Iwasawa theory of class field towers

Abstract

Let $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $\mu$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $\delta>0$ such that $d_n=\delta p^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\geq2$, as $n\to\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.