Prime Decomposition and the Iwasawa mu-invariant

TL;DR

This paper explores the construction of number field towers with large Iwasawa mu-invariants using various uniform pro-p groups, extending Iwasawa's original work beyond the case of =_p.

Contribution

It demonstrates that certain uniform pro-p groups, especially those with fixed-point-free automorphisms, can realize arbitrarily large mu-invariants in Galois towers over number fields.

Findings

Large mu-invariants linked to primes splitting completely in towers.

Existence of fixed-point-free automorphisms influences mu-invariant size.

Extension of Iwasawa's construction to broader classes of pro-p groups.

Abstract

For $\Gamma=\mathbb{Z}_p$, Iwasawa was the first to construct $\Gamma$-extensions over number fields with arbitrarily large $\mu$-invariants. In this work, we investigate other uniform pro-$p$ groups which are realizable as Galois groups of towers of number fields with arbitrarily large $\mu$-invariant. For instance, we prove that this is the case if $p$ is a regular prime and $\Gamma$ is a uniform pro-$p$ group admitting a fixed-point-free automorphism of odd order dividing $p-1$. Both in Iwasawa's work, and in the present one, the size of the $\mu$-invariant appears to be intimately related to the existence of primes that split completely in the tower.