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On Iwasawa's class number formula for $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions

Sohei Tateno

Abstract

Let $p$ be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated $p$-torsion Iwasawa modules, which are closely related to class numbers. We also construct some $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions whose Iwasawa $μ$-invariant is nonzero. At the end of this paper, we calculate the determinants of some matrices that are related to the groups $\mathbb{Z}_p\rtimes\mathbb{Z}_p$.

On Iwasawa's class number formula for $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions

Abstract

Let be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated -torsion Iwasawa modules, which are closely related to class numbers. We also construct some -extensions whose Iwasawa -invariant is nonzero. At the end of this paper, we calculate the determinants of some matrices that are related to the groups .

Paper Structure

This paper contains 5 sections, 16 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Some properties of $\Lambda(G)$-modules
  3. The main result
  4. Construction of $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions with $\mu_G\neq0$
  5. Determinants related to $\mathbb{Z}_p\rtimes\mathbb{Z}_p$

Key Result

Theorem 1

Let $K_\infty /K$ be a $\mathbb{Z}_p$-extension and $K_n$ be the subfields corresponding to the subgroups $p^n\mathbb{Z}_p$ of $\mathbb{Z}_p$. If we denote $e_n$ for the $p$-exponent of the class number $h(K_n)$ of $K_n$, then there exist some $\mu,\lambda\geq0$ and $\nu\in\mathbb{Z}$ such that for sufficiently large $n$.

Theorems & Definitions (33)

  • Theorem : Iwasawa, Theorem 4.
  • Theorem : CM, Theorem I.
  • Theorem : Perbet, Corollary 3.4.
  • Theorem : Lei, Corollary 5.3.
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 23 more