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 $\mathbb {Z}_p$. First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $\lambda (G)$, of $G$. We also prove some results about certain localisation of $\lambda (G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty }$ 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 $F_{\infty }/F$ can be deduced from certain congruences between abelian $p$-adic zeta functions of Deligne 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 $\mathbb {Z}_p\rtimes \mathbb {Z}_p$-extensions.