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Asymptotic growth patterns for class field towers

Arindam Bhattacharyya, Vishnu Kadiri, Anwesh Ray

Abstract

Let $p$ be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a $\mathbb{Z}_p$-tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain $\mathbb{Z}_p$-extensions in which a given prime above $p$ is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for $p$-primary class numbers of the number fields in a $\mathbb{Z}_p$-tower.

Asymptotic growth patterns for class field towers

Abstract

Let be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain -extensions in which a given prime above is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for -primary class numbers of the number fields in a -tower.
Paper Structure (7 sections, 9 theorems, 73 equations)

This paper contains 7 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Motivation from Iwasawa theory
  3. Main results
  4. Organization
  5. Acknowledgment
  6. Pro-$p$ groups and their Hilbert series
  7. Growth asymptotics for split prime $\mathbb{Z}_p$-extensions

Key Result

Theorem 1.2

Assume that the following conditions are satisfied Then, there exists a $\mathbb{Z}_p$-extension $\mathds{L}_\infty/\mathds{L}$ in which $\mathfrak{p}_1$ is totally split. Moreover, there exists a constant $C>0$ (independent of $n$ and $k$) and $n_0\in \mathbb{Z}_{\geq 0}$, such that for all $n\geq n_0$ and $k\geq 1$, we have that Let $\mu(\mathds{L}_\infty/\mathds{L})$ denote the Iwasawa $\mu$-

Theorems & Definitions (27)

  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.3: Vinberg's criterion
  • ...and 17 more