English

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

Number Theory 2019-06-04 v3

Abstract

Let pp be an odd prime and F,F_{\infty,\infty} a pp-adic Lie extension of a number field FF with Galois group isomorphic to ZprZp\mathbb{Z}_p^r\rtimes\mathbb{Z}_p, r1r\geq 1. Under certain assumptions, we prove an asymptotic formula for the growth of pp-exponents of the class groups in the said pp-adic Lie extension. This generalizes a previous result of Lei, where he establishes such a formula in the case r=1r=1. An important and new ingredient towards extending Lei's result rests on an asymptotic formula for a finitely generated (not necessarily torsion) Zp[[Zpr]]\mathbb{Z}_p[[\mathbb{Z}_p^r]]-module which we will also establish in this paper. We then continue studying the growth of pp-exponents of the class groups under more restrictive assumptions and show that there is an asymptotic formula in our noncommutative pp-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