Asian Journal of Mathematics

Volume 23 (2019)

Number 3

Analogues of Iwasawa’s $\mu = 0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension

Pages: 383 – 400

DOI: https://dx.doi.org/10.4310/AJM.2019.v23.n3.a2

Authors

Junhwa Choi (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Yukako Kezuka (Fakultät für Mathematik, Universität Regensburg, Germany)

Yongxiong Li (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Abstract

Let $K = \mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, and let $\mathcal{O}$ be the ring of integers of $K$. The prime $2$ splits in $K$, say $2\mathcal{O} = \mathfrak{pp}^{\ast}$, and there is a unique $\mathbb{Z}_2$-extension $K_{\infty}$ of $K$ which is unramified outside $\mathfrak{p}$. Let $H$ be the Hilbert class field of $K$, and write $H_{\infty} = HK_{\infty}$. Let $M(H_{\infty})$ be the maximal abelian $2$-extension of $H_{\infty}$ which is unramified outside the primes above $\mathfrak{p}$, and put $X(H_{\infty}) = \mathrm{Gal}(M(H_{\infty}) / H_{\infty})$. We prove that $X(H_{\infty})$ is always a finitely generated $\mathbb{Z}_2$-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak $\mathfrak{p}$-adic Leopoldt conjecture for the compositum $J_{\infty}$ of $K_{\infty}$ with arbitrary quadratic extensions $J$ of $H$. We also prove some new cases of the finite generation of the Mordell–Weil group $E(J_{\infty})$ modulo torsion of certain elliptic curves $E$ with complex multiplication by $\mathcal{O}$.

Keywords

Iwasawa theory, weak Leopoldt conjecture, Iwasawa $\mu$-invariant, elliptic curves, complex multiplication

2010 Mathematics Subject Classification

Primary 11R23. Secondary 11G05, 11G15.

Received 1 November 2017

Accepted 22 November 2017

Published 9 July 2019