On the Iwasawa asymptotic class number formula for ${\mathbb{Z}}_p^r⋊{\mathbb{Z}}_p$-extensions

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 ${\mathbb{Z}}_p^r⋊{\mathbb{Z}}_p$, $r\ge 1$. Under certain assumptions, we prove an asymptotic formula for the growth of $p$-exponents of the class groups in the said $p$-adic Lie extension. This generalizes a previous result of Lei, where he establishes such a formula in the case $r=1$. An important and new ingredient towards extending Lei's result rests on an asymptotic formula for a finitely generated (not necessarily torsion) ${\mathbb{Z}}_p[[{\mathbb{Z}}_p^r]]$-module which we will also establish in this paper. 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 is for the commutative extension) in a special case.

17 pages. Corrected the proof of Proposition 2.2.1