## On the Iwasawa asymptotic class number formula for $\mathbb {Z}_{p}^r\rtimes \mathbb {Z}_{p}$-extensions

### Volume 189 / 2019

#### Abstract

Let $p$ be an odd prime and $F_{\infty,\infty}$ a $p$-adic Lie extension of a number field $F$ with Galois group isomorphic to $\mathbb{Z}_{p}^r\rtimes\mathbb{Z}_{p}$, $r\geq 1$. Under certain assumptions, we prove an asymptotic formula for the growth of $p$-exponents of the class groups in the $p$-adic Lie extension. This generalizes a previous result of Lei, who established such a formula for $r=1$. A new ingredient towards extending Lei’s result is an asymptotic formula for a finitely generated (not necessarily torsion) $\mathbb{Z}_{p}[\![ \mathbb{Z}_{p}^r]\!]$-module. We then continue studying the growth of $p$-exponents of the class groups under more restrictive assumptions and show that there is an asymptotic formula in our noncommutative $p$-adic Lie extension analogous to a refined formula of Monsky (which concerns the commutative extension) in a special case.