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

TL;DR

This paper extends the Iwasawa asymptotic class number formula to certain noncommutative $p$-adic Lie extensions, generalizing previous results and establishing new asymptotic formulas for class group growth.

Contribution

It introduces a new asymptotic formula for $p$-exponents of class groups in $Z_p^r times Z_p$-extensions, generalizing Lei's work for $r=1$, and develops a novel approach for finitely generated modules.

Findings

Established an asymptotic formula for class group growth in noncommutative $p$-adic Lie extensions.

Extended Lei's result from $r=1$ to higher $r$ cases.

Derived an asymptotic formula analogous to Monsky's in a special noncommutative setting.

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 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.