On the main conjecture of Iwasawa theory for certain non-cyclotomic ${\mathbb{Z}}_p$-extensions

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}{\mathfrak{p}}^{\ast }$. Let $H_{\mathrm{\infty }}=H{K}_{\mathrm{\infty }}$ where $K_{\mathrm{\infty }}$ is the unique ${\mathbb{Z}}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_{\mathrm{\infty }})$ for the maximal abelian $p$-extension of $H_{\mathrm{\infty }}$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_{\mathrm{\infty }})=\mathrm{G}\mathrm{a}\mathrm{l}(M(H_{\mathrm{\infty }})\mathrm{/}{H}_{\mathrm{\infty }})$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_{\mathrm{\infty }})$. As a consequence, we have that if $X(H_{\mathrm{\infty }})=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_{\mathrm{\infty }})$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.