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 $K_{\mathrm{\infty }}$ of $K$, which is unramified outside $\mathfrak{p}$. Let $H$ be the Hilbert class field of $K$, and write $H_{\mathrm{\infty }}=H{K}_{\mathrm{\infty }}$. Let $M(H_{\mathrm{\infty }})$ be the maximal abelian $2$-extension of $H_{\mathrm{\infty }}$, which is unramified outside the primes above $\mathfrak{p}$, and put $X(H_{\mathrm{\infty }})=\mathrm{G}\mathrm{a}\mathrm{l}(M(H_{\mathrm{\infty }})\mathrm{/}{H}_{\mathrm{\infty }})$. We prove that $X(H_{\mathrm{\infty }})$ 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 $J_{\mathrm{\infty }}$ of $K_{\mathrm{\infty }}$ with arbitrary quadratic extensions $J$ of $H$. We also prove some new cases of the finite generation of the Mordell-Weil group $E(J_{\mathrm{\infty }})$ modulo torsion of certain elliptic curves $E$ with complex multiplication by $\mathcal{O}$.