Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases

Source abstract

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