Asymptotic growth of Iwasawa invariants in Noncommutative towers of number fields

TL;DR

This paper investigates the asymptotic behavior of Iwasawa invariants, specifically the growth of the lambda-invariant, in noncommutative towers of number fields with certain Galois group properties.

Contribution

It provides new insights into the growth patterns of Iwasawa lambda-invariants in noncommutative Galois extensions under specific conditions.

Findings

Lambda-invariant growth is characterized asymptotically.

Mu-invariant remains zero in the studied extensions.

Class groups are isomorphic to a direct sum of p-divisible groups.

Abstract

Let $p$ be an odd prime, $F$ be a number field and consider a uniform infinite pro-$p$ extension $F_\infty$ of $F$ with Galois group $G=Gal(F_\infty/F)$. Let \[G=G_0\supset G_1\supset\dots \supset G_n\supset G_{n+1}\supset \dots\] be the descending $p$ central series of $G$ and set $F_n:=F_\infty^{G_n}$. Assume that $G$ is uniform and that $F_\infty$ contains the cyclotomic $\mathbb{Z}_p$-extension of $F$. Denote by $A_n$ the $p$-primary part of the class group of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$. The $\lambda$-invariant of $F_n$ coincides with the corank of $A_n$ as a $\mathbb{Z}_p$-module. Assume that the Iwasawa $\mu$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $F$ equal to $0$. Then, the $\mu$-invariant of the cyclotomic $\mathbb{Z}_p$-extension of $F_n$ is $0$ as well and $A_n$ is isomorphic to $\left(\mathbb{Q}_p/\mathbb{Z}_p\right)^{\lambda_n}$. We study the asymptotic growth of $\lambda_n$ as $n$ goes to $\infty$.