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 $G\mathrm{/}H$ is isomorphic to the additive group of $p$-adic integers ${\text{mathbbZ}}_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $\mathrm{\Lambda }(G)$, of $G$. We also prove some results about certain localisation of $\mathrm{\Lambda }(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\mathrm{\infty }}$ 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 $F_{\mathrm{\infty }}\mathrm{/}F$ 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 $$\mathbb{Z}$_p\rtimes