On the Iwasawa asymptotic class number formula for $\mathbb{Z}_p^r\rtimes\mathbb{Z}_p$-extensions
Abstract
Let be an odd prime and a -adic Lie extension of a number field with Galois group isomorphic to , . Under certain assumptions, we prove an asymptotic formula for the growth of -exponents of the class groups in the said -adic Lie extension. This generalizes a previous result of Lei, where he establishes such a formula in the case . An important and new ingredient towards extending Lei's result rests on an asymptotic formula for a finitely generated (not necessarily torsion) -module which we will also establish in this paper. We then continue studying the growth of -exponents of the class groups under more restrictive assumptions and show that there is an asymptotic formula in our noncommutative -adic Lie extension analogous to a refined formula of Monsky (which is for the commutative extension) in a special case.
Keywords
Cite
@article{arxiv.1803.06095,
title = {On the Iwasawa asymptotic class number formula for $\mathbb{Z}_p^r\rtimes\mathbb{Z}_p$-extensions},
author = {Dingli Liang and Meng Fai Lim},
journal= {arXiv preprint arXiv:1803.06095},
year = {2019}
}
Comments
17 pages. Corrected the proof of Proposition 2.2.1