On the Iwasawa asymptotic class number formula for -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.
