Proceedings in Mathematics & Statistics

  • DOI: https://doi.org/10.1007/978-3-642-32199-3

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2013

  • Hardcover ISBN: 978-3-642-32198-6Published: 19 October 2012

  • Softcover ISBN: 978-3-642-44335-0Published: 09 November 2014

  • eBook ISBN: 978-3-642-32199-3Published: 19 October 2012

  • Series ISSN: 2194-1009

  • Series E-ISSN: 2194-1017

  • Edition Number: 1

  • Number of Pages: XII, 208

  • Topics: Number Theory, Algebraic Geometry, K-Theory

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