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

Volume 189 / 2019

Dingli Liang, Meng Fai Lim Acta Arithmetica 189 (2019), 191-208 MSC: Primary 11R23; Secondary 11R29, 11R20. DOI: 10.4064/aa180412-12-7 Published online: 13 May 2019


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.


  • Dingli LiangSchool of Mathematics and Statistics
    Wuhan University
    Wuhan, Hubei, 430072, P.R. China
  • Meng Fai LimSchool of Mathematics and Statistics &
    Hubei Key Laboratory of Mathematical Sciences
    Central China Normal University
    Wuhan, Hubei, 430079, P.R. China

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