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# 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

#### 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.

The second author Y.K. is supported by the SFB 1085 “Higher invariants” (University of Regensburg) funded by the DFG. The third author Y.L. is supported by NSFC grant A010102-11671380.

Received 1 November 2017

Accepted 22 November 2017

Published 9 July 2019