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

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 $\mathcal{O}$.