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Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

Josue Avila

Abstract

For a real quadratic field $K=\mathbb{Q}(\sqrt{D})$, let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$. Greenberg conjectured that the corresponding Iwasawa module $X_{\infty}$ is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when $X_{\infty}$ is cyclic and the prime is $p=2$. Furthermore, we find a fundamental system of units for certain biquadratic fields of the form $\mathbb{Q}(\sqrt{2}, \sqrt{D})$ and show how to use it to calculate the order of $X_{\infty}$.

Iwasawa module of the cyclotomic $\mathbb{Z}_{2}$-extension of certain real quadratic fields

Abstract

For a real quadratic field , let denote the cyclotomic -extension of . Greenberg conjectured that the corresponding Iwasawa module is finite. Building on the work of Mouhib and Movahhedi, we provide new examples of real quadratic fields for which the conjecture holds, when is cyclic and the prime is . Furthermore, we find a fundamental system of units for certain biquadratic fields of the form and show how to use it to calculate the order of .

Paper Structure

This paper contains 5 sections, 19 theorems, 38 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. First Case
  4. Second Case
  5. Third Case

Key Result

Theorem 2.1

fukudamain Let $K_{\infty}/K$ be a $\mathbb{Z}_{p}$ extension such that $K_{\infty}/K$ is totally ramified. If $\left| A_1 \right| = \left| A_{0} \right|$, then $\left| A_{n}\right|=\left| A_{0}\right|$ for all $n \geq 0$.

Figures (1)

  • Figure :

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Lemma 4.1
  • proof
  • ...and 29 more