Asymptotic growth of Iwasawa invariants in Noncommutative towers of number fields

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 $\infty$.

arXiv comment10 pages