On uniqueness in equivariant Iwasawa theory

Abstract

This sequel to our paper `On the ``main conjecture'' of equivariant Iwasawa theory' studies the possibility of the vanishing of $S{K}_1(QG)$, when $QG$ is the total ring of fractions of the Iwasawa algebra $\Lambda G=Z_p[[G]]$, with $G$ the Galois group of the Galois extension $K/k$ of loc. cit. The vanishing is equivalent to $S{K}_1(D)=0$ for all division algebras $D$ in the Wedderburn components of $QG$, so can be studied via the classification $D_{r,s,F}$ of these $D$'s, which provides a crossed product order $\Delta$ in $D$ and the valuation $v_{\bullet }$ with valuation ring ${\Delta }_{\bullet }$. $\Delta$ contains a special element $\Pi$, which generates a maximal subfield of $D$ over its centre with $V_{\bullet }(\Pi )=1$ and $\Pi \Delta {\Pi }^{-1}=\Delta$. The induced $\Pi$-filtration on $\Delta$ enables a study of the reduced norm built on a congruence for $nr(d)$ mod $\Pi \Delta$ for $d\in \Delta$. Given $d\in {\Delta }^{\times }$ with $nr(d)=1$, and $n\ge 1$ maximal with $d\in 1+{\Pi }^n\Delta$ (called the level of $d$), the main problem is to find a suitable commutator product $c\equiv d$ mod ${\Pi }^{n+1}\Delta$. Then $c^{-1}d\equiv 1$ mod ${\Pi }^{n+1}\Delta$ hence, setting $d^{\prime }=c^{-1}d$, has $nr(d^{\prime })=1$ and level $n^{\prime }>n$. Repetition ends in $[{\Delta }^{\times },{\Delta }^{\times }]$ when the level gets sufficiently large.

108 pages