Equivariant Iwasawa theory: an example

TL;DR

This paper demonstrates the validity of the equivariant main conjecture in Iwasawa theory for certain Galois extensions with specific group structures, assuming the vanishing of the Iwasawa -invariant.

Contribution

It provides a concrete example where the equivariant main conjecture holds under particular conditions on the Galois group and -invariant.

Findings

The conjecture holds for Galois groups with dimension 1 pro- Lie groups.

Validity depends on the vanishing of the -invariant.

The result extends understanding of Iwasawa theory in non-abelian settings.

Abstract

The equivariant `main conjecture' of Iwasawa theory is shown to hold for a Galois extension $K/k$ of number fields with Galois group an $l$-adic pro-$l$ Lie group of dimension 1 containing an abelian subgroup of index $l$, provided that Iwasawa's $\mu$-invariant $\mu(K/k)$ vanishes.