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Noncommutative Iwasawa Main Conjectures over Totally Real Fields - Munster, April 2011 (Paperback, 2013 ed.)
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Noncommutative Iwasawa Main Conjectures over Totally Real Fields - Munster, April 2011 (Paperback, 2013 ed.)
Series: Springer Proceedings in Mathematics & Statistics, 29
Expected to ship within 10 - 15 working days
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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
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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