Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases

Fix an odd prime p. Let G be a compact p-adic Lie group containing a closed, normal, pro-p subgroup H which is abelian and such that G/H is isomorphic to the additive group of p-adic integers Zp. First we assume that H is finite and compute the Whitehead group of the Iwasawa algebra, Λ(G), of G. We also prove some results about certain localisation of Λ(G) needed in Iwasawa theory. Let F be a totally real number field and let F∞ be an admissible p-adic Lie extension of F with Galois group G. The computation of the Whitehead groups are used to show that the Main Conjecture for the extension F∞/F can be deduced from certain congruences between abelian p-adic zeta functions of Delige and Ribet. We prove these congruences with certain assumptions on G. This gives a proof of the Main Conjecture in many interesting cases such as Zp ⋊ Zp-extensions.