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- Title
The main conjecture of Iwasawa theory for totally real fields.
- Authors
Kakde, Mahesh
- Abstract
Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ. We give an explicit description of K 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 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.
- Subjects
IWASAWA theory; FORMALLY real fields; LIE groups; ISOMORPHISM (Mathematics); NONCOMMUTATIVE algebras; CYCLOTOMY
- Publication
Inventiones Mathematicae, 2013, Vol 193, Issue 3, p539
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-012-0436-x