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

Abstract

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 $H_{\infty }=H{K}_{\infty }$ where $K_{\infty }$ is the unique ${\mathbb{Z}}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_{\infty })$ for the maximal abelian $p$-extension of $H_{\infty }$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_{\infty })=\mathrm{Gal}(M(H_{\infty })/H_{\infty })$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_{\infty })$. As a consequence, we have that if $X(H_{\infty })=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_{\infty })$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

arXiv comment32 pages. Comment added after Corollary 5.4

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