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

TL;DR

This paper extends Iwasawa theory to non-cyclotomic $Z_2$-extensions of imaginary quadratic fields, proving finite generation of certain Galois modules and establishing the weak Leopoldt conjecture in new cases.

Contribution

It introduces an elliptic analogue of Sinnott's cyclotomic argument to prove finite generation of Galois modules in non-cyclotomic $Z_2$-extensions and verifies the weak $rak{p}$-adic Leopoldt conjecture for these extensions.

Findings

Proves $X(H_ty)$ is finitely generated as a $Z_2$-module.

Establishes the weak $rak{p}$-adic Leopoldt conjecture for the compositum $J_ty$.

Provides new cases of finite generation of Mordell-Weil groups of CM elliptic curves.

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 $2\mathcal{O} = \mathfrak{p} \mathfrak{p}^\ast$, 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 $\mathcal{O}$.