Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the unramified Iwasawa module of a $\mathbb{Z}_p$-extension generated by division points of a CM elliptic curve

Tsuyoshi Itoh

Abstract

We consider the unramified Iwasawa module $X (F_\infty)$ of a certain $\mathbb{Z}_p$-extension $F_\infty/F_0$ generated by division points of an elliptic curve with complex multiplication. This $\mathbb{Z}_p$-extension has properties similar to those of the cyclotomic $\mathbb{Z}_p$-extension of a real abelian field, however, it is already known that $X (F_\infty)$ can be infinite. That is, an analog of Greenberg's conjecture for this $\mathbb{Z}_p$-extension fails. In this paper, we mainly consider analogs of weak forms of Greenberg's conjecture.

On the unramified Iwasawa module of a $\mathbb{Z}_p$-extension generated by division points of a CM elliptic curve

Abstract

We consider the unramified Iwasawa module of a certain -extension generated by division points of an elliptic curve with complex multiplication. This -extension has properties similar to those of the cyclotomic -extension of a real abelian field, however, it is already known that can be infinite. That is, an analog of Greenberg's conjecture for this -extension fails. In this paper, we mainly consider analogs of weak forms of Greenberg's conjecture.

Paper Structure

This paper contains 17 sections, 10 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Our questions
  3. Organization of the present paper
  4. Changes from the previous version
  5. Criteria for the questions (Q1) and (Q2)
  6. Preliminaries
  7. Criteria for (Q2)
  8. Criteria for (Q1)
  9. Examples for the questions (Q1) and (Q2)
  10. Software used in the example calculation
  11. Fukuda-Komatsu's examples
  12. Example for (Q2) with $K = \mathbb{Q} (\sqrt{-1})$ and $p=5$
  13. Examples with $K = \mathbb{Q} (\sqrt{-11})$ and $p=3$
  14. Similar questions for the case when $p=2$
  15. Questions and results
  16. ...and 2 more sections

Key Result

Lemma 2.2.1

Assume that $K$, $p$, $E$ satisfy (C1), (C2), (C3). If $(\mathcal{U}^1 / \mathcal{E}^1)^\chi$ is not trivial, then $\mathrm{Gal} (M (F_\infty) / L (F_\infty))^\chi$ is not trivial.

Theorems & Definitions (40)

  • Remark 1.1.1
  • Remark 1.1.2
  • Remark 1.1.3
  • Remark 2.1.1
  • Lemma 2.2.1: cf. e.g., Kra, O-T95
  • proof
  • Proposition 2.2.2
  • proof
  • Remark 2.2.3
  • Remark 2.2.4
  • ...and 30 more