The main conjecture of Iwasawa theory for totally real fields

By Mahesh Kakde. Inventiones Mathematicae, 2010

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Let p be an odd prime. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}$\end{document} be a compact p-adic Lie group with a quotient isomorphic to ℤp. We give an explicit description of K1 of the Iwasawa algebra of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}$\end{document} in terms of Iwasawa algebras of Abelian subquotients of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}$\end{document}. We also prove a result about K1 of a certain canonical localisation of the Iwasawa algebra of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{G}$\end{document}, 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-pp-adic Lie extension of Abelian extensions of ℚ.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f89c375698326100601def3bfe1e9ed06525fd1