The main conjecture of Iwasawa theory for totally real fields

Abstract

Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of $\mathcal{G}$ in terms of Iwasawa algebras of Abelian subquotients of $\mathcal{G}$ . We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of $\mathcal{G}$ , 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.