Title
Noncommutative Iwasawa Main Conjectures Over Totally Real Fields: Mnster, April 2011 (Springer Proceedings In Mathematics & Stat,Used
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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 noncommutative Iwasawa theory for many motives. Noncommutative Iwasawa theory has emerged dramatically over the last decade, culminating in the recent proof of the noncommutative main conjecture for the Tate motive over a totally real padic 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 noncommutative main conjecture were discovered by Venjakob, and were then systematically developed in the subsequent papers by CoatesFukayaKatoSujathaVenjakob and FukayaKato. 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 nonabelian 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 nonabelian 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 selfcontained 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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