The main conjecture of Iwasawa theory for totally real fields

Abstract

Let p be an odd prime. Let G be a compact p-adic Lie group with a quotient isomorphic to ℤp. We give an explicit description of K 1 of the Iwasawa algebra of G in terms of Iwasawa algebras of Abelian subquotients of G. We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of G, which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa's conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ. © 2012 Springer-Verlag Berlin Heidelberg.