Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures for Totally Real Fields

Abstract

These notes are aimed at providing a not too technical introduction to both the background from classical Iwasawa theory for, as well as a detailed discussion of, the principal result (see Theorem 5.1) of Mahesh Kakde's fundamental paper [K1] proving, subject to the Iwasawa conjecture, the non-commutative main conjecture for totally real p-adic Lie extensions of a number field. Kakde's work is the beautiful development of ideas initiated by Kazuya Kato in his important paper [KA]. The material covered roughly corresponds to the oral lectures given by one of us at the Workshop. We have not attempted here to discuss the detailed methods of proof used either by Kakde in his paper, or by Ritter and Weiss in their important related work [RW], leaving all of this to the written material of the subsequent lecturers at the Workshop. We would also like to particularly thank R. Greenberg and K. Ardakov for some very helpful comments which have been included in the present manuscript. In particular, we are very grateful to Greenberg for providing us with a detailed explanation of his observation (Theorem 4.5) that Wiles' work (Theorems 4.3 and 4.4) on the abelian main conjecture for totally real number fields, can be extended to include the case of abelian characters,whose order is divisible by p. © Springer-Verlag Berlin Heidelberg 2013.